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+[    {        "title": "1612.02692v2.Synthesis_and_pressure_and_field_dependent_magnetic_properties_of_the_Kagome_bilayer_spin_liquid_Ca___10__Cr__7_O___28__.pdf",        "content": "Synthesis and pressure and \feld dependent magnetic properties of the\nKagome-bilayer spin liquid Ca 10Cr7O28\nAshiwini Balodhi and Yogesh Singh\nIndian Institute of Science Education and Research Mohali,\nSector 81, S. A. S. Nagar, Manauli PO 140306, India\n(Dated: November 9, 2018)\nWe report synthesis of polycrystalline samples of the recently discovered spin liquid material\nCa10Cr7O28and present measurements of the ambient and high pressure magnetic susceptibility \u001f\nversus temperature T, magnetization Mversus magnetic \feld Hat variousT, and heat capacity\nCversusTat various H. The ambient pressure magnetic measurements indicate the presence\nof both ferromagnetic and antiferromagnetic exchange interactions with dominant ferromagnetic\ninteractions and with the largest magnetic energy scale \u001810 K. The \u001f(T) measurements under\nexternally applied pressure of up to P\u00191 GPa indicate the robust nature of the spin-liquid state\ndespite relative increase in the ferromagnetic exchanges. C(T) shows a broad anomaly at T\u00192:5 K\nwhich moves to higher temperatures in a magnetic \feld. The evolution of the low temperature\nC(T;H) and the magnetic entropy is consistent with frustrated magnetism in Ca 10Cr7O28.\nI. INTRODUCTION\nMost local moment magnets undergo a transition from\na high temperature paramagnetic state to a magnetically\nlong range ordered state below some critical temperature.\nThe ordered state is like a solid while the paramagnetic\nstate is like the gaseous state. The state analogous to a\nliquid is elusive in magnets. That is because most mag-\nnetic solids have a unique state with the lowest energy\nand are able to freeze into that solid-like long range or-\ndered state at su\u000eciently low temperatures. If one is able\nto suppress this tendency to order, one could attain a liq-\nuid like state of spins where they would be strongly en-\ntangled and yet dynamically \ructuating down to T= 0.\nThis can be achieved by enhancing quantum \ructuations\nwhich can melt the magnetic solid. One way of construct-\ning such a quantum spin-liquid (QSL) is by arranging\nthe magnetic moments on low dimensional or geometri-\ncally frustrated lattices. Indeed, QSL's were \frst demon-\nstrated to exist for quasi-one-dimensional spin chains like\nKCuF 3and Sr 2CuO 3(see Refs. 1 and 2 for experimental\nreviews ).\nThe quest for spin liquid realizations in higher di-\nmensions has led to a \rurry of activity in the last two\ndecades resulting in the discovery of quite a few candidate\nspin liquid materials3,4. The best established candidates\nare the quasi-two-dimensional Kagome lattice quantum\nmagnet Herbertsmithite ZnCu 3(OH) 6Cl2,5,6the triangu-\nlar lattice organic magnets \u0014{(BEDT-TTF) 2Cu2(CN) 3\n(Refs. 7 and 8) and EtMe 3Sb[Pd(dmit) 2]2,9and the re-\ncently discovered Yb based triangular lattice magnet\nYbMgGaO 410. There are some other materials which\nshowed promise either because of their geometrically frus-\ntrated lattice, like the 3-dimensional hyper-kagome iri-\ndate Na 4Ir3O8, or because of novel frustration mech-\nanisms like the Kitaev QSL candidates A2IrO3. Both\nthese material families however, showed freezing or mag-\nnetic order at low temperatures11{14.\nRecently a new quasi two-dimensional quantum mag-net Ca 10Cr7O28has been discovered with a Kagome bi-\nlayer structure15. Using bulk measurements like mag-\nnetic susceptibility and heat capacity, and microscopic\nmeasurements like \u0016SR and neutron scattering this ma-\nterial has been reported to show all the expected sig-\nnatures of a gapless quantum spin liquid (QSL)15. The\nspin-liquid state in Ca 10Cr7O28has been shown to de-\nvelop from a novel frustration mechanism where compet-\ning ferro- and anti-ferromagnetic exchange interactions\nwithin a Kagome-bilayer suppress the possibility of long-\nranged magnetic order. How this spin-liquid state evolves\nif the balance between the competing interactions is dis-\nturbed by external perturbations like hydrostatic pres-\nsure or magnetic \feld, is the question we address in this\nwork.\nHere we report synthesis and structure of poly-\ncrystalline samples of the recently discovered QSL\nCa10Cr7O28. We present a detailed study of ambient\nand high pressure magnetic susceptibility \u001fversus tem-\nperatureT, magnetization Mversus magnetic \feld Hat\nvariousT, and heat capacity CversusTat variousH.\nII. EXPERIMENTAL DETAILS\nPolycrystalline samples of Ca 10Cr7O28were prepared\nby solid state synthesis. The starting materials CaCO 3\n(99.99%, Alfa Aesar) and Cr 2O3(99.99%, Alfa Aesar)\nwere taken to make the Ca:Cr ratio 10 :5 : 7 and mixed\nthoroughly in an agate mortar and pelletized. The pel-\nlet was placed in a covered Al 2O3crucible, heated in air\nat 750oC for 24 hrs for calcination and then heated to\n1000oC for 48 hrs, and then quenched in Argon to room\ntemperature. After the initial heat treatment, the mate-\nrial was reground and pressed into a pellet and given two\nheat treatments of 24 h each at 1100oC, with an interme-\ndiate grinding and pelletizing step. The pellet was always\nbrought to room temperature by quenching in Argon.\nHard well sintered pellets were obtained which are dark\ngreen in color. Powder x-ray di\u000braction (PXRD) pat-arXiv:1612.02692v2  [cond-mat.str-el]  19 Jul 20172\nterns were obtained at room temperature using a Rigaku\ndi\u000bractometer with Cu K \u000bradiation, in the 2 \u0012range\nfrom 10oto 90owith a step size of 0 :02o. Intensity data\nwere accumulated for 5 s/step. The ambient pressure\nmagnetization Mversus temperature Tand magnetic\n\feldHwere measured using a VSM option of a Quan-\ntum Design physical property measurement system (QD-\nPPMS), and the heat capacity Cas a function of Tand\nHwas measured using a QD-PPMS. The high pressure\n\u001fversusTfor pressures upto P\u00191 GPa were measured\nin a SQUID magnetometer for Cryogenics Limited (CL-\nSQUID) using a piston-clamp based high pressure cell.\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s41\n/s50 /s32/s40/s100/s101/s103/s41/s32/s73/s40/s111/s98/s115/s41\n/s32/s73/s40/s99/s97/s108/s99/s41\n/s32/s79/s98/s115/s45/s67/s97/s108/s99\n/s66/s114/s97/s103/s103/s32/s112/s101/s97/s107/s115/s67/s97\n/s49/s48/s67/s114\n/s55/s79\n/s50/s56/s32\n/s32\nFIG. 1. (Color online) Rietveld re\fnement of powder x-ray\ndi\u000braction data for Ca 10Cr7O28. The solid circles represent\nthe observed data, the solid lines through the data repre-\nsent the \ftted pattern, the vertical bars represent the peak\npositions, and the solid curve below the vertical bars is the\ndi\u000berence between the observed and the \ftted patterns.\nIII. RESULTS\nA. Structure\nThe powder x-ray di\u000braction (PXRD) data is shown in\nFig. 1. The PXRD data could be re\fned starting with the\nstructural model recently reported for Ca 10Cr7O2815,16.\nThe Rietveld re\fnement results are also plotted in Fig. 1.\nThe lattice parameters obtained from the \ft are a=\nb= 10:7885(3) \u0017A,c= 38:163(1) \u0017A,\u000b=\f= 90o,\r=\n120o, and the cell volume V= 3846:79(20) \u0017A3. The\nWycko\u000b atomic positions, site occupancies, and thermal\nparameters obtained from the \ft are given in Table I.\nThese parameters match well with recently reported data\non single crystals and con\frm the single phase synthesis\nof polycrystalline Ca 10Cr7O28.15,16\nSix out of the seven Cr ions per formula unit are in\nFIG. 2. (Color online) Structure of Ca 10Cr7O28viewed per-\npendicular (top panel) to the c-axis. The 4 inequivalent CrO 4\ntetrahedra are shown in di\u000berent colors. Planes formed by\nthese CrO 4tetrahedra are stacked along the c-axis. A view\napproximately down (bottom panel) the c-axis showing the\nbi-layer Kagome network of Cr ions.\nthe Cr5+valence state while one is in the Cr6+valence\nstate15. The Cr5+ions are magnetic with S= 1=2 while\nthe Cr6+ion is non-magnetic. The magnetic results dis-\ncussed below are consistent with this distribution. Fig-\nure 2 shows the arrangements of the magnetic Cr ions.\nThese ions sit in oxygen tetrahedra. There are 4 inequiv-\nalent tetrahedra shown as the 4 di\u000berent colors in Fig. 2.\nThe structure is made up of layers of these di\u000berent tetra-\nhedra alternating along the c-axis as shown in the top\npanel of Fig. 2. Within the layers the tetrahedra are ar-\nranged on a Kagome lattice as can be seen in the lower\npanel in Fig. 2. The magnetic connectivity between the3\nKagome layers is such that the blue and orange Kagome\nlayers interact with each other but are isolated from the\nother layers. Similarly, the yellow and purple Kagome\nlayers interact but are isolated from the others. Thus,\nmagnetically, Ca 10Cr7O28can be viewed as a layered bi-\nlayer Kagome system.\nTABLE I. Atomic parameters obtained by re\fning x-ray pow-\nder di\u000braction for Ca 10Cr7O28with a space group 167, R3c.\nThe lattice constants are a=b= 10:788(5) \u0017A,c= 38:163\u0017A,\n\u000b=\f= 90 and\r= 120o\nAtom Wyck x y z Occ B (\u0017A)\nCa1 36f 0.292(4) 0.165(8) -0.070(6) 0.97 0.0058\nCa2 36f 0.182 -0.204(3) -0.008(5) 0.96 0.0071\nCa3 36f 0.374 0.157(7) 0.022 1 0.0086\nCa4 12c 0.666(7) 0.333 0.087 1 0.0099\nCr1 36f 0.316(2) 0.154(5) 0.124 1 0.0012\nCr2 36f 0.164(7) -0.142 -0.107 0.95 0.0143\nCr3A 12c 0.000 0.000 -0.018(4) 0.64(5) 0.0031\nCr4B 12c 0.000 0.000 -0.007(5) 0.33 0.0064\nO3A 12c 0.000 0.000 -0.023(2) 0.65(9) 0.001\nO3B 12c 0.000 0.000 -0.364 0.0256(9) 0.0900\nO1 36f 0.300(9) 0.189(8) 0.054(9) 0.84(8) 0.0192\nO2 36f 0.209 0.173(4) 0.182(5) 1.000 0.0110\nO3 36f 0.116(9) -0.0331 0.130(7) 1.000 0.0395\nO4 36f 0.367 0.147(6) 0.131 1.000 0.0008\nO5 36f 0.1239 -0.155(7) -0.062 0.95(4) 0.033\nO6 36f 0.231(7) -0.267(6) -0.124(4) 1.000 0.0081\nO8 36f -0.010(3) -0.227(7) -0.118(4) 0.86(5) 0.0094\nO9 36f -0.106 0.011(5) 0.006(6) 0.91(3) 0.0131\nB. Ambient Pressure Magnetic Results\nFigure 3 (a) shows the magnetic susceptibility \u001f=\nM=H versus temperature Tof Ca 10Cr7O28measured in\na magnetic \feld H= 1 and 20 kOe. Curie-Weiss like local\nmoment paramagnetism is clearly evident. At low tem-\nperatures\u001fincreases rapidly and reaches values which\nare quite large and similar to values seen in ferromagnetic\nmaterials. However, no anomaly signalling long ranged\nmagnetic order is observed down to T= 1:8 K.\nThe 1=\u001f(T) data at high temperatures T\u0015200 K were\n\ft by the Curie-Weiss expression \u001f=\u001f0+C\nT\u0000\u0012, where\u001f0\nis aTindependent contribution, Cis the Curie constant,\nand\u0012is the Weiss temperature. The 1 =\u001f(T) data and\nthe \ft are shown in Fig. 3 (b). The \ft gave the values\n\u001f0=\u00005:2(4)\u000210\u00005cm3/Cr,C= 0:30(2) cm3K/Cr,\nand\u0012= 4:1(6) K. The value of Cis smaller than the\nvalue 0:375 cm3K/Cr expected for S= 1=2 with ag-\nfactorg= 2. However, out of the 7 Cr ions in each\nformulae unit of Ca 10Cr7O28, 6 are in Cr+5valence state\nwithS= 1=2 and 1 Cr ion is in Cr+6valence state and\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s77/s47/s72/s32 /s40/s99/s109 /s51\n/s32/s47/s32/s67/s114 /s41\n/s84 /s32/s40/s75/s41/s32/s53/s48/s48/s32/s79/s101\n/s32/s50/s48/s32/s107/s79/s101/s40/s97/s41\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48/s54/s48/s48/s55/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s72/s47/s77/s32 /s40/s99/s109/s45/s51\n/s32/s67/s114 /s41\n/s84 /s32/s40/s75/s41\n/s40/s98/s41/s72/s47/s77/s32/s40 /s99/s109 /s45/s51\n/s32/s67/s114/s41\n/s84 /s32/s40/s75/s41FIG. 3. (Color online) (a) Magnetization Mdivided by mag-\nnetic \feldHversus temperature Tfor Ca 10Cr7O28measured\ninH= 1 and 20 kOe. (b) H=M versusTmeasured at\nH= 20 kOe. The solid curve through the data is a \ft by\nthe Curie-Weiss expression. The inset shows the H=M data\nbelowT= 50 K to highlight the deviation of the data from\nthe Curie-Weiss \ft. The data is presented per Cr in the for-\nmula unit (which is 7). (see text for details).\nis expected to be non-magnetic with S= 0. Therefore,\nthe value of the Curie constant per magnetic Cr ion will\nbeC= 7=6\u0002the value 0:30(2) cm3K/Cr found above.\nThis givesC\u00190:35(2) which in turn leads to an e\u000bective\nmagnetic moment \u0016eff\u00191:68(5)\u0016Bwhich is close to the\nvalue 1:73\u0016Bexpected for spin S= 1=2 withg-factor\nequal to 2.\nThe value of \u0012= 4:1(6) K is small and positive indi-\ncating weak ferromagnetic exchange interactions. This is\nconsistent with the huge increase in \u001fat lowTseen in\nFig. 3 (a). The magnetic exchange interactions are how-\never, more complex as can be inferred from the deviation\nbelow about T= 20 K of 1 =\u001f(T) from the Curie-Weiss \ft\nas can be seen in Fig. 3 (b) inset. The 1 =\u001f(T) data devi-\nates upwards of the Curie-Weiss \ft which means that the\n\u001f(T) data becomes smaller than expectation from the \ft.\nThis indicates the presence of antiferromagnetic interac-\ntions. Thus the magnetic susceptibility data suggests the\npresence of both ferromagnetic and antiferromagnetic ex-4\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56/s77/s32 /s40\n/s66/s47/s32/s67/s114 /s41\n/s111/s72 /s32/s40/s107/s79/s101/s41/s32/s51/s48/s48/s32/s75\n/s32/s49/s48/s48/s32/s75\n/s32/s53/s48/s32/s75\n/s32/s50/s53/s32/s75\n/s32/s53/s32/s75\n/s32/s49/s46/s56/s32/s75\nFIG. 4. (Color online) Magnetization Mversus magnetic \feld\nHfor Ca 10Cr7O28measured at various temperatures T. The\ndata is presented per Cr in the formula unit (which is 7).\nchange interactions with the ferromagnetic interactions\nbeing the dominant ones leading to a net positive Weiss\ntemperature \u0012\u00194 K.\nThe magnetization Mversus magnetic \feld Hdata\nmeasured at various temperature Tare shown in Fig. 4.\nForT\u001550 K,M(H) isotherms are linear in H. For the\nM(H) data atT= 25 K one observes a slight curva-\nture. However, the M(H) isotherms at T= 5 and 1:8 K\nshow clear curvature with tendency of saturation. This is\nconsistent with net ferromagnetic interactions which are\nweak. The data at T= 1:8 K, although still increasing\nwithH, are near saturation to values close to \u001980% of\nthe value expected ( M= 1\u0016B=Cr) forS= 1=2 moments.\nThis is again consistent with only 6 out of 7 Chromium\nions being magnetic. The fact that the magnetic mo-\nments can be saturated at magnetic \felds \u00185 T also\nsuggests that the energy scale of the largest magnetic\nexchange interactions in Ca 10Cr7O28is small \u001810 K.\nC. High Pressure Magnetic Susceptibility\nIf the spin-liquid state in Ca 10Cr7O28is stabilized by\na delicate balance of several magnetic exchanges15then\npressurizing the material may disturb this balance and\nlead to a destruction of the spin-liquid state and may in\nturn lead to the stabilization of a magnetically ordered\nstate. Additionally, the Kagome-bilayers, which are mag-\nnetically isolated at ambient pressure15may start inter-\nacting if brought closer. With this motivation we have\nperformed high pressure measurements of the magnetic\nsusceptibility \u001fversusTat various applied hydrostatic\npressuresP. Figure 5 shows the high pressure \u001f(T) data\nfor Ca 10Cr7O28. From the main panel we can see that\nalthough the magnitude of \u001fat the lowest temperature\nincreases slightly, the basic behaviour of \u001f(T) doesn't\nchange upto the highest pressures used in our measure-\nmentsP\u00191 GPa. For typical transition metal oxides\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s48/s46/s49/s48/s48/s46/s50/s48/s48/s46/s51/s48/s48/s46/s52/s48\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48/s49/s48/s48/s48/s32/s49/s47 /s32/s40/s99/s109/s45/s51\n/s32/s109/s111/s108/s45/s67/s114 /s41\n/s84/s32/s40/s75/s41/s32/s48/s32/s71/s80/s97\n/s32/s48/s46/s57/s53/s32/s71/s80/s97\n/s32/s32\n/s32/s48/s46/s57/s53/s32/s71/s80/s97\n/s32/s48/s46/s55/s56/s32/s71/s80/s97\n/s32/s48/s46/s52/s51/s32/s71/s80/s97\n/s32/s48/s46/s50/s49/s32/s71/s80/s97\n/s32/s48/s32/s71/s80/s97/s99/s109/s51\n/s47/s109/s111/s108/s45/s67/s114\n/s84/s32/s40/s75/s41FIG. 5. (Color online) Magnetic susceptibility \u001fversusTfor\nCa10Cr7O28measured in various externally applied pressures\nPin a magnetic \feld H= 1 T. The inset shows the 1 =\u001f(T)\nversusTdata forP= 0 andP\u00191 GPa. The solid curve\nthrough the data in the inset are \fts to a Curie-Weiss expres-\nsion. The data is presented per Cr in the formula unit (which\nis 7).\nthis pressure amounts to a contraction in the unit cell vol-\nume of about 1%. This is a large change in the unit-cell\nsize. The fact that the \u001f(T) doesn't show any signi\fcant\nchange suggests that the spin-liquid state in Ca 10Cr7O28\nis quite robust and does not hinge on some special val-\nues of the exchange parameters. To make a quantitative\nanalysis of the change in \u001f(T) we have \ft the high tem-\nperature data to a Curie-Weiss behaviour. The 1 =\u001f(T)\ndata forP= 0 and \u00191 GPa are shown in the inset in\nFig. 5 and the \fts to the Curie-Weiss expression are also\nshown here as solid curves through the data. In these\n\fts the e\u000bective magnetic moment was \fxed at its ambi-\nent pressure value. The value of the Weiss temperature\nchanges from \u0012\u00194 K atP= 0 GPa to \u0012\u00197 K at\nP= 1 GPa suggesting an increase in the relative im-\nportance of the existing ferromagnetic exchanges. This\nis consistent with the increased magnitude of \u001fat the\nlowest temperatures in applied pressures.\nD. Heat Capacity\nHeat capacity Cversus temperature Tdata for\nCa10Cr7O28measured between T= 1:8 and 40 K in var-\nious magnetic \felds Hare shown in the main panel in\nFig. 6 (a). The inset shows the same data plotted as\nC=T versusT. The \frst thing to note is that the H= 0\ndata below T= 10 K shows an upturn and approximately\nsaturates below T= 3 K. This might suggest an onset\nof long-ranged order. On application of a magnetic \feld\nthis upturn moves up in temperature and develops into\na complete peak at H= 3 T. The peak then moves to\nhigher temperatures for larger H. The broad peak is very\nunlike that expected for a second order phase transition5\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s32/s67/s47/s84/s32/s40/s74/s47/s109/s111/s108/s32/s75/s50\n/s41\n/s84 /s32/s40/s75/s41\n/s32/s32/s67/s32/s40/s74/s47/s109/s111/s108/s32/s75/s41\n/s84 /s32/s40/s75/s41/s32/s48/s32/s84\n/s32/s49/s32/s84\n/s32/s51/s32/s84\n/s32/s53/s32/s84\n/s32/s55/s32/s84\n/s32/s108/s97/s116/s116/s105/s99/s101\n/s67/s97\n/s49/s48/s67/s114\n/s55/s79\n/s50/s56/s40/s97/s41\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48/s48/s46/s48/s48/s46/s50/s48/s46/s53/s48/s46/s55/s48/s46/s57/s49/s46/s50\n/s32/s32/s83/s47/s67/s114/s32/s40/s82/s108/s110/s50/s41\n/s84 /s32/s40/s75/s41/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55/s48/s49/s50/s51/s52/s53/s54/s55/s56\n/s32/s32/s84\n/s80/s32/s40/s75/s41\n/s72/s32/s40/s84/s41\n/s32/s32/s67/s32/s40/s74/s47/s67/s114/s32/s109/s111/s108/s32/s75/s41\n/s84 /s32/s40/s75/s41/s32/s48/s32/s84\n/s32/s49/s32/s84\n/s32/s51/s32/s84\n/s32/s53/s32/s84\n/s32/s55/s32/s84/s40/s98/s41\nFIG. 6. (Color online) (a) Heat capacity Cversus tempera-\ntureTfor Ca 10Cr7O28measured in various magnetic \felds H.\nThe inset shows the C=T versusTdata. (b) The di\u000berence\nheat capacity \u0001 C=C\u0000lattice versus Tfor various magnetic\n\felds. Inset shows the magnetic entropy estimated from the\n\u0001C(T) data at various H. The inset in the inset shows the\npeak position TPversus magnetic \feld H. The solid curve\nthrough the data is a linear \ft.\nwhere a\u0015-like anomaly is usually seen in the heat capac-\nity. Thus we believe that this anomaly does not signal\na magnetic phase transition. This is supported by the\nentropy recovered under the peak as we now describe.\nFrom Fig. 6 (a) inset it is clear that the magnetic \felds\nhave a pronounced e\u000bect at low temperatures. However,\nthe data above about T= 30 K is independent of H\nsuggesting that it is mostly of non-magnetic origin. We\ntherefore use the data above T= 30 K and extrapolate it\nto lower temperatures to get an approximate estimation\nof the lattice heat capacity. This data is shown as the\nsolid curve in Fig. 6 (a). Using this we can get the mag-\nnetic contribution to the heat capacity at various mag-\nnetic \felds by subtracting the approximate lattice con-\ntribution from the C(T;H) data. This has been done\nand the resulting di\u000berence data \u0001 C(T;H) are shown in\nFig. 6 (b) for all magnetic \felds. The \u0001 C(T;H) thus\nobtained can be used to estimate the magnetic entropy\nby integrating \u0001 C=T versusTdata. The temperature\ndependence of the entropy S(T) estimated in this wayis shown in Fig. 6 (b) inset for all magnetic \felds. The\nS(T) data is presented in units of Rln2 per magnetic Cr.\nWe note that S(T) forH= 0 reaches about 85% Rln2\natT\u001930 K. It is therefore unlikely that a long ranged\nmagnetic order will occur at lower temperatures. If such\na transition does occur it can involve only 15% Rln2 en-\ntropy which would mean a much reduced moment order-\ning. This is consistent with previous C(T) measurements\non single crystalline Ca 10Cr7O28down toT= 0:3 K in\nH= 0 that have revealed the absence of long ranged\nmagnetic ordering15.\nIn a magnetic \feld, the anomaly in the magnetic con-\ntribution moves to higher temperatures although the\nmagnitude of the peak does not change as can be seen\nmost clearly in Fig. 6 (b). The temperature of the\npeakTPas a function of magnetic \feld His plotted\nin the smaller inset in Fig. 6 (b). We \fnd an al-\nmost linear dependence of TPonHas can be seen by\nthe solid curve through the data which is the relation\nTP= 1:986 + 0:63H. The entropy S(T) associated with\nthe peak in the magnetic contribution to C, shown in\nFig. 6 (b) inset, is also pushed up in temperatures with\nincreasingHand we recover the full Rln2 atH\u00153 T.\nThis behaviour of the entropy in a magnetic \feld is a\nhallmark of geometrically frustrated magnets. The frus-\ntration suppresses the tendency for long ranged order\nleading to the accumulation of the entropy of the un-\norderedS= 1=2 moments at lower temperatures. The\nmagnetic \feld leads to partial alignment of the disordered\nmoments at higher temperatures than at H= 0 leading\nto magnetic entropy being recovered to higher tempera-\ntures. The observation that \felds of H\u00147 T a\u000bect the\nmagnetic heat capacity and are able to move the mag-\nnetic entropy to higher temperatures again suggests that\nthe magnetic energy scales in Ca 10Cr7O28are\u001810{20 K.\nIV. SUMMARY AND DISCUSSION:\nWe have synthesized polycrystalline samples of the re-\ncently discovered Kagome bilayer spin liquid material\nCa10Cr7O28and studied in detail its temperature depen-\ndent magnetic susceptibility at ambient and high pres-\nsure, isothermal magnetization, and temperature and\nmagnetic \feld dependent heat capacity measurements.\nThe ambient pressure magnetic measurements indi-\ncate the presence of both ferromagnetic (FM) and an-\ntiferromagnetic (AFM) exchange interactions with the\nFM interactions dominating. The net magnetic scale is\nabout \u001810{15 K as evidenced by the near saturation\nof the magnetization at T= 1:8 K in a magnetic \feld\nofH= 5 T. The Curie constant is consistent with one\nout of the 7 Chromium ions per formula unit being non-\nmagnetic. The magnetic Cr ions form a bilayer Kagome\nlattice. High pressure magnetic susceptibility measure-\nments up to P\u00191 GPa reveal that the spin-liquid state\nat ambient pressure is quite robust and may not depend\non a delicate balance between any speci\fc values of com-6\npeting exchange interactions. Additionally, our results\nindicate that at high pressure the relative strength of fer-\nromagnetic interactions increases as evidenced by an in-\ncrease in the value of the Weiss temperature from \u0012= 4 K\natP= 0 to\u0012= 7 K atP= 1 GPa.\nThe heat capacity in H= 0 shows an incomplete\nanomaly peaked around \u00192 K. The entropy recov-\nered between T= 1:8 K andT= 30 K in H= 0 is\nclose to 85% Rln2 suggesting absence of magnetic or-\ndering for lower temperatures, consistent with a spin-\nliquid state. This broad anomaly peaked in Cis con-\nsistent with previous zero-\feld measurements on single\ncrystal Ca 10Cr7O28and was associated with the onset\nof coherent quantum \ructuations15. A similar broad\nanomaly has been observed for several other QSL can-\ndidates. For example, the organic triangular lattice spin\nliquid EtMe 3Sb[Pd(dmit) 2]2,9shows an anomaly in the\nheat capacity at \u00196 K while the recently discovered QSL\ncandidate YbMgGaO 4shows a heat capacity anomaly at\n\u00192:4 K10. This anomaly for QSLs is understood to be a\ncrossover from a thermally disordered state to a quantum\ndisordered state. A low temperature anomaly in the heatcapacity which moves up in temperatures on the applica-\ntion of magnetic \feld is a hallmark of frustrated magnets\nin general. For example, in addition to the above materi-\nals, the pyrochlore spin-ice material Pr 2Zr2O7also shows\nan anomaly in the heat capacity at \u00192 K which is at-\ntributed to the formation of a collective spin-ice state.\nThis anomaly has been shown to move approximately\nlinearly to higher temperatures with magnetic \felds17.\nThe heat capacity anomaly for Ca 10Cr7O28occurs at\n2:4 K atH= 0 and moves to higher temperatures in a\nmagnetic \feld approximately linearly. Since the \felds of\nour measurements are much smaller than the saturation\n\felds of 12{13 T15,16, the linear dependence of the peak\ntemperature with His intriguing. Since the ferromag-\nnetic exchange is dominant in Ca 10Cr7O28it is possible\nthat in-plane short-ranged order develops and is strength-\nened in a \feld. However, future measurements would be\nneeded to understand this observation.\nAcknowledgments.{ We thank the X-ray facility at IISER\nMohali. YS acknowledges DST, India for support\nthrough Ramanujan Grant #SR/S2/RJN-76/2010 and\nthrough DST grant #SB/S2/CMP-001/2013.\n1M. Yamashita, T. Ishii, and H. Matsuzaka, Coord. Chem.\nRev.198, 347 (2000).\n2P. Lemmens, G. Gu ntherodt, and C. Gros, Phys. Rep.\n375, 1 (2003).\n3F. Mila, Eur. J. Phys. 21, 499 (2000).\n4L. Balents, Nature (London) 464, 199 (2010).\n5J. S. Helton, K. Matan, M. P. Shores, E. A. Nytko, B. M.\nBartlett, Y. Yoshida, Y. Takano, A. Suslov, Y. Qiu, J.-H.\nChung, D. G. Nocera and Y. S. Lee, Phys. Rev. Lett. 98,\n107204 (2007).\n6T. .-H. Han, J.S. Helton. S. Chu, D.G. Nocera, J.A.R.\nRievera, C. Broholm and Y. S. Lee, Nature (London) 492,\n406 (2012), and references therein.\n7S. Yamashita, Y. Nakazawa, M. Oguni, Y. Oshima, H.\nNojiri, Y. Shimizu, K. Miyagawa and K. Kanoda, Nat.\nPhys. 4, 459 (2008).\n8M. Yamashita, N. Nakata, Y. Kasahara, T. Sasaki, N.\nYoneyama, N. Kobayashi, S. Fujimoto,, Nat. Phys. 5, 44\n(2009).\n9M. Yamashita, N. Nakata, Y. Senshu, M. Nagata, R. Kato,\nT. Shibauchi and Y. Matsuda,, Science 328, 1246 (2010).\n10Y. Li, H. Liao, Z. Zhang, S. Li, F. Jin, L. Ling, L. Zhang, Y.\nZou, L. Pi, Z. Yang, J. Wang, Z. Wu, Q. Zhang, Scienti\fcReports 5, 16419 (2015).\n11Yogesh Singh, Y Tokiwa, J Dong, and P Gegenwart, Phys.\nRev. B 88, 220413 (2013).\n12A. C. Shockley, F. Bert, J-C. Orain, Y. Okamoto, and P.\nMendels Phys. Rev. Lett. 115, 047201 (2015).\n13Yogesh Singh and P Gegenwart, Phys. Rev. B 82, 064412\n(2010).\n14Yogesh Singh, S Manni, J Reuther, T Berlijn, R Thomale,\nW Ku, S Trebst, and P. Gegenwart, Phys. Rev. Lett. 108,\n127203 (2012).\n15C. Balz, B. Lake, J. Reuther, H. Luetkens, R. Schnemann,\nT. Herrmannsdrfer, Y. Singh, A. T. M. Nazmul Islam, E.\nM. Wheeler, J. A. Rodriguez-Rivera, T. Guidi, G. G. Sime-\noni, C. Baines, and H. Ryll, Nat. Phys. 12, 942 (2016).\n16C. Balz, B. Lake, A. T. M. N. Islam, Y. Singh, J. A.\nR.-Rivera, T. Guidi, E. M. Wheeler, G. G. Simeoni and\nH.Ryll, Phys. Rev B 95, 174414 (2017)\n17S. Petit, E. Lhotel, S. Guitteny, O. Florea, J. Robert, P.\nBonville, I. Mirebeau, J. Ollivier, H. Mutka, E. Ressouche,\nC. Decorse, M. Ciomaga Hatnean, and G. Balakrishnan,\nPhys. Rev. B 94, 165153 (2016)."    },    {        "title": "1612.06047v2.Sensitivity_of_Fields_Generated_within_Magnetically_Shielded_Volumes_to_Changes_in_Magnetic_Permeability.pdf",        "content": "Sensitivity of Fields Generated within Magnetically\nShielded Volumes to Changes in Magnetic Permeability\nT. Andalibb, J.W. Martina,b,\u0003, C.P. Bidinostia,b, R.R. Mammeia,b,\nB. Jamiesona,b, M. Langb, T. Kikawac\naPhysics Department, The University of Winnipeg, 515 Portage Avenue, Winnipeg, MB,\nR3B 2E9, Canada\nbDepartment of Physics and Astronomy, University of Manitoba, Winnipeg, MB R3T 2N2,\nCanada\ncTRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada\nAbstract\nFuture experiments seeking to measure the neutron electric dipole moment\n(nEDM) require stable and homogeneous magnetic \felds. Normally these ex-\nperiments use a coil internal to a passively magnetically shielded volume to\ngenerate the magnetic \feld. The stability of the magnetic \feld generated by\nthe coil within the magnetically shielded volume may be in\ruenced by a number\nof factors. The factor studied here is the dependence of the internally generated\n\feld on the magnetic permeability \u0016of the shield material. We provide measure-\nments of the temperature-dependence of the permeability of the material used\nin a set of prototype magnetic shields, using experimental parameters nearer\nto those of nEDM experiments than previously reported in the literature. Our\nmeasurements imply a range of1\n\u0016d\u0016\ndTfrom 0-2.7%/K. Assuming typical nEDM\nexperiment coil and shield parameters gives\u0016\nB0dB0\nd\u0016= 0:01, resulting in a tem-\nperature dependence of the magnetic \feld in a typical nEDM experiment of\ndB0\ndT= 0\u0000270 pT/K for B0= 1\u0016T. The results are useful for estimating the\nnecessary level of temperature control in nEDM experiments.\nKeywords: Magnetic Shielding, Neutron Electric Dipole Moment, Magnetic\nField Stability\n\u0003Corresponding author\nEmail address: j.martin@uwinnipeg.ca (J.W. Martin)\nPreprint submitted to Nuclear Instruments and Methods in Physics Research AJune 1, 2017arXiv:1612.06047v2  [physics.ins-det]  30 May 20171. Introduction\nThe next generation of neutron electric dipole moment (nEDM) experiments\naim to measure the nEDM dnwith proposed precision \u000edn.10\u000027e\u0001cm [1,\n2, 3, 4, 5, 6, 7, 8]. In the previous best experiment [9, 10] which discovered\ndn<3:0\u000210\u000026e\u0001cm (90% C.L), e\u000bects related to magnetic \feld homogene-\nity and instability were found to dominate the systematic error. A detailed\nunderstanding of passive and active magnetic shielding, magnetic \feld gener-\nation within shielded volumes, and precision magnetometry is expected to be\ncrucial to achieve the systematic error goals for the next generation of experi-\nments. Much of the research and development e\u000borts for these experiments are\nfocused on careful design and testing of various magnetic shield geometries with\nprecision magnetometers [11, 12, 13, 14, 15].\nIn nEDM experiments, the spin-precession frequency \u0017of neutrons placed in\nstatic magnetic B0and electric E\felds is measured. The measured frequencies\nfor parallel \u0017+and antiparallel \u0017\u0000relative orientations of the \felds is sensitive\nto the neutron electric dipole moment dn\nh\u0017\u0006= 2\u0016nB0\u00062dnE (1)\nwhere\u0016nis the magnetic moment of the neutron.\nA problem in these experiments is that if the magnetic \feld B0drifts over\nthe course of the measurement period, it degrades the statistical precision with\nwhichdncan be determined. If the magnetic \feld over one measurement cycle\nis determined to \u000eB0= 10 fT, it implies an additional statistical error of \u000edn\u0018\n10\u000026e\u0001cm (assuming an electric \feld of E= 10 kV/cm which is reasonable for\na neutron EDM experiment). Over 100 days of averaging, this would make a\n\u000edn\u001810\u000027e\u0001cm measurement possible. Unfortunately the magnetic \feld in\nthe experiment is never stable to this level. For this reason, experiments use\na comagnetometer and/or surrounding atomic magnetometers to measure and\ncorrect the magnetic \feld to this level [9, 11, 12]. Drifts of 1-10 pT in B0may be\n2corrected using the comagnetometer technique, setting a goal magnetic stability\nfor theB0\feld generation system in a typical nEDM experiment.\nIn such experiments, typically B0= 1\u0016T is used to provide the quantiza-\ntion axis for the ultracold neutrons. The B0magnetic \feld generation system\ntypically includes a coil placed within a passively magnetically shielded volume.\nThe passive magnetic shield is generally composed of a multi-layer shield formed\nfrom thin shells of material with high magnetic permeability (mu-metal). The\nouter layers of the shield are normally cylindrical [1, 4] or form the walls of a\nmagnetically shielded room [16, 17]. The innermost magnetic shield is normally\na specially shaped shield, where the design of the coil in relation to shield is\ncarefully taken into account to achieve adequate homogeneity [9, 3, 5].\nMechanical and temperature changes of the passive magnetic shielding [18,\n19], and the degaussing procedure [19, 17, 20] (also known as demagnetization,\nequilibration, or idealization), a\u000bect the stability of the magnetic \feld within\nmagnetically shielded rooms. Active stabilization of the background magnetic\n\feld surrounding magnetically shielded rooms can also improve the internal\nstability [18, 12, 21]. The current supplied to the B0coil is generated by an\nultra-stable current source [11]. The coil must also be stabilized mechanically\nrelative to the magnetic shielding.\nOne additional e\u000bect, which is the subject of this paper, relates to the fact\nthat theB0coil in most nEDM experiments is magnetically coupled to the\ninnermost magnetic shield. If the magnetic properties of the innermost magnetic\nshield change as a function of time, it then results in a source of instability of\nB0. In the present work, we estimate this e\u000bect and characterize one possible\nsource of instability: changes of the magnetic permeability \u0016of the material\nwith temperature.\nWhile the sensitivity of magnetic alloys to temperature variations has been\ncharacterized in the past [22, 23], we sought to make these measurements in\nregimes closer to the operating parameters relevant to nEDM experiments. For\nthese alloys, it is also known that the magnetic properties are set during the\n\fnal annealing process [24, 25, 23]. In this spirit we performed our measure-\n3ments on \\witness\" cylinders, which are small open-ended cylinders made of the\nsame material and annealed at the same time as other larger shields are being\nannealed.\nThe paper proceeds in the following fashion:\n\u000fThe dependence of the internal \feld on magnetic permeability of the inner-\nmost shielding layer for a typical nEDM experiment geometry is estimated\nusing a combination of analytical and \fnite element analysis techniques.\nThis sets a scale for the stability problem.\n\u000fNew measurements of the temperature dependence of the magnetic per-\nmeability are presented. The measurements were done in two ways in\norder to study a variety of systematic e\u000bects that were encountered.\n\u000fFinally, the results of the calculations and measurements are combined\nto provide a range of temperature sensitivities that takes into account\nsample-to-sample and measurement-to-measurement variations.\n2. Sensitivity of Internally Generated Field to Permeability of the\nShieldB0(\u0016)\nThe presence of a coil inside the innermost passive shield turns the shield\ninto a return yoke, and generally results in an increase in the magnitude of B0.\nThe ratio of this \feld inside the coil in the presence of the magnetic shield to\nthat of the coil in free space is referred to as the reaction factor C, and can be\ncalculated analytically for spherical and in\fnite cylindrical geometries [26, 27].\nThe key issue of interest for this work is the dependence of the reaction factor\non the permeability \u0016of the innermost shield. Although this dependence can be\nrather weak, the constraints on B0stability are very stringent. As a result, even\na small change in the magnetic properties of the innermost shield can result in\nan unacceptably large change in B0.\nTo illustrate, we consider here the model of a sine-theta surface current on\na sphere of radius a, inside a spherical shell of inner radius R, thickness t,\n4and linear permeability \u0016. The uniform internal \feld generated by this ideal\nspherical coil is augmented by the reaction factor in the presence of the shield,\nbut is otherwise left undistorted. The general reaction factor for this model is\ngiven by Eq. (38) in Ref. [26]. In the high- \u0016limit, with t\u001cR, the reaction\nfactor can be approximated as\nC'1 +1\n2\u0010a\nR\u00113\u0012\n1\u00003\n2R\nt\u00160\n\u0016\u0013\n; (2)\nwhich highlights the dependence of B0on the relative permeability \u0016r=\u0016=\u0016 0\nof the shield.\nFig. 1 (upper) shows a plot of B0versus\u0016rfor coil and shield dimensions\nsimilar to the ILL nEDM experiment [9, 28]: a= 0:53 m,R= 0:57 m, and\nt= 1:5 mm. In addition to analytic calculations, we also include the results\nof two axially symmetric simulations conducted using FEMM [29] to assess the\ne\u000bects of geometry and discretization of the surface current. The di\u000berences\nare small, suggesting that the ideal spherical model of Ref. [26] and the high- \u0016\napproximation of Eq. 2 provide valuable insight for the design and analysis of\nshield-coupled coils.\nIn the \frst simulation, the same spherical geometry was used as for the\nanalytic calculations. However, the surface current was discretized to 50 in-\ndividual current loops, inscribed onto a sphere, and equally spaced vertically\n(i.e. a discrete sine-theta coil). A square wire pro\fle of side length 1 mm was\nused. As shown in Fig. 1, this simulation gave excellent agreement with the\nanalytic calculations. In the second simulation, a solenoid coil and cylindrical\nshield (length/radius = 2) were used with the same dimensions as above. Simi-\nlarly, the coil was modelled as 50 evenly spaced current loops, with the distance\nfrom an end loop to the inner face of the shield endcap being half the inter-\nloop spacing. In the limit of tight-packing (i.e., a continuous surface current)\nand in\fnite \u0016, the image currents in the end caps of the shield act as an in\f-\nnite series of current loops, giving the ideal uniform \feld of an in\fnitely long\nsolenoid [30, 31]. As shown in Fig. 1, the result is similar to the spherical case,\nwith di\u000berences of order one part per thousand and a somewhat steeper slope\n5Exact calculation-sphereApproximation-sphereSimulation-sphereSimulation-solenoid0.9900.9951.0001.005B0(0,0,0)(μT)\n10 00020 00030 00040 00050 0000.0000.0050.0100.0150.020\nShield permeabilityμrμ/B0*dB0/dμFigure 1: Upper: Magnetic \feld at the coil center as a function of magnetic permeability\nof the surrounding magnetic shield for a geometry similar to the ILL nEDM experiment as\ndiscussed in the text. Lower:\u0016\nB0dB0\nd\u0016vs. permeability. The solid curve is the exact calculation\nfor the ideal spherical coil and shield from Ref. [26]; the dashed curve is the approximation\nof Eq. 2. The circles and squares are the FEMM-based simulations for the spherical and\nsolenoidal geometries with discrete currents. Since the spherical simulation was in agreement\nwith the calculation, it is omitted from the lower graph. For the exact calculation and the\ntwo simulations, currents were chosen to give B0= 1\u0016T at\u0016r= 20;000.\nofB0(\u0016r).\nFig. 1 (lower) shows the normalized slope\u0016\nB0dB0\nd\u0016of the curves from Fig. 1\n(upper). In ancillary measurements of shielding factors (discussed brie\ry in\nSection 3.1), we found \u0016r= 20;000 to o\u000ber a reasonable description of the\nquasistatic shielding factor of our shield. Using this value as the magnetic\n6permeability of our shield material, Fig. 1 (lower) shows that\u0016\nB0dB0\nd\u0016varies by\nabout 20% (from 0.008 to 0.01) for the spherical vs. solenoidal geometries. We\nadopt the value\u0016\nB0dB0\nd\u0016= 0:01 as an estimate of this slope in our discussions in\nSection 4, acknowledging that the value depends on the coil and shield design.\nFor a high- \u0016innermost shield, the magnetic \feld lines emanating from the\ncoil all return through the shield. This principle can be used to estimate the\nmagnetic \feld Bminside the shield material, and in our studies gave good\nagreement with FEA-based simulations. For the solenoidal geometry previously\ndescribed and used for the calculations in Fig. 1, Bmis largest in the side\nwalls of the solenoidal \rux return, attaining a maximum value of 170 \u0016T. If\nwe assume \u0016r=20,000, the Hm\feld is 0.007 A/m. Typically the shield is de-\ngaussed (idealized) with the internal coil energized. After degaussing, Bmmust\nbe approximately the same, since essentially all \rux returns through the shield.\nHowever, the Hm\feld may become signi\fcantly smaller because after degauss-\ning, it must fall on the ideal magnetization curve in Bm\u0000Hmspace. (For a\ndiscussion of the ideal magnetization curve, we refer the reader to Ref. [25].)\nIn principle, the Hm\feld could be reduced by an order of magnitude or more,\ndepending on the steepness of the ideal magnetization curve near the origin.\nThusBm= 170\u0016T andHm<0:007 A/m set a scale for the relevant values for\nnEDM experiments. Furthermore, the \feld in the nEDM measurement volume,\nas well as in the magnetic shield, must be stable for periods of typically hun-\ndreds of seconds (corresponding to frequencies <0:01 Hz). This sets the relevant\ntimescale for magnetic properties most relevant to nEDM experiments.\n3. Measurements of \u0016(T)\n3.1. Previous Measurements and their Relationship to nEDM Experiments\nPrevious measurements of the temperature dependence of the magnetic prop-\nerties of high-permeability alloys have been summarized in Refs. [22, 25, 32].\nThese measurements are normally conducted using a sample of the material\nto create a toroidal core, where a thin layer of the material is used in order to\n7avoid eddy-current and skin-depth e\u000bects [32, 23]. A value of \u0016is determined by\ndividing the amplitude of the sensed Bm-\feld by the amplitude of the driving\nACHm-\feld (similar to the method described in Section 3.3). Normally the\nfrequency of the Hm-\feld is 50 or 60 Hz. The value of \u0016is then quoted either at\nor near its maximum attainable value by adjusting the amplitude of Hm. De-\npending on the details of the Bm\u0000Hmcurve for the material in question, this\nnormally means that \u0016is quoted for the amplitude of Hmbeing at or near the\ncoercivity of the material [22, 23], resulting in large values up to \u0016r= 4\u0002105.\nIt is well known that \u0016measured in this fashion for toroidal, thin metal\nwound cores depends on the annealing process used for the core. There is a\nparticularly strong dependence on the take-out or tempering temperature after\nthe high-temperature portion of the annealing process has been completed [32,\n23, 22]. Such studies normally suggest a take-out temperature of 490-500\u000eC.\nThis ensures that the large \u0016r= 4\u0002105is furthermore maximal at room\ntemperature. Slight variations around room temperature, and assuming the\ntake-out temperature is not controlled to better than a degree, imply a scale\nof possible temperature variation of \u0016of approximately\f\f\f1\n\u0016d\u0016\ndT\f\f\f'0:3-1%/K at\nroom temperature [22, 23].\nA challenge in applying these results to temperature stability of nEDM ex-\nperiments is that, when used as DC magnetic shielding, the high-permeability\nalloys are usually operated for signi\fcantly di\u000berent parameters ( Bm,Hm, and\nfrequencies).\nFor example, when used in a shielding con\fguration, the e\u000bective perme-\nability is often measured to be typically \u0016r= 20;000 rather than 4 \u0002105. This\narises in part because Hmis well below the DC coercivity. As noted in Sec-\ntion 2, a more appropriate Hmfor the innermost magnetic shield of an nEDM\nexperiment is <0:007 A/m, whereas the coercivity is Hc= 0:4 A/m [23]. The\nfrequency dependence of the measurements could also be an issue. Typically,\nnEDM experiments are concerned with slow drifts at <0:01 Hz timescales\nwhereas the previously reported \u0016(T) measurements are performed in an AC\nmode at 50-60 Hz.\n8The goal of our experiments was to develop techniques to characterize the\nmaterial properties of our own magnetic shields post-annealing, in regimes more\nrelevant to nEDM experiments.\nWe created a prototype passive magnetic shield system in support of this\nand other precision magnetic \feld research for the future nEDM experiment to\nbe conducted at TRIUMF. The shield system is a four-layer mu-metal shield\nformed from nested right-circular cylindrical shells with endcaps. The inner\nradius of the innermost shield is 18.44 cm, equal to its half-length. The radii and\nhalf-lengths of the progressively larger outer shields increase geometrically by a\nfactor of 1.27. Each cylinder has two endcaps which possess a 7.5 cm diameter\ncentral hole. A stove-pipe of length 5.5 cm is placed on each hole was designed to\nminimize leakage of external \felds into the progressively shielded inner volumes.\nThe design is similar to another smaller prototype shield discussed in Ref. [33].\nThe magnetic shielding factors of each of the four cylindrical shells, and of\nvarious combinations of them, were measured and found to be consistent with\n\u0016r\u001820;000.\nIn our studies of the material properties of these magnetic shields, two dif-\nferent approaches to measure \u0016(T) were pursued. Both approaches involved\nexperiments done using witness cylinders made of the same material and an-\nnealed at the same time as the prototype magnetic shields. We therefore expect\nthey have the same magnetic properties as the larger prototype shields, and\nthey have the advantage of being smaller and easier to perform measurements\nwith.\nThe two techniques employed to determine \u0016(T) were the following:\n1. measuring the low-frequency AC axial magnetic shielding factor of the\nwitness cylinder as a function of temperature, and\n2. measuring the temperature-dependence of the slope of a minor B-H loop,\nusing the witness cylinder as a transformer core, similar to previous mea-\nsurements of the temperature dependence of \u0016, but for parameters closer\nto those encountered in nEDM experiments.\n9We now discuss the details and results of each technique.\n3.2. Axial Shielding Factor Measurements\nIn these measurements, a witness cylinder was used as a magnetic shield.\nThe shield was subjected to a low-frequency AC magnetic \feld of \u00181 Hz.\nThe amplitude of the shielded magnetic \feld Bswas measured at the center\nof the witness cylinder using a \ruxgate magnetometer. Changes in Bswith\ntemperature signify a dependence of the permeability \u0016on temperature. The\nrelative slope of \u0016(T) can then be calculated using\n1\n\u0016d\u0016\ndT=\u00001\nBsdBs\ndT\n\u0016\nBsdBs\nd\u0016: (3)\nThe numerator was taken from the measurements described above. The denom-\ninator was taken from \fnite-element simulations of the shielding factor for this\ngeometry as a function of \u0016.\nThis measurement technique was su\u000eciently robust to extract the tempera-\nture dependence of the shielding factor with some degree of certainty. Possible\ndrifts and temperature depends of the \ruxgate magnetometer o\u000bset were miti-\ngated by using an AC magnetic \feld. Any temperature coe\u000ecients in the rest\nof the instrumentation were controlled by performing the same measurements\nwith a copper cylindrical shell in place of the mu-metal witness cylinder.\nThis technique is quite di\u000berent than the usual transformer core measure-\nments conducted by other groups. As shall be described, it o\u000bers an advantage\nthat considerably smaller BmandHm\felds can be accessed. Measuring the\ntemperature dependence of the shielding factor is also considerably easier than\nmeasuring the temperature dependence of the reaction factor, since the sensi-\ntivity to changes in \u0016(T) is considerably larger in magnitude for the shielding\nfactor case where\u0016\nBsdBs\nd\u0016\u0018 \u00001 compared to the reaction factor case where\n\u0016\nB0dB0\nd\u0016\u00180:01.\n3.2.1. Experimental Apparatus for Axial Shielding Factor Measurements\nThe witness cylinder was placed within a homogeneous AC magnetic \feld.\nThe \feld was created within the magnetically shielded volume of the prototype\n10magnetic shielding system (described previously in Section 3.1) in order to pro-\nvide a controlled magnetic environment. A short solenoid inside the shielding\nsystem was used to produce the magnetic \feld. The solenoid has 14 turns with\n2.6 cm spacing between the wires. The solenoid was designed so that the \feld\nproduced by the solenoid plus innermost shield approximates that of an in\fnite\nsolenoid. The magnetic \feld generated by the solenoid was typically 1 \u0016T in\namplitude. The solenoid current was varied sinusoidally at typically 1 Hz.\nThe witness cylinder was placed into this magnetic \feld generation system\nas shown schematically in Fig. 2. The cylinder was held in place by a wooden\nstand.\nA Bartington \ruxgate magnetometer Mag-03IEL70 [34] (low noise) mea-\nsured the axial magnetic \feld at the center of the witness cylinder. The \ruxgate\nwas a \\\rying lead\" model, meaning that each axis was available on the end of a\nshort electrical lead, separable from the other axes. One \rying lead was placed\nin the center of the witness cylinder, the axis of the \ruxgate being aligned with\nthat of the witness cylinder. The \ruxgate was held in place rigidly by a plastic\nmounting \fxture, which was itself rigidly mounted to the witness cylinder.\nTo increase the resolution of the measured signal from the \ruxgate, a Bart-\nington Signal Conditioning Unit (SCU) was used with a low-pass \flter set to\ntypically 10-100 Hz and a gain set to typically >50. The signal from the SCU\nwas demodulated by an SR830 lock-in ampli\fer [35] providing the in-phase and\nout-of-phase components of the signal. The sinusoidal output of the lock-in\nampli\fer reference output itself was normally used to drive the solenoid gener-\nating the magnetic \feld. The time constant on the lock-in was typically set to\n3 seconds with 12 dB/oct rollo\u000b.\nAs shall be described in Section 3.2.2, a concern in the measurement was\nchanges in the \feld measured by the \ruxgate that could arise due to motion of\nthe system components, or other temperature dependences. This could generate\na false slope with temperature that might incorrectly be interpreted as a change\nin the magnetic properties of the witness cylinder.\nTo address possible motion of the witness cylinder with respect to the \feld\n11Figure 2: (color online) Axial shielding factor measurement setup. The witness cylinder with\nan inner diameter of 5.2 cm and a length of 15.2 cm is placed inside a solenoid (shown in red)\nwith a diameter of 30.8 cm and a length of 35.5 cm, containing 14 turns. The thickness of the\nwitness cylinder is 1 =1600= 0:16 cm. The loop coil (shown in blue) is mechanically coupled\nto the witness cylinder and has a diameter of 9.7 cm.\n12generation system, another coil (the loop coil, also shown in Fig. 2) was wound\non a plastic holder mounted rigidly to the witness cylinder. The coil was one\nloop of copper wire with a diameter of 9.7 cm. Plastic set screws in the holder\n\fxed the loop coil to be coaxial with the witness cylinder.\nSystematic di\u000berences in the results from the two coils (the solenoidal coil,\nand the loop coil) were used to search for motion artifacts. As well, some\ndi\u000berences could arise due to the di\u000berent magnetic \feld produced by each\ncoil, and so such measurements could reveal a dependence on the pro\fle of the\napplied magnetic \feld. This is described further in Section 3.2.2.\nThe temperature of the witness cylinder was measured by attaching four\nthermocouples at di\u000berent points along the outside of the cylinder. This allowed\nus to observe the temperature gradient along the witness cylinder. To reduce\nany potential magnetic contamination, T-type thermocouples were used, which\nhave copper and constantan conductors. (K-type thermocouples are magnetic.)\nThermocouple readings were recorded by a National Instruments NI-9211\ntemperature input module. The magnetic \feld (signi\fed by the lock-in ampli\fer\nreadout) and the temperature were recorded at a rate of 0.2 Hz.\nTemperature variations in the experiment were driven by ambient temper-\nature changes in the room, although forced air and other techniques were also\ntested. These are described further in Section 3.2.2.\n3.2.2. Data and Interpretation\nAn example of the typical data acquired is shown in Fig. 3. For these data,\nthe \feld applied by the solenoid coil was 1 \u0016T in amplitude, at a frequency\nof 1 Hz. Fig. 3(a) shows the temperature of the witness cylinder over a 70-\nhr measurement. The temperature changes of 1.4 K are caused by diurnal\nvariations in the laboratory. The shielded magnetic \feld amplitude Bswithin\nthe witness cylinder is anti-correlated with the temperature trend as shown in\nFig. 3(b). Here, Bsis the sum in quadrature of the amplitudes of the in-phase\nand out-of-phase components (most of the signal is in phase). The magnetic \feld\nis interpreted to depend on temperature, and the two quantities are graphed as a\n13function of one another in Fig. 3(c). The slope in Fig. 3(c) has been calculated\nusing a linear \ft to the data. The relative slope at 23\u000eC was found to be\n1\nBsdBs\ndT=\u00000:75%/K.\nFigs. 3(d), (e), and (f) show the same measurement with essentially the same\nsettings, when the mu-metal witness cylinder is replaced by a copper cylinder.\nA similar relative vertical scale has been used in Figs. 3(e) and (f) as Figs. 3(b)\nand (c). This helps to emphasize the considerably smaller relative slope derived\nfrom panel (f) compared to panel (c). A variety of measurements of this sort\nwere carried out multiple times for di\u000berent parameters such as coil current.\nRunning the coil at the same current tests for e\u000bects due to heating of the coil,\nwhereas running the coil at a current which equalizes the \ruxgate signal to\nits value when the mu-metal witness cylinder is present tests for possible e\u000bects\nrelated to the \ruxgate. For all measurements the temperature dependence of the\ndemodulated magnetic signal was <0:1%/K, giving con\fdence that unknown\nsystematic e\u000bects contribute below this level.\nSome deviations from the linear variation of BswithTcan be seen in the\ndata, particularly in Figs. 3(a), (b), and (c). For example, when the tempera-\nture changes rapidly, the magnetic \feld takes some time to respond, resulting\nin a slope in Bs\u0000Tspace that is temporarily di\u000berent than when the tem-\nperature is slowly varying. This is typical of the data that we acquired, that\nthe data would generally follow a straight line if the temperature followed a\nslow and smooth dependence with time, but the data would not be linear if\nthe temperature varied rapidly or non-monotonically with time. We also tried\nother methods of temperature control, such as forced air, liquid \rowing through\ntubing, and thermo-electric coolers. The diurnal cycle driven by the building's\nair conditioning system gave the most stable method of control and the most\nreproducible results for temperature slopes.\nAs mentioned earlier, data were acquired for both the solenoid coil and the\nloop coil. A summary of the data is provided in Table 1. Repeated mea-\nsurements of temperature slopes using the loop coil fell in the range 0.4%/K <\nj1\nBsdBs\ndTj<1.5%/K. Similar measurements for the solenoidal coil yielded 0.3%/K <\n14Figure 3: Ambient temperature and shielded magnetic \feld amplitude, measured over a 70\nhour period. (a) temperature of the witness cylinder as a function of time. (b) magnetic\n\feld amplitude measured by \ruxgate at center of witness cylinder vs. time. (c) magnetic \feld\nvs. temperature with linear \ft to data giving1\nBsdBs\ndT=\u00000:75%/K (evaluated at 23\u000eC). In\npanels (d), (e), and (f), the same quantities are shown for a 20-hour run with a copper cylinder\nin place of the witness cylinder with the linear \ft giving1\nBsdBs\ndT=\u00000:03%/K.\n15Trial1\nBsdBs\ndTCoil\n# (%/K) type\n1 -0.32 solenoid\n2 -0.30 solenoid\n3 -0.33 solenoid\n4 -1.53 loop\n5 -0.42 loop\n6 -1.30 loop\n7 -0.74 solenoid\n8 -1.05 loop\n9 -0.73 solenoid\n10 -1.23 loop\n11 -0.75 solenoid\n12 -1.12 loop\nTable 1: Summary of data acquired for the AC axial shielding factor measurements, in chrono-\nlogical order. Data with an applied \feld of \u00181\u00006\u0016Tand a measurement frequency of 1 Hz\nare included. Data which used daily \ructuations of the temperature from 21-24\u000eC over a\n10-80 hour period are included. Other data acquired for systematic studies are not included\nin the table.\nj1\nBsdBs\ndTj<0.8%/K.\nIn general, the slopes measured with the loop coil were larger than for the\nsolenoidal coil. This is particularly evident for measurements 6-12, which were\nacquired daily over the course of a few weeks alternating between excitation\ncoils but all used the same witness cylinder and otherwise without disturbing\nthe measurement apparatus. A partial explanation of this di\u000berence is o\u000bered\nby the \feld pro\fle generated by each coil, and its interaction with the witness\ncylinder. This is addressed further in Section 3.2.3.\nThe other di\u000berence between the loop coil and the solenoidal coil was that\nthe loop coil was rigidly mounted to the witness cylinder, reducing the possibility\nof artifacts from relative motion. Given that this did not reduce the range of\n16the measured temperature slopes we conclude that relative motion was well\ncontrolled in both cases.\nSeveral other possible systematic e\u000bects were considered, all of which were\nfound to give uncertainties on the measured slopes <0:1%/K. These included:\nthermal expansion of components including the witness cylinder itself, temper-\nature variations of the magnetic shielding system within which the experiments\nwere conducted, degaussing of the witness cylinder, and temperature slopes of\nvarious components e.g. the \ruxgate magnetometer and the lock-in ampli\fer.\nAs mentioned earlier in reference to Fig. 3(d), (e), and (f), the stability of\nthe system was also tested by replacing the mu-metal witness cylinder with a\ncopper cylinder and in all cases temperature slopes <0:1%/K were measured,\ngiving con\fdence that other unknown systematic e\u000bects contribute below this\nlevel.\nBased on the systematic e\u000bects that we studied, we conclude that they do\nnot explain the ranges of values measured for1\nBsdBs\ndT. We suspect that the range\nmeasured is either some yet uncharacterized systematic e\u000bect, or a complicated\nproperty of the material. We use this range to set a limit on the slope of \u0016(T)\n3.2.3. Geometry correction and determination of \u0016(T)\nTo relate the data on Bs(T) to\u0016(T), the shielding factor of the witness\ncylinder as a function of \u0016must be known. Finite element simulations in FEMM\nand OPERA were performed to determine this factor. The simulations are also\nuseful for determining the e\u000bective values of BmandHmin the material, which\nwill be useful to compare to the case for typical nEDM experiments when the\ninnermost shield is used as a \rux return.\nFor closed objects, such as spherical shells [26, 27], the shielding factor ap-\nproaches in\fnity as \u0016!1 , andj\u0016\nBsdBs\nd\u0016j!1. Because the witness cylinders are\nopen ended, the shielding factor asymptotically approaches a constant rather\nthan in\fnity in the high- \u0016limit, and as a result j\u0016\nBsdBs\nd\u0016j<1 here. From the\nsimulations the ratio\u0016\nBsdBs\nd\u0016was calculated. A linear model of the material was\nused where Bm=\u0016Hmwith\u0016constant.\n17j\u0016\nBsdBs\nd\u0016jj1\nBsdBs\ndTj(%/K)1\n\u0016d\u0016\ndT(%/K)\n(simulated) (measured) (extracted)\nSolenoidal Coil 0.42-0.50 0.3-0.8 0.6-1.9\nLoop Coil 0.56-0.65 0.4-1.5 0.6-2.7\nTable 2: Summary of OPERA and FEMM simulations and shielding factor measurements,\nresulting in extracted temperature slopes of \u0016.\nThe simulations di\u000bered slightly in their results, dependent on whether\nOPERA or FEMM was used, and whether the solenoidal coil or loop coil were\nused. Based on the simulations, the result is j\u0016\nBsdBs\nd\u0016j= 0:42\u00000:50 for the\nsolenoidal coil, with the lower value being given by FEMM and the upper value\nbeing given by a 3D OPERA simulation, for identical geometries. This is some-\nwhat lower than the value suggested by Ref. [36] with \fts to simulations per-\nformed in OPERA, which we estimate to be 0.6. We adopt our value since it is\ndi\u000ecult to determine precisely from Ref. [36]. For the loop coil, we determine\nj\u0016\nBsdBs\nd\u0016j= 0:56\u00000:65, the range being given again by a di\u000berence between\nFEMM and OPERA.\nCombining the measurement and the simulations, the temperature depen-\ndence of the e\u000bective \u0016(at\u0016r= 20;000 which is consistent with our measure-\nments) can be calculated by equation (3). The results of the simulations and\nmeasurements are presented in Table 2. Combining the loop coil and solenoidal\ncoil results, we \fnd 0.6%/K <1\n\u0016d\u0016\ndT<2:7%/K to represent the full range for\nthe possible temperature slope of \u0016that observed in these measurements.\nAs stated earlier, the simulations also provided a way to determine the typ-\nicalBmandHminternal to the material of the witness cylinder. According to\nthe simulations, the Bmamplitude was typically 100 \u0016T and theHmamplitude\nwas typically 0.004 A/m. These are comparable to the values normally encoun-\ntered in nEDM experiments, recalling from Section 2 that Hm<0:007 A/m for\nthe innermost magnetic shield of an nEDM experiment. A caveat is that these\nmeasurements were typically conducted using AC \felds at 1 Hz, as opposed to\n18the DC \felds normally used in nEDM experiments.\n3.3. Transformer Core Measurements\nAn alternative technique similar to the standard method of magnetic mate-\nrials characterization via magnetic induction was also used to measure changes\nin\u0016. In this measurement technique, the witness cylinder was used as the core\nof a transformer. Two coils (primary and secondary) were wound on the witness\ncylinder using multistranded 20-gauge copper wire. The windings were made as\ntight as possible, but not so tight as to potentially stress the material. The wind-\nings were not potted in place. Three witness cylinders were tested. Data were\nacquired using di\u000berent numbers of turns on both the primary and secondary\ncoils (from 6 to 48 on the primary, and from 7 to 24 on the secondary).\nFig. 4 shows a picture of one of the witness cylinders, wound as described.\nIt also shows a schematic diagram of the measurement setup, which we now use\nto describe the measurement principle.\nI(t)\n1 \nV(t)\nFigure 4: Photograph of a witness cylinder showing transformer windings (left) and schematic\nof the transformer measurement (right). The primary coil was driven by the sine-out of an\nSR830 lock-in ampli\fer, which was also used to demodulate induced voltage V(t) in the\nsecondary coil. The driving current I(t) was sensed by measuring the voltage across a stable\n1 \n resistor.\nThe primary coil generated an AC magnetic \feld as a function of time H(t),\nwhile the secondary coil was used to measure the emf induced by the time-\nvarying magnetic \rux proportional to dB(t)=dt. To a good approximation\nHm(t) =NpI(t)\n2\u0019R(4)\n19whereNpis the number of turns in the primary, I(t) is the current in the\nprimary, and Ris the radius of the witness cylinder, and\ndBm(t)\ndt=_Bm(t) =V(t)\nb`(5)\nwhereV(t) is the voltage generated in the secondary, and band`are the thick-\nness and length of the witness cylinder respectively. For a sinusoidal drive\ncurrentI(t), and under the assumption that Bm(t) =\u0016Hm(t) with\u0016being a\nconstant, the voltage generated in the secondary V(t) should be sinusoidal and\nout of phase with the primary current.\nThe internal oscillator of an SR830 lock-in ampli\fer was used to generate\nI(t). This was monitored by measuring the voltage across a 1 \n resistor with\nsmall temperature coe\u000ecient in the primary loop. The lock-in ampli\fer was\nthen used to demodulate V(t) into its in-phase VXand out-of-phase VYcom-\nponents (or equivalently _Bm(t) being demodulated into _Bm;Xand _Bm;Y, as in\nequation (5)). The experiment was done at 1 Hz with Hm(t) as small as possi-\nble, typically 0.1 A/m in amplitude, to measure the slope of the minor Bm\u0000Hm\nloops near the origin of the Bm\u0000Hmspace.\nThe temperature of the core was measured continuously using the same ther-\nmocouple arrangement described previously. Measurements of VYas a function\nof temperature would then signify a change in \u0016with temperature. In general,\nwe used ambient temperature variations for the measurements, similar to the\nprocedure used for our axial shielding factor measurements.\nThe naive expectation is that the out-of-phase VYcomponent should signify\na non-zero \u0016, and the in-phase VXcomponent should be zero. In practice, due\nto a combination of saturation, hysteresis, eddy-current losses, and skin-depth\ne\u000bects, the VXcomponent is nonzero. It was found experimentally that keeping\nthe amplitude of Hm(t) small compared to the apparent coercivity ( \u00183 A/m for\nthe 0.16 cm thick material at 1 Hz frequencies) ensured that the VYcomponent\nwas larger than the VXcomponent. This is displayed graphically in Fig. 5, where\nthe dependence of _Bm;Yand _Bm;X on the amplitude of the applied Hm(t) is\ndisplayed, for a driving frequency of 1 Hz. Clearly the value of _Bm;X can be\n20considerable compared to _Bm;Y, for larger Hmamplitudes near the coercivity.\nAt larger amplitudes, the material goes into saturation. Both _Bm;Yand _Bm;X\neventually decrease as expected at amplitudes much greater than the coercivity.\nTo understand the behavior in Fig. 5, a theoretical model of the hysteresis\nbased on the work of Jiles [37] was used. The model contains a number of ad-\njustable parameters. We adjusted the parameters based on our measurements of\nBm\u0000Hmloops including the initial magnetization curve. These measurements\nwere performed separately from our lock-in ampli\fer measurements, using an\narbitrary function generator and a digital oscilloscope to acquire them. The\nmeasurements were done at frequencies from 0.01 to 10 Hz. It was found that\nthe frequency dependence predicted by Ref. [37] gave relatively good agreement\nwith the measured Bm\u0000Hmloops once the \fve original (Jiles-Atherton [38])\nparameters were tuned.\nFor the parameters of the (static) Jiles-Atherton model, we used Bs=\n0:45 T,a= 3:75 A/m,k= 2:4 A/m,\u000b= 2\u000210\u00006,c= 0:05, which were\ntuned to our Bm\u0000Hmcurve measurements. For classical losses, we used the\nparameters \u001a= 5:7\u000210\u00007\n\u0001m,d= 1:6 mm (the thickness of the material),\nand\f= 6 (geometry factor). These parameters were not tuned, but taken\nfrom data. For anomalous losses we used the parameters w= 0:005 m and\nH0= 0:0075 A/m, which we also did not tune, instead relying on the tuning\nperformed in Ref. [37].\nThese parameters were then used to model the measurement presented in\nFig. 5, including the lock-in ampli\fer function. As shown in Fig. 5, trends in the\nmeasurements and simulations are fairly consistent. The sign of _Bm;Xrelative\nto_Bm;Yis also correctly predicted by the model (we have adjusted them both\nto be positive, for graphing purposes). We expect that with further tuning of\nthe model, even better agreement could be achieved.\nThe model of Ref. [37] makes no prediction of the temperature dependence of\nthe parameters. Ideally, the temperature dependence of _Bm;Yand _Bm;Xunder\nvarious conditions could be used to map out the temperature dependence of the\nparameters. However, this is beyond the scope of the present work.\n21 (A/m)\nmH1−101 10 Demodulated Signal (T/s)4\n−103−102−101−101Data-XSimulation-X\nData-Y\nSimulation-Y\nFigure 5: _Bm;X and _Bm;Y as a function of amplitude of the applied Hm\feld at 1 Hz. Points\nshow the acquired data. Curves display the simulation based on the model described in the\ntext.\nWe make the simplifying assumption that temperature dependence of _Bm;Y\nmay be approximately interpreted as the temperature dependence of a single\nparameter\u0016, i.e. that\n1\n_Bm;Yd_Bm;Y\ndT=1\n\u0016d\u0016\ndT: (6)\nThis is justi\fed in part by our selection of measurement parameters (the am-\nplitude ofHm= 0:1 A/m and a measurement frequency of 1 Hz) which ensure\nthat _Bm;Ydominates over _Bm;X.\nWe assign no additional systematic error for this simpli\fcation, and all our\nresults are subject to this caveat. We comment further that in our measurements\nof the axial shielding factor (presented in Section 3.2), the same caveat exists. In\nthat case the in-phase component dominates the demodulated \ruxgate signal.\nIn a sense, measuring \u0016(T) itself is always an approximation, because it is\nactually the parameters of minor loops in a hysteresis curve which are measured.\n22In reality, our results may be interpreted as a measure of the temperature-\ndependence of the slopes of minor loops driven by the stated Hm.\nMeasurements of1\n_Bm;Yd_Bm;Y\ndTas a function of Twere made. In general, the\ndata mimicked the behavior of the axial shielding factor measurements, giving a\nsimilar level of linearity with temperature as the data displayed in Fig. 3. Other\nsimilar behaviors to those measurements were also observed, for example: (a)\nwhen the temperature slope changed sign, _Bm;Ywould temporarily give a dif-\nferent slope with temperature, (b) the measured value of1\n_Bm;Yd_Bm;Y\ndTdepended\non a variety of factors, most notably a dependence on which of the three witness\ncylinders was used for the measurement, and on di\u000berences between subsequent\nmeasurements using the same cylinder.\nTable 3 summarizes our measurements of the relative slope1\n_Bm;Yd_Bm;Y\ndTfor\na variety of trials, witness cylinders, and numbers of windings. The data show\na full range of 0 :03\u00002:15%/K for1\n\u0016d\u0016\ndT=1\n_Bm;Yd_Bm;Y\ndT, again naively assuming\nthe material to be linear as discussed above. The sign of the slope of \u0016(T) was\nthe same as the axial shielding factor technique.\nA dominant source of variation between results in this method arose from\nproperties inherent to each witness cylinder. One of the cylinders (referred to as\n\fin Table 3) gave temperature slopes consistently larger1\n\u0016d\u0016\ndT\u00180:88\u00002:15%/K\nthan the other two1\n\u0016d\u0016\ndT\u00180:03\u00000:77%/K (referred to as \u000band\r, with some\nevidence that \rhad a larger slope than \u000b). We expect this indicates some\ndi\u000berence in the annealing process or subsequent treatment of the cylinders,\nalthough to our knowledge the treatment was controlled the same as for all\nthree cylinders. Since our goal is to provide input to future EDM experiments\non the likely scale of the temperature dependence of \u0016that they can expect, we\nphrase our result as a range covering all these results.\nDetailed measurements of the e\u000bect of degaussing were conducted for this\ngeometry. The ability to degauss led us ultimately to select a larger number\nof primary turns (48) so that we could fully saturate the core using only the\nlock-in ampli\fer reference output as a current source. A computer program\nwas used to control the lock-in ampli\fer in order to implement degaussing.\n23Trial1\n_Bm;Yd_Bm;Y\ndTcore\n# (%/K) used\n1 0.15 \u000b\n2 0.03 \u000b\n3 0.04 \u000b\n4 0.06 \u000b\n5 1.07 \f\n6 0.93 \f\n7 0.88 \f\n8 0.88 \f\n9 0.09 \u000b\n10 1.23 \f\n11 2.15 \f\n12 1.85 \f\n13 1.20 \f\n14 0.77 \r\nTable 3: Summary of data acquired for the transformer core measurements. Three di\u000berent\nwitness cylinders, arbitrarily labeled \u000b,\f, and\r, were used for the measurements. A 1 Hz\nexcitation frequency was used with amplitudes for Hmranging from 0.1 to 0.3 A/m. Fluc-\ntuations in the temperature ranged from 21-24\u000eC and measurement times over a 10-80 hour\nperiod are included. Other data acquired for systematic studies are not included in the table.\n24A sine wave with the measurement frequency (typically 1 Hz) was applied at\nthe maximum lock-in output power. Over the course of several thousand os-\ncillations, the amplitude was decreased linearly to the measurement amplitude\n(\u00180:1 A/m). After degaussing with parameters consistent with the recommen-\ndations of Refs. [19, 17], the measured temperature slopes were consistent with\nour previous measurements where no degaussing was done.\nOther systematic errors found to contribute at the <0:1%/K level were:\nmotion of the primary and secondary windings, stability of the lock-in ampli\fer\nand its current source, and stability of background noise sources.\nTo summarize, the dominant systematic e\u000bects arose due to di\u000berent simi-\nlarly prepared cores giving di\u000berent results, and due to variations in the mea-\nsured slopes in multiple measurements on the same core. The second of these\nis essentially the same error encountered in our axial shielding factor measure-\nments. We expect it has the same source; it is possibly a property of the\nmaterial, or an additional unknown systematic uncertainty.\n4. Relationship to nEDM experiments\nNeutron EDM experiments are typically designed with the DC coil being\nmagnetically coupled to the innermost magnetic shield. As discussed in Sec-\ntion 2, if the magnetic permeability of the shield changes, this results in a\nchange in the \feld in the measurement region by an amount\u0016\nB0dB0\nd\u0016= 0:01.\nThe temperature dependence of \u0016has been constrained by two di\u000berent\ntechniques using open-ended mu-metal witness cylinders annealed at the same\ntime as our prototype magnetic shields. We summarize the overall result as\n0.0%/K<1\n\u0016d\u0016\ndT<2.7%/K, where the range is driven in part by material prop-\nerties of the di\u000berent mu-metal cylinders, and in part by day-to-day \ructuations\nin the temperature slopes.\nWe note the following caveats in relating this measurement to nEDM exper-\niments:\n\u000fAlthough the measurement techniques rely on considerably larger frequen-\n25cies and di\u000berent Hm-\felds than those relevant to typical nEDM experi-\nments, we think it reasonable to assume the temperature dependence of\nthe e\u000bective permeability should be of similar scale. For frequency, both\ntechniques typically used a 1 Hz AC \feld, whereas for nEDM experiments\nthe \feld is DC and stable at the 0.01 Hz level. Furthermore, in one mea-\nsurement technique the amplitude of Hmwas\u00180:004 A/m and in the\nother was\u00180:1 A/m. For nEDM experiments Hm<0:007 A/m and is\nDC.\n\u000fBoth measurement techniques extract an e\u000bective \u0016that describes the\nslope of minor loops in Bm\u0000Hmspace. A more correct treatment would\ninclude a more comprehensive accounting of hysteresis in the material,\nwhich is beyond the scope of this work.\nAssuming our measurement of 0.0%/K <1\n\u0016d\u0016\ndT<2:7%/K and the generic\nEDM experiment sensitivity of\u0016\nB0dB0\nd\u0016= 0:01 results in a temperature de-\npendence of the magnetic \feld in a typical nEDM experiment ofdB0\ndT= 0\u0000\n270 pT/K. To achieve a goal of \u00181 pT stability in the internal \feld for nEDM\nexperiments, the temperature of the innermost magnetic shield in the nEDM\nexperiment should then be controlled to the <0:004 K level if the worst-case de-\npendence is to be taken into account. This represents a potentially challenging\ndesign constraint for future nEDM experiments.\nAs noted by others [39], the use of self-shielded coils to reduce the coupling\nof theB0coil to the innermost magnetic shield is an attractive option for EDM\nexperiments. The principle of this technique is to have a second coil structure\nbetween the inner coil and the shield, such that the net magnetic \feld generated\nby the two coils is uniform internally but greatly reduced externally. For a\nperfect self-shielded coil, the \feld at the position of the magnetic shield would\nbe zero, resulting in perfect decoupling, which is to say a reaction factor that\nis identically unity. For ideal geometries, such as spherical coils [40, 41, 42]\nor in\fnitely long sine-phi coils [43, 44, 45], the functional form of the inner\nand outer current distributions are the same, albeit with appropriately scaled\n26magnitudes and opposite sign. More sophisticated analytical and numerical\nmethods have been used extensively in NMR and MRI to design self-shielded\ngradient [46, 47], shim [48, 49], and transmit coils [45, 50], and should be of\nvalue in the context of nEDM experiments, as well. We are also pursuing novel\ntechniques for the design of self-shielded coils of any arbitrary \feld pro\fle and\ngeometric shape [51].\n5. Conclusion\nIn the axial shielding factor measurement, we found 0.6%/K <1\n\u0016d\u0016\ndT<\n2:7%/K, with the measurement being conducted with a typical Hm-amplitude\nof 0.004 A/m and at a frequency of 1 Hz. In the transformer core case, we\nfound 0.0%/K <1\n\u0016d\u0016\ndT<2:2%/K, with the measurement being conducted with\na typicalHm-amplitude of 0.1 A/m and at a frequency of 1 Hz.\nThe primary caveat to these measurements is that both measurements (trans-\nformer core and axial shielding factor) do not truly measure \u0016. Rather they\nmeasure observables related to the slope of minor hysteresis loops in Bm\u0000Hm\nspace. They would be more appropriately described by a hysteresis model like\nthat of Jiles [37], but to extract the temperature dependence of all the parame-\nters of the model is beyond the scope of this work. Instead we acknowledge this\nfact and relate the temperature dependence of the e\u000bective \u0016measured by each\nexperiment.\nWe think it is interesting and useful information that the two experiments\nmeasure the same scale and sign of the temperature dependence of their respec-\ntive e\u000bective \u0016's. This is a principal contribution of this work.\nIn future work, we plan to measure B0(T) directly for nEDM-like geometries\nusing precision atomic magnetometers. We anticipate based on the present work\nthat self-shielded coil geometries will achieve the best time and temperature\nstability.\n276. Acknowledgments\nWe thank D. Ostapchuk from The University of Winnipeg for technical sup-\nport. We gratefully acknowledge the support of the Natural Sciences and Engi-\nneering Research Council Canada, the Canada Foundation for Innovation, and\nthe Canada Research Chairs program.\nReferences\nReferences\n[1] A. P. Serebrov et al. , JETP Lett. 99, 4 (2014).\n[2] A. P. Serebrov et al. , Phys. Procedia 17, 251 (2011).\n[3] K. Kirch, AIP Conf. Proc. 1560 , 90 (2013).\n[4] C. A. Baker, et al. , Phys. Procedia 17, 159 (2011).\n[5] I. Altarev, et al. , Nuovo Cim. C 35, 122 (2012).\n[6] R. Golub and S. K. Lamoreaux, Phys. Rept. 237, 1 (1994).\n[7] T. M. Ito (for the nEDM Collaboration), J. Phys. Conf. Ser. 69012037,\n2007.\n[8] R. Picker (for the TRIUMF Japan-Canada UCN Collaboration), in the pro-\nceedings of MENU2016, July 25-30, 2016, Kyoto, Japan, arXiv:1612.00875\n[physics.ins-det].\n[9] C. A. Baker, et al. , Phys. Rev. Lett. 97, 131801 (2006).\n[10] J. M. Pendlebury et al. , Phys. Rev. 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Crawford, private communication .\n30"    },    {        "title": "1612.08805v1.Magnetic_structure_of_Cu2MnBO5_ludwigite__thermodynamic__magnetic_properties_and_neutron_diffraction_study.pdf",        "content": "Magnetic structure of Cu 2MnBO 5 ludwigite: thermodynamic, magnetic properties and \nneutron diffraction study \n \nEvgeniya Moshkina 1, 2, Clemens Ritter 3, Evgeniy Eremin 1, 4 , Svetlana Sofronova 1, \nAndrey Kartashev 1, 5 , Andrey Dubrovskiy 1, and Leonard Bezmaternykh 1 \n \n1Kirensky Institute of Physics, Federal Research Cen ter KSC SB RAS, Krasnoyarsk,  \n660036 Russia \n2Siberian State Aerospace University, Krasnoyarsk, 6 60014 Russia \n3Institut Max von Laue - Paul Langevin, BP 156, F-38 042 Grenoble Cedex 9, France \n4Siberian Federal University, Krasnoyarsk, 660041 Ru ssia \n5Krasnoyarsk State Pedagogical University, Krasnoyar sk, 660049 Russia  \n \nAbstract —We report on the thermodynamic, magnetic propertie s and the magnetic structure of \nludwigite-type Cu 2MnBO 5. The specific heat, the low-field magnetization an d the paramagnetic \nsusceptibility were studied on a single crystal and  combined with powder neutron diffraction \ndata. The temperature dependence of the specific he at and the neutron diffraction pattern reveal a \nsingle magnetic phase transition at T=92 K, which c orresponds to the magnetic ordering into a \nferromagnetic phase. The cation distribution and th e values and directions of magnetic moments \nof ions in different crystallographic sites are est ablished. The magnetic moments of Cu 2+  and \nMn 3+  ions occupying different magnetic sites in the fer rimagnetic phase are pairwise antiparallel \nand their directions do not coincide with the direc tions of the principal crystallographic axes.The \nsmall value of the magnetic moment of copper ions o ccupying site 2a is indicative of partial \ndisordering of the magnetic moments on this site. T he magnetization measurements show a \nstrong temperature hysteresis of magnetization, whi ch evidences for field-dependent transitions \nbelow the phase transition temperature. \n \nI. Introduction \nCu 2MnBO 5 belongs to the family of quasi-two-dimensional oxy borates with the ludwigite \nstructure. Ludwigites have a complex crystal struct ure, which involves quasi-low-dimensional \nelements (zigzag walls and three-leggedladders) for med by metal-oxygen octahedral [1−3]. The \nludwigite unit cell contains four formula units and  includes divalent and trivalent cations or \ndivalent and tetravalent ones. In this structure, m etal cations are distributed over four \nnonequivalent positions. \nThe complex crystallographic structure and the pres ence of four nonequivalent positions \noccupied by magnetic cations lead to the formation of complex magnetic structures in the \nludwigite-type crystals. In view of this, it is com plex and often impossible to determine the \nconfiguration of magnetic moments using macroscopic  magnetization measurements. In \naddition, the ludwigite structure is characterized by the large number of triangular groups formed \nby metal cations, which sometimes leads to the occu rrence of frustrations and spin-glass-like \nstates [2−12]. \nTo date, the microscopic magnetic structure has bee n experimentally determined only for \nthe monometallic ludwigites Co 3BO 5 and Fe 3BO 5 [10-12]. An important feature of the Co 3BO 5 \nand Fe 3BO 5 ludwigites is the division of the magnetic structu re into two subsystems. In Fe 3BO 5, \nwhich sees a charge ordering transition just below room temperature, the magnetic subsystems order at different temperatures with mutually ortho gonal magnetic moments [11]. The Co 3BO 5 \nludwigite displays a single magnetic transition wit h the presence of an ordered arrangement of \nlow spin and high spin states of the Co 3+  ions (S Co3+ =0) [12]. These features occur most likely to \nweaken the frustrations in the system. The magnetic  structure of ludwigites containing different \nmagnetic cations have not yet been experimentally i nvestigated; however, from the behavior of \ntheir physical properties it was concluded that the  magnetic ordering could possibly not involve \nall subsystems and that in some compounds the magne tization of different sublattices could order \nat different temperatures and point in different di rections [4, 13]. \nThe existence of Mn−Cu ludwigites was reported just  recently [6]. Single-crystal samples \nwere synthesized and the primary structural and mag netic characterization was performed for the \ncomposition Mn:Cu=1:1 (Cu 1.5 Mn 1.5 BO 5). Similar to other Cu-containing ludwigites, the \nsynthesized compound has a monoclinically distorted  ludwigite structure [7]. Due to the \npresence of quasi-low-dimensional elements in the s tructure, many ludwigites in the ordered \nphase are characterized by a strong magnetic anisot ropy between the directions H|| c and H ┴c, \nwhere c is the hard magnetization axis [4, 8, 9]. However, in the Cu 1.5 Mn 1.5 BO 5 ludwigite, the \nanisotropy is weak and the difference between the m agnetic moment values is only \nM(H||c):M(H ┴c)=1.5. This represents a fundamental difference fr om other ludwigite-type \ncompounds. In addition, in contrast to other Mn-con taining ludwigites, the Cu 1.5 Mn 1.5 BO 5 \ncompound has a large magnetic moment, which exceeds  e.g. tenfold the magnetic moment of \nNi 1.5 Mn 1.5 BO 5 [6]. \nHere we report on thorough investigations of the ph ysical properties of the Cu 2MnBO 5 \nludwigite with a different cation ratio. In contras t to the previously investigated Cu 1.5 Mn 1.5 BO 5 \ncompound, manganese ions in this ludwigite are main ly in the state with valence 3+, which \nreduces the probability of admixing divalent mangan ese to the Cu 2+  ions. In our previous study \n[5], we synthesized the Cu 2MnBO 5 ludwigite single crystals by the flux technique. I t was the \nfirst study on this compound, where its structural and magnetic properties were investigated; in \nparticular, the composition was refined, the struct ure was clarified, the magnetic transition \ntemperature was determined, the strong hysteresis i n the field-cooling (FC) and zero field-\ncooling (ZFC) modes was established, and an anomaly  in the magnetization curves near 75 K \nwas found. The group theoretical analysis was perfo rmed, the indirect exchange interactions \nwere calculated in the framework of the Anderson−Za vadsky model, and a model of the \nmagnetic structure was proposed.  \nTo shed light on the microscopic nature of the magn etic behavior and clarify the \nmechanisms of the magnetic phase transition, we stu died the magnetic structure of the \nCu 2MnBO 5 ludwigite using powder neutron diffraction, measure d and interpreted the \ntemperature dependence of specific heat of the crys tal, established orientational field-\ntemperature dependences of magnetization, and analy zed temperature dependences of the \nmagnetic susceptibility. \n \nII. Experimental Details \n \nThe Cu 2MnBO 5 ludwigite single crystals were grown by the flux t echnique. The \ncrystallization conditions were described in detail  in [5]. \nMagnetic measurements of the Cu 2MnBO 5 single crystal were performed on a Physical \nProperty Measurements System PPMS-9 (Quantum Design ) at temperatures of T=3−300 K in \nmagnetic fields of up to 80 kOe. Specific heat was measured using an original adiaba tic calorimeter with three screens at \ntemperatures from ~64 K (slightly below the nitroge n melting point) to ~320 K [14]. At low \ntemperatures (down to 2 K), the measurements were p erformed ona PPMS facility (Quantum \nDesign). The specific heat determination error was no more than 1% in both cases. \nThe investigated sample was a crystal set with a to tal mass of 244.7 mg. Specific heat of \nthe auxiliary elements (heating pad, lubricant, etc .) was determined separately. \nPowder neutron diffraction data were recorded at th e Institut Laue Langevin, Grenoble, \nFrance, on a D2B high resolution powder diffractome ter with a neutron wavelength of λ = 1.594 \nÅ at room temperature. Due to the fact that the sam ple had been prepared through crushing of \nsingle crystals, strong texture effects became visi ble in the high resolution neutron powder data. \nThis texture had disappeared only after powdering t he sample down to a grain size below 100 \nµm. The sample was placed in a cylindrical double-w all vanadium container in order to reduce \nthe absorption resulting from the B10 isotope. The temperature dependence of the neutron \ndiffraction pattern was measured on a D20 high-inte nsity powder diffractometer, as well situated \nat the Institut Laue Langevin,  with λ = 2.41 Å between 1.6 K and 150 K taking spectra of  5 min \nevery degree. Additional data were taken at base te mperature (1.6 K) and at 110 K with the \nlonger acquisition time of 45 min. As the absorptio n of the sample is stronger at λ = 2.41 Å than \nat λ = 1.594 Å, the sample had to be additionally dilut ed for these measurements by adding \naluminum powder. All neutron data were analyzed usi ng the Rietveld refinement program \nFULLPROF [15]. The aluminum powder was refined as a  second phase. Magnetic symmetry \nanalysis was performed using the program BASIREPS [ 16, 17]. \n \nIII. Magnetic Properties \n \nFigure 1 shows the temperature dependences of magne tization of the investigated \nCu 2MnBO 5 single crystal, which were obtained in the FC (coo ling in nonzero magnetic field) \nand FH (sample heating after precooling in nonzero magnetic field) regimes at H=200 Oe (H||a). \nAt a temperature of T≈90−92 K, both curves reveal the sharp magnetization  growth \ncorresponding to the phase transition from the para magnetic to the magnetically ordered state. In \nthe vicinity of the phase transition temperature, o ne can observe a small hysteresis of the FC and \nFH dependences with a value of ∆T1≈0.8 K. At lower temperatures, the dependences exhib it an \nanomalously strong temperature hysteresis in the ra nge of T≈46−85 K with a value of ∆T2≈14 K \nat H=200 Oe. To study this phenomenon, temperature dependences of the magnetization were \nmeasured as well in fields of H=20, 50, and 1000 Oe . The measurements show that the width of \nthe hysteresis depends nonlinearly on the applied m agnetic field; specifically, at H=50 Oe, we \nhave ∆T2≈5 K and at H=20 and 1000 Oe, the temperature hyster esis is less than ∆T2≈1 K. \nWhen measuring the orientational dependences of the  sample magnetization, we used a \ncrystal with the natural habit in the form of a qua drangular prism. Magnetization was measured \nalong the x, y and  z geometrical axes of the prism. The z axis coincided with the a \ncrystallographic axis and the x and y axes corresponded to the (1 1 0) and (-1 1 0) \ncrystallographic directions. \n  \nFIG. 1. Temperature dependences of magnetization ob tained in the FC (cooling at H=200 Oe) \nand FH (sample heating in a field of H=200 Oe after  precooling at H=200 Oe) regimes (H||c). \n \nFigure 2 presents the orientational dependences of magnetization of the Cu 2MnBO 5 sample \nobtained in a magnetic field of H=1 kOe. All the cu rves contain the broad asymmetrical \nmaximum, which evidences for the existence of the d omain structure in the crystal. The position \nof this maximum changes depending on the magnetic f ield direction; in the direction H|| x, one \ncan observe a shelf (constant magnetic moment regio n) in the temperature range of 5−15 K. \n \n \nFIG. 2. Temperature dependences of magnetization ob tained in a magnetic field of H=1000 Oe \napplied in the macroscopic directions H|| x, H||y, and H||z of the single-crystal samples with the \nnatural habit. \n \nThe temperature dependences of the inverse molar su sceptibility for H|| x, H||y, and H||z in \nthe temperature range of T=2−300 K are presented in Figure 3. It can be seen that above the \nmagnetic transition temperature the experimental da ta obtained with different magnetic field \ndirections do not coincide; i.e., the paramagnetic phase is characterized by anisotropy. This \nanisotropy can result from the strong g-factor anis otropy caused by the coexistence of two Jahn–\nTellerions, Cu 2+ and Mn 3+ . Not being part of this study of the low temperatu re behavior we will \ncheck this later by studying the electron spin reso nance (ESR) spectra. \n \n \nFIG. 3. Inverse susceptibility of Cu 2MnBO 5 (H||x, H||y, and H||z). \n \nIV. Powder Neutron Diffraction \nFigure 4 shows the refinement of the high resolutio n data taken at room temperature. The \ncompound crystallizes in the space group P21/c as already proposed by Bezmaternykh et al.[6] \nfor a compound with composition Cu 1.5 Mn 1.5 BO 5. In this structure, the Mn and Cu cations are \ndistributed over four different sites. Due to the s trongly differing neutron scattering lengths for \nMn ( bcoh  = −3.73 fm) and Cu ( bcoh  = 7.72 fm), it is possible to determine precisely the cation \ndistribution over these four sites.  \n \nFIG. 4. Observed (dots, red), calculated (black, li ne), and difference pattern of Cu 1.94 Mn 1.06 BO 5 \nat 295 K. The tick marks indicate the calculated po sition of the nuclear Bragg peaks. \nTable 1 gives the lattice parameters, atom coordina tes, and the occupations resulting from \nthe refinement. It can be seen that there is a clea r site preference with the Mn 3+  cation occupying \nalmost exclusively one of the 4 e sites (labelled 4 e2 in Table 1), while the Cu 2+  ion is found at a \n90% level on the 4 e1 and the 2 d and 2 a sites. The refined stoichiometry corresponds to a \nCu 1.94(1) Mn 1.06(1) BO 5compound. Bond valence calculations using the deter mined interatomic \ndistances confirm the assumed valences of +3 for Mn  and +2 for Cu. This structure is \nmonoclinically distorted with respect to the struct ure of the closely related Fe 3BO 5 compound, \nwhich crystallizes in space group Pbam at room temperature [11]. Fe 3BO 5 sees depending on \ntheir valence a strong site preference for Fe 3+  and Fe 2+ cations: while Fe 2+  resides on sites 4 g and \n2a (space group Pbam), Fe 3+  is preferentially found on sites4 h and 2 d. This can be compared to \nthe situation in our Mn 1.06 Cu 1.94 BO 5 compound, where Mn 3+  is mostly found on site 4 e2, which \ncorresponds to site 4 h in Pbam. \n \nTABLE I. Results of the Rietveld refinement of the high-resolution neutron diffraction data at \n295 K for Cu 1.94 Mn1.06 BO 5 in P21/c . \nP21/c X Y Z Occ. Mn/Cu  \n2a 0 ½ ½ 0.090(4)/0.910(4) \n2d ½ 0 ½ 0.068(4)/0.932(4) \n4e1 0.0638(6) 0.9877(2) 0.2790(1) 0.102(4)/0.898(4) \n4e2 0.576(2) 0.7324(5) 0.3785(4) 0.877(2)/0.123(2) \nB 0.4057(8) 0.2640(2) 0.3670(2)  \nO1 0.0038(8) 0.0953(2) 0.1454(2)  \nO2 0.1492(8) 0.8725(2) 0.4118(2)  \nO3 0.4661(8) 0.1187(2) 0.3654(2)  \nO4 0.6091(8) 0.6597(2) 0.5369(2)  \nO5 0.6419(7) 0.8332(2) 0.2337(2)  \na [Å] 3.13851(4)    \nb [Å] 9.4002(1)    \nc [Å] 12.0204(1)    \nβ [°] 92.267(1)    RBragg  3.7    \n \nFigure 5 shows the low-angle region of the thermal dependence of the neutron diffraction \npattern of Cu 1.94 Mn1.06 BO 5. A transition is clearly visible at about 90 K, wh ere an increase in the \nintensity of several Bragg reflections can be disce rned. In accordance with the magnetic data, \nthis transition is identified as a transition to a magnetically ordered, most probably ferromagnetic \nstate. Down to the lowest temperatures, there is no  further transition.  \n \n \nFIG. 5. Thermal dependence of the neutron diffracti on pattern of Cu 1.94 Mn1.06 BO 5 between 2 K \nand 140 K. Only every third spectrum of the origina l measurement is shown. \nUsing the program K-search, which is a part of the FULLPROF suite of refinement \nprograms, the magnetic propagation vector κ = 0 was confirmed. Fitting the intensity of the \nBragg peak having the most intense magnetic contrib ution, a transition temperature of T C = 92 K \nwas established. Magnetic symmetry analysis using t he program BASIREPS was used to \ndetermine for κ = 0 the allowed irreducible representations (IR) an d their basis vectors (BV) for \ncation sites 4 e, 2 d, and 2 a; they are listed in Table 2. \nTABLE II. Basis vectors (BV) of the allowed irreduc ible representations (IR) for κ = 0 for the \nWykoff  positions 4 e, 2 d and 2 a of space group P21/c \n IR1 IR2 IR3 IR4 \n4e BV1 BV2  BV3 BV1 BV2 BV3 BV1  BV2 BV3  BV1 BV2 BV3 \nx, y, z 1 0 0 0 1 0  0 0 1 1 0 0 0 1 0 0 0 1 1 0 0  0 1 0 0 0 1  1 0 0 0 1 0 0 0 1 \n-x, y+½, -z+½  -1 0 0  0 1 0  0 0 -1 -1 0 0  0 1 0 0 0 -1  1 0 0  0 -1 0  0 0 1  1 0 0 0 -1 0  0 0 1 \n-x, -y, -z 1 0 0 0 1 0  0 0 1 -1 0 0  0 -1 0  0 0 -1 1 0 0  0 1 0 0 0 1  -1 0 0  0 -1 0  0 -1 0  \nx, -y+½, z+½ -1 0 0  0 1 0  0 0 -1 1 0 0 0 -1 0  0 0 1 1 0 0  0 -1 0  0 0 1  -1 0 0  0 1 0 0 -1 0  \n             \n2d, 2 a             \nx, y, z 1 0 0 0 1 0  0 0 1    1 0 0  0 1 0 0 0 1     \nx, -y+½, z+½ -1 0 0  0 1 0  0 0 -1    1 0 0  0 -1 0  0 0 1     \nFor the determination and refinement of the magneti c structure, a difference data set \ncreated by subtracting the high intensity data set taken with long counting times within the \nparamagnetic phase at 110 K from the data set at 1. 6 K was used. This allows refining solely the \nmagnetic contribution and increases thereby the pre cision of the magnetic moment determination. The fixed scalefactor needed for per forming this type of purely magnetic \nrefinement gets first evaluated from the refinement  of the 110 K data set. Atomic positions were \nfixed to the values resulting from the refinement o f the high-resolution refinement (Table 1). \nTesting all the allowed IRs, it is found that the m agnetic structure sees a ferromagnetic alignment \nof spins along the a and c unit cell directions corresponding to IR3, which c orresponds to the one \nproposed already in [5].There is no contribution co ming from BV2 of this IR3, there is therefore \nno antiferromagnetic component present in the magne tic structure. Figure 6 shows the results of \nthe refinement or the difference data set 2 K – 110  K.  \n \n \n \nFIG. 6. Refinement of the difference spectrum 2 K –  110 K of Cu 1.94 Mn 1.06 BO 5. Observed (dots, \nred), calculated (line, black), and difference patt ern. The tick marks indicate the calculated \npositions of the magnetic Bragg peaks. Two regions at 2 θ ~ 50° and ~ 54° were excluded due to \nthe presence of strong up/down features at the posi tions nuclear Bragg peaks of the added Al – \nphase. \n \n \nFIG. 7. Magnetic structure of Cu 1.94 Mn 1.06 BO 5  at 2 K; the numbers correspond to the different \ncation sites: 4 e2 mainly occupied by Mn (4), 4 e1 mainly occupied by Cu (3), 2 d (2) and 2 a (1) as \nwell both mainly occupied by Cu. \nWhile the 4 e2 site, which is mainly occupied by manganese, posse sses a magnetic moment \nof about 2.7 µ B, the 4 e1 site and the 2 d and 2 a sites, which are mainly occupied by copper, have \n− as expected for a Cu 2+  ion – lower moment sizes of about 0.9, 1.1 and 0.4  µB, respectively. The \nspin directions on the different sites are not para llel, but form an arrangement comprising strong \nferrimagnetic elements. Figure 7 displays the magne tic structure where the lengths of the arrows \nreflect the relative size of the magnetic moments. Table 3 gives details of the refined magnetic \ncomponents. The corresponding Shubnikov or magnetic  space group was determined to P2 1′/c ′ \nusing the programs of the Bilbao Crystallographic S erver and of the Isotropy software package \n[18, 19]. \n \nTABLE III. Results of the refinement of the magneti c structure using BV1 and BV3 of IR3. \nMagnetic components were determined using the Mn 3+  and the Cu 2+  magnetic formfactors for \nthe different cation sites depending on which catio n occupies predominantly the concerned site. \nThe total magnetic moments µ Tot  are given in µ B. The numbering corresponds to the one used in \nFigure 7 and in the main text. \n BV1 BV3 µ Tot.  \n(1) Cuon 2 a 0.09(8) -0.44(9)  0.45(10)  \n(2) Cuon 2 d 0.60(8) 0.97(6) 1.12(9) \n(3) Cuon 4 e1 -0.23(3)  0.91(5) 0.93(6) \n(4) Mn on 4 e2 -1.93(2)  -1.91(6)  2.66(6) \nRMagn.  5.3   \n \nThe four different sublattices only possess ferroma gnetic interactions, a fact which can be \ndirectly linked to the site specific occupation by either Mn 3+  or Cu 2+  ions. 90° superexchange \ninteractions M-O-M should in fact be ferromagnetic between cations of the same type having the \nsame valence following the Goodenough−Kanamouri [20 ] rules. The reduced value of the \nmagnetic moment found for Mn 3+  - 2.7 µ B instead of the theoretical 4.0 µ B – can be related to the \nnon-negligible amount of Cu 2+  (12%) occupying the 4 e2 site which will hinder an equivalent \namount of neighboring Mn 3+ cations to adopt a ferromagnetic alignment and coul d even lead \nlocally to some antiferromagnetic Mn 3+  - Cu 2+  interactions. \n \nV. Thermodynamic Properties \nFigure 8 illustrates the specific heat measurements  in the entire temperature rangeinzero \nmagnetic field (T=2–320 K, H=0). One can observe an  anomalous behavior with a temperature \npeak at T c=88.1 K. The lattice specific heat was determined u sing linear combinations of the \nDebye–Einstein functions with the characteristic te mperatures found to be T D= 331 K and T E= \n780 K. It can be seen that the low temperature regi on isnot correctly interpolated. The same \nbehavior was previously observed in another ludwigi te crystal, Ni 5GeB 2O10  [13]. Subtracting the \nlattice contribution to the specific heat from the experimental data, we found the excess specific \nheat and the phase transition entropy ∆S= 0.6 J/(mol*K). Under the assumption that the mag netic \nmoments order completely in the crystal, the maximu m possible entropy of the magnetic phase \ntransition can be calculated from the formula: )/( 2 . 25 ) 1)(2ln( ) 1)(2ln( ∆ ∆∆2 3Kmol J Cu SRn Mn SRnSSSCu Mn Cu Mn ⋅ =+ ++ =+=+ +(1) \nWhere n Mn  and nCu  are the ion concentrations, S(Mn 3+ )=2 and S(Cu 2+ )=1/2 are the spin magnetic \nmoments of ions, and R is the universal gas constant. The magnetic phase t ransition entropy \nobtained using formula (1) exceeds by far the exper imental value. This difference is indicative of \nthe absence of complete ordering of the magnetic mo ments at this magnetic phase transition, \nwhich agrees with the resultsfrom the neutron magne tic scattering data.The partial ordering of \nthe magnetic moments is characteristic of heteromet allic ludwigites, which contain two or more \nmagnetic ions [2, 3]. The homometallic ludwigites F e 3BO 5 [10, 11, 21] and Co 3BO 5 [12, 21] are \ncharacterized, on the contrary, by the long-range m agnetic order. \nIn addition, we studied the temperature dependence of specific heat in an external magnetic \nfield of H=4.7 kOe (inset a of Figure 8). It can be  noted that the temperature of the magnetic \nphase transition changes only weakly in the applied  magnetic field while the specific heat peak is \nsignificantly spread. A similar behavior was observ ed on the completely magnetically ordered \nludwigites Co 3BO 5[21] and Co 5SnB 2O10  [22]. This behavior is indicative of the presence of \nantiferromagnetic interactions in the crystal [21].  \nWe attribute the anomaly of the excess specific hea t at Т=23 K (inset b of Figure 8) to \nadditional contributions to the lattice specific he at, which are ignored in the Debye−Einstein \nmodels. Although the compound under study is dielec tric, at temperatures close to zero, the \nspecific heat decreases in accordance with the line ar law, which were observed for all \ninvestigated ludwigites [13, 22]. \n \nFIG. 8. Specific heat curves (H=0). The black line shows experimental data and the red line, the \nlattice contribution to specific heat. Inset (a): s pecific heat curves at H=0 and H=4.7 kOe. Inset \n(b): residual specific heat. \n \nIn Section III, devoted to the magnetic properties of the investigated ludwigite, we found a \ntemperature hysteresis of the magnetization in the heating and cooling modes in magnetic fields \nof up to H=1 kOe. The dependences of magnetization contain inflection points below the phase \ntransition temperature. To study this effect, we ca lculated the temperature dependences of the \ntemperature derivative of the squared magnetization  (Figure 9), since, according to the molecular \nfield theory, the magnetic contribution to the spec ific heat is proportional to the squared \nspontaneous magnetization [23]. \n \nFIG. 9. Temperature dependences of the normalized t emperature derivative of the squared \nmagnetization at H=20, 50, 200, and 1000 Oe. \n \nFigure 9 shows the dM 2/dT(T) dependences obtained at H=20, 50, 200, and 1 000 Oe.All \ncurves show a peak independent of the external magn etic field which corresponds to the \nmagnetic phase transition at T ≈ 89 K. This temperature is consistent with the phas e transition \ntemperature determined from the specific heat measu rements and with the neutron diffraction \ndata. However, below the transition temperature T ≈89 K, the dM2/dT (T) dependences show a \nsecond peak, whose position and shape depend, to a great extent, on the applied magnetic field. \nAs the magnetic field H is increased, the peak sign ificantly spreads and shifts to lower \ntemperatures. \nAccording to the neutron diffraction data obtained,  the Cu 2MnBO 5 ludwigite undergoes the \nonly magnetic phase transition at a temperature of Tc≈92 K. However, the neutron scattering \nexperiment was carried out at H=0 and, according to  the temperature behavior of the derivative \nof the squared magnetization, in magnetic fields cl ose to zero we can expect the coincidence of \nthe position of the second peak with the phase tran sition temperature. \nThe inset in Figure 8 shows the temperature depende nce of specific heat in the range of \nT=82−96 K, which involves the phase transition regi on. It can be seen that the specific heat peak \nis fairly broad even without external magnetic fiel d (according to the temperature dependence of \nthe excess specific heat, the peak width attains ∆T≈15 K), which can suggest, e.g., the gradual \npartial ordering of the moments in the 2 а site, which manifests itself as a hysteresis in the  \nmagnetization curves. \nThe dependence of specific heat obtained at H=5 kOe  also does not exclude such an \ninterpretation due to the large field value. It can  be seen in Figure 9 that at H=1 kOe, the \nmaximum of the derivative significantly broadens an d shifts toward lower temperatures. In other \nwords, according to the temperature extrapolation o f the center position and peak shape, in a \nmagnetic field of H=5 kOe this peak can be absent. \nSuch a field dependence of the temperature anomaly peak position is observed in systems \nwith the spin-reorientation transition (see, for ex ample, [24]). As the magnetic field is increased, \nthe temperature of spin reorientation lowers. \n \n \nVI. Discussion \nTo date, the magnetic \nCo 3BO 5 [12] and Fe 3BO 5\nsomewhat different, but the main peculiarities are identical \ntwo subsystems where the first one comprises the Fe ions on sites 4 \none those of the Fe ions on sites 4 \ndifferent three leg ladders (3LL) \nperpendicular directions [1 1\nthe same two subsystems  \nmagnetic moments are directed \nand along the b axis in the second subsystem \nludwigite the magnetic order \nHowever the second subsystem \n1-3 subsystem. But, formed by 4 \nions due to the nonmagnetic low spin state of the Co \nconnected with the 3-1-3 3 LL \nIn the compound investigated by us, the magnetic mo ments lie in \nHowever, there is a certain similarity with the \nshows the magnetic moments of ions \ncommon reference point. It \nand in positions 3 and 1 lie almost in one straight and are antiferromagneti cally oriented \ntwo straights make an angle \nmagnetic subsystem is divided in the same two subsy stems, but the angle between the magnetic \nmoments amounts to about  \nFIG. 10. \nThe difference in the orientation of the magnetic m oments in \ncan be caused by the Jahn \noxygen octahedra surrounding iron lie \ndistorted due to the Jahn −Teller effect \nThe estimation of the \nthat in Fe 3BO 5 there are many frustrating interactions, since magnetic  structure has been determined only for  \n5 [10, 11]. The results obtained by [10] and \nsomewhat different, but the main peculiarities are identical : the magnetic system is divided in \nwhere the first one comprises the Fe ions on sites 4 h \none those of the Fe ions on sites 4 g and 2 a (Pbam  setting). The two subsystems form two \nleg ladders (3LL)  [4] which order in Fe 3BO 5 at different temperatures in \n1]. In the case of Co 3BO 5, the magnetic system is \n which order, however, at the same temperature \ndirected  along the c axis in the first subsystem formed by the triad 4 \nthe second subsystem  formed by the triad 3 -\norder  of the 3-1-3 subsystem is the same as in \nsubsystem  4-2-4, unlike the Fe 3BO 5, has almost the \nformed by 4 -2-4 triads, this 3LL consists only of the \ndue to the nonmagnetic low spin state of the Co 3+  ions positioned on site 4 \nLL  by super-superexchange interactions Co- O\nIn the compound investigated by us, the magnetic mo ments lie in \nthere is a certain similarity with the magnetic structure of \nshows the magnetic moments of ions on each crystallographic site; for convenience, they  have \nIt  can be seen that the magnetic moments of  \nlie almost in one straight and are antiferromagneti cally oriented \nangle  of 60°. Thus, in the crystal under study \nmagnetic subsystem is divided in the same two subsy stems, but the angle between the magnetic \n 60° with the moments lying in the ac  plane. \n Orientations of the magnetic moments (NPD data) \n \nThe difference in the orientation of the magnetic m oments in Cu \ncan be caused by the Jahn −Teller effect; as mentioned in  [5], in Fe 3BO \nsurrounding iron lie  in the bc  plane, while in Cu 2MnBO \n−Teller effect  and the long axes are turned in the \nthe exchange interactions using the Anderson −\nthere are many frustrating interactions, since the metal ions in the ludwigite  monometallic ludwigites \n[10] and [11] for Fe 3BO 5 are \nmagnetic system is divided in \n and 2 d while the second \nsetting). The two subsystems form two \nat different temperatures in \n, the magnetic system is as well divided into \ntemperature  [11]. In Fe 3BO 5 the \nformed by the triad 4 -2-4 \n-1-3 [25]. In the Co 3BO 5 \nin  the Fe 3BO 5 ludwigite. \nthe  same direction as the 3-\nthe  chains of the position 2 \npositioned on site 4 . These chains are \nO-B-O-Co. \nIn the compound investigated by us, the magnetic mo ments lie in a different plane– ac. \nmagnetic structure of  Fe 3BO 5 [11]. Figure 10 \nn each crystallographic site; for convenience, they  have a \n ions in positions 2 and 4 \nlie almost in one straight and are antiferromagneti cally oriented . The \nstudy , similar to Fe 3BO 5, the \nmagnetic subsystem is divided in the same two subsy stems, but the angle between the magnetic \n \n \n(NPD data) . \nCu 2MnBO 5 and in Fe 3BO 5 \nBO 5 the long axes of the \nMnBO 5 the octahedra are \nthe a direction as well. \n−Zavadsky model shows \nmetal ions in the ludwigite structure form triangular groups and most of them c ouple in triads with each other [26, 27 ]. The \nmagnetic moments of the two subsystems arrange orth ogonally, possibly, to reduce the \nfrustrations [26]. In Cu 2MnBO 5, part of the exchange interactions between the sub systems is also \nfrustrated and the other are very weak [5], which l eads, as in Fe 3BO 5, to the nonparallel \norientation of the moments in the subsystems.  \nSuch a separation of the magnetic system in two sub systems oriented nonparallel is \napparently characteristic of all ludwigites; howeve r, up to now the magnetic structure have been \nonly studied for Fe 3BO 5, Co 3BO 5 and now Cu 2MnBO 5. This idea is in directly confirmed by \ninvestigations of the magnetization of single cryst als of FeCo 2BO 5 and Ni 5GeB 2O10  [4, 13], \nwhich also evidence the occurrence of magnetization  in two directions. \nOne more specific feature of is the small magnetic moment of a copper ion in site 1 ( 2a ). \nThe calculation of exchange interactions showed tha t the exchange interactions with ions in site \n4 (4e 2) are weakly antiferromagnetic and the exchange int eractions with the two nearest ions on \nsite 3 (4e 1) are different: one is weakly ferromagnetic and th e other, antiferromagnetic.  \nAt the magnetic phase transition, ions in site 1 ( 2a ) are apparently weakly coupled by the \nexchange interaction with the rest ions and order i ncompletely. The FH and FC temperature \ndependences of magnetization reveal the above-discu ssed hysteresis, which can be related to the \nincomplete ordering of the magnetic moments of ions  in site 1 ( 2a ) and, as we stated above, the \nbehavior of specific heat does not contradict the p roposed model. \nVII. Conclusions \nThe structural, magnetic, and thermodynamic propert ies of the ludwigite Cu 2MnBO 5, a \nnew compound in the family of quasi-low-dimensional  oxyborates with the ludwigite structure, \nhave been studied. The quasi-two-dimensional crysta l structure and the presence of a large \nnumber of magnetic ions on different sites in the u nit cell lead to a magnetic structure which is \ndifficult to establish by macroscopic magnetic stud ies. The Cu 2MnBO 5 ludwigite is the first \nheterometallic representative of the family of ludw igites whose microscopic magnetic structure \nwas experimentally determined by neutron powder dif fraction. Similar studies had been carried \nout earlier for the monometallic Fe 3BO 5 and Co 3BO 5 ludwigites. Combining the new results on \nCu 2MnBO 5 with the results on Fe 3BO 5 and Co 3BO 5 it appears as a common feature of the \nludwigites that the magnetic structure is divided i nto two subsystems of three leg ladders labelled \n4-2-4 and 3-1-3 where the numbers represent the dif ferent magnetic cation sites forming the \nladders. This characteristic of the magnetic struct ure is linked to the specific geometry of the \ncrystal structure and occurs to weaken the frustrat ion in the system. The magnetic structure of \nCu 2MnBO 5 is more complex than in Fe 3BO 5 – the directions of all the four magnetic moments \ndo not coincide with the principal crystallographic  directions in the crystal, which is most likely \ncaused by the Jahn−Teller effect. 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"    },    {        "title": "1702.07439v1.Angle_dependence_and_optimal_design_for_magnetic_bubblecade_with_maximum_speed.pdf",        "content": "1 \n Angle-dependence and optimal de sign for magnetic bubblecade wit h \nmaximum speed \nDuck-Ho Kim,1† Kyoung-Woong Moon,2 Sang-Cheol Yoo,1,3 Dae-Yun Kim,1 Byoung-Chul \nMin,3 Chanyong Hwang,2 and Sug-Bong Choe1★ \n1Department of Physics and Institu te of Applied Physics, Seoul N ational University, Seoul, \n08826, Republic of Korea. \n2Center for Nanometrology, Korea Research Institute of Standards  and Science, Daejeon, \n34113, Republic of Korea. \n3Center for Spintronics, Korea In stitute of Science and Technolo gy, Seoul, 02792, Republic of \nKorea. \n†Present Address: Institute for  Chemical Research, Kyoto Univers ity, Kyoto, Japan \n★Correspondence to: sugbong@snu.ac.kr \n \nUnidirectional magnetic domain-wall motion is a key concept und erlying next-\ngeneration application devices. Such motion has been recently d emonstrated by applying \nan alternating magnetic field, resulting in the coherent unidir ectional motion of magnetic \nbubbles. Here we report the optimal configuration of applied ma gnetic field for the \nmagnetic bubblecade—the coherent unidirectional motion of magne tic bubbles—driven \nby a tilted alternating magnetic field. The tilted alternating magnetic field induces \nasymmetric expansion and shrinkage of the magnetic bubbles unde r the influence of the \nDzyaloshinskii-Moriya interaction,  resulting in continuous shif t of the bubbles in time. 2 \n By realizing the magnetic bubbleca de in Pt/Co/Pt films, we find  that the bubblecade speed \nis sensitive to the tilt angle w ith a maximum at an angle, whic h can be explained well by \na simple analytical form within  the context of the domain-wall creep theory. A simplified \nanalytic formula for the angle for maximum speed is then given a s  a  f u n c t i o n  o f  t h e  \namplitude of the alternating magn etic field. The present observ ation provides a practical \ndesign rule for memory and logi c devices based on the magnetic bubblecade. \nMagnetic domain-wall (DW) motion has been intensively studied a s a test body of the \nemerging spin-dependent phenomena [1–5] as well as a building b lock of the potential \napplication devices [6–9]. Such D W motion has been achieved by the spin-orbit [2–4,10] or \nspin-transfer [11–14] torques through injection of the spin-pol arized current. Fairly recently, \nMoon et al . [9] proposed another scheme to generate a similar motion by a pplying an \nalternating magnetic field to chiral DWs. The coherent unidirec tional bubble motions generated \nby this scheme is referred as a “magnetic bubblecade,” which en ables the demonstration of \nmulti-bit bubble memory operation. The key concept underlying t his scheme relies on the \nbroken symmetry caused by the Dzya loshinskii–Moriya interaction  (DMI), which induces the \nasymmetric expansion and shrinkage of magnetic bubbles [9,15,16 ]. This scheme was further \nrevealed to induce the coherent unidirectional motion of the DW s and skyrmions [17]. Here, \nwe investigate the optimal angle and magnitude of the external alternating magnetic field for \nthe magnetic bubblecade. For this study, the magnetic bubblecad e is realized in Pt/Co/Pt films \nwith sizable DMI [15], which ha ve a strong perpendicular magnet ic anisotropy (PMA) [18,19]. \nThe bubblecade speed is then examined with respect to the tilt angle and magnitude of the \nexternal alternating magnetic field. A clear angular dependence  is observed and explained using \nDW creep theory, which provides an optimal design rule for the magnetic bubblecade. 3 \n Results \nSchematic Diagram of  the Bubble Motion. \nFigure 1 shows a magnetic bubble (up domain) with the Néel DW c onfiguration caused \nby a positive DMI [15,16,20]. The magnetization (red arrows) in side the DW is pointing \nradially outward in all directi ons. By applying a tilted altern ating magnetic field, a bubblecade \nalong the + ݔ direction (yellow arrow) was generated [9], of which the measu red bubblecade \nspeed ݒ was measured by a magneto-optical Kerr effect (MOKE) microscop e with respect to \nthe tilt angle ߠ and the amplitude ܪ of the alternating magnetic field. The tilt angle of the \nelectromagnet is defined from the + z direction to the + x direction is shown in Fig. 1. \nAngle Dependence of Bubble Motion. \nFigure 2(a) plots the measured ݒ w i t h  r e s p e c t  t o  ߠ  under several fixed ܪ  as \ndenoted inside the plot. It is clear from the figure that each ݒ curve exhibits a maximum at an \nangle ߠ  as indicated by the purple arrow. Hereafter, ߠ  will denote the angle for the \nmaximum ݒ .The measured ߠ is plotted with respect to ܪ in Fig. 2(b). The inset of Fig. 2(b) \nshows that the DW speed ܸDW exactly follows the DW creep criticality by showing the linear  \ndependence with respect to ܪିଵ/ସ for the case of ߠൌ0 . \nIn the creep regime, ݒ can be described by an equation [Supplementary information \nin Ref. 9] as ݒൌܪߚ ௫ܪ௭ିଵ/ସexpൣെߙ ܪ௭ିଵ/ସ൧, where ߚ is a constant related to the asymmetry \nin the DW motion and ߙ is the creep scaling constant for ܪ௫ൌ0. The validity of the present \nformula was confirmed by experiment for the range of ܪ௫ smaller than 50 mT. By replacing \nthe strengths of the magnetic field as ܪ௭≡ܪc o sߠ  and ܪ௫≡ܪs i nߠ , the equation can be 4 \n rewritten as a function of ܪ and ߠ as given by \n\t\t\t\tݒሺߠ,ܪሻൌܪߚଷ/ସsinߠሺcosߠሻିଵ/ସexpൣെߙ ܪିଵ/ସሺcosߠሻିଵ/ସ൧.\t \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tሺ1ሻ  \nThe solid lines in Fig. 2(a) show  the best fits with Eq. (1). I n this fitting, the experimental value \nof ߙ (=6.7 T1/4) is used, which was determined from an independent measurement  of the DW \ncreep criticality [18,19,21–24]. Therefore, the fitting was don e with a single fitting parameter \nߚ .The good conformity supports the  validity of the present equa tion. \n For a given ܪ ,ߠ0 can be obtained from the maximization condition with respect t o \nߠ i.e. ∂/ݒ∂ߠ|ఏୀఏ0ൌ0, as  \n\t\t\t\tሺcosߠ0ሻଵ/ସሺ4cotଶߠ01ሻൌߙܪିଵ/ସ.                                     ( 2 )  \nSince the variation of ሺcosߠ0ሻଵ/ସ is negligibly small in comparison to ሺ4cotଶߠ01ሻ a s  \nshown by Fig. 3(a), it is sufficient to approximate ሺ4cotଶߠ01ሻ≅ߙܪିଵ/ସ for the range \nof the experimental ߠ0 ( ~ 3 0), leading to \n\t\t\t\tߠ0ൌa c o tቂඥሺߙܪିଵ/ସെ1ሻ/4ቃ.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tሺ3ሻ  \nThe solid line in Fig. 2(b) shows the numerical evaluation of E q. (3). Though the experimental \ndata appears scattered in compari son to the small variation of ߠ0, the solid line accords well \nwith the experimental data. Pl ease note that Eq. (3) does not c ontain any fitting parameters, \nbecause ߙ was determined from an independent measurement. \nFrom the creep equation ܸDWൌܸexpൣെߙሺܪ௫ሻܪ௭ିଵ/ସ൧, one can find a logarithmic \ndependence ߙሺܪ௫ሻܪ௭ିଵ/ସൌl nሺܸ/ܸDWሻ, where ܸ is the characteristic DW speed and ߙ i s  \nthe creep scaling constant. Due to the logarithmic dependence, ߠ0 in Eq. (3) is basically a 5 \n slowly-varying function of ܸDW . Please note that, even if ܸDW  varies by 10 times, ߠ0 \nchanges by only a few degrees , as we discuss later. \nTwo-Dimensional Contour Map of Bubble Motion with respect to ࣂ a n d  ࡴ. \nTo further check the validity of the present theory, we measure  the two-dimensional \ncontour map of ݒሺߠ,ܪሻ, which is plotted with respect to ߠ and ܪ as shown by Fig. 3(b). \nThe colour contrast is scaled with the value of logሺݒሻ as the scale bar shown on the right lower \nend. For this plot, ݒ  was experimentally measured for each values of ߠ  and ܪ  over the \nrange of ߠ from 10 to 65  with 5 step and the range of ܪ from 16 to 24 mT with 1-mT step. \nIn the map, each colour traces each equi-speed contour. Several  equi-speed contours are \nhighlighted by the circular symbols, of which the position ሺߠ,ܪሻ indicates the values of ߠ \nand ܪ of the same speed for each equi-speed contour. The purple soli d lines plot the prediction \nfrom the present model. It is cl ear from the figure that the mo del prediction matches well with \nthe experimental results, confir ming the validity of the presen t theory. \nDependence of ࣂ0 o n  ࡴ and ࢻ. \nFigure 4 examines the dependence of ߠ0  on ܪ  a n d  ߙ . The circular symbols are \nobtained by solving Eq. (2) numer i c a l l y  a n d  t h e  s o l i d  l i n e s  a r e  from Eq. (3). The good \nconformity between the symbols a nd lines verifies again the val idity of Eq. (3). Figure 4(a) \nplots ߠ0 with respect to ܪ for several fixed ߙ over the practical range for Pt/Co/Pt films \n[13,15,18,19,23,24]. The figure shows that, for all the values of ߙ, ߠ0 increases drastically \nas ܪ increases up to about 3 mT and then, exhibits a slow variation  as ܪ increases further. \nFigure 4(b) plots ߠ0 with respect to ߙ for several fixed ܪ .It is also seen that ߠ0 is greatly \nreduced for the range of small ߙ, but slow variation for the range of large ߙ. The present 6 \n observations provide a general guideline for the optimal ߠ0  to be about 30  f o r  p r a c t i c a l  \nexperimental conditions. \nDiscussion   We would like to mention that ݒ can be affected by the asymmetries caused by other \nmechanisms such as chiral damping [16,25,26] or DW width variat ion [27]. Since the DW \nwidth has a dependence on ܪ and ߚ is proportional to the DW width, the value of ߚ a l s o  \nvaries with respect to ܪ .However, it is confirmed for the present films that the DW wi dth \nvariation is small (< 30%) [28] a nd that the chiral damping can  be ignored owing to the \nexperimental observation of parabolic ݒ dependence on ܪ\n௫ [15]. The good conformity of the \npresent model to the experimental results reciprocally verifies  that the asymmetry of the present \nfilms is mainly governed by the DW energy variation and thus, t he present films provide a good \ntest system to examine the magnetic bubblecade caused by the DM I-induced asymmetries. \nOther films with large effects from the other mechanisms [24,25 –27,29] require further \ninvestigation for each o ptimal configuration. \n Finally, we like to mention that the bubblecade can be realize d even in the depinning \nand flow regimes, where a similar asymmetric DW motion appears [16]. Though we expect \nthat there also exists angular d ependence of the maximum bubble cade speed, here we limit our \nanalytic description only in the creep regime, since the origin  of the asymmetry in the other \nregimes is not fully understood yet. However, a similar approac h can be applicable to those \nregimes, once the analytic formula on ܪ௫ dependence is uncovered. \nIn conclusion, we examined the optimal configuration of the ext ernal magnetic field \nfor the magnetic bubblecade. Fro m the clear angular dependence of the bubblecade speed, the 7 \n o p t i m a l  a n g l e  f o r  t h e  m a x i m u m  s p e e d  w a s  d e t e r m i n e d  e x p e r i m e n t a l ly and explained \ntheoretically by a model based on  the DW creep theory. The opti mal angle is finally given by \na simple equation of the amplitude of the alternating magnetic field. Our findings directly \nelucidate the major factors on the dynamics in the magnetic bub blecade, enabling the design \nof the optimal device configuration. \nMethods \nSample preparations.  For this study, Pt/Co/Pt films with strong perpendicular magne tic \nanisotropy (PMA) were prepared [18]. The detailed layer structu re is 5.0-nm Ta/2.5-nm Pt/0.3-\nnm Co/1.0-nm Pt, which was deposited on a Si wafer with 100-nm SiO 2 by use of dc magnetron \nsputtering. All the film s exhibit clear circular domain expansi on with weak pinning strength \n[9,18]. This film has sizeable DMI , which induces asymmetric DW  motion. \nMeasurement of the bubble speed.  The magnetic domain images were then observed by use \nof a MOKE microscope equipped with a charge-coupled device came ra. To apply a tilted \nmagnetic field onto the films, a Ferris-wheel-like electromagne t is mounted to the microscope, \nsuch that it revolves on the x-z plane around the focal point of the microscope. The magnetic \nfield can be varied up to 35 mT on the focal plane. The tilt an gle of the electromagnet can be \ncontrolled from 0 to 90  in 5 steps. To measure the bubblecade speed ݒ ,a magnetic bubble \nwas initially created by use of the thermomagnetic writing tech nique [9,14,19]. To apply \nalternating magnetic field, a magnetic field pulse of ܪ with a duration time ∆ݐ is applied \nwith an angle ߠ and successively, a reversed magnetic field pulses of െܪ with the same ∆ݐ \nand ߠ is applied. After application of each field pulse, the domain image is captured by the \nMOKE microscope. The bubblecade speed is calculated by measurin g the center position of \nthe bubble in each image.  8 \n References \n1. Thomas, L. et al. Oscillatory d ependence of current-driven m agnetic domain wall motion on \ncurrent pulse length, Nature 443, 197–200 (2006).  \n2. Miron, I. M. et al. Current-d riven spin torque induced by th e Rashba effect in a ferromagnetic \nmetal layer, Nat. Mater. 9, 230–234 (2010). \n3. Miron, I. M. et al. Fast current-induced domain-wall motion controlled by the Rashba effect, \nNat. Mater. 10, 419–423 (2011). \n4. Ryu, K.-S., Thomas, L., Yang, S.-H. & Parkin, S. Chiral spin  torque at magnetic domain \nwalls, Nat. Nanotech. 8, 527–533 (2013). \n5. Thomas, L. et al. Resonant amplification of magnetic domain- wall motion by a train of \ncurrent pulses, Science 315, 1553–1556 (2007). \n6. Parkin, S. S. P., Hayashi, M. & Thomas, L. Magnetic domain-w all racetrack memory, \nScience 320, 190–194 (2008). \n7. Allwood, D. A. et al. Magnetic  domain-wall logic, Science 390, 1688–1692 (2005). \n8. Franken, J. H., Swagten, H. J.M. & Koopmans, B. Shift regist ers based on magnetic domain \nwall ratchets with pe rpendicular anisot ropy, Nat. Nanotech. 7, 499–503 (2012). \n9. Moon, K.-W. et al. Magnetic bubblecade memory based on chira l domain walls, Sci. Rep. 5, \n9166 (2015). \n10. Haazen, P. P. J. et al. Domai n wall depinning governed by t he spin Hall effect, Nat. Mater. \n12, 299–303 (2013). 9 \n 11. Zhang, S. & Li, Z. Roles of nonequilibrium conduction elect rons on the magnetization \ndynamics of ferromagnets, Phys. Rev. Lett. 93, 127204 (2004). \n12. Beach, G. S. D., Knutson, C., Nistor, C., Tsoi, M. & Erskin e, J. L. Nonlinear domain-wall \nvelocity enhancement by spin-pol arized electric current, Phys. Rev. Lett. 97, 057203 (2006). \n13. Lee, J.-C. et al. Universality classes of magnetic domain w all motion, Phys. Rev. Lett. 107, \n067201 (2011). \n14. Moon, K.-W. et al. Distinct universality classes of domain wall roughness in two-\ndimensional Pt/Co/Pt films, Phys. Rev. Lett. 110, 107203 (2013). \n15. Je, S.-G. et al. Asymmetric magnetic domain-wall motion by the Dzyaloshinskii-Moriya \ninteraction, Phys. Rev. B 88, 214401 (2013). \n16. Kim, D.-H., Yoo, S.-C., Kim, D.-Y ., Min, B.-C. & Choe, S.-B . Universality of \nDzyaloshinskii-Moriya interaction effect over domain-wall creep  and flow regimes, \narXiv:1608.01762 (2016).  \n17. Moon, K.-W. et al. Skyrmion motion driven by oscillating ma gnetic field, Sci. Rep. 6, \n20360 (2016). \n18. Kim, D.-H. et al. Maximizing domain-wall speed via magnetic  anisotropy adjustment in \nPt/Co/Pt films, Appl. Phys. Lett. 104, 142410 (2014). \n19. Kim, D.-H. et al. A method for compensating the Joule-heati ng effects in current-induced \ndomain wall motion, IEEE Trans. Magn. 49(7) , 3207–3210 (2013). 10 \n 20. Kim, D.-Y ., Kim, D.-H., Moon, J. & Choe, S.-B. Determinatio n of magnetic domain-wall \ntypes using Dzyaloshinskii–Moriya-interaction induced domain pa tterns, Appl. Phys. Lett. 106, \n262403 (2015). \n21. Lemerle, S. et al. Domain wall creep in an Ising ultrathin magnetic film, Phys. Rev. Lett. \n80, 849 (1998). \n22. Kim, K.-J. et al. Interdimensional universality of dynamic interfaces, Nature 458, 740–742 \n(2009). \n23. Metaxas, P. J. et al. Creep and flow regimes of magnetic do main-wall motion in ultrathin \nPt/Co/Pt films with perpendicular anisotropy, Phys. Rev. Lett. 99, 217208 (2007). \n24. Lavrijsen, R. et al. Asymmetric magnetic bubble expansion u nder in-plane field in Pt/Co/Pt: \nEffect of interface engineering, Phys. Rev. B 91, 104414 (2015). \n25. Jué, E. et al. Chiral dampi ng of magnetic domain walls, Nat . Mater. 15, 272–277 (2016). \n26. Akosa, C. A., Miron, I. M., Gaudin, G. & Manchon, A. Phenom enology of chiral damping \nin noncentrosymmetric magnets, Phys. Rev. B 93, 214429 (2016). \n27. Kim, D.-Y ., Kim, D.-H. & Choe, S.-B. Intrinsic asymmetry in  chiral domain walls due to \nthe Dzyaloshinskii–Moriya interaction, Appl. Phys. Express 9, 053001 (2016) \n28. Je, S.-G. et al. Drastic emergence of huge negative spin-tr ansfer torque in atomically thin \nCo layers, arXiv:1512.03405 (2015). \n29. Lau, D., Sundar, V ., Zhu, J.-G. & Sokalski, V . Energetic mo lding of chiral magnetic bubbles, \nPhys. Rev. B 94, 060401(R) (2016). 11 \n Figure Captions \nFigure 1. Schematic descriptions of the unidirectional bubble m otion induced by tilted \nalternating magnetic field.  Illustration of a bubble domain (bright circle) and the DW (gr ey \nring), surrounded by a domain of opposite magnetization (dark a rea). The red symbols and \narrows indicate the direction of the magnetization inside the D W and domains. The dashed \ncircles represent the previous bubble positions and the yellow arrow indicates the direction of \nthe bubble motion. \nFigure 2. Angle dependence of magnetic bubble motion (a) Measured ݒ with respect to ߠ \nfor several ܪ( symbols). The solid lines are best fits with Eq. (1). The pur ple arrow represents \nߠ. The inset plots ܸDW with respect to ܪିଵ/ସ for the case of ߠൌ0 . (b) ߠ with respect to \nܪ .The solid line is the numerical evaluation of Eq. (3) with th e experimental value of ߙ \n(=6.7 T1/4). \nFigure 3. Simplification of ሺࣂܛܗ܋0ሻ/ሺܜܗ܋ࣂ0ሻ and two-dimensional contour \nmap of ࢜ሺࡴ,ࣂሻ  a s  a  f u n c t i o n  o f  ࣂ  and ࡴ. ( a) Numerical calculations of ሺcosߠ0ሻଵ/ସ, \nሺ4cotଶߠ01ሻ , and ሺcosߠ0ሻଵ/ସሺ4cotଶߠ01ሻ as a function of ߠ0 . (b) Two-dimensional \ncontour map of ݒሺܪ,ߠሻ plotted with respect to ߠ and ܪ .The colour contrast represents the \nvalue of logሺݒሻ with scale bar on the right lower end. The purple solid line i s the numerical \nevaluation of Eq. (3). \nFigure 4. The dependence of ࣂ0 as a function of ࡴ and ࢻ. (a) ߠ0 with respect to ܪ for \nvarious ߙ and (b) ߠ0 with respect to ߙ for various ܪ .The circular symbols are obtained \nby solving Eq. (2) num erically and the solid lines are from Eq.  (3). The open symbols indicate \nthe experimental data. 12 \n Acknowledgements  \nThis work was supported by a National Research Foundations of K orea (NRF) grant that was \nfunded by the Ministry of Science, ICT and Future Planning of K orea (MSIP) \n(2015R1A2A1A05001698 and 2015M3D1A 1070465). D.-H.K. was support ed by a grant \nfunded by the Korean Magnetics Society. K.W.M. and C.H. were su pported from the National \nScience Foundation Grant No. DMR-1504568, Future Materials Disc overy Program through \nthe National Research Foundation of Korea (Grant No. 2015M3D1A1 070467). B.-C.M. was \nsupported by the KIST institutional program and Pioneer Researc h Center Program of \nMSIP/NRF (2011-0027905). \nAuthor contributions \nD.-H.K. and K.-W.M. planned and designed the experiment and S.- B.C. supervised the study. \nD.-H.K. and D.-Y .K. carried out the measurement. S.-C.Y . and B. -C.M. prepared the samples. \nD.-H.K., K.-W.M., S.-B.C., and C.H. performed the analysis and D.-H.K. and S.-B.C. wrote \nthe manuscript. All authors disc ussed the results and commented  on the manuscript.  \nAdditional information \nCorrespondence and request for mat erials should be addressed to  S.-B.C. \nCompeting financial interests \nThe authors declare no competing financial interests. "    },    {        "title": "1702.08347v1.Basics_of_the_magnetocaloric_effect.pdf",        "content": "Basics of the magnetocaloric e\u000bect\nVittorio Basso\nIstituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135 Torino, Italy\n(Dated: April, 7, 2013)\nThis chapter reviews the basic physics and thermodynamics that govern magnetocaloric materials.\nThe thermodynamics of magnetic materials is discussed by introducing relevant free energy terms\ntogether with their microscopic origin leading to a discussion of the sources of entropy that can\nchange in an applied magnetic \feld. Such entropies account for measurable magnetocaloric e\u000bects,\nespecially in the vicinity of magnetic phase transitions. Particular attention is devoted to \frst\norder magnetic transitions that involve the coupling of spin, lattice, electronic and anisotropic\nmagneto-crystalline degrees of freedom. The problem of irreversibility and hysteresis, present in\nmagnetocaloric materials with \frst order phase transitions is discussed in the context of out-of-\nequilibrium thermodynamics and hysteresis modeling.\nContents\nI. Introduction 1\nII. Thermodynamics and magnetocaloric e\u000bect 2\nA. Equilibrium thermodynamics 2\n1. Thermodynamic potentials and equations of state 2\n2. Demagnetizing e\u000bects 3\nB. Magnetocaloric e\u000bect 4\n1. \u0001sisoand \u0001Tad 4\n2. Thermodynamics of \frst order phase transitions 5\nIII. Second order magnetic transitions 7\nA. Entropy of magnetic moments 7\n1. Statistical mechanics of a paramagnet 7\n2. Magnetic moment and electron spin 9\nB. Ferromagnets 10\n1. Mean \feld theory of a ferromagnet 10\n2. Magnetocaloric e\u000bect around the second order critical point 11\nIV. First order magnetic transitions 12\nA. Coupling between magnetism and structure 13\n1. Exchange energy 13\n2. Structural and electronic free energy 14\nB. First order transition due to magneto-elastic coupling 16\n1. The Bean-Rodbell model 17\n2. Magnetocaloric e\u000bect around the \frst order phase transition 18\nC. Magnetocrystalline anisotropy energy 19\nV. Hysteresis and modeling 20\nA. Hysteresis and entropy production 20\nB. Equivalent driving force 21\nC. Preisach-type models 22\nReferences 24\nI. INTRODUCTION\nThe recent discoveries of a so-called \\giant\" magnetocaloric e\u000bect (MCE) in alloys such as Gd 5Si2Ge2, La(Fe,Si) 13\nand (Mn,Fe) 2(P,Z), have driven strong research e\u000borts focused on its maximization. A principal aim has been thearXiv:1702.08347v1  [cond-mat.mtrl-sci]  27 Feb 20172\ndevelopment of magnetic cooling devices working around room temperature, which will use such alloys as their\nrefrigerant. In this chapter, we review the basics of the magnetocaloric e\u000bect by \frst considering the general physics\nof magnetic materials [1{7] and the relevant thermodynamics [8{11] that governs magnetocaloric properties [12{18].\nThis chapter is organized as follows. In Section II we introduce the magnetocaloric e\u000bect in the context of equilibrium\nthermodynamics. In Sections III and IV we introduce the thermodynamics of magnetic materials by discussing the\nmicroscopic origin of the relevant free energy terms. We are particularly interested to review the mechanisms that give\nrise to magnetic phase transitions and the sources of a magnetic \feld-induced entropy change in their vicinity, since\nthe largest MCE is found around such transitions. Particular attention is devoted to \frst order magnetic transitions\nwhich involve the coupling of di\u000berent degrees of freedom including: spin, lattice, electronic and magneto-crystalline\nanisotropy.\nThis is an active \feld of research where di\u000berent approaches and interpretations have been proposed and are\ncurrently widely discussed in the literature [12, 18]. We introduce the basic ideas underlying di\u000berent approaches\nwith the aim of presenting their conceptual basis together with their intrinsic limitations. We describe: in Section\nIII the second order magnetic phase transition of a ferromagnet in the mean \feld theory and in Section IV, the \frst\norder phase transition in the Bean-Rodbell model of magneto-elastic coupling [19]. In both cases we discuss the\nconsequences for the magnetocaloric e\u000bect. In Section V we touch the problem of the irreversibility and hysteresis,\nbecause most of the magnetocaloric materials with \frst order phase transitions display temperature hysteresis as well\nas magnetic \feld hysteresis. We refer to concepts and models that have been developed to describe magnetic hysteresis\nin ferromagnets and can be extended to phase transitions [20]. Through a basic understanding of the mechanisms\nof \frst order magnetic phase transitions we hope to fully exploit the cooling potential of magnetic materials and to\nmake magnetic refrigeration at room temperature a viable alternative to conventional refrigeration technologies.\nII. THERMODYNAMICS AND MAGNETOCALORIC EFFECT\nA. Equilibrium thermodynamics\n1. Thermodynamic potentials and equations of state\nEquilibrium thermodynamics, or Gibbs thermostatics, is a theory that applies to systems which are uniquely de\fned\nby the values of their state variables [10]. It is natural to take as state variables the set of its extensive properties: the\ninternal energy U, the volume V, the magnetic moment m, the entropy Sand so on. The system is then de\fned once\na relation connecting the state variables are known. This expression is called the fundamental equation and consists\nof the expression relating internal energy Uto all the other state variables U(m;V;S;::: ). The corresponding intensive\nconjugated variables are de\fned by the derivatives of the internal energy. For example: the pressure is p=\u0000@U=@V ,\nthe magnetic \feld is \u00160H=@U=@m , where\u00160is the permeability of free space, the temperature is T=@U=@S , and\nso on. When dealing with material properties it is useful to introduce speci\fc quantities as volume densities or mass\ndensities. For solid magnetic materials it is reasonable to assume that mass is conserved and to allow the volume to\nchange. Hence for magnetocaloric materials it is then appropriate to use speci\fc extensive variables, calculated as\nmass densities. We here introduce: the speci\fc internal energy u, the speci\fc volume v, the magnetization M, the\nspeci\fc entropy sand so on. In order to have an explicit dependence on the conjugated intensive variables rather\nthan on the extensive ones, the speci\fc free energy fand Gibbs free energy ghave also to be introduced. The free\nenergyf(M;v;T ) depends explicitly on the temperature Tand has the following relation with the internal energy\nf=u\u0000Ts. Consequently, the derivatives of the speci\fc free energy f(M;v;T ) gives the magnetic \feld\n\u00160H=@f\n@M(1)\nthe pressure\np=\u0000@f\n@v(2)\nand the speci\fc entropy\ns=\u0000@f\n@T(3)3\nThe relations obtained by these derivatives are the state equations. They express the dependence of H,pandson the\nindependent variables M,vandT. The speci\fc Gibbs free energy g(H;p;T ) which is related to the speci\fc free energy\nbyg=f\u0000\u00160HM +pvand depends only on the intensive variables. For a magnetic material the speci\fc Gibbs free\nenergy is particularly useful because the intensive variables magnetic \feld H, pressurepthe temperature Tare often\nthe externally controlled variables in experiments. The derivatives of the Gibbs potential gives the magnetization:\n\u00160M=\u0000@g\n@H(4)\nthe volume\nv=@g\n@p(5)\nand the speci\fc entropy\ns=\u0000@g\n@T(6)\nSince the three state equations are obtained by the derivatives of the same function g(:), it turns out that they are\nnot independent of each other. Due to the properties that an equilibrium thermodynamic potential must satisfy [10],\nthe second mixed derivatives of g(:) coincide, with the consequent relations being known as Maxwell relations:\n\u00160@M\n@T\f\f\f\f\nH;p=@s\n@H\f\f\f\f\np;T(7)\n@v\n@T\f\f\f\f\nH;p=\u0000@s\n@p\f\f\f\f\nH;T(8)\n\u00160@M\n@p\f\f\f\f\nH;T=\u0000@v\n@H\f\f\f\f\np;T(9)\n2. Demagnetizing e\u000bects\nWhen the previous de\fnitions are extended to vector quantities, each component of the magnetic \feld vector is\ngiven by the derivative with respect to the relative magnetization component: \u00160Hx=@u=@M xand so on. If the\nmagnetic system consists of a body of \fnite size, we have also to take explicitly into account the energy term associated\nwith the creation of a magnetostatic \feld ~HMgenerated by the magnetization distribution in space [3, 5]. ~HMis\ngiven by the solution of the magnetostatic Maxwell equations r\u0001~HM=\u0000r\u0001~Mvandr\u0002~HM= 0 and the energy\nof the magnetostatic \feld is given by the integral extending over the magnetic body volume V:\nUM=\u0000\u00160\n2Z\nV~HM\u0001~Mvd3r (10)\nwhere~Mvis the magnetization vector as volume density. The magnetostatic energy depends on the internal distri-\nbution of the magnetization, however when it is reasonable to consider the magnetization as uniform inside the body,\nthe problem is greatly simpli\fed because the magnetostatic \feld is due to the distribution of the magnetization at\nthe sample surface only. A simplifying case is when the sample is ellipsoidal; then the magnetostatic \feld is spa-\ntially uniform inside the body. By taking the reference frame along the tree main axis ( a;b;c ) of the ellipsoid, the\nmagnetostatic energy is\nUM=V1\n2\u00160\u0000\nNaM2\nvx+NbM2\nvx+NcM2\nvz\u0001\n(11)4\nwhere the dimensionless proportionality factors are the demagnetizing coe\u000ecients which depend only on the aspect\nratios of the ellipsoid and have the property Na+Nb+Nc= 1. In the case of spatial uniformity the magnetostatic \feld is\nalso called demagnetising \feld ~Hd=~HM, because, as it can be seen by taking the derivatives of Eq.11, it is proportional\nto the magnetization components, but oriented in the opposite direction ~Hd=\u0000(NaMvx+NbMvy+NcMvz. In presence\nof both an applied \feld (i.e. applied by suitable coils) ~Haand the demagnetizing \feld ~Hd, the two contributions\nsuperpose to give the magnetic \feld ~H:~H=~Ha+~Hd.\nOne of the problems of the thermodynamics of magnetism is if the magnetostatic energy has to be included in\nthe thermodynamic internal energy Uor not [3, 5]. As often in thermodynamics, the choice is left to the analysis\nof the experimental constraints. In fact the result is that, if the magnetostatic energy term is not included in U,\nthen the intensive variable coupled to the magnetization is the \feld ~H, while if it is included in U, the intensive\nvariable coupled to the magnetization results to be the applied \feld ~Ha. In many experimental situations, it may be\nrelatively easy to control the applied magnetic \feld ~Ha, while the control of Hmay require a detailed knowledge of the\ndemagnetizing coe\u000ecients and a feedback control on the sources of ~Ha. If the applied \feld ~Hais used a \feld variable,\nall the thermodynamic relations derived in the previous section are still valid, but one has to bear in mind that the\ninternal energy of the system and all the thermodynamic potentials will contain also the energy of the demagnetizing\n\feld. This mans that the corresponding thermodynamics will depend on the shape of the sample through its aspect\nratio.\nB. Magnetocaloric e\u000bect\n1.\u0001sisoand\u0001Tad\nThe magnetocaloric e\u000bect is de\fned as the adiabatic temperature change \u0001 Tador the isothermal entropy change\n\u0001sisodue to the application of the magnetic \feld Hat constant pressure [12]. For systems in thermodynamic\nequilibrium, the two quantities are derived by the entropy state equation s(H;T) at constant pressure as shown in the\nsketch of Fig.1a. The isothermal entropy change \u0001 sisois the di\u000berence between two curves at the same temperature,\nT:\n\u0001siso(H;T) =s(H;T)\u0000s(0;T) (12)\nwhile the adiabatic temperature change is the di\u000berence between two curves at the same entropy s(Fig.1):\n\u0001Tad(H;s) =T(H;s)\u0000T(0;s) (13)\nThe \u0001Tadcan also be expressed as a function of the temperature T=T(0;s) at zero magnetic \feld, giving \u0001 Tad(H;T),\nas it is commonly done in experiments. The two quantities \u0001 siso(H;T) and \u0001Tad(H;T) are not independent because\nthey are related to the slope of the s(H;T) curve as a function of Tand therefore to the speci\fc heat:\ncp(H;T) =T@s\n@T\f\f\f\f\nH;p(14)\nMagnetic refrigeration cycles can be drawn in the ( s;T) diagram as shown in Fig.1b. Without going into the details\nof the speci\fc magnetic thermodynamic cycles employed in magnetic refrigeration (see [21] and other chapters in\nthis book), we simply observe that cycles of high cooling power and large temperature span can be realized by the\nmaximization of both \u0001 sisoand \u0001Tadof the magnetic material. For example in a magnetic Carnot cycle ABCD,\nQc=Tc\u0001sDAis the heat extracted from the cold bath and \u0001 TABis the di\u000berence between the hot and cold bath\ntemperatures Th\u0000Tc= \u0001TAB. Such quantities can be derived from the entropy state equation s(H;T) of the magnetic\nmaterial which in turn can be constructed by the integration of the magnetic \feld-dependent experimental speci\fc\nheat [22, 26, 27]. An example of the inter-relation of magnetocaloric properties is shown in Fig.2 for La(Fe-Co-Si) 13.\nThe Maxwell relation of Eq.(7) is particularly important for the MCE, because it relates the entropy s(H;T)\nequations of state with the the magnetization M(H;T) (all at the same constant pressure). The magnetic \feld-\ninduced entropy change s(H1)\u0000s(H0) can be often be computed from magnetization measurements:\ns(H1)\u0000s(H0) =\u00160ZH1\nH0@M\n@TdH (15)5\ns\nTH=0Hcp = T ds/dTΔsisoΔTadABCDs\nTΔsisoH=0a)b)c)HlimΔTad\nFIG. 1: a) Entropy state equation s(H;T). b) Magnetic Carnot cycle in the ( s;T) diagram. c) Entropy state equation s(H;T)\nfor an ideal \frst order transition.\nand the temperature-induced magnetization change M(T1)\u0000M(T0) can be computed from entropy change measure-\nments:\nM(T1)\u0000M(T0) =1\n\u00160ZT1\nT0@s\n@HdT: (16)\nMagnetic materials with a second order phase transition where, at the critical temperature, the magnetic system\npasses from an ordered ferromagnetic state to a disordered paramagnetic state can be considered to be always in\nthermodynamic equilibrium. Then the equilibrium relations derived in the previous section apply well. Equation\n(15) gives a practical way to determine the entropy change without the need of calorimetric setups [22, 23, 25], while\nEq.(16) has been used in the past to arrive at an accurate determination of the saturation magnetization close to\nteh critical temperature of magnetic materials [24]. Refs.[22, 26] show that in magnetic materials with second order\ntransitions the entropy change \u0001 s(H;T) constructed experimentally by either the integration of cp(H;T) or by the\nMaxwell relation Eq.(15) are in good agreement as expected.\n2. Thermodynamics of \frst order phase transitions\nThe equilibrium thermodynamics developed so far requires that the system state corresponds to a global potential\nenergy minimum. When this is not true, the free energy has more then one global minimum, leading to a \frst order\nphase transition [10]. In Fig. 3a, a free energy potential fLwith two minima as a function of the magnetization M\nis shown as an example. Here we use the subscript L(Landau) to denote that the potential is a non-equilibrium\none. When computing the magnetic \feld state equation \u00160H=@fL=@M corresponding to this example potential,\none \fnds that M(H) has an s-shaped curve (Fig. 3b). If the magnetic \feld His used as controlling variable, there\nare multiple values of Mcorresponding to the same H, a result which is not compatible with the assumptions made\nfor uniquely de\fned, equilibrium states. The thermodynamic behavior of such a system characterized therefore has\nan intrinsically out-of-equilibrium character in the s-shaped region.\nHere we are interested on how the system may pass from one minimum to the other by making an abrupt phase\ntransition since such transitions are associated with the largest single changes in entropy and temperature. If one\nconsiders the local stability of the energy minima, the evolution of the system state follows a global instability\ncorresponding to the dashed lines of Fig.3b. There are two contrasting cases. The \frst is the completely out-of-\nequilibrium picture in which the system transforms into in the new state only at a critical \feld H=Hcrat which\nthe original minimum is completely unstable. Such a situation also generates hysteresis and is generally followed only\nif there are no other energetically favorable ways to pass to the low energy minimum before the instability occurs.\nHowever, the macroscopic system always possesses many internal degrees of freedom by which, with the contribution6\n2702 802 903 003 1001232702 802 903 003 105006007008009002\n852 902 953 003 0550607080902\n702 802 903 003 1002468L\na(Fe-Co-Si)13 \n0.5T \n1.0T \n1.5TΔT(K)T\n(K)1.5 T1.0 T0.5 T0.0 TLa(Fe-Co-Si)13 \n 0.0 T \n 0.5 T \n 1.0 T \n 1.5 Tcp(Jkg-1K-1)T\n (K)0.0 T0\n.5 T1\n.0 T1\n.5 TLa(Fe-Co-Si)13 \n 0.0 T \n 0.5 T \n 1.0 T \n 1.5 Ts-s0 (Jkg-1K-1)T\n(K)L\na(Fe-Co-Si)13 \n0.5T \n1.0T \n1.5 T \n0.5T iso \n1.0T iso-Δs (Jkg-1K-1)T\n(K)0.5 T1.0 T1.5 T1.5 T1\n.0 T0\n.5 T\nFIG. 2: Magnetocaloric e\u000bect in La(Fe 1\u0000x\u0000yCoySix)13withx= 0:077 andy= 0:079 [57]. Top left: cp(H;T); top right:\ns(H;T); bottom left: \u0001 siso(H;T); bottom right: \u0001 T(H;T). After Ref.[119].\nof spontaneous \ructuations, they are generally able to transform to the new phase before the global instability. From\nthis idea of a phase transition a second situation arises when it is possible to use the Maxwell convention in which the\nsystem may spontaneously select the minimum with the lowest global Gibbs free energy gL=fL\u0000\u00160HM (Fig.3c). If\nso, then equilibrium behavior is recovered as the selection of the lowest minimum has the e\u000bect to remove the e\u000bects\nof the energy barrier. At the \feld H=Heqwhen the two minima have the same energy level (dashed line of Fig.3\nright) the system can be indi\u000berently in one phase or the other or in a phase coexistence state at no additional energy\ncost. The corresponding phase transformation (Fig.3b) is a vertical line without hysteresis.\nA limit case which is of interest for magnetic refrigeration is the state equation s(H;T) for an ideal \frst order\nequilibrium phase transition in which the entropy has discontinuous change (see Fig.1c). The temperature at which\nthe transition occurs depends on the magnetic \feld Hand its derivative is given by the Clausius-Clapeyron equation:\ndT\ndH=\u0000\u00160\u0001M\n\u0001s(17)\nwhere \u0001Mand \u0001sare the discontinuous changes of the magnetization and the entropy at the transition. If \u0001 Mand\n\u0001sare constant values, then one obtains that, for a magnetic \feld variation from 0 to H, the transition temperature\nchanges by an amount \u0001 T=\u00160H\u0001M=\u0001s. The adiabatic temperature change is limited by the speci\fc heat value.\nBy taking the ratio ( cp=T) as a constant value, the upper limit is \u0001 Tad= \u0001s=(cp=T) as can be seen from Fig. 1a.7\nThe energy product \u0001 siso\u0001\u0001Tadis equal to\u00160H\u0001MifH <H limand to (\u0001s)2=(cp=T) ifH >H limwith\u00160Hlim=\n(\u0001s)2=(\u0001Mcp=T). From these simpli\fed relations, one obtains that in an ideal \frst transition, at a given magnetic\n\feldH, the maximization of \u0001 Mgives the maximum energy product, while the ratio of the \u0001 sand (cp=T) determines\nthe upper limit of the adiabatic temperature change [28, 29].\nM\nHfL\nMgL\nMHcrHeqHcrHeqa)b)c)\nFIG. 3: a) Free energy potential fL(M) with two minima. b) Magnetic \feld state equation M(H). c) Gibbs free energy\ngL=fL(M)\u0000\u00160HM\nIII. SECOND ORDER MAGNETIC TRANSITIONS\nAs we have seen in Section II, the thermodynamics of a magnetic material can be fully determined by performing\nexperiments. However, it is useful to understand the microscopic physical mechanisms that are at the origin of the\nmagneto-thermal behavior. While the thermodynamics of solids, with the main aim of the prediction of the speci\fc\nheat, is a well developed classical branch of solid state physics [8, 9], the thermodynamics of magnetic solids, with the\naim of the prediction of the magnetocaloric e\u000bect, has been the subject of detailed studies only in recent years [12, 18].\nWhile many of the theoretical predictions of electronic structure, the formation of atomic magnetic moments,\nexchange interactions, the kind of magnetic order and so on, are now based on \frst principle calculations [7], the\nthermodynamics of the magnetocaloric e\u000bect rely on statistical mechanics [18]. The reason is that the free energy\nof a magnetic material is the consequence of the presence of several contributions to the system entropy including:\nthe atomic magnetic moments (due to electron spin and orbital momentum), the lattice vibrations and the electronic\nstates.\nA. Entropy of magnetic moments\nThe e\u000bect of the magnetic \feld on the entropy due to atomic magnetic moments can be appreciated by considering\nthe thermodynamics of an ensemble of non interaction magnetic moments that give rise to paramagnetic behavior [1{\n3, 6]. We now examine in detail the statistical mechanics of an an ensemble of magnetic moments and discuss how\nhow much this simple model may represent real magnetic materials.\n1. Statistical mechanics of a paramagnet\nWe consider a system composed by magnetic moments localized at the atom sites. The atomic magnetic moment\nis due to the total angular momentum of the electrons and its projection malong the direction of the magnetic \feld\nism=\u0000gmJ\u0016BwheremJis a number that can assume 2 J+ 1 discrete values between + Jand\u0000J, whileJis the\ntotal angular momentum quantum number due to contribution of the orbital and spin momentum and gis the Land\u0013 e\ng-factor.\u0016Bis the Bohr magneton \u0016B=e~=(2me),eis the electron charge, meis the electron mass, ~is the Planck\nconstanthdivided by 2 \u0019. Their values are given in Table I.8\nBoltzmann constant kB1:381\u000210\u000023J K\u00001\nAvogadro constant NA 6:022\u00021023mol\u00001\nPlanck constant h 6:626\u000210\u000034m2kg s\u00001\nelementary charge e 1:602\u000210\u000019A s\nelectron mass me9:109\u000210\u000031kg\nBohr magneton \u0016B 9:27\u000210\u000024A m2\nkB=\u0016B 1.49 T K\u00001\nkBNA 8.31 J K\u00001mol\u00001\n\u0016BNA 5.58 Am2mol\u00001\nTABLE I: Values of physical constants\nThe energy of the magnetic moment in the magnetic \feld His\nE0=\u00160gmJ\u0016BH (18)\nwhere\u00160= 4\u0019\u000210\u00007is the permeability of free space. Being the atomic moment distinguishable, the partition\nfunctionZis given by the sum over the states of the Boltzmann weight\nZ=+JX\nmJ=\u0000Jexp\u0012\n\u0000E0\nkBT\u0013\n(19)\nwherekBis the Boltzmann constant. The speci\fc Gibbs free energy for an ensemble nmagnetic moments per unit\nmass isg=\u0000nkBTlnZand gives\ng=\u0000nkBT\u0014\nln\u0014\nsinh\u00122J+ 1\n2Jx\u0013\u0015\n\u0000ln\u0014\nsinh\u00121\n2Jx\u0013\u0015\u0015\n(20)\nwhere the variable xis de\fned as\nx=\u00160gJ\u0016BH\nkBT: (21)\nThe magnetization is given by Eq.(4) and is\nM=M0MJ(x) (22)\nwhereM0=nm 0=ngJ\u0016 Bis the saturation magnetization at T= 0, andMJ(x) is the Brillouin function:\nMJ(x) =2J+ 1\n2Jcoth\u00122J+ 1\n2Jx\u0013\n\u00001\n2Jcoth\u00121\n2Jx\u0013\n: (23)\nThe entropy is given by Eq.(6) and is\ns=nkBsJ(x) (24)\nwheresJ(x) is the Brillouin entropy function\nsJ(x) = ln\u0014\nsinh\u00122J+ 1\n2Jx\u0013\u0015\n\u0000ln\u0014\nsinh\u00121\n2Jx\u0013\u0015\n\u0000xMJ(x) (25)\nBy expressing sJas a function of the normalized magnetization m=M=M 0, the \frst terms of the power series\nexpansion are:9\nsJ(m) =\u0014\nln(2J+ 1)\u00001\naJ\u00121\n2m2+bJ\n4m4+O(m6)\u0013\u0015\n(26)\nwhere\naJ=J+ 1\n3J(27)\nand\nbJ=3\n10[(J+ 1)2+J2]\n(J+ 1)2(28)\nFrom Eq.(26) one \fnds that the entropy of the ensemble of magnetic moments has its maximum at m= 0, and its\nvalue iss(0) =nkBln(2J+ 1), the upper limit for the entropy associated with the atomic magnetic moments with\n2J+ 1 discrete levels. It is important to notice that the expression for the entropy of the ensemble of magnetic\nmoments derived here is the direct consequence of the discrete number of energy levels of the magnetic moment is\na magnetic \feld and therefore of the electronic origin of the atomic magnetic moment. The thermodynamics of a\nmagnetic moment taken as a classical vector with continuous orientation would lead to unphysical results as shown\nin Ref.[30].\n2. Magnetic moment and electron spin\nLocalised electrons . A particularly nice example of magnetism due to localised magnetic moments is given by the\npartial \flling of the 4 fshell in the rare earth elements. Although the simple atomic model presented in Section I\nwould apply only to isolated atoms, it turns out that several magnetic solid compounds, in which the interaction\nbetween the 4 felectrons and the surrounding atoms is small, follow theoretical predictions very well [7, 31, 32].\nThe same occurs for salts containing transition metal elements with 3 delectrons. The main di\u000berence is that in 3 d\nelements only the spin momentum contributes to the magnetic moment. This occurs because the wavefunctions of 3 d\nelectrons are spatially extended and the orbital momentum is said to be quenched, i.e. suppressed, by the presence\nof the crystal \feld of the surrounding atoms [6].\nNon-localised electrons . The situation is much more complex when the magnetic moment is due to partially de-\nlocalised electrons, as for example in ferromagnetic metals with 3 delements. In the case of metals [8], electrons\ncan travel from one atom to the other and the wavefunctions are not limited to atomic sites. As a consequence the\nmagnetic moment of one atom is not necessarily a multiple of the electron spin1\n2and there is no simple theory pro-\nviding an expression for the entropy of magnetic moments. By considering the electrons contributing to the magnetic\nmoment, the correspondent entropy can be approached by two complementary viewpoints.\nFrom one point of view, the magnetic electrons can be considered as delocalised and \flling the appropriate energy\nbands, and obey Fermi-Dirac statistics. This means that the contribution to the entropy comes from those electrons\nlying in an energy band of amplitude kBTaround the Fermi level. This way of looking at the spin entropy has been\napplied to magnetocaloric materials [18], however one generally expects a small entropy contribution as this entropy\nis essentially that of the electrons in Pauli paramagnet [6]. The other way to look at the problem is to consider\nthat, based on experimental observations, the atomic magnetic moment of itinerant ferromagnets does not disappear\nabove the Curie point [6] as in a Pauli paramagnet under zero magnetic \feld. This means that in a ferromagnetic\nmaterial the magnetic moment, independently of the localised or delocalised nature of the electrons and of the thermal\n\ructuations, is formed at the atom site. This argument is supported by the fact that the collective wave functions\ngiving rise to parallel alignment of spins are of the spatially anti-symmetric (anti-bonding) type. These anti-bonding\nwavefunctions are characterised by high probability densities only at the atom site, because the wavefunction changes\nsign between adjacent atoms. Conversely, the wavefunctions giving rise to anti-parallel alignment of spins are of the\nspatially symmetric (bonding) type. These bonding states, with widespread wavefunctions, have lower energy with\nrespect to the anti-bonding ones and \fll the low levels of the energy band. Therefore they do not essentially contribute\nto ferromagnetism [6]. By this observation one may associate the magnetic moment to the atom site and be justi\fed\nin using a Boltzmann-Gibbs statistical weight for counting the spin states rather then the Fermi-Dirac one.\nWhen the atomic magnetic moment is proportional to an atomic spin Swhich is a multiple of the electron spin1\n2,\nthe counting of spin states for each atom can be done by sum rules for the spin. In the case of metals where the10\nmoment is a non-integer multiple of1\n2the sum rules for the spin do not apply. An analytical continuation of the\nWeiss-Brillouin theory has been used to evaluate the entropy for localized magnetic moments [12]. Further re\fnements\nfor a theory of the entropy associated with the magnetic moment are obtained by considering the space correlation of\nthe spin \ructuations giving rise to spin waves [33]. This contribution has the same origin of the Bloch law, giving a\nlow temperature correction to the temperature dependence of the saturation magnetization. It is therefore expected\nto be relevant at low temperatures [6].\nB. Ferromagnets\nA simple model for a second order transition is now given in terms of the molecular \feld theory of ferromagnetism\n[1{3, 6].\n1. Mean \feld theory of a ferromagnet\nThe ferromagnet is characterised by an exchange interaction between spins which gives an energy term that is\nminimum for parallel magnetic moments. In the mean \feld model, the interaction is associated with a molecular \feld\nWM which has the dimension of a magnetic \feld and is proportional to the magnetization M. The free energy of a\nferromagnet is then:\nfL=\u00001\n2W\u0016 0M2\u0000TsM (29)\nwhere the \frst term is the exchange energy, Wis the Weiss molecular \feld coe\u000ecient and sMis the entropy associated\nwith the magnetic moments. By using the expression previously derived for the paramagnet, Eq.(24) for the entropy\nsM, and by introducing the normalised magnetic \feld h=H=H 0, whereH0=WM 0, and the normalized temperature\nt=T=Tc, whereTcis the Curie temperature given by\nTc=aJ\u00160gj\u0016BWM 0\nkB; (30)\nwe \fnd the Weiss equation for ferromagnetism:\nh=\u0000m+taJM\u00001\nJ(m): (31)\nThe stability of the PM and the FM solutions is determined by the condition @h=@m> 0. By expanding the inverse\nof the Brillouin function (Eq.(23)) as a power series:\naJM\u00001\nJ(m) =m+bJm3+O(m5); (32)\nwe have from Eq.(31) that the paramagnetic state m= 0 is stable for t <1 while the ferromagnetic state m > 0 is\nstable fort<1. The normalized free energy ^fL=fL=(\u00160H0M0) is:\n^fL=^f0\u00001\n2m2+aJtsJ(m) (33)\nand the \frst terms of the power expansion in mare\n^fL=^f0\u00001\n2m2+t\u00121\n2m2+bJ\n4m4+O(m6)\u0013\n: (34)\nFig.4 (left) shows the free energy of Eq.(33) for di\u000berent values of t. In the example J= 1=2 for whichM\u00001\n1=2(m) =\ntanh\u00001(m) ands1=2(m) = ln 2\u0000(1=2)(1 +m) ln(1 +m)\u0000(1=2)(1\u0000m) ln(1\u0000m).11\nmm2nd order1st orderfree energy\nfree energyt=1tt>1\nt<1t=1\nFIG. 4: Left: Free energy for a second order transition. Right: Free energy for a \frst order transition\n2. Magnetocaloric e\u000bect around the second order critical point\nOnce Equation (31) is solved (giving the value of m), the entropy can be computed by Eq.(24) with the argument\nxgiven byx=M\u00001\nJ(m). Fig. 5 shows: the reduced magnetisation mgiven by the numerical solution of Eq.(31),\nthe reduced entropy ^ s=sJof Eq.(25), the normalized Gibbs free energy ^ gL=^fL\u0000hm, and the entropy change\n^s(h)\u0000^s(0) forJ= 7=2 where the di\u000berent lines correspond to the magnetic \feld hgoing from h= 0 toh= 0:05 in\nsteps of 0.01. The magnetic \feld induced entropy change has a maximum at the Curie temperature, t= 1. The mean\n\feld theory allows us also to derive approximate expressions for the magnetic \feld induced entropy change ^ s(h)\u0000^s(0)\naround the Curie temperature t= 1 and for small m. In the paramagnetic state with t >1, from Eq.(31), to \frst\norder in small mone \fnds the Curie-Weiss law of the magnetization:\nm'h\nt\u00001: (35)\nFor smallmthe entropy of Eq.(25) is proportional to m2and the entropy change has a quadratic dependence on\nmagnetic \feld:\n^s(h)\u0000^s(0)'\u00001\n2aJ\u0012h\n1\u0000t\u00132\n: (36)\nWe can see that as t!1 entropy change increases. In the ferromagnetic state, t <1 and there is a spontaneous\nmagnetization for h= 0. For small mwe have\nm'\u00121\u0000t\nbJ\u00131=2\n; (37)\nand the entropy change varies linearly with the \feld:\n^s(h)\u0000^s(0)'\u00001\n2aJhp\nbJ(1\u0000t)(38)\nand increases for t!1. At the Curie point t= 1 from Eq.(31) we \fnd that the mean \feld value of the so-called\ncritical exponent with respect to \feld:12\nm'\u0012h\nbJ\u00131=3\n: (39)\nThe entropy change is maxised at t= 1 and varies as the 2 =3 power of the \feld:\n^s(h)\u0000^s(0)'\u00001\n2aJ\u0012h\nbJ\u00132=3\n: (40)\nThe mean \feld theory of ferromagnetism presented here can be applied to describe the magnetocaloric e\u000bect around\nthe Curie temperature [34]. The MCE has been studied and described with success in ferromagnetic alloys containing\nrare earth elements [12] and the agreement of the theory with experiments can be further improved by taking into\naccount the crystal \feld of the surrounding atoms [18]. The magnetic \feld dependences of the entropy change found by\nthe mean \feld theory around the second order phase transition corresponds well to the exponents found in amorphous\nalloys [35]. Refs.[36{38] have discussed and extended the entropy change around the Curie temperature in relation\nto the mean \feld laws. A more re\fned approach is obtained by using the theory of critical phenomena around the\nsecond order transition. The experimental \u0001 s(H;T) values follow the scaling laws of critical phenomena [35, 39] very\nwell in the case of second order Curie transitions.\nmagnetization m\nentropy ŝGibbs free energy ĝtemperature t\nentropy change ŝ(h)-ŝ(0)j=7/2\ntemperature ttemperature t\ntemperature t\nFIG. 5: Normalized magnetization m, entropy ^s, Gibbs free energy ^ g=^f\u0000hmand entropy change ^ s(h)\u0000^s(0) for the mean\n\feld theory of ferromagnetism with J=7\n2. The lines correspond to the magnetic \feld hgoing from h= 0 toh= 0:05 in steps\nof 0.01.\nIV. FIRST ORDER MAGNETIC TRANSITIONS\nA phase transition is classi\fed as \frst order when the order parameter (i.e. the magnetization) changes discon-\ntinuously. While most magnetic materials have a second order transition at the Curie point, most of the recently13\ndeveloped magnetocaloric materials have discontinuous transitions. Relevant examples include Gd-Si-Ge [40{42],\nMn-As [43{47], Fe-Rh [48{50], Mn-Fe-P-As [51, 52], Co-Mn-Si [53], La-Fe-Si [54{58], mangnanites [59, 60] and the\nHeusler alloys Ni-Mn-X [61{65]. First order magnetic phase transitions are the consequence of the coupling between\nthe magnetic moments and the exchange interaction with the electronic and structural degrees-of-freedom.\nA. Coupling between magnetism and structure\n1. Exchange energy\nIndirect exchange . Much of the recent interest in the MCE in \frst order magnetic phase transitions has been\ngenerated by the discovery of the high values of the magnetic \feld induced entropy change \u0001 s(H) in Gd 5Si2Ge2[40].\nThis e\u000bect has been called giant magnetocaloric e\u000bect (GMCE) for being larger than the standard material Gd [66].\nIn Gd 5Si2Ge2the magnetic moment is due to the 4 felectrons of the Gd ions which are ferromagnetically coupled by\nan indirect exchange [67]. The material has a \frst order phase transition around 270 K between a low temperature\northorhombic Gd 5Si4-type structure and a high temperature monoclinic Gd 5Si2Ge2-type structure [68]. The phases\nhave di\u000berent magnetic properties and the transition can be driven either by the temperature and the magnetic \feld\n[69]. Each phase appears to have a di\u000berent Curie temperature [70, 71]. This behavior is con\frmed by the fact that he\nmagneto-structural transition is accompanied by the braking of covalent bonds between Si and Ge, causing a decrease\nof the exchange interaction between Gd moments. At the transition temperature the variation of the exchange is\nsu\u000ecient to destroy the ferromagnetic order, giving rise to a \frst order transition [67] and to a sudden change of the\nspin entropy. However, it was demonstrated that the change of spin entropy is accompanied by a change of structural\nentropy, i.e. the monoclinic phase is at higher entropy with respect to the orthorhombic one, giving a structural\nenhancement of the \u0001 s[71{73].\nVolume dependence of direct exchange . Magnetic \frst order phase transitions due to the coupling between mag-\nnetism and structure were originally discussed to understand MnAs [19, 74]. The magnetic moments of Mn, due to the\n3delectrons, would have a natural tendency to align antiferromagnetically, because of a negative exchange integral.\nHowever, the exchange is found to increase with the interatomic distance between Mn and, in alloys where the Mn\natoms are found at large interatomic distances, it may become positive, giving rise to ferromagnetic order (Ref. [7]\np.395). MnAs has a \frst order phase transition around 312 K between a low temperature hexagonal NiAs-type struc-\nture and a high temperature orthorhombic MnP-type structure. The low temperature phase is ferromagnetic while\nthe high temperature phase is paramagnetic. Bean and Rodbell were able to explain the \frst order phase transition\nby considering that the ferromagnet exchange depends explicitly on the speci\fc volume [19]. This assumption is\nreasonable when the change in the unit cell at the transition is re\rected in a global volume change. The conclusion\nis that the low temperature high volume FM phase may collapse into a high temperature low volume PM phase and\nthe order of the transition is governed by the dimensionless parameter \u0011which depends on magnetic and structural\nparameters such as the isothermal compressibility of the lattice \u0014T. When\u0011is larger than a critical value \u0011cthe tran-\nsition becomes of \frst order. The Bean and Rodbell model is particularly interesting as it involves the contribution\nof the structure to the total entropy. The idea of the model has been developed further in Refs. [18, 75{87].\nElectronic energy Electronic energy is particularly important for itinerant electrons in metals. In magnetic metals,\nas for example in transition metals, the 3 delectrons contributing to the atomic magnetic moment are not localized\nat an atomic site. To satisfy the Pauli Exclusion principle, the electronic states of the collective wavefunctions must\npopulate the energy bands of the crystal rather than the atomic levels. In ferromagnetic metals, the energy bands are\nsplit into spin-up and spin-down sub-bands and, because some of the electrons are constrained to be spin-polarized as\nthey contribute to the magnetic moment, the spin-up and spin-down energy sub-bands are asymmetrically \flled. The\nasymmetric \flling has an energy cost, that has to be balanced by the energy gain due to the ferromagnetic exchange\ninteraction. In a power expansion of the electron energy as a function of the reduced magnetization mone \fnds the\nm2as the \frst term. This term is inversely proportional to the density of states at the Fermi level n(\u000fF).\nThe result of the energy balance with the exchange energy (which is also proportional to m2, but with the minus\nsign), is known as the Stoner criterion. It says that ferromagnetism due to itinerant electrons can exist only if exchange\nenergy dominates over the electronic energy. By considering a higher order in the power expansion, m4, Wohlfarth\nand Rodes [88] demonstrated that if the Stoner criterion is not veri\fed (i.e. if the system is PM), but the coe\u000ecient\nof them4order is strong and negative, the system may exhibit a stable coexistence of both PM and FM states at\nthe same temperature. This condition corresponds to the itinerant electron metamagnetic (IEM) transition because\na magnetic \feld may induce a \frst order phase transition from PM to FM state [33, 88{90].\nThe condition of Wohlfarth and Rodes is realised when the density of states is small for the equal \flling of spin-up\nand spin down (PM state) and high for the asymmetric \flling of the spin-up band with respect to the spin-down band\n(FM state). The \frst order phase transition of La(Fe 1\u0000xSix)13alloy and its hydrogenation have been explained by14\ninvoking this mechanism [91{94]. Further contributions to the electronic energy are also expected by the fact that the\ndensity of states may depend on the interatomic distance. In particular, close atomic arrangements give rise to wide\nenergy bands characterized by low densities of states. This e\u000bect would give rise to an energy contribution in which\nthe e\u000bective exchange would depend on volume in a way similar to the magnetovolume argument discussed before in\nthe speci\fc case of Mn [95].\n2. Structural and electronic free energy\nIn solids, the main contributions to the free energy, other than magnetic, are elastic, phononic and electronic bands.\nThe sum of these three terms give rise to the equation of state of a solid for its lattice and electronic parts [8{11]. It\nis worth considering them in detail to have an approximate description of the related free energy terms.\nElastic energy . The elastic energy term felarepresents the potential energy related to interatomic forces between\nthe atoms in the lattice and depends on the strain tensor. To a \frst approximation one may consider isotropic e\u000bects\nand take the elastic energy to be a function of the speci\fc volume change !. By using a power expansion we have\nfela(!) =v0\n2\u00140!2+O(!3); (41)\nwhere!= (v\u0000v0)=v0is related to the speci\fc volume vand to the speci\fc volume in absence of any pressure v0.\nPhonon energy . The phonon term of the structural free energy is due to thermal vibration modes of the atoms\nin the lattice. In a classical approach, because of the law of the equipartition of the energy, one would have kBT\ncontribution for each degree-of-freedom of the atom (i.e. 3 for an atom in a solid). In a quantum approach one has to\nconsider the atomic masses as quantum harmonic oscillators and take the spectrum of the vibration modes. A good\napproximation is given by the Debye model in which the phonon spectrum is taken as isotropic in the wave-vector\nspace with a maximum frequency de\fned as the Debye frequency \u0017D. In the Debye model the free energy of the\nphonons ([9] p.275) is\nfD=f(0) + 3nkBT\u0014\nln [1\u0000exp (\u0000y)]\u00001\n3D(y)\u0015\n(42)\nwith\ny=TD\nT; (43)\nwhereTDis the Debye temperature related to the Debye frequency by h\u0017D=kBTDandD(y) is the Debye function:\nD(y) =3\ny3Zy\n0x3\nexp(x)\u00001dx: (44)\nThe Debye model gives a very good description of the speci\fc heat of solids at constant volume, cv\ncv=T@s\n@T\f\f\f\f\nv(45)\nwheres=\u0000@fD=@T is the entropy. We thus obtain\ncv= 3nkBC(y) (46)\nwhere\nC(y) = 4D(y)\u00003y\nexp (y)\u00001(47)15\nis a function that for T > T Dgivescv'3nkB, which is the law of Dulong and Petit. The Debye temperature\nTDis the only parameter in Eq.(46) and is a characteristic of the solid. To describe the thermal expansion in the\ncontext of the Debye theory one has to introduce the presence of anharmonic e\u000bects of the atomic potential [96].\nWhen atoms change their interatomic distances, the non linearities of the potential give rise to slight changes of the\nphonon vibration frequencies. In the quasi-harmonic approximation one still considers harmonic waves, but allows the\nfrequencies to change with the volume v. In the Debye model this is introduced through the Gr uneisen parameter, \r:\n\r=\u0000@ln\u0017D\n@lnv(48)\nwhich expresses the volume dependence of the Debye frequency \u0017D. From the de\fnition of the Gr uneisen parameter\nthe Debye temperature TD(!) is found to be dependent of the reduced volume !, introducing a volume dependence\nin thefDterm.\nElectronic band energy . In metals, due to the Fermi-Dirac statistics, the contribution of the \ructuations of the\nelectronic states in energy bands is limited to an energy region of amplitude kBTaround the Fermi level. The result\nof this statistics is found in solid state textbooks [8] and the leading order of a power expansion as a function of the\ntemperature gives the term:\nfele=\u0000\u00192\n6n(kBT)2n(\u000fF) (49)\nwhere n(\u000fF) is the density of states of the unsplit band at the Fermi level and the integral of the density of states n(\u000f)\nup to the Fermi level gives the number of valence electrons per atom. The contribution of such states to the entropy\nis\nsele=\u00192\n3nk2\nBTn(\u000fF): (50)\nThis contribution is linear in Tand is often much smaller than the other contributions to the entropy. It is normally\nrelevant to the speci\fc heat only at low temperatures. There can be exceptions at room temperature, however, when\nthe change of electronic density of states is large during, for example an IEM transition [97, 98].\nState equation of a solid . By using the above approximations to describe an isotropic solid we arrive at an expression\nfor the free energy which is a function of !andT:\nfS(!;T) =fela(!) +fD(!;T) +fele(T): (51)\nThe corresponding equations of state are given by applying Eqs.(2) and (3). By considering the behavior around\n!= 0 andT=T0one has the linear state equations:\np=\u00001\n\u0014T!+\u000bp\n\u0014T(T\u0000T0) (52)\nand\nsS=sS0+v0\u000bp\n\u0014T!+bv(T\u0000T0); (53)\nwhich satisfy the Maxwell relation v0@!=@T =\u0000@s=@p . The parameter bv=dsS=dTjvis the speci\fc entropy capacity\nat constant volume for the solid related to the speci\fc heat at constant volume by bv=cv=T0. We take the elastic term\nas the \frst term of the power expansion and a linear expansion of the volume dependence of the Debye temperature\nTD(!) =TD0(1\u0000\r!), de\fning y0=TD0=T0. We can then derive the values of the parameters appearing in the\nequations of state (52) and (53) as a function of the parameters of the elastic, Debye and electronic free energies. The\ninverse of the isothermal compressibility is\n1\n\u0014T=@\n@!\u00121\nv0@fS\n@!\u0013\n(54)16\nand yields\n1\n\u0014T=1\n\u00140\u0000cv(y0)T0\r2\nv0: (55)\nThe thermal expansion is obtained by:\n\u000bp=\u0014T\nv0@\n@!\u0012\n\u0000@fS\n@T\u0013\n(56)\nand is\n\u000bp=\u0014T\rcv(y0)\nv0: (57)\nThe speci\fc entropy capacity at constant volume is given by bv=cv=T0where\ncv= 3nkBC(y0) +\u00192\n3nk2\nBT0n(\u000fF) (58)\nis the speci\fc heat at constant volume. By taking the three parameters \u0014T\u000bpandbvas constants, the free energy\nfS(!;T) can be expressed as a power expansion around the values != 0 andT=T0[11]:\nfS(!;T) =fS0(0;T0) +v0\n\u0014T!2\n2\u0000\u0014\u000bpv0\n\u0014T!+s0\u0015\n(T\u0000T0)\u0000bv1\n2(T\u0000T0)2: (59)\nThe linear state equations for the reduced volume !and the entropy of the structural part sS, valid around p= 0\nandT=T0, are:\n!=\u0000\u0014Tp+\u000bp(T\u0000T0) (60)\nsS\u0000sS0=\u0000v0\u000bpp+bp(T\u0000T0) (61)\nwheresS0is a reference entropy value (at T=T0andp= 0) andbpis related to bvby the expression bv=bp\u0000\u000b2\npv0=\u0014T\nand is related to the speci\fc heat at constant pressure by bp=cp=T0. The corresponding Gibbs free energy the\nstructural lattice is \fnally\ngS(p;T) =gS(0;T0)\u00001\n2v0\u0014Tp2+ (v0p\u000bp\u0000s0)(T\u0000T0)\u00001\n2bp(T\u0000T0)2(62)\nB. First order transition due to magneto-elastic coupling\nThe paradigm for a \frst order magnetic transition is arguably the Bean and Rodbell model of magneto-elastic\ncoupling [19]. The basic idea of the model is to describe a ferromagnet in which the the interatomic distance in\ruences\nthe exchange interaction. If the change of interatomic distance is re\rected in a global volume change the ferromagnetic\nexchange depends explicitly on the volume and one has a coupling between the elastic and magnetic parts of the free\nenergy. The \frst order nature of the transition is revealed by the minimization of the total free energy due to the sum\nof these free energy terms. The Bean and Rodbell model is a paradigm example for the magnetocaloric e\u000bect because\nit shows how the entropy of the crystal lattice may be involved in the magnetic \feld-induced total entropy change.17\n1. The Bean-Rodbell model\nThe speci\fc Landau free energy fL(M;!;T ) is:\nfL=\u00001\n2W(!)\u00160M2\u0000TsM(M) +fS(!;T) (63)\nwhere the \frst two terms on the right hand side are the free energy of the ferromagnet, Eq.(29), and fS(!;T) is\nthe free energy describing the structural lattice. The molecular \feld coe\u000ecient Wis assumed to depend linearly on\nthe reduced volume as W(!) =W0(1 +\f!) where\fis a dimensionless coe\u000ecient. The basic result of the Bean\nand Rodbell model can be obtained by the approximated fS(!;T) of Eq.(59), giving linear equations of state for the\nstructural part of the system. The state equations for the magneto-elastically coupled magnetic material are given by\nimposing both Eq.(2)\n1\nv0@fL\n@!=\u0000p (64)\nand Eq.(1)\n@fL\n@M=\u00160H (65)\nBy imposing the \frst condition we obtain the equilibrium value of !:\n!=\u0000\u0014T\u0012\np\u0000\u0011\n3\f\u0014Tm2\u0013\n+\u000bp(T\u0000T0); (66)\nwhere we have introduced the dimensionless parameter \u0011of Bean and Rodbell [19]:\n\u0011=3\n2\f2\u0014T\u00160M2\n0W0\nv0: (67)\nBy comparing Eq.(66) with Eq.(60) we see that the volume dependence of the ferromagnetic exchange gives rise to\nan exchange magnetostriction term which appears as an equivalent pressure pW=\u0000\u0011m2=(3\f\u0014T). This depends on\nthe square of the magnetization, m2. By imposing the second condition we obtain the equation\n\u0000NW(!)\u00160M\u0000T@sM\n@M=\u00160H: (68)\nBy substituting !from Eq.(66) and dividing all terms by \u00160H0=\u00160M0W0we have\nh=\u0000[1 +\f(\u000bp(T\u0000T0)\u0000\u0014Tp)]m\u00001\n3\u0011m3\u0000taJ\nnkB@sM\n@m(69)\nwheret=T=Tc0andTc0is given by Eq.(30). The temperature T0of Eqs. (60) and (61) is arbitrary. Then, by\ntakingT0=Tc0, we may write the linear mterm as\u0000[1 +\u0010(t\u00001)\u0000\u0019]mwhere we de\fne the dimensionless pressure\n\u0019=\f\u0014Tpand the dimensionless parameter \u0010(zeta),\n\u0010=\u000bp\fTc0 (70)\nwhich takes into account the role of the thermal expansion of the lattice. The normalized Landau free energy ^fL(m;t)\nas a function of mis obtained by the integral of Eq. (69):\n^fL=\u00001\n2\u0012\n[1 +\u0010(t\u00001)\u0000\u0019]m2+1\n6\u0011m4\u0013\n\u0000taJ\nnkBsM(m): (71)18\nThe magnetization mis given by the solution of Eq.(69):\nh=\u0000[1 +\u0010(t\u00001)\u0000\u0019]m\u00001\n3\u0011m3+taJM\u00001\nJ(m): (72)\nThe number of possible stable solutions of Eq.(72) is evaluated by taking the power expansion. One obtains:\nh= [(t\u00001)(1\u0000\u0010) +\u0019]m+\u0010\ntbJ\u0000\u0011\n3\u0011\nm3+tO(m5): (73)\nBy de\fning\ntP= 1\u0000\u0019\n1\u0000\u0010(74)\nwe have that, for h= 0 the PM state with m= 0 is always a solution. But the PM state is an energy minimum only\nfort > t P, while for t < t Pthere is always one stable solution with m > 0, i.e. a FM state [76]. The order of the\ntransition is determined by the sign of ( tPbJ\u0000\u0011=3). By de\fning the critical value \u0011c= 3bJtPwe have that when\nthe PM solution is marginally stable ( t=tP), there is a FM solution if \u0011 > \u0011 c. This means that the PM and FM\nstates may coexist and the transition is \frst order. The normalized Landau free energy ^fL(m;t) as a function of m,\nEq. (71), is shown in Fig. 4 for J=1\n2,tP= 1,\u0011c= 1 and\u0011= 2 for di\u000berent values of t, showing the coexistence of\nPM and FM states. If \u0011<\u0011 cthere is no possible coexistence and the transition is instead second order.\n2. Magnetocaloric e\u000bect around the \frst order phase transition\nThe entropy is given by\ns=\u0000@fL\n@T\f\f\f\f\nm;!(75)\nBy taking the derivative of Eq.(63) with respect to Tand substituting Eq.(66) we obtain\ns=sM(m) +sW(m) +sS(p;T) (76)\nwheresM(m) is the magnetic entropy of Eq.(24), sS(p;T) is the structural lattice entropy of Eq.(61) and\nsW(m) =nkB\n2aJ\u0010m2(77)\nis the magneto-elastic entropy, a term of structural lattice origin, induced by the ferromagnetic exchange forces\nthrough the magneto-elastic interaction. The magnetic entropy sM(m) has a maximum at m= 0 and it decreases\nto zero for m= 1. The magneto-elastic entropy depends on the parameter \u0010and is proportional to m2. To analyse\nthe competition between sMandsW, the two terms that depend on m, we introduce the normalized entropy, ^ s(m) =\n(sM(m)+sW(m))=(nkB). The maximum di\u000berence is between the entropy at m= 0 andm= 1, \u0001^smax= ^s(0)\u0000^s(1)\nand is:\n\u0001^smax= ln(2J+ 1)\u00001\n2aJ\u0010: (78)\nBy using the power expansion of Eq.(26) for sMwe obtain\n^s= ln(2J+ 1)\u00001\n2aJ\u0014\n(1\u0000\u0010)m2+bJ\n2m4+O(m6)\u0015\n(79)\nwhere we see that the total entropy may be increased or decreased depending on the sign of \u0010. When\u0011 > \u0011 cthe\ntransition is \frst order and there is a discontinuous jump of the magnetization m. At the transition temperature\nbetween the low temperature phase (LT) and the high temperature phase (HT), the entropy ^ sincreases discontinuously\nwith a jump \u0001 s=sHT\u0000sLT>0. We therefore have the following cases:19\n\u000ffor\u0010 <1 the transition is from LT-FM ( m6= 0) to HT-PM ( m= 0) and the magnetic entropy change is positive,\n\u0001sM>0:\n{For\u0010 <0 the magneto-elastic entropy change is positive, \u0001 sW>0, and there is an enhancement of the\ntotal entropy change with respect to the magnetic contribution \u0001 s>\u0001sM.\n{For 0<\u0010 < 1 the magneto-elastic entropy change is negative, \u0001 sW<0, and there is reduction of the total\nentropy change with respect to the the magnetic contribution \u0001 s<\u0001sM.\n{For\u0010!1, to order m2the two contributions oppose one other \u0001 sW!\u0000\u0001sM, and \u0001s!0\n\u000fFor\u0010 > 1 the transition is from LT-PM ( m= 0) to HT-FM ( m6= 0) and the magnetic entropy change is\nnegative \u0001sM<0, but the magneto-elastic entropy change is positive. For \u0010 >\u0010 c, where\u0010J= 2aJln(2J+ 1)\nis the critical value at which the entropy of the m= 0 andm= 1 are the same, we have that \u0001 sW>\u0000\u0001sM,\nand at the transition the total entropy change is lower than the magneto-elastic contribution \u0001 s<\u0001sW[76].\nIn a \frst order transition the equilibrium is determined by the Maxwell convention in which the system is allowed\nto select the minimum with the lowest Gibbs free energy. The Gibbs potential is gL=fL\u0000\u00160HM +pv0!. By taking\nthe di\u000berence gL\u0000gS, and dividing by \u00160H0M0we obtain the normalized potential ^ gL=^f\u0000hm, where ^fis given\nby Eq.(71). Fig. 6 shows the magnetic \feld induced entropy change computed for J=1\n2, underp= 0 for which\nM\u00001\nJ(m) = tanh\u00001(m),aJ= 1,bJ= 1=3 and\u0011c= 1. The values of the parameters are \u0011= 2,\u0010=\u00000:5;0:0;0:5. The\nmagnetic \feld his in the range 0 <h< 0:04 inhsteps of 0.004. Eq. (72) is solved numerically and the transition is\ntaken at the temperature at which ^ gFM= ^gPM. Fig. 6 shows that the entropy change increases when the contribution\nfrom the structure is positive (at negative \u0010) and that the the transition temperature ttdependence on the magnetic\n\feldhdecreases. This is due to the Clausius-Clapeyron equation which, in normalized form, is dtt=dh= \u0001m=\u0001^s,\nwhere \u0001mis the discontinuous change of the magnetization. At a given value of \u0001 ma lower \u0001^scorresponds to a\nhigherdtt=dhwhich may result in a higher adiabatic temperature change.\ntζ=0ζ=0.5ζ=-0.5\ntt-Δs\nFIG. 6: Magnetic \feld induced entropy change \u0001 s(h;t) for the Bean Rodbell model computed for J= 1=2. The lines correspond\nto magnetic \feld hfrom 0 to 0.04 in steps of 0.004.\nThe coupling between volume and magnetism has been studied by Jia et al. [95], who noticed that if at the transition\nthe volume changes discontinuously, then the total entropy change, including the magnetic contribution, is a\u000bected.\nTo sustain this idea, in Refs.[95, 99, 100] the authors have represented the experimental entropy change \u0001 sof several\nmagnetocaloric alloys (Gd 5SixGe1\u0000x, LaFe 13\u0000xSixand Ni 2:15Mn 0:85\u0000xCuxGa) as a function of the magnetization\nchanges. By subtracting from this data the theoretical magnetic entropy, they were able to determine the sign and the\namplitude of the structural contribution. In Gd 5Si2Ge2and Ni 2:15Mn 0:85\u0000xCuxGa the lattice contribution increases\nthe entropy change at the transition while in LaFe 13\u0000xSixand MnAs [19], even if the magnetic entropy change is\nlarge, the role of the lattice entropy is that of decreasing the \u0001 s. This has not necessarily negative implication for\nthe magnetocaloric e\u000bect, because, as shown in the example of Fig.6 the temperature width of the \u0001 sincreases.\nC. Magnetocrystalline anisotropy energy\nThe description of the anisotropic nature of the MCE has received particular attention in the research literature [12,\n18, 101]. The rotation of the magnetization away from its easy axis has an associated entropy change which is due to the20\ndi\u000berent spin entropy along di\u000berent directions and is phenomenologically described by the temperature dependence\nof the anisotropy constants. For example by taking the uniaxial anisotropy to \frst order\nfAN=K1(T) sin2\u0012 (80)\nwhere\u0012is the angle formed by the magnetization vector with the easy axis. The entropy term associated with the\nanisotropy will be\nsAN=\u0000dfAN\ndT=\u0000dK1\ndTsin2\u0012 (81)\nand therefore the entropy di\u000berence between the \u0012=\u0019=2 and the\u0012= 0 directions will be \u0001 sAN=sAN(\u0019=2)\u0000\nsAN(0) =\u0000dK1=dT. In ferromagnets with an easy axis ( K1>0) the entropy change associated with the rotation of\nthe magnetization along the hard direction (hard plane) gives an increase of the entropy if dK1=dT < 0.\nThe underlying physical phenomenon is that the entropy of the spin system is larger if the magnetization is directed\nalong a hard direction. A basic understanding of this phenomenon can be obtained by considering the Callen and\nCallen law of the anisotropic magnetization [102]. If the total magnetization is constrained along a direction of hard\nmagnetocrystalline anisotropy, the atomic magnetic moments will tend to fan out around the hard axis in order to\nminimize the total energy. This e\u000bect gives rise to an averaging of the magnetocrystalline anisotropy and a decrease\nof the saturation magnetization value along the hard axis (anisotropic magnetization). If we simply associate spin\ndisorder with spin entropy, we then obtain that the spin entropy will be larger along an hard axis. First principles\nevaluation of this anisotropic contribution to the entropy requires speci\fc theoretical developments [18, 103, 104].\nThe Callen and Callen argument may help us to understand the magnetocaloric e\u000bect in spin reorientation transi-\ntions in the presence of two magnetic sublattices, as for example in Er 2Fe14B [105{109] and NdCo 5[110] and in other\nalloys [111{114]. If the moments of the two sublattices are rigidly coupled (ferro or antiferro), then minimization\nof the total energy will select which sublattice will satisfy its local anisotropy. At high temperature the system will\nalways be in the state that yields the highest entropy. For both Er 2Fe14B and NdCo 5it is the RE moments that\ndominate the entropy contribution at high temperature, probably because they are loosely coupled to each other [115].\nThe reorientation transition can be discontinuous (from plane to axis) as well as continuous (through an intermediate\neasy cone) depending on the high order anisotropy constants [109, 110, 116].\nV. HYSTERESIS AND MODELING\nIn the previous sections we have considered equilibrium \frst order phase transitions by using the Maxwell convention\nin which the system selects the energy minimum of lowest energy. This is however only an idealized limit situation.\nAs a matter of fact, real systems do not follow either the equilibrium transition or the completely out-of-equilibrium\npicture given by the global instabilities of the dashed lines of Fig. 3 (center). Instead they behave in an intermediate\nway [117]. The \frst order transition occurs by the spontaneous formation of domains of the new phase within the old\nphase. The domains will be separated by phases boundaries and the phase transformation may occur by the motion\nof these boundaries in a phase coexistence state.\nIn real systems many internal non-intrinsic contributions play a major role. These e\u000bects give rise to: i) a smooth\ntransition between the phases rather than the vertical slope of the equilibrium Maxwell construction, and ii) a\nhysteresis with smaller amplitude with respect to the jumps of the global instability picture. The formation of\nthe nuclei of the new phase is somehow spread around the Maxwell construction because phase coexistence may\ncontribute to the minimization of space-dependent energy terms such as elastic energy, related to the internal stresses,\nand magnetic energy, related to internal magnetostatic \felds. These e\u000bects are therefore related to the presence\nof structural defects and disorder. The distribution of disorder also gives rise to localized energy barriers for the\nnucleation and the motion of the phase boundaries which are smaller than the energy barrier separating the two\nminima of the free energy.\nA. Hysteresis and entropy production\nThe entropy in the presence of a \frst order, hysteretic transition is sketched as a function of temperature in Fig. 7a.\nThe presence of hysteresis has the peculiar e\u000bect, making the magnetocaloric properties history-dependent. Both\nthe entropy change and the temperature change depend on the history of the HandTvariables in preparing the21\nexperimental material sample [118, 119]. The presence of reversible and irreversible e\u000bects is clearly revealed in the\nmeasurement of the speci\fc heat, which is di\u000berent if it is measured by temperature scanning experiments or by ac\nexperiments [120]. The reason is that in phase transitions with hysteresis there is a superposition of irreversible and\nreversible processes. While the scanning experiments catch all processes the ac methods select only the reversible ones.\nFig. 7b shows how the change of the direction of the temperature variation corresponds to tracing of a new entropy-\ntemperature curve and a further reversal produce a minor hysteresis loop [5]. The direct application of equilibrium\nrelations such as the Maxwell relations to \frst order phase transitions with hysteresis may then create ambiguous\nresults as discussed widely in the literature [121{128]. These problems can be avoided by using direct calorimetric\nmethods [129{136].\nA second point worthy of discussion here is the fact that, in an out-of-equilibrium process we also have to deal with\nthe non-conservation of the entropy [10, 20]. For an out-of-equilibrium process the second law of thermodynamics is\nstated as\u000es=\u000ees+\u000eiswhere\u000es, the di\u000berential of the entropy state variable, equals the sum of \u000eesthe di\u000berential of\nthe entropy exchanged with the surrounding thermal bath, and \u000eis, the di\u000berential of the entropy produced internally\nby irreversible processes. The entropy exchanged with the thermal bath can be estimated from a direct measurement\nbecauseT\u000ees=dt =dq=dt is the heat \row with the thermal bath. For the evaluation of the entropy sof a material with\nhysteresis one should be able to evaluate both the instantaneous entropy production \u000eisand the exchanged entropy\n\u000ees. The entropy production \u000eis, is de\fnite positive as a consequence of the second principle of thermodynamics\nbut it can only be measured in a cyclic process. In a closed cycle transformation we haveH\n\u000es= 0, therefore the\nentropy produced over one entire loop is \u0001 is=H\n\u000eis=\u0000H\n\u000ees. The di\u000berential \u000eiscannot be determined by purely\nexperimental means and a physical theory separating the exchanged and produced entropy is needed in order to\ncomputesfrom measured heat \rux [137]. To have an order of magnitude of the two, we note that the amplitude of\n\u0001isis independently given by the heat dissipated in a svs.Thysteresis loop which is given by the loop areaH\nsdT.\nIf we approximate the svs.Tloop as a parallelogram of height \u0001 sand width \u0001 Thyst(see Fig. 7c), we have that\nthe entropy production over the entire loop is approximately \u0001 is= \u0001s\u0001Thyst=TwhereTis the average temperature\nof the transition. As the entropy production is de\fnite positive, the measurable integral \u0001 es=R\n(\u000es\u0000\u000eis)dtwill\nhave the shape shown in Fig. 7c. The entropy produced in the entire loop depends on the ratio \u0001 Thyst=Twhich,\nfor magnetocaloric materials with transitions around room temperature T'300 K and small temperature hysteresis\nThyst<1 K, is a small contribution that may be disregarded to a \frst approximation [138{140].\ns\nTPM phase\nFM phaseH=0H=Hmaxa)b)s\nTΔesΔisc)\nFIG. 7: a) Entropy as a function of temperature and magnetic \feld in a \frst order transition with hysteresis. b) Branching\nexample. c)Top: entropy of a squared hysteresis loop. Bottom: integral of the exchanged entropy in an idealized heating and\ncooling experiment.\nB. Equivalent driving force\nIn order to arrive at a model of the hysteresis in the \frst order phase transition we \frst consider a non-equilibrium\nGibbs free energy gL(M;H;p;T ) that, in a certain range of its intensive parameter ( p,H, orT), is characterized\nby two distinct energy minima as a function of the magnetization M(taken here as the order parameter). All the\nother extensive variables, the volume vand the entropy s, are related to Mas in the example of Section III.A. The\ntwo minima of the function gL(M;H;p;T ) correspond to the two stable phases that we can call the low temperature22\nphase (LT) and the high temperature phase (HT), depending on their relative stability with respect to T. We may\nthen consider the thermodynamics of each of the phases separately.\nWe examine the non-equilibrium Gibbs free energy at each minimum, i.e. gLT=gL(MLT;H;T) andgHT=\ngL(MHT;H;T). In a limited range of HandT, the energies gLT(H;T) andgLT(H;T) can be considered as the\nequilibrium potential. This occurs as soon as there exists an energy barrier separating the two minima. Once we have\nde\fned this initial hypothesis, we consider the phase transition between the LT phase and the HT phase driven by\neither of the intensive variables ( p,H, orT). In thermodynamic equilibrium the Maxwell construction would apply\nand the system would select the state for which the Gibbs free energy is minimum. In presence of two phases LT and\nHT with di\u000berent Gibbs potentials gLTandgHTrespectively, the sign of the di\u000berence gLT\u0000gHTwhich will decide\nwhich of the two phases is globally stable. If gLT\u0000gHT<0 the system will be in the LT phase, while if gLT\u0000gHT>0\nthe system will be in the HT one. The free energy di\u000berence gLT\u0000gHTtakes then the role a driving force of the\ntransformation, encapsulating the action of temperature, pressure and magnetic \feld [141, 142].\nC. Preisach-type models\nThe presence of disorder gives rise to a complex hysteresis relationship characterized by smoothed, rather than\nabrupt, properties and the phenomenon of branching at the turning points of the input variable [5]. Hysteresis has\nbeen studied in detail in particular by using a Preisach-type model in which the output is due to the superposition of\nmany bistable units [20]. To describe a \frst order phase transformation in terms of bistable contributions we consider\nas a driving force the half di\u000berence z(H;T) = (gLT\u0000gHT)=2 [141]. Each unit has switching thresholds at z=gu\u0006gc\nwhere the + sign refers to the switch from 0 !1 and the\u0000sign to 1!0. The values of guandgcare properties of\nthe individual unit (Fig. 8a). The units are distributed according to two parameters: the width gcand the shift gu.\nHere we suppose that guandgcare independent of the intensive variables, re\recting the e\u000bects of structural disorder\nonly. The disorder in the material is re\rected in a statistical distribution of the units, p(gc;gu). At a given instant of\ntime, the state (0 or 1) of each bistable unit can be represented in the ( gc;gu) plane and the regions of the plane in\nthe 0 or 1 state are determined by the temporal history of z(t) only. The approach to the out-of-equilibrium phase\ntransformation just described turns out to be perfectly equivalent to the Preisach model of hysteresis. The p(gc;gu)\ndistribution is then called the Preisach distribution and all the mathematical results of that model can be applied to\nthe present case. In particular, in the plane ( gc;gu) the 0 and 1 regions are separated by the borderline function b(gc)\n(Fig. 8d) which is determined by the temporal history of z(t) (Fig. 8c) by the inequality jb(gc)\u0000z(t)j\u0014gcat each\ntime instant. This borderline function fully characterizes the non-equilibrium phase-coexistence state of the material.\ng(i)\nzagugcg(LT)g(HT)x\nz01gugcgcgu( LT )( HT )b(gc)z1ztimez3z2a)b)c)d)\nFIG. 8: a) A bistable unit of the phase transformation of the phase fraction xas a function of e\u000bective force z= (gLT\u0000gHT)=2.\ngcandgurepresent the e\u000bect of structural disorder. b) Energy of the bistable unit. c) Temporal hstory of the input z(H;T).\nd) State line b(gc) in the Preisach plane ( gc;gu) representing the state of an ensemble of bistable units.\nThe out-of-equilibrium thermodynamics of the system is derived by starting from the assumption that the non-\nequilibrium Gibbs free energy g(H;T;b (gc)) of the system is a function of the intensive variables HandTand of the\ninternal variable, the function b(gc). Its expression is given by the superposition of the bistable contributions (Fig.8b):23\n01234567050100150200250\nμ0H (T)Magnetization (Am2kg-1)80 K85 K74 K68 K90 KTemperature (K)60708090100020406080entropy chage -Δs (Jkg-1K-1)0 - 2 T0 - 5 T0 - 7 T2 T - 05 T - 07 T - 001234567050100150200250\nμ0H (T)Magnetization (Am2kg-1)\nTemperature (K)60708090100020406080entropy chage -Δs (Jkg-1K-1)80 K85 K74 K68 K90 Kfrom Maxwell relations\nmodelmodelexperimentGd5(SixGe1-x)4  x=0.082\nFIG. 9: A model of Gd 5(SixGe1\u0000x)4withx= 0:082. Top left: experimental M(H;T) curves after Ref. [42]. Top right: \u0001 s\ncomputed from the Maxwell relation using the experimental data, after Ref. [42]. Bottom left: model of M(H;T). Bottom\nright: prediction of \u0001 sfrom the model fter Ref.[123]. The model does not predict the unphysical spikes obtained with Maxwell\nrelations.\ng(H;T;b (gc)) =a(H;T) +Z1\n0dgc\"Zb(gc)\n\u00001(gu\u0000z)p(gc;gu)dgu\u0000Z1\nb(gc)(gu\u0000z)p(gc;gu)dgu#\n; (82)\nwherea(H;T) is the half sum a(H;T) = (gLT+gHT)=2. The phase fraction per unit mass xof HT phase is given by:\nx=Z1\n0dgcZb(gc)\n\u00001p(gc;gu)dgu: (83)\nThe previous expression corresponds to the the Preisach model integral with zas input variable and the phase fraction\nxas output variable. For a description of the thermodynamic state of the system the aforementioned state-line b(gc)\ntakes the role of an internal thermodynamic variable not explicitly coupled to intensive variables [20]. We make then\nuse of the results known for thermodynamics with internal variables. The extensive variables, magnetization Mand\nspeci\fc entropy s, are given by the expressions:\nM=\u0000@g\n@H\f\f\f\f\nT;b(gc)(84)24\n260265270275280–20–15–10–505\nT (K)s-sA(Jkg-1K-1)^012345050100150\nμ0H (T)M (Am2kg-1)250 K270 K280 K285 KmodelGd5Si2Ge2μ0H=2TABCDB'D'AMR cycle Gd5Si2Ge2μ0H=0modelirreversiblereversible\nFIG. 10: Modeling Gd 5Si2Ge2. Left: comparison of experimental and modelled M(H;T). Right: model prediction of an active\nmagnetic regenerative refrigeration cycle. After Ref. [140]\nand\ns=\u0000@g\n@T\f\f\f\f\nH;b(gc)(85)\nwhere the internal variable, the function b(gc), is kept constant. The rate of entropy production di^s=dt is given by\nTdis\ndt=\u0000Z1\n0\u000eg\n\u000eb(gc)\f\f\f\f\nH;T@b\n@tdgc (86)\nwhere we have made use of the function derivative. By the fact that the system is not in the equilibrium state every\ntransformation with a change in the state line corresponds to an internal generation of entropy. This is the original\nand non obvious result obtained by the use of the internal variable thermodynamics. By taking the distribution\np(gc;gu) independent of HandTthe previous expressions are easily computed, giving:\nM=xMHT+ (1\u0000x)MLT (87)\nand\ns=xsHT+ (1\u0000x)sLT (88)\nwhereMHT=\u0000@gHT=@H,MLT=\u0000@gLT=@H,sHT=\u0000@gHT=@T,sLT=\u0000@gLT=@T andxis the phase fraction\ngiven by the Preisach model expression Eq.(83) with z(H;T) as input. The rate of entropy production di^s=dt is thus\ngiven by:\nTdis\ndt= 2Z1\n0[z\u0000b(gc)]p(gc;b(gc))@b\n@tdgc: (89)\nThe previous expressions can be easily computed by analytic or numerical means once the Preisach distribution\np(gc;gu) and the Gibbs free energies of the pure phases, gLT(H;T) andgHT(H;T) are known. 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However, \nmost methods used for c hang ing magnetism are inefficient  or destructive to the magne tic material . Here we report high -\nperformance magnetic control of a gas -responsive single -molecule ma gnet (SMM). The results exhibit that the magnetic properties \nof the SMM can be significantly changed according to the gas envir onment  it is in and some of the magnetic states can be \nreversibl y transformed or coexist ent in the SMM through artificial control. More importantly, the single crystalline structure of the \nSMM remains unchanged during the transformation  process except  for slight change of the la ttice constant. Th us, th is work opens \nup new insights into the stimuli -responsive magnetic materials which have great prospects for application  in artificial design \nmagnetic network and also highlight  their potential as smart mater ials. \nMaterials that change their magnetic properties in re-\nsponse to the external stimuli have long been of interest for \ntheir potential applicability in magnetic storage device, \nspintronics and smart magnetic materials1-5. Organic materials \nare suitable candidates for such materials due to their chemical \ndiversity, flexibility and designab ility. However, the most \ncommon methods6-17 to tune the magnetism of organic mat eri-\nals provide poor controllability and even destroy the mat erials. \nFor these reasons, finding a proper tu nable magnetic material \nand method has become a challenge for the scientific commu-\nnity. One pro mising candidate is a chemical responsive organ-\nic magnet in which the component associate d with the mag-\nnetism can be flexibly substituted with very little da mage to \nthe crystal structure in response to external che mical stimuli.  \nHere we report a gas -responsive single -crystal single -\nmolecule magnet  (SMM) [Mn 3O(Et 3-\nsao) 3(ClO 4)(MeOH) 3]18,19,20 (hereafter Mn 3-CH 3OH for short , \nbecause it has methanol as  a ligand s). Mn 3-CH 3OH complex \ncrystallizes in the trigonal system, space group of R -3 reported \nby Inglis et al. It has a  honeycomb -like magnetic  network  and \nexhibits slow relaxation and quantum tunneling of magnetiz a-\ntion (QTM) at low temperature due to it s high anisotropy en-\nergy barrier and large ground state spin21,22. Distinct from  iso-\nlated SMMs sy stem23and dimer  SMMs system24, Mn 3-CH 3OH \nhas i dentical  intermolecular antiferromagnetic (AFM) ex-\nchange interaction in its honeycomb -like magnetic ne twork18 \nand the AFM exchange interaction depends on the i ntermolec-\nular hydrogen bond formed by the ligand on the Mn 3-CH 3OH \nmolecule18,19,20. Recently, some of us has reported that the \nmethanol lig ands can be replaced when the Mn 3-CH 3OH crys-\ntal is exposed in the air f or a few days25,26,27, while the space \ngroup and the crystal symmetry remain unchanged18,25,26. The \nchange  of methanol ligand s induces a signif icant change of the magnetism , such as AFM exchange interaction and the QTM \neffect.  Interestingly, the methanol l igand s can be rever sibly \ngrafted back to the molecule when the sample is exposed in \nmethanol gas25. However, the nature of the reaction has not \nbeen verified in the pr evious work.  \nIn order to understand the chemical mechanism of the rea c-\ntion and employ it to achieve control of magnetism wit hout \ndestroy ing the crystal structure, we carr ied out more detailed \nstudy on the transformation of Mn 3-CH 3OH. The prototype of \nMn 3-CH 3OH complex is synthesized according to the method \nreported by Inglis et al18. The singl e crystal picture is shown in \nFig.1a. The intermol ecular hydrogen bond18 which associates \nwith the AFM exchange int eraction is formed by the H atom \non the methanol  ligan d and the O atom on the neighbo ring \nmolecules, leading to a two -dimensional honeycomb -like \nmagnetic network (in ab plane) as shown in Fig.1b and Fig.1c. \nIn the c direction (easy axis), the intermolecular intera ction is \nvan der Waals ’ force, thus Mn 3-CH 3OH can be regarded as a \ntwo-dimensional magnetic system.  \nFirst, t he synthesized samples of  Mn 3-CH 3OH from the \nsame batch are divide d into four groups and store in air, ox y-\ngen dried by sulphuric acid, water vapor and water vapor with \na certain concentration of oxygen for about two weeks, respe c-\ntively. We find that a ligand  displacement reaction take place. \nThe methanol ligand s on Mn 3-CH 3OH are replaced by water \nmolecule  for sa mples store in air , water vapor and water vapor \nwith oxygen , while samples in the oxygen dried by sulphuric \nacid keep unchanged  (the sample with water ligands is named \nas Mn 3-H2O, the crystal structure of it is available free of \ncharge via www.ccdc.cam.ac.uk/data_request/cif  with the \nnumber CCDC -1475184 ). And samples store in environmen t \nwith oxygen change faster. That indicates the oxygen plays a \nrole of catalyst during the rea ction.   \n \n 2 \n \nFigure  1. (a) The picture of the single -crystal sample of the three \nkinds of Mn 3. (b) The hydrogen bond between Mn 3-NH 3 mole-\ncules. (c) The ab plane structure of Mn 3-NH 3. The three kind s of \nMn 3 have the same lattice structure.  Colour code: Mn, purple; O, \nred; C, black; N, olive green; Cl, green; H, cyan. Most carbon and \nhydrogen atoms are neglec ted for clarity.  \nBased on this conclusion, we first propose a gas -responsive \nscheme to attain the goal of non -destructive ma gnetism control \nof SMM Mn 3. Both meth anol ligand s and water ligand s link \nwith Mn ions via Mn -O coordination bond, which  is relatively \nweaker than other kinds of coordination bonds such as Mn -N \ncoordination bond. Hence, the met hanol ligand s and the water \nligand s should be change d to some kind of l igand s contain  N \nwhen Mn 3-CH 3OH and Mn 3-H2O are placed in a suitable ni-\ntrogen -rich atmosphere . In consideration of the fact that the \nsize of the gas molecule should be small enough to i nfiltrate \ninside the crystal, we choose ammonia  for the di splacement \nreaction. We expose d the single -crystal of Mn 3-CH 3OH and \nMn 3-H2O in the atmosphere of dried a mmonia  gas. The dried \nammonia gas is obtained by warming  up 30  mL aqueous  am-\nmonia  at 40 C and  then dried by soda lime.  The results indi-\ncate that the methanol ligands on Mn 3-CH 3OH and water lig-\nands on Mn 3-H2O are indeed replaced by amm onia ligands \n(we name the new molecule as Mn 3-NH 3, the molec ular struc-\nture is shown in Fig. 2). The whole transmissi on process is \nshown in Fig.2. What ’s more, the ligand s displac ement  rate \nsignificantly  increases in the presence of oxygen. As expected, \nthe single -crystal structure of Mn 3-NH 3 was not destroyed \nduring the reaction. The single -crystal X -ray diffra ction \n(SXRD) results exhibit that the Mn 3-NH 3 still crystallizes in \nthe trigonal system, space group R -3 while the lattice co n-\nstants show a slight variation18,25 (the crystal structure of it is \navailable via www.ccdc.cam.ac.uk/data_request/cif  with the \nnumber CCDC -1475183 ). On the other hand, the Mn 3-NH 3 \ncannot be change d back to Mn 3-CH 3OH and Mn 3-H2O when \nexposed to methanol gas and vapor, indicat ing that  the Mn 3-\nNH 3 molecule is more stable . In order to pro ve this point, we \ncalculate d the coordination bond energy of th ese three kinds of \nMn 3 molecules by density functional  theory (DFT). The mo-\nlecular structures of the three kinds of Mn 3 for our calculations \nare extracted from the corresponding experimental cr ystal \nstructures. The single -point energies with different spin mu lti-\nplicities are calculated by DFT  at the B3LYP/6 -31G** level, \nwhile atom Mn is treated with LANL2DZ pseudo potential. \nAll the calculations  are performed with Gaussian -09 pr ogram.   The resul ts show that the Mn -O coordination bond energy \nof Mn 3-CH 3OH and Mn 3-H2O are 0.54  eV and 0.57  eV, while \nthe Mn -N coordination bond energy of Mn 3-NH 3 is 0.76  eV  \nThus, t he large bond energy gap between Mn -N and Mn -O \ninhibits  the backward reaction of Mn 3-NH 3 cannot take place, \nwhereas the bond energy of Mn 3-CH 3OH and Mn 3-H2O are \nquite close enabling  mutual tran sform ation . Fig.1c exhibits the \nmutual transformation process of these three kinds of Mn 3.As \ndepicted in Fig.2 , the molecular structure of Mn 3-CH 3OH, \nMn 3-H2O and Mn 3-NH 3 are the same except for the ligand s. \nHence , the total spin of the mol ecule, which  depends on the \nintramolecular structure , is e xpected to be the same for  the \nthree kinds of Mn 3. This is verified by DFT calculations which \nshowed that the g round state spin for all these mol ecules is the \nsame with S=6. Also,  the spin densities exhibit ed a considera-\nble localization on the Mn3+ ions ( see supplementary material ). \nIt thus ind icates that the single molecular magnetism of the \nthree kinds of Mn 3 are the same. On the other hand, the diffe r-\nent ligand leads to a change of the lattice constants and the \nlength of hydrogen bond (L). For Mn 3-CH 3OH, \na=b=13.4446 Å, c=34.4519 Å, L=2.0225 Å; for Mn 3-H2O, \na=b=13.1471 Å, c=34.501 Å, L=1.9657 Å; for Mn 3-NH 3, \na=b=12.9806 Å, c=34.7392 Å, L=2.0081 Å. The small change \nof the length of hydrogen bond has a great influence on the \nmacromagnetism and the QTM effect of the three kinds of \nMn 3, as discussed below.   \n \nFigure 2. The mutual  transformation process of these three kinds \nof M n3. The essence of the process is substitution  of the ligand s \nthat are labeled  by red circle. Colour code: Mn, purple; O, red; C, \nblack; N, olive green; Cl, green; H, cyan.  \nTo illustrate the effects of the different ligand s on the mag-\nnetic  properties, we p erform ed direct current (dc) , alternating \ncurrent (ac) susceptibility and heat capacity on single -crystal \nof Mn 3-CH 3OH, Mn 3-H2O and Mn 3-NH 3, respectively. The dc \nmagnetic  measurements are performed on 7 T SQUID -VSM \n(Quantum  Design)  at 1.8  K. The single -crystal sample s are \nfirst cut into the re gular shape, t hen they are  well oriented and \nfixed on a home -made Teflon cubic, which is glued  on the \nsample holder. The magnetic easy axis is ensured to be  paral-\nlel to the applied magnetic field. The ac magnetic measu re-\nments and heat capacity  measur ements are performed carried  \n \n 3 on 14  T PPMS  (Quantum  Design)  equipped with standard ac \nmagnetometer system option  and standard heat capacity option .  \nRecent researches25 show that Mn 3-CH 3OH exhibit hyst e-\nresis loops with QTM ef fect and has no phase -transition due to \nthe weak AFM intermolecular exchange interaction while the \nMn 3-H2O exh ibits an AFM phase transition at TN=6.5 K. For \nMn 3-NH 3, Fig. 3a exhibits the M/H-T curves during field -\ncooling (FC) process. The magnetization fir st shows an ascent \nas temperature d ecreas es then starts to drop at about 7.5  K, \nwhich suggests that there is an AFM phase -transition. To fu r-\nther dete rmine the phase -transition temperature, we fall back \non the heat capacity ( CP-T) curves in different  magnet ic fields \nshown in Fig. 3b. It is seen that the peak shifts to low temper a-\nture as the magnetic field is increasing, which is a feature of \nAFM phase -transition. And the phase -transition temperature is \nseen to  be TN=5.5 K at zero field which is consistent wi th the \ntemperature where the M/H-T curves drop steeply (see sup-\nplementary material ). Fig. 3c shows the sketch map of the \nAFM structure of Mn 3. The AFM long range correlation is \nformed in ab plane. In c direction, there is no magnetic corr e-\nlation between mo lecules. Thus, it can be regarded as a stack \nstructure of independent AFM layers. According to the mean -\nfield theory, the phase -transition temperature is determined by \nthe strength of the intermolecular exchange intera ction and the \nnumber of the nearest ne ighboring molecules. As mentioned \nabove, the three kinds of Mn 3 have the same lattice structure, \nthus the phase -transition temperature should only proportional \nto the strength of intermolecular exchange interaction. Cons e-\nquently, we can make a conclusion t hat the AFM intermolec u-\nlar exchange interaction of Mn 3-NH 3 is stronger than Mn 3-\nCH 3OH and weaker than Mn 3-H2O.  \n \nFigure 3. (a) The susceptibility vs temperature curves of \nMn 3-NH 3 during field cooling process. The applied magnetic field \nis H=100  Oe and is parallel to c axis. (b) Heat capacity vs tempe r-\nature curves of the same sample at different magnetic fields. (c) \nThe sketch map of the AFM structure of Mn 3. The light pink and \nlight blue ball s represent  molecules with spin down and spin up, \nrespectively. T he direction of the spin is  parallel to the c axis.  \nTo get more information of the AFM intermolecular e x-\nchange interaction of the three kinds of Mn 3, we perform ed a \ndc hysteresis experiment. Fig. 4a and Fig. 4b depict  the no r-\nmalized hyst eresis loops and th e derivative curves  of these \nthree kinds of Mn 3 at T=1.8K . It can be seen that all of the \nloops exhibit QTM steps. However, the shape of the loops and the position  of the QTM peaks shown in Fig. 3b of them are \ndifferent, indicat ing that  the height of the a nisotropy energy \nbarrier and the strength of intermolecular AFM exchange in-\nteraction  are different.  \nAdditionally, r ecent work has reported that the Mn 3-\nCH 3OH could transform into Mn 3-H2O when the sample is \nplaced in air for about a month27. Hence, t o check  the stability \nof Mn 3-NH 3 in air, we perform ed the dc susceptibility  exper i-\nments on the same single -crystal Mn 3-NH 3 sample after stor-\ning in air for a month. The hysteresis loops shown in Fig.4c \nsuggest that the macromagnetism of Mn 3-NH 3 remains  un-\nchanged, which also means the lattice structure and the easy \naxis are unchanged. Besides, the position of the main QTM \nresonant fields shown in Fig.4d are the same indicating that  \nthe intermolecular AFM exchange interaction is inv ariant.  \n \nFigure 4. (a) Normalized  magnetization hysteresis loops of Mn 3-\nCH 3OH (black line), Mn 3-H2O (red line) and Mn 3-NH 3 (blue line) . \n(b) The deri vatives curves from -3 T to 3  T of the three kinds of \nMn 3. They are shifted along y axis for clarity. c , The hysteresis \nloops of Mn 3-NH 3 at Day 1 and Day 30. d, The deri vative curves \nfrom -3 T to 3  T of Mn 3-NH 3 at Day 1 and Day 30. The applied \nmagnetic field is along c axis and the sweep field rate is 50  Oe/s. \nMeanwhile , the ac susceptibility experiments of single -\ncrystal of Mn 3-NH 3 were  also p erformed. The in phase co m-\nponent curves shown in Fig. 5a first exhibit ascent as the te m-\nperature is decreased then drop s at about 7.5  K, which is coi n-\ncident with the results of dc magnetization experiments . \nMoreover, the peaks do not move with the change o f freque n-\ncy – a characteristic of AFM phase -transition. Meanwhile, it is \nseen that the peaks of out of phase components shown in Fig. \n5b move to higher temperature as the fr equency is increas ed, \nwhich is the typical characteristic for the spin -flip relaxat ion \nof SMMs. Using the Arrhenius law28: the effe ctive energy \nbarrier of Mn 3-NH 3 is figured out Ueff = 46.85  K±1.91 K, with \nthe fitting curve shown in the supplementary material . The \neffective energy barrier of Mn 3-NH 3 is larger than Mn 3-H2O26 \nand smaller t han Mn 3-CH 3OH.    \n        \n0exp( )\nBU\nkTeff  (1)  \n \n 4 As just stated above, Mn 3-CH 3OH can be transformed into \nMn 3-H2O while Mn 3-H2O can be changed into Mn 3-NH 3 when \nthey are exposed in air or dried ammonia gas for a per iod of \ntime. Hence, the coexistent s tate of Mn 3-CH 3OH and Mn 3-H2O, \nMn 3-H2O and Mn 3-NH 3 should be achieved when we control \nthe exposure time of Mn 3-CH 3OH and Mn 3-H2O in air or dried \nammonia gas  respect ively. Fig. 6a exhibits the out of phase \ncomponent of Mn 3-CH 3OH sample exposed in air for 6 days. \nIt is clear that there are two distinguishable peaks  in the curves \n(see supplementary material ) and both of them move to higher \ntemperature as frequency increasing. It indicates that there are \ntwo spin -flip relaxation process es in the system which is  a \nproof that Mn 3-CH 3OH and Mn 3-H2O molecules coexist in  the \nsame sample. Meanwhile , Fig. 6b shows the out of phase \ncomponent of Mn 3-H2O sample which is exposed in dried \nammonia gas for 6 days. An env eloping line can be observed. \nBecause the Ueff of Mn 3-H2O26 and Mn 3-NH 3 is close to each \nother, the enveloping line should be formed by two  unimodal \ncurves (see supplementary material ) overlapped which belong \nto the spin -flip r elaxation process of Mn 3-H2O and Mn 3-NH 3, \nrespectively. T herefore, the Mn 3-CH 3OH and Mn 3-H2O, Mn 3-\nH2O and Mn 3-NH 3 can be coexistent in the same sample  if we \nappropriately control the time that the sample is exposed in air \nor in a mmonia gas.  \n \nFigure 5. (a), The in phase component vs temperature curves of \nMn 3-NH 3 at different frequencies.  (b), The out of phase comp o-\nnent vs temper ature curves of Mn 3-NH 3 at different frequencies.  \nIn order to quantitatively analyze the magnetic properties \nof these three kinds of Mn 3, we utilize the equation of QTM \nfields of SMM with identical intermolecular  exchange intera c-\ntion from | -S to |S-l described as27: \n  \n00/ ( ) /Z B BH lD g n n JS g                                                  \nwhere D is the magnetic anisotropy parameter21,22,25,27 leading \nto the energy barrier U=DS2, J is antiferromagnetic intermo-\nlecular  exchange interaction parameter, n and n stand for  the \nnumber of a  tunnel ing molecule's neighboring molecules o c-\ncupy ing the |-S and |S state, respectively . S represents the \nground state spin, g is Lande factor, μB is Bohr magn eton, μ0 is \npermeability of vacuum. Using  this equation and the specific \nposition of the QTM peaks shown in Fig. 2b we can figure out \nthe value of D and J of SMMs27. For Mn 3-CH 3OH, D=0.98  K, \nJ=-0.041  K27; for Mn 3-H2O, D=0.925  K, J=-0.132  K25; and for \nMn 3-NH 3, D=0.884  K, J=-0.094  K. The results indicate that \nthe anisotropy energy barrier ( U=DS2) of Mn 3-NH 3 is the \nsmallest which is in contradiction with the effe ctive energy  barrier Ueff given by ac magnetic e xperiment. And it should \nalso be not ed that the value of the Ueff is bigger than U. The \nreason is that U=DS2 is the single particle energy barrier, but, \nthere is the multi -body spin -flip process29 in these systems due \nto the existence of intermolecular exchange interaction. The \nmulti -body sp in-flip process will significantly increase the \nheight of the e ffective energy barrier;  however, for Mn 3-H2O \nand Mn 3-NH 3 the intermolecular exchange interaction is \nstrong enough to lead to the AFM phase transition which will \nsuppress the multi -body spin -flip process. Thus, the height of \nthe effective energy barrier will decrease as the intermolec ular \nexchange interaction increasing.  \nOn the other hand, the AFM intermolecular exchange inte r-\naction of Mn 3-H2O and Mn 3-NH 3 is about three times and two \ntimes stron ger than Mn 3-CH 3OH, respectively. Hence, the \nAFM phase -transition  only can be observed in th ese two sy s-\ntems . As mentioned above, the intermolecular exchange inte r-\naction depends on the hydrogen bond between molecules. The \nresults show that the relationship between the strength of the \ninteraction and the length of hydrogen bond is monotonic as \nshown in Fig. 3c, the shorter the length of hydrogen bond the \nstronger is the interaction. The reason is that the intermolec u-\nlar exchange interaction is formed by super -exchange pat h-\nways30 which is determined by the overlap integral  of the wave \nfunction. Shorter distance leads to larger  overlapping of the \nwave function resulting  in bigger exchange interaction p aram-\neter J31,32.  \n \nFigure 6. (a), The out of phase component  vs temperature curves \nof Mn 3-CH 3OH and Mn 3-H2O coexistent state at different fr e-\nquencies.  (b), The out of phase component vs temperature curves \nof Mn 3-NH 3 and Mn 3-H2O coexistent state  at different freque n-\ncies. \n (2)  \n \n 5 In summary, we have achieved the mutual  transformation \nof ligand in monocrystal single -molecule magnet Mn 3 by ex-\nternal stimuli (different chemical gas atmosphere ). The meth a-\nnol ligand s on Mn 3-CH 3OH can be substituted by water l ig-\nands when the sample is exposed in air for a few days. Int er-\nestingly, t he water  ligand s on Mn 3-H2O can also be replaced \nby ammonia using the same method. Importantly, the hydr o-\ngen bond between molecules depends  on these ligands which \ndetermine the intermolecular  exchange interaction. As a result, \nthese three kinds of Mn 3 exhibit different dc hysteresis loops, \nQTM effect and ac spin -flip effective energy barrier at low \ntemper ature. Moreover, Mn 3-H2O and Mn 3-NH 3 exhibit an \nAFM phase transition at TN= 6.5K and TN= 5.5K respectively, \nthat is rel ative high in the field of transitio n metal SMMs. On \nthe other hand, we also obtain the coexistent state of Mn 3-\nCH 3OH and Mn 3-H2O, Mn 3-H2O and Mn 3-NH 3 when we con-\ntrol the time that the sample ex posed in air or in ammonia gas,  \nthat indicates the magnetic properties of Mn 3 is tunable. More \nimportantly, the single -crystal structure of Mn 3 keeps u n-\nchanged during the whole transformation process. Therefore, \nour results open up an avenue  for exploring  the non -\ndestructive pr oduction of two -dimensional SMM -network by \nexternal stimuli. We also believe  that the gas -responsive Mn 3 \nwill have great potential in the multifunctional  magnetic m ate-\nrial. \nSee supplementary materials for the results of dc and ac \nmagnetic  measurements of th ree kinds of Mn 3 and correspon d-\ning theoretical ana lysis. \nAuthor to whom correspondence should be addresse d. E-\nmails : luojj@ime.ac.cn  \n \nThis work was financially supported by the National Na t-\nural Science Foundation of China (No. 61404161 ).  The a u-\nthors  gratefully acknowledged S. S. K.  for the assistance with \nmagnetic  measure ments at Institute  of Physics Chinese Aca d-\nemy of Sciences.  \nREFERENCES  \n(1) L. Bogani,  W. Wernsdorfer,  Nat. Mater. 7, 179(2008 ). \n(2) A. R. Rocha, V. M. Garcí a -suá re z, S. W. Bailey, C. J. Lambert, J. \nFerrer, S. Sanvito, Nat. Mater.  4, 335(2005). \n(3) L. Zhu, K. L. Yao, Z. L. Liu,  L. Lu, T. Zheng, Q. Wu, A. M.  \nSchneider, D. Zhao, L. Yu, Appl. Phys. Lett.  96, 082115 (2010) . \n(4) C. Romeike, M. R. 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China  \n3Department of Physics and High Pressure Science and Engineering Center, University of \nNevada, Las Vegas, Nevada 89154, USA  \n4Department of Physics and Astro nomy,  Shanghai  Jiao Tong University, Shanghai 200240, \nP . R. China  \n5Collaborative Innovation Center of Advanced Microstructures, Nanjing  210093 , P . R. \nChina  \n*Corresponding author: qmzhang@ruc.edu.cn  \n \n \nAbstract  \n \nDirac materials exhibit  intriguing  low-energy carrier dynamics  that offer a fertile \nground for novel physics discovery. O f particular interest is the in terplay of Dirac \ncarriers with  other quantum phenomena , such as magnetism. Here we report on  a \ntwo-magnon Raman scattering study o f AMnBi 2 (A=Ca,  Sr), a prototypical  magnetic \nDirac system comprising  alternating Dirac -carrier and magnetic layers . We present  \nthe first  accurate determination of the  exchange energies  in these compounds  and,  \nby comparison  to the reference compound BaMn 2Bi2, we show that the Dirac -carrier \nlayers in AMnBi 2 significantly enhance the exchange coupling between the magnetic \nlayers , which in turn drives  a charge -gap opening along the Dirac locus . Our findings \nbreak new grounds in unveiling the fundamental physics of  magnetic Dirac materials , \nwhich offer  a novel platform for probing a  distinct  type of spin-Fermion  interaction. \nThe outstanding  properties of these  materials allow  a delicate manipulation of the \ninteraction between the Dirac carriers and magnetic moments , thus holding  great \npromise for applications in magnetic Dirac devices.   Introduction  \n Recent years have seen the emergence of a new class of materials whose \nlow-energy carrier dynamics obey the Dirac equation, instead of the Schrodinger \nequation that describes most condensed matter systems. These so -called  Dirac \nmaterials exhibit linear carrier dispersion and massless ch iral excitations  that \ngive rise to  novel quantum phenomena such as the ultra -high electron mobility \nand quantum Hall effect1-4. So-far identified Dirac materials  include graphene2,3, \ntopological insulators5,6, and  d-wave and iron -pnictide superconductors7,8. \n One of the most intriguing aspects of Dirac materials is the interplay of their \nunique carrier dynamics with  other quantum phenomena. A prominent case is \nmagnetism, which  may significantly change the electronic band structures of \nDirac materials , as demonstrated by  an antiferromagnetic (AF) long -range order  \nin grap hene observed in a recent experiment9. Further  studies on the mutual \ninfluence of the Dirac -type electronic excitations  and magnetism are  impeded  by \nthe small size of available graphene samples (~7 nm in width). A suitable model \nmaterial system that possesses both Dirac car riers and magnetic order is  \nessential t o further  explo ration of  novel physics and innovative device concepts  \nin magnetic Dirac materials . The recent discovery  of coexisting  linear Dirac \nbands and long -range magnetic order in SrMnBi 2 and CaMnBi 2 provides an \nexciting platform for the study of magnetic Dirac materials . \n Transport measurements reveal  linear band dispersions near the Fermi \nenergy  in both AMnBi 2 (A=Sr, Ca) compounds10-13. First -principles calculations \nindicate that such Dirac -type linear dispersions come  from  the 6px and 6py orbits \nof the Bi ions in the intercalated Ca(Sr )Bi layers , which are slight ly hybridiz ed \nwith the d  orbits of the Ca or Sr ions10,12,14,15 . The calculated band structure has \nbeen verified by angle- resolved photoemission ( ARPES ) experiments10,16,17 .  \nThe transport measurements also indicate  an AF transition around 290 K10-13. \nThe ground state predict ed by first -principles calculations has  a checkerboard AF \norder of Mn2+ spins, with a spin moment of ~4  µB14,15. Subsequent neutron \ndiffraction measurements18 confirm ed the AF transition and  the estimated  size of the magnetic  moment , and  the experiment also  demonstrated  that CaMnBi 2 and \nSrMnBi 2 have the C-type and G -type AF structures, respectively . Moreover,  it is \nshown  that magnetic order ing could open a n energy gap in Dirac fermion band in \nCaMnBi 2 but not in SrMnBi 2 at the mean- field level18. The AMnBi 2 compounds \ncomprise alternating Ca(Sr)Bi -layers accommodating  2D Dirac elec trons and \nMnBi -layers  containing a long -range magnetic order , and this  confi guration is  \nsimilar to the case  reported in the magnetic graphene.  The availability  of \nlarge -size AMnBi 2 crystals allows experimental explorations of  the interplay \nbetween the coexisting magnetic order  and Dirac electrons.  \n In this work , we report on Raman scattering measurements of  two-magnon \nexcitations in SrMnBi 2 and CaMnBi 2. From  the measured and calculated Raman \nspectra, we have determined , for the first time, the  nearest- neighbor  (J1), \nnext -nearest- neighbor  (J2) and interlayer  (Jc) exchange energ y. By comparison to \na reference system BaMn 2Bi2, we find an unusual enhancement of the interlayer \nexchange coupling s between neighboring magnetic layers via the intervening \nDirac -carrier layers. W e further examined  the effect of the magnetism  on the \nDirac electron band  structure  using  a spin- fermion model . Our results show  that \nthe enhanced interlayer exchange coupling drives a charge- gap opening along the \nDirac locus in CaMnBi 2. Unlike the effect of the spin -orbit coupling (SOC), the gap \nopen ed by the AF order ing allows  both the upper and lower branches of the \nDirac locus  to cross  the Fermi level , which well explains  recent ARPES \nmeasurements16. The present study addresses a fundamental issue in magnetic \nDirac materials, i.e., the interplay between Dirac carriers and magnetism. The  \nmaterials systems studied here provide a novel platform for probing a new type \nof charge -moment interaction where the magnetic and conducting layers  are \ncoupled but well separated , making  these materials  a new family of  prototypes  \nwell described by the spin -Fermion model . Th is unusual separation of \nconducting (Dirac) charge and magnetic moments allows a delicate manipulation \nof the interaction between the charge and moment subsystems, which can be \nexplored for innovative spintronic applications .  Results  \n \nIdentification of  two -magnon Raman spectra \nThe three compounds  CaMnBi 2, SrMnBi 2 and BaMn 2Bi2 share  similar crystal and \nmagnetic structures, as shown in Fig. 1a- 1c, and their Raman spectra exhibit a \ncommon feature around 5 00~800  cm-1 (Fig. 1d). The measured spectral weights \nshift  towards larger wavenumbers following the order of the ionic radii of Ca, S r \nand Ba, and  the overall  spectral feature remains unchanged under  different \nexcitation sources (see the inset in Fig. 1d), which indicates that these spectra \ncome from the Raman process rather than a photoluminescence process. A \nmulti -phonon process is also unlikely  because there are no strong phonon \nexcitations above 300 cm-1. Moreover, w e analyzed  the two -magnon Raman \nprocess  in BaMn 2Bi2, using  the exchange couplings S J1=21.7(1.5) meV, \nSJ2=7.85(1.4) meV, S Jc=1.26(0.02) meV determined by  the neutron scattering \nmeasurements19. Ou r calculations (see Supplementary  Note 1 for the method of \ncalculation ) produced  two-magnon excitations that peak  around 650 cm-1, in \ngood agreement with our experimental  observation.  This  agreement between \nexperiment and theory further demonstrates  that the spectral feature around \n500-800 cm-1 in these Mn-Bi compounds originates  from the two -magnon Raman \nprocess.  \nEvolution of  two -magnon peak with temperature \n  We show in Fig. 2 the evolution of the two-magnon peak with temperature.  \nThe spectra from all three compounds exhibit the same trend s with increasing \ntemperature, including  a shift to lower wavenumbers in peak position, a gradual \nbroadening  in peak width , and a reduction in intensity , which nevertheless \nremains  visible above the transition temperatures  (T\nN). These trends are typical \nfor the two-magnon Raman process . The shift of the peak position reflects  the \nenergy changes of the large -q magnons  and magnon- magnon interactions , the \npeak broadening  indicat es a decrease of the magnon lifetime, a nd the visibility of the peak structure even above T N follows  a general feature o f the two-magnon \nprocess, since the peak is dominated  by the magnons at the Brill ouin zone \nboundary  where  the magnetic  correlation remains viable  even far above T N20.  \nThe peak at ~500 cm-1 is likely associated with a process involving  a phonon and  \na magnon, considering  its temperature evolution is similar to that of the \ntwo-magnon spectra and there is a strong spin -orbit coupling in the syst em21.  \nThe anomalies in resistivity and susc eptibility reported at 260 K and 50 K in \nSrMnBi 2 and CaMnBi 2, respectively , were  attributed to spin  realignments or a \nslight structural  change  without any  anomal ies in specific heat12-14,18. We \nobserved no anomalies in the two -magnon spectra at these temperatures (Fig s. \n2a& b). Our  results thus rule  out spin -related processes as the origin of these \nanomalies since two-mag non spectra are highly sensitive to  variations of the \nmagnetic order in the system . \nDeter mination of exchange energies  \n \nBy comparing the characteristic energies extracted from the spectra  to the \ncalculated two- magnon density of states (DOS) , we have determined, for the first \ntime,  the exchange energies in SrMnBi 2 and CaMnBi 2, as summarized in Table I \n(See Supplementary Notes 2 -7 , Supplementary  Figs. 1- 3 and Supplementary  \nTables 1 -3 for details ). We also have extracted the exchange energies for \nBaMn 2Bi2, and the values  are in good  agreement with those determined by the \nneutron scattering measur ements19, which  confirms the reliability of the results \nfrom the two -magnon Raman spectra. The small difference between the J c valu es \nextracted by the two methods  may have result ed from different sample sources \nand/or experimental methods.  \nThe very high energy resolution of  Raman scattering (~ 0.1 meV)  combined \nwith  the sharp two -magnon features observed  in all  three crystals  are expected \nto produce  an accurate determination of the magnetic exchange energy25. There \nis also appropriate verification on the validity of the theoretical model adopted in \nthe present work, which predicts  that the characteristic points in two symmetry channels  are exactly the same (Fig. 3, A 1g and B 1g, see below) . This predicted  \ncoincidence is indeed  seen in the experimental spectra. T here are four available \ncharacteristic p oints in the two -magnon spectra, and a ny three of them can  \nprovide  the same exchange parameters.  We have examined  the accuracy of the \nextracted exchange energies  by inputting much  larger error bars for the  raw data \n( Supplementary Notes 6 and 7, Supplementary Figs 2 and 3 ). Particularly for the \nsmall Jc, its value is exactly proportional to the width of the plateau between ω 1 \nand ω 2 ( Supplementary Note 2 ). The value of J c read out f rom Fig. 3  well \ncoincides with those listed in Table I.  It should be emphasized  that the same \nmodeling is equally applied to all  three systems studied in the present work  \nwithout any additional constraints . This means that the exchange energies \nrelative to each other are consistently comparable even if their  magnitudes may \nhave deviations.  \nBased on the obtained  exchange energies, we have calculated the \ntwo-magnon spectra in the A1g and B 1g channels  (see  Supplementary Note 1 ), \nwhere  the irreducibl e representations of the D4h point group , A1g and B 1g, denote  \ndifferent symmetry channels and  can be separated  by configuring the \npolarizations of incident and scattered light . The results ( Fig. 3) show  that the \nmagnon- magnon interactions have little influence on the A 1g spectra but \ncontribute a sharp  resonant peak in the B1g channel . Magnon- magnon scattering \ndrives a spectral weight transfer to low er energies and consequently causes such \na magnetic ex citon -like resonance peak, whose position corresponds to the \nexciton energy. The simulations including  the magnon- magnon interactions \nproduce results in much better agreement with the experim ental data for \nBaMn 2Bi2 compared to the results of the non- interacting calculations.  Ho weve r, \nthe non-interacting results agree better with the experimental spectra in AMnBi 2. \nIt should be noted that the non-interacting  calculations in AMnBi 2 have some \ndiscrepancies with the experimental data in some spectral details.  The presence  \nof itinerant electrons, particularly in CaMnBi 2 and SrMnBi 2, may be responsible \nfor such  deviations. The itinerant electrons tend to reduce the effective intensities of the incident light and contribute to a relatively low signal -to-noise \nratio. Furthermo re, itinerant electrons  also bring higher -order corrections to the \nlinear spin- wave model . The oretical  calculations for such corrections  are \nextremely complicated  and have not been archived in the literature; however, \nsuch corrections are not expected to a ffect the main features and characteristic \npoints  in the spectra, although they  can possibly modify some spectral details.   \nDiscussion s \nThe strong suppression of the resonant peak in Ca(Sr)MnBi 2 is associated \nwith the strong SOC effect in the Ca(Sr)Bi layer ,16 which leads to  an easy -axis \nanisotropy (along the Sz direction) of the exchange couplings, as well as an \nanisotropic damping of the magnon- magnon interactions  that can suppress the \nresonant peak in the B 1g channel of the Raman spectra. Although the resonance \nstems from magnon -magnon interactions as mentioned above and is strongly \nsuppressed by the SOC effect, the resonance peak intensities may not be  a good \nmeasure of the interaction or SOC strengths. This is because many basic  factors \nsuch as dist inguished crystal symmetries ( the presence of a horizontal mirror \nplane in SrMnBi 2 but absent in CaMnBi 2), magnetic structures (G -type for \nSrMnBi 2 but C-type for CaMnBi 2) and car riers (Dirac electrons dominant in \nSrMnBi 2 but Dirac plus ordinary electrons in CaMnBi 2) are not taken into account. \nFurther insights into this important issue require additional  experim ental and \ntheoretical investigation.  \nThe in- plane exchange couplings  J1 and J2 in the three compounds studied \nhere are well correlated with the in- plane lattice constants . SrMnBi 2 has the \nlargest lattice parameter along its a  axis and the smallest J 1 and J2; BaMn 2Bi2 and \nCaMnBi 2 have similar a  values and their J 1 and J2 are quite close to each other \n(see Table I). This close correlation suggests that the in- plane magnetism is well \ndescribed by the super -exchange mechanism.  \n \n \nTable I  The m agnetic exchange energies extracted from the  Raman spectra. Here a is the in -plane lattice constant and d  the distance between the neighboring MnBi -layers.  \n SJ1 (meV) SJ2 (meV) SJc (meV) a (Å) d (Å) \nCaMnBi 2 20.77(0. 79) 7.29(0. 48) -1.31(0.10) 4.50a 11.07 a \nSrMnBi 2 16.00 (0.30) 4.75(0.17) 2.92(0.09) 4.58a 11.57 a \nBaMn 2Bi2 21.45 (0.32) \n/ 21.7(1.5)b 6.26(0.20) /  \n7.85(1.4)b 0.78(0.0 8) / \n1.26(0.02)b 4.49b 7.34 b \na: ref. 18; b: ref. 19  \n \n  In sharp contrast , the interlayer coupling Jc exhibits highly anomalous \nbehavior. The distances between the neighboring MnBi- layers in  CaMnBi 2 and \nSrMnBi 2 (11.07 Å and 11.57 Å, respectively ) are much larger  than that in \nBaMn 2Bi2 (7.34 Å) because of the intercalation of the additional Bi- layers in the \ntwo magnetic Dirac compounds . At such large interlayer distances, the interlayer \ncoupling Jc is usually expected to  be negligible as suggested  by recent neutron \nmeasurements and calculations15,18. In fact, negligible Jc values have  been \nreported in many other layered compounds with magnetic inter layer distance \n≳0.7 nm, such as K 2NiF 4, K 2MnF 4 and Rb 2MnF 423,24. Surprisingly, the extracted Jc \nvalue for SrMnBi 2, which has the largest MnBi -inter layer distance, is 3.6 times \nthat in BaMn 2Bi2, which has  the smallest MnBi- inter layer distance. Meanwhile, \nCaMnBi 2 also has a  larger Jc about 1.7 times  that of BaMn 2Bi2. Th is unusual \nenhancement of J c is apparently beyond  the standard  super -exchange mechanism \nand indicates novel physics in these magnetic Dirac materials . A key s tructural  \nfeature in the AMnBi 2 compounds is a Dirac -carrier  Ca(Sr)Bi layer between the \nneighboring magnetic MnBi- layers ; in contrast,  there is only  a single layer of Ba2+ \nions between the neighboring MnBi- layers in BaMn 2Bi2. This  structural contrast  \nsuggests that the enhanced interlayer magnetic coupling stems from the \ninterplay of magnetism and  the Dirac carrier s in the intervening  Ca(Sr)Bi laye r.  \nThe spin -fermion system s studied here provide unique insights into the novel physics of the composite magnetic and Dirac electron systems where the layers \naccommodating itinerant carrier s are sandwiched by ordered and insulating \nmagnetic layers . On the other hand, the present material systems do not provide \nan adequate platform to clearly identify the role of the ordinary electrons. A \ndefinitive resolution of this issue requires the synthesis of appropriately structured spin- fermion system s and additional theoretical exploration, which is \nbeyond the scope of our present work.  \n To understand the interplay of the Dirac carriers and magnetism in SrMnBi\n2 \nand CaMnBi 2, especially the enhancement of the coupling J c between neighboring \nmagnetic lay ers and the modifications of the electronic band structure in the \nDirac -carrier layers, we consider the following effective spin -fermion model  \ndescribing both the itinerant electrons in the Bi 6 px and 6 py orbit s and \ninterac ting local magnetic moments on the  Mn ions : \nH=∑𝑡ijαβlciαl+\ni,j,α,β,lcjβl+𝜆SO∑ ciαl+ciβl′iαβll′ +𝐽K\n2∑ ciαl+𝛔ll′cjαl′∙𝐒i±z�iαll′ +\n∑𝐽i′j′H𝐒i′∙𝐒j′i′j′  (1) \nwhere ciαl+ creates an itinerant electron at site i in orbit  α with spin index l in the \nCa(Sr)Bi Dirac -carrier layer, 𝐒i′ refers to the local moment of the Mn ion below \nor above the Ca(Sr)Bi layer , 𝑡ijαβl is the hopping integral of the itinerant electrons , \n𝜆SO is the spin- orbit coupling, J K is the Kondo coupling between the itinerant \nelectrons and local moments, and JH is the super -exchange coupling between the \nlocal moments . Here w e use t he exchange energies determined from our \ntwo-magnon Raman spectra and  treat the local moments of the Mn ions as \nclassical spins , which is justified by the large moment of ~4 μB per Mn  at low \ntemperatures , and the magnetic interaction in the AFM state is  treated in a \nmean- field approximation.   \nThe results of our model calculations show  that both  SrMnBi 2 and CaMnBi 2 \npossess anisotropic Dirac bands, but the processes for the gap opening between \nthe upper and lower Dirac bands are very different in these two compounds . As \nalready noticed in a previous study15, the different arrangement of the Ca or Sr cations leads to a gap opening along a general direction in SrMnBi 2, but not in \nCaMnBi 2. As a result , there are four isolated anisotropic Dirac points along the \nΓ-M direction in SrMnBi 2, but a line of continuous Dirac points is present in \nCaMnBi 2 (see Supplementary Note 8, Supplementary Fig. 4 and Supplementary \nTable  4). We exam ine the effects of SOC and magnetic order on  the gap opening \nin the Dirac bands. For SrMnBi 2, as shown in Fig. 4(a) -(b), the SOC opens a small \ngap (~0.01 eV for λSO=0.6 eV) at the Dirac band along the Γ -M direction, and it \nslightly enhances  the existing  gap b etween the lower and upper Dirac bands . The \nmagnetic order has no net effect on  the band  structure at the mean -field level \nsince the influence coming from the upper and l ower Mn layers exactly cancel  out \ndue to the  G-AFM order. For CaMnBi 2, the effect of SOC is similar , which  opens a \nsmall gap of ~0.01 eV between the upper and lower Dirac bands. This  gap, \nhowever,  is much smaller than observed in a recent ARPES measurement, which \nis about 0.05 -0.1 eV.  \nSurprisingly , we find that in CaMnBi 2, the C -AFM order  introduces  a mass ive \nterm proportional to the sublattice magnetization,  𝐽K|𝑚z|, which acts on \nitinerant electrons , and  this term  is highly effective in opening  a gap in the Dirac \nbands  (Fig. 4( c)-(d)). Taking  JK=0.01  eV, the gap increases five- fold to 0.05 eV, \nwhich is  consistent with the value observe d in recent ARPES  experiment16. \nMeanwhile, we also estimated  the effective Jc in CaMnBi 2 driven by the RKKY \ninteraction (see  Supplementary Note 9 ). At the same  JK=0.01 eV, we obtain ed \n|Jc|~1 meV , which  is in good agreement with the value obtained independently \nfrom fitting  the Raman spectra (Table I) . These  results show consistently that the \ncoupling between the  Dirac electrons and  local moments has a profound impact \non both the effective interlayer magnetic interaction and the Dirac electronic \nband structure.  This finding highlights a powerful characteristic of these \nmagnetic Dirac systems  and raises exciting prospects of manipulating these key \nproperties  by tuning  the interlayer exch ange coupling in magnetic Dirac \nmaterials.  The spin- Fermion model traditionally applies to systems with \nmagnetism and conducting carrier s coexisting in the same lattice. The MnBi compounds studied here present a new environment where the magnetic \nmoments  and conducting carriers are well separated in different sub systems . \nThis allows an accurate description of the spin -Fermion interaction in these \nmaterials , and the results offer new insights into the fundamental physics that \nmay inspire innovative design concepts for spintronic  applications . \n In summary,  we have performed a systematic two -magnon Raman study of \nmagnetic Dirac compounds SrMnBi 2 and CaMnBi2 . Our measurements combined \nwith model calculations produced, for the first time, an accurate determination of \nthe exchange energies, which allow  a quantitative understanding of the novel \nphysics in these materials. A  comparison with  the reference compound BaMn 2Bi2 \nreveals  that the interlayer exchange coupl ings are significantly enhanced and the \nmagnon- magnon interactions are suppressed  by the Dirac -carrier layers . We \nfurther investigated the effects of magnetism on the band  structure of Dirac \ncarriers  and found that the magnetic order has drastic effects on  the gap opening \nin the Dirac bands in CaMnBi 2, whi ch explains  recent ARPES measurements. The  \ndiscovery of the intriguing interplay of Dirac carriers  and magnetism sheds new \nlight on the rich physics in magnetic Dirac materials . Our reported work sets key \nbenchmarks for these distinct systems containing coupled but well separated \nmagnetic and Dirac -carrier layers that can be accurately described by  the \nspin -Fermion model . These results unveil new fundamental physics and  pave the \nway for innovative design and development of magnetic Dirac devices for \nspintronic  applications .  \n \n  \nMethods  \nHigh -quality crystals of BaMn\n2Bi2, SrMnBi 2 and CaMnBi 2 were grown by self -flux \nmethod. The details of crystal growth can be found elsewhere.11,14 ,18,22 The \nantiferro magnetic transition temperatures of SrMnBi 2 and CaMnBi 2 could be find \nin Ref. 18, where magnetic susceptibilities  and resistivities  were measured in the \nsame batch of crystals as  used in our measurements.  Raman measurements were performed with a Jobin Yvon HR800 single -grating -based micro -Raman system \nequipped with a volume Bragg grating low -wave number suite, a liquid -nitrogen \ncooled back -illuminated CCD detector and a 633 nm laser (Melles  Griot). The \nlaser was focused into a spot of ~5 μ m in diameter on sample surface, with a \npower less than100  μW to avoid overheating .  \n \nReferences  \n1.Vafek, O. &  Vishwanath, A. Dirac f ermions in solids -from high T c cuprates and graphene to \ntopological insulators and Weyl semimetals.  Annu.  Rev. Condens . Matter Phys . 5, 83-112 (2014).  \n2. Novoselov, K. S. et al.  Electric field effect in atomically thin carbon films.  Science 306, 666–669 \n(2004).  \n3. Castro Neto, A. H., Guinea, F., Peres, N. M., Novoselov, K. S. & Geim, A. K. The electronic \nproperties of graphene.  Rev. Mod. Phys . 81, 109–162 (2009).  \n4. Zhang, Y. , Tan, Y. W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall \neffect and Berry 's phase in graphene. Nature  438, 201–204 (2005).  \n5. Hasan, M. & Kane, C. Colloquium: Topological insulators. Rev. Mod . Phys.  82, 3045– 3066 (2010).  \n6. Qi, X. & Zhang, S. Topological insulators and superconductors. Rev. Mod. Phys.  83, 1057 –1110 \n(2011).  \n7. Orenstein, J. & Millis, A. J. Advances in the physics of high -temperature  superconductivity. Science \n288, 468–474 (2000).  \n8. Richard, P. et al.  Observation of Dirac cone electronic dispersion in BaFe 2As2. Phys. Rev. Lett.  104, \n137001 (2010).  \n9. Magda, G. Z. et  al. Room -temperature magnetic order on zigzag edges of narrow graphene \nnanoribbons. Nature 514, 608 -611 (2014) . \n10. Park, J. et al.  Anisotropic Dirac f ermions in a Bi square net of SrMnBi 2. Phys. Rev.  Lett. 107, \n126402 (2011).  \n11. Wang, K., Graf, D., Lei, H., Tozer, S. W. &  Petrovic C. Quantum transport of two -dimensional \nDirac fermions in SrMnBi 2. Phys. Rev. B  84, 220401(R) (2011).  \n12. Wang, K. et al.  Two-dimensional Dirac fermions and quantum  magnetoresistance  in CaMnBi 2. \nPhys. Rev. B  85, 041101(R) (2012).  \n13. He, J. B., Wang, D. M. & Chen, G. F. Giant magnetoresistance in layered manganese pnictide \nCaMnBi 2. Appl. Phys. Lett.  100, 112405 (2012).  \n14. Wang, J. et al.  Layered transition -metal pnictide SrMnBi 2 with metallic blocking  layer. Phys. Rev. \nB 84, 064428 (2011).  \n15. Lee, G. , Farhan, M. A., Kim, J. S. & Shim, J. H. Anisotropic  Dirac electronic st ructures of AMnBi 2 \n(A=Sr, Ca). Phys.  Rev. B  87, 245104 (2013).  \n16. Feng, Y. et al.  Strong anisotropy of Dirac cones in SrMnBi 2 and CaMnBi 2 revealed by  \nangle-resolved photoemission  spectroscopy. Sci. Rep.  4, 5385 (2014).  \n17. Jia, L. L. et al.  Observation of  well-defined quasiparticles at a wide energy range in a \nquasi -two-dimensional system . Phys.  Rev. B  90, 035133 (2013).  \n18. Guo, Y. F.  et al.  Coupling of magnetic order to planar Bi electrons in the anisotropic  Dirac metals AMnBi 2 (A = Sr, Ca) . Phys. Rev. B  90, 075120 (2014).  \n19. Calder, S.  et al.  Magnetic structure and spin excitations in BaMn 2Bi2. Phys. Rev. B  89, 064417 \n(2014) . \n20. Cottam, M. G. & Lockwood, D.  J. Light Scattering in Magnetic Solids  Ch. 6 (John Wiley & Sons., \nNew York,  1986).  \n21. Cardona, M. &  Güntherodt, G. Light Scattering in Solids IV  Ch. 4 (Springer -Verlag, Berlin, 1984).  \n22. Saparov, B. &  Sefat, A. S. Crystals, magnetic and electronic properties of a new ThCr 2Si2-type \nBaMn 2Bi2 and K -doped compositions.  J. Solid State Chem. 204, 32-39 (2013).  \n23. Legrand, E. & Plumier,  R. Neutron diffraction investigation of a ntiferromagnetic K2NiF 4. Phys. \nStatus Solidi B 2, 317-320 (1962) . \n24. de Wijn, H. W., Walker, L. R. &  Walstedt, R. E.  Spin-wave analysis of the q uadratic -layer \nantiferromagnets  KNiF 4, KMnF 4 and RbMnF 4. Phys. Rev. B  8, 285-295 (1973).  \n25. Devereaux , T. P. & Hackl , R. Inelastic Light Scattering from Correlated Electrons . Rev. Mod. \nPhys.  79, 175 ( 2007).  \nAcknowledgements  \nThis work was supported by the Ministry of Science and Technology of China (Grant \nNos.: 2016YFA0300504, 2016YFA0300501 and 2016YFA0300604) and the NSF of \nChina. C. F. C. was supported in part by DOE under Cooperative Agreement No. \nDENA0001982. Y.G.S was su pported by the Strategic Priority Research Program (B) \nof the Chinese Academy of Sciences (Grant No. XDB07020100). Q.M.Z., A.M.Z. and \nT .L.X were supported by the Fundamental Research Funds for the Central \nUniversities and the Research Funds of Renmin Unive rsity of China (10XNI038, \n14XNLF06, 14XNLQ07).  \n \nAuthor contributions Q.M.Z conducted the whole study and wrote the paper. A.M.Z made Raman measurements, data analysis and wrote the paper. C.L.L made numerical calculations and data analysis. R.Y made theoretical modeling and calculations, and wrote the paper. C.J.Y and Y.G.S grew SrMnBi\n2 and CaMnBi 2 single crystals. \nG.H.Z and T.L.X grew BaMn 2Bi2 single crystals. C.F.C, X.Q.W. and J.T.J made data \nanalysis and paper revising and checking.  \n \n \n \n \n  \n \n \n \n \n \n \n  \nFigure 1 C rystal and magnetic structures and two -magnon Raman spectra of \nCaMnBi 2, SrMnBi 2 and  BaMn 2Bi2. (a)- (c) The c rystal and magnetic st ructures of \nthe three compounds18,19. (d) Their t wo-magnon excitations measured at 10 K. \nThe inset s hows the two -magnon spectra at  different excitation energies. The \nspectra were collected in an unpolarized configuration to obtain a better \nsignal -to-noise ratio.  \n  \n \nFigure  2 Temperature evolution of the two-magnon spectra.  Measured \nRaman spectra of  (a) CaMnBi 2, (b) SrMnBi 2, and  (c) BaMn 2Bi2.  Highlighted are \nthe spectra at 50  K and 300 K in (a), 2 60 K and 30 0 K in (b) and at 100 K in (c) \nwhere resistivity and susceptibility anomalies or  antiferromagnetic transition s \nwere  observed in measurements using other techniques as discussed in the text, \nbut no anomalies are visible in the two -magnon spectra at these temperatures . \nThe spectra were collected in an unpolarized configuration to obtain a better \nsignal -to-noise ratio.  \n \nFigure  3 Measured and calculated two -magnon  Raman  spectra.  The \nlow-temperature (10 K) two -magnon Raman spectra in the A 1g and B 1g channels \n(circles) compared to the calculated results based on the linear spin- wave theory \nwithout considering the magnon- magnon interactions (red lines) and  the results  \ntaking into account the magnon- magnon interactions  (black lines) .  The vertical \ndashed lines mark the characteristic frequencies  that are independent of the \nsymmetries and the magnon- magnon interactions. The se two characteristic \nfrequencies, in combinatio n with the cut- off frequencies, determine the exchange \nenergies J1, J2 and Jc (see  Supplementary Note 2  and Supplementary Fig. 1 for \ndetails ).  \n \n  \n \nFigure  4 Anisotropic Dirac bands in SrMnBi 2 and CaMnBi 2.  Panels (a) and (b) \nshow the dispersion of the Dirac bands (black lines) along the Γ-M direction of \nthe Brilluion zone and the locus of the crossing point energy of the Dirac bands \n(red lin es) without or  with the  spin -orbit coupling λ SO for SrMnBi 2. The lower and \nupper branches ar e shown in solid and dashed lines. The i nset in panel (b) shows \nthe isolated Dirac points (black dots  D) in SrMnBi 2 and continuous Dirac points \n(Dirac locus, red line) in CaMnBi 2. Panels (c) and (d) show the corre sponding \nDirac bands (black lin es) along th e Γ-M direction and the Dirac locus (red lin es) \nwith the spin -orbit coupling λ SO for CaMnBi 2 at two different values of the \nexchange coupling J K. Here,  the gap size between the lower and upper  Dirac \nbands are dominated by the exchange coupling J K instead of the spin- orbit \ncoupling  λSO. Here Δk=|k-kD|, where kD is the moment of the Dirac point. The Γ-M \ndirection and Dirac locus  direction are along  the black line and red curve in the \ninset in panel (b) , respectively . \n \n          \nSupplementary Information \n \n \nSupplementary Figure 1 : The three characteristic frequencies ω 1, ω2 and ω3 in the \ntwo-magnon Raman spectra  and van -Hove singularities in the two -magnon DOS.  \n  \n \n  \n \n \n  \nSupplementary Figure 2 : Determination of characteristic spectral points in \nBaMn 2Bi2 (upper), SrMnBi 2 (medium) and CaMnBi 2 (lower) . The red dashed boxes \nin all the insets indicate the error ranges, as described in the above text. The black \nlines are the baselines given by fitting the background at the high -frequency end.  \n \n \n \n  \n \n  \n \n  \n \n  \n \n  \n \nSupplementary Figure 3: Determination of characteristic spectral points in Sr MnBi 2 \nusing the unpolarized spectra . The red dashed boxes in the insets indicate the error \nranges.  \n \n \n \nSupplementary Figure 4: The energy gaps of anisotropic Dirac bands with various J k \nin CaMnBi 2. The figure shows the dispersion of the Dirac bands (black lines) along \nthe Γ-M direction of the Brilluion zone and the locus of the crossing point energy of \nthe Dirac bands ( magenta lines) for CaMnBi 2. The lower and upper branches are \nshown in solid and dashed lines.  \n \n \n \n \n  \nSupplementary  Table 1: Three characteristic frequencies  obtained from  the \nexperimental  spectra and the extracted exchange parameters with the error bars \nobtained through the error estimation described in Supplementary Note 3. \n CaMnBi 2 SrMnBi 2 BaMn 2Bi2 \n𝜔1 (cm-1) 561.2(2.3) 569.3(1.8) 722. 6(3.2)  \n𝜔2 (cm-1) 680.2(7.8) 708.1(1.9) 758.7(2.3)  \n𝜔3 (cm-1) 954.9 (16.9 ) 825.2(5.8) 1005.5(7.4)  \nSJ1  (meV)  20.77(0. 79) 16.0 0(0.30) 21.4 5(0.32)  \nSJ2  (meV)  7.29(0.48) 4.75(0.17) 6.26(0.20)  \nSJc  (meV)  -1.31(0.10) 2.92(0.09) 0.78(0.08)  \n \nSupplementary  Table 2:  Comparison of the parameters in SrMnBi 2 extracted from \nthe polarized and unpolarized spectra, respectively.  \n \nSrMnBi 2 Polarized (A 1g)    Unpolarized  \n𝜔1 (cm-1) 569.2(1.8) 568.7(1. 8) \n𝜔2 (cm-1) 708.1(1.9) 707.3(1.8) \n𝜔3 (cm-1) 825.2(5.8) 826.2(5.3) \nSJ1  (meV)  16.0 0(0.30) 16.0 8(0.27) \nSJ2  (meV)  4.75(0.17) 4.79(0.15) \nSJc  (meV)  2.92(0.09) 2.89(0.08) \n \nSupplementary  Table 3:  Comparison of exchange interactions for three materials \nwith/without single -ion anisotropy.  \n \n  CaMnBi 2 SrMnBi 2 BaMn 2Bi2 \n𝐷\n𝐽1=0.046  SJ1  (meV)  20.85(0.81) 15.97 (0.39) 21.24(0.32)  \nSJ2  (meV)  7.76(0.51) 4.97(0.23) 6.51(0.21)  \nSJc  (meV)  -1.23(0.10)  2.70(0.10) 0.72(0.07)  \n𝐷\n𝐽1=0 SJ1  (meV)  20.77(0.79)  16.00(0.30)  21.45(0.32)  \nSJ2  (meV)  7.29(0.48)  4.75(0.17)  6.26(0.20)  \nSJc  (meV)  -1.31(0.10)  2.92(0.09)  0.78(0.08)  \n \nSupplementary  Table 4:  The hopping parameters and the chemical potential  µ \nModel parameters  SrMnBi 2 (eV) CaMnBi 2 (eV) \n𝑡1σx 2.00  2.00  \n𝑡1πx -0.50  -0.50  \n𝑡2x 0.12  0.26  \nµ 0.00  0.11   \n \nSupplementary Note 1: Method of calculation for two -magnon Raman scattering \nspectra  \n \nThe standard theory of magnetic R aman scattering is based on the \nElliott -Fleury -London  theory1. The Raman scattering operator is given by  \n          𝑂 �~∑𝐽𝑖𝑗�𝐞�𝑖𝑛∙𝐝̂𝑖𝑗��𝐞�𝑜𝑢𝑡∙𝐝̂𝑖𝑗� 𝐒𝑖 𝑖𝑗 ∙𝐒𝑗.                        (1) \nHere 𝐽𝑖𝑗 is the spin exchange interaction between the local moments at site i and \nsite j, 𝐞�𝑖𝑛 and 𝐞�𝑜𝑢𝑡 are unit polarization  vectors of the incoming and scattering \nlight , respectively, and 𝐝̂𝑖𝑗 is the  vector connecting site i and site j. The Raman cross \nsection at zero temperature is given by the imaginary  part of the correlation function \n𝐼(𝜔)=−𝑖∫d𝑡 𝑒𝑖𝜔𝑡〈T𝑡𝐎�+(𝑡)𝐎�(0)〉0, where ⟨...⟩0 represents the quantum \nmechanical average over the g roun d state, and T t is the time ordering operator.  \nWe use the spin -wave approach within the framework of perturbative theory to \ncalculate Raman spectra2. By introducing the Holstein -Primako ff transformation, we \nexpress spins in A (spin up) and B (spin down)  sublattices in terms of H-P boson \noperators  𝑎𝑖 and 𝑏𝑗. The system  Hamiltonian H=𝐽1∑𝐒𝑖∙𝐒𝑗 〈𝑖𝑗〉 +𝐽2∑𝐒𝑖∙𝐒𝑗 〈〈𝑖𝑗〉〉 +\n𝐽𝑐∑𝐒𝑖∙𝐒𝑗 〈〈〈𝑖𝑗〉〉〉  is expanded in powers of 1/ S as: \n            H ≈S2�𝐸0+1\nS𝐻�0+1\nS2𝐻�1+⋯�                          (2) \nHere  𝐸0 is a constant  classical energy , 𝐻�0 contains quadratic terms of magnons \nand represents  the linear spin -wave  (LSW)  correction to the classical energy , and 𝐻�1 \ncontains magnon quartic terms, representing  two-body magnon -magnon \ninteractions . The higher order terms  are ignored in the present treatment.  \nTo calculate the correlation function  𝐼(𝜔), we first apply Fourier and \nBogoliubov transformations to diagonalize the quadratic LSW part 𝐻�0 in terms of \nBogoliubov magnons α𝐤 and β−𝐤. The 𝐻�1 part then contains Bogoliubov magnons \nin quartic order,  and the magnon- pair- scattering term α𝐤+β−𝐤+β−𝐤′α𝐤′ in 𝐻�1 is \ntreated within the ladder  approximation. The interaction vertex is reserved to the \nlowest (1/S)0 order, and the ladder diagrams are su mmed up exactly.  \n \nSupplementary Note 2:  Determin ation of exchange energies  J1, J2 and Jc \n \nWe determine the exchange couplings J 1, J2 and Jc from three characteristic \nfrequencies  ω1, ω2 and ω3, which, as shown in Supplementary Figure  1, correspond \nto the frequencies of the shoul ders and cut -off of the A1g Raman spectra . These \nfrequencies are associated with the van -Hove singularities in  the two -magnon \ndensity of states ( DOS ), and are not  shifted by the magnon -magnon  interactions \nwithin our approximations ( Supplementary Figure  1). In the linear spin -wave  theory, \nthese  frequencies take the following analytic al form : \n 1) G-type AFM  \n                   ω 1=4𝑆(2𝐽1+𝐽𝑐)�𝐽1−2𝐽2\n𝐽1+2𝐽2                        (3) \n                  ω 2=4𝑆[2𝐽1(𝐽1−2𝐽2)+𝐽𝑐(𝐽1+2𝐽2)]\n�𝐽12−4𝐽22                      (4) \n           ω 3≳4S�(2𝐽1−2𝐽2+𝐽𝑐)[2(𝐽1−𝐽2)(𝐽12−2𝐽1𝐽2+2𝐽22)+𝐽𝑐(𝐽1−2𝐽2)2]\n𝐽12−2𝐽1𝐽2+𝐽2(2𝐽2−𝐽𝑐)         (5) \n               ω 2−ω1=SJc×16J2/J1\n�1−(2J2/J1)2                    (6) \n \n \nwhich holds when Jc<4J2 and 𝐽𝑐<2𝐽1(𝐽1−2𝐽2)\n𝐽1+2𝐽2; \n2) C- type AFM  \n                  𝜔 1=8𝑆𝐽1�𝐽1−2𝐽2\n𝐽1+2𝐽2                               (7) \n                  𝜔 2=8𝑆𝐽1(𝐽1−2𝐽2−𝐽𝑐)\n�𝐽12−4𝐽22                                                                   (8) \n                  𝜔 3= 8𝑆(𝐽1− 𝐽2−𝐽𝑐)                                                    (9) \n               𝜔 2−𝜔1=|𝑆𝐽𝑐|×8\n�1−(2𝐽2/𝐽1)2                    (10)  \n \nwhich holds when |𝐽𝑐|<𝐽1(𝐽1−2𝐽2)\n2𝐽2, and these constraints for both cases are valid in \nthe cases  studied in the present work . From the experimentally determined \nfrequencies, we can extract the three exchange couplings J 1, J2 and Jc using  the \nabove formulas . \n \nSupplementary Note 3:  Error estimation of the exchange energies  \n \nThe characteristic frequencies can be expressed as  \n              𝛚 =𝛚(𝐉)                                     (11)  \nThe experimental ly determined frequencies  have errors  around their averages  \n               𝛚 =𝛚�+𝛿𝛚                                  (12) \nThe corresponding exchange energies can then be written as  𝐉=𝐉̅+𝛿𝐉, where 𝐉̅ is \ndetermined via the equation 𝛚�=𝛚(𝐉̅), and 𝛿𝐉 is obtained through the expansion  \n              𝛿 𝛚=�𝜕𝛚\n𝜕𝐉�|𝐉=𝐉̅ 𝛿𝐉,                               (13)  \nwhere  �𝜕𝛚\n𝜕𝐉�|𝐉=𝐉̅ is the Jacobi’s determinant, which leads to  \n                𝛿 𝐉=[�𝜕𝛚\n𝜕𝐉�|𝐉=𝐉̅ ]−1 𝛿𝛚.                            (14)   \n Supplementary Note 4:  Effects of p ossible spin anisotropy on the exchange \nenergies in SrMnBi\n2 and CaMnBi 2 \n \nIn Sr(Ca)MnBi 2 materials Dirac carriers in Ca(Sr )Bi layers  are subjected to SOC \neffect which may introduce spin anisotropy in the exchange couplings between Mn \nions. To take into account the spin anisotropy effect we consider the following model  \nH=𝐽1∑ [𝑆𝑖𝑧\n〈𝑖𝑗〉𝑆𝑗𝑧+𝜂\n2(𝑆𝑖+𝑆𝑗−+𝑆𝑖−𝑆𝑗+)]+𝐽2∑𝐒𝑖∙𝐒𝑗 〈〈𝑖𝑗〉〉 +𝐽𝑐∑𝐒𝑖∙𝐒𝑗 〈〈〈𝑖𝑗〉〉〉 ,  (15)  \nwhere η  is the parameter describing the anisotropy, and 𝜂<1 ensures that spins \nare aligned along the z direction.  \nTo determine  the parameters  J1, J2, Jc and η, we make use of an additional \ncharacteristic point  in the measured Raman spectra, namely  the absorption edge in \nthe 𝐵1𝑔 channel.  Following  the same procedure  described  above, we find that \ncompared to the isotropic model t he difference in the exchanges energies c aused by \nthe anisotropy is less than 10%, which indicates that the results obtained using the \nisotropic model are quite reasonable.  \n \nSupplemen tary Note 5 : Effects of single -ion anisotropy in CaMnBi 2, SrMnBi 2 and \nBaMn 2Bi2 \n \nIt has been reported in inelastic neutron scattering experiments that in \nBaMn 2Bi2 materials there is a spin gap of 16 meV4. The magnetic excitations can be \nwell fit to a Heisenberg model plus single -ion anisotropy terms4 \nH=∑𝐽𝑖𝑗𝐒𝑖∙𝐒𝑗 𝑖𝑗 −𝐷∑(𝑆𝑖𝑧)2\n𝑖 ,                    (31)  \nwhere 𝐷>0 ensures that spins are aligned along the z direction.  \nWe find that all characteristic frequencies will be shifted in presence of finite D \nterms. However, the relation  \n𝜔2−𝜔1=𝑆𝐽𝑐×16𝐽2/𝐽1\n�1−(2𝐽2/𝐽1)2                     (32)  \nfor G-type AFM and  \n𝜔2−𝜔1=|𝑆𝐽𝑐|×8\n�1−(2𝐽2/𝐽1)2                     (33)  \nfor C-type AFM still remains unchanged. This implies that the width of the \nshoulders in 𝐴1𝑔 spectra still has no direct relations with D terms. Using the \nvalue 𝐷/𝐽1 =0.046  extracted in INS experiments for BaMn 2Bi2 materials4, we \ncan calculate the exchange interactions as is shown in Supplementary  Table 3 . \n    We can see that in presence of D terms, the exchange interactions are \nslightly changed. The interchange coupling S Jc becomes slightly smaller in \npresence of D, but the difference is less than 10%. It is not surprising that D \nterms can generate a relatively big spin gap but do not strongly affect the exchange interaction results extracted from Raman experiments, since D terms \nmainly affect low energy magnetic ex citations, but Raman techniques mainly \nprobe high energy physics.  \n \nSupplementary Note 6 : Determination of c haracteristic frequencies and their \nerrors  \n \nThe spectra for BaMn 2Bi2 and SrMnBi 2 have a good signal -to-noise level, and  \nthe frequencies can be directly read out from the raw spectra (ω 2 in BaMn 2Bi2 and ω 1 \nin SrMnBi 2) or obtained from taking the first derivative (ω 1 in BaMn 2Bi2 and ω 2 in \nSrMnBi 2). For CaMnBi 2, one must be more careful as the original polarized spectra \nhave a higher noise level, making it hard to accurately determine the characteristic \nspectral points. On the other hand, spin -wave calculations indicate that both \npolarized spectra (A 1g and B 1g) have exactly the same characteristic points. This \nmeans that the combined A 1g and B 1g spec tra, i.e., the unpolarized data, possess the \nsame characteristic spectral points. This enables an alternative scenario, where we have collected many unpolarized spectra at 10 K with a much better signal- to-noise \nratio, as shown in Supplementary Figure 2 . Using the unpolarized spectra (A\n1g+B1g), \none can easily determine the characteristic frequencies and their errors, in the same \nway as done above for BaMn 2Bi2 and SrMnBi 2. We have examined the validity of the \nmethod using SrMnBi 2, in which there is little di fference between the parameters \nderived from the polarized spectra and the unpolarized ones, respectively \n(Supplementary Note 7 ). \nThen we can extract the frequency parameters and quantitatively estimate the \nassociated errors in the following procedure (see  Supplementary Figure 2 ). 1) A \ngeneral linear fitting was made for a linear region selected from the raw spectra or the first -derived data. This gives the standard deviations in intensity (or its derivative); \n2) For ω\n1 and ω 2 associated with the local maxi mums/minimums of the raw spectra \nor the first -derivatives, we can identify a region which starts  from the \nmaximums/minimums and vertically extends by double standard deviations (the heights of the red dashed error boxes). All the data points in the region are possible \nmaximums/minimums. This simply fixes the corresponding deviations in frequency (the widths of the error boxes) and provides the standard errors; 3) For ω\n3 not \nassociated with a local extreme, we first determined the baselines (black lines) by fitting the background at the high -frequency end. The left bound of the error box is \nreached when the positive intensity deviations from a baseline begin to exceed the standard deviations estimated in the first step. Similarly the right bound is defined at  \nthe position where the intensity deviations approach the standard ones. This gives the frequency parameter ω\n3 and the associated errors. The characteristic frequencies obtained from the experimental spectra strictly following the above procedure and \nthe e xtracted exchange parameters with the error bars, are listed in Supplementary \nTable  1. \n \nSupplementary Note 7 : Comparison of the parameters extracted from the \npolarized and unpolarized spectra in SrMnBi 2 \n \nFollowing the procedure described in Supplementary Note 6 , we can also obtain \nthe frequency parameters in SrMnBi 2 with the unpolarized spectra (see \nSupplementary Figure 3). And the obtained characteristic frequencies and the \nextracted exchange parameters from the pol arized and unpolarized  spectra, are \nlisted in  Supplementary Table  2 for comparison. There is little difference between \nboth cases.  This demonstrates that the unpolarized spectra work well as the \npolarized ones in obtaining the characteristic frequency points.  \n \n \n Supplementary N ote 8: Electronic b and structures and Dirac point s in SrMnBi\n2 and \nCaMnBi 2 \n \nWe consider the spin-fermion Hamiltonian introduced in the main text  \nH=∑𝑡ijαβlciαl+\ni,j,α,β,l cjβl+𝜆SO∑ ciαl+ciβl′iαβll′ +𝐽K\n2∑ ciαl+𝛔ll′cjαl′∙𝐒i±z� iαll′ +\n∑𝐽i′j′H𝐒i′∙𝐒j′i′j′  (16)  \nThe first term is a two -orbital tight -binding model for the itinerant electrons in Bi 6 px \nand 6py orbits (in  the Sr (Ca)Bi  layer).  The second term contains the atomic spin -orbit \ncoupling.  𝐒i±z� refers to the local moment of Mn in layers above or  below the Bi site \ni. For simplicity, we do not consider the in fluence of the  Bi 6pz orbit . Since the \nobserved magnetic moment s in these materials are about 4  μB per Mn, we treat  Si’ \nas classical spins. We take the 𝐽i′j′H values obtained from the Raman measurements, \nso that the ground state of the model has either a G -AFM (SrMnBi 2) or a C -AFM \n(CaMnBi 2) order. We then treat the effects of the AFM order on  the band  structure \nof the itinerant electrons at the mean -field level . \n    Within above approximations, the Mn local moments serve as local magnetic fields that couple to the itinerant electr ons and modify their dispersion. The \nHamiltonian is then written as  \n                 H  ≈H\nTB+ HS,                                   (17)  \nwhere  \n          𝐻 TB=∑𝑡ij𝛼𝛽lci𝛼l+\nij𝛼𝛽l cj𝛽l+𝜆SO∑ ci𝛼l+ci𝛽l′i𝛼𝛽ll′ ,               (18)  \n            𝐻 S=𝐽𝐾\n2∑𝑚𝑛(𝑐𝑛𝛼↑+𝑐𝑛𝛼↑−𝑐𝑛𝛼↓+𝑐𝑛𝛼↓)𝑛𝛼 ,                   (19)  \nand 𝑚𝑛=〈𝑆𝑛𝑧〉 is the sublattice  magnetic moment for n = A, B sublattice . For \nSrMnBi 2, the magnetic order is G -AFM , where the Mn ions in upper and lower layers \nbelong to the A and B sublattices , respectively, and at the mean -field level, the effect \nfrom the two layers cancels out exactly, hence H ≈HTB. But for CaMnBi 2, the C -AFM \norder induce s an uncompensated magnetic field, which acts as a mass term since it \nhas different signs on the two subla ttices . \nWe rewrite  HTB into a matrix form3: \n      𝐻 TB=�𝐻𝐴𝐴+𝐻𝑠𝑜𝐻𝐴𝐵\n𝐻𝐵𝐴𝐻𝐵𝐵+𝐻𝑠𝑜�,                             (20)  \nwhere  Hso and Hnn' are 2×2 matrices. The spin -orbit coupling takes the form  \n        𝐻 𝑠𝑜=�0−𝑖𝜆𝑠𝑜\n𝑖𝜆𝑠𝑜 0�  ,                                     (21)  \nwhere  𝜆𝑠𝑜 is the coupling constant. In our calculations , we consider two cases,  \n𝜆𝑠𝑜=0, and 𝜆𝑠𝑜=0.6 eV.3 \nFor the hopping integrals, w e assume that the dominant terms  are the \nintraorbital ones, and we neglect the interorbital hopping . We then obtain the \nfollowing hopping matrices for SrMnBi 2 and CaMnBi 2: \n𝐻𝐴𝐴𝑆𝑟=�2𝑡2𝑥𝑠cos(𝑘𝑥+𝑘𝑦)−𝜇 0\n0 2𝑡2𝑥𝑠cos(𝑘𝑥+𝑘𝑦)−𝜇�,               (22)  \n𝐻𝐵𝐵𝑆𝑟=�2𝑡2𝑥𝑠cos(𝑘𝑥−𝑘𝑦)−𝜇 0\n0 2𝑡2𝑥𝑠cos(𝑘𝑥−𝑘𝑦)−𝜇�,               (23)  \n𝐻𝐴𝐵𝑆𝑟=𝐻𝐵𝐴𝑆𝑟=�2(𝑡1𝜎𝑥𝑠cos𝑘𝑥+𝑡1𝜋𝑥𝑠cos𝑘𝑦) 0\n0 2(𝑡1𝜎𝑥𝑠cos𝑘𝑦+𝑡1𝜋𝑥𝑠cos𝑘𝑥)�,   (24) \n𝐻𝐴𝐴𝐶𝑎=𝐻𝐵𝐵𝐶𝑎=�4𝑡2𝑥𝑐cos����𝑥cos𝑘𝑦−𝜇 0\n0 4𝑡2𝑥𝑐cos𝑘𝑥cos𝑘𝑦−𝜇�,         (25)  \n𝐻𝐴𝐵𝐶𝑎=𝐻𝐵𝐴𝐶𝑎=�2(𝑡1𝜎𝑥𝑐cos𝑘𝑥+𝑡1𝜋𝑥𝑐cos𝑘𝑦) 0\n0 2(𝑡1𝜎𝑥𝑐cos𝑘𝑦+𝑡1𝜋𝑥𝑐cos𝑘𝑥)�.   (26)  \nHere the hopping parameters and the chemical potential  µ are determined by fitting \nto the DFT band structure , and  their values are summarized in Supplementary Table \n4. Note that due to the buckling of the Sr  cations, 𝐻𝐴𝐴𝑆𝑟≠𝐻𝐵𝐵𝑆𝑟 for general k. The \nband  structure can be obtained by diagonalizing  the mean -field Hamiltonian . \nWithout the spin- orbit coupling,  for SrMnBi 2, the energy in each band reads  \n𝐸1(2),±𝑆𝑟=\n2𝑡2𝑥𝑠cos𝑘𝑥cos𝑘𝑦±�4𝑡2𝑥𝑠sin2𝑘𝑥sin2𝑘𝑦+4�𝑡1𝜎(𝜋)𝑥𝑠cos𝑘𝑥+𝑡1𝜋(𝜎)𝑥𝑠cos𝑘𝑦�2 (27)  \nBy requiring  �4𝑡2𝑥𝑠sin2𝑘𝑥sin2𝑘𝑦+4�𝑡1𝜎(𝜋)𝑥𝑠cos𝑘𝑥+𝑡1𝜋(𝜎)𝑥𝑠cos𝑘𝑦�2=0, we obt ain four Dirac \npoints . For 𝑡1𝜎𝑥𝑠>𝑡1𝜋𝑥𝑠, they are located at  𝑘𝑥=0, 𝑘𝑦=±arccos  (−𝑡1𝜋𝑥𝑠𝑡1𝜎𝑥𝑠⁄ ), and \n𝑘𝑦=0,𝑘𝑥=±arccos  (−𝑡1𝜋𝑥𝑠𝑡1𝜎𝑥𝑠⁄ ). Focusing on one Dirac point 𝑘𝑦=0, 𝑘𝑥=\n±arccos  (−𝑡1𝜋𝑥𝑠𝑡1𝜎𝑥𝑠⁄ ), the dispersion is very anisotropic, as shown in Fig. 4( a) in the \nmain text . Turning on the spin-orbit coupling pushes apart  the Dirac points as shown \nin Fig.  4(b).  \nFollowing the same procedure, we  obtain the electron ic band  structure for \nCaMnBi 2. Without the spin-orbit coupling,  \n𝐸1(2),±𝐶𝑎=2𝑡2𝑥𝑐cos𝑘𝑥cos𝑘𝑦±�𝐽̃𝑘2+4�𝑡1𝜎(𝜋)𝑥𝑐cos𝑘𝑥+𝑡1𝜋(𝜎)𝑥𝑐cos𝑘𝑦�2       (28)  \n \nwhere  𝚥̃𝑘=𝐽𝑘|𝑚𝑛| is the effective magnetic field to itinerant electrons due to the \nC-AFM order. If we neglect the magnetic coupling, the band structure has \ncontinuous Dirac points along  the lines  𝑘𝑥=±arccos (−𝑡1𝜋𝑥𝑐𝑡1𝜎𝑥𝑐⁄ cos𝑘𝑦) and \n𝑘𝑦=±arccos (−𝑡1𝜋𝑥𝑐𝑡1𝜎𝑥𝑐⁄ cos𝑘𝑥). The magnetic field term 𝚥̃𝑘 acts as a mass term \nand opens a gap between the upper and lower branches of the Dirac bands. Note that a finite spin- orbit coupling may also opens a gap, but it is m uch smaller \ncompared to the one associated with the AFM order (see Fig.4 (c)-(d), Supplementary \nFigure 4). \n  \nSupplementary Note 9 : Estimat e of the RKKY interaction  \n \nIn the above spin-fermion  model, the coupling between itinerant electrons and \nlocal moments mediate s an RKKY interaction among the local moments:  \n𝐻𝑅𝐾𝐾𝑌 (𝐑𝑖−𝐑𝑗)=1\n2𝐽(𝐑𝑖−𝐑𝑗)𝐒𝑖∙𝐒𝑗,                    (29)  \nwith the RKKY coupling  given by  \n              𝐽 (𝐑)=2(𝐽𝐾\n𝑁)2∑Θ(𝐸𝐤)−Θ�𝐸𝐤′�\n𝐸𝐤−𝐸𝐤′𝐤,𝐤′ 𝑒𝑖(𝐤−𝐤′)∙𝐑,                 (30)  \nwher e 𝐽(𝐑)=𝐽�𝐑𝑖−𝐑𝑗�,Θ(𝐸𝐤) is the Fermi distribution function. A  simple \nestimate based on the perturbation theory  gives  the induced RKKY coupling at the \norder of ( JK)2/EF, where E F is the Fermi energy of the relevant conduction band. \nTaking the value of J K~10-20 meV , which is estimated from the size of the gap \nbetween the two Dirac bands , and EF~0.1 eV  from the fitting to the DFT results , we \nget that the RKKY -induced interlayer coupling | J(𝑧̂)|~1 -4 meV. These Jc values are \ncompatible with those obtained from our Raman measurements  (see Table I in the \nmain text) . This agreement with the experimental results suggests  the model used in \nour analysis  is valid in  understanding the fund amental mechanisms underlying the \nnovel phenomena observed in our measured Raman spectra. A more accurate \nquantitative description of the size and sign of the RKKY interaction  would require \nmore sophisticated band -structure calculations, which are beyond the scope of the present work.  \n \n \nSupplementary References  \n1. Fleury, P. A. & Loudon, R. Scattering of light by one- and two-magnon excitations. Phys. Rev.  \n166, 514-530 (1968).  \n2. Luo, C., Datta , T. & Yao, D. X. Spectrum splitting of bimagnon  excitations in a spatially frustrated \nHeisenberg antiferromagnet revealed by resonant inelastic x -ray scattering. Ph ys. Rev. B 89, \n165103 (2014) . \n3. Lee, G. , Farhan, M. A., Kim, J. S. & Shim, J. H. Anisotropic  Dirac electronic structures of \nAMnBi 2 (A=Sr, Ca). Phys.  Rev. B  87, 245104 (2013).  \n4. Calder , S. et al.  Magnetic structure and spin excitations in BaMn 2Bi2. Phys. Rev.  B 89, 064417  \n(2014).  \n \n      "    },    {        "title": "1703.08431v1.Phase_Diagram_of__α__RuCl__3__in_an_in_plane_Magnetic_Field.pdf",        "content": "Phase Diagram of \u000b-RuCl 3in an in-plane Magnetic Field\nJ. A. Sears,1Y. Zhao,2, 3Z. Xu,2, 3J. W. Lynn,2and Young-June Kim1,\u0003\n1Department of Physics, University of Toronto, 60 St. George St., Toronto, Ontario, M5S 1A7, Canada\n2NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland, 20899, USA\n3Department of Materials Science and Engineering,\nUniversity of Maryland, College Park, Maryland 20742, USA\n(Dated: March 27, 2017)\nThe low-temperature magnetic phases in the layered honeycomb lattice material \u000b-RuCl 3have\nbeen studied as a function of in-plane magnetic \feld. In zero \feld this material orders magnetically\nbelow 7 K with so-called zigzag order within the honeycomb planes. Neutron di\u000braction data show\nthat a relatively small applied \feld of 2 T is su\u000ecient to suppress the population of the magnetic\ndomain in which the zigzag chains run along the \feld direction. We found that the intensity of\nthe magnetic peaks due to zigzag order is continuously suppressed with increasing \feld until their\ndisappearance at \u0016oHc=8 T. At still higher \felds (above 8 T) the zigzag order is destroyed, while\nbulk magnetization and heat capacity measurements suggest that the material enters a state with\ngapped magnetic excitations. We discuss the magnetic phase diagram obtained in our study in the\ncontext of a quantum phase transition.\nThe transition metal halide \u000b-RuCl 3has a crystal\nstructure made up of stacked honeycomb layers of edge-\nsharing RuCl 6octahedra. Plumb et al. [1] found that\nspin orbit coupling in this material is substantial, lead-\ning to a j eff=1\n2state description of the Ru3+valence\nelectrons. Since this material is built up with edge-\nsharing RuCl 6octahedra, its spin Hamiltonian is believed\nto include a signi\fcant bond-dependent Kitaev interac-\ntion [2, 3], making \u000b-RuCl 3a material of great interest in\nthe ongoing search for a Kitaev spin liquid ground state\n[4{23]. Although \u000b-RuCl 3orders magnetically at low\ntemperature with zigzag magnetic order [24{27], this ma-\nterial has shown some signatures of spin-liquid physics,\nsuch as a broad continuum of magnetic excitations iden-\nti\fed in both Raman scattering [28] and inelastic neutron\nscattering measurements [25, 29].\nWhen a magnetic \feld is applied within the honey-\ncomb plane, previous bulk measurements have reported\nthat\u000b-RuCl 3undergoes a number of transitions [26, 30{\n32], including low \feld transitions resembling spin-\rop\ntransitions occurring at 1 T and 6 T, followed by the\napparent loss of zigzag magnetic order at 8 T. In con-\ntrast, when a magnetic \feld is applied perpendicular to\nthe honeycomb planes the zigzag magnetic order appears\nto be robust up to \felds of 14 T [31]. The high \feld\nphase above the loss of zigzag magnetic order has been\nthe subject of particular interest recently [26, 30{33]. It\nhas been proposed that this phase may be a simple po-\nlarized paramagnetic state [26], however this does not\naccount for the lack of saturation in the magnetization\n[30]. The high \feld phase has also been characterized by\nNMR measurements [32] which show that the magnetic\nexcitations develop an energy gap. The gap size was sim-\nilar for the two \feld directions measured, a result di\u000ecult\nto reconcile with the physics of a polarized paramagnetic\n\u0003Electronic address: yjkim@physics.utoronto.castate. This \fnding of gapped excitations in the high \feld\nphase is in contrast to recent thermal conductivity mea-\nsurements [33], which suggested the presence of gapless\nexcitations in the high \feld phase.\nIn this paper, we have characterized these \fnite \feld\ntransitions using magnetic neutron di\u000braction, and bulk\nheat capacity and magnetization measurements on the\nsame samples. Neutron di\u000braction measurements show\nthat at low \feld (2 T) the di\u000bracted intensity due to\none of the zigzag domains disappears, suggesting that re-\ndistribution in domain population occurs in this rather\nlow \feld range. We found that the zigzag magnetic or-\nder temperature Tcis continuously suppressed with ap-\nplied \feld, and eventually disappears above the critical\nin-plane \feld of \u0016oHc= 8 T.[39] The high \feld phase\nabove the critical \feld is characterized by a magnetic ex-\ncitation gap, \u0001, which can be extracted from the speci\fc\nheat data. The energy scales both below ( Tc) and above\n(\u0001) the critical \feld exhibit power law scaling as a func-\ntion of in-plane magnetic \feld, indicating the presence\nof a quantum critical point. We note that the quan-\ntum phase transition due to transverse-\feld in the Ising\nmodel provides a reasonable phenomenological descrip-\ntion of the observed phase diagram.\nSingle crystals of \u000b-RuCl 3were grown from commer-\ncial RuCl 3powder (Sigma-Aldrich, Ru content 45-55%)\nby vacuum sublimation in sealed quartz tubes. This re-\nsulted in \rat, plate-like crystals with typical dimensions\n1-2 mm2and mass 1-5 mg. The crystallographic c direc-\ntion (hexagonal notation) was found to be perpendicular\nto the large surface of the crystal. Throughout this paper\nwe will use the hexagonal crystallographic notation with\na= 5:96\u0017A andc= 17:2\u0017A, in which the a-b plane coin-\ncides with the honeycomb layers. The crystals have well\nde\fned facets at 120\u000eangles, and it was found that the\nfacets coincide with the hexagonal (1,1,0) type directions.\nNeutron di\u000braction measurements were carried out us-\ning the BT-7 triple axis spectrometer at the NIST Center\nfor Neutron Research (NCNR) [35]. The neutron di\u000brac-arXiv:1703.08431v1  [cond-mat.str-el]  24 Mar 20172\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5\nµo˜H (T)0.00.51.01.52.02.5Intensity (normalized)\n(0.5,0,1) (0,0.5,2)\n(0,0.5,2)\n(0.5,0,1)\n0.48 0.50 0.52\nH (rlu)\n0.48 0.50 0.52\nK (rlu)\n0 T\n1 T\n1.6 T\n4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5\nµo˜H (T)010002000300040005000600070008000(0,0.5,2) Intensity (arb. units)\nT=2K\nT=5K\nT=6K\n(a) (b)\nFIG. 1: (a) Low \feld magnetic peak intensity at 2 K as a function of the in-plane component of magnetic \feld ( \u0016o~H). The\nintensity is normalized to the value at zero \feld. The inset shows individual scans of (0.5,0,1) and (0,0.5,2) Bragg peaks at 0,\n1, and 1.6 T (in-plane \feld) and 2 K. (b) High \feld intensity of the (0,0.5,2) peak at 2 K, 5 K, and 6 K. Solid lines are \fts\nwith\u0018(H\u0000Hc)2\f\u0003to extract the critical \feld. Same critical exponent \f\u0003= 0:28 was used for all three curves. Error bars\nwhere indicated represent one standard deviation.\na*b*\nH\nDomain 1 Domain 2 Domain 3\nFIG. 2: Magnetic structures and Bragg peak positions in the\n\frst Brillouin zone for each of the three possible zigzag mag-\nnetic domains. In a vertical magnetic \feld the intensities due\nto Domain 1 disappeared and intensities due to Domain 2\nincreased. Note that the moments are shown pointing along\nthe zigzag direction for illustrative purposes only. Drawings\nof magnetic structure were done in VESTA 3 [34].\ntion data were collected using a crystal array of 60 crys-\ntals, with a mass of 100 mg. The incident neutron energy\nwas 14.7 meV, and measurements were conducted in the\n(H0L) plane as well as the plane containing the (0, 0.5,\n2) and (-0.5, 0.5, 2) magnetic Bragg peaks. In both cases\nmagnetic \felds up to 15 T were applied perpendicular to\nthe scattering plane using either a 10 T or a 15 T verti-\ncal \feld superconducting magnet. In order to gain access\nto the (0,0.5,2) magnetic peak it was necessary to rotatethe sample such that the angle between the magnetic\n\feld and and the honeycomb plane was approximately\n35\u000e. In this case, we quote the in-plane component of the\n\feld, ~H, rather than the total \feld applied. For all the\nother measurements, magnetic \feld was applied within\nthe honeycomb plane.\nMagnetization and heat capacity was measured as a\nfunction of temperature using a Physical Property Mea-\nsurement System (PPMS) with \felds up to 14 T. The\nmagnetization measurements were conducted on a collec-\ntion of six crystals mounted with the \feld applied along\nthe in-plane (-1,2,0) direction. The heat capacity mea-\nsurements were done with a single crystal mounted ver-\ntically on an aluminum oxide mount in the same orien-\ntation as that used for the magnetization measurements.\nThe phonon contribution to heat capacity was subtracted\nusing the non-magnetic isostructural \u000b-IrCl 3[36].\nWe have investigated the magnetic transitions directly\nby measuring the magnetic Bragg peak intensity as a\nfunction of \feld. When the magnetic \feld was applied\nperpendicular to the H0L plane, all the magnetic peaks\nin this plane { ( \u00060:5;0;l) withl= 1;2;4 { decreased\nin intensity and disappeared at the relatively low mag-\nnetic \feld of 2 T. The sample was then rotated to gain\naccess to the (0,0.5,2) magnetic peak, which was found\nto increase in intensity over this \feld range, as shown\nin Fig. 1(a). The critical \feld for this transition is in\nrough correspondence with the low \feld transition ob-\nserved in bulk measurements and previously interpreted\nas a spin-\rop type transition, which traditionally refers\nto a re-orientation of spins perpendicular to the applied3\n(a) (b)\n(c) (d)\nFIG. 3: Magnetic heat capacity as a function of temperature\nand in-plane magnetic \feld (a) below the 8 T transition and\n(c) above 8 T. The phonon contribution was removed by sub-\ntracting the heat capacity of isostructural \u000b-IrCl 3. The solid\nlines in (c) are \fts to an exponential expression for a gapped\nsystem (Ae\u0000\u0001=T). (b) Magnetization divided by magnetic\n\feld as a function of temperature for magnetic \felds rang-\ning from 5.2 to 7.4 T in steps of 0.2 T. (d) Magnetization as\na function of temperature for magnetic \felds above the 8 T\ntransition.\n\feld in an antiferromagnet [30]. In our experiment, the\ndirection perpendicular to the applied \feld corresponds\nto the hexagonal (1,0,0) direction, which is not one of\nthe easy-axes. We also note that a spin-\rop transition\nstill preserves the magnetic ordering wave vector, even\nthough magnetic Bragg peak intensities will be modi-\n\fed. Therefore, spin-\rop transition is not compatible\nwith our observation of the disappearance of allmag-\nnetic Bragg peaks in the H0L plane. This unexpected\n\fnding can be explained as a result of a change in mag-\nnetic domain population. Zigzag magnetic order can be\ndescribed as ferromagnetic zigzag chains, running along\nthe so-called zigzag direction of a honeycomb lattice, cou-\npled antiferromagnetically. Due to the 3-fold symmetry\nof the lattice, zigzag magnetic order may occur in one\nof three possible directions, resulting in three magnetic\ndomains that contribute to di\u000braction intensity in di\u000ber-\nent regions of reciprocal space as shown in Fig. 2. The\ndisappearance of the peaks in the H0L plane is well ex-plained by the disappearance of domain 1 as shown in\nFig. 2. The increase in intensity for the (0,0.5,2) magnetic\npeak belonging to domain 2 is expected for a redistribu-\ntion of domain population from domain 1 into domains\n2 and 3. We con\frmed that the domain 3 population in-\ncreases with \feld as well (not shown). We note that this\n\\domain-reorientation\" occurs gradually with \feld, and\nreaches equilibrium above about 2 T. The observed grad-\nual\feld-dependence is also consistent with this change\ncoming from domain population change as spin-\rop tran-\nsitions tend to be \frst order when the \feld is parallel to\nthe spin direction. Above this \\domain-reorientation\"\ntransition, the magnetic Bragg peak shows little change\nin intensity up to 6 T. Above 6 T the intensity begins to\ndecrease and disappears entirely above \u0016oHc\u00198 T, di-\nrectly con\frming that zigzag magnetic order disappears\nabove a critical in-plane \feld of approximately 8 T. This\ntransition is continuous as a function of magnetic \feld.\nThe zigzag order parameter,p\nI, whereIis the inten-\nsity of the (0,0.5,2) peak, exhibits power law behaviorp\nI\u0018(H\u0000Hc)\f\u0003with\f\u0003= 0:28\u00060:05. This power\nlaw behavior seems to hold for higher temperature data\nas well, although the critical \feld H cshifts to lower \feld\nwith increasing temperature.\nHeat capacity and magnetization data collected at zero\nmagnetic \feld both show signatures of the zigzag mag-\nnetic ordering at low temperature. The heat capacity at\nzero magnetic \feld shows a sharp feature at 6.5 K and\na second, smaller feature around 9 K [Fig. 3(a)]. The\nmagnetic Bragg peaks observed by neutron di\u000braction in\nour samples show an ordering temperature of about 7-\n8 K [24], so we attribute the lower temperature feature\nto this zigzag ordering. The nature of the 9 K feature\nseen in our samples is not known, but Cao et al. re-\nported that stacking disorder in \u000b-RuCl 3can increase\nthe ordering temperature to approximately 14 K [25, 27]\nand it is plausible to suppose that the 9 K transition ob-\nserved in our sample arises from a grain with a di\u000berent\nstacking order.\nAs \feld is increased, the sharp feature in the heat\ncapacity decreases in size before shifting to lower tem-\nperature and becoming di\u000ecult to resolve as shown in\nFig. 3(a). Magnetization data at low \feld show a sharp\ndrop upon decreasing temperature below 7 K as the crys-\ntal enters the ordered phase. Figure 3(b) shows that\nthis drop becomes smaller in size and eventually disap-\npears at high \feld once the zigzag magnetic ordering has\ndisappeared. In the high \feld phase the heat capacity\nno longer shows any sharp feature, but instead shows a\nbroad feature that increases in temperature with increas-\ning magnetic \feld [Fig. 3(c)]. The low temperature heat\ncapacity data were \ft using an expression for activated\nbehavior (Ae\u0000\u0001=T) to extract the magnetic excitation\ngap \u0001. The magnetization in the high \feld phase shown\nin Fig. 3(d) increases gradually with decreasing temper-\nature, reaching \feld-dependent saturation values at low\ntemperature.\nThe experimental results are summarized in Fig. 4,4\nFIG. 4: In-plane \feld { temperature phase diagram. ZZ3:\nzigzag magnetic order with three equal domain populations;\nZZ2: zigzag magnetic order with redistributed (two) domain\npopulation; QPM: quantum disordered phase with gapped\nmagnetic excitations; PM: paramagnetic phase. The phase\nboundary between ZZ2 and PM is the transition temperature\nTcobtained from heat capacity and neutron measurements.\nThe thick solid line is from the transverse \feld Ising model,\nand the thin solid line is \ft with a power law as described in\nthe text. The value of \u0001 found from the heat capacity data\nis also shown (right-hand axis) and the dashed line is a linear\n\ft to the gap size \u0001.\nwhich combines neutron and bulk measurements to de-\ntermine the phase diagram. The low \feld transition was\nfound to be a change in magnetic domain population,\nseparating phases made up of 3 and 2 magnetic domains\n(phases ZZ3 and ZZ2 respectively). The loss of magnetic\norder above the high \feld transition was also con\frmed,\nalthough the nature of the high \feld phase remains to be\nclari\fed. The magnetic excitation gap in the high \feld\nphase was characterized by \ftting low temperature heat\ncapacity data. The gap size scales with magnetic \feld,\ngoing to zero at \fnite \feld rather than at zero \feld as\nwould be expected for a simple polarized paramagnetic\nstate. This \fnding is consistent with the NMR measure-\nments reported previously [32], but contrasts with the\nresults of thermal conductivity measurements which sug-\ngested the presence of gapless excitations [33].\nThe observation of vanishing energy scales towards a\ncritical \feld in both high and low \feld regimes is strongly\nsuggestive of quantum critical behavior. Although de-\ntailed analysis of the spin Hamiltonian of \u000b-RuCl 3is be-\nyond the scope of this paper, the phase diagram could be\nunderstood heuristically by comparing our results with\none of the simplest models that goes through a quan-tum phase transition: the transverse-\feld Ising model\n(TFIM). There is also physical motivation for our choice\nof transverse \feld Ising model. \u000b-RuCl 3does show a\nlarge uniaxial anisotropy and the magnetic \feld in our\nexperimental setup has a large component transverse to\nthe easy axis. This is a result of the domain-reorientation\ntransition which favors domains in which the zigzag chain\ndirections are perpendicular to the \feld direction. The\nmoment direction has been found to point along the\nzigzag direction (neglecting a small out-of-plane compo-\nnent) [27], resulting in a phase with magnetic \feld nearly\nperpendicular to the moment directions.\nIn Fig. 4, we compare the phase boundary with the\nTFIM mean \feld result and \fnd that they are in good\nagreement in the region close to Hc. We could also \ft\nTc(H) using a power law with Tc(H)\u0018(Hc\u0000H)0:18as\nshown in the \fgure. Above the critical \feld, the gap fol-\nlows a power law scaling \u0001 \u0018(H\u0000H\u0003\nc)z\u0017withz\u0017\u00191.\nNote that the critical \feld value extrapolated from this\nscalingH\u0003\nc\u00196:5 T is slightly di\u000berent from the criti-\ncal \feld\u0016oHc\u00198 T. This discrepancy may be due to\nthe complex nature of the Hamiltonian of the real mate-\nrial, or indicates the necessity of another parameter that\nneeds to be tuned to reach the quantum critical point\nthat exists away from the T\u0000Hplane. We note that\nthe critical exponent relation z\u0017= 1 is consistent with\nthed= 2 Ising model [37]. In addition, in Fig. 1(b), the\nmagnetic order parameter could be \ftted well using the\ncritical exponent \f\u0003= 0:28, which is close to the theo-\nretical value of 0.32 [38]. Finally, the low-temperature\nsaturation behavior observed in Fig. 3(d) is naturally ex-\nplained by the temperature dependence of the transverse\nmagnetization in the TFIM.\nIn conclusion, we have determined the high \feld phase\ndiagram for \u000b-RuCl 3using neutron di\u000braction, magneti-\nzation, and heat capacity measurements. We have con-\n\frmed the loss of zigzag order in the high \feld phase and\nfound that the material enters into a phase with gapped\nmagnetic excitations. 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Pfeuty, Journal of Physics C: Solid State Physics 9,\n3993 (1976), URL http://stacks.iop.org/0022-3719/\n9/i=21/a=018 .\n[39] We con\fne our discussion only to magnetic \feld ap-\nplied within the honeycomb plane. Justi\fcation for this\nis purely empirical. Large anisotropy in magnetic suscep-\ntibility measurements and high-\feld magnetization indi-\ncates that easy axis is within the honeycomb plane. How-\never, a recent neutron di\u000braction study reported that the6\nordered moment has components perpendicular to the\nplane. We are only concerned with magnetic behaviorprojected to the honeycomb plane in this paper."    },    {        "title": "1705.10349v3.Magnetic_hexadecapole_order_and_magnetopiezoelectric_metal_in_Ba___1_x__K__x_Mn__2_As__2_.pdf",        "content": "Magnetic hexadecapole order and magnetopiezoelectric metal state in Ba 1\u0000xKxMn 2As2\nHikaru Watanabe\u0003and Youichi Yanase\nDepartment of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan\nWe study an odd-parity magnetic multipole order in Ba 1\u0000xKxMn2As2and related materials. Al-\nthough BaMn 2As2is a seemingly conventional Mott insulator with G-type antiferromagnetic order,\nwe identify the ground state as a magnetic hexadecapole ordered state accompanied by simultaneous\ntime-reversal and space-inversion symmetry breaking. A symmetry argument and microscopic calcu-\nlations reveal the ferroic ordering of leading magnetic hexadecapole moment and admixed magnetic\nquadrupole moment. Furthermore, we clarify electromagnetic responses characterizing the magnetic\nhexadecapole state of semiconducting BaMn 2As2and doped metallic systems. A magnetoelectric\ne\u000bect and antiferromagnetic Edelstein e\u000bect are shown. Interestingly, a counter-intuitive current-\ninduced nematic order occurs in the metallic state. The electric current along the z-axis induces\nthexy-plane nematicity in sharp contrast to the spontaneous nematic order in superconducting\nFe-based 122-compounds. Thus, the magnetic hexadecapole state of doped BaMn 2As2is regarded\nas a magnetopiezoelectric metal . Other candidate materials for magnetic hexadecapole order are\nproposed.\nI. INTRODUCTION\nMultipole moment, a concept established in the clas-\nsical electromagnetism, characterizes the anisotropy of\nelectric and magnetic charge distribution. Emergent mul-\ntipole order in condensed matter physics has attracted\nfundamental interests for more than three decades [1].\nFerroic and antiferroic order of multipole moment has\nbeen observed in many d- andf-electron systems. Al-\nthough previous studies have focused on the even-parity\nmultipole order [1], recent studies point to the odd-parity\nmultipole order which may be realized in locally non-\ncentrosymmetric systems [2{8]. Experimentally, several\nmaterials have been identi\fed [9], which can be traced\nback to Cr 2O3[10].\nLocally noncentrosymmetric systems preserve global\ninversion symmetry in the crystal structure although the\nlocal site symmetry lacks inversion symmetry. Then, the\nantisymmetric spin-orbit coupling (ASOC) entangles var-\nious degrees of freedom such as spin, orbital, and sublat-\ntice [11, 12]. The peculiar electronic structure may cause\nintriguing phenomena characterizing odd-parity multi-\npole order, such as magnetoelectric (ME) e\u000bect [3, 4].\nAlthough previous theoretical works of odd-parity mag-\nnetic multipole order are based on toy models [3{6], in\nthis paper we show the complete classi\fcation of mag-\nnetic multipole order in tetragonal systems and identify\nthe magnetic hexadecapole order in BaMn 2As2. Char-\nacteristic electromagnetic responses in the magnetic hex-\nadecapole state are clari\fed.\nBaMn 2As2is an isostructural compound of BaFe 2As2,\na parent compound of Fe-based high-temperature su-\nperconductors (the space group is No.139, I4=mmm ).\nHowever, physical properties of BaMn 2As2and doped\nBaMn 2As2are quite di\u000berent from the Fe-based com-\npounds; BaMn 2As2undergoes the G-type antiferromag-\n\u0003watanabe.hikaru.43n@st.kyoto-u.ac.jp\nFIG. 1. (Color online) Contrast of magnetic structure of\nBaMn 2As2and BaFe 2As2.\nnetic (AFM) transition below high N\u0013 eel temperature\nTN= 625 K and shows semiconducting behaviors [13{\n15]. On the other hand, BaFe 2As2is a metallic com-\npound with a stripe magnetic structure [15{17] shown in\nFigure 1. Neither superconductivity nor structural tran-\nsition, which have been observed in Fe-based 122 com-\npounds [15{17], occurs in doped BaMn 2As2.\nThe ground state of BaMn 2As2seems to be a conven-\ntional Mott insulator with AFM order [18, 19], analogous\nto cuprate high-temperature superconductors. However,\nwe notice unusual symmetry of the AFM state, namely,\nunbroken translation symmetry. This is, indeed, be-\ncause of a locally noncentrosymmetric crystal structure\nof BaMn 2As2. The two Mn sites are crystallographically\nnonequivalent even in the paramagnetic state. In the\nfolded Brillouin zone, the wave vector of magnetic or-\nder isq= 0, and therefore, a ferroic order parameter\nmay characterize the seemingly \"AFM order\". Because\nthe space-inversion (SI) symmetry is broken instead of\nthe translation symmetry, an odd-parity multipole mo-\nment may be a relevant order parameter specifying the\nground state of BaMn 2As2. Interestingly, BaMn 2As2can\nbe metalized by doping hole carriers (Ba 1\u0000xAxMn2As2,\nA= K, Rb) [20{24] or applying high pressure [25]. Then,arXiv:1705.10349v3  [cond-mat.mtrl-sci]  28 Aug 20172\nthe AFM order is robust in the hole-doped regime [20, 22{\n24, 26]. Hence, unconventional responses characteristic\nof itinerant odd-parity magnetic multipole state are ex-\npected, for which studies may open a new paradigm of\nmultipole physics.\nThe paper is organized as follows. In Sec. II, we classify\nthe magnetic multipole order by group theory and iden-\ntify the candidates of order parameter in BaMn 2As2. A\ncomplete classi\fcation based on irreducible representa-\ntions (IRs) of a given point group is carried out as done\nfor unconventional superconductors [27]. In Sec. III, we\nmicroscopically evaluate the magnetic multipole moment.\nThere remains an ambiguity of the de\fnition for odd-\nparity magnetic multipole moment in crystals, similar\nto electric polarization in a bulk system. To avoid this\ndi\u000eculty, we propose a unique de\fnition of odd-parity\nmagnetic multipole moment by di\u000berence from a refer-\nence state. In Sec. IV, we introduce an e\u000bective single-\nband model Hamiltonian for studies of electromagnetic\nresponses. In Sec. V, we demonstrate ME e\u000bect arising\nfrom the magnetic hexadecapole order and its enhance-\nment in the metallic state. The AFM Edelstein e\u000bect\nis also shown. In Sec. VI, we show a counter-intuitive\ncurrent-induced nematic order, the in-plane ( xy-plane)\nrotational symmetry breaking by out-of-plane electric\ncurrent (Jk^z). This response is a manifestation of odd-\nparity magnetic order in the metallic system. In Sec. VII,\na brief summary is given, and we propose other mag-\nnetic hexadecapole compounds showing magnetic struc-\nture similar to BaMn 2As2.\nII. GROUP-THEORETICAL CLASSIFICATION\nIn general, a phase transition leads to symmetry reduc-\ntion, such as the time-reversal (TR) symmetry breaking\nby ferromagnetic order. The crystal symmetry of the sys-\ntem is represented by point group, and thus phase transi-\ntions can be characterized by the reduction of the point\ngroup. In the framework of the group theory, physical\nquantities are classi\fed into IRs of a given point group,\nand symmetry constraints for emergent responses are ob-\ntained. The order parameter of the phase transition has\nto belong to the totally-symmetric IR of the point group\nin the ordered state, but not in the normal state. This\nscheme is supported by Landau's symmetry argument of\nsecond-order phase transitions [28].\nIn BaMn 2As2, the crystallographic point group D4h\ndescends to the sub-group D2dby the AFM transition.\nIRs in the normal state are reduced to those in the or-\ndered state as shown in Table I. The IRs of D2ddo\nnot have subscripts g=u, which indicate the SI symme-\ntry breaking by the AFM order. Hence, it is suggested\nthat the seemingly conventional G-type AFM order of\nBaMn 2As2is identi\fed as a parity-violating odd-parity\nmultipole order. According to Table I, only the B1uIR\nin the normal state is reduced to the totally-symmetric\nA1IR in the AFM state. Following the group-theoreticalframework, we conclude that a basis function belonging\nto theB1uIR is a relevant order parameter of BaMn 2As2.\nTABLE I. Reductions of IRs D4h!D2d. The two-fold rota-\ntional symmetry axes of D2dare thex/yaxes ofD4h.\nD4hA1gA2gB1gB2gEgA1uA2uB1uB2uEu\nD4h#D2dA1A2B1B2E B 1B2A1A2E\nNow, we classify magnetic multipole moments in the\ntetragonal system with the D4hsymmetry, and make a\nlist of possible magnetic multipole order. The magnetic\nmultipole moments are written as [29]\nMlm=\u0016BX\ni\u00122l(i)\nl+ 1+ 2s(i)\u0013\n\u0001ri \nrl\nir\n4\u0019\n2l+ 1Y\u0003\nlm(\u0012i;\u001ei)!\n;\n(1)\nwhere\u0016B,l,s, andYlmare respectively Bohr mag-\nneton, orbital angular momentum, spin, and spherical\nharmonics. The label irepresents electrons in the unit\ncell and (ri;\u0012i;\u001ei) are polar coordinates of the i-th elec-\ntron from a reference point. The phase factor satis\fes\nY\u0003\nlm= (\u00001)lYl\u0000m(Condon-Shotley phase). When we\ndiscuss multipole moments in a lattice system, it is con-\nvenient to use cubic harmonics Z\u0006\nlmde\fned by\nZ+\nlm=(\u00001)m\np\n2(Ylm+Y\u0003\nlm);\nZ\u0000\nlm=(\u00001)m\nip\n2(Ylm\u0000Y\u0003\nlm);(2)\nfor 0< land 0< m\u0014l. Whenm= 0, we denote\nZl0=Yl0. Accordingly, the multipole moment in the\nCartesian coordinates is denoted by\nM+\nlm=(\u00001)m\np\n2(Mlm+M\u0003\nlm);\nM\u0000\nlm=(\u00001)m\nip\n2(Mlm\u0000M\u0003\nlm):(3)\nIn our classi\fcation, we treat the spin and orbital an-\ngular momentum as a classical axial-vector, ( ^x;^y;^z)\u0011\n\u0016B(2l=(l+ 1) + 2s), since we take thermodynamical and\nquantum mechanical expectation values. Table II shows\nthe classi\fcation of multipole moments of low rank ( l\u0014\n4) in theD4hpoint group symmetry, revealing candidates\nof order parameter of the AFM state in BaMn 2As2. Up\nto rank-4, magnetic multipole moments belonging to the\nB1uIR of theD4hpoint group are\nM+\n22(Quadrupole) :p\n3 (x^x\u0000y^y); (4)\nM+\n42(Hexadecapole) :8\n><\n>:3p\n5z\u0000\nx2\u0000y2\u0001^z\n+p\n5\n2\u0000\n7z2\u0000r2\u0001\n(x^x\u0000y^y)\n\u0000p\n5\n2\u0000\nx2\u0000y2\u0001\n(x^x+y^y):\n(5)\nThese basis functions are certainly invariant under all\nsymmetry operations in the AFM state. Thus, the AFM3\norder may be identi\fed as magnetic quadrupole order\nor magnetic hexadecapole order. In Sec. III, we micro-\nscopically evaluate multipole moments and show that the\nmagnetic hexadecapole moment is the leading order pa-\nrameter.\nIn Table III, we show the complete classi\fcation of\nmagnetic multipole order parameter in the D4hpoint\ngroup. The TR odd basis functions in both real space\nand momentum space are listed. In the real space repre-\nsentation, the basis functions are nothing but the mag-\nnetic multipole moments. On the other hand, the mo-\nmentum space representation looks quite di\u000berent fromthe real space representation for the odd-parity magnetic\nmultipole order. This is because the parity under TR\noperation is opposite between randk. It should be no-\nticed that the odd-parity basis functions in the momen-\ntum space are \"spin-independent\". They indicate spin-\nindependent corrections to the energy spectrum, which\nare characteristic feature of odd-parity magnetic multi-\npole states. Although both TR and SI symmetries are\nbroken, the combined PTsymmetry is preserved. There-\nfore, the Kramers degeneracy at each momentum is en-\nsured, and the deformation of band structure has to be\nspin-independent.\nTABLE II. List of magnetic multipoles up to rank-4. First and second column show a rank and symbol of magnetic multipole,\nrespectively. Third column shows IR in the point group D4h. Fourth column shows a representation by local magnetic moment\n[Eq. (1)]. We also show toroidal dipole moment Tiand magnetic monopole moment r\u0001s, which are not represented by any\nlinear combination of magnetic multipole moment.\nlMlm IR basis function\nl= 1M10A2g ^z\nM+\n11Eg ^x\nM\u0000\n11Eg ^y\nl= 2M20A1u 2z^z\u0000x^x\u0000y^y\nM+\n21Eup\n3 (x^z+z^x)\nM\u0000\n21Eup\n3 (y^z+z^y)\nM+\n22B1up\n3 (x^x\u0000y^y)\nM\u0000\n22B2up\n3 (y^x+x^y)\nTxEu z^y\u0000y^z\nTyEu x^z\u0000z^x\nTzA2u y^x\u0000x^y\nMonopoleA1u x^x+y^y+z^z\nl= 3M30A2g3\n2\u0000\n3z2\u0000r2\u0001^z\u00003z(x^x+y^y)\nM+\n31Eg 2p\n6zx^z+p\n6\n4\u0000\n5z2\u0000r2\u0001^x\u0000p\n6\n2x2^x\u0000p\n6\n2xy^y\nM\u0000\n31Eg 2p\n6yz^z+p\n6\n4\u0000\n5z2\u0000r2\u0001^y\u0000p\n6\n2y2^y\u0000p\n6\n2xy^x\nM+\n32B2gp\n15\n2\u0000\nx2\u0000y2\u0001^z+p\n15z(x^x\u0000y^y)\nM\u0000\n32B1gp\n15xy^z+p\n15z(y^x+x^y)\nM+\n33Eg3p\n10\n4\u0000\nx2\u0000y2\u0001^x\u00003p\n10\n2xy^y\nM\u0000\n33Eg3p\n10\n4\u0000\nx2\u0000y2\u0001^y+3p\n10\n2xy^x\nl= 4M40A1u 2z\u0000\n5z2\u00003r2\u0001^z\u00003\n2\u0000\n5z2\u0000r2\u0001\n(x^x+y^y)\nM+\n41Eu3p\n10\n4x\u0000\n5z2\u0000r2\u0001^z+p\n10\n4z\u0000\n7z2\u00003r2\u0001^x\u00003p\n10\n2zx(x^x+y^y)\nM\u0000\n41Eu3p\n10\n4y\u0000\n5z2\u0000r2\u0001^z+p\n10\n4z\u0000\n7z2\u00003r2\u0001^y\u00003p\n10\n2yz(x^x+y^y)\nM+\n42B1u3p\n5z\u0000\nx2\u0000y2\u0001^z+p\n5\n2\u0000\n7z2\u0000r2\u0001\n(x^x\u0000y^y)\u0000p\n5\n2\u0000\nx2\u0000y2\u0001\n(x^x+y^y)\nM\u0000\n42B2u 6p\n5xyz^z+p\n5\n2\u0000\n7z2\u0000r2\u0001\n(y^x+x^y)\u0000p\n5xy(x^x+y^y)\nM+\n43Eup\n70\n2x\u0000\nx2\u0000y2\u0001^z\u0000p\n70\n4x\u0000\nx2+y2\u0001^z+3p\n70\n4z\u0000\nx2\u0000y2\u0001^x\u00003p\n70\n2xyz^y\nM\u0000\n43Eup\n70\n2y\u0000\nx2\u0000y2\u0001^z+p\n70\n4y\u0000\nx2+y2\u0001^z+3p\n70\n4z\u0000\nx2\u0000y2\u0001^y+3p\n70\n2xyz^x\nM+\n44A1up\n35\n4\u0000\nx2+y2\u0001\n(x^x+y^y) +p\n35\n4\u0000\nx2\u0000y2\u0001\n(x^x\u0000y^y)\u00003p\n35\n2xy(y^x+x^y)\nM\u0000\n44A2up\n35\n2\u0000\nx2\u0000y2\u0001\n(y^x+x^y) +p\n35xy(x^x\u0000y^y)\nIn the same manner, we can classify the electric mul- tipole order. The electric multipole moment is given by\nQlm=eX\nirl\nir\n4\u0019\n2l+ 1Y\u0003\nlm(\u0012i;\u001ei): (6)4\nTABLE III. The TR odd basis functions of IRs in D4h. Basis are represented both in real space and in momentum space. The\ntotally-symmetric IR ( A1g) is not shown.\nIR Basis in real space Basis in momentum space\nA2gM10 ^z ^z\nM30 z(x^x+y^y);z2^z kz(kx^x+ky^y);k2\nz^z\nB1gM\u0000\n32 xy^z;z(y^x+x^y) kxky^z;kz(ky^x+ky^y)\nB2gM+\n32\u0000\nx2\u0000y2\u0001^z;z(x^x\u0000y^y)\u0000\nk2\nx\u0000k2\ny\u0001^z;kz(kx^x\u0000ky^y)\nEgM\u0006\n11 [^x;^y] [^x;^y]\nM\u0006\n31 [zx^z;yz^z];\u0002\nz2^x;z2^y\u0003\n[kzkx^z;kykz^z];\u0002\nk2\nz^x;k2\nz^y\u0003\n\u0002\nx2^x;y2^y\u0003\n;[xy^x;xy^y]\u0002\nk2\nx^x;k2\ny^y\u0003\n;[kxky^x;kxky^y]\nM\u0006\n33 [xy^x;xy^y];\u0002\u0000\nx2\u0000y2\u0001^x;\u0000\nx2\u0000y2\u0001^y\u0003\n[kxky^x;kxky^y];\u0002\u0000\nk2\nx\u0000k2\ny\u0001^x;\u0000\nk2\nx\u0000k2\ny\u0001^y\u0003\nA1uM20 2z^z\u0000x^x\u0000y^y\nkxkykz\u0000\nk2\nx\u0000k2\ny\u0001Monopole x^x+y^y+z^z\nM40 z3^z;z2(x^x+y^y)\nM+\n44\u0000\nx2+y2\u0001\n(x^x+y^y);\u0000\nx2\u0000y2\u0001\n(x^x\u0000y^y)\nxy(y^x+x^y)\nA2uTz y^x\u0000x^ykzM\u0000\n44\u0000\nx2\u0000y2\u0001\n(y^x+x^y);xy(x^x\u0000y^y)\nB1uM+\n22 x^x\u0000y^y\nkxkykz M+\n42 z\u0000\nx2\u0000y2\u0001^z\nz2(x^x\u0000y^y);\u0000\nx2\u0000y2\u0001\n(x^x+y^y)\nB2uM\u0000\n22 y^x+x^y\nkz\u0000\nk2\nx\u0000k2\ny\u0001\nM\u0000\n42 xyz^z\nz2(y^x+x^y);xy(x^x+y^y)\nEuM\u0006\n21 [x^z+z^x;y^z+z^y]\n[kx;ky]Tx;Ty [z^y\u0000y^z;x^z\u0000z^x]\nM\u0006\n41\u0002\nz2x^z;z2y^z\u0003\n;\u0002\nz3^x;z3^y\u0003\n;[zx(x^x+y^y);yz(x^x+y^y)]\nM\u0006\n43\u0002\nx\u0000\nx2\u0000y2\u0001^z;y\u0000\nx2\u0000y2\u0001^z\u0003\n;\u0002\nx\u0000\nx2+y2\u0001^z;y\u0000\nx2+y2\u0001^z\u0003\n\u0002\nz\u0000\nx2\u0000y2\u0001^x;z\u0000\nx2\u0000y2\u0001^y\u0003\n;[xyz^x;xyz ^y]\nWe introduce expressions in the Cartesian coordinates as\nQ+\nlm=(\u00001)m\np\n2(Qlm+Q\u0003\nlm);\nQ\u0000\nlm=(\u00001)m\nip\n2(Qlm\u0000Q\u0003\nlm);(7)\nfor 0< land 0< m\u0014l. Table IV shows the classi-\n\fcation of electric multipole order in tetragonal system\nbased on the point group symmetry D4h. For example,\nbasis functions of B1uandB2uIRs in real space repre-\nsent the electric octapole order, which has been studied\nin Sr 3Ru2O7and a bilayer Rashba system [2, 7]. On\nthe other hand, A2uandEuIRs correspond to the ferro-\nelectric order, and A1uIR shows electric dotriacontapole\norder. Electric multipole moment is invariant under the\nTR operation, and therefore, the odd-parity electric mul-\ntipole order parameter in k-space has \\spin-dependent\"\nform consistent with Fermi liquid theory by Fu [30]. For\ninstance, the electric octapole order is regarded as spinnematic order in k-space[2]. This is in sharp contrast to\nthe spin-independent form of odd-parity magnetic multi-\npole order in k-space.\nIII. MAGNETIC HEXADECAPOLE ORDER\nPrevious studies of multipole order have mainly fo-\ncused on even-parity multipole formed by localized elec-\ntrons [1]. Then, the local multipole is represented by\ntotal angular momentum multiplets, which can be sys-\ntematically treated with the aid of Stevens' operator-\nequivalent method [29, 31, 32]. With this method, even-\nparity multipole moment operators are recast by angular\nmomentum operators with the use of the Wigner-Eckart\ntheorem. On the other hand, the expectation value of\nodd-parity multipole moment operators vanishes when\nthe local basis has the SI parity. Therefore, the operator-\nequivalent method cannot be used to evaluate odd-parity\nmultipole moments. Hence, we should adopt local basis5\nTABLE IV. The TR even basis functions of IRs in D4h. Basis are represented both in real space and momentum space. In the\nreal space representation, electric multipole moments up to rank-4 are shown. The rank-5 dotriacontapole Q\u0000\n54is also shown\nforA1uIR.\nIR Basis in real space Basis in momentum space\nA1gQ20 z2k2\nz\nQ40 z4k4\nz\nQ\u0006\n44x4\u00006x2y2+y4k4\nx\u00006k2\nxk2\ny+k4\ny\nxy(x2\u0000y2) kxky(k2\nx\u0000k2\ny)\nA2gQ\u0000\n44 xy(x2\u0000y2) kxky(k2\nx\u0000k2\ny)\nB1gQ+\n22 x2\u0000y2k2\nx\u0000k2\ny\nQ+\n42 (x2\u0000y2)(7z2\u0000r2) (k2\nx\u0000k2\ny)(7k2\nz\u0000k2)\nB2gQ\u0000\n22 xy kxky\nQ\u0000\n42 xy(7z2\u0000r2) kxky(7k2\nz\u0000k2)\nEgQ\u0006\n21 [zx;yz ] [kzkx;kykz]\nQ\u0006\n41\u0002\nzx(7z2\u00003r2);yz(7z2\u00003r2)\u0003\u0002\nkzkx(7k2\nz\u00003k2);kykz(7k2\nz\u00003k2)\u0003\nQ\u0006\n43\u0002\nzx(x2\u00003y2);yz(x2\u00003y2)\u0003 \u0002\nkzkx(k2\nx\u00003k2\ny);kykz(k2\nx\u00003k2\ny)\u0003\nA1u(Q\u0000\n54)xyz\u0000\nx2\u0000y2\u0001 kx^x+ky^y+kz^z\nkz^z\u0000kx^x;kz^z\u0000ky^y\nA2uQ10 zky^x\u0000kx^y\nQ30 z(5z2\u00003r2)\nB1uQ\u0000\n32 xyz kx^x\u0000ky^y\nB2uQ+\n32 z\u0000\nx2\u0000y2\u0001\nky^x+kx^y\nEuQ\u0006\n11 [x;y]\n[kx^z;ky^z];[kz^x;kz^y] Q\u0006\n31\u0002\nx(5z2\u0000r2);y(5z2\u0000r2)\u0003\nQ\u0006\n33\u0002\nx(x2\u00003y2);y(3x2\u0000y2)\u0003\nwith mixed SI parity, which are formed by hybridization\nof even- and odd-parity orbitals.\nIn BaMn 2As2, the magnetic moment is formed mainly\nby Mndorbitals, and hybridization with As porbitals\ngives rise to the anisotropic magnetic charge distribu-\ntion, namely, magnetic multipole moments. The d-phy-\nbridization leads to the SI parity mixing. Hence, the odd-\nparity magnetic multipole moments [Eq. (1)] are eval-\nuated by calculating local magnetic multipole moments\n(LMMMs) of Mn-As clusters.[33] Since the magnetic unit\ncell of BaMn 2As2is the same as the crystal unit cell,\nthe unit cell contains two nonequivalent Mn-As clusters\nshown in Figure 2. Thus, we here evaluate LMMMs on\nthe two Mn-As clusters.\nIn this section we consider magnetic multipole mo-\nment induced by spin angular momentum, for simplic-\nity. With the use of the linear combination of atomic or-\nbitals method (LCAO method), local basis is expressed\nby superposition of atomic orbitals on Mn and As atoms.\nThen, a hybridized d-porbital mainly consists of Mn d\norbital and contains As porbitals [34]. With such hybrid\nlocal basis, we evaluate odd-parity magnetic multipole\nmoments.\nFIG. 2. (Color online) Two nonequivalent Mn-As clusters in\nBaMn 2As2. In the AFM state, the magnetization is opposite\nbetween two clusters. Since one cluster is transformed to the\nother under the PToperation, the global PTsymmetry is\npreserved. In the left \fgure, As atoms surrounding Mn atoms\nare labeled by As(1)-As(4) for discussions.\nA. Undoped BaMn 2As2\nHere, we calculate LMMMs of the Mn-As cluster in un-\ndoped BaMn 2As2. The formal valence of the Mn atom\nis +2 with \fve 3 delectrons and the spin con\fguration\nis the completely high-spin state [35]. Thus, the or-\nbital angular momentum quenches in the Mn atom. The\nleading odd-parity magnetic multipole moment comes\nfrom the observed z-component of spin magnetic mo-6\nment [13, 14]. Therefore, among the candidates [Eqs. (4)\nand (5)] the magnetic hexadecapole moment M+\n42;z\u0011\n3p\n5z\u0000\nx2\u0000y2\u0001^zis naturally the multipole order param-\neter of BaMn 2As2.\nThe expectation value of M+\n42;zis given by contribu-\ntions of \fve electrons in the Mn-As cluster,\n\nM+\n42;z\u000b\nL=\u0016BD\n6p\n5z\u0000\nx2\u0000y2\u0001\nszE\nL\n=\u0016B5X\nj=1h j\ndpj6p\n5z\u0000\nx2\u0000y2\u0001\nj j\ndpihszij;\n(8)\nwhere the subscript hiLindicates average on the local\nbasis of the Mn-As cluster and we used ^z= 2\u0016Bsz. Or-\nbital wave functions j j\ndpirepresent \fve Mn 3 dorbitals\nhybridized with As 4 porbitals, andhszijdenotes spin\npolarization of j-th orbital ( j= 1\u00005).\nThe two nonequivalent Mn-As clusters have the same\nhexadecapole moment\nM+\n42;z\u000b\nL, since both the octapole\nelectric charge distribution h j\ndpj6p\n5z\u0000\nx2\u0000y2\u0001\nj j\ndpi\nand the magnetic moment hszijare opposite between\nthe clusters. Thus, the hexadecapole moment is a ferroic\norder parameter as we expected. This is furthermore\nensured by the symmetry; the odd-parity magnetic mul-\ntipole moment operators have the even parity for the PT\nsymmetry and hybrid d-porbitals of two nonequivalent\nMn-As clusters are transformed to each other under the\nPToperation. Therefore, the expectation value of the\nhexadecapole moment M+\n42;zis equivalent between the\ntwo clusters.\nNow we evaluate the magnetic hexadecapole moment\nby focusing on a Mn-As cluster withP5\nj=1hszij>0\nwithout loss of generality. Considering quenched orbital\nangular momentum, we approximate the hybrid d-por-\nbital by the hybrid s-sorbitalj ssifor a rough estima-\ntion. In the s-sorbital, the sorbitals of four As atoms,\njs;As(i)i, are perturbatively hybridized with the sor-\nbital of Mn atom, js;Mni. The As atoms are labeled by\nthe indexi(see Figure 2). Denoting the hopping energy\nbetween Mn and As atoms as \u0000tand the level splitting\nas \u0001dp<0, and assuming jt=\u0001dpj\u001c 1, we obtain the\nwave function of the hybrid orbital,\nj ssi=js;Mni+t\n\u0001dp4X\nijs;As(i)i: (9)\nThe hybridized componentP\nijs;As(i)iis not an eigen-\nstate of the SI symmetry, indicating the local SI sym-\nmetry breaking in BaMn 2As2. This is an essential in-\ngredient of the odd-parity LMMMs. We calculate the\nhexadecapole moment up to O(t=\u0001dp) as\n\nM+\n42;z\u000b\nL=h ssj3p\n5z\u0000\nx2\u0000y2\u0001\nj ssimz;\n= 24p\n5IEOt\n\u0001dpmz+O(t2=\u00012\ndp); (10)whereIEO=hs;As(1)jz(x2\u0000y2)js;Mniis a matrix\nelement of the electric octapole moment and mz=\n2\u0016BP5\nj=1hszijis the total spin magnetic dipole moment.\nWe here adopt Slater-type orbitals [36, 37], in which\norbital wave functions are approximated by those of\nhydrogen-like atoms parametrized by e\u000bective princi-\npal quantum number n\u0003, orbital and magnetic quantum\nnumbers (l;m), and shielding factor \u000b:\n n\u0003;l;m;\u000b (r) =Nrn\u0003\u00001e\u0000\u000brYlm(^r); (11)\nwithNbeing a normalized factor. E\u000bective parameters\nn\u0003and\u000bare determined by the Slater rule [36]. Real\nSlater-type orbitals for l>0 are represented by using the\ncubic harmonics Z\u0006\nlminstead ofYlm. Using parameters,\n(n\u0003;l;m;\u000b ) = (3;0;0;3:52\u0017A\u00001) forjs;Mni;\n(n\u0003;l;m;\u000b ) = (3:7;0;0;2:68\u0017A\u00001) forjs;As(1)i;(12)\nand position of the As(1) atom ( x;y;z ) = (a=2;0;c0) with\nlattice parameters a= 4:15\u0017A andc0= 1:49\u0017A [13, 14],\nwe obtainIEO= 0:025\u0017A3. Then, the local magnetic\nhexadecapole moment is evaluated as\n\nM+\n42;z\u000b\nL'\u00000:66\u0016B\u0017A3; (13)\nfort=\u0001dp=\u00000:1 andmz= 5\u0016B.\nB. Hole-doped BaMn 2As2\nLightly hole-doped Ba 1\u0000xKxMn2As2shows metallic\nbehaviors, and doping hole carriers gives the rigid band\nshift in the band structure [20]. Then, the magnetic\nstructure remains to be the AFM state with a large mag-\nnetic moment 4 :21\u0016Bforx= 0:05 [20, 26]. Thus, the hex-\nadecapole moment\nM+\n42;z\u000b\nis robust in the hole-doped\nregime. On the other hand, the hole doping changes\nthe \flling of Mn 3 dorbitals and partially restores the\norbital angular momentum, implying non-negligible ef-\nfects of LS-coupling (spin-orbit coupling). This results\nin anisotropic distribution of magnetic charge in the xy-\nplane and induces magnetic quadrupole moment without\nsuppressing the ^ z-collinear AFM order.\nIn a heavily hole-doped region, Ba 1\u0000xKxMn2As2also\nundergoes the ferromagnetic transition and the ferromag-\nnetic moment is aligned in the xy-plane [21, 23, 24, 38].\nThe X-ray magnetic circular dichroism experiment iden-\nti\fed that the ferromagnetic moment arises from the As\nporbitals and coexists with the AFM moment of Mn\natoms [38]. Although the interplay of the magnetic hex-\nadecapole order and the ferromagnetic order would be an\ninteresting subject, it is left for a future study. In this\npaper we focus on the G-type AFM state, which realizes\nin the lightly hole-doped region, x < xc\u00180:19 [21], al-\nthough we also show some numerical results beyond this\ndoping region.7\nARPES study [39] and DFT+DMFT calculations [40]\nhave shown that the valence band of BaMn 2As2mainly\nconsists of Mn 3 dx2\u0000y2and As 4pzorbitals. The doped\nholes occupy the hybridized d-porbital, whose wave func-\ntion is obtained by the LCAO method,\nj dp;\u00061\n2i=jdx2\u0000y2;Mn;\u00061\n2i+t\u000b\n\u0001dpjp\u000b;\u00061\n2i\n+t\f\n\u0001dpjp\f;\u00061\n2i\u0006i\u0015\n\u00011jdxy;Mn;\u00061\n2i\n\u0000i\u0015\n2 (\u0001 2\u00072h)jdyz;Mn;\u00071\n2i\u0006\u0015\n2 (\u0001 2\u00072h)jdzx;Mn;\u00071\n2i;\n(14)\nwitht\u000b(t\f) being the hopping parameter between the Mn\njdx2\u0000y2iand Asjp\u000bi(jp\fi) orbitals. Figure 3 illustrates\nthep\u000bandp\forbitals, which are given by linear combi-\nnations ofporbitals of four As atoms. The orbital wave\nfunctions are explicitly written as\njp\u000bi=jpx;As(1)i\u0000jpy;As(2)i\u0000jpx;As(3)i+jpy;As(4)i;\n(15)\njp\fi=jpz;As(1)i+jpz;As(2)i+jpz;As(3)i+jpz;As(4)i;\n(16)\nwhich are compatible with the symmetry of the Mn\ndx2\u0000y2orbital. Energy levels of As porbitals, Mn dxy\norbital, and Mn dyz(dzx) orbital from the level of Mn\ndx2\u0000y2orbital are denoted by \u0001 dp, \u00011, and \u0001 2, respec-\ntively. The AFM molecular \feld, \u0000h\u001bz(h > 0 for the\nMn-As cluster with mz>0), has been introduced for\nMndorbitals, and \u0015is the LS-coupling constant which\nis generally small in 3 dtransition metal ions.\nFIG. 3. (Color online) Sketch of p\u000bandp\forbitals in a Mn-\nAs cluster. (a) The p\u000borbital is directed to the xy-plane,\nand (b) the p\forbital consisting of pz-orbitals extends in the\nz-direction.\nIn the hole-doped BaMn 2As2, the LS-coupling in-\nduces the local magnetic quadrupole moment M+\n22=\n\u0016B2p\n3(xsx\u0000ysy) [Eq. (4)] which belongs to the same IR\nas the magnetic hexadecapole moment M+\n42. We here cal-\nculate the expectation value of M+\n22as follows. First, the\ncontribution of one hole in the hybrid d-porbitalj dp;1\n2i\nis evaluated,\n\nM+\n22\u000b\nL=\u0000\u0016BD\n2p\n3 (xsx\u0000ysy)E\nL(17)\n=\u0000\u0016B4p\n3\u0015\n\u00012\u00002h\u0012\nI\u000bt\u000b\n\u0001dp+I\ft\f\n\u0001dp\u0013\n;(18)where\nI\u000b=hpx;As(1)jxjdzx;Mni+hpx;As(1)jyjdyz;Mni;\n(19)\nI\f=hpz;As(1)jxjdzx;Mni+hpz;As(1)jyjdyz;Mni:\n(20)\nAssuming Slater-type orbitals with e\u000bective parameters\n(n\u0003;l;m;\u000b ) = (3;2;\u00061;3:52\u0017A\u00001); (21)\nfor the Mn dyzanddzxorbitals, and\n(n\u0003;l;m;\u000b ) = (3:7;1;\u00061(0);2:68\u0017A\u00001); (22)\nfor the Asporbitals, we obtain\nI\u000b=\u00000:241\u0017A; I\f=\u00000:0563 \u0017A; (23)\nfor the lattice constant of Ba 1\u0000xKxMn2As2(x= 0:05),\na= 4:16\u0017A andc0= 1:49\u0017A [26]. When we take t\u000b=\u0001dp=\nt\f=\u0001dp=\u00000:1 and\u0015=(\u00012\u00002h) =\u00000:01, the magnetic\nquadrupole moment induced by one hole per Mn atom is\nestimated as,\n\nM+\n22\u000b\nL= 1:0\u000210\u00003\u0016B\u0017A: (24)\nThen, the magnetic quadrupole moment of hole-doped\nBa1\u0000xKxMn2As2is obtained as,\n\nM+\n22\u000b\nL=x\u00025:0\u000210\u00004\u0016B\u0017A: (25)\nThe magnitude of the magnetic quadrupole moment\nM+\n22\u000b\nLis reduced by small factors \u0015=(\u0001\u00002h) and\nx. Therefore, the magnetic hexadecapole moment re-\nmains to be the leading order parameter of hole-doped\nBaMn 2As2.\nC. Order parameter of odd-parity magnetic\nmultipole order in crystals\nLMMMs specify microscopic distribution of magnetic\ncharge around magnetic atoms or clusters, as we have\nstudied in previous subsections. However, there are am-\nbiguities in the de\fnition of macroscopic odd-parity mul-\ntipole moment in crystal systems. In order to avoid the\nambiguity, we here introduce a unique de\fnition by re-\nmoving an irrelevant component which does not break\nthe SI symmetry.\nFirst, operators of multipole moment de\fned by\nEq. (1) may depend on the origin of coordinates. Al-\nthough later this ambiguity is resolved by subtracting the\nirrelevant component, it is convenient to choose an inver-\nsion center as the origin. Then, the magnetic unit cell is\nde\fned so that its center is the inversion center. The in-\nversion center is no longer an inversion center in the AFM\nstate, because the SI symmetry is spontaneously broken.\nHowever, it still remains to be an origin of the PTop-\neration preserved in the odd-parity magnetic multipole\nstate.8\nFIG. 4. (Color online) (a) A unit cell of BaMn 2As2. The red\npoints show the inversion centers P1,P2,P3, andP4. (b)-(e)\nThe magnetic unit cell corresponding to each inversion center.\nCon\fguration of neighboring Mn atoms is shown.\nSecond, there remains an ambiguity for the choice of an\ninversion center and a magnetic unit cell. Actually, the\ncrystal structure of BaMn 2As2contains four nonequiva-\nlent inversion centers, namely, P1,P2,P3, andP4in Fig-\nure 4. Coordinates of Mn atoms depend on the choice of\ninversion center and corresponding unit cell. The mag-\nnetic multipole moment, indeed, depends on the inver-\nsion center when it is simply de\fned by the expectation\nvalue of Eq. (1) in the unit cell. For instance, let us \frst\nchoose the inversion center P1. Coordinates originating\nfrom the inversion center ( X;Y;Z ) are related to the co-\nordinates (x;y;z ) used in previous subsections for Mn-As\nclusters; (X;Y;Z ) = (x+a=2;y;z +c=4) for the Mn(1)\natom at (X;Y;Z ) = (a=2;0;c=4). Then, the expectation\nvalue ofZ(X2\u0000Y2)szfor electrons in the Mn-As cluster\nis decomposed into LMMMs and evaluated as\n\nZ(X2\u0000Y2)sz\u000b\nMn(1)=\nz(x2\u0000y2)sz\u000b\nMn(1)\n+a2c\n16hsziMn(1); (26)\nbecause the symmetry-adapted LMMM operators are\nonly the hexadecapole moment M+\n42;zand the dipole mo-\nmentM10/sz. Summing up contributions from two Mn\natoms in the unit cell, we obtain the multipole moment,\n\nM+\n42;z\u000b\nP1= 2\nM+\n42;z\u000b\nL+3p\n5a2c\n4mz: (27)\nSimilarly, we obtain\n\nM+\n42;z\u000b\nP2= 2\nM+\n42;z\u000b\nL\u00003p\n5a2c\n4mz; (28)\n\nM+\n42;z\u000b\nP3= 2\nM+\n42;z\u000b\nL; (29)\n\nM+\n42;z\u000b\nP4= 2\nM+\n42;z\u000b\nL; (30)\nwhen we choose the inversion center P2,P3, andP4, re-\nspectively. We here notice that the contribution from\nthe local magnetic dipole moment \u00063p\n5a2c\n4mzcauses the\nambiguity.To resolve the ambiguity, we rede\fne the magnetic\nmultipole moment by di\u000berence from a reference state,\nfollowing procedures used for electric dipole moment [41,\n42], magnetic monopole moment [43, 44], and magnetic\ntoroidal moment [45, 46]. For this purpose, we consider\nthe virtual crystal structure illustrated in Figure 5. In\nthe virtual crystal structure [Figure 5(a)], the As atoms\nlie in the same plane as Mn atoms and Ba atoms have\nbeen removed. Then, the D4hsymmetry is preserved\neven in the AFM state, since the Mn atoms are inversion\ncenters. However, the magnetic hexadecapole moment\nde\fned by Eq. (1) remains \fnite for the inversion cen-\nterP1andP2due to the irrelevant terms, \u00063p\n5a2c\n4mz.\nThus, we de\fne the order parameter of odd-parity mag-\nnetic multipole order hMlmiby subtracting the irrelevant\ncomponent,\nhMlmi=hMlmi\u0000\u0000hMlmi0\n\u0000; (31)\nwhere \u0000 denotes an inversion center and hi0\n\u0000indicates\nthe expectation value in the virtual crystal structure,\nnamely, the reference state. Although the multipole mo-\nmenthMlmi0\n\u0000in the reference state depends on an inver-\nsion center, the odd-parity magnetic multipole moment\nde\fned by di\u000berence from the reference state is unique in\nthe sense that it is independent of the choice of inversion\ncenter and unit cell.\nFIG. 5. (Color online) (a) Virtual crystal structure whose SI\nsymmetry is recovered in the AFM state. (b) Real crystal\nstructure of BaMn 2As2.\nThe local magnetic hexadecapole moment vanishes in\nthe virtual crystal, namely,\nM+\n42;z\u000b0\nL= 0, since the Mn-\nAs clusters preserve the local SI symmetry. In other\nwords, the macroscopic magnetic hexadecapole moment\nde\fned above is given by the LMMM,\n\nM+\n42;z\u000b\n= 2\nM+\n42;z\u000b\nL: (32)\nSimilarly, we obtain\n\nM+\n22\u000b\n= 2\nM+\n22\u000b\nL; (33)\nfor the magnetic quadrupole moment. Thus, the macro-\nscopic magnetic multipole moment in BaMn 2As2is given\nby the LMMMs investigated in Secs. IIIA and IIIB. It is\nagain stressed that the local SI symmetry breaking in the\ncrystal structure plays an essential role for the odd-parity\nmagnetic multipole order.9\nThe procedure used in this subsection can be applied to\nnot only BaMn 2As2but also various odd-parity magnetic\nmultipole states. First, a magnetic and centrosymmetric\ncrystal structure is considered as a reference state. Sec-\nond, an irrelevant component which is \fnite in the cen-\ntrosymmetric state is evaluated. Then, the odd-parity\nmagnetic multipole moment in real crystals is uniquely\nde\fned by di\u000berence from the reference state. The refer-\nence state is not uniquely determined in general. How-\never, it is reasonable to consider the virtual structure in\nFigure 5 for BaMn 2As2as a reference state which re-\nstores the local SI symmetry of magnetic sites. Using\nthis framework, we are able to estimate odd-parity mag-\nnetic multipole moment more precisely by \frst-principles\ncalculations [43, 44, 47{49]. The \frst principles study of\nBa1\u0000xKxMn2As2is an important future work.\nFor calculations of the multipole moment, additional\ncare is needed for the multivalued problem [41, 44, 45].\nWhen evaluating electric dipole moment by using the\nBerry phase formulation [41], we may obtain the elec-\ntric dipole moment with the arbitrariness of neR, where\nnis an integer and Ris the minimal lattice vector along\nthe polarization axis. The physically meaningful dipole\nmoment should be smaller than the arbitrary term. Sim-\nilar multivalued problem may also occur in calculations\nof higher-order multipole moment. The arbitrary term\nof magnetic hexadecapole moment, namely, the quan-\ntum unit of magnetic hexadecapole moment \u0001 M+\n42;zis\nroughly evaluated as\n\u0001M+\n42;z\u0018a2cmz; (34)\nwhich is in the same order as the irrelevant terms\n\u00063p\n5a2c\n4mzin Eqs. (27) and (28). Our evaluation of\nthe magnetic hexadecapole moment\nM+\n42;z\u000b\n\u00181\u0016B\u0017A3\n[Eqs. (13) and (32)] is much smaller than the quantum\nunit \u0001M+\n42;z\u0018102\u0016B\u0017A3, and therefore our calculation\ndoes not su\u000ber the multivalued problem.\nIV. EFFECTIVE MODEL\nIn the following part of this paper, we show characteris-\ntic properties induced by odd-parity magnetic multipole\norder. For this purpose, we introduce a tight-binding\nHamiltonian for the valence band of BaMn 2As2mainly\nconsisting of Mn dx2\u0000y2orbital [39, 40].\nBy projecting the \fve-orbital model to the valence\nband (Appendix A), the e\u000bective Hamiltonian is obtained\nas\nH=Hhop+HASOC +HAFM =X\nkcy\nkH(k)ck;(35)\nH(k)=\u0012\n\u000f(k) + [gA(k)\u0000hA]\u0001\u001bVAB(k)\nVAB(k)\u000f(k) + [gB(k)\u0000hB]\u0001\u001b\u0013\n;\n(36)\nwhere\u001b= (\u001bx;\u001by;\u001bz) is the Pauli matrix and ck=\n(ck;A;+;ck;A;\u0000;ck;B;+;ck;B;\u0000)Tis a vector representationof annihilation operators labeled by momentum k, sub-\nlattice index \u001c=A;B, and spin \u001b=\u0006. The kinetic\nenergy term is given by\nHhop=X\nk;\u001c;\u001b\u000f(k)cy\nk;\u001c;\u001bck;\u001c;\u001b\n+X\nk;\u001b\u0010\nVAB(k)cy\nk;A;\u001bck;B;\u001b+h:c:\u0011\n; (37)\nwhere\n\u000f(k) =\u00002t1(coskx+ cosky)\u00008t2coskx\n2cosky\n2coskz\n2;\n(38)\nVAB(k) =\u00004~t1coskx\n2cosky\n2\u00002~t2coskz\n2; (39)\nare intra-sublattice and inter-sublattice hopping energy,\nrespectively. The G-type AFM structure of BaMn 2As2\nis taken into account by the molecular \feld term HAFM.\nSince the magnetic moment is parallel to the z-axis and\nchanges its sign between the A and B sublattices, the\nAFM molecular \feld is given by hA=h^zandhB=\u0000h^z.\nThekdependent Zeeman terms gA(B)(k)\u0001\u001boriginate\nfrom the LS-coupling and the inter-orbital hybridization\nbetween the Mn dx2\u0000y2orbital and other Mn dorbitals,\nas we show the derivation from the \fve-orbital model in\nAppendix A. The PTsymmetry preserved in the AFM\nstate ensures the staggered structure gA(k) =\u0000gB(k)\u0011\ng(k). Theg-vectorg(k) is decomposed into the odd-\nparity and even-parity parts, g(k) =g0(k) +g00(k). The\nodd-parity component represents the ASOC term by\ng0(k) =0\nB@\u000b1sinky+\u000b2coskx\n2sinky\n2coskz\n2\n\u000b1sinkx+\u000b2sinkx\n2cosky\n2coskz\n2\n\u000b3sinkx\n2sinky\n2sinkz\n21\nCA:(40)\nThis term arises from the local SI symmetry breaking of\nMn atoms, and therefore, all the coe\u000ecients \u000b1,\u000b2, and\n\u000b3are \fnite in both paramagnetic and magnetic hex-\nadecapole states. In contrast, the additional component\ng00(k) denotes an even-parity spin-orbit coupling, called\nas symmetric spin-orbit coupling (SSOC). The deriva-\ntion from the \fve-orbital model gives the expression (see\nAppendix A),\ng00(k) =0\n@\fsinkx\n2cosky\n2sinkz\n2\n\u0000\fcoskx\n2sinky\n2sinkz\n2\n01\nA: (41)\nThe SSOC term breaks the TR symmetry, although it\nbreaks neither the local nor global SI symmetry. There-\nfore, the SSOC term disappears in the paramagnetic\nstate. The broken TR symmetry by the AFM order gives\nrise to the SSOC term.\nDiagonalizing the Bloch Hamiltonian H(k), we obtain\nthe energy spectrum\nEk=\u000f(k)\u0006q\nVAB(k)2+jg(k)\u0000h^zj2; (42)10\nwith double degeneracy protected by the PTsymmetry.\nIn the undoped system, the Fermi level lies in the gap of\nthe two bands. Then, the system shows insulating behav-\niors. Doping hole carriers lowers the Fermi level without\nreconstruction of the band structure [20]. Then, the par-\ntially \flled valence band leads to metallic behaviors.\nIn the following sections, we investigate electromag-\nnetic responses resulting from the SI symmetry break-\ning. Then, the SSOC term does not play an important\nrole since it does not break local or global SI symme-\ntry, as discussed in Appendix A. Thus, we set \f= 0\nfor simplicity, and assume the parameters, t1=\u00000:1,\nt2=\u00000:05,~t1= 0:05,~t2= 0:01 for the kinetic en-\nergy term, h= 1 for the AFM molecular \feld, and\n\u000b1=\u00000:005,\u000b2= 0:001,\u000b3= 0:01 for the ASOC\nterm, unless mentioned otherwise. The interlayer cou-\npling is moderate in BaMn 2As2compared with a related\nquasi-two-dimensional compound LaMnAsO [40]. Thus,\nmoderate interlayer hopping integrals are assumed. We\nadopt the unit for the lattice parameter, a=c= 1.\nV. MAGNETOELECTRIC EFFECT\nA. Uniform magnetoelectric e\u000bect\nIn the previous sections, the AFM state of BaMn 2As2\nhas been identi\fed as an odd-parity magnetic multipole\nstate, where the SI and TR symmetry are broken whereas\nthe combined PTsymmetry is preserved. Then, the ME\ncoupling is allowed in a free energy expansion in accor-\ndance with group-theoretical discussions [50]. The result-\ning ME e\u000bect, M= ^\u000bE, that is, the electric \feld-induced\nmagnetization has been observed in experiments [10].\nThe symmetry argument tells us that the ME response\nis attributed to rank-2 magnetic multipole orders listed\nin Table II. Decomposing the ME tensor ^ \u000b= (\u000b\u0016\u0017) into\nisotropic, antisymmetric, and traceless symmetric terms,\nwe have\n^\u000b=1\n3(Tr^\u000b)^1 +1\n2\u0000\n^\u000b\u0000^\u000bT\u0001\n+\u00141\n2\u0000\n^\u000b+ ^\u000bT\u0001\n\u00001\n3(Tr^\u000b)^1\u0015\n;\n(43)\ncorresponding to magnetic monopole moment,P\nixi^xi,\nmagnetic toroidal dipole moment,P\nj;k\u000fijkxj^xk, and\nmagnetic quadrupole moment, xi^xj+xj^xiandxi^xi\u0000\nxj^xjfori6=j. In accordance with the symmetry of mag-\nnetic quadrupole moment, M+\n22/x^x\u0000y^y, the ME e\u000bect\ncharacterized by the ME tensor\n^\u000b=0\n@\u000b0 0\n0\u0000\u000b0\n0 0 01\nA; (44)\nis allowed in BaMn 2As2.\nTo demonstrate the ME e\u000bect, we calculate the MEcoe\u000ecient by Kubo formula,\n\u000b\u0016\u0017=eg\u0016B~\n2iNX\nk;p;q[\u001b\u0016(k)]pq[v\u0017(k)]qp\nEp(k)\u0000Eq(k) +i\u000ef(Ep)\u0000f(Eq)\nEp(k)\u0000Eq(k);\n(45)\nwherepandqlabel the band indices, Nis the number\nof unit cell, \u000eis a scattering rate, and f(E) is the Fermi\ndistribution function. [ \u001b\u0016(k)]pqand [v\u0017(k)]qpare respec-\ntively the band representation of spin operator \u001b\u0016and\nvelocity operator v\u0017(k) =@H(k)=@k\u0017.\nWe plot the ME coe\u000ecient as a function of the chemi-\ncal potential \u0016in Figure 6. Our numerical result is con-\nsistent with the symmetry argument. Only the ME co-\ne\u000ecients\u000bxx=\u0000\u000byyare \fnite, corresponding to the\nmagnetic quadrupole moment M+\n22. Dark background\nin the \fgure represents the metallic region where the\nchemical potential lies in the valence band or conduc-\ntion band. Otherwise, the chemical potential lies in the\ngap, and the system is insulating. Interestingly, the ME\ne\u000bect is signi\fcantly enhanced in the metallic region.\nThe magnitude of the magnetoelectric coupling in the\ninsulating phase ( j\u000bxxj=j\u000byyj\u001810\u00004) corresponds to\n\u001810\u00003ps\u0001m\u00001when we takejt1j= 100 meV. This mag-\nnetoelectricity is much smaller than that of the proto-\ntypical magnetoelectric material Cr 2O3(j\u000b?j=j\u000bxxj=\nj\u000byyj\u001810\u00001ps\u0001m\u00001,j\u000bkj=j\u000bzzj\u00181 ps\u0001m\u00001) [51],\nbecause only a small magnetic quadrupole moment is in-\nduced by the LS-coupling term. However, precise es-\ntimation of the magnetoelectric coupling requires more\nelaborate works. For instance, calculations based on the\nmultiorbital model, estimation of orbital magnetoelec-\ntricity [52, 53], and DFT calculations are desired.\nFIG. 6. (Color online) ME coe\u000ecients \u000bxx(red triangles) and\n\u000byy(blue circles) as a function of the chemical potential \u0016.\nWe assume the temperature T= 0:01 and the scattering rate\n\u000e= 0:01, and choose the unit eg\u0016B~=2 = 1.11\nB. Antiferromagnetic Edelstein e\u000bect\nNext, we show the AFM Edelstein e\u000bect, namely, the\nAFM spin polarization induced by the electric current.\nThis characteristic response of locally noncentrosymmet-\nric systems [3] is attracting recent interest for application\nto antiferromagnetic spintronics [54{56].\nThe operator of AFM spin moment is de\fned as \u001bAF\n\u0016=\n\u001b\u0016\u001czwith the Pauli matrix \u001cacting on the sublattice\nspace. The ASOC term is uniform between sublattices\nwhen it is represented by the AFM spin operator,\nHASOC =X\nk;\u001c;\u001b;\u001b0g(k)\u0001\u001bAFcy\nk;\u001c;\u001bck;\u001c;\u001b0: (46)\nHence, the AFM spin-momentum locking occurs in lo-\ncally noncentrosymmetric systems, that is analogous to\nthe spin-momentum locking in globally noncentrosym-\nmetric systems. The above representation of the ASOC\nterm indicates the staggered ME e\u000bect represented by\nMAF= ^\u000bAFE; (47)\nin analogy to the Edelstein e\u000bect [57], that is, the spin\npolarization due to the current-induced shift of Fermi\nsurface. Since the ASOC term contains an in-plane com-\nponent,/kx\u001bAF\ny+ky\u001bAF\nx, which is a basis function of\nthe totally-symmetric A1gIR of theD4hpoint group, we\nhave \fnite staggered ME coe\u000ecients, \u000bAF\nyx=\u000bAF\nxy. The\nASOC term is derived from only the local SI symmetry\nbreaking and does not require the TR symmetry break-\ning. Therefore, the staggered ME e\u000bect occurs in both\nparamagnetic state and AFM state.\nReplacing\u001b\u0016with\u001bAF\n\u0016in Eq. (45), we calculate the\nstaggered ME coe\u000ecient \u000bAF\n\u0016\u0017. Figure 7 shows numerical\nresults of\u000bAF\nyx=\u000bAF\nxyin the AFM state ( h= 1) and the\nparamagnetic state ( h= 0). Although the staggered ME\ne\u000bect is caused by the in-plane component of the ASOC\nterm,g0\nx(k) andg0\ny(k), this component is suppressed in\nthe AFM state (see Appendix A) since the spin polariza-\ntion along the z-axis suppresses the spin-\ripping process.\nThus, we assume the in-plane components in the para-\nmagnetic state, \u000b1(para)=\u00000:05 and\u000b2(para)= 0:01,\nwhich are larger than those in the AFM state. Indeed,\nthe staggered ME coe\u000ecient is smaller in the AFM state\nthan in the paramagnetic state.\nIn contrast to the uniform ME e\u000bect, the staggered\nME e\u000bect is essentially induced by the electric current.\nThe shift of Fermi surface under the current results in the\nAFM spin polarization, like in the Edelstein e\u000bect [57].\nThus, we call Eq. (47) the AFM Edelstein e\u000bect.\nThe di\u000berence between the uniform ME response and\nstaggered ME response comes from the PTparity of spin\noperators. The PTparity is even for the AFM spin mo-\nment\u001bAF, while the uniform spin operator \u001bisPTodd.\nSince the velocity operator vhas evenPTparity, the\nME coe\u000ecient \u000b\u0016\u0017is purely determined by interband\ne\u000bects, whereas the staggered ME coe\u000ecient \u000bAF\n\u0016\u0017by in-\ntraband e\u000bects. Thus, \u000b\u0016\u0017/\u001c0while\u000bAF\n\u0016\u0017/\u001c1with\nFIG. 7. (Color online) Staggered ME coe\u000ecient \u000bAF\nxy(=\u000bAF\nyx)\nas a function of the chemical potential \u0016in the AFM state\n(h= 1, red triangles) and in the paramagnetic state ( h= 0,\nblue circles). The shaded area indicates the metallic region\nforh= 1. We assume T= 0:01 and\u000e= 0:01 and adopt the\nuniteg\u0016B~=2 = 1.\nrespect to the lifetime of quasiparticles, indicating the\nelectric \feld-induced uniform ME e\u000bect and the electric\ncurrent-induced AFM Edelstein e\u000bect. Indeed, the latter\ndoes not occur in the insulating state. In Appendix B,\nwe prove the lemma for Kubo formula supporting these\ndiscussions.\nHere, we show a simpli\fed expression of \u000bAF\nxy. The\nmatrix element of the velocity operator is obtained as\n[v\u0016(k)]pq=@Ep(k)\n@k\u0016\u000epq\n+ (Ep(k)\u0000Eq(k))\u001c\nup(k)\f\f\f\f@uq(k)\n@k\u0016\u001d\n;(48)\nwherejup(k)idenotes Bloch states satisfying\nH(k)jup(k)i=Ep(k)jup(k)i. Summing up intra-\nband contributions, we obtain\n\u000bAF\nxy'eg\u0016B~\niNX\nk;p\u0002\n\u001bAF\nx(k)\u0003\npp\ni\u000e@Ep(k)\n@ky@f(E)\n@E\f\f\f\nEp\n=\u0000eg\u0016B~\n\u000eNX\nk;p\u0002\n\u001bAF\nx(k)\u0003\npp@f(Ep)\n@ky: (49)\nBecause\u001c= 1=\u000e, we con\frm \u000bAF\nxy/\u001c1. At low tem-\nperatures, the staggered ME coe\u000ecient \u000bAF\nxyis deter-\nmined by quasiparticles near the Fermi surface. Since\nthe AFM spin moment is locked to momentum due to\nthe ASOC term, the deformation of Fermi surface rep-\nresented by @f(E)=@kygives rise to the \fnite AFM mo-\nmentMAF\nx=\nTrP\nk\u001bAF\nx(k)\u000b\n, as schematically shown\nin Figure 8.\nThe AFM Edelstein e\u000bect enables electrical switching\nof AFM domain [54, 55], pointing to the AFM spintron-12\nFIG. 8. (Color online) (a) The in-plane component of g-vector\n[g0\nx(k),g0\ny(k)] in the ASOC term. The momentum depen-\ndence on the kz= 0 plane is shown by arrows. The electric\n\feld along the y-direction shifts the Fermi surface (circle with\na solid line). (b) The k-dependent AFM spin polarization on\nthe Fermi surface. Summation for the momentum leads to a\nmacroscopic AFM spin polarization in the x-direction.\nics [56]. However, the seemingly AFM structure is clas-\nsi\fed into the odd-parity magnetic multipole. In other\nwords, the \\AFM domain\" switched by the electric cur-\nrent is, indeed, the domain of ferroic odd-parity mag-\nnetic multipole from the viewpoint of multipole physics.\nAlthough it may be expected that the magnetic hexade-\ncapole moment of BaMn 2As2can be switched by inject-\ning an electric current, it is unlikely at least in the linear\nresponse region. The e\u000bective AFM Zeeman \feld driven\nby electric \feld is con\fned to the xy-plane and the AFM\nEdelstein e\u000bect cannot switch the z-collinear AFM do-\nmains of BaMn 2As2. However, an uniaxial strain along\nthe in-plane direction reduces the site symmetry of Mn\natoms and accordingly induces another AFM Edelstein\ne\u000bect characterized by a \fnite coe\u000ecient \u000bAF\nzz. Then,\nthe magnetic hexadecapole moment may be switched by\nthe electric current along the z-direction: M+\n42>0$\nM+\n42<0. Furthermore, the electric current along the\nz-direction induces the strain \feld in the xy-plane, as\nwe show in Sec. VI. Therefore, the nonlinear e\u000bect of the\nelectric current gives the e\u000bective AFM Zeeman \feld and\nmay switch the magnetic hexadecapole domain.\nVI. CURRENT-INDUCED NEMATICITY\nWe here show a counter-intuitive response in the metal-\nlic magnetic multipole state. The electric current along\nthez-axis induces the nematicity in the xy-plane.\nAs we show in Table III, the order parameter of\nodd-parity magnetic multipole order represented in k-\nspace indicates spin-independent asymmetric modulation\nof the band structure, which has been demonstrated in\nseveral models [3, 4, 6, 8]. In BaMn 2As2, the order pa-\nrameter of the B1uIR iskxkykzink-space. The corre-\nsponding cubic asymmetry in the energy spectrum results\nfrom the coupling of the AFM molecular \feld and the\nz-axis component of the ASOC term, which is indeed,\n\u0000hg0\n3(k)/kxkykz, in the long-wavelength limit. Thisterm induces a tetrahedral modulation of Fermi surfaces\nas shown in Figure 9. The same modulation also arises\nfrom the coupling between the ASOC and SSOC terms,\ng0(k)\u0001g00(k), although this term is negligible as we dis-\ncussed in Appendix A.\nFIG. 9. (Color online) The tetrahedral modulation of Fermi\nsurface is shown with the solid lines. Fermi surfaces in the (a)\nk[110]\u0000kzplane and (b) k[110]\u0000kzplane are shown. k[110]\nandk[110]are momentum along the [110] and [ 110] direction,\nrespectively. The chemical potential is set to \u0016=\u00000:8 and a\nlarge ASOC \u000b3= 0:3 is assumed for emphasizing the tetrahe-\ndral modulation. Fermi surfaces for \u000b3= 0 are plotted with\nthe dashed lines for a comparison.\nThe electric current along the z-axis induces \fnite ex-\npectation value of kz. Then, the tetrahedral modulation\nleads tokxkykz!kxkyhkzi, indicating the nematicity\nin the xy-plane resulting from the nematic modulation\nof Fermi surface. This is an intuitive explanation of the\ncurrent-induced nematicity shown below.\nThe modulation of Fermi surface may be quanti\fed by\nthe weighted density operator nf[58],\nnf=1\nNX\nk;\u000bfkcy\nk;\u000bck;\u000b; (50)\nwherefkis the weighting function and the index \u000bspec-\ni\fes the internal degree of freedom such as spin and sub-\nlattice. For example, fk= coskx\u0000coskyrepresents the\ndx2\u0000y2-wave modulation, and then, the spontaneous or-\ndering ofhnfiis called the dx2\u0000y2-wave Pomeranchuk\ninstability. The tetrahedral modulation of kxkykztype\nin BaMn 2As2is given by the following weighted density\noperator,\nnT=1\nNX\nk;\u001c;\u001bTkcy\nk;\u001c;\u001bck;\u001c;\u001b; (51)\nwhere\nTk= sinkx\n2sinky\n2sinkz\n2: (52)\nAs we have discussed, the expectation value of nTcan be\nregarded as an order parameter of B1umagnetic multi-\npole order in the metallic state. In Figure 10, we plot\nhnTiin the AFM state, which is indeed \fnite in the\nmetallic region. The sign of the tetrahedral modulation13\nis naturally opposite between the upper and lower bands\nbecause of the sign \u0006of the energy spectrum [Eq. (42)].\nFIG. 10. (Color online) The tetrahedral modulation of Fermi\nsurface,hnTi. Parameters are T= 0:01 and\u000e= 0:01. The\nshaded area indicates the metallic region. hnTiis \fnite in the\npresence of a Fermi surface.\nWhen we look at the Fermi surface on a kz= constant\nplane, the diagonal modulation kxkyappears. However,\nthekxkymodulation is opposite between the kz=cplane\nand thekz=\u0000cplane, and therefore, the summation\nforkzresults in vanishing in-plane nematic order in the\nequilibrium state. Now we notice that the applied electric\n\feld perpendicular to the xy-plane causes the imbalance\nbetweenkz=candkz=\u0000c, which gives rise to the\ndiagonal nematicity in the stationary state. A schematic\nillustration is shown in Figure 11.\nFIG. 11. (Color online) Schematic \fgure for the mechanism of\ncurrent-induced nematicity. (a) The tetrahedral modulation\nof a Fermi surface is illustrated. The diagonal nematicity\nofkxkytype is canceled out by the kz-summation. (b) The\nelectric \feld along the z-direction breaks the balance between\nthekz>0 region and kz<0 region, giving rise to the kxky-\ndiagonal nematic order.\nTo investigate the current-induced nematic order, we\nde\fne a nematic operator nDas follows [59],\nnD=1\nNX\nk;\u001bDkcy\nk;A;\u001bck;B;\u001b+h:c: ; (53)with\nDk= sinkx\n2sinky\n2: (54)\nThe current-induced nematicity represented by\nhnDi=\u001fDEz; (55)\nis calculated by using Kubo formula,\n\u001fD=\u0000ie~\nNX\nk;\u0016:\u0017[nD(k)]\u0016\u0017[vz(k)]\u0017\u0016\nE\u0016(k)\u0000E\u0017(k) +i\u000ef(E\u0016)\u0000f(E\u0017)\nE\u0016(k)\u0000E\u0017(k);\n(56)\nwherenD(k) is the band representation of nD.\nFIG. 12. (Color online) The nematic susceptibility to the\nelectric \feld. Parameters are T= 0:01 and\u000e= 0:01, and we\ntake the unit e~= 1.\nFigure 12 shows the numerical result of nematic sus-\nceptibility \u001fD. It is indeed shown that the current-\ninduced nematic order occurs in the metallic region.\nSince the nematic operator nDisPTeven, the nematic\nsusceptibility \u001fDis determined by intraband contribu-\ntions as\u000bAF\n\u0016\u0017is. Thus, the nematicity is essentially\n\\current-induced\" and it does not occur in the insulating\nstate. This means that the current-induced nematic or-\nder is a response characterizing the odd-parity magnetic\nmultipole order in itinerant systems.\nBy using the lemma proved in Appendix B, the ne-\nmatic susceptibility to electric \feld is obtained as\n\u001fD'\u0000e~\n\u000eVX\nk;p[nD(k)]pp@f(Ep)\n@kz: (57)\nThus,\u001fD/\u001c1as expected. Note that the nematic sus-\nceptibility appears with the same sign between the upper\nand lower bands, in contrast to the tetrahedral modula-\ntion (Figure 10). This is because the nematic operator is\nde\fned by the inter-sublattice hopping. By using opera-\ntors for bonding and anti-bonding orbitals [60],\n(\nbk;\u001b=1p\n2(ck;A;\u001b+ck;B;\u001b) for bonding orbital,\nak;\u001b=1p\n2(ck;A;\u001b\u0000ck;B;\u001b) for anti-bonding orbital,\n(58)14\nthe nematic operator is recast to\nnD=1\nNX\nk;\u001bDk\u0010\nby\nk;\u001bbk;\u001b\u0000ay\nk;\u001bak;\u001b\u0011\n: (59)\nOwing to the negative sign in front of ay\nk;\u001bak;\u001b, the trans-\nlation of Fermi surface by electric current induces the ne-\nmaticity with the same sign in the upper band and the\nlower band.\nAlthough we have discussed an electronic nematic or-\nder so far, the nematicity induces a structural deforma-\ntion through electron-lattice couplings. Thus, the elec-\ntric current along the z-axis induces the lattice struc-\ntural deformation in the xy-plane illustrated in Figure 13.\nThe structural nematic order which has been observed in\nFe-based 122-compounds [15{17] is essentially di\u000berent\nfrom the current-induced nematic order proposed by this\nwork. In BaFe 2As2, the orthorhombic transition sponta-\nneously occurs at low temperatures. On the other hand,\nin the odd-parity magnetic multipole state of BaMn 2As2,\nthe nematicity is induced by the external electric cur-\nrent. Furthermore, the SI symmetry is not broken in the\northorhombic stripe AFM state of BaFe 2As2, while the\nspontaneous SI symmetry breaking plays an essential role\nin BaMn 2As2. As expected from an intuitive explanation\nfor the current-induced nematic order, the electric cur-\nrent along the x-axis ( y-axis) also induces the structural\ntransition of yz-type ( zx-type).\nFIG. 13. (Color online) Sketch of the current-induced struc-\ntural transition. The electric \feld along the z-axis in-\nduces the electronic nematicity in the xy-plane, which leads\nto the tetragonal-orthorhombic structural transition through\nelectron-lattice couplings.\nThe structural deformation driven by the electric \feld\ncan be regarded as a (inverse) piezoelectric e\u000bect. For\ninsulators, a piezoelectric-coupling constant is given by\neijk=@Pi\n@\u000fjk\f\f\f\nE=0; (60)\nwherePis an electric dipole moment and \u000fjkis a strain\ntensor. Alternatively, it is recast,\nsij=X\nkekijEk; (61)wheresijis a stress tensor and we assume \u000fij= 0. The\nD2dsymmetry allows piezoelectric couplings\nexyz=exzy=eyxz=eyzx;\nezxy=ezyx;(62)\nandezxyandezyxrepresent the stress in the [110]-\ndirection or the [ 110]\u0000direction induced by the electric\n\feld along the [001]-direction. This piezoelectric e\u000bect is\nsimilar to the current-induced nematic order studied in\nthis work. However, there are signi\fcant di\u000berences in\ntheir mechanism, symmetry, and manifestation. In in-\nsulators, the piezoelectric deformation is mainly caused\nby ionic displacements induced by electric \feld. Then,\nthe polar rank-3 tensor eijkhas the even parity under\nthe TR operation [61]. On the other hand, the piezo-\nelectricity we propose, namely, the current-induced ne-\nmaticity, is characteristic of metallic systems, and then\nthe \"piezoelectric\" tensor ~ eijkhas the odd parity under\nthe TR operation [62]. In other words, the direction of\nthe strain, the [110]-direction or the [ 110]-direction, is\nreversed by applying the TR operation. Therefore, the\ninverse piezoelectric e\u000bect is switchable by changing the\nAFM domain. Thus, the metallic magnetic hexadecapole\nstate may be called \\magnetopiezoelectric metal\". Inter-\nestingly, hole-doped BaMn 2As2realizes such an exotic\nstate which may be useful for device applications.\nWe have con\frmed that the conventional piezoelectric-\nity does not occur in the magnetopiezoelectric metal from\nthe viewpoint of symmetry. Both of eijkand ~eijkare po-\nlar tensors, and require the SI symmetry breaking. The\nTR even piezoelectric tensor eijkis forbidden in the mag-\nnetic hexadecapole state, since the PTsymmetry is pre-\nserved. Then, the electric \feld does not directly couple\nto the strain, but indirectly couples through the electric\ncurrent. The response is represented by the same form\nas Eq. (61), although the response tensor is replaced by\nthe TR odd one ~ eijk\nRecently, a related phenomenon has been proposed by\nRef. 63. The authors have revealed the electric current\ngeneration by a time-dependent strain in metallic sys-\ntems where both of SI symmetry and TR symmetry are\nbroken. This is a dynamical and inverse response of the\ncurrent-induced nematic order which we reveal in this\nwork.\nVII. SUMMARY AND DISCUSSION\nIn this paper, we investigated the odd-parity magnetic\nmultipole order in BaMn 2As2and clari\fed characteristic\nresponses. The obtained results are summarized below.\nFirst, we have classi\fed the magnetic multipole order\non the basis of the IRs of point group symmetry, similar\nto the classi\fcation of unconventional superconductivity\nby Sigrist and Ueda. [27] The symmetry argument in-\ndicates the odd-parity magnetic multipole order in the\nAFM state of BaMn 2As2, which belongs to the B1uIR15\nofD4hpoint group. Possible multipole moments are\nmagnetic quadrupole moment M+\n22and magnetic hexade-\ncapole moment M+\n42.\nNext, the microscopic analysis of seemingly conven-\ntional collinear G-type AFM state in undoped BaMn 2As2\nand hole-doped Ba 1\u0000xKxMn2As2reveals the leading\nmagnetic hexadecapole order. In the hole-doped metallic\nsystem, the orbital angular momentum of Mn 3 delec-\ntrons is partially restored, and then the LS-coupling in-\nduces the magnetic quadrupole moment M+\n22as an ad-\nmixed odd-parity magnetic order parameter. The micro-\nscopic study implies that the local SI symmetry breaking\nat magnetic sites plays an essential role for the odd-parity\nmagnetic multipole order. Furthermore, we propose a\nde\fnition of macroscopic order parameter of odd-parity\nmagnetic multipole order, in which ambiguities due to\nthe choice of unit cell are removed.\nThen, we have introduced an e\u000bective Hamiltonian and\nshown electromagnetic responses induced by the odd-\nparity magnetic multipole order. The ME e\u000bect oc-\ncurs in accordance with the existence of the magnetic\nquadrupole moment. The AFM Edelstein e\u000bect has\nalso been shown, and the electrical switching of mag-\nnetic multipole moment has been discussed. Interest-\ningly, the metallic odd-parity magnetic multipole state,\nwhere both of TR and SI symmetry are spontaneously\nbroken, shows an asymmetric modulation of Fermi sur-\nface. The tetrahedral modulation of kxkykztype oc-\ncurs in doped BaMn 2As2, and induces a counter-intuitive\ncurrent-induced nematic order. The in-plane nematic or-\nder is induced by the out-of-plane electric current. Thus,\nthe itinerant magnetic hexadecapole state is identi\fed\nas magnetopiezoelectric metal. These exotic phenomena\nare derived from the ASOC term arising from local SI\nsymmetry breaking.\nAlthough odd-parity multipole order has been dis-\ncussed for only a few crystalline materials so far, a\nvariety of magnetic compounds may be identi\fed as\nodd-parity magnetic multipole state. Indeed, we have\nrevealed that a seemingly conventional AFM state of\nBaMn 2As2is identi\fed as the magnetic hexadecapole\nstate. This work is a proposal of magnetic hexadecapole\norder, although magnetic monopole, toroidal dipole,\nand magnetic quadrupole compounds have been stud-\nied [10, 46, 64, 65]. From our analysis of BaMn 2As2, we\nimmediately notice that many other compounds show the\nmagnetic hexadecapole order with the magnetic struc-\nture similar to BaMn 2As2. For instance, we iden-\ntify other Mn-based 122-systems [BaMn 2Pn2(Pn=P,\nBi)], [66, 67] Cr-based 122-systems [ RCr2Si2(R=Ho, Er,\nTb) and AeCr2As2(Ae=Ba, Sr)] [68{71], Mn-based 112-\nsystems [ XMnBi 2(X=Ca, Sr, Eu)] [72, 73], Mn-based\n111-systems [KMn Pn(Pn=As, Sb, Bi)] [74, 75], and\nMn-based 1111-systems [LaMnPO and RMnAsO ( R=La,\nNd)] [76, 77], as magnetic hexadecapole compounds. The\nlocal SI symmetry breaking of magnetic sites and stag-\ngered alignment of magnetic moment are satisfactory\ncondition for the odd-parity magnetic multipole order.This condition may be satis\fed in various magnetic sys-\ntems we have not noticed. More elaborate study of odd-\nparity multipole order will re\fne understanding of spon-\ntaneous parity violation and resulting exotic phenomena\nin condensed matter.\nACKNOWLEDGMENTS\nThe authors are grateful to T. Nomoto and S. Sumita\nfor fruitful discussions and comments. This work was\nsupported by a Grant-in-Aid for Scienti\fc Research on\nInnovative Areas \\J-Physics\" (JP15H05884) and \\Topo-\nlogical Materials Science\" (JP16H00991) from the Japan\nSociety for the Promotion of Science (JSPS), and by\nJSPS KAKENHI Grants (Numbers JP15K05164 and\nJP15H05745).\nAppendix A: Derivation of the e\u000bective model\nFive-orbital tight-binding Hamiltonian for Mn 3 dor-\nbitals is represented by\nH=Heven+Hodd+HLS+HCEF+HAFM;(A1)\nwhereHevenandHoddare hopping terms with even- and\nodd-parity under the SI operation, respectively. The LS-\ncoupling term is written as\nHLS=\u0015X\ni;\u001cli;\u001c\u0001si;\u001c; (A2)\nwhere\u0015is the coupling strength, l(s) is orbital (spin)\nangular momentum operator, and the label iand\u001cin-\ndicate the site and sublattice index, respectively. The\ncrystalline electric \feld term HCEFwhich mainly arises\nfrom the ligand \feld due to As atoms (Figure 2) gives\nrise to the level splitting of Mn dorbitals. The dlevels\nare classi\fed by the local point group D2dof Mn sites,\ndz2|{z}\nA1+dxy|{z}\nB1+dx2\u0000y2|{z}\nB2+dyz;dzx|{z}\nE; (A3)\nwhere the IR of the point group is indicated for each\ndlevel. We here neglect electron correlation e\u000bects in\nthe AFM state and take into account the molecular \feld\nterm,\nHAFM =X\ni;\u001c\u00002h\u001czsz\ni;\u001c; (A4)\nwhere\u001cis the Pauli matrix acting on the sublattice\nspace.\nNow we derive the single band Hamiltonian for the\nvalence band. Because the LS-coupling is small in 3 d\nelectron systems, the Russell-Saunders picture is appro-\npriate. The crystalline electric \feld is much larger than\nthe LS-coupling. Therefore, we can perturbatively treat16\nthe LS-coupling term. Then, the eigenstate of atomic\nHamiltonianHCEF+HLS+HAFM for mainlydx2\u0000y2or-\nbital is obtained as\nj\u001bz=\u0006;\u001czi=jdx2\u0000y2;\u001bz;\u001czi+i\u0015\u001bz\n\u00011jdxy;\u001bz;\u001czi\n\u0000i\u0015\n2 (\u0001 2\u00002h\u001bz\u001cz)jdyz;\u0000\u001bz;\u001czi\n+\u0015\u001bz\n2 (\u0001 2\u00002h\u001bz\u001cz)jdzx;\u0000\u001bz;\u001czi;(A5)\nwheres=1\n2\u001b, and \u0001 1(\u00012) is the energy level of dxy\norbital (dyzanddzxorbitals) from the level of dx2\u0000y2\norbital.\nProjecting the \fve-orbital model [Eq. (A1)] to the\nHilbert space spanned by j\u001bz;\u001czi, we obtain the projected\nHamiltonian as Eq. (35). The coupling constants of the\nASOC term and the SSOC term for the A sublattice,\ngA\u0001\u001b, are obtained as\n\u000b1=2tzx;1\u0015\u00012\n\u00012\n2\u00004h2; (A6)\n\u000b2=4\u0010\nt(1)\nzx;2\u0000t(2)\nzx;2\u0011\n\u0015\u00012\n\u00012\n2\u00004h2; (A7)\n\u000b3=16txy;3\u0015\n\u00011; (A8)\n\f=\u00008\u0010\nt(1)\nzx;2+t(2)\nzx;2\u0011\n\u0015h\n\u00012\n2\u00004h2: (A9)\nThe hopping integrals are written as\n\u0000tzx;1=hdzx;(a;0;0)jHkinjdx2\u0000y2;(0;0;0)i;(A10)\n\u0000t(1)\nzx;2=hdzx;(a\n2;a\n2;c\n2)jHkinjdx2\u0000y2;(0;0;0)i;(A11)\n\u0000t(2)\nzx;2=hdzx;(\u0000a\n2;\u0000a\n2;\u0000c\n2)jHkinjdx2\u0000y2;(0;0;0)i;\n(A12)\n\u0000txy;3=hdxy;(a\n2;a\n2;c\n2)jHkinjdx2\u0000y2;(0;0;0)i;(A13)\nwhereHkin=Heven+Hodd,jd\r;(x;y;z )idenotes the or-\nbital wave function of Mn 3 d\rorbital on the A sublattice,\nand (x;y;z ) is the Cartesian coordinates.\nThe entanglement of spin and orbital due to the LS-\ncoupling results in the k-dependent spin-orbit coupling\nterms. According to Eqs. (A6)-(A9), the in-plane com-\nponents of the g-vector originate from the hybridiza-\ntion ofjdx2\u0000y2;\u001bziwithjdzx(dyz);\u0000\u001bzi. Therefore, the\ncoupling constants specifying the in-plane component,\nnamely,\u000b1,\u000b2and\f, are suppressed by the large AFM\nmolecular \feld. On the other hand, the out-of-plane com-\nponent of the ASOC term, g0\nz(k)\u001bz, is robust against the\nAFM order.\nThe SSOC term, g00(k)\u0001\u001b, is induced by the AFM\norder, although it disappears in the paramagnetic state.\nThis term does not play an essential role for the electro-\nmagnetic responses studied in this paper. Because theSSOC term remains \fnite in the reference state shown in\nFigure 5(a), it does not induce the ME e\u000bect character-\nistic of the odd-parity magnetic multipole state. From\nEq. (42), we notice that the coupling between the ASOC\nterm and the SSOC term also induces the tetrahedral\nmodulation of the band structure, which is given by\ng0(k)\u0001g00(k) =\n2\u000b1\f\u0012\nsinkysinkx\n2cosky\n2\u0000sinkxsinky\n2coskx\n2\u0013\nsinkz\n2:\n(A14)\nAlthough the current-induced nematicity studied in\nSec. VI also arises from this term, it is a higher order cor-\nrection with respect to the LS-coupling constant \u0015. Since\nthe LS-coupling is much smaller than the AFM molecular\n\feld and the crystalline electric \feld, Eq. (A14) is negli-\ngible compared with the leading order term proportional\ntoh\u000b3. Therefore, we neglect the SSOC term in Secs. V\nand VI.\nAppendix B: Lemma for Kubo formula\nElectromagnetic responses in the linear response region\nare generally given by Kubo formula. When the Hamil-\ntonian is represented by a quadratic form of one-body\noperators, the response function for uniform and static\nperturbation is obtained as a simple form,\n\u001fAB=CX\nk;p;q[A(k)]pq[B(k)]qp\nEp(k)\u0000Eq(k) +i\u000ef(Ep)\u0000f(Eq)\nEp(k)\u0000Eq(k);\n(B1)\nwhereCis a constant factor, pandqare band indices,\n[A(k)]pqand [B(k)]qpare band representation of uniform\noperatorsAandB, respectively. The band energy is\ndenoted by Ep(k),\u000eis a constant scattering rate, and\nf(E) is the Fermi distribution function. For instance, the\nelectric conductivity tensor \u001b\u0016\u0017is obtained by assigning\nthe current operators j\u0016andj\u0017toAandB, respectively.\nHere, we consider the Hamiltonian which preserves the\nPTsymmetry. The PTsymmetric system has at least\ndouble degeneracy at each momentum k(Kramers pair),\nand single particle states are labeled by \u001bz=\u0006with\nPauli matrix acting on the degenerate Hilbert space. The\nBloch states of the Kramers pair are transformed to each\nother by the PToperation,\nPTjn;k;\u001bi= (i\u001b2)\u001b0\u001bjn;k;\u001b0i; (B2)\nwherejn;k;\u001biis denoted by the band index n, crystal\nmomentumk, and (pseudo-)spin \u001b. The band index p\n(andq) is speci\fed by the combination of nand\u001b(n0and\n\u001b0). The intraband contributions to the response func-\ntion Eq. (B1) come from pairs ( p;q) withn=n0. Thus,\nintraband contributions are regarded as intra-Kramers\npair contributions. On the other hand, interband con-\ntributions are given by bands with nonequivalent energy,\nn6=n0.17\nNow we show a lemma about the relation between the\nPTparity ofAandBand the response function:\n\u000fWhen the product of PTparity of operators Aand\nBis odd, the response function \u001fABis determined\nby the interband contributions.\n\u000fWhen the product of PTparity is even, the re-\nsponse function is given by the intraband contribu-\ntions.\nTo prove the lemma, we consider two Kramers pairs pro-\ntected by the PTsymmetry,jn;k;\u001bi,jm;k;\u001bi. By in-\nserting the PToperator the matrix element [ A(k)]pqfor\n(p;q) = [(n;\u001b);(m;\u001b0)] is transformed as\n[A(k)](n;\u001b);(m;\u001b0)(B3)\n=h\n~A(k)i\n(m;\u001b00);(n;\u001b000)(i\u001b2)y\n\u001b0\u001b00(i\u001b2)\u001b000\u001b(B4)\n= [A(k)](m;\u001b00);(n;\u001b000)(i\u001b2)y\n\u001b0\u001b00(i\u001b2)\u001b000\u001b(\u00001)PA;(B5)\nwhere ~A= (PT)A(PT)\u00001andPAdenotes the PTparity\nof the Hermitian operator A. Here, we use ( PT)k=k\nand the anti-Hermitian property of PT-operator\nh\u001ejAj i=h~ j(PT)Ay(PT)\u00001j~\u001ei; (B6)\nj~\u001ei=PTj\u001ei;j~ i=PTj i: (B7)\nBy using the relation Eq. (B5), the summation for the\nband index in Eq. (B1) simpli\fed.\nFirst, the intra-Kramers pair contributions [ p;q =\n(n;\u001b);(n;\u001b0)] to the response function are given by\n\u001f(intra)\nAB =\nX\nn;kC\ni\u000e@f(E)\n@E\f\f\f\nEn(k)X\n\u001b;\u001b0[A(k)](n;\u001b);(n;\u001b0)[B(k)](n;\u001b0);(n;\u001b):\n(B8)\nThe r.h.s is transformed by\nX\n\u001b;\u001b0[A(k)](n;\u001b);(n;\u001b0)[B(k)](n;\u001b0);(n;\u001b)(B9)\n=X\n\u001b;\u001b0[A(k)](n;\u001b0);(n;\u001b)[B(k)](n;\u001b);(n;\u001b0)(\u00001)PAB;\n(B10)\nwherePAB=PA+PB. Hence, we obtain\n\u001f(intra)\nAB = (\u00001)PAB\u001f(intra)\nAB: (B11)\nSimilarly, the inter-Kramers pair contributions [ p;q=\n(n;\u001b);(m;\u001b0) withn6=m] are simpli\fed as follows. We\ndivide the inter-Kramers pair contributions\n\u001f(inter)\nAB =\u001f(inter;odd)\nAB +\u001f(inter;even)\nAB; (B12)with\n\u001f(inter;odd)\nAB =X\nn;m;k\u000bf(En)\u0000f(Em)\n\u0001nm\u0001nm\n\u00012nm+\u000e2\n\u0002X\n\u001b;\u001b0[A(k)](n;\u001b);(m;\u001b0)[B(k)](m;\u001b0);(n;\u001b);\n(B13)\n\u001f(inter;even)\nAB =X\nn;m;k\u000bf(En)\u0000f(Em)\n\u0001nm\u0000i\u000e\n\u00012nm+\u000e2\n\u0002X\n\u001b;\u001b0[A(k)](n;\u001b);(m;\u001b0)[B(k)](m;\u001b0);(n;\u001b);\n(B14)\nand \u0001nm\u0011En(k)\u0000Em(k). Then, we obtain\n\u001f(inter;odd)\nAB =\u0000(\u00001)PAB\u001f(inter;odd)\nAB; (B15)\n\u001f(inter;even)\nAB = (\u00001)PAB\u001f(inter;even)\nAB: (B16)\nAn arbitrary scattering rate \u000eis set to zero for inter-band\ncontributions, as usual. Thus, we have\n\u001f(inter)\nAB =\u0000(\u00001)PAB\u001f(inter)\nAB: (B17)\nEqs. (B11) and (B17) clarify the relation between the\ncombinedPTparity ofAandBand the response func-\ntion.\n\u001f(intra)\nAB = 0 for odd PAB; (B18)\n\u001f(inter)\nAB = 0 for even PAB: (B19)\nThus, the response function is given by the intraband\ncontributions for even PAB, while it is given by the in-\nterband contributions for odd PAB. This is the lemma\nwhich is proved in this section. Our results in Secs. V\nand VI have been discussed on the basis of the lemma.\nThe scaling with respect to the scattering rate is also\nobtained from the proof.\n\u001fAB/\u000e\u00001for evenPAB; (B20)\n\u001fAB/1 for odd PAB: (B21)\nThe response function \u001fABfor evenPABoriginates from\nthe deformation of Fermi surface, and therefore, disap-\npears in the insulating state lacking Fermi surface. On\nthe other hand, for odd PAB, the response function comes\nfrom the deformation of wave function. Then, a \fnite re-\nsponse function may be obtained even in the insulating\nstate.\nThe lemma is an extension of Onsager's reciprocity re-\nlation. It is straightforward to prove other relations for\nresponse functions, for instance ensured by TR symme-\ntry.\nAppendix C: Nematic operators\nIn this paper we adopt the nematic operator nDin\nEq. (53). However, the nematic operator quantifying the18\ndeformation of Fermi surface is not unique. Indeed, we\nmay consider another nematic operators\nnD1=1\nNP\nk;\u001c;\u001bD0\nkcy\nk;\u001c;\u001bck;\u001c;\u001b; (C1)\nnD2=1\nNP\nk;\u001c;\u001bD00\nkcy\nk;\u001c;\u001bck;\u001c;\u001b; (C2)\nwith weighting functions D0\nk= sinkxsinkyandD00\nk=\nsin (kx=2) sin (ky=2) cos (kz=2). These nematic operatorscorrespond to the next-nearest-neighbor intralayer hop-\nping and nearest-neighbor interlayer hopping, respec-\ntively. The nematic order parameters hnD1iandhnD2i\nbelong to the same IR ( B2g) of theD4hpoint group as\nhnDi. 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Phys. 105, 093916 (2009)."    },    {        "title": "1706.01840v1.Designing_magnetism_in_Fe_based_Heusler_alloys__a_machine_learning_approach.pdf",        "content": "Designing magnetism in Fe-based Heusler alloys: a machine learning approach\nMario \u0014Zic, Thomas Archer,\u0003and Stefano Sanvito\nSchool of Physics, AMBER and CRANN Institute, Trinity College, Dublin 2, Ireland\n(Dated: June 7, 2017)\nCombining material informatics and high-throughput electronic structure calculations o\u000bers the\npossibility of a rapid characterization of complex magnetic materials. Here we demonstrate that\ndatasets of electronic properties calculated at the ab initio level can be e\u000bectively used to identify\nand understand physical trends in magnetic materials, thus opening new avenues for accelerated\nmaterials discovery. Following a data-centric approach, we utilize a database of Heusler alloys\ncalculated at the density functional theory level to identify the ideal ions neighbouring Fe in the\nX2FeZHeusler prototype. The hybridization of Fe with the nearest neighbour Xion is found to\ncause redistribution of the on-site Fe charge and a net increase of its magnetic moment proportional\nto the valence of X. Thus, late transition metals are ideal Fe neighbours for producing high-moment\nFe-based Heusler magnets. At the same time a thermodynamic stability analysis is found to restrict\nZto main group elements. Machine learning regressors, trained to predict magnetic moment and\nvolume of Heusler alloys, are used to determine the magnetization for all materials belonging to\nthe proposed prototype. We \fnd that Co 2FeZalloys, and in particular Co 2FeSi, maximize the\nmagnetization, which reaches values up to 1 :2 T. This is in good agreement with both ab initio\nand experimental data. Furthermore, we identify the Cu 2FeZfamily to be a cost-e\u000bective materials\nclass, o\u000bering a magnetization of approximately 0 :65 T.\nI. INTRODUCTION\nHeusler alloys, a vast family of ternary compounds, are\noften considered an ideal platform for engineering and\ndesigning novel functional materials. Such class includes\nboth metals and insulators, and among them supercon-\nductors, topological insulators, thermoelectric alloys, and\nboth optical and magnetic materials1,2. As such, the pos-\nsibility of using alloys of this family for \fne tuning and\ncontrolling the electronic structure and the magnetic or-\nder is tantalizing. However, despite several decades of\nintense investigation and accumulated understanding on\nthe Heuslers compounds3, the tuning of their properties\nstill proceeds via chemical intuition in a slow trial-and-\nerror mode. It is then an intriguing prospect to explore\nmore high-throughput methods for materials screening\nand understand whether these can identify novel design-\ning rules.\nReliable and low-cost computational methods now al-\nlow one to perform systematic investigations of large\nregions of the chemical space. This is known as the\ncomputational high-throughput approach4,5. The anal-\nysis of the generated data has lead data-mining and\nmachine-learning techniques to become part of the ma-\nterial science toolbox6,7(for a non-exhaustive list of cur-\nrently available materials databases see [8{13]). Mate-\nrials can be classi\fed by using descriptors14{16, simple\nproxies for sometime complex materials characteristics,\nand system properties estimated via machine leaning re-\ngression and classi\fcation17{20. The latter are particu-\nlarly useful when a direct calculation is prohibitive. An\napproach based on the machine learning uses statistical\ninference for predicting properties of a given system with-\nout performing an actual electronic structure calculation.\nThis enables a fast, objective and cost-e\u000bective analysis\nof large amounts of multi-dimensional data, making ma-\nFIG. 1. The local coordination of the atomic sites in a\nHeulser alloy. The neighbours of the central atom form two\nshells of di\u000berent symmetry. Atoms belonging to the nearest\nneighbour shell, shown in blue and magenta, coordinate the\ncentral atom tetrahedrally. The next nearest neighbour shell\nis made out of six (green) atoms and has octahedral symmetry.\nchine learning a natural extension of the computational\nhigh-throughput strategy.\nHere we use machine learning (ML) techniques to\nstudy the magnetism of Fe-containing Heusler alloys.\nIron o\u000bers a large magnetic moment, second only to Mn,\nbut in contrast to Mn that is known to maintain a high-\nspin state in Heusler compounds21, Fe is more suscepti-\nble to changes in the local chemical environment. This\nmakes it an ideal choice for exploring the predictive power\nof ML techniques, when applied to magnetic materials.\nIn this work we show that knowing the composition of\nthe \frst two Fe coordination shells is su\u000ecient to accu-\nrately estimate its magnetic moment using ML regres-arXiv:1706.01840v1  [cond-mat.mtrl-sci]  6 Jun 20172\nsion. By combining ML and density functional theory\n(DFT) data we are able to identify and explain trends\nin the compounds magnetic moment. Finally, the ideal\nprototype for an Fe-based regular Heusler alloy is pro-\nposed. The ML regression is used to rapidly characterise\nand rank all possible alloys of the proposed prototype,\naccording to the maximal attainable magnetization. We\nnote that all predictions are made by only using a list\nof Fe neighbours, without any need for additional ab ini-\ntiocalculations. This demonstrates the potential of the\nmachine learning approach for developing a fast, high-\nvolume, method for screening magnetic materials.\nThe paper is organized as follows. In the next sec-\ntion we describe the methods at the foundation of the\nmachine-learning process and the general attributes that\nenter into the description of magnetism in Heusler al-\nloys. Then we introduce our results focussing on the role\nof the nearest neighbour and of next nearest neighbour\ncoordination, and de\fning the physical origin of the mag-\nnetic moment trends. Then we use our machine learning\nscheme to identify magnets with large magnetization. Fi-\nnally we conclude.\nII. METHOD\nA. General Considerations\nStructure-to-property relations are at the heart of all\nproblems in the material science. These are implicitly\ndetermined by the electronic structure of any given com-\npound, which nowadays is routinely computed by using\nab initio methods. Accurate information about various\nmaterial properties can then be extracted solely from the-\nory. Machine learning (ML) allows us to take a rational\napproach to large-scale material investigation. The un-\nderlying assumption is that once there is enough materi-\nals data available, an answer to the structure-to-property\nquestion should be already implicitly contained in the\ndata. We can thus speak about \\learning from the data\".\nThe ML methods are built speci\fcally for this task, pro-\nviding us with a practical mean to construct approximate\nstructure-to-property relationship maps with a well de-\n\fned domain of validity. The latter, however, needs to be\nestablished through tests. A major advantage of the ML\napproach is that it thrives on large datasets, o\u000bering a\nhigh-throughput, objective analysis of the material prop-\nerties. The data are never discarded, but instead they\nare continuously integrated to re\fne the predictions and\ncan be reused to address new questions. Thus, it is an\ninductive, data-driven, approach of performing material\nresearch. The trade-o\u000b is that the ML results are usu-\nally less accurate than those obtained by using ab initio\nmethods.\nHere we focus on \\supervised learning\" methods,\nwhich include regression and classi\fcation algorithms.\nThe advantage of such class of schemes is that the quality\nof the ML predictions can be evaluated, for instance in\nPROBLEM\nTEST\nML Algo\nFeatures\nData \nSelection\nBuild Model\nAPPLICATIONa)\nb)c)\nd)\ne)FIG. 2. Schematic of the supervised learning strategy. The\napproach consists of 5 steps: a)de\fning the problem, b)\nselecting and pre-processing the data, c)building the ML\nmodel, d)testing the model and, \fnally e)applying the ML\nmodel to the problem of interest.\nthe case of the regression by calculating a mean-square\nerror. The purpose of the training procedure is then that\nof minimizing the risk of making incorrect predictions.\nThe idea is that a well-trained algorithm may produce\na signi\fcant error for an individual system but it shall\nperform in a satisfactory manner for the entire data set.\nAn outline of the general procedure used in this work\nis shown in \fgure 2. The process starts by de\fning the\nproblem and by selecting suitable data to describe it.\nThe data may come from di\u000berent sources and is usu-\nally combined into suitable input features. For example,\nin \fgure 1 we use a cluster of atoms to de\fne the mag-\nnetic moment of the central atom and various atomic\ndata (see equation 1) are considered to describe the clus-\nter further. Next, the available data are split into the\ntraining and the test dataset, according to the output\nproperty that one wishes to evaluate. Here it is crucial\nto preserve the underlying property distribution when\nthe data is split, otherwise one may end up with a biased\ntraining set. If the training set is biased, the ML model\nmay not be predictive for data outside it, and further-\nmore the \fdelity of the algorithm may be erroneously es-\ntimated. In brief the ML model trained on such a dataset\nwill usually perform badly on new data, i.e. it will not\nbe predictive. The test step is used as an independent\ncheck of the ML accuracy and its ability to generalize to\nnew data. The test dataset is never used for building the\nML model.\nThe model building phase, depicted as a single step\nin Fig. 2, is actually an iterative two-step procedure.\nFirst a set of input features, namely an input vector\nencoding a number of chemical/physical properties, is\nconstructed from the raw input data and then di\u000ber-\nent ML algorithms are trained using that input. The\nlatter step includes the choice of the, e.g., regression al-\ngorithm, and the optimization of the hyperparameters.\nThese two steps are repeated until a satisfactory accu-\nracy is achieved.\nThe choice of the input data and its transformation\ninto a useful set of input features is the most impor-3\ntant aspect of the entire process. In this step we use\nour domain knowledge to convert the raw data into de-\nscriptive features, which correlate with the output. This\ndescribes an inductive approach for constructing a ML\nmodel, which is feasible when we have some understand-\ning of the underlying processes correlating the input to\nthe output. In material science this should often be the\ncase. A minimal number of input features, which are\nincluded in the model, leaves a possibility of its inter-\npretation. Alternatively, one needs to follow a deductive\napproach. In this case one starts from a as-large-as pos-\nsible number of input features and performs an input\nreduction analysis, eliminating the variables that do not\ncorrelate with the output.\nThere is usually no a unique way to choose the input\nfeatures and interpret the ML model. If the dimension\nof the input space is not large one may seek to explore\nthe signi\fcance of the individual input features. This\ncan help in deepening the knowledge of the system un-\nder investigation and guide further ab initio calculations.\nHowever, such step is not always possible especially when\nthe dimension of the input space is large. The impor-\ntance of having a working ML description of the system\nis not diminished by this feature. It is often much more\npractical to explore the data using a ML algorithm than\nworking with the raw data. For example, in the case of a\nML regression, one deals with a single function (a map),\ninstead of a large database. Exploring the connection\nbetween the variables is thus much simpler and faster.\nThe added bene\ft is that the ML algorithm will often\naccurately interpolate where data is missing.\nB. Machine Learning Model for Magnetism\nIn this work we use data extracted from an in-house-\nmade Heusler alloy database, named Materials Mine8.\nAll calculations have been performed using the PAW22\npseudopotential implementation of DFT contained in the\nVASP code23{27and the generalized gradient approx-\nimation of the exchange and correlation functional as\nparametrized by Perdew, Burke and Ernzerhof28. For\neach chemical composition, the ground state is calcu-\nlated for di\u000berent site occupations, structural parameters\nand sublattice magnetic order. Furthermore, we comple-\nment the DFT data by various atomic properties infor-\nmation obtained from a wide range of sources in litera-\nture, including both experimental and theoretical data.\nThe sources will be properly cited individually whenever\nused.\nThe knowledge of the crystal cell volume, namely\nthe inter-atomic distance, is vital for studying electronic\nstructure properties and magnetism in particular. It is\nreasonable to expect that this quantity will repeatedly\nappear in all ML models and we wish to be able to pre-\ndict it without relying on the DFT data. We have then\ntrained a volume regressor using the DFT data for the\n229 fully relaxed full-Heusler structures, having the low-est energy for a given composition and site occupation.\nWe note that the number of di\u000berent compositions in the\ndatabase is larger than 229, but we restrict out choice\nto this dataset since the same was used with success for\nother investigations. In any case we will show that this\nchoice does not a\u000bect the quality of the \fnal result.\nThe ML model was built using the ridge regressor algo-\nrithm as implemented in Scikit-learn package29. We have\nused 30 % of the dataset for the test and the rest was used\nas training set. The input vector was constructed by in-\ncluding the atomic numbers, the atomic volumes, and\nthe atomic radii of the three nonequivalent ions. The\natomic volumes were obtained from Mentel30and then\nthe atomic radius was calculated for each element. Here\nwe assume atoms to be homogeneous solid spheres. A\nroot mean square (RMS) error on the test data was cal-\nculated to be 3 :16\u0017A3. In comparison, the mean volume of\nalloys in the dataset was \u001963\u0017A3. The attained accuracy\nof the ML algorithm is thus comparable to the precision\nof the DFT calculations, namely is a good predictor for\nthe volume.\nWe now wish to estimate the magnetic moment of Fe-\ncontaining Heusler alloys by using the ML approach. The\nmagnetic moment of 3 dtransition metals is well localized\nand can be understood as an atomic moment, which gets\nmodi\fed by its local surrounding. It is therefore inter-\nesting to try to relate the environment of an atom to\nits moment. The DFT data was used to construct clus-\nters of atoms representing the local environment of the\ncentral atom, as shown in \fgure 1. We construct one\ncluster for each atomic site of the parent alloy, namely\nwe construct 4 clusters per DFT calculation (per Heusler\nprototype). The data corresponding to the lowest energy\nstates having a given formula unit was selected from the\ndatabase. Note that here we consider a much larger, and\nless constrained, set of calculations than before. In fact,\nwe construct 18,268 clusters, of which \u00197,000 were used\nfor the test. The site projected magnetic moment of the\ncentral atom, obtained from the DFT results, was used\nas the target property. The input vector was constructed\nas\n~ vin= (fZig;R0;alat;fr0ig;fNig;S0); (1)\nwherefZig(i= 0;1;2;3) andRiare the atomic number\nand the atomic radius of the i-th atom, respectively, and\nNiis the valence ( i= 0;1;2). The atomic positions are\nlabeled as in Fig. 1. Here r0iis the distance between 0-\nth and the i-th atom, scaled by the sum of their atomic\nradii. The \\e\u000bective\" cubic lattice constant, alat, is cal-\nculated from the volume of the parent Heusler structure,\nwhich in turn is estimated using the previously discussed\nregression model. Finally, S0is the Stoner parameter of\nthe central atom, obtained from Janak31.\nAs for as the regression is concerned, we have found the\nRandom Forest Regression, as implemented in the Scikit-\nlearn library29, to give the best results. The RMS error\nmeasured on the test dataset is 0 :4\u0016B, with the RMS4\nerror on the subset having Fe as central atom being mea-\nsured somewhat higher, \u00180:54\u0016B. We did not notice\nany improvement in the RMS error when we performed\nthe training by only using Fe-centered clusters. We be-\nlieve there are two reasons for this \fnding. The \frst is\nthe reduced dataset size utilized. We \fnd that \u00187000\nclusters are barely su\u000ecient to converge the algorithm\nlearning curve. The second is the that the larger dataset\nalso contains non-magnetic atoms, whose magnetic mo-\nment is trivial to estimate. We note that in some cases\nthe errors are much larger than the RMS value, which\nseems to be at least partly related to the observed con-\nvergence issues in the high-throughput calculations. For\nexample, the DFT data is found to exhibit a large varia-\ntion in the magnetic moment of Fe (see Fig. 3). Reliable\nmethods for data cleaning are needed, however, in this\nwork we treat such anomalous calculations simply as a\nnoise.\nIII. RESULTS\nA. Role of the Coordination Shells\nIn the Methods section we have presented a ML model\nfor predicting the magnetic moment of an atom embed-\nded in a Fe-containing regular Heusler alloy (see \fgure 1).\nThe model estimates the magnetic moment based on four\nkey variables: the three atomic numbers specifying the\ncoordination of the central atom, fZig, and the lattice\nconstant of the parent Heusler alloy. The latter can be\nestimated by using a ML regression (see Table I). The\natomic properties, needed to construct the input vector\n(Eq. 1), are easily obtained. This makes the method com-\npletely free of input ab initio parameters. We then use\nthis model to explore and gain a deeper understanding of\nour DFT data. Here we focus on X2FeZalloys, where the\ncentral atom is Fe, with the cubic L2 1crystal symmetry\nand the corresponding tetragonal structures32. In this\ncase the two inequivalent sites in the \frst coordination\nshell are occupied by identical atoms (i.e. Z1=Z2=X\nin Fig. 1), a fact that allows us to explore the e\u000bects of\nthree structural parameters.\n1. Role of the Next-Nearest Neighbour\nWe \frst look at the composition of the second coor-\ndination shell, i.e. the e\u000bect of Z3of the magnetic mo-\nment of Fe, mFe(values are provided in unit of Bohr\nmagneton,\u0016B). The DFT dataset was sampled at a con-\nstant volume, namely the Wigner-Seitz radius was set\ntoRWS= 2:7a0, and the 8 nearest neighbour atoms\nwere \fxed to Fe. This Wigner-Seitz radius roughly cor-\nresponds to a Heusler lattice constant of 5 :8\u0017A. The cor-\nresponding data and the ML estimate of the magnetic\nmoment are shown in Fig. 3(a).\na)mFe  ( μ B )\n0.5\n1\n1.5\n2\n2.5\n3\n3.5\nb)\nc / a\n0.8\n1\n1.2\n1.4\n1.6\n1.8mFe  ( μ B )\n0.5\n1\n1.5\n2\n2.5\n3\n3.5\nZ\n3\n10\n20\n30\n40\n50FIG. 3. Estimate of the magnetic moment of Fe, mFe. a)mFe\n(in\u0016B) as a function of the next-nearest-neighbour atomic\nnumber,Z3, for a Wigner-Seitz radius of 2 :7a0. Thec=aratio\nof the parent Heusler alloy is color coded. A machine learning\nestimate is shown by a solid blue line. b) The same data as\nin a), where now the size of the symbolsais proportional\nto the calculated enthalpy of formation, \u0001 H. Large circles\ncorrespond to more stable alloys.\naSee Ref. [33] for details.\nVisual inspection of the data reveals that transition\nmetals (21 < Z 3<30, 39< Z 3<48) and main group\nelements (13 < Z 3<16, 31< Z 3<34, 49< Z 3<52)\nmake two distinct classes of next-nearest-neighbours. On\nthe one hand, transition metals tend to increase the mag-\nnetic moment of Fe proportionally to their valence. On\nthe other hand, main group elements tend to cluster and\nyield a maximal magnetic moment. In particular, for a\ngiven nearest neighbour and volume mFeis only weakly\na\u000bected by the choice of the main group element at the\nZ3site. This can be seen, for example, in \fgure 4, where\nthe magnetic moment of Fe appears not to be correlated\nto the atomic number of the next-nearest neighbour, Z3.\nHere we would like to point out that this does not contra-\ndict the well established Slater-Pauling rule3,34. In par-\nticular, for Co 2XYalloys this rule would imply that the\nnet cell moment, mcell, scales linearly with the valence\nof the alloy, NV, and reaches maximum when NV= 30.\nAs an example, for Co 2FeAl, Co 2FeSi and Co 2FeP alloys\n(NV= 29, 30 and 31) we \fnd the DFT moments, mcell,\nto be 5:10\u0016B, 5:48\u0016Band 4:68\u0016B(perf:u:), respec-\ntively. However, the corresponding Fe moments change\nonly slightly, namely we \fnd mFe= 2:77\u0016B, 2:80\u0016Band\n2:65\u0016B.\nWe note that the observed ML trend is volume de-5\npendent, giving a spurious representation of the valence\ntrend across the transition metal series. The problem,\nhowever, does not a\u000bect the main group elements. The\nML magnetization trend captured at smaller volumes,\nRWS\u00192:4a0, is qualitatively di\u000berent from the trend at\nlarger ones, RWS\u00192:8a0. In contrast, the DFT data\ntrends shown in Fig. 3 remain by large una\u000bected by\nthe volume change. We, therefore, \fnd it necessary to\ncombine the DFT data and the ML approach to obtain a\ncomplete picture. In spite of this, the numerical precision\nof the ML estimate is always within the limits established\nby the regressor test procedure, \u00180:5\u0016B.\nFrom a material design perspective, the learning is that\ntheZ3element can be chosen to ensure the stability of\na Heusler alloy without compromising the magnetic mo-\nment. Figure 3(b) shows the enthalpy of formation for\nthe same set of Heusler alloys shown in \fgure 3(a). The\nenthalpy of the alloy is calculated with respect to the\ndecomposition into the most stable elemental phases, so\nthat does not provide a strict stability criterion, but sim-\nply a guideline for stability35. With a small number of\nexceptions we \fnd that only the main group elements at\ntheZ3site have a good chance to yield thermodynam-\nically stable alloys. Such a result could be anticipated\nbased on the known Heusler chemistry3. The freedom\nof the choice of Z3opens up the possibility to tune the\nvolume of the alloy, and to control the critical tempera-\nture36,37. In Table I we show that Heusler alloy volume\ncan be accurately estimated using the ML regression.\n2. Role of the Nearest Neighbour\nIn the previous discussion we have shown that the mag-\nnetic moment of Fe is independent of the choice of Z3as\nlong asZ3is a main group element. We now study the\nmagnetic moment as a function of the nearest neighbour\nion,Z1, keeping a main group element at the Z3site\nand a \fxed volume (see Fig. 4). The trend with volume\nofmFedepends on the choice of Z 1and it is di\u000ecult to\nqualify. The ML regression, however, can be used to take\nthe volume e\u000bects into account with a good level of pre-\ncision. We \fnd that the valence of the nearest neighbour\nions determines the magnetic moment of Fe for the en-\ntire range of volumes investigated. When the valence of\nthe nearest neighbour ion is less than 8 the moment de-\ncreases, and conversely, when it is larger it increases. The\nsame trend is found for all nearest neighbours belonging\nto the 3dand the 4dtransition metal series, as shown\nin Fig. 4. The maximal moment of Fe is obtained when\nNi or Pd constitute the \frst coordination shell. The ML\ntrend, shown with a blue line, reproduces the magnetiza-\ntion trend for all possible nearest neighbours. For main\ngroup neighbours the magnetization shows a strong vari-\nation with the valence, taking a minimum value in the\nmiddle of the series. The overall magnetic moment is\nthen reduced when compared to the situation with tran-\nsition metal elements . In conclusion, we have found that\n13\n14\n15\n16\nZ\n3mFe (μ B )\n0.5\n1\n1.5\n2\n2.5\n3\n3.5\n4\nZ\n1\n10\n20\n30\n40\n50\n20\n30\n40\nZ\n3\nZ\n1\n10\n15\n20FIG. 4. Left panel - Magnetic moment moment of Fe, mFe\n(in\u0016B), for a wide range of nearest neighbours at a constant\nWigner-Seitz volume ( RWS= 2:7a0). The atomic number of\nthe next nearest neighbour, Z3, is color coded, while here we\nplot data as a function of the \frst nearest neighbour atomic\nnumber,Z1. We can notice a linear increase of the mag-\nnetic moment across the transition metal series which does\nnot depend on Z3. The symbol are the DFT data while the\ncorresponding machine learning trend is shown with the blue\nline. Right panel - a data sample containing a wider range of\nmain group elements. The data elucidates the origin of the\noscillation in the machine learning trend throughout the main\ngroup series.\nlate transition metals: Co, Ni, Cu, Rh, Pd, and Ag, make\nthe most desirable nearest neighbours of Fe, since they\nmaximize its local magnetic moment.\n3. Physical Origins of the Magnetic Moment Trends\nIn order to understand the physical origin of the es-\ntablished trend in the transition metal series, we select\na subset of the compounds shown in Fig. 4 for further\nanalysis. In particular we look at the following Fe-\ncontaining alloys: Sc 2FeSi, Ti 2FeAl, V 2FeSi, Cr 2FeSi,\nMn2FeS, Fe 2FeAl, Co 2FeAl, Ni 2FeSi and Cu 2FeSi, which\nall possess a cubic L2 1structure. We note that, in gen-\neral, the tetragonal distortion does not change the main\ntrend, so that it is not considered here. An analysis of the\nsite projected density of states (PDOS) reveals that both\nthe Fe and the nearest neighbor atom remain charge neu-\ntral throughout the series. This means that Fe is always\noccupied by 6 electrons and the origin of the magnetic\nmoment trend is solely due to the on-site charge redistri-\nbution, as shown in \fgure 5. By integrating the orbital\nresolved Fe PDOS we \fnd that for early TMs ( ZTM\u001425)\nthe amount of charge transferred from the minority t2g\nband to the egspin bands is proportional to the near-\nest neighbour valence, with the charge of the majority6\na)\nm\nFe\nm\ne\ng\nm\nt\n2g m (μ B )\n0\n1\n2\n3\n4\nZ\nTM\n22\n24\n26\n28\nb)\n-\nΔ\n Q\nt\n2g\n  ↓\n \nΔ\n Q\ne\ng\n \nΔ\n Q\nt\n2g\n  ↑\n \nΔ\n Q\nFeΔ Q (e )\n-1.0\n-0.5\n 0.0\n 0.5\n 1.0\nZ\nTM\n22\n24\n26\n28\nFIG. 5. a)Orbital resolved magnetic moment of Fe, mFe,\nfor TM 2FeZ clusters, where the selected TMs are 3 delements\n(ZTM= 21 toZTM= 29). The magnetic moment of the e g\nband,meg, is roughly constant throughout the series, while\nthe total moment, mFe, increases following the increase of\nthe t 2gmoment,mt2g.b)Orbital and spin resolved change\nof the Fe site projected charge, \u0001Q Fe, throughout the TM\nseries. As a reference we take the site projected charges of\nSc2FeSi (ZTM= 21). For the t 2gminority spin band we plot\nthe charge loss, \u0000\u0001Qt#\n2g, to clearly show its correlation with\nthe charging of the Fe egband, \u0001Q eg.\nt\n2gDensity of States  (States / eV)\n−4\n−2\n0\n2\n4\nE - E\nf\n  (eV)\n−4\n−2\n0\n2\n4\ne\ng\nZ\n1\n = 21\nZ\n1\n = 23\nZ\n1\n = 27\nE - E\nf\n  (eV)\n−4\n−2\n0\n2\n4\nFIG. 6. Projected density of states of Fe in the TM 2FeZ\nclusters, where the selected TMs are Sc ( Z1= 21), V (Z1=\n23) and Co ( Z1= 27). The t 2gband is shown in the left-hand\nside panel and the e gin the right-hand side one. The spin up\nand down channels are shown as the positive and the negative\nvalues, respectively.\nt2gband, located deep below the Fermi level, remaining\nroughly the same. As a result, the Fe net magnetic mo-\nment increases proportionally to the valence of the near-\nest neighbour. For Mn, Fe and Co nearest neighbour we\nobserve a reverse charge transfer, from the minority eg\nband to the majority t2gone, \u0001Qt\"\n2g\u00190:4e, resulting in\nan noticeable kink in the magnetic moment trend.\nOur understanding of such charge re-distributionTABLE I. The volume and the magnetic moment of Fe for\na number of Heusler alloys. The machine learning (ML) esti-\nmates are compared to the DFT results. Volumes are given\nfor the primitive unit cell containing 4 atoms. The length\n(volume) is expressed in \u0017A (\u0017A3) and the magnetic moments\nare in\u0016B.\nCompound ajjc=a VDFT VML MDFT MML\nCo2FeSi 5.63 1.2 44.49 45.86 2.79 2.69\nCu2FeAl 5.51 1.2 50.20 45.44 2.52 2.67\nRh2FeSn 5.89 1.2 61.21 59.74 3.13 3.12\nNi2FeAl 5.39 1.0 47.02 43.77 2.69 2.69\nNi2FeGa 5.38 1.2 46.70 47.31 2.73 2.81\nmechanism is the following. At the begining of the transi-\ntion metal series, the nearest neighbour atoms hybridize\nweakly with the t2gband of Fe, resulting in a narrow and\nstrongly spin split Fe t2gband (see Fig. 6). As such the\nenergy overlap of the Fe t2gspin bands is initially very\nsmall. By increasing the valence of the nearest neighbour\nion one leads to a stronger hybridization with the Fe t2g\nspin bands, which then get wider. Consequently, their\noverlap increases, giving rise to an increasingly strong\nrepulsive Coulomb interaction. It is therefore energeti-\ncally more favourable to transfer a part of the minority\nt2gcharge to the egband, which is strongly spin-split and\n5 eV to 6 eV wide in energy, indicating spatially delocal-\nized orbitals. For late TM neighbours, e.g. Co ( Z1= 27),\nthe hybridization with Fe results in a wide majority d-\nband, which can now accommodate extra electrons since\nthe Coulomb repulsion is reduced by the band broad-\nening. At the same time the energy cost of adding ex-\ntra electrons to the egband is increased, as the band is\nnearly full. For Mn neighbours ( Z1= 25) we \fnd the Fe\negcharge to be Q eg\u00192:5e. The reverse charge transfer\nthus reduces the Coulomb energy of the egband. The\nd-band electronic structure of Fe in Sc 2FeSi, V 2FeSi and\nCo2FeSi alloys, which clearly illustrate the mechanism\njust described, is shown in Fig. 6.\nB. Application\n1. Screening of High-Magnetic-Moment Heulser alloys\nusing ML Methods\nThe analysis carried out on the dependence of the Fe\nmagnetic moment on the local chemical environment en-\nables us to propose LTM 2FeMG as an optimal chemical\ncomposition for a ternary Fe-based Heusler alloy with\nmaximummFe(LTM stands for late transition metal and\nMGfor main group element). We stress that our asser-\ntion applies only to L2 1-type Heusler alloys, but notably\na well-established preferential site occupation rule32, sug-\ngests that the proposed stoichiometry will crystallize in\nthe required regular Heusler phase. There exists a num-\nber of Heusler alloys reported in literature, which belong7\nm\nFe\nm\ncellmML (μ B )\n0\n1\n2\n3\n4\n5\n6\nm\nDFT\n  (\nμ\nB\n)\n0\n1\n2\n3\n4\n5\n6\nFIG. 7. Comparison between the magnetic moments of\nLTM 2FeMG Heusler alloys predicted by the ML regression,\nmML, and those calculated with DFT, mDFT. The magnetic\nmoment of Fe, mFe, is predicted directly using the regression\n(blue dots). The total magnetic moment per cell, mcell, (green\ndots) is estimated by using the empirical correction scheme\ndescribed in the text. The red line denotes perfect agreement,\nmML=mDFT.\nto the proposed prototype. For example, Co 2FeSi is a\nwell known ferromagnetic half-metal with a critical tem-\nperature of 1100 K38. Other examples of related Heusler\nalloys include: Co 2FeAl, Cu 2FeAl, Ni 2FeAl, Ni 2FeGa\nand Rh 2FeSn39{43, proving that the proposed prototype\nhas a good chance to yield thermodynamically stable al-\nloys.\nThe structural and the magnetic properties of these\nalloys can be predicted by using the machine learning\nregression. In Table I we compare the ML results for\nthe volume and the Fe magnetic moment for the afore-\nmentioned alloys with the corresponding DFT values ex-\ntracted from the DFT database8, demonstrating indeed\na good agreement. We note that the method presented\nhere can only be used to directly evaluate the magne-\ntization of alloys containing a single Fe atom and no\nother magnetic elements. When other magnetic ions are\npresent in the composition the ML method will in gen-\neral underestimate the total cell magnetic moment, and\nit will need to be extended to take the magnetic ordering\ninto account. The fact that most of these Heusler type\npresent ferromagnetic ordering (a ferromagnetic ground\nstate is found to be stable over the entire range of vol-\numes, namely RWS= 2:3a0to 3:0a0) allows us to easily\naccount for the additional magnetic atoms. In fact the\ntotal moment per cell can be obtained by simply adding\nthe average magnetic moment of the LTMs to the mag-\nnetic moment of Fe.\nHere among the LTMs only Co, Ni, Rh, and Ir are\nfound to have appreciable magnetic moments and their\naverage values, \u0016 m, have then been estimated by using the\nsite-projected magnetic moment data of various Heusler\nalloys found in the database. We estimate the follow-\ning average moments: \u0016 mCo= 1:19\u0016B, \u0016mNi= 0:37\u0016B,\n\u0016mRh= 0:34\u0016Band \u0016mIr= 0:35\u0016B. The remaininglate transition metals tend to be either non-magnetic or\nweakly magnetic, leaving Fe as the only source of mag-\nnetic moment.\nWe have then used the method described to char-\nacterize all the possible compounds of the proposed\nLTM 2FeMG prototype. The main group elements have\nbeen chosen among: Al, Si, P, Ga, Ge, As, In, Sn and Sb,\nand the late transition metals among: Co, Ni, Cu, Rh,\nPd, Ag, Ir, Pt and Au. The magnetization has been cal-\nculated for each Heusler alloy by using the ML estimate\nof the magnetic moment and the volume. The results\nhave been compared a posteriori with the DFT ones and\nfound to be in a good agreement, see Fig. 7. The error\nfor the magnetic moment is below 0 :5\u0016Bf:u:\u00001for all\nthe alloys considered. We have also found that Rh and\nIr are the two ions, which allow one to maximize the Fe\nmagnetic moment, reaching out a value of 3 \u0016B=atom.\nHowever, the maximal cell magnetization of 1 :2 T was\nachieved in Co 2-based magnets, with Ni 2- and the Cu 2-\nbased based alloys following and having a magnetization\nof 0:83 T and 0:65 T, respectively.\nIV. CONCLUSIONS\nIn conclusion we have investigated the magnetic mo-\nment of Fe in Heusler alloys and its dependence on the\nlocal chemical environment. We have identi\fed the va-\nlence of the Fe neighbours as the key parameter govern-\ning the moment. The LTM 2FeMG prototype has been\nfound to be the ideal for an Fe-based ternary Heusler\nalloy with maximum magnetization. By using machine\nlearning algorithms we have estimated the volume and\nthe magnetic moment for the entire family of such com-\npounds, and the alloys have been ranked according to\ntheir performance, namely the maximal magnetization.\nWe \fnd Co 2FeSi and Co 2FeAl at the top of our list.\nThese are a well known high-performance magnets for\nspintronics38,39. For large-scale production or in appli-\ncations as permanent magnets, where the performance is\nmeasured in magnetization per dollar, Cu 2-based mag-\nnets, such as Cu 2FeAl40, become the best choice. Fi-\nnally, we have demonstrated that machine learning can\nbe used as a cost-e\u000bective and reliable method for mate-\nrial characterization. We have also shown that combining\nmaterial informatics and high-throughput DFT calcula-\ntions makes a powerful platform for accelerated materials\nresearch.\nACKNOWLEDGMENTS\nThis work has been funded by Science Foundation\nIreland (Grant No. 14/IA/2624 and AMBER Cen-\nter). 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Suits, Solid State Comun. 18, 423 (1976)."    },    {        "title": "1706.05177v1.Observation_of_Various_and_Spontaneous_Magnetic_Skyrmionic_Bubbles_at_Room_Temperature_in_a_Frustrated_Kagome_Magnet_with_Uniaxial_Magnetic_Anisotropy.pdf",        "content": "  Submitted to  \n1 \n \nDOI: 10.1002/((please add manuscript number))  \nArticle type: Communication  \n \nObservation of  Various  and Spontaneous Magnetic Skyrmionic Bubbles at Room -\nTemperature in a Frustrated Kagome Magnet with Uniaxial Magnetic Anisotropy  \n \nZhipeng Hou*, Weijun Ren*,Bei Ding*, Guizhou Xu, Yue Wang, Bing Yang, Qiang Zhang, \nYing Zhang,  Enke Liu,  Feng Xu, Wenhong Wang, Guangheng Wu, Xi -xiang Zhang, Baogen \nShen, Zhidong Zhang  \n Dr. Z. P. Hou, B. Ding, Y. Wang, Dr. Y. Zhang, Dr. E. K. Liu, Prof. W. H. Wang, Prof. G. H. \nWu, Prof. B. G. Shen  \nBeijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China  \nE-mail: wenhong.wang@iphy.ac.cn (W. H. Wang)  \nDr.W. J. Ren, Dr. B. Yang, Prof. Z. D. Zhang  \nShenyang Materials Science National Laboratory, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China  \nE-mail:wjren@imr.ac.cn  (W. J . Ren)  \nDr. G. Z. Xu , Prof. F. Xu  \nSchool of Materials Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China  \nDr. Qiang Zhang, Prof. X. X. Zhang King Abdullah University of Science and Technology (KAUST), Physical Science and \nEngineering  (PSE) , Thuwal 23955- 6900, Saudi Arabia  \n [*] Z.P.H, W.J.R, and B.D contributed equally to this work.  Keywords:  skyrmion ic bubbles , topological spin textures , kagome  magnet , Fe\n3Sn2, spintronic \ndevices  \n \n \nThe quest for materials hosting topologically protected nanometric spin textures, so- called \nmagnetic skyrmions or  magnetic skyrmionic  bubbles, continues to be fuelled by the promise \nof novel devices .[1-5] The skyrmionic spin textures  have  been mostly  observed in non-\ncentrosymmetr ic crystals , such as  the cubic chiral magnets MnSi,[6-9] (Mn,Fe)Ge,[10,11]  \nFeCoSi ,[12] Cu2OSeO 3,[13-15] and also  the polar magnet  GaV 4S8,[16] where Dzyaloshinskii -\nMoriya interaction (DMI)  is active . A number of intriguing electromagnetic phenomena, \nincluding the topological Hall effect,[7] skyrmion magnetic resonance,[17] thermally induced \nratchet motion,[18] and effective magnetic monopoles,[19] have been demonstrated  to be closely \nrelated to the topologically nontrivial spin texture of skyrmions. T hese novel topological \nproperties , together with nanoscale dimensions, a stable particle -like feature, and an ultralow \nthreshold for current -driven motion, make magnetic skyrmions  fundamentally promising for \napplications  in next -generation high -density and low-dissipation memory devices .[1-5]   Submitted to  \n2 \n \nHowever, although a very recent  publication by Tokunage et al .[20]  reported the observation \nof skyrmion lattices above room temperature  (RT) in β–type Cu -Zn-Mn alloys, skyrmions in \nthe bulk chiral magnets  have been mostly observed below RT .[6-16] Moreover , the spin texture \nof skyrmions  that were stabilized by DMI  in chiral  magnets  is quite limited.  The s kyrmion s \nwith variable spin textures  may be more attractive for further  technological applications  \nbecause they can adapt to various external stimuli act ing as information carriers in  spintronic \ndevices . Therefore,  one particularly important current research direction aims at the discovery \nof new materials that host  skyrmions  with variable  spin textures  at room temperature.  \nIn addition to the non- centrosymmetric chiral magnets  in which the magnetic  skyrmions  are \nstabled  by DMI , the centrosymmetric materials with uniaxial magnetic anisotropy (UMA)  are \nanother family of materials that can host skyrmions.[21-26] In these materials, the competition \nbetween the magnetic dipole interaction and uniaxial easy axis anisotropy is the key force  in \nthe formation of skyrmion s.[21-26] The skyrmions in centrosymmetric materials  are \ntopologically equivalent to those  in the chi ral magnets[21-26] but posse ss two degrees of \nfreedom, i.e. helicity and vorticity .[5] Since their internal  degree of freedom  is similar to that  \nin the topologically trivial magnetic bubbles , they are also called skyrmionic bubbles .[24] The \nmost interesting physics  in skyrmionic bubbles  is that the vorticity of the ir spin textures  varies  \nwith the internal structure of the Bloch lines (BLs) , resulting  in a variety of spin textures .[22-26] \nFor instance,  a new type of spin texture formed by two skyrmions  with opposite spin vorticity \n(the topological number  equals 2), called  a biskyrmion, has been experimentally discovered in \nthe centrosymmetric  tetragonal magnetite La1-xSrxMnO 3(x=0.315)[23] and hexagonal (Mn 1-\nxNix)65Ga35(x=0.5)[25] at temperatures  around 60 K and 300K, respect ively. More recently , Yu \net al.[24] have found a variety of spin textures  in magnetic skyrmionic bubbles in orthorhombic \nmagnetite  La1-xSrxMnO 3(x=0.175)  at 100K. Thus, the  multifarious topological nature  of \nskyrmionic bubbles offers us an opportunity  to manipulate  their topological spin textures  \nthrough external stimuli.  \nRecently, the domain structure s in the  frustrated magnet with tunable UMA  were studied \nnumerically by Leonov et al [27]. They sh owed that the UMA strongly affects spin ordering. \nThey also predicted that different spin structures, including isolated magnetic skyrmions, may \ncoexist in a frustrated magnet,  and that the isolated magnetic skyrmion s also possess \nadditional degrees of freedom  (spin vorticity  and helicity ), similar to that in the magnetic \nskyrmionic bubbles . Their predictions are not only extremely interesting, but also point to \nfurther investigat ion of  the variable topological spin textures  in the frustrated magnets . \nFurthermore,  Pereiro  et al .[28] have theoretically  shown that Heisenberg and DMI   Submitted to  \n3 \n \ninteractions in kagome  magnets can overcome  the thermal fluctuation  and stabilize  the \nskyrmions at relatively high temperature s, even at room temperature . Based on the theoretical \ninvestigations above, we revisit the frustrated magnet s with kagome lattice to search for the \npossible skyrmionic bubbles that are able to stabilize  at room temperature.   \nOne of the promising materials is Fe 3Sn2, suggested  by Pereiro et al.,[28] which  has a \nlayered rhombohedral structure with alternate stacking of  the Sn layer  and the Fe -Sn bilayers  \nalong the c -axis, as shown in Figure 1 a. The Fe atoms form bilayers of offset kagome \nnetworks, with Sn atoms  throughout the kagome layers as well as between the kagome \nbilayers. Very importantly, this material is a non- collinear frustrated ferromagnet with a high \nCurie temperature T c of 640K, and shows a spin reorientation that  the easy  axis rotates \ngradually from the  c-axis to the ab- plane as the temperature decreases.[29-32] Recently, a large \nanomalous Hall effect was  observed in this material, which is strongly related to the frustrated \nkagome bilayer of Fe atoms.[33,34] In this communication, we report that  magnetic skyrmionic  \nbubbles with various spin textures  can indeed be realized in  the single crystals of Fe 3Sn2 at \nroom temperature . The emergence of skyrmionic bubbles and the magnetization dynamics \nassociated with the transition of different bubbles via the field -driven motion of the Bloch \nlines are  revealed by in -situ Lorentz transmission electron microscopy  (LTEM) , and further  \nsupported by the  micromagnetic simulations and magnetic transport measurements. The se \nresults demonstrate that Fe 3Sn2 facilitates a unique magnetic control of topological spin \ntextures at room temperature, making it a promising candidate for further  skyrmion -based \nspintronic devices.  \nHigh -quality single crystals of Fe 3Sn2 were synthesized by a Sn -flux method, as describ ed \nin the Methods section. These crystals are layered and exhibit mirror -like hexagonal faces \nwith a small thickness  (see Figure S1 , SI). By using single -crystal X -ray diffraction (SXRD), \nthe crystal parameters were identified as a = b = 5.3074Å  and c = 19.7011Å , with respect to \nthe rhombohedral unit cell (space group R -3m), agreeing well with the previous studies.[32-34] \nHaving determined the unit lattice parameters and orientation matrix, we then found that the  \nhexagonal face was  normal to [001] with the (100), (010), and (110) faces around (see Figure \nS2, SI). In addition, as shown in Figure S3 , both the  temperature -dependent  in-plane and out -\nof-plane magnetization curves measured on the bulk crystal indicate that the ferromagneti c \ntransition temperature  Tc is about 640K . \nFigure 1b  shows the temperature -dependent UMA coefficient  (Ku), which was estimated \nby the approximation of K u=HkMs/2, where M s is the saturation magnetization  and Hk is the \nanisotropy field  defined as the critical field above which the difference in magnetization   Submitted to  \n4 \n \nbetween the two magnetic field directions (H//ab and H// c) becomes smaller than 2% (see \nFigure S4 , SI). One can notice that the value  of Ku  increases  monotonically  with the decreas e \nof temperature. Simultaneously, the Fe moments gradually rotate from the c -axis towards the \nab-plane (see the inset of Figure  1b), demonstrating a gradual transformation from the \nmagnetically easy axis (uniaxial magnetic anisotropy)  into a  magnetically eas y plane with \ndecreasing the temperature. The most important message conveyed to us by Figure 1b  is that \nthe magnetic domain configuration in Fe 3Sn2 may vary over  a very wide temper ature range of \n80-423K , because the domain structure is strongly  affected  by the magnetic anisotropy. \nTherefore, Fe 3Sn2 should be a good platform for us to explore the correlation between the spin \ntexture and K u in a very wide temper ature range.  \nWe then imaged the magnetic domain structure s using LTEM under zero magnetic field in \nthe temperature range of 300K ~130K, as shown in Figure 1c- f. The corresponding selected-\narea electron diffraction (SAED) patterns suggest that the sample is normal to the [001] \ndirection (see the inset of Figure 1b ). Nanosized stripe domains with an average periodicity \nof ~150nm  were clearly observed . Notably, the value of periodicity  is comparable to that in \nthe bulk (Mn 1-xNix)65Ga35(x=0.5) ,[25] but nearly two times larger than that  in the La 1-\nxSrxMnO 3(x=0.175).[24] The sharp contrast between the dark stripe domains and the bright \nwalls  suggests that the doma ins possess out -of-plane magnetizations and are separated by \nBloch domain walls. With the decrease of temperature, the stripes ’ periodicity l widened \nwhile the d omain wall thickness D  remained  almost a constant (see Figure S5 , SI). \nInterestingly, we found that when the temper ature was lower than 130K, the stripe domains  \ndisappeared and vortex domains formed, indicating that the spin starts to lie into the ab- plane \nbelow 130K . The critical temperature of the LTEM sample co incides with that  of the bulk \nsample , but with a slight deviation .[32-34] This feature suggests that  the spin texture of domains \nin the bulk and LTEM samples show little differences. Th at can be attributed to the fact that \nthe magnetic anisotropy in Fe 3Sn2 is high enough to over come the spin rearrangement effect \nresulting from the increase of demagnetizing energy in the thin LTEM sample. To understand \nthe physics behind this domain structure t ransformation, we performed numerical simulations \nwith estimated parameters of exchange constant ( A) and anisotropy energy  (Ku⊥) associated \nwith the perpendicular component of the anisotropy field (see Methods). As shown in Fig ure \n1g, the stripe domain gradually transformed into a vortex domain with decreasing  Ku⊥, \nagreeing well with the LTEM images. This feature suggests that the domain morphology  in \nFe3Sn2 is mainly governed by the anisotropy perpendicular  to the ab-plane , which is \nconsistent with the simulated results based on a frustrated magnet .[27] It is well known that  the   Submitted to  \n5 \n \nmagnetic domain structure also depends greatly on an ext ernal magnetic field . Therefore , we \nhave simulated the domain structure under different magnetic fields that are perpendicular to \nthe ab-plane  (as shown in Figure 1h). It is interesting  to note that the stripe domains \ngradually transformed into bubbles with increasing the external field. Mo re strikingly, isolated \nmagnetic skyrmions formed when the magnetic field increased to 400 mT.  \nTo experimentally observe the domain structure variation with a n external  magnetic field \npredicted by the simulation, we imaged the domain structure under different magnetic fields \nat room temper ature using LTEM. Figure 2 a-d show s the over-focused  LTEM images under \ndifferent out -of-plane magnetic fields (the corresponding zero- field image is shown in Figure \nS6, SI). The gradual transformation from a  stripe domain structure into magnetic skyrmionic  \nbubbles is clearly observed as the magnetic field increases from 0 to 860mT. In Figure 2a,  we \nshow a snapshot of the transformation process under a magnetic field  of 300mT . One can \nnotice that  the stripe  domains, dumbbell -shaped domains and magnetic bubbles coexist in the \nimage . We actually observed that, during the evolution of the domain structures, the stripe  \ndomains  gradually broke into the dumbbell -like domains first before evolving into the \nmagnetic bubbles. When the magnetic field increased above 800m T, all the stripe s and \ndumbbell -like domains completely transformed into magnetic bubbles. One should note that \nwe o nly changed the external magnetic field and kept all other conditions constant in Fig. 2a-\nd. Therefore, the changes in domain structure can be ent irely ascribed to the exter nal field \neffect.  A closer analysis  of the entire transformation process reveals that the spin texture of \nthe magnetic bubbles changes dramatically  with increasing the magnetic field, as shown in \nFigure  2e-h (the bubbles with different spin textures are notated by different numbers). The \nstruct ural evolution of the magnetic bubbles with increasing external magnetic field should be \nclosely related to the change of topology in  the bubble s, as previously observed.[24,26]  \nTo characterize the topological spin textures of the magnetic bubbles, a transport -of-\nintensity equation (TIE) was employed to analyze the over - and under -focused LTEM images. \nFigure 2i -l display s the spin textures of the bubble domains shown in Figure 2e- h. The white \narrows show the directions of the in- plane  magnetic inductions , while the black regions \nrepresent the domains with out -of-plane magnetic inductions . Bubble “1”, composed of a pair \nof open Bloch lines (BLs), is characteristic of the domain structure com monly observed in \nferromagnetic compounds with uniaxial magnetic anisotropy. In this bubble, the topological \nnumber N is determined to be 0, because the magnetizations of the BLs are nonconvergent. \nFurther increase of the magnetic field induced the formation of bubble “2”, which has two arc-shaped walls with opposite helicity, separated by two BLs. Similar to that of  bubble  “1”,   Submitted to  \n6 \n \nthe spin texture  of bubble “2” is  also not convergent, leading to the same topological number, \nN=0. H owever,  when the  magnetic fi eld increased above 800mT, the topological number of \nbubbles “3” and “4” transforms from 0 to 1, being equal to that of skyrmions. As shown in \nFigure 2k , bubble “3” has a pendulum structure with two BLs, in which the spin textures \nbecome conver gent. Compared with the recent  results obtained in La 1-xSrxMnO 3(x=0.175) by \nYu et al,[24] bubble “3” can also be regarded as a specific type of rarely observed skyrmionic \nbubble. Bubble “4” is the most orthodox  skyrmionic bubble , possessing the same domain \nstructure as the skyrmions observed in chiral magnets. One important feature of this bubble is \nthat the thickness of the Bloch wall is comparable to the radius of the bubble, leading to a small core region. Following previous reports,\n[24] we understand that bubble “3” transformed \ninto bubble “4” through the motion of BLs driven by the magnetic field. Therefore, although \nthe magnetic textures in bubbles “3” and “4” are strikingly different, they are homeomorphi c \nin topology. To understand the above transformations of spin texture in more detail, we \nsuccessfully recorded the transformation process from the topologically nontrivial magnetic \nbubbles to the topologically protected skyrmions in a Fe 3Sn2 (001) thin- plate using in -situ \nLTEM  (see Supplementa ry Movie , SI). Figure 3 a-f presents several  snapshots of the \ntransformation process, selected from a  movie taken by LTEM at 300 K and under different \nout-of-plane magnetic fields. We observed a  pair of BLs move  along the bubble wall and \neventually die out  within 8.3 seconds , as the field increased from 840mT to 850mT.  These \nresults demonstrate clear ly that isolated  magnetic  skyrmions can be realized  in the frustrated  \nFe3Sn2 magnet through BL motion by tuning the external magneti c field, even at room \ntemperature.  \nIn addition to the observation of various spin textures  of skyrmionic  bubbles during BL \nmotion, we further found that the magnetic  domain configurations after turning  off the \nexternal fields depend strongly on the strength of the external fields. If the thin- plate s ample is \nfirst magnetized to saturation  (i.e. the sample is in  single -domain state ), then the domain will \nreturn to the multi- stripe  state after turning off the external magnetic field ( see Figure S7 , SI). \nHowever, if the sample  is magnetized to an  intermediate state  (for example, at a field of \n700mT  as shown in Figure S8 , SI), the domain structure evolves differently  after turning off \nthe external magnetic field.  Figure 4 a shows the under -focused LTEM image taken at 300K  \nafter turning off the external field of  700mT by which the sample was magnetized to an \nintermediate state. The coexistence of different types of magnetic domains, e.g. magnetic \nbubbles, stripes and dumbbell -like domains is clearly observed. After careful analysis of the \nimage using TIE, the detailed spin texture of the domain enclosed by the  square  in Figure  4a   Submitted to  \n7 \n \nis shown in Figure 4e. Unexpectedly, the magnetic bubbles possess three concentric rings . It \nis particularly interesting that the winding directions of  the inner and outer rings are opposite \nto that of  the middle ring, indicating that the helicity reverses inside the bubbles. These  \nspontaneous bubbles  can be considered skyrmions, analogous to those  observed in  BaFe 12-x-\n0.05ScxMg 0.05O19.[22] \nTo explore the evolution of the domain structure with temper ature, the domain structures  in \nthe same region as shown in Figure 4a  were also imaged at several lower temperatures  \nwithout changing the external  field ( Figure 4b -d). These results help us to understand how \nthe domain structure evolve s under the influence of different magnetic energies, i.e. magnetic \nanisotropy, exchange energy , and static magnetic energy.  As the temperature decreases , the \nsize of the  spontaneous bubbles change slightly  and the stripes gradually transform into \nskyrmions. Consequently, the number of spontaneous bubble s increases  with decreasing \ntemperature. The maximum density of the bubbles appear s at 250K. The spontaneous bubbles \nvanish as the temperature further decrease below 100K , due to the transformation from \nuniaxial, out -of-plane anisotropy into in -plane anisotropy ( Figure  1b). The corresponding \nspin textures of the skyrmions  marked in Figure 4b -d were  also extracted by using  TIE \nanalysis, as shown in Figure 4f -h. It was found that the size of the innermost ring  increases \nwith decreasing the temperature. Micromagnetic simulations are currently ongoing  in order to \nexplore the interplay  among the different magnetic parameters behind t hese features.  \nThe formation of  magnetic skyrmionic  bubbles and isolated skyrmion spin textures  in the \nbulk Fe 3Sn2 single crystals were further studied and confirmed by magnetic and magneto -\ntransport measurements, similar to previous studies of other skyrmion -based \nmaterials .[7,20,25,35] It should be noted here that t he sample  for magnetic and magnetotransport \nmeasurements is from the same crystal s used for LTEM  observations . In Figure  5a, we  show \nthe dependence of magnetoresistance (MR) on the magnetic field ( H) that is applied normal  to \nthe ab-plane in the temperature range of 100 -400K. The inset shows the details of the MR -H \ncurve at 3 00K,  in which two broad peaks are clearly seen at about 200 mT and 800mT , \nrespectively. Based on the LTEM resul ts shown in Figure  2 and our analysis, it can be  \nconclude d that the peak at 200mT  (Ha) reflects the starting point of the transformation from \nstripes to magnetic bubbles, whereas the peak  at 800mT ( Hm) represents the starting point of \nthe transformation from  magnetic bubbles to ins ulated skyrmions. When the magnetic field \nincreases above 900mT ( Hc), the sample reaches the magnetically saturated state (see the M -H \ncurves in Figure  S4) in which  the spins are aligned along the field direction , and \nconsequent ly all the skyrmions  die out . A close inspection of the MR -H curves in the range of   Submitted to  \n8 \n \n400-100K reveals that all three critical fields shift monotonically with temperature as \nindicated by the dotted lines. As shown in Figure 5b , we can also identify the three critical \nfields in the field -dependent AC-susceptibility, though the critical fields are slightly lower \nthan those observed in the MR results and show slightly weaker  temper ature dependence. \nBased on the LTEM, MR, and AC -susceptibility results, we can roughly create a magnetic \nphase diagram as depicted in Figure  5c. Based on the phase diagram, we predict that the \ntopological spin texture states in Fe 3Sn2 may extend to a much higher temperature, perhaps up \nto Curie temperature (~640K). However, due to the technical limitations of our current \nmeasurements, we could not perform the experiments at a temperature higher than 400K. It is \nwell known that  stable skyrmion ic states at hig h temperatures are critical for technical \napplications in magnetic storage and spintronic s devices. Therefore, the observation of a \nskyrmion state in Fe 3Sn2, not only in a very wide temper ature range but also at a high \ntemper ature, strongly suggest s that Fe 3Sn2 is a very promising material for both sky rmion \nphysics and potential technical applications of magnetic skyrmions.  \nThe ongoing  and future studies will include electrically prob ing various exciting \nphenomena  in this material, such as the skyrm ion Hall effect ,[36, 37] and quantiz ing the \ntransport of magnetic skyrmions,[38,39] similar to the study of emergent electrodynamics of \nskyrmions in bulk chiral materials .[40] \n \nExperimental Section  \nSample Preparation : Single crystals of Fe 3Sn2 were synthesized by using the Sn -flux method \nwith a molar ratio of Fe : Sn = 1:19. The starting materials were mixed together and placed in \nan aluminum  crucible with higher melting temperatures at  the bottom. This process was \nperformed  in a glove box fill ed with a rgon gas. To avoid the influence of volatilization of Sn \nat high temperatures, the whole assembly was first sealed inside a tantalum (Ta) tub e under \nproper Ar pressure. The Ta tube was then sealed in a quartz tube filled with 2 mbar Ar \npressure. The crystal growth was carried out in a furnace by heating the tube from room \ntemperature up to 1150 ℃ over a period of 15  h, holding at this temperature for 72 h, cooling \nto 910℃ over  6 h, and subsequently cooling to 800℃ at a rate of 1.5 K/h. The excess S n flux \nwas removed by spinning the tube in a centrifuge at 800℃. After the centrifugation process, \nmost of the flux contamination was removed from the surfaces of crystals  and the remaining \nflux was polished. \nMagnetic and Transport Measurements: The magnet ic moment was measured by using a \nQuantum Design physical properties measurement system (PPMS)  between 10K and 400K,   Submitted to  \n9 \n \nwhereas the magnetic moment above 400K was measured by using a vibrating sample \nmagnetometer  (VSM).  To measure the (magneto -) transport properties, several single crystals \nwere milled into a bar  shape with a typical size of about 0.6 × 0.4 × 0.05 mm3. Both \nlongitudinal and Hall resistivity we re measured using a standard four -probe method on a \nQuantum Design PPMS. The field dependence of the Hall resistivity was obtained after \nsubtracting the longitudinal resistivity component. \nLTEM Measurements: The thin plates for Lorentz TEM observations were cut from bulk \nsingle -crystalline samples and thinned  by mechanical polishing and argon- ion milling. The \nmagnetic domain contrast was observed by using Tecnai F20 in the Lorentz TEM mode  and a \nJEOL -dedicated Lorentz TEM, both equipped with liquid- nitro gen, low-temperature holders \n(≈100 K) to study the temperature dependence of the magnetic domains. The magnetic \nstructures were  imaged directly in the electron microscope. To determine the spin helicity of \nthe skyr mions, three sets of images  with under -, over -, and just (or zero) focal lengths were \nrecorded by a charge- coupled device (CCD) camera, and then the high -resolution in-plane \nmagnetic induction distribution mapping was obtained by QPt software, based on the \ntransport of the intensity equation (TIE) . The inversion of magnetic contrast can be seen by \ncomparing  the under - and over -focused images. The colors and arrows depict the magnitude \nand orientation of the in- plane  magnetic induction . The objective lens was turned off when the \nsample holder was inserted, and the perpendicular magnetic field was applied to the stripe  \ndomains by increasing the objective lens gradually in  very small increments . The specimens \nfor the TEM observations were prepared by polishing, dimpling, and subsequently ion milling.  \nThe crystalline orientation of the crystals was determined by selected -area electron diffraction  \n(SAED).  \nMicromagnetic  Simulations : Micromagnetic  simulations were carried out with three -\ndimensional object oriented micromagnetic  frame work (OOMMF) code,  based on the LLG \nfunction.[41] Slab geometries of dimensions 2000 nm×  2000 nm × 100 nm were used, with \nrectangle mesh of  size 10nm×10nm×10nm. We used a damping constant α=1 to ensure quick \nrelaxation to the equilibrium state. The material parameters were chosen according to the \nexperimental values of Fe 3Sn2, where the saturation magnetization M s = 5.66×105 A/m at \nroom temperature, and the uniaxial magnetocrystalline anisotropy constant K u=1.8×105 J/m3. \nAs the magnetic moment aligned obliquely along the c -axis, we defined K u⊥as the anisotropy \nenergy associated with the perpendicular component of the anisotropic field, i.e. 𝐾𝐾u⊥=\n�1\n2�𝐻𝐻k⊥𝑀𝑀s. The exchange constant A  was estimated to be 1.4×10-11 A/m by  𝐷𝐷=π�𝐴𝐴𝐾𝐾u⁄, \nwhere D is the domain wall width obtained from the LTEM results. These three parameters   Submitted to  \n10 \n \nvary with temperature, hence we investigated the dependence of domain morphology  on \nexchange energy ( A) and anisotropy energy as s hown in Figure 1g  by fixing the M s. The \nequilibrium states are all obtained by fully relaxing the randomly distributed magnetization.  \nThe simulations on the field -dependent domain structures at 300K (as shown in Figure 1h ) \nwere conducted  at zero temperature (no stochastic field) but the values o f parameters \ncorrespond to 300K, because the change tendency of magnetic do main  alters little by \ntemperature.  \nSupporting Information  \nSupporting Information is available from the Wiley Online Library or from the author.  \n \nAcknowledgements  \nThe authors thank  Jie Cui and Dr. Yuan Yao  for discussions and thei r help in L TEM \nexperiments . This work is supported by the National Natural Science Foundation of China \n(Grant Nos. 11474343, 11574374, 11604148, 51471183, 51590880, 51331006 and 5161192), \nKing Abdullah University of Science and Technology (KAUST) Office of Sponsored \nResearch (OSR) under Award No: CRF -2015- 2549- CRG4, China Postdoctoral Science \nFoundation NO. Y6BK011M51, a project of the Chinese Academy of Sciences  with grant \nnumber KJZD -EW-M05 -3, and Strategic Priority Research Program B of the Chinese \nAcademy of Sciences under the grant No. XDB07010300. \n \nReferences:  \n[1] T.H.R. Skyrme, Nucl. Phys . 1962, 31, 556. \n[2]K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno, A. Thiaville,T. Ono, Nat. \nMater. 2007, 6, 269. \n[3] R. Hertel, C. M. Schneider, Phys. Rev. Lett. 2006, 97, 177202. \n[4] A. Fert, V. Cros, J. Sampaio, Nat. Nanotechnol. 2013, 8, 152. [5]N. Nagaosa, Y.Tokura, Nat. Nanotechnology . 2013, 8, 899. \n[6] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, P. \nBoni, Science 2009, 323, 915. \n[7] A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. Niklowitz, P. Boni,Phys. Rev. \nLett. 2009, 102, 186602. \n[8] C. Pappas, E. Lelivre -Berna, P. Falus, P.M. Bentley, E. Moskvin, S. Grigoriev, P. Fouquet, \nB. Farago, Phys. Rev. Lett . 2009, 102, 197202.   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Loidl, Nat. \nMater . 2015, 14, 1116. \n[17] Y. Onose, Y. Okamura, S. Seki, S. Ishiwata, Y . Tokura ,Phys. Rev. Lett . 2012, 109, \n037603. \n[18] M. Mochizuki, X.Z. Yu, S. Seki, N. Kanazawa, W. Koshibae, J. Zang, M. Mostovoy, Y. \nTokura, N. Nagaosa,Nat. M ater. 2014, 13, 241. \n[19] P. Milde, et al., Science2013,340, 1076.  \n[20]Y. Tokunaga, X.Z.Yu, J.S.White, H.M. Rǿnnow, D. Morkawa, Y. Taguchi, Y. Tokura, \nNat. Commun . 2015, 6, 7638. \n[21]Y.S. Lin, J. Grundy,E. A. Giess, Appl. Phys. Lett . 1973, 23, 485. \n[22]X. Z. Yu, M. Mostovoy, Y. Tokunaga, W. Z. Zhang, K. Kimoto,Y. Matsui, Y. Kaneko, N. \nNagaosa, Y. Tokura, Proc. Natl. Acad. Sci.USA 2012, 109, 8856. \n[23] X.Z.Yu, Y. Tokunaga1, Y. Kaneko1, W.Z. Zhang, K. Kimoto, Y. Matsui, Y. Taguchi , Y. \nTokura1,Nat. Commun. 2014, 5, 3198. [24] X.Z.Yu, Y. Tokunaga, Y. Taguchi, Y. Tokura, Adv. Mater. 2017, 29, 1603958. \n[25] W. H. Wang, Y. Zhang, G. Xu, L. Peng, B. Ding, Y. Wang, Z. Hou,X. Zhang, X. Li, E. \nLiu, S. Wang, J. Cai, F. Wang, J. Li, F. Hu, G. Wu,B. Shen, X. Zhang, Adv. Mater. 2016, 28, \n6887. [26]C. Phatak,  O. Heinonen, M . D. Graef,  A. P-. Long , Nano Lett . 2016, 16, 4141. \n[27]A. O. Leonov, M. Mostovoy, Nat. Commun. 2015, 6, 8275.   Submitted to  \n12 \n \n[28]M. Pereiro, D. Yudin, H. Chico, C. Etz, O. Eriksson, A. Bergman, Nat. Commun. 2014, 5, \n4815. \n[29] B. Malaman, D. Fruchart, G. L. Caër, J. Phys. F. 1978, 8, 2389. \n[30] G. L. Caër, B. Malaman, B. Roques, J. Phys. F. 1978,8, 323. \n[31]G. L. Caër, B.Malaman, L. Häggstr öm, and T. Ericsson, J. Phys.F.1979 , 9, 1905. \n[32] L. A. Fenner, A. A. Dee, and A. S. Wills, J. Phys.: Condens. Matter . 2009, 21, 452202. \n[33] T. Kida, L. A. Fenner, A. A. Dee, I. Terasaki, M. Hagiwara, A. S. Wills, J. Phys.: \nCondens. Matter. 2011, 23, 112205. [34] Q. Wang, S. S. Sun, X. Zhang, F. Pang, H. C. Lei, Phys. Rev. B. 2016, 94, 075135. \n[35] H. F. Du, J.P. DeGrave, F. Xue, D. Liang, W. Ning, J.Y.Yang, M.L. Tian, Y.H. Zhang, S. \nJin, Nano Lett . 2014,14, 2026. \n[36] T. Schulz, R. Ritz, A. Bauer, M. Halder, M. Waner, C. Frznz, C. Pfleiderer, K. Everschor, \nM. Garst, A. Rosch, Nat. Phys . 2012, 8, 301. \n[37]W. J. Jiang, X. C. Zhang, G. Q. Yu, W. Zhang, X. Wang, M. B. Jungfleisch, J. E. Pearson, \nX. M. Cheng, O. Heinonen, K . L. Wang, Y . Zhou, A . Hoffmann , S. G. E. te  Velthuis , Nat. \nPhys . 2017, 13, 162. \n[38] K. Litzius, I. Lemesh, B. Kr üger, P. Bassirian, L. Caretta, K. Richter, F. B üttner, K. Sato, \nO. A. Tretiakov, J. F örster, R. M. Reeve, M. Weigand, L. Bykova, H. Stoll, G. Schütz, G. S. \nD. Beach, M. Kl äui, Nat. Phys . 2017, 13, 170. \n[39] S. Z.Lin, C. Reichhardt, C. D. Batista,  A. Saxena,  Phys. Rev. Lett.  2013, 110, 207202. \n[40] C.Reichhardt, D.Ray,   C. J. O.Reichhardt, Phys. Rev. B . 2015, 91, 104426. \n[41]M. J. Donahue , D. G. Porter, NISTIR 1999 6376;http://math.nist.gov/oommf . \n    Submitted to  \n13 \n \n \n \n  \n \n  \n \n  \n \n  \n \n  \n \n \nFigure 1 . Structure, magnetic properties, and micromagnetic stimulations of Fe\n3Sn2. a) The \ncrystal structure of Fe 3Sn2 (up), a top view of the  kagome layer of Fe atoms (down, left ) and a \npossible spin (arrows)  configuration of the Fe atoms (down, right ). b) The temperature \ndependence of anisotropy constant ( Ku) in the temperature range 10- 400K. The insets (from \nright to left) show the schematic of the angle between the magnetic easy axis and the c-axis at \n300, 150, and 6K, respectively. c-f) The representative images of the domain structures of \nsame area in a Fe 3Sn2 thin-plate taken by Lorentz tr ansmission electron microscopy (LTEM) \nwith an electron beam perpendicular to the ab- plane of Fe 3Sn2 when the sample temper ature \nwas lowered from 300 K to 130K in  zero external magnetic field. The inset  of (c) shows the \ncorresponding selected -area electron diffraction (SAED) pattern. g)  The plan view  of \nequilibrium states under different ratios of  exchange constant A  and perpendicular component \nfor the magnetocrystalline  magnitude  𝐾𝐾u⊥for a fixed  Ms=5.7×105 A/m and thickness (100nm) \nin zero external magnetic field. The magnetization along the z -axis (m z) is represented by \nregions in red (+m z) and blue ( -mz), whereas the in -plane magnetization (m x, m y) is \nrepresented by the white regions. h ) The simulated  field-dependent domain morphology under \nseveral magnetic fields that capture the domain evolution from stripes to type -Ⅱbubbles and \nskyrmionic bubbles.  \n  Submitted to  \n14 \n \n \n \n \n \n  \n \n  \n \n \n \n \n \n \nFigure 2. Magnetic field dependence of the magnetic domain morphology imaged using \nLTEM at  300K . a-d) The over-focused LTEM images under different out -of-plane magnetic \nfields at 300K. The regions in the white boxes show the different types of magnetic bubble \ndomains. e-h) Enlarged LTEM images of the  white boxes in (a -d) showing the  magnetic \nbubble domains . i-l) Corresponding  spin textures for the bubble domains shown in (e -h), \nextracted from the analysis usin g TIE. Colors (the inset  of (i) shows the color wheel) and \nwhite arrows represent the direction of in -plane magnetic induction, respecti vely, whereas the \ndark color represents the magnetic induction along the out -of-plane direction. ( e) and (f) \ndisplay the type -Ⅱ bubbles, and (k) and  (l) show different types of skyrmionic bubbles.  \n \n \n \n \n \n \n \n \n \n \n \n  Submitted to  \n15 \n \n \n             \n \n \n \n \n \n \nFigure 3.  Evolution of the  magnetic bubble  through Bloch line (BL) motion, induced by the \nmagnetic field. a- f) Series of LTEM images of bubble “3” observed at different times and \nfields applied along the c -axis. The field was increased  slowly for 8 second s from 840 to \n850mT. \n  \n  Submitted to  \n16 \n \n \n \nFigure 4.  LTEM images of the magnetic domain structures taken at different temper atures  \nafter turning off  the external magnetic field applied along c -axis. The field of 700mT is not \nstrong enough to saturate the sample magnetically.  a-d) Stripe domains and skyrmionic \nbubbles coexist in the sample  over the temper ature range of 300K to 170K, after turning off \nthe magnetic field. The regions enclosed in the squares are the skyrmionic bubbles. e -h) The \ncorresponding magnetization textures obtained from the TIE analysis for the bubble domains \nin the squares in (a -d). The inset  of (e) shows the color wheel. The spontaneous skyrmionic \nbubbles with triple -ring structures and random helicities are observe d.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  Submitted to  \n17 \n \n \n \n \n  \n \n \n \n  \nFigure  5. Field dependence of magnetoresistance (MR), AC -susceptibility and the magnetic \nphase diagrams of Fe\n3Sn2. a) The magnetic field ( H) dependence of MR obtained in the \ntemperature range of 100K -400K, with magnetic fields  applied normal to the ab- plane. The \ninset shows the details of the MR -H curve measured at 3 00K.  b)The magnetic field \ndependence of the real part of AC -susceptibility χ in the temperature range of 100K -400K. c) \nMagnetic phase diagram of bulk sample in the magnetic field versus the temperature plane, as \ndeduced from the temperature dependence of MR and χ curves. The three fully filled circles  \nindicate the experimental data obtained from the in situ LTEM observations. The error bars \nwere added based on the results measured on three different samples.  \n \n \n(b) (c) (a)   Submitted to  \n18 \n \nVarious  and spontaneous magnetic  skyrmionic bubbles are experimental ly observed for \nthe first time, at room temperature in a frustrated  kagome magnet Fe 3Sn2 with unixial \nmagnetic anisotropy. The magnetization dynamics were investigated using in-situ Lorentz \ntransmission electron microscopy,  revealing that the transformation between different \nmagnetic bubbles and domains are via the motion of Bloch lines driven by applied external \nmagnetic field. The results demonstrate that Fe 3Sn2 facilitates a unique magnetic control of \ntopological spin textures  at room temperature , making it a promising candidate for further  \nskyrmion -based spintronic devices.  \n \nKeywords: Skyrmio nic bubbles; Topological spin textures; Kagome magnet ; Fe 3Sn2; \nSpintronic devices  \n \n \nZhipeng Hou*, Weijin Ren *,Bei Ding*, Guizhou Xu, Yue Wang, Bing Yang, Qiang Zhang, \nYing Zhang, Enke Liu, Feng Xu, Wenhong Wang, Guangheng Wu, Xi -xiang Zhang, Baogen \nShen, Zhidong Zhang  \n \nObservation of Multiple  and Spontaneous Skyrmionic Magnetic Bubbles at Room \nTemperature in a Frustrated Kagome Magnet with Uniaxial Magnetic Anisotropy  \n \n \nTOC  Figure  \n \n \n \n  \n \n \n \n  \n \n  \n \n \n \n \n \n \n \n \n \n \n. \n \n \nFrustrated kagome lattice @ Fe3Sn2 b c \na \na b \nVarious  magnetic bubbles @ RT    Submitted to  \n19 \n \nSupplementary Information to  \nObservation of  Various  and Spontaneous Magnetic Skyrmionic Bubbles at Room -\ntemperature in a Frustrated Kagome Magnet with Uniaxial Magnetic Anisotropy \n \nZhipeng Hou*, Weijin Ren *, Bei Ding *, Guizhou Xu, Yue Wang, Bing Yang, Qiang Zhang, \nYing Zhang, Enke Liu, Feng Xu, Wenhong Wang, Guangheng Wu, Xi -xiang Zhang, Baogen \nShen, Zhidong Zhang  \n \nSingle -crystal X -ray diffraction (SXRD) was performed on the  crystal shown in Figure  S1 \nwith a Bruker APEX II diffractometer using Mo K -alpha radiation (lambda = 0.71073 A) at \nroom temperature. Exposure time was 10 seconds with a detector distance of 60 mm. Unit cell \nrefinement and data integration were performed with Bruke r APEX3 software. A total of 180 \nframes were collected over a total exposure time of 2.5 hours. The crystal lattice parameters were established to be a = b = 5.3074Å, c = 19.7011Å with respect to the rhombohedral unit \ncell (space group R -3m), agreeing well  with the previous studies. As shown in Figure  S2, to \nascertain  the crystal orientations, the sample was mounted on a holder and Bruker APEX II \nsoftware was used to indicate the face normal of the crystal , after the unit cell and orientation \nmatrix were determined. One can notice that the hexagonal face is normal to [001] with the \n(100), (010), and (110) faces around.  \n \n             \n \nFigureS1 . X-ray diffraction pattern of a Fe\n3Sn2 single crystal along the perpendicular \ndirection of  hexagonal surface, which indicates that the hexagonal surface is parallel to the \nab-plane and perpendicular to the c -axis. Inset: The typical photograph of Fe 3Sn2 single \ncrystal placed on a millimeter grid. The crystal is 0.3 mm × 0. 3 mm × 0. 2 mm in size and \npossesses hexagonal  mirror -like surfaces . \n \n \n  Submitted to  \n20 \n \n \n \n \n \n \n \n \nFigure S2. (a) Single -crystal X -ray diffraction processi ng image of the (00l ) plane in the \nreciprocal lattice of Fe 3Sn2 obtained on the crystal mentioned above. No diffuse scattering is \nseen and all the resolved spots fit the crystal lattice structure established for Fe 3Sn2. (b) \nVarious crystal planes and their corresponding normal directions of Fe 3Sn2 single crystal. The \nyellow striped region is the glue to af fix the crystal to the holder . \n \n \n   \n \n \n   \n \n \n \n \n \n \n \n \nFigure S3. Temperature dependence of magnetization with the field -cooling (FC) model in an \nexternal  magnetic field of 500Oe between 5K and 700K. As is shown in Fig ure S3, the Curie \ntemperature T\nc is established to be 660K, which is similar  to previous reports. When the \ntemperature falls below Tc, the magnetization  for both magnetic fields ( H//c and H//ab)  first \ndecreases and then starts to increase at 420K , reaching a maximum at 80K. When the \ntemperature decreases below 80K, the slight decr ease in magnetization  can be attributed  to the \nentrance of the spin glass state (SGS).    \n \n \n  Submitted to  \n21 \n \n \n  \n \n    \n \n   \n \n \nFigure S4. a) The magnetic  field dependence of magnetization in fields parallel to the c -axis \n(black line and  black symbol ) and ab- plane (red  line and  red symbol ) in the temperature range \nof 400K -6K. b)  The temperature dependence of the saturation magnetization  M\ns. \n \n \n \nFigureS 5. a) The temperature dependence of domain wall thickness D . The error bar denotes \nthe deviation of three individual width measurements. One can notice that the value of D  is \nnearly independent from  the change in  temperature. The exchange stiffness constant A  can be \nestablished by using the equation , 𝐴𝐴=𝐷𝐷𝐾𝐾𝑢𝑢2\n𝜋𝜋2.b) The temperature  dependence of A. By \ndecreasing the temperature, the value of A  increases correspondingly .  \n(a) (b)   Submitted to  \n22 \n \n \nFigureS 6. a-b) The over- and under -focused LTEM images under  zero magnetic field at \n300K. c-f) Corresponding under -focused TEM ima ges for Figure 2 (a -d). The boxed regions  \ncorrespond to the  magnetic bubbles  shown in  Figure 2 (a -d). \n \n \n \nFigure S7. The under -focused LTEM  images after a saturated magnetization. When the \nsample is magnetized to a saturated state, the domain reverts to the stripe.  \n \n  \n \n \n  Submitted to  \n23 \n \n \n \n \nFigure S8. The corresponding  over-focused  (a, b, c, d)  and under -focused (e, f, g, h)  LTEM  \nafter an unsaturated magnetization in a different region from that in Figure 3. If the sample is \nmagnetized to an intermediate state, then the  skyrmionic bubbles with concentric rings  appear \nafter the magnetic field decreases to zero.  \n \n \n \n"    },    {        "title": "1706.08861v1.Magnetic_properties_of_photosynthetic_materials___a_nano_scale_study.pdf",        "content": "Magnetic properties of photosynthetic materials - a nano scale study\nAbhishek Bhattacharya1, Su\f O Raja1, Md. A Ahmed2, Sudip Bandyopadhyay2, Anjan Kr. Dasgupta1\n1Department of Biochemistry, University of Calcutta, 35, Ballygunge circular road,\nKolkata 700019, India2Department of Physics, University of Calcutta,\nRajabajar Science College, 92 A.P.C. road, Kolkata-700009, India.\n(Dated: November 14, 2018)\nPhotosynthetic materials form the basis of quantum biology. An important attribute of quantum\nbiology is correlation and coherence of spin states. Such correlated spin states are targets of static\nmagnetic \feld. In this paper, we report magnetic properties and spectroscopically realizable static\nmagnetic \feld e\u000bect in photosynthetic materials. Two classes of nano-scale assembly of chlorophyll\n(NC) are used for such a study. Magnetic measurements are made using a superconducting quantum\ninterference device (SQUID). Both ferromagnetic and superparamagnetic states are observed in NC\nalong with a blocking temperature around 250 K. Low temperature quantum (liquid nitrogen)\nspectroscopy is employed to see how optical transitions are a\u000bected in presence of static magnetic\n\feld. Plausible practical application aspects of magnetic properties of this optically active material\nare discussed in the text.\nPACS numbers: PACS numbers go here. These are classi\fcation codes for your research. See\nhttp://publish.aps.org/PACS/ for more info.\nI. KEYWORDS\nPhotosynthetic Material, Nanoscale, Magnetic Property, Blocking Temperature, Superparamagnetism, Ferromag-\nnetism, Spin, Nano chlorophyll (NC), Magnetic Memory, FACS, NIRF, Nano Materials.\nII. INTRODUCTION\nPhotosynthetic pigment, Chlorophyll a(Chla) acts as a primary photon receptor molecule for light harvesting and\nphoto chemistry. Additionally, multiple other components such as chlorophyll b(other forms are also reported) (1),\naccessory pigments like beta-carotene, pheophytins, xanthophylls, secondary derivative molecules (protoporphyrins,\nchlorins, pyrochlorophylls, etc.), proteins (D1, D2, psb, RubisCO etc.), protein bound pigment complexes in mem-\nbranes (LHC) satisfy the photosynthetic light reactions to collect (like an antenna), transfer (near unity rate) and\npreserve the photon energy. Chl ais an aromatic and macro-cyclic natural dye. Central magnesium (Mg2+) atom is\nlinked to nitrogen atoms of the poly pyrrole ring to serve as a site for the excitation energy distribution and control for\nthe excitation transfer reactions (2-3). These hydrophobic molecules tend to aggregate among themselves in contact\nto water (4-7) and interact among themselves by weak \u0019-stacking interactions in polar environments (8-13).\nNano scale stabilization and interaction study of similar chemical dyes exemplify the signi\fcance and importance\nof such a photo receptor (14-20). In the present context, the impact of nano scaling and colloidal stabilizations on\nthe properties of the \ruorophore molecule and its interaction study had been reported. Interestingly, it may be noted\nthat novel properties may indeed be emergent at the nano scale (21) such as a crystalline CoSi nano wire exhibit\nemergent ferromagnetism at the nano scale contrary to its diamagnetic bulk phase (22).\nMagnetic transitions and memory are generally attributed to and are intrinsic to the magnetic materials such\nas metals (Fe, Co, Ni), magnetic nanoparticles, magnetically susceptible metal nano structures (nano wires (23),\nmagnetic core-shell nanoparticles etc.). However, some reports infer to a pi-stacking dependent magnetic transitions\n(24-26). Additionally, establishment of the fact that pi-pi interaction is strong enough to beat the cationic repulsion\nin metalloporphyrins also support the postulate (27) and may mediate assembly stabilizations (28-29). While size\ndependent alterations in the magnetic transitions of any material is reported to follow the nano scaling laws (30-31),\nit may be interesting to study how the control over size of such a macro-cyclic \ruorophore stabilized as a nano-bio\nassembly impact to its intrinsic properties and spin chemistry (32).\nThe primary question addressed in this paper is that, can we replicate magnetic memory e\u000bects in cell free systems\nsuch as, nano-assembly of chl a. The question concerns about the mechanism for the magnetic memory component of\nsuch an assembly, if any. Instead of looking at a continuous and prior exposure to \feld, we changed our approach in\nwhich we make zero \felds and with \feld magnetic measurements, which would directly predict the magnetic nature\nof such an assembly. Self-assembly in turn could create grounds for the stabilization of long range magnetic domains,\nprospering the possibility for the emergence of memory. A\u000ermative answer to this question would in turn opens\nup possible implications of this versatile sensory tool to address novel optical and magnetic problems and materialsarXiv:1706.08861v1  [q-bio.BM]  10 Jun 20172\ncharacterization studies. Potentiality for the development of novel opto-magnetic technologies and coherent methods\nhad also been discussed. Lastly, applicability of the NC as a \row cytometry compatible nano-bio and imaging probe\nhad been examined.\nIII. MATERIALS AND METHODS\nIsolation and puri\fcation of Chl Fresh mesophyll tissue from the green leaves of spinach (Spenacea oleracea) was\ncollected for pigment extraction. The tissue are then frozen in liquid nitrogen (-196oC) for freeze drying and grinding\nof the leaves. The pigments were extracted in 2:1:1, methanol: petroleum ether: diethyl ether solvent system. All\nthe procedure was conducted at dark and or dim light conditions. Upper ether layer containing most of the pigments\n(Chlorophylls and accessory pigment molecules such as pheophytin, carotenoids, xanthophylls, chlorins etc.) was\ncollected carefully. The lower methanol layer containing pigment-protein complexes, membrane remnants, accessory\npigments and debris are discarded. Puri\fcation of chl afrom a mixture of pigment analytes was performed using simple\nchromatographic techniques. Chromatographic separations of the crude pigment mixture was conducted by passing\nit through a vertical silica gel column (silica gel, SRL India, 100-200 mesh) and/or by thin-layer chromatography\n(TLC silica gel 60254 or DC kieselgel 60F254 Merck, USA). A liquid/gas phase mixture of mobile phase consisting\nof n-hexane, n-hexane: acetone (1:1 to 1:0.1. respectively), acetone and acetone: methanol (v/v) had been used for\ncolumn chromatography (\row rate = 1 ml/min) and a volatile mixture of 60% petroleum ether, 16% cyclo-hexane/\nn-hexane, 10% ethyl acetate, 10% acetone and 4% methanol (v/v) had been utilized as a mobile phase in case of\nplanar chromatographic separations (33). The Rfvalues for each of the pigments had been determined and the value\nfor free chl was found to be 0.4. Major experimental data were validated by comparing it to a commercial spinach\nchlafrom Sigma-Aldrich, USA. Chl ais extremely sensitive and prone to degradations to its secondary derivative\nmolecules, photo bleaching, oxidation and thermal damage.\nNano chlorophyll (NC) stabilizations Chlaat a solvent perturbed bulk or free state are phase transferred to\nabsolute polar solvent water. The tetra-pyrrole moiety of the chl atends to self-aggregate by securing the hydrophobic\ncore by\u0019\u0000\u0019stacking self-interactions. Size controlled nanoparticle stabilization follows as a result of such solvent\nphase perturbation under ultra-sonication. Citrate was used as a reducing, stabilizing and capping agent for size\ncontrolled nanoparticle synthesis. Precipitation of the macro-cyclic aggregates occurs frequently in water without any\nstabilizing or capping agent. Di\u000berent concentration of tri-sodium citrate was used to stabilize chl ain water based\nbu\u000bers and for the synthesis of desired size of chl ananoparticle (NC). 1 \u0016g/ml puri\fed chl adissolved in organic\nsolvent was added drop wise to the tri sodium citrate solution placed in a hot water bath (set at boiling point of the\norganic solvent) under continuous sonication with a probe sonicator. A \row rate of 0.1.5 ml/min was maintained. The\nsonication was implied with an amplitude frequency set at 80 with 0.5 cycles of interval for 10 to 30 minutes and the\nresulting colloidal solution containing stable nanoparticle NC were stored in freezer for further characterization studies\nand applications (34). The hydrodynamic diameter (size) and the relative \ruorescence emission of the colloidal NC\nwere found to rely directly on the size of the particles, concentration of the stabilizer and pH of the medium. NC was\nthen diluted in biological bu\u000bers such as phosphate bu\u000ber (pH 7.2.), DMSO for further experimental measurements.\nNotably, chl ais prone to degradations by thermal damage, oxidation etc. and exert frequent photo-bleaching whereas\nthe colloidal NC is photo stable at a liquid state or at an immobilized \flm state.\nColloidal stability measurements and zeta potential The colloidal stability of the synthesized nanoparticle\n(NC) was measured in a Beckman Coulter zeta instrument with metal electrode cuvette. The instrument measures\nthe surface potential using an electrophoretic mobility shift assay.\nUV-Vis spectroscopy Absorbance measurements were conducted in a Thermo-Vision Evolution 300 spectropho-\ntometer. A spectral range of 350 nm to 800 nm was scanned for chl a. A xenon lamp was illuminated for absorbance\nmeasurements. Nitric oxide sensing and dose dependent experiments were conducted using 96-well plates and plate\nreaders which acquire data at a \fxed \u0015corresponding to the O.D maxof NC at 25oC.\nFluorescence measurements Fluorescence measurements were performed in a PTI \ruorescence spectrophotome-\nter (QuantamasterTM40, USA). Excitation wavelength was set at 430nm with excitation and emission monochromators\nand emission was collected to a perpendicular direction between 650nm to 750nm for chlorophyll a. The bandwidth\nwas set at 5nm. Time kinetics data was collected at \fxed excitation and emission wavelengths set at 430nm and\n665nm respectively. The temperature dependent experiments were performed with a temperature controlled peltier\nsystem. Synchronous \ruorescence measurements were conducted in a 350nm to 750nm wavelength region with si-\nmultaneous excitation and emission scanning. A range of di\u000berent solvent interacting environments (polarity shifts)\nand their e\u000bect on the \ruorescence was examined for puri\fed chl a. Polarization study was conducted utilizing the\ncorresponding excitation and emission polarizer placed between the light paths.\nCryogenic \ruorescence measurements Low temperature \ruorescence emission spectroscopy was conducted\nin a Hitachi F-7000 spectrophotometer, Japan. The excitation monochromator was set to 480nm (slit width = 53\nnm). Emission was scanned between 650 and 800 nm (slit width = 10 nm). Free chl dissolved in solvent and NC at\nphosphate bu\u000ber (pH 7.2) was diluted in presence of cryoprotectant glycerol ( >60%) before cryogenic measurements.\n77K \ruorescence spectra were baseline corrected and the \ruorescence ratios for P723=P700was calculated and plotted\nagainst wavelength. Fluorescence time kinetics data was acquired at a \fxed excitation (480nm) and emission (700nm\nor 723nm) wavelength against time. Again, a Mg-porphyrin speci\fc excitation of 590nm result in multiple emission\nbands adjacent to the near-infrared region (NIRF).\nDynamic light scattering (DLS/PCS) DLS was used as the primary characterization tool for the determination\nof size and shape of the synthesized nanoparticle NC. The instrument measures the hydrodynamic properties of the\nnano scale particles depending on their scattering pro\fle and di\u000busion behavior. Measurements was performed in a\nMalvern Nano ZS80 (UK) dynamic light scattering set up equipped with a 532nm excitation laser source. All the\nmeasurements were conducted at 4oC and 25oC as mentioned in the text accompanied by a peltier system. The\nauto-correlation pro\fle (g(2)-1) against the correlation delay times ( \u0016S) infer the liquid state di\u000busion pattern and\nthe number of scatterers present at a time point.\nAtomic force microscopy (AFM) Surface topographical imaging of the nanoparticles (NC) was conducted in\nNanoscope IVa (Vecco/Digital Instruments Innova, Santa Barbara, CA, USA). Soft silicon probes (RTESPA) was used\nwith a tip radius of 8 nm, mounted on a single-beam cantilever. Cantilever (115-135 \u0016m) de\rections were recorded\nwith a cantilever frequency (f0) of 240-308 KHz, horizontal scan rate of 1.2 Hz and 512 samples per line. Spring\nconstant of the cantilevers was set at 20-80 N/m. Scanning was conducted at 25oC in air. Data was analyzed by\nNanoscope software (Version 5.1.2r3). Phosphate bu\u000ber (pH 7.2) soluble NC was then immobilized on to a hydrophilic\nglass chip. The chips were pre-treated with piranha solution in heat bath for surface functionalization of the glass\nsurface with hydroxyl group (-OH).\nScanning electron microscopy (SEM) Scanning electron microscopy was conducted in an Evo18 special edition\nCarl Zeiss system with EHT of 15kV. The samples were pre-\fxed and dried in acetone washed glass chips. The samples\nwere gold sputtered (4nm x 2) before measurements to induce surface conductivity to the samples.\nTransmission electron microscopy (TEM) Transmission electron microscopy was conducted in a JEM 2100\nHR-TEM system with EHT of 200kV. The samples were pre-\fxed and dried in a copper grid and were negatively\nstained with uranyl acetate. The sample was cooled to liquid nitrogen temperatures during the measurements to\nprotect it from burning by harsh electron beam energy.\nMagnetic measurements (SQUID) Magnetic characterizations, \feld and temperature dependent magnetization\nmeasurements was performed in a magnetic properties measurement system (MPMS), superconducting quantum\ninterference device, vibrating sample magnetometer (SQUID VSM, Make: Quantum Design). Chl a was air dried in\na heat chamber and NC was lyophilized to dryness before magnetic measurements. The dry weight of the samples\nwas noted for further calculations.\nFluorescence spectroscopy in presence and absence of static magnetic \feld A moderate to low strength\nstatic magnetic \feld (SMF) of 100 to 500mT was exposed to the samples. Notably, the static magnetic exposure\nwas imparted before experimental measurements and not during the measurements. 10 minutes of magnetic (SMF)\nincubation at 4oC was imparted to the samples before steady state and time kinetic \ruorescence measurements at\nroom temperature as well as at cryogenic temperatures. A 0.01 to 0.05 \u0016g/ml of sample was diluted in relevant\ndissolving medium (solvent or bu\u000ber) and used for further study as mentioned in the text. Higher concentrations of\nthe \ruorophore often exhibit self-quenching e\u000bects and hence lower concentrations are selected for the experiments.\nRelevant objects for characterizations study\na. GNP synthesis GNP synthesis was carried out using an accepted method (35), with minor modi\fcations (36).\nAn aqueous solution of HAuCl 4(20 mM, 25ml) was brought nearly to boiling condition and stirred continuously with a\nstirrer. Freshly prepared tri-sodium citrate solution (38.8.mM) was added quickly at a time. The citrate concentration\nis related to the particle size, resulting in a change in color from pale yellow to deep red. The temperature was brought\ndown to normal and the colloidal solution was stirred for an additional few minutes with excess citrate for volume make\nup. Typical plasmonic resonance for GNP was found at 530nm, which according to standard literature corresponds\nto a size close to 40nm. The \fnal atomic concentration of gold was calculated to be 200 \u0016M.\nb. SNP synthesis Silver nanoparticle was synthesized by reducing silver nitrate 20 mM (AgNO 3) solution in\npresence of sodium borohydrate (100mM). Citrate acts as a stabilizer in the preparation. The stirring was performed\nat normal temperatures leading to a change in the color of the solution from transparent to yellow rendering SNP\nformation. After 15 minutes of stirring at room temperature the prepared SNP was incubated in ice for half an hour\nbefore use. The plasmon resonance was found in the range 420nm, corresponding to a size 40nm. The \fnal atomic\nconcentration of silver was calculated to be 200 \u0016M.\nc. Carbon nano materials MWNT and amorphous graphene was prepared as described previously (37). A mi-\nnuscule amount of dry and powered carbon nano material or a 1 \u0016g/ml amount of carbon nano tube and graphene\nwas examined.4\nd. Nitric oxide Pure nitric oxide donor DETA-NONOate (Cayman Chemicals, USA) was prepared as stock\nsolution of 100 mM in 0.1.(N) NaOH or phosphate bu\u000ber saline (PBS; pH 7.4). GSNO was prepared freshly by\nmixing 1M aqueous solution of sodium nitrite (NaNO 2, Merck, USA) with 1M solution of reduced glutathione (GSH;\nSigma Chemical Co., USA) in 1(N) HCl (HCl, Merck, USA) in 1:1 (v/v) ratio. GSNO formed was protected from\nlight and placed on ice immediately. Further working dilutions (nM, \u0016M and mM ranges) was prepared freshly before\nexperimental measurements. Sample was incubated with desired amount of sources for 15 to 30 minutes at 37oC\nbefore measurements.\nAll the other reagents and chemicals used are of analytical grade Sigma-Aldrich (USA), Merck (USA) and SRL\n(India) products of >99.9% purity.\nThermal imaging Thermal \ructuation of any surface (liquid, colloid or solid) immobilized to NC was determined\nusing a FLIR-IR camera. A low power 658nm red laser source of 25mW was used to perturb the NC for the thermal\nmeasurements. The experiments were conducted in a temperature controlled enclosed room to avoid any unwanted\nthermal \ructuations which often hamper the measurements and start to calibrate the instrument.\nFlow cytometry Flow cytometry measurements were conducted in a BD-INFLUX \row system dedicated to nano\nparticle science. The instrument is equipped with a multiple of \fve laser excitation sources and multiple detector\nchannels to register the forward scattering (FSC), side scattering (SSC), polarization status and \ruorescence emission\nof the samples under a \row cell. The optical polarization and depolarization status had been exploited in the context\nof SSC channels (38). Data analysis was performed using FlowJo V10 \row cytometry analysis software. Additional\nadvantages of utilization of near infrared \ruorescent (NIRF) window with optical \flters (cut o\u000b, band pass or long\npass \flters) may improve the sensitivity and applicability of the instrument. Laser excitation channels of 488nm and\n561nm wavelength had been utilized to illuminate the samples at a \row cell. Corresponding emissions were acquired\nat 692 \u000640nm channel for 488nm laser and 670 \u000630nm and 750nm LP (long pass channel) for 561nm excitation.\nIV. RESULTS\nAFM TEM and SEM of NC Topographical, ultra-structural and surface properties of the particles was measured\nusing higher end atomic force microscopy (AFM), high-resolution transmission electron microscopy (HR-TEM) and\nscanning electron microscopy (SEM) respectively (Figure 1).\nFIG. 1: Nano scale characterization of NC: A, Hydrodynamic properties of NC dissolved in phosphate bu\u000ber (pH 7); B, 3-D\nheight pro\fle of the immobilized NC imaged by tapping mode atomic force microscope; C, Surface scanning image of the\nimmobilized andgold sputtered NC; D, Ultra-structural transmission electron microscopic analysis of NC.\nSemi-spherical and bean shaped particles bearing a size of 70nm with a three dimensional topography with uneven\nsurface morphology had been con\frmed by tapping mode atomic force measurements.\nSurface scanning microscopy (39) again correlate to the force microscopic results (Figure 1). Surface scanning\nexperiments con\frmed particulate nano-cone structure formed due to higher order self-assembly. The particles exhibit\nelongated bi-axial morphology with multi partite arrangement.\nHigher resolution TEM images (HR-TEM) however decipher the actual arrangement of the self-assembled macro-\ncyclic backbones (Figure 2). Finely assembled self-stacked rings stabilized as higher order extensive domains consisting5\nFIG. 2: HR-TEM analysis of NC \fxed at a copper grid. The scale bar is 100nm and 10nm for sub-\fgure A and B respectively.\nof ultra-structural patchy lattice structures are prominent. Ultra-structural and morphological fringe pattern may\noriginate from self-assembled nano domains. TEM images of NC evidentially con\frm the presence of particulate NC\nof 12-24 nm size.\ntextbfCharge Distribution of NC Self-assembly derived stabilizations of the photosynthetic dye chl awere found to\ncorrelate directly to the presence of appropriate stabilizing and capping conditions. A negative surface zeta potential\nof -20.6 mV signify the colloidal stability of the particles formed in water or any biological bu\u000ber such as phosphate\nbu\u000ber of pH 7. Precipitation of the macro-cycles due to un-controlled self-aggregation had been observed in water\nwithout any stabilizing agent. A critical concentration for the stabilizer citrate had been established. 12 to 15mM of\ncitrate was found to be su\u000ecient to stabilize NC with a hydrodynamic size of 86 to 100nm.\nNIRF Fluorescence emissions from NC NIRF window had been tested for the colloidal assembly. A gain in\nthe dual emission ratio ( P723=P700) for NC implies a profound impact of nano scaling on the low energy emission band\nat 77K. Notably, a marked degree of stokes shifts had been observed for membrane bound chl a(Arachis hypogaea),\nfree chl a(solvent extracted from Spenacea oleracea, Arachis hypogaea) and NC at cryogenic conditions.\nSMF e\u000bects on 77K emission spectrum Primarily, an external SMF source of 100 to 500mT strength was\nused to investigate the translation of magnetic transitions (41) to a photonic and spectroscopic output (\ruorescence\nemission). Brie\ry the SMF e\u000bects were observed only at cryogenic temperatures (77K) and not at room temperature\n(Figure 3). In other words the SMF e\u000bect is absent in presence of thermal noise. The plausible inference that follows\nis that there is a strong dependence of SMF e\u000bect at particular assembly size that was stabilized. The di\u000berence of\nSMF e\u000bect on chl a and NC (panels A and D of Figure 3) further support this point. In the former case the SMF\nenhances the quantum yield in the NIRF peak, the reverse being true for NC. The lower decay rate of the NIRF at\n\u0015max= 723nm in presence of SMF (as compared to the same in absence of SMF) implies stabilization of the exited\nstates and memory.\nFIG. 3: Fluoresence Spectroscopyexcitation at 430nm: A, 77 K emission spectrum -Dual peaks of free chl a(green) and SMF\nexposed chl a(red) SMF enhancing the NIRF emission B, Fluorescence decay kinetics of the NIRF peak with increasing\nstatic magnetic \feld strengths showing lesser decay rates or higher stabilization of the excited states C, SMF e\u000bect at room\ntemperatures; D, 77K spectra of NC in presence and absence of SMF - SMF attenuating the NIRF emission (reciprocal to\npanel A - implying the SMF e\u000bect is assembly dependent) .6\nFluorescence life time e\u000bects Fluorescence lifetime decay analysis (Nano LED, Horiba scienti\fc) of chl aand\nNC shows similar SMF response. Comparable lifetime values were reported previously for the free \ruorophore at a\nsolvent perturbed state (5.6 ns) as well as in bu\u000ber solutions (0.17-3ns) (42-44). Nano scale stabilizations of free chl a\n(3 ns) result in a substantial gain in the excited state lifetimes of NC to >4 ns values. Lifetime decay measurements\n(in the order of 10\u000009seconds) again a\u000erm presence of magnetic memory (Table 1).\nTABLE I: Fluorescence lifetime decay analysis of chl aand NC at room temperatures. \u001cfdenote \ruorescence decay time in\nthe order of nanoseconds (ns). \u001cfvalues represent an average of at least two distinct experimental lifetime values. The lifetime\nvalues were derived from the multi-exponential \ftting of the decay kinetic data points; Ex. and Em. denotes the excitation\nand emission maxima of the \ruorophore and n/a denote not applicable.\nFluorophore Lifetime ( \u001cf, ns) Ex. max (nm) Em. max (nm) Solvent\nFree Chla from spinach 2.859 370 665 acetone\nNC 4.458 370 665 phosphate bu\u000ber (pH 7.2)\nFree Chla 4.37 370 665 methanol\nFree Chla + SMF 3.45 370 665 acetone\nFree Chla + SMF 3.56 370 665 methanol\nNC + SMF 3.18 370 665 phosphate bu\u000ber (pH 7.2)\nFree chl aat acetone dissolved state exhibit a gain in \ruorescence lifetime ( \u001cf) upon SMF exposure. While free\nchladissolved in methanol exhibit a fast decay of \u001cfafter spin perturbations due to elevated secondary degradation\nreactions of chl ain methanol. However, \u001cfwas found to be less for NC (4.4.5ns) after SMF incubations (4.2.ns). A\ngain in the value of \u001cffor acetone soluble free chl aand a fast decay of the excited state lifetime for NC upon SMF\ntreatment may be attributed to as spin mixed excited state interactions (intra or inter), spin driven re-alignment or\nre-orientation or altered energy transfer dynamics or a nano scale spin quantum phenomena. Altered excited state\n\ruorescence lifetimes for free chl aand NC again infer to a direct relation to the \ruorescence decay rate kinetics,\nthe size, shape, atomic packing, assembly pattern and stabilizations of the assemblies correlated to the structural,\nspectrophotometric and magnetic measurements. Notably, an external magnetic \feld sensing ability of such a nano\nassembly had been explored. Polarity screening based on hydrophobic self-interactions depending on the immediate\nsurrounding solvent molecules may be utilized to design a green polarity meter.\nMagnetic properties of photosynthetic material To address the problem of magnetic \feld sensing by such\na green nano assembly (NC), and in search for a probable mechanistic basis for the SMF mediated spin interactions\nand memory, direct magnetic transitions of the samples was measured. Field dependent magnetization (M-H and\nM-T) curves for the nano particulate NC indicate profound magnetic transitions correlated to its nano scale ultra-\nstructural properties, NIRF emissions at 77K and excited state lifetime analysis. M-H curves for NC at 300K, 150K\nand 5K indicate dominant diamagnetism for all of the curves with very typical nature (Figure 4). In the \feld region\nof\u0006400 Oe, all the M-H curves show magnetic hysteresis with positive magnetization and magnetization increases\nwith increasing \feld. In the \feld region above \u0006400 Oe but below \u00062000 Oe, the M-H curves exhibit magnetic\nhysteresis with positive magnetization and magnetization decreases with increasing \feld. Further, in the \feld region\nabove \u00062000 Oe but below \u00066000 Oe, the M-H curves show magnetic hysteresis with negative magnetization and\nmagnetization decreases with increasing \feld. This kind of behavior of M-H curves above \u0006400 Oe and below \u00066000\nOe is quite typical. This type of magnetic behavior manifested may be attributed to the presence of minuscule\nferromagnetic domains, embedded in the highly diamagnetic matrix. Alternatively, a \fnite size dependency of the\nassembly at a nano range (superparamagnetism) may have adequate logic to address the problem.\nFIG. 4: NC characterization: Magnetic hysteresis (M-H curve) Cyan, red and green lines represent magnetization at 5K, 150K\nand 300K respectively.7\nNano dimensional stabilization of the \ruorophore results in a decrease of MSand an increase of H C. Less M Sat the\nnano scale may be accounted for either by a transition from multi to single domain or to a superparamagnetic range\nof particles or may be dependent on surface spin canting e\u000bects (45). For extreme small nano particles such kind of\ninteractions may seem prominent but for larger particles these e\u000bects render insigni\fcant. These results re-con\frm\nthe fractal nature of arrangement and ultra-structural stabilization of particles by higher order self-assembly. Uneven\nsurface boundaries again may lead to generation of loose and disordered free electron spins at the particle surface other\nthan particles net magnetic moment which often tend to cancellation of the net magnetization and lead to a lower\nvalue of M Sin case of NC. However, the coercive force (H C) was found to increase in case of NC with respect to the\nfree state as with an extension of domain size from superparamagnetic (single-domain) to multi-domains, individual\nmoment of each domain may not add up to orient in a particular direction. The M Scan be improved to the bulk or\nfree level by using di\u000berent surface functionalization strategies to the particles which need to be explored further and\nis a subject for the future prospects of the work.\nFIG. 5: M-T measurements of the \ruorophore and NC: Upper and lower panels illustrates the paired responses (zero \feld and\nwith \feld) at two respective \feld strengths 200 Oe (left panel) and 500 Oe (right panel). Upper panel shows the response of\nchla and the lower panel illustrates the response of NC. Notably, the blocking temperature (T B) was only attained for NC\n(arrow).\nAdditionally, temperature dependent magnetization (M-T) measurements (Figure 5) under zero \feld cool (ZFC)\nand \feld cool (FC) conditions were performed for both the free and nano chl at 200 Oe and 500 Oe \feld strengths. The\nM-T curves under ZFC and FC conditions for NC initially exhibit a profound dip at very low temperature (below 10K)\nand then magnetization increases with increasing temperature. M-T curves under ZFC condition for the bulk or free\nchl however exhibit no such dip at low temperatures and in the entire temperature range magnetization increases with\nincreasing magnetic \feld. Actually thermal energy acts as an agent of ordering energy of magnetism for the free and\nnano chl (except up to 10K). So far as the M-T curves under FC condition for the free chl is concerned both the curves\nat 200Oe and 500Oe indicate gradual increasing nature as a function of increase in temperature. Whereas for the nano\nassembled NC, both the curves at 200Oe and 500Oe indicated gradual decreasing nature with increase in temperature\nup to 100K and thereafter maintain a near constancy or rather a very small decreasing trend. The noteworthy feature\nin the M-T measurement is that the M-T curves under ZFC and FC conditions for the bulk chl do not intersect each\nother up to room temperature (300K). Whereas those same curves for the NC merges at around 250K. This type of\nnature of M-T curves is quite common for a superparamagnetic system. Stabilizations of NC may cause emergence\nof such nano scale magnetic property. In such a superparamagnetic system at blocking temperature (T B), maximum\nvalue of magnetization was attained and M-T curves under ZFC and FC conditions intersect each other. Results are\nquite evident as TBfor the bulk or free state is above 300K as M-T curves under ZFC and FC conditions do not\ncollapse at 200Oe and even at 500Oe \feld strengths. Whereas in case of NC, maximum magnetization was attained\nand the M-T curves under ZFC and FC conditions converge at around 250K. Categorically, T Bwas attained only\nfor the NC around 250K and no (or rather negligible) ferromagnetic interaction was present at 300K. Notably, NC\nexhibits profound and elevated magnetic transitions and memory limited by the size of the nano assembly with respect\nto the diamagnetic free \ruorophore.\nSome practical applications The nano-scale photosynthetic material can be subjected to a simple practical\napplication. Using polarization enable \row cytometry we are able to classify di\u000berent classes of carbon nano tube and8\ngraphene illustrated in Figure 6. The chlorophyll nanoparticle NC can rapidly discriminate and identify MWNT and\ngraphene at a liquid and \row state by SSC-Pol and SSC-Depol phase analysis (complexity or granularity channel).\nAdditionally, a liquid state sensing of nitric oxide had been examined. NC exert coherent heating by a low power\n(25mW) red laser of 658nm. Lastly, the response of NC to polarized light was measured which impart additional\nadvantage to the nano NIRF tool.\nFIG. 6: Applications of NC: Detection of MWNT (cyan) and graphene (red) by SSC-Pol (x-axis) and SSC-Depol (y-axis) at a\n\row cell at room temperatures.\nV. CONCLUSIONS\nThe chl a in non-polar solvents and NC in aqueous solvent show a varying range of spectral and magnetic properties\nwith the former (spectral response) sensitive to presence of static magnetic \feld. The sensitivity to magnetic \feld\nsuggests possibility of alignment of the porphyrin rings as the delocalized pi electrons therein can impart magnetic\nmoment. Studies with apo-systems (porphyrin stripped) may provide some clue of the stated hypothesis and work in\nthis direction is presently in progress.\nLastly, Schr odinger in his seminal book \"What is life\" indicated that the negative entropy from the Sunlight as\nabsorbed by the photoreceptor molecules may solve the thermodynamic riddle. As the higher entropic drive predicted\nby Boltzmann is apparently contradicted in the Darwin's theory which implies generation of order from disorder. The\nsecret of the negative entropy seems to originate from the extreme high e\u000eciency with which the light is converted\ninto chemical energy. The 'quantum material' that does this job is chlorophyll. Magnetic characterization of this\nmolecule naturally elevates the intricacy of the problem to new heights. It implies that the conversion may actually\nbe integrated in nature. Interestingly, magnetic hysteresis of such a nano assembled bio probe has the advantage to\naddress novel applications and methods development. Bio-compatible NC as a \ruorescence (NIRF) enabled sensory\ntool may be exploited to address novel detection techniques, object characterizations, quantum measurements and as\na \row cytometry probe. Quantum biological phenomena may be more common than what is known till today (e.g.\nquantum beats in photosynthesis or avian magneto-sensing).\nVI. ACKNOWLEDGMENTS\nThe authors like to thank Department of Biotechnology, Govt. of India (BT/PR3957/NNT/28/659/2013) for\nproviding funds. We thank DBT-CU-IPLS and CRNN-CU for providing high end infrastructural facilities and in-\nstruments and research. The authors show their gratitude to CoE, CU. We thank Dr. Maitree Bhattacharyya and\nTurban Kar for their support regarding the lifetime experiment.9\nVII. ABBREVIATIONS\nChla, Chlorophyll a; NC, nano chlorophyll; MWNT, multi wall carbon nano tube; GSNO, S-nitroso-glutathione;\nDETA-NONOate, (Z)-1-(N-(2-aminoethyl)-N-(2-ammonioethyl) amino) diazen-1-ium-1,2-diolate; GNP, gold nanopar-\nticle; SNP, silver nanoparticle; O.D, optical density; FACS, \ruorescence-activated cell sorting; FSC, forward scattering;\nSSC, side scattering; FSC (par), parallel forward scattering; FSC (prep), perpendicular forward scattering; LP, long\npass \flter; mT, milli-tesla; Oe, oersted; MS, saturation magnetization; HC, coercivity; ZFC, zero \feld cool; FC, \feld\ncool; SMF, static magnetic \feld; NIRF, near infra-red \ruorescence, TB, blocking temperature.\nVIII. REFERENCE\n1. A. W. Larkum, M. Kuhl: Chlorophyll d: the puzzle resolved. Trends Plant Sci 10 (8) 355-357.1. (2005).\ndoi:10.1016/j.tplants.2005.06.005\n2. N. 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Spaldin2\n1Max Planck Institute for the Structure and Dynamics of Matter, 22761 Hamburg, Germany\n2Materials Theory, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Z ¨urich, Switzerland\n3Institut f ¨ur Physik, Martin-Luther-Universit ¨at Halle-Wittenberg, 06099 Halle/Saale, Germany\n4Forschungszentrum J ¨ulich GmbH, Helmholtz Institute Erlangen-N ¨urnberg for Renewable Energy (IEK-11), 90429 N ¨urnberg, Germany\n5Department of Materials Science and Engineering, McCormick School of Engineering,\nNorthwestern University, 2200 Campus Drive, Evanston, IL 60208, USA\n(Dated: November 13, 2018)\nUsing a combination of first-principles and magnetization-dynamics calculations, we study the effect of the\nintense optical excitation of phonons on the magnetic behavior in insulating magnetic materials. Taking the\nprototypical magnetoelectric Cr 2O3as our model system, we show that excitation of a polar mode at 17 THz\ncauses a pronounced modification of the magnetic exchange interactions through a change in the average Cr-Cr\ndistance. In particular, the quasi-static deformation induced by nonlinear phononic coupling yields a struc-\nture with a modified magnetic state, which persists for the duration of the phonon excitation. In addition, our\ntime-dependent magnetization dynamics computations show that systematic modulation of the magnetic ex-\nchange interaction by the phonon excitation modifies the magnetization dynamics. This temporal modulation\nof the magnetic exchange interaction strengths using phonons provides a new route to creating non-equilibrium\nmagnetic states and suggests new avenues for fast manipulation of spin arrangements and dynamics.\nI. INTRODUCTION\nThe field of non-linear phononics, in which high-intensity\nterahertz (THz) optical pulses are used to drive phonon ex-\ncitations, is of increasing interest1,2. The non-linear pro-\ncesses triggered by the strong phonon excitations have been\nshown repeatedly to introduce complex structural modifica-\ntions in materials, which in turn cause striking and often un-\nexpected changes in properties. Examples include the stimu-\nlation of insulator to metal transitions in correlated oxides3–5\nand the enhancement of superconducting properties in high- Tc\ncuprates6,7and other materials8. In all cases, theoretical stud-\nies combining density functional theory with phenomenolog-\nical modeling have been invaluable in interpreting the exper-\nimental results2,9–13and even in predicting new phenomena,\nsuch as the recent switching14and creation15of ferroic states,\nahead of their experimental observation16.\nIn addition to modifying electronic properties, there are\na number of examples of THz phonon excitation triggering\nmagnetic phenomena on a picosecond (ps = 10\u000012s) time-\nscale. Early results indicate that selective phonon excitations\ncan induce demagnetization processes17,18, and two-phonon\nexcitation19has been shown to excite magnons by the stimu-\nlated rotational motion of atoms13,15. We note that these be-\nhaviors are distinct from ultrafast femtosecond (fs = 10\u000015s)\nspin-flip relaxation processes induced by optical frequency\npulses, such as the pioneering experiments of Refs. [20–\n22], which heat the electronic/lattice sub-system. They are\nalso distinct from the THz excitation of electro-magnons in\nmultiferroics23, in which the electric field of the light pulse\ncouples directly to the dipole moment of the electron-magnon\nquasiparticle.\nIn this work, we address theoretically how the structural\nchanges triggered by the non-linear phononic processes af-\nfect the magnetic energy landscape. We are particularly in-\nterested in the situation in which an excited infra-red-active\nphonon mode couples quadratic-linearly to a Raman-activemode, causing a shift in the average structure that persists for\nthe duration of the phonon excitation. We show that the in-\nduced structure can have a different magnetic ordering from\nthe equilibrium structure, so that the lattice excitation can\ncause a spin-state transition. In addition, we explore the spin\ndynamics induced by the phonon coupling, and show that\nvarious complex spin-flip patterns can be selectively excited\nthrough appropriate choice of the phonon driving frequency.\nIn the next section we review the now well-established\ntheory of non-linear phononics. We then present a model\nthat combines the non-linear phononics formalism with the\nHeisenberg Hamiltonian to describe spin-phonon coupling\nthrough the changes in magnetic exchange interactions that\nare induced by changes in structure. In Section III, we ap-\nply the model to the prototypical magnetoelectric material,\nCr2O3(Fig. 1), using first-principles calculations to obtain all\nthe material-specific parameters. In Section IV we present\nand discuss the analytical solution of the non-linear phononic\nHamiltonian for the lattice dynamics and in Section V the nu-\nmerical simulations of the magnetization dynamics based on\nthe Landau-Lifshitz Gilbert equation24,25. The implications of\nour findings and suggestions for future work are discussed in\nthe Summary.\nII. THEORY\nHere we describe separately the modeling of phononic and\nmagnetic lattice systems before outlining our approach to\nmodeling their coupling. We begin with the description of\nlattice anharmonicity.\nFor large atomic displacements, such as those induced by\nintense optical pulses, the usual harmonic description of lat-\ntice phonons breaks down and higher order anharmonicities\nbecome relevant. The lattice Hamiltonian can then be writtenarXiv:1707.03216v3  [cond-mat.str-el]  5 Mar 20182\nas2,10\nHvib(\u0018IR;\u0018R) =!2\nIR\n2\u00182\nIR+!2\nR\n2\u00182\nR+g\u0018R\u00182\nIR\n+\rIR\n4\u00184\nIR+\rR\n4\u00184\nR;(1)\nwhere!IR,!Rare the frequencies of the infrared (IR) and Ra-\nman (R) modes, \u0018IRand\u0018Rare distortions, \rIR=Rare fourth\norder anharmonic constants and grepresents the coupling be-\ntween two phonon modes. (Terms in \u00183\nRare small and so\nare neglected for conciseness.) The dominant anharmonic re-\nsponse to optical pumping comes from the third-order \u0018R\u00182\nIR\nterm, which has been shown to cause a shift in the potential\nenergy to a finite value of the Raman normal mode coordinate,\ncreating in turn a quasi-static change in the structure2,9,10. For\na single optical pulse, this structural distortion decays and\nthe system relaxes back to the ground state, whereas con-\ntinous driving yields a combination of time-dependent and\ntime-independent structural distortions. We will discuss these\ndistortions later based on the analytical solution of Eqn. (1).\nTo model the magnetic structure we consider a Heisenberg\nHamiltonian with\nHmag=X\nhi;jiJi;j(Si\u0001Sj) +DNX\ni=1(Sz\ni)2; (2)\nwhereSiis a localized spin magnetic moment, Ji;jare the\nmagnetic exchange interactions between spins iandj, and\nDis the uniaxial magneto-crystalline anisotropy (MCA) en-\nergy. We introduce the coupling of the local spin moments\nFIG. 1: (a) Unit cell of Cr 2O3with the red arrows indicating the\nground state antiferromagnetic spin magnetic order. (b) Schematics\nof the phonon-driven change in magnetic ground state: The excita-\ntion of a polar phonon mode ( \u0018IR) induces an increase in the nearest-\nneighbor Cr-Cr bond length by the square-linear anharmonic phonon\ncoupling. The longer bond length results in ferromagnetic exchange\ninteraction between the Cr ions creating a transient change in the\nmagnetic state for the duration of the phonon excitation.contained in the Heisenberg Hamiltonian, to the distortion \u0018of\nthe lattice Hamiltonian by expanding the magnetic exchange\ninteractions with respect to the distortion10,26. For an expan-\nsion up to second order we obtain the following spin-phonon\ncoupling Hamiltonian:\nHsp=X\nhi;ji@Ji;j\n@\u0018(Si\u0001Sj)\u0018+X\ni;j@2Ji;j\n@\u00182(Si\u0001Sj)\u00182:(3)\nNote that the first derivatives of exchange with respect to\nmode\u0018can be zero for certain mode symmetries, and in gen-\neral only the second order derivatives are non-zero. For the\nsymmetry-conserving Raman modes, \u0018R, however, the first\norder spin-lattice coupling is non-zero; note also that these are\nthe modes that have a quadratic-linear lattice coupling with\nthe IR modes in Eqn. (1). Since the \u0018Rdistortion is symmetry\nconserving we can directly write the exchange interaction as\na function of the mode amplitude as\nJi;j(\u0018R) =Ji;j+@Ji;j\n@\u0018R\u0018R+@2Ji;j\n@\u00182\nR\u00182\nR+::: ; (4)\nwith the same labeling as in Eqn. (2). (For a phonon mode of\ngeneral symmetry, either Raman or IR active, the situation is\nmore complicated since the symmetry breaking can split de-\ngenerate exchange interactions, resulting in an increased total\nnumber of inequivalent exchange interaction parameters). In\nprinciple the MCA energy term, D, is also a function of the\nmode amplitude. However, we find that its variation is negli-\ngible for Cr 2O3.\nA. Computational details\nTo calculate the structure, phonons and magnetic ex-\nchange interactions of Cr 2O3we use density functional the-\nory with the local spin density approximation plus Hubbard\nU(LSDA +U) exchange-correlation functional. We use pa-\nrametersU=4 eV andJ=0.5 eV on the Cr- dorbitals and treat\nthe double counting correction within the fully-localized limit.\nThese parameters have been shown to give a good descrip-\ntion of Cr 2O3in earlier work27–29. We use the Vienna ab-\ninitio simulation package (V ASP)30within the projector aug-\nmented wave (PAW) method31using default V ASP PAW pseu-\ndopotentials generated with the following valence-electron\nconfigurations: Cr ( 3s23p64s13d5), O ( 2s23p4). We sam-\nple the Brillouin zone in our total energy calculations us-\ning 11\u000211\u000211 and 9\u00029\u00025k-point meshes for the primi-\ntive rhombohedral and hexagonal cells respectively, and use\na plane-wave energy cutoff of 600 eV . Finally, for computing\nthe MCA energy of Cr 2O3we use an increased k-point grid\nof 14\u000214\u000214 within the rhombohedral cell.\nPrevious theoretical studies of Cr 2O3have addressed the\nmicroscopic origin of the magnetoelectric effect28,32,33and\nthe magnetic properties27,29using a combination of first-\nprinciples density functional theory (DFT) calculations and\neffective Hamiltonian approaches. These studies demon-\nstrated that magnetoelectric properties, phonon frequencies3\nand magnetic exchange interactions, all key quantities in this\nwork, are well described by DFT calculations with technical\ndetails similar to those chosen here.\nWe calculate the atomistic spin-dynamics by solving the\nLandau-Lifshitz Gilbert equation numerically using the Heun\nmethod34with an integration time step that is one thousandth\nof the fasted period of the oscillations (\u0019\u0016\n\rD10\u00003\u00194fs).\nIII. Cr 2O3\nCr2O3crystallizes in the corundum structure which is com-\nposed of a combination of edge- and face-sharing CrO 6oc-\ntahedra. The magnitude 3 \u0016Bspin magnetic moments on\nthed4Cr3+ions order antiferromagnetically below the N ´eel\ntemperature, TN=307 K, in a collinear “ (\";#;\";#)” pattern\nwith magnetic space group R30c(161) that breaks inversion\nsymmetry (Fig. 1)35,36. The primitive unit cell, with its four\nchromium and six oxygen atoms is shown in Fig. 1(a). As a\nresult of its simultaneous breaking of time-reversal and space-\ninversion symmetry, Cr 2O3exhibits the linear magnetoelec-\ntric effect, in which a magnetic/electric field induces an elec-\ntric/magnetic polarization. Indeed, Cr 2O3is considered to be\nthe prototypical magnetoelectric, being the material in which\nthe effect was first predicted37and subsequently measured38.\nA. Calculated lattice properties of Cr 2O3\nWe begin by calculating the lowest-energy structure of\nCr2O3by relaxing its rhombohedral unit cell to obtain a force-\nfree DFT reference structure. We initialized our computations\nusing data from the experimental study of Ref. [39] and op-\ntimized the structure until the forces on each atom were less\nthan 0.01 meV/ ˚A. The resulting structure has a unit cell vol-\nume of 96.46 ˚A3, with the coordinates x= 0:152for Cr and\nx= 0:304for O at the Wyckoff positions 4cand6e, respec-\ntively, in good agreement with literature experimental39and\ntheoretical28values.\nNext, we compute the phonon frequencies and eigenvec-\ntors of our ground-state structure using density functional per-\nturbation theory43. Light radiation only excites polar phonon\nmodes close to the center of the Brillouin zone, q= (0;0;0).\nConsequently, we do not calculate the full phonon band struc-\nture but only the modes at this special point in reciprocal\nspace. Since the primitive cell of Cr 2O3contains 10 atoms,\nthere are 27 non-translational zone-center phonon modes,\nwhich span the irreducible representatives of the 30mpoint\ngroup: 2A1g\n3A2g\n2A1u\n2A2u\n10Eg\n8Eu. Of these\nmodes only the A2uandEumodes are polar, with the dipole\nmoments of the A2umodes pointing along the long rhombo-\nhedral axis ( a+b+c) and those of the Eumodes perpendicular\nto it. TheA1gmodes, which are not directly excitable by light,\nhave the symmetry of the Cr 2O3point group and consequently\nexhibit a square-linear coupling to the polar modes in the an-\nharmonic potential. We list in Tab. I the computed frequencies\nFIG. 2: Displacement pattern of the Cr 2O3phonon modes relevant\nin this work. (a) shows symmetry-conserving A1gmodes, and (b)\ndisplays IR-active A2umodes. The grey arrows show the displace-\nment direction of each atom for the specific mode, with the indicated\ndirections defining positive displacement amplitudes. The notation\nindicates the irreducible representation for the mode symmetry fol-\nlowed in brackets by the calculated mode frequency in THz, rounded\nto the nearest integer.\nofA1gand optically active modes together with available ex-\nperimental frequencies from the literature40–42, and find good\nagreement.\nIn Fig. 2 we show the displacement patterns of the A2u\nandA1gmodes with the grey arrows indicating the direction\nof displacement of the atoms for positive mode amplitude.\nWithin the Cr 2O3structure the 9.3 Thz A1g(A1g(9)) mode\nmodulates the Cr-Cr distance, whereas the higher frequency\n17.3 ThzA1g(17) mode modulates the Cr-O-Cr bond-angles\nvia a rotation of the oxygen octahedra around the rhombohe-\ndral axis. Both polar A2umodes exhibit a collective motion\nof the oxygens along the rhombohedral axis, with the 17.2 Thz\nA2u(17) mode involving the larger relative movement of the\nCr and oxygen atoms. The Cr-Cr bond lengths are unchanged\nby the movement patterns of the polar modes.\nWith our calculated phonon eigenvectors as the starting\npoint, we next compute the anharmonic phonon coupling con-\nstants by mapping the potential of Eqn. (1) onto total en-\nergy calculations of Cr 2O3structures, distorted by appropri-4\nTABLE I: Phonon frequencies of symmetry conserving Raman and\ninfrared-active modes of Cr 2O3in THz. The experimental values\n(EXP) are taken from Refs. [40–42]. The displacement patterns of\ntheA1gandA2umodes are shown in Fig. 2 (a,b).\nsym. DFT EXP\nA1g 9.3 9.0\nA1g 17.3 16.5\nA2g 8.0 –\nA2g 13.8 –\nA2g 20.7 –\nEg 9.2 8.7\nEg 10.7 10.5\nEg 12.4 12.0\nEg 16.1 15.6\nEg 19.2 18.5\nA2u 12.2 12.1\nA2u 17.2 16.0\nEu 9.3 9.1\nEu 13.5 13.2\nEu 17.0 16.1\nEu 19.0 18.2\nate superpositions of the phonon eigenvectors as in previous\nwork10,13. We are primarily interested in the quadratic-linear\ncoupling of Eqn. (1), which is only nonzero if the linear com-\nponent has the full point group symmetry, which is A1gfor\nCr2O3. For convenience, we assume that the radiation is ori-\nented along the rhombohedral axis such that only A2umodes\nare directly excited, then we compute the 2D-potential of\nEqn. (1) for all combinations of polar A2uandA1gmodes.\nIn Fig. 3 (a) we show the computed potential landscape for\nthe combination of the A1g(9) andA2u(17) phonon modes.\nDisplacement of the A2u(17) mode causes a shift of the po-\ntential minimum of the A1g(9) mode, as shown in the cuts of\nthe 2D-potential in Fig. 3 (b). The red dashed line in Fig. 3 (a)\nshows the position of the A1g(9) mode minimum within the\n2D potential landscape. For negative and positive amplitudes\nof theA2u(17) mode, the potential minimum position shifts to\npositive amplitudes of the A1g(9), corresponding to a negative\nsign of the square-linear coupling. We quantify this observa-\ntion by fitting the complete potential landscape using Eqn. (1),\nto extract all anharmonic coupling constants and repeat the\ncalculation for all combinations of A2uandA1gmodes. The\ncomputed anharmonic constants are given in Tab. II.\nWe find that the nominal value of the quadratic-linear an-\nharmonic coupling gvaries from 6 to 101 meV/(pu˚A)3and\nexhibits positive or negative sign, so that modulations of the\nCr2O3structure with positive and negative amplitudes of the\nA1gmodes can be induced by exciting the appropriate polar\nmode. (Note that the opposite choice of sign in the definition\nof the phonon eigenvectors would reverse the sign of g; the\nsigns given in Tab. II correspond to the phonons as defined in\nFIG. 3: (a) Calculated two-dimensional potential surface of the\nanharmonic phonon-phonon interaction between \u0018R=A1g(9) and\n\u0018IR=A2u(17). The red line in the three dimensional plot shows\nthe position of the potential minimum. (b) selected cuts through the\ntwo dimensional potential surface shown in (a). Note that we plot\n\u0001V(\u0018IR;\u0018R) =V(\u0018IR;\u0018R)\u0000V(\u0018IR;\u0000g\u00182\nIR=(2!2\nR)), so that the\nminimum is set to 0 meV .\nTABLE II: Upper panel: Anharmonic coupling constants g, in units\nof [meV/(pu˚A)3], between symmetry conserving A1gand IR active\nphonon modes of A2usymmetry. Lower panel: quartic anharmonic\nconstants,\r, in units of [meV/(pu˚A)4].\nmodes A2u(12) A2u(17)\nA1g(9) 6 -86\nA1g(17) -38 101\nmodes A1g(9)A1g(17)A2u(12)A2u(17)\n\rIR 1 4 4 14\nFig. 2).\nMinimization of Eqn. (1) gives the amount of induced struc-\ntural distortion to be \u0018R\u0019 \u0000g\u00182\nIR=!2\nR. Consequently, for\nthe combination of polar A2u(12) andA1g(9) modes, excita-\ntion of the polar mode induces, due to the positive coupling\nconstantg, a negative amplitude of the A1g(9) mode which\nresults in a decrease in the nearest-neighbor Cr-Cr distance.5\nIn contrast, the A2u(17) mode couples with a negative cou-\npling constant gto theA1g(9) mode and so the induced quasi-\nequilibrium structure has an increased Cr-Cr distance. The\nA1g(17) mode changes the oxygen octahedral rotation angles\naround the Cr ions. Its negative coupling to the A2u(12) mode\nresults in a decreased rotational angle, whereas the positive\ncoupling to the A2u(17) mode increases the rotational angle\nin the quasi-equilibrium structure.\nB. Calculated magnetic properties of Cr 2O3\nThe fact that the transient structure generated through the\nquadratic-linear coupling of the optically excited polar modes\nto theA1g(9) Raman mode has a modified Cr-Cr distance sug-\ngests that it might also have a different magnetic ground state.\nTo explore this possibility, we next calculate the energy dif-\nference between the AFM ground-state ordering (\";#;\";#)\nand two other magnetic orderings of the Cr spins – ferromag-\nnetic (FM) (\";\";\";\")and another antiferromagnetic (AFM 1)\n(\";\";#;#)– as a function of the A1g(9) distortion amplitude.\nFor the equilibrium structure, we find that the AFM 1state\nis 67 meV and the FM state 162 meV in energy above the\nAFM ground state. Modulating the structure with the pattern\nof atomic displacements corresponding to the A1g(9) phonon\nmode in the positive direction, so that the Cr-Cr nearest-\nneighbor distance, ( dCr\u0000Cr), is increased, significantly low-\ners both of these energy differences. For positive ampli-\ntudes larger than \u0018R\u00150.75pu˚A, corresponding to a stretch-\ning ofdCr\u0000Cr=0.06 ˚A, the energy of the AFM 1state be-\ncomes lower than the AFM ground state; at larger amplitudes\n(\u0018R\u00151.9pu˚A) the FM state becomes lower in energy than\nthe original ground state, but remains higher in energy than\nthe AFM 1state. We therefore predict that a crossover to\nthe AFM 1state should be achievable through quadratic-linear\ncoupling with appropriate choice of the polar mode excitation\nfrequency and intensity. (Note that modulating the structure\nwith a negative amplitude of A1g(9), which decreases the Cr-\nCr nearest-neighbor bond, increases the relative energies of\nthe FM and AFM 1states). In contrast, modulating the struc-\nture along the eigenvector of the second A1gmode at 17 THz,\nor along those of the polar A2umodes has only a small effect\non the magnetic energy landscape.\nTo explore the magnetic behavior further, we next calcu-\nlate the magnetic exchange interactions of the ground-state\nstructure using the Heisenberg Hamiltonian of (2), including\nmagnetic exchange interactions, Jn, up to fifth nearest neigh-\nbors, as shown in Fig. 4; this Hamiltonian has been shown to\ngive an accurate theoretical description of the magnetoelec-\ntric effect and magnetic transition temperature of Cr 2O327,28.\nSpecifically, our Heisenberg Hamiltonian for the magnetic ex-changes reads:\nHexch\nCr2O3=J1(S1\u0001S2+S3\u0001S4)\n+ 3J2(S1\u0001S4+S2\u0001S3)\n+ 3J3(S1\u0001S2+S3\u0001S4) (5)\n+ 6J4(S1\u0001S3+S2\u0001S4)\n+J5(S2\u0001S3+S1\u0001S4);\nwith theJnas shown in Fig. 4, and the labeling of spins as\nin Fig.1. We extract the magnetic exchange interactions from\nthe total energy differences between four distinct magnetic ar-\nrangements within the non-primitive hexagonal cell, using the\napproach of Ref. [44]. The resulting magnetic exchange inter-\nactions are listed in Tab. III and are in agreement with earlier\ntheoretical works27–29. We find the nearest and next-nearest\nneighbor interactions, J1andJ2, to be strongly antiferromag-\nnetic, whereas J3andJ4favor ferromagnetic arrangements.\nThe furthermost exchange interaction that we consider, J5, is\nweakly antiferromagnetic. Finally, we note that, in contrast to\nother magnetic insulators45, higher-order magnetic exchanges\nsuch as four-body interactions are not required for the descrip-\ntion of the magnetism in Cr 2O329.\nNext, we compute how the modulation of the Cr 2O3struc-\nture by the phonon mode eigenvectors changes the magnetic\nexchange interactions, using the same approach to extract the\nexchange interactions as we used above for the ground-state\nstructure. (For the A2umodes we neglect the small splittings\ninJvalues that result from the lowered symmetry.) Our cal-\nculated coefficients of the expansion of Eqn. (4), listed up\nto quadratic order in \u0018in Table III, are a measure of the\nspin-phonon coupling for each mode. In Fig. 4 (b,c), we\nplot the five nearest-neighbor magnetic exchange constants\nas a function of the A1g(9) andA2u(17) phonon mode am-\nplitudes. We find that the A1g(9) mode significantly changes\nthe nearest-neighbor exchange interaction, whereas the longer\nrange magnetic exchange interactions are less affected by the\nstructural modulation. An intriguing result is the sign change\nof the nearest-neighbor exchange interaction J1at amplitudes\n\u0018\u00150.75pu˚A, corresponding to an increase of 0.06 ˚A in the\nCr-Cr bond length, consistent with the crossover to AFM 1or-\ndering that we found above. In contrast to the A1g(9) mode\nwe see that the A2u(17) mode has minimal direct effect on\nthe magnetic exchange interactions. The other A1gandA2u\nmodes (not shown) also have minimal effect on the exchange\ninteractions. The spin-phonon coupling constants obtained by\nfitting these results to Eqn. (4) are listed in Tab. III; as ex-\npected the coefficients of J1for theA1g(9) mode are large.\nWe can understand the strong J1response by analyzing the\ndisplacement pattern of the A1g(9) mode in the context of the\norigin of the J1magnetic exchange interaction that has been\ndiscussed in the literature. Earlier analysis of the magnetic\ninteractions in the ground-state of Cr 2O327showed that the\nmain contribution to J1arises from an antiferromagnetic di-\nrect exchange interaction between the nearest Cr atoms com-\nbined with a small ferromagnetic superexchange component\nfrom the 82\u000eCr-O-Cr interaction. For positive amplitudes of\ntheA1g(9) mode, the Cr-Cr distance increases thus decreas-\ning the antiferromagnetic direct exchange interaction. At the6\nFIG. 4: (a) Illustration of the magnetic exchange interactions in Cr 2O3, from first to fifth nearest neighbor. (b,c) Changes in the magnetic\nexchange interactions due to structural modifications by the A1g(9) andA2u(17) modes. Note that for the A1g(9) mode the nearest-neighbor\nmagnetic exchange ( J1) changes sign for negative phonon mode amplitudes.\nTABLE III: Upper panel: Magnetic exchange interactions (meV) for\nthe ground-state structure of Cr 2O3. Lower panel: Spin-phonon cou-\npling constants (units meV/(pu˚A) and meV/(pu˚A)2for first/second\norder) for the A1gandA2umodes of Cr 2O3.\nJ1 J2 J3 J4 J5\n25.4 21.2 -3.9 -3.3 4.2\nn 1 2 3 4 5\nA1g(9)\n@Jn=@\u0018 -57.9 -4.4 -0.1 -0.6 -1.0\n@2Jn=@\u0018214.5 0.2 0.1 0.0 0.1\nA1g(17)\n@Jn=@\u0018 12.4 1.4 -0.0 0.1 0.3\n@2Jn=@\u001823.9 0.4 0.1 0.0 0.0\nA2u(12)\n@2Jn=@\u001820.5 0.1 -0.1 0.0 0.2\nA2u(17)\n@2Jn=@\u00182-0.7 0.1 0.0 0.0 0.0\nsame time, the Cr-O-Cr angle becomes closer to 90\u000eenhanc-\ning the ferromagnetic superexchange. The result is a change\nin sign ofJ1. We note that this observation is possibly con-\nnected to the findings of Ref. [27], in which strong modula-tions of magnetic energies induced by small changes of the\nCr2O3ground-state structure were reported. Moreover, since\nthe direct magnetic exchange interaction only affects J1, the\nmagnetic exchange interactions Jnwithn\u00152are less af-\nfected by the structural distortion.\nFinally, we calculate the MCA energy of Cr 2O3, from\nthe energy difference between alignment of the Cr spin mo-\nments along ( Ejj) and perpendicular ( E?) to the rhombohe-\ndral axis, including the spin-orbit interaction in our calcula-\ntions. We obtain an energy difference Ejj\u0000E?= -27\u0016eV; the\nexperimental35,46,47values range from -12 \u0016eV to -16\u0016eV . We\nalso calculate the change in MCA energy when the structure\nis modulated by the A1gorA2uphonon modes and find no\nsignificant change (a mode amplitude of \u0018=\u00062pu˚A lowers\nthe MCA energy by \u001910 %). In particular, the rhombohedral\neasy axis is preserved upon structural modulation. This find-\ning justifies our omission of MCA terms in our spin-phonon\nHamiltonian, Eqn. (3).\nTo summarize this section, we find a strong dependence\nof theJ1nearest-neighbor magnetic exchange interaction on\nthe structural distortion associated with the A1g(9) mode,\nwith positive mode amplitude, corresponding to increased\nCr-Cr distance, inducing a change in sign. This depen-\ndence leads to a crossover between antiferromagnetic states.\nSince theA1g(9) mode couples quadratic-linearly to the A2u\nmodes, this crossover can be induced by optical excitation of\nthe polar modes. Following the classical considerations de-\nrived in Refs. [10,13], we estimate that a pulse fluence of7\n\u001840 mJ/cm2at a frequency 17 THz should be sufficient to in-\nduce this crossover transition. A similar fluence was reported\nin Ref. [16] without damaging the sample.\nIV . NON-LINEAR LATTICE DYNAMICS\nHaving established that the structural modification induced\nvia non-linear phononic coupling can lead to a change in mag-\nnetic ordering, we next evaluate the dynamical behavior asso-\nciated with driving a phonon. We begin by calculating the\nnon-linear lattice dynamics using the vibrational crystal po-\ntential given in Eqn. (1), followed by the resulting spin dy-\nnamics. We study the case in which an IR mode is excited\nby a sinusoidal driving force F(t)with amplitude Edrive with\nfrequency \nand calculate the resulting dynamics of the cou-\npled R mode, focussing particularly on the combination of the\nA2u(17) andA1g(9) which yields a negative amplitude A1g(9)\ndisplacement and possible ferromagnetism. The time evolu-\ntion of the system described by the potential of Eqn. (1) is\nthen governed by the following set of differential equations:\n\u0018IR+!2\nIR\u0018IR+\rIR\u00183\nIR= 2g\u0018IR\u0018R+F(t); (6)\n\u0018R+!2\nR\u0018R+\rR\u00183\nR=g\u00182\nIR; (7)\nF(t) =Edrivesin(\nt): (8)\nWe derive a closed analytical solution of the dynamic equa-\ntions in the limit in which the coupling and the anharmonicity\nare small relative to the frequency, that isg\rIR\rR\n!IR!R\u001c1by\nfollowing the approach of Ref. [48]. For the case of Cr 2O3,\nourab initio values, provided in Table I and II, indicate that\nthe combination of A1gwith polarA2umodes fulfills this cri-\nterion. The dynamics of the IR mode are then given by:\n\u0018IR(t) =AIRsin[e!IRt] +A\nsin[\nt]\n+gAIRAR\n2[!2\nIR\u0000(!IR\u0000!R)2]cos[(e!IR\u0000e!R)t]\n+gAIRAR\n2[!2\nIR\u0000(!IR+!R)2]cos[(e!IR+e!R)t](9)\n+gA\nAR\n2[!2\nIR\u0000(\n\u0000!R)2]sin[(\n\u0000e!R)t]\n+gA\nAR\n2[!2\nIR\u0000(\n +!R)2]sin[(\n + e!R)t];\nand those of the R mode by:\n\u0018R(t) =\u0018R0+ARcos[e!Rt]\n+gA2\nIR\n4[!2\nR\u00004!2\nIR]cos[2e!IRt]\n+gA2\n\n4[!2\nR\u00004\n2]cos[2\nt] (10)\n+gAIRA\n2[!2\nR\u0000(\n +!IR)2]sin[(\n + e!IR)t]\n+gAIRA\n2[!2\nR\u0000(\n\u0000!IR)2]sin[(\n\u0000e!IR)t]:The time-independent displacement of the R mode oscillation\nis given by\u00180=g(A2\nIR+A2\n\n)=(4!2\nR), with the amplitude fac-\ntors,AIRandARdepending on the initial amplitudes, \u0018R(0)\nand\u0018IR(0)andA\n= 1=(!2\nIR\u0000\n2). We indicate frequencies\nwith a tilde which have been renormalized by the anharmonic\ncoupling, as given by Eqns. (A.1) and (A.2) in the Appendix.\nThe solution shows that the anharmonic potential and the\ncoupling between the phonon modes induce motions of the\noscillators which display several components given by cosine\nand sine terms. Each of these terms corresponds to a single\ncomponent of the motion with a specific amplitude and fre-\nquency – either the renormalized original frequency of each\noscillator, indicated by the tilde, or sums or differences of the\noriginal frequencies. We emphasize that these motions arise\nfrom a single mode, which exhibits multiple frequencies be-\ncause of its anharmonicity and coupling.\nNext we analyze the frequencies and amplitudes of each\nterm in Eqn. 10 for the A1g(9) R mode. In Fig. 5 (a,b) we\nshow the frequencies and relative amplitudes of the R mode\nmotions as a function of the drive frequency \n, obtained us-\ning the parameters for the A2u(17) (IR) –A1g(9) (R) coupled\nphonon modes. Note that the only effect of the external driv-\ning amplitude, Edrive, is to scale the amplitude of the mo-\ntion. We see that for drive frequencies close to the 17 THz\neigenfrequency of the IR mode, the frequencies of the R mode\ncomponents range from sub THz to 40 THz (note the logarith-\nmic scale in the lower part of Fig. 5 (a)), with the highest fre-\nquency components at around 34 THz being twice the renor-\nmalized IR mode eigenfrequency (blue line), twice the driver\nfrequency (green dashed line) and the sum of the renormal-\nized IR mode and driver frequencies (red dashed-dotted line).\nThe renormalized R mode frequency is close to 10 THz, and\nlike the renormalized IR mode frequency is independent of the\ndrive frequency. The frequency of the lowest frequency com-\nponent of the motion is given by the difference between the\nfrequency of the driver and the IR mode eigenfrequency, and\nas a result it has a strong dependence on the drive frequency,\nbecoming small as the drive frequency approaches the eigen-\nfrequency of the IR mode (note that the divergence when the\ndrive frequency equals the eigenfrequency of the IR mode is\nnot physical, and arises because of the absence of damping in\nour simulations.)\nIn Fig. 5 (b) we show the relative amplitudes of each fre-\nquency component normalized to the time-independent dis-\nplacement\u0018R0of\u0018R(t)which we set to 100 %. We see a large\nspread in amplitudes for the different components of the mo-\ntion, with the motions with the renormalized IR and R eigen-\nfrequencies having the largest amplitudes, in the order of 10 to\n20 % of\u0018R0, as well as minimal dependence on the drive fre-\nquency. The other three motion components, whose frequen-\ncies depend explicitly on the drive frequency, have strongly\ndrive-frequency-dependent amplitudes, as expected. Of these,\nthe high-frequency 2 \nmotion has the smallest amplitude fol-\nlowed by the \n +e!IRmotion, with the slow \n\u0000e!IRmo-\ntion having the largest amplitude, becoming similar in size\nto thee!IRande!Rmotions in the vicinity of the eigenfre-\nquency of IR mode. Again, the divergence when the mode\nfrequency matches the driver frequency results from the ab-8\nFIG. 5: (a) Frequencies, !, and (b) relative amplitudes, \u0018(normal-\nized to\u0018R0), of the five separate parts of the R mode motion as a\nfunction of the external driving frequency. Note the separation and\nthe linear and logarithmic scales in (a).\nsence of damping in our model, and so we do not analyze this\npoint in detail.\nTo summarize this section, we find that, in addition to the\ntime-independent offset \u0018R0of the R mode induced by its\nquadratic-linear coupling to the IR mode, the R mode has a\ncomplex oscillatory motion made up of different frequencies.\nThe largest amplitude motions have high frequencies, set by\nthee!IRande!Rfrequencies. Close to resonance between the\ndrive and IR-mode frequencies, an additional component of\nthe motion with a slower frequency \n\u0000e!IRalso develops a\nsignificant amplitude. This slow motion is particularly inter-\nesting since it is tunable in amplitude and frequency by the\nexternal driver; in the next section we will explore how it can\nbe exploited to engineer the spin dynamics.\nV . SPIN DYNAMICS\nNext we discuss how the structural modulations we de-\nscribed above drive the spin dynamics, by combining our\nfindings for the structural dynamics with those for the spin-\nphonon coupling. The time-dependent exchange modulationinduced by the structural modulation is obtained by combin-\ning Eqn. (4) for Jn(\u0018)with Eqn. (10) for \u0018(t)to yield the\nJn(\u0018(t)). We include only the modulations caused by the\nA1g(9) R mode. While this mode is driven by the excita-\ntion of theA2u(17) (IR) mode, the latter has negligible ef-\nfect on the exchange interactions, and so the time-dependent\nmagnetic exchange modulations are dominated by Jn(\u0018R(t)).\nFor the spin-dynamics we consider a single Cr 2O3unit-cell\nwith four magnetic Cr sites and periodic boundary conditions.\nRewriting the Heisenberg Hamiltonian of Eqn. (5) we obtain\nHmag;exch\nfac(\u0018R(t)) = ~J1(\u0018R(t))(S1\u0001S2+S3\u0001S4)\n+~J2(\u0018R(t))(S1\u0001S4+S2\u0001S3)(11)\n+~J3(\u0018R(t))(S1\u0001S3+S2\u0001S4);\nwhere the net magnetic exchange interactions ~Jiare given by\n~J1(\u0018R(t)) =J1(\u0018R(t)) + 3J3(\u0018R(t))\n~J2(\u0018R(t)) =3J2(\u0018R(t)) +J5(\u0018R(t)) (12)\n~J3(\u0018R(t)) =6J4(\u0018R(t))\nandS1toS4are the four classical spins in the unit cell as\nshown in Fig. 1 (a). Our full magnetic Hamiltonian is then\nHmag(t) =Hmag;exch\nfac(\u0018R(t)) +D4X\ni=1(Sz\ni)2(13)\nwhere\u0018R(t))denotes the specific time-dependent exchange\ninteraction strength (see Eqn. (12) and Table III) and Dis\nthe MCA energy which we fixed to the computed equilibrium\nvalue.\nWe next calculate the classical magnetization dynamics\nusing the Landau-Lifshitz-Gilbert equation24,25,49within an\natomistic approach50,51\ndSi\ndt=\u0000\r\n1 +\u000b2h\nSi\u0002He\u000b\ni(t)i\n\u0000\u000b\r\n1 +\u000b2h\nSi\u0002h\nSi\u0002He\u000b\ni(t)ii\n:(14)\nHereHe\u000b\ni(t) =\u00001\n\u0016B\u000eHmag;exch\nfac(t)\n\u000eSiwith\u0016Bthe Bohr magne-\nton,\ris the electron gyromagnetic ratio and \u000bis the Gilbert\ndamping. We take the value of Gilbert damping for Cr 2O3,\n\u000b\u0019\r\n2\u0019\u0017\u0001Bpp\u00190:07, estimated from spectral line widths\nmeasured at room temperature using electron paramagnetic\nresonance36.\nNext, we calculate ~Ji(\u0018R(t))when the structure is modified\nby the quadratic-linear coupling of the A1g(9) andA2u(17)\nphonon modes. We excite the latter using a continuous field\nof strength of Edrive =0.6 MV/cm oscillating at a frequency\nof 16.9 THz; the result is shown in Fig. 6 (a). Note that we\ninclude a noise corresponding to a temperature of 0.1 K in our\nspin-dynamics simulation to prevent the system from becom-\ning stuck in shallow metastable minima52.\nBefore the mode is excited (at t= 0), all ~Jiare constant,\nwith ~J1and~J2positive and ~J3negative. When the oscillat-\ning field is applied, the frequency-dependent induced struc-\ntural changes described in the previous section change ~Jicor-\nresponding to the changes in bond lengths and angles. We9\nFIG. 6: Spin dynamics in the phonon-driven state of Cr 2O3. (a) Mod-\nulation of the magnetic exchange interaction ~Ji(\u0018R(t)) with\u0018R(t)\nderived from Eqn. (10). The laser field, E(t), is switched on at time\nt= 0, withEdrive =0.6 MV/cm at \n =16.900 THz. The change in\nsign of the average exchange for ~J1is clearly visible; note that be-\ncause of the fast oscillating components ( !\u001530 THz) of the \u0018Rmo-\ntion the time-dependent exchange interaction can not be resolved on\nthis scale. (b,c) Time dependence of the z-components (normalized\nto their ground-state values) of the four spin magnetic moments in\nthe Cr 2O3unit cell with the labeling corresponding to Fig. 1 (a). The\nblue spheres (Cr atoms) and red arrows (spins) represent the Cr 2O3\nmagnetic ground state.\nsee that, while the magnetic exchange interactions oscillate,\ntheaverage magnetic exchange interactions ~J1change sign\nto negative, reflecting a net ferromagnetic interaction between\nthe nearest neighbor sites. Since the next-nearest neighbor in-\nteraction ~J2still prefers an AFM alignment the system does\nnot become fully FM but instead adopts the AFM 1state with\nits(\";\";#;#)ordering of magnetic moments on the Cr sites.\nThis is consistent with the cross-over to the AFM 1state with\nincreasingA1g(9) amplitude that we saw in the first part of\nthis paper.\nThe remaining panels of Fig. 6 show the response of the\nspin system to this modification of ~Ji, with (b) and (c) show-\ning the time evolution of the z-component of magnetization\nof the individual Cr ions and (d) that of the total spin mo-\nment of the unit cell, Stot;z(t) =1\nNPN\ni=1Si;z(t). The spins\nreact to the change in their average exchange modulation\nand reorient on the same time scale as the J-oscillations into\nthe new AFM arrangement. Note that this process is at the\nsame speed of the period of the exchange excitation oscilla-\ntions, 2\u0019=e!R\u00190.1 ps. The AFM arrangement then achieves a\nsteady state without further dynamical evolution provided that\nthe displacement of the A1g(9) continues by excitation of the\nFIG. 7: Spin dynamics in the phonon-driven state of Cr 2O3. (a)\nModulation of the magnetic exchange interaction ~Ji(\u0018R(t))with\n\u0018Rderived from Eqn. (10). The laser field E(t)is switched on at\ntimet= 0, withEdrive =0.6 MV/cm at \n = 16.995 THz. Setting\nthe excitation frequency closer to resonance induces a slow, large-\namplitude modulation component in the \u0018Rmotion, which becomes\nresolvable in the time-dependent magnetic exchange. (b,c) Time de-\npendence of the z-component of the four spin magnetic moments in\nthe Cr 2O3unit cell with the labeling corresponding to Fig. 1 (a) and\nthe spin magnitudes normalized to their static ground-state values.\nWe illustrate the Cr 2O3magnetic ground state by the blue spheres\nrepresenting the Cr atoms with the arrows showing the magnetic mo-\nments.\nA2u(17) phonon mode.\nNext, we exploit our finding from Section IV that a compo-\nnent of the R mode motion can be tuned to low frequency with\nan increased amplitude by selecting a drive frequency \nclose\nto resonance. In Fig. 7 (a) we show the time dependence of ~Ji,\ncalculated for the combination of A1g(9) andA2u(17) modes\nusing the analytical solution of Eqn. (10), this time with the\ndriving frequency, \n = 16.995 THz, close to resonance. It is\nclear that the oscillation frequency of the exchange interac-\ntions develops a significant slow component with a frequency\naround 10GHz. The resulting spin dynamics are depicted\nin the lower panels of Fig. 7. In contrast to the case shown\nin Fig. 6, a steady AFM 1state is not achieved on pumping,\nand instead the spins exhibit a flipping between up and down\nalignment. Again the spin dynamics behavior persists as long\nas the phonon mode is driven.\nIn conclusion, our calculations indicate that the quadratic-\nlinear coupling between the A1g(9) andA2u(17) modes leads\nto a reversal of the average value of the nearest-neighbor ex-\nchange between the Cr ions when the optical A2u(17) mode\nis continuously excited with sufficiently large amplitude. De-\npending on the closeness of the excitation laser frequency to10\nthe eigenfrequency of the A2u(17) mode, the additional oscil-\nlatory component of ~Ji(t)can be either fast or slow. In the\nfirst case the system responds with a steady-state change in its\nmagnetization to a ferromagnetic state; in the second an alter-\nnating switching occurs on a tens of picoseconds time scale.\nWe note that the two limits shown here represent a small frac-\ntion of spin-dynamic possibilities, with the tuning of the drive\nfrequency relative to the resonance, as well as on-off schemes\nfor the excitation, offering the potential to modulate the ex-\nchange interactions in multiple complex ways.\nVI. SUMMARY\nWe calculated the structural and magnetic responses of\nchromium oxide, Cr 2O3, to intense excitation of its optically\nactive phonon modes. Using a general spin-lattice Hamil-\ntonian, with parameters calculated from first principles, we\nshowed that the quasi-static structural distortion introduced\nthrough the non-linear phonon-phonon interaction can change\nthe magnetic state from its equilibrium antiferromagnetic to a\nnew antiferromagnetic ordering with ferromagnetically cou-\npled nearest-neighbor spins. This transition is driven by\nthe change in nearest-neighbor magnetic exchange interaction\nwhen the Cr-Cr separation is modified through non-linear cou-\npling of the optical phonons to a symmetry-conserving A1g\nRaman-active mode.The new antiferromagnetic ground state\npersists for as long as the system is continuously excited, pro-\nvided that the excitation frequency is faster than the magneticrelaxation time.\nRegarding dynamics, we find that the motion of the ex-\ncited optical modes and coupled Raman-active mode can be\ndecomposed into several different frequencies which depend\nstrongly on the difference between the excitation and reso-\nnance frequencies. This sensitivity of the response to the input\nfrequency allows selection of complex vibrational frequency\npatterns which can lead to additional components in the spin\ndynamics, for example flips of the Cr spin lattice.\nWe emphasize that we explored in this work a minimal\nmodel of phonon-driven spin dynamics, and we expect that\nextensions of the model will reveal yet more complex physics,\nsuch as dynamically frustrated or spin-spiral states. We\nhope that our work will inspire additional theoretical and ex-\nperimental studies to uncover the rich behavior of coupled\nmagneto-phononic systems.\nVII. ACKNOWLEDGMENTS\nThis work was supported financially by ETH Zurich, the\nERC Advanced Grant program, No. 291151 (MF, CK and\nNAS), the ERC under the European Union’s Seventh Frame-\nwork Programme (FP7/2007-2013) / ERC Grant Agreement\nn\u000e319286 (Q-MAC) and by the DFG through SFB762 and\nTRR227. 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Detailed expressions for the renormalized frequencies\nThe explicit expressions for the mode frequencies renor-\nmalized by the anharmonic coupling are\ne!IR=!IR\u0000g2A2\nR\n8!2\nR!IR+g2A2\n\n4!2\nR!IR\u0000g2AIR\n16!IR[!2\nR\u00004!2\nIR]\u0000g2A2\n\n4!IR[!2\nR\u0000(\n +!IR)2]\u0000g2A2\n\n4!IR[!2\nR\u0000(\n\u0000!IR)2]\n\u0000g2A2\nR\n8!IR[!2\nIR\u0000(!IR+!R)2]\u0000g2A2\nR\n8!IR[!2\nIR\u0000(!IR\u0000!R)2]+3\rIRA2\nIR\n8!IR+3\rIRA2\n\n4!IR; (A.1)\nfor the IR mode, and\ne!R=!R\u0000g2A2\nIR\n8!2\nR[!2\nIR\u0000(!IR+!R)2]\u0000g2A2\nIR\n8!R[!2\nIR\u0000(!IR\u0000!R)2]\u0000g2A2\n\n8!R[!2\nIR\u0000(\n\u0000!R)2]+3\rRA2\nR\n8!R; (A.2)\nfor the R mode."    },    {        "title": "1707.04496v1.Electric_field_controlled_magnetic_exchange_bias_and_magnetic_state_switching_at_room_temperature_in_Ga_doped_α_Fe2O3_oxide.pdf",        "content": "1 \n Electric field controlled magnetic exchange bias and magneti c state switching at \nroom temperature in Ga doped α-Fe2O3 oxide  \nR.N. Bhowmik*, and Abdul Gaffar Lone  \nDepartment of Phys ics, Pondicherry University, R. V. Nagar, Kalapet, Pondicherry -\n605014, India  \n*Corresponding author: Tel.: +91 -9944064547; Fax: +91 -413-2655734  \nE-mail: rnbhowmik.phy@pondiuni.edu.in  \nAbstract  \nWe have developed a new magnetoelectric material based on Ga doped α-Fe2O3 in \nrhombohedral phase . The material is a  canted ferromagnet  at room  temperature  and showing \nmagneto -electric properties . The experimental results of electric field controlled magnetic \nstate provide d a direct evidence of room temperature magnetoelectric coupling in Ga doped \nα-Fe2O3 system . Interestingly, (un -doped) α-Fe2O3 system does not exhibit any electric field \ncontrolled magnetic exchange bias shift, but Ga doped α-Fe2O3 system has show n an \nextremely high electric field induced magnetic exchange bias shift up to the value  of 1120 Oe  \n(positive) . On the other hand, i n a first time, we report the electric field controlled magnetic \nstate switching both in α-Fe2O3 and in Ga doped α-Fe2O3 systems . The switching of magneti c \nstate is highly sensitive to ON and OFF modes, as well as to the change of polarity of applied \nelectric voltage during in -field magnetic relaxation experiments . The switching of magnetic \nstate to upper level  for positive electric field and to down level for negative electric field \nindicates that electric and magnetic orders are coupled in the Ga dop ed hematite system. Such \nmaterial is of increasing demand in today for multifunctional  applications in next generation \nmagnetic sensor, switch ing, non -volatile memory and spintronic devices.  \nKeywords: Ga doped hematite,  Rhombehedral structure, Exchange bias, Room temperature \nmagneto -electric s, Electric field controlled magneti c state . \n \n 2 \n 1. Introduction  \nThe conventional spintronics devices  uses spin-transfer torque technique , where spin -\npolarized current or magnetic field controls  magnetic state and switching of magnetization  \n[1]. This is high power consuming process . The main problem of the current or magnetic \nfield controlled magnetization switching is the Joule heating effect during operation of the \nspintronics devices . Alternatively , electric field controlled switching of magnetization is an \nenergy -efficient technique (less Joule heating effect ) for the development of low power \nconsumption spintronic devices with additio nal non -volatile functionality [2]. The electric \nfield controlled magnetization state can be achieved in a special class of magneto -electric s, \nknown as m ultiferroelectrics , where  magnetization  and electric polarization  are strongly \ncoupled in a crystal structure. Although ferroelectric order (needs empty d shell ions) and \nmagnetic order (needs partially filled d shell ions) are mutually exclusive in nature in a \ncrystal structure, some of the magnetic oxides (BiFeO 3, TbMnO 3, CoCr 2O4) showed signature \nof multi -ferroelectric properties [ 3-4]. Among them , only few oxides have shown direct or \nindirect evidences of magneto -electric coupling where magnetic state has been controlled by \nelectric field [ 5]. The concept of magneto -electric effect was emerged from the land mark \ndiscoveries  of induced magnetism under ele ctric field and induced electric polarization under \nmagnetic field in a moving dielectric material [ 6]. Antiferromagnetic Cr 2O3 was the first \nmaterial  that showed linear magneto -electric effect [ 7-8], which is not strong enough for \nroom temperature applica tion. The materials with room temperature magneto -electric  \nproperties are of increasing interest in the field of spintronics, non -volatile memory devices, \ndata storage  (electric ally writing and magnetic ally reading) , and sensor applications [ 9-11].  \nRecently, s ome hetero -structured materials, either naturally exist  (β-NaFeO 2 [12]) or \ndesigned superlattice s ((LuFeO 3)9/(LuFe 2O4)1 [13], Ti0.8Co0.2O2/Ca 2Nb3O10/Ti0.8Co0.2O2 [14]) \nor theoretically  predicted (R 2NiMnO 6/La 2NiMnO 6 [15]) exhibit ed electric field controlled 3 \n magnetic exchange bias and switching at room temperature. These engineered magneto -\nelectrics are mostly multi -layered structure of ferromagnetic( FM)/antiferromagnetic ( AFM ) \nfilms on ferroelectric (FE) substrate. BiFeO 3 is a well known single -phased oxide  that \nestablished electric field controlled magnetic state near to room temperature. Recent reports \n[15-17] attributed the observed strong magnetoelectric coupling in the thin film of BiFeO 3 to \nlarge strain induced anisotropy generated at the interfaces of substrate  and film. Otherwise , \nmagnetoelectric coupling in bulk BiFeO 3 is very poor. Some ferrites, known as hexaferrite s, \nwith hexagonal  structure (M -type: BaFe 10.2Sc1.8O19 [18] and SrFe 12O19 [19], Z -type \nSr3Co2Fe24O41 [20]) indicated  electric field controlled magnetic state at room temperature  and \nin most of the cases at low er temperatures  [21]. At this point, we mention that derivatives of \nhematite (α -Fe2O3), which  is an anisotropic electrical insulator , could be interesting for \ndeveloping material with electric field controlled magnetic switching  properties . Some of t he \nhematite derived systems (GaFeO 3, FeTiO 3) in orthorhombic phase showed magnetoelectric  \nproperties  at low er temperature s [22-24] and some compositions in thin film form (e.g., Mn \ndoped Ga0.6Fe1.4O3:Mg [25] and GaFeO 3 [26]) showed magnetoelectric properties at room \ntemperature . Further, hematite (α -Fe2O3) derivatives in corundum (rhombohedral) structure \noffers electrically polar and room  temperature multiferroic materials [ 27-28]. \nIn an attempt of searching new magnetoelectrics,  we have taken an extensive research \nprogram to develop materials with room temperature ferromagnetism and electric field \ncontrolled magnetic state  based on rhombohedral phase of metal doped hematite system . We \ndeveloped Ga doped hematite system in rhombohedral structure ( α-Fe2-xGaxO3) with canted \nferromagnetic state at room temperature and good signature of ferroelectric polarization  [29-\n31], although electric field dependence polarization curves are not completely free from \nleakage due  to relatively high conductivity of the samples [32-33]. In this work , we establish \nGa doped α-Fe2O3 system  as a new magnetoelectric  material, where magnetic state at room 4 \n temperature is controlled by external  electric field and switching of magnetic state is sensitive \nto ON and OFF modes, as well as to the change of polarity of applied electric field .  \n2. Experimental   \nWe have prepared t he Ga doped α-Fe2O3 (α-Fe2-xGaxO3: x = 0.2 -1.0) system by \nmechanical alloying of the fine powders of α -Fe2O3 and β -Ga2O3 oxides . The alloying time \nwas increased up to 100 h  depending on the composition . We determined  structural phase \nevolution of the alloyed samples using X-ray diffraction (XRD) pattern, recorded using Cu \nKα radiation (  = 1.54054 Å). The samples with higher Fe content (x = 0.2) produced single \nphased structure by mechanical alloying itself. The samples with higher Ga content (x = 0.8) \ndo not produce single  phased structure by mechanical alloying alone and it needed special \nheat treatment under vacuum, as described in [ 29, 31 ]. To maintain identical heat treatment \ncondition, we directly heated the mechanical alloyed powder at 800 0C under high vacuum \n(10-5 mbar) for 2 -6 h with fast heating and cooling rate  during change of temperature . The \nsingle phased rhombohedral structure (space group  R3̅C) was confirmed from XRD pattern \nand supported by Micro -Raman spectra (Fig.  1) for the samples used in this work with \ncompositions α-Fe1.8Ga0.2O3, α-Fe1.6Ga0.4O3 and α-Fe1.2Ga0.8O3, respectively . The band at \naround 1320 cm-1 (magnon -phonon mode) in Micro -Raman spectra confirmed a strong spin-\nlattice coupling in rhombohedral structure  of Ga doped hematite system . The structural and \nmagnetic properties for some of the Ga doped hematite samples in rhombohedral phase have \nbeen discussed in earlier works  [29, 31-32]. On the other hand, mechanical alloying and \nsubsequent vacuum annealing were not enough to produce singl e phased rhombohedral \nstructure for the composition α-FeGaO 3. The mechanical alloyed powder of this composition \nwas heated at 1250 oC in air to stabilize orthorhombic structure. The orthorhombic phase  was \ntransformed into rhombohedral phase by mechanical mill ing of the  heated sample up to 50 h \nand the rhombohedral phase was refined  by vacuum annealing of the sample at 700 0C (2 h). 5 \n The vacuum annealing of milled sample at 800 0C indicated re-appearance of orthorhombic \nphase. In this work, we used the rhombohedral phased sample (Ga10MM50 V7) that was \nsubjected to 50 h milling and vacuum annealed at 700 0C (2 h) . A brief  description of the \nsamples (Ga02MA100 V8, Ga04MA50 V8, Ga04MA100 V8, Ga08MA25 V8, Ga10MM50 V7) \nand their structural information are provided in Table 1 . The prepared material can be \nconsidered as a solid solution of Ga atoms in to α-Fe2O3 structure, where non -magnetic Ga \natoms have been dissolved in the lattice sites of magnetic Fe atoms. Variation of the lattice \nparameters in the samples is attributed to the differences in grain size and incorporation of \nGa3+ with smaller ionic radius (0.62 Å) into the lattice sites of Fe3+ ions with larger radius \n(0.645Å). The notable feature is that peak intensity of IR-active E u(LO) mode of lattice \nvibration  (at about ~ 667 cm-1) significantly increased and displaced to higher wave number \nfor Ga10MM50V7 sample. The increase of the intensity of IR-active E u(LO) mode reflects an \nincreasing local disorder in the lattice structure of hematite  at higher Ga content . The \ndisplacement of the peaks to higher wave number is related to the decrease of cell parameter \nin the lattice structure of hematite  at higher Ga content. A pellet shaped sample of typical \ndimension 3  mm x 2 mm x 1 mm was placed between two thin Pt sheets , which were  \nconnected to 2410 -C meter using thin Pt wires for applying dc electric voltage during \nmeasurement of dc magnetization with magnetic field and time  by using vibrating sample \nmagnetometer (LakeShore 7404, USA) . The sample sandwiched between Pt electrodes was \nplaced on the flat surface of the sample holder (Kel -F) and tightly fixed by Teflon tape. The \nproper electrical contact has been checked from identical values of current on reversing the \napplied v oltage at ±5 V.  \n3. Experimental results  \nFig. 2(a)  show s the  M(H) loop for hematite sample  at room temperature , measured \nunder different values of applied electric voltage (0 -300 V). The inset of Fig. 2(a) (magnified 6 \n loop at 0 V and 300 V) confirms the absence of any electric field induced shift of  M(H) loop  \nin hematite sample. The time dependence of magnetization in the presence of 5 kOe magnetic \nfield ( in-field magnetic relaxation ) in hematite sample (Fig. 2(b)) shows a typical character of  \nan antiferr omagnet or canted antiferromagnet ic system  [34-35]. Interestingly,  magneti c state \nof hematite sample, irrespective of the increase or decrease of in-field magnetization with \ntime (Fig. 2(c-d)), is switchable by external  electric voltage. The magnetic state  is highly \nsensitive to the change of polarity, as well as to ON and OFF switching modes of the external \nelectric voltage. One can see a n instant  magnetization jump to higher state (lower state) by \nswitching ON the electric voltage +100 V ( -100 V) and retu rned back to original magnetic \nstate by switching OFF the electric voltage. The magnetic switching is repeatable and the \nchange of magnetization is nearly 0.87 -1.00 %  by applying electric voltage at ±100 V .  \nOn the other hand, electric field controlled magnetic properties in Ga doped hematite \nsystem are remarkably different from hematite sample. We present details experimental \nresults for the compositions α-Fe1.8Ga0.2O3 (Ga02MA100V8) , α-Fe1.6Ga0.4O3 (Ga04MA50V8, \nGa04MA100V8),  α-Fe1.2Ga0.8O3 (Ga08MA25V8)  and α-FeGaO 3 (Ga10MM50V7) , \nrespectively . The M(H) loops of Ga02MA100V8 sample  (Fig. 3(a)) showed a shift by \nincreasing the applied voltage up to 200 V. Fig. 3(b) shows that M(H) loop of the sample \nshifted along the direction of po sitive magnetic field and negative magnetization for applying \nboth positive and negative bias voltage. The negative bias voltage has little effect on \nchanging the nature of the loop shift. But, M(H) loop at -200 V has slightly shifted towards \nnegative magn etic field direction in comparison to the loop measured at +200 V. The \nmagnetic exchange bias field (H exb) has been calculated from the shift of the center of the \nM(H) loop under electric voltage with respect to the center of the M(H) loop at 0 V. Fig. 3(c) \nshows that magnetic exchange bias field (H exb) of the Ga02MA100 sample can be increased \nup to + 1226 Oe for applied voltage +200 V and Hexb was found up to +1190 Oe for applied 7 \n voltage -200 V.  On the other hand, magnetic state of the Ga02MA100 V8 sampl e during in -\nfield magnetic relaxation process is instantly jumps to higher magnetization state by applying \n+100 V (Fig. 3(d) and to lower magnetization state by applying -100 V (Fig. 3(f), in addition \nto the natural magnetic relaxation process (decreasing trend) of the spins with time. Despite \nthe fact of a rapid change of magnetization with time during ON and OFF modes of electric \nvoltage, the in -field magnetic relaxation confirms a step -wise increment/decrement of \nmagnetization of the sample with the step -wise (100 V) increment/decrement of applied \nvoltage up to ±500 V. This resulted in the change of magnetization by 0.96% and 0.60 % for \nthe applied voltage +500 V and -500 V, respectively with respect to 0V reference level.   \nFig. 4(a) shows the M(H) loop f or Ga04 MA100 V8 sample, measured in the magnetic \nfield range ±16 kOe and in the presence of constant electric voltage 0 V to + 200 V. The \nM(H) loops under positive electric  voltage shifted towards the positive magnetic field axis \nand negative magnetization direction  in comparison to the M(H) loop measured at 0 V.  The \nelectric field induced shift of M(H) loop is clearly visible from the plot within small magnetic \nfield range (Fig. 4(b)). It may be noted that m agnetization vectors in the M(H)  curve revers ed \nupon reversing the magnetic field directions. Fig. 4(c) shows that shift of the M(H) loop \ncertainly depends on the polarity of applied electric voltage.  The M(H) loop shifted towards \npositive magnetic field direction upon application of + 200 V (positive).  The loop shift under \nnegative voltage ( -200 V) is not exactly the mirror image of the loop at + 200 V with \nreference to the loop at 0 V. On reversing the electric field from + 200 V to -200 V, the \nmagnetization shifted towards negative magnetic field and positive magnetization directions  \nwhen measured at -200 V in comparison to the M(H) loop measured at + 200 V. However, \nboth the M(H) loops under opposite polarity of electric voltage remained in the same positive \nside of the magnetic field with respect to the loop at 0 V. This results in a decrease of positive \nmagnetic exchange bias field by nearly -75 Oe during measurement at - 200 V (with H exb = 8 \n +267 Oe) with reference to the exchange bias field (H exb = +392 Oe) during measurement at \n+ 200 V. The results are similar to that observed in the synthetic multiferroelectric film , \nconsisting of La0.67Sr0.33MnO 3 (ferromagnet)  and BaTiO3 (ferroelectric) [35]. Fig. 4(d) shows \na rapid increase of the magnetic exchange bias field at the initial increment (up to 20 V) of \nelectric voltage and subsequently slowed down at higher electric voltages to achieve the \nvalue that falls in the range of magnetic coercivity (362 ± 10 Oe) for Ga04 MA100 V8 sample. \nThe magnetic coercivity (H C) has  been calculated from the average of the values in positive \nand negative magnetic field axis of the M(H) loop.  The Ga04 MA50 V8 sample also exhibited \nsimilar kind of electric field controlled M(H) loops (Fig. 5(a)) and a rapid increase of \nmagnetic exchange fi eld (H exb) for applied electric field < 20 V and then slowed down to \nachieve a saturation value for electric voltage above 50 V (inset of Fig. 5(a)). The magnetic \ncoercivity (H C) of the sample at higher electric voltages approaches  to the value 353 Oe \nobserved at 0 V, whereas Ga04 MA50 sample has achieved the H exb up to + 370 Oe for \nelectric voltage at + 200 V.  Fig.5(b) shows that  M(H) loop shift of the Ga04MA50  sample \ndepends on polarity of the applied electric voltage . The M(H) loop shifts towards the nega tive \nmagnetic field axis when measured at -200 V in comparison to the M(H) loop measured at \nelectric voltage at + 200 V.  Subsequently, magnetic  exchange bias shift  for negative electric \nvoltages ( -10 Oe at -100 V and - 40 Oe at -200 V) is significant with respect to the values at \npositive voltages (+ 100 V and + 200 V).  The H exb(V) curve for positive and negative electric  \nfield variation is nearly symmetric about a reference line (inset of Fig. 5(b)), which lies in \nbetween H exb(+V) and H exb(-V) curves, and  of course, not with respect to H C(0 V) line.  \nWe have examined the electric field controlled magnetic state at room temperature for \ntwo samples of α-Fe1.6Ga0.4O3 (Fig.6 for Ga04 MA100 V8 and Fig.7 for Ga04 MA50 V8) \nthrough in -field magnetic relaxation  experim ent ( time dependence of magnetization at \nconstant magnetic field 5 kOe ). The in -field magnetic relaxation behaviour in both the 9 \n samples is more or less similar  in character (i.e., M(t) decreases in the presence of 5 kOe \nfield) as in hematite sample (Fig. 2(b)). Fig. 6(a) shows the example of magnetic relaxation of \nGa04 MA100 V8 sample in the presence of 5 kOe and at 0 V. The magnetization in both the \nsamples is highly sensitive and switchable under ON -OFF mode s and reversal of the polarity \nof applied electric  voltage, as measured at + 100 V (Fig. 6(b), Fig. 7(a)), at -100 V (Fig. 6(c), \nFig. 7(b)), at cyclic order of the polarity change with sequence 0 V + 100 V 0 V  -100 \nV (Fig. 6(d), Fig. 7(c)). As shown in Fig. 7(d -e), magnetization in both the samples showed a \nsudden jump in response to the change of applied voltage either from ON to OFF or OFF to \nON modes. A complete reversal of the magnetization vectors (negativ e to positive or vice \nversa) is not observed  macroscopic level upon reversing the polarity of electric voltage \nduring in -field magnetic relaxation of spins (coloured symbol), but the samples instantly \njump ed to higher magnetic state at the time of applying  positive voltage ( M  1.36 %) and \nto lower magnetic state at the time of negative voltage ( M  1.14 %) with respect to \nmagnetization state at 0 V. The change of magnetization was calculated using the formula \nM (%) = (M(V)−M(0))∗100\nM(0). The in -field ma gnetization being in the meta -stable state relaxed \nslowly with time  even in  the presence of electric field, irrespective of positive or negative \nsign, towards achieving the magnetization state at 0 V. However, there exists a gap ( M ~ \n0.13-0.26 %) between the relaxed magnetic state just before and after switching the ON/OFF \nmodes of electric voltage. It indicates that switched magnetic state after relaxation is different \nfrom the relaxed state at 0 V. The existence of magnetization gap during M(t) measureme nt is \nirrespective of the magnitude, polarity and cycling of electric voltage. The relaxation of \nmagnetization under simultaneous presence of electric and magnetic fields may be related to \nthe kinetic energy transfer of charge -spin carriers.  Fig. 6(e -f) showed that the relaxation rate \nis relatively fast at higher electric voltage (±  200 V), where in -field magnetization under 5 10 \n kOe relaxed rapidly to the magnetization state either at 0 V preceding to the application of \nelectric voltage or even continued to r elax.  Such rapid relaxation of magnetization under high  \nelectric field may be affected by magneto -conductivity that cannot retain the switche d \nmagnetization state for long time . Fig. 7(f) shows the in -field magnetic relaxation where \napplied electric voltage has been increas ed step-wise (size 50 V) up to +  500 V for every 50 s  \ninterval during measurement time up to 600 s. This experiment confirms a systematic \nincrease of magnetization (magnetic spin order) with the increase of positive electric voltage. \nFig. 7(g)  shows that  M of the base and peak values of M(t) data under electric voltage \ncontinuously increased with the increase of electric voltage in comparison to the M(t) data at \n0 V (during f irst 50 s of the measurement). The M is found to be 2.51 % and 1.23 % for the \npeak and base values in the presence of electric voltage 500 V during last 50 s of the \nmeasurement time. The gap between M (%) at peak and base line increased with applied \nvoltage, which could be related to relaxation of the induced magnetization at higher voltage. \nThe composition α-Fe1.2Ga0.8O3 (Ga08MA25 V8) also exhibited electric field controlled \nmagnetic exchange bias shift (Fig. 8) and magnetic state switching (Fig. 9) . Similar to other \ncompositions of Ga doped hematite system, M(H) loop of the Ga08MA25 V8 sample (Fig. \n8(a)) shifted towards the positive magnetic field axis and negative magnetization direction \nwhen measured under positive electric voltage in comparison to t he M(H) loop measured at 0 \nV. The M(H) loop shift also depends on the polarity of applied electric voltage  (Fig. 8(b)). \nThe M(H) loop under negative voltage ( -200 V) has shifted towards positive magnetic field \ndirection in comparison to the loop measured u nder + 200 V (positive).  This results in a \ndecrease of magnetic exchange bias from +85 Oe during measurement at + 200 V to +76 Oe \nduring measurement at - 200 V.  The inset of Fig. 8(b) shows that magnetic coercivity of the \nGa08MA25 V8 sample is limited within the range 210 ±5 Oe for applying the electric voltage \nin the range 0 V to 300 V. However, magnetic exchange bias shift increases with the increase 11 \n of applied electric field and reached to + 152 Oe for the M(H) loop that was measur ed under \nelectric voltage +300 V.  Fig. 9 shows the in -field magnetic relaxation at 5 kOe magnetic field \nof the Ga08MA25 sample under different modes of the application of electric voltages. The \nmagnetic state of the sample is highly sensitive and switchabl e under the ON-OFF mode s and \nreversal of the polarity of applied electric voltage with respect to the in -field magnetic \nrelaxation state at 0 V (Fig. 9(a)). The switching of magnetic state of the sample has been \ntested by a cyclic change of electric voltag e with a sequence 0 V + 200 V 0 V  + 200 \nV (Fig. 9(b), 0 V - 200 V 0 V  - 200 V  (Fig. 9(c)), 0 V + 200 V 0 V  - 200 V \n0 V (Fig. 9(d)), and a step -wise increment of voltage up to + 500 V and -500 V separately  \nwith step size 100 V ( Fig. 9(f))), at cyclic order of the polarity change with sequence 0 V + \n100 V 0 V  -100 V (Fig. (e)). The concrete information from Fig. 9 is that magneti c state \nof the sample is highly switchable under the application of electric field. The magnetic state \ninstantly jump s to higher magnetization level in response to positive voltage and to lower  \nmagnetization level in response to negative voltage, irrespective of the intermediat e in-field \nmagnetic relaxation of the sample. The magnetization in the presence of electric voltage with \nrespect to 0 V line systematically increases/decreases by the increment/decrement of the \nvoltage up to +500 V/ -500 V with step size 100 V and the chang e was up to 2 % (Fig. 9(f)). \nFig. 10(a) shows that rhombohedral structured sample (Ga10MM50V8) of the composition α-\nFe1.8Ga0.2O3 is a weak ferromagnet at room temperature. This sample in the rhombohedral \nphase also  exhibits (Fig. 10(b)) electric field indu ced magnetic exchange bias shift (up to at \n14 Oe at +200 V), although the ferromagnetic order in this composition has been diluted due \nto more amount of non -magnetic Ga content and lattice disorder. We have shown in Fig. \n10(c) that the exchange bias shift for different composition of α-Fe2-xGaxO3 system is close or \nless than the values of magnetic coercivity of the corresponding samples . Both these \nmagnetic parameters have decreased with increase of Ga content in hematite structure.  12 \n  4. Discussion  \nThe electric field controlled magnetic state has been reported mainly in thin -films  of \nFM semiconductors, multiferroics, and multi -layered structure consisting of ferroelectric  (FE) \nand ferromagnetic (FM) materials [ 1-2]. It has been proposed that change of charge carrier \ndensity by electric field changes the magnetic exchange interactions in FM semiconductor. \nOn the other hand, charge (electric polarization) and spin (magnetization) coupling by \napplied electric field can modif y the magnetic state in multif erroic material. The electric field \ncontrolled magnetic exchange bias was first explored at the heterostructured interface of \nCr2O3 (Co/Pt) 3 [37], where electric f ield controlled magnetism was attributed to strain -\nmediated magnetoelectric coupling at the heterojunction of FM and FE layers  [36, 38]. The \npresent Ga doped hematite system is not in thin film form  and the observed electric field \ninduced magnetic exchange bias shift can be excluded  from strain mediated electro -magnetic \ncoupling effect . Also, magnetic coercivity in the present samples with specific Ga content is \nnearly independent of applied electric field, unlike the increase of magnetic exchange bias \nshift with applied electric vol tage. This means the magnetic anisotropy, related to spin–orbit \ninteraction of electrons  or interfacial anisotropy as proposed for magnetic tunnel junctions in \nFM semiconductors [ 1], is practically insensitive to external  electric field  in Ga doped \nhematite system . Hence, the electric field induced magnetic state in rhombohedral structure \nof Ga doped hematite system over a wide range of Ga content  could be originated from \ndifferent sources . Since the material is new and there is not any straight -forward mechanism \nor theoretical model available for explaining the electric field induced magnetic state , we \nunderstand the observed electric field induced magnetic state using models proposed for \ndifferent systems and seems to be reasonable  for the Ga doped hematite system . In the \nabsence of a significant role of the strain mediated interfacial coupling  and spin -orbit \ninteractions for showing electric field controlled magnetic state  in Ga doped hematite 13 \n samples , we focus on electric field dependent perturbation in the magnetic spin order at the \ndomain walls as proposed for many artificially designed multi -layered materials [ 38-39]. \nWe noted that layer kind lattice structure  of material , whether single phased (BiFeO 3) \n[16] or multi -phased (BaTiO 3/La0.67Sr0.33MnO 3 [36]), seems to be more sensitive for showing \nelectric field controlled magnetic exchange bias . R. Moubah et al. [ 40] directly captured the \nimag e of coupled FE and AFM  domains in multiferroic BiFeO3 single crystal . It was \nobserved that several AFM domains coexist  inside one single FE domain . Skumryev et al. \n[41] explained the magnetization reversal and magnetic exchange bias (EB) shift in FM \nNi81Fe19 (Py) film deposited on AFM multiferr oic (LuMnO3) single crystal as the effect of \nelectric -field driven decoupling between FE and AFM  domain walls. They proposed the \nexistence of clamped and unclamped AFM domain walls (AF -DWs) at the interfaces with FE \ndomains. A coupling between clamped AFM -DWs and FE domains exerts electric field \ncontrolled torque on FM moments, where as unclamped AF -DWs do not play significant role \nin the electric field controlled magneti c switching.  In fact , electric field induced magnetic \nexchange bias field in our materi al (reached up to 1220 Oe at 200 V for Ga02MA100 V8 \nsample) is remarkably large in comparison to the reported values ( 100-300 Oe) in hetero -\nstructured multiferroic films [37 , 41]. However, a similar variation of exchange bias shift on \nincreasing the applied electric voltage  was noted in the Ga doped hematite samples . The \nswitching  of (canted)ferro magnetic state by electric field indicates the creation of additional \nmagnetic exchange interaction s in the system.  It may be mentioned that hematite ( α-Fe2O3) \ndoes not show any signature of ferroelectric properties and electric field induced magnetic \nexchange bias. On the other hand, Ga doped hematite system exhibited  electric field induced \nmagnetic exchange bias effect.  Although leakage of polarization is no t completely overcome \ndue to high conductivity of the samples, but existence of ferroelectric polarization in Ga \ndoped hematite samples with rhombohedral structure has been realized. However, there is a 14 \n further scope of improvement of ferroelectric polariz ation in present material by preparing in \nthin film form, as described in some thin films of Ga doped hematite in orthorhombic phase \n[25-26]. The important point to be noted is that electrical polarization in the polycrystalline α-\nFe1.6Ga0.4O3 samples [ 33] is responded to magnetic field and polarization curve is well \ncomparable quantitatively and qualitatively to that in Mn doped orthorhombic structured \nGa0.6Fe1.4O3:Mg thin films  [25]. Even, the shape of the polarization curve in our Ga doped \nhematite samp le is far better than that observed in M-type hexaferrite  SrFe 12O19 [19], an \nemerging multi -ferroelectric material.  At the same time, magnetic coercivity decreased with \nthe increase of Ga content in hematite structure.  These features indicated a modified magnetic \nstructure in Ga doped hematite system [ 29, 31-32]. \nWe propose a  schematic diagram (Fig. 11) to understand the electric field controlled \nmagnetic spin order (exchange bias and switching) in Ga doped hematite  system . As sketched \nin Fig. 11(a), t he hematite system in rhombohedral structure with R3̅C space group is \nmagnetically multi -layered spins structure, where in -plane Fe3+ spins form FM order and Fe3+ \nspins in alternating planes (say, A and B) along off -plane direction form AFM order by \nsuperexchange interactions ( Fe3+A-O2--Fe3+B). The weak ferromagnetism arises due to canting \namong AFM  aligned spins by Dzyaloshinskii –Moriya (DM) interaction s [ D⃑⃑ .(Sn⃑⃑⃑⃑ ×Sn+1⃑⃑⃑⃑⃑⃑⃑⃑ )]. \nThe replacement of magnetic Fe atom by non -magnetic Ga atom increases magnetic non-\nequivalen ce between A and B planes. The enhancement of spin -lattice coupling and small \natomic displacement within rhombohedral structure of Ga doped hematite are indicated from \nMicro -Raman spectra. The dielectric peak in the temperature regime (290 K -310 K), where \nFe3+ spins started flipping from in -plane direction (canted FM state) to out of plane direction \n[32, 42], indicated (canted) spin order induced magneto -electric coupling in Ga doped \nhematite system. We restrict our discussion for the origin of electric polarization in Ga doped \nhematite system based on the models of spin induced ferroelectricity  [5, 24]. The spin 15 \n induced ferroelectricity/multiferroic properties (electric field controlled magnetism) has been \nobserved at lower tem perature in a family of orthoferrites RFeO 3 (R = Gd, Tb, Dy) [ 43]. In \nsuch spin canted systems, S⃑ i×S⃑ j ≠ 0 where S i and S j represent spins at two canted spin sites . \nAccording to inverse Dzyaloshinskii -Moriya (DM) model [18, 39 ], the antisymmetric spin \nexchange interactions êij×(S⃑ i×S⃑ j), where êij is the unit vector connecting spins at sites i and j, \ncan produce macroscopically observable polarization ( Pij⃑⃑⃑ ) under the influence of relativistic \nspin–orbit coupling, despite the lack of non -centrosymmetric displacement of cations, as in \nBaTiO 3. The magnitude of such spin order induced polarization depends on the displacement \nof intervening ligand ions (O-2) that favour the DM interaction and a net non -zero induced \nstriction. In spin induced ferroelectrics, a non -centrosymmetric magnetic spin order breaks \nthe inversion symmetry , a pre -requisite for generating electric polarization [ 44]. The increase \nof magnetic in -equivalence and in-equivalent exchange strictions between the spin s order in \nA and B planes and change of superexchange path lengths (Fe3+-O2--Fe3+) in rhombohedral \nstructure of Ga doped hematite system is helpful for breaking of inversion symmetry of spin \norder along off -plane direction . Second source of the breaking of inversion symmetry of spin \norder could be the magnetically in -equivalent core -shell spin structure in nano -sized grains of \nGa doped hematite system (Fig. 11(b)). In a typical AFM system with layered spin structure, \nthe net magnetic moment (< >) between two adjacent layers A and B is zero . The net \nmoment is non -zero in the case of canted FM/AFM spin order (Fig. 11(c). In spin canted \nsystem, the orientation of magnetic moment vector in the presence of external magnetic field \n(Hext) with respect to  local easy axis (EA) is determined by the resultant free energy E = E FM+ \nEAFM + E coupling , where E FM, EAFM, and E coupling  represent the free energy terms for FM layer, \nAFM layer and interlayer spin coupling, respectively [4 5]. When exchange field correspo nds \nto AFM interactions (H AFM) dominates and greater than H ext, the in-field magnetization of the \nsystem, as we observe at 5 kOe, can relax with time (in -field magnetic relaxation) to achieve 16 \n a magnetic state near to bulk AFM (core part of the grains); rat her than showing an increase \nof magnetization with time as in the case of a disordered FM [34]. On the other hand, \ninterlayer spin exchange coupling energy ( Ecoupling ) is largely responsible for electric field \ninduced magnetic exchange bias and magnetic state switching. In the materials like BiFeO 3, \nwhich is structurally and magnetically similar to Ga doped hematite system, a direct 1800 \nreversal of DM vector by ferroe lectric polarization is forbidden from symmetry point of view. \nHowever, first -principle calculations [1 6] predicted a deterministic reversal of the DM vector \nand canted spin moments by electric field. The coupling between DM vector and induced \npolarization  can switch the magnetization by 180 0. A complete reversal of magnetization \n(180 0 rotation) by electric field has been predicted in a perpendicularly magnetized thin film \n[46]. The response of local magnetization vector M(x, t) in spin canted systems, as applicable \nfor the present Ga doped hematite system, to external electric field depends on a spatial \ndistribution of spin  magnetization vector components inside the magnetic domains and on \nthe temporal evolution of the magnetization spin vector, determine d by Landau -Lifshitz -\nGilbert equation: ∂M⃑⃑⃑ \n∂t= −γ0(M⃑⃑⃑ ×H⃑⃑ eff)+ α\nMS(M⃑⃑⃑ ×∂M⃑⃑⃑ \n∂t), where γ0 is the gyromagnetic ratio \nand α are the Gilbert damping factor. A competition between the spin torque (first term) and \ndamping torque (second term) determines the orientation of the magnetization vectors either \nin one side of the interfacial plane or complete reversal about the plane. If we look carefully \nthe magnetization switching in M(H) and M(t) curves of our samples , it is understood that all  \nthe spins in the magnetic domains are not  participating in spin reversal process under electric \nfield. A fraction (roughly  1-2 %) of the total sp ins in a magnetic domain, most probably at \nthe domain walls or shell part of the grains (Fig. 11(e-f)), reverses their magnetic spin \ndirections (180 0) upon reversal the polarity of applied electric voltage and rest of the spins \n(interior to the domain) re luctantly participat e in the reversal process. We understand that a \nunique coupling between magnetic spin order and ferroelectric polarization at the domain 17 \n walls [36, 41, 43, 47 ] or core -shell interface s plays a dominant role in exhibiting magnetic \nexchange bias effect  in Ga doped hematite system. In case of M(H) measurement,  core part \nof the spin structure also contributes in magnetic domain wall motion/domain rotation \nprocess. The application of electric field induces a strong magnetic exchange cou pling  at the \ninterfaces of domain walls or core -shell interfaces. This results in an irreversible shift of the \nM(H) loop in the presence of electric field . On the other hand, FM coupling among the spins \nin the presence of magnetic field is stronger than th e induced exchange coupling. Hence, the \nshift of M(H) loop under positive or negative electric voltage s is always along positive \nmagnetic field direction with respect to the loop at 0 V. A minor shift of the loop along \nnegative magnetic direction under neg ative electric field in comparison to the loop under \npositive electric field confirms that only a fraction of interfacial spins participates in electric \nfield controlled 180 0 reversal process . A complete 180 0 reversal (switching) of the se \ninterfacial  magneti c spin  vectors by reversing the  electric field polarity is macroscopically \nobserved  in Ga doped hematite system during M(t) measurement  with reference to the \nmagnetization state at 0 V , although the net magnetization is always positive under positi ve 5 \nkOe magnetic field and magnetic state switches to higher level and lower level for application \nof positive and negative electric voltage, respectively.  \n5. Conclusions  \n \nThe replacement of magnetic Fe atoms by non-magnetic Ga atoms in rhombohedral \nstructure of α-Fe2O3 enhances magnetoelectric  properties  and magnetic order has been diluted \nat considerably higher value of Ga content . The ferromagnetic loop, being a universal feature \nirrespective of metal, semiconductor or insulator, un der external electric field was used to \ndetect the existence of magneto -electric coupling in Ga doped hematite system. The Ga  \ndoped hematite system exhibited similar lattice  (rhombohedral) and magnetic (spin canting)  \nstructure as in BiFeO 3 and both are lead free system. The magneto -electric coupling in Ga 18 \n doped hematite system (rhombohedral structure) is free from strain mediated coupling effect \nand considered to be intrinsic  of the material . The magnetic exchange bias  shift, generally a \nlow temperature  phenomenon in magnetically bi -layered (FM /AFM)  systems after high \nmagnetic field cooling from higher temperature , has been observed in Ga doped hematite \nsystem at room temperature itself without magnetic field cooling and controlled absolutely by \nelectric  field. A complete reversal of the magnetization vectors (negative to positive or vice \nversa) is not observed at macroscopic level upon reversing the polarity of electric voltage \nduring in -field magnetic relaxation of spins , but the samples instantly jumpe d to higher \nmagnetic state at the time of applying positive voltage ( M  2-2.5 %) and to lower magnetic \nstate at  the time of applying negative voltage. The electric field controlled magnetic exchange \nbias and switching behaviour can be understood in terms of spin induced ferroelectricity  \n/multiferroic properties at the domain walls or exchange coupling at the interf aces of core -\nshell spin structure. I t is understood that a fraction (roughly  1-2 %) of the total spins at the \ndomain walls or core-shell interface reverses their magnetic spin directions (180 0) upon \nreversal the polarity of applied electric voltage and rest of the spins (interior to the domain  or \ncore) reluctantly participat e in the reversal process.  A complete 180 0 reversal (switching) of \nthe interfacial  magneti c spin  vectors by reversing the  electric field polarity is macroscopically \nconfirmed  during M(t) measurement  with reference to the magnetization state at 0 V . The \nelectric field controlled magnetic state in Ga doped hematite system at room temperature can \nhave a tremendous impact on developing spintronics based technology , non-volatile me mory \nand electro -magnetic switching devices.  \nAcknowledgment  \nAuthors thank to CIF, Pondicherry University for providing some experimental facilities. \nRNB acknowledges research grants from Department of Science and Technology (NO. 19 \n SR/S2/CMP -0025/2011) and Co uncil of Scientific and Industrial Research (No. 03(1222)/12/ \nEMR_II), Gov ernmen t of India for carrying out th is experimental work.  \n \nReferences  \n[1] F. Matsukura, Y. Tokura, and H.Ohno, Nature Nanotech. 10, 209 (2015).  \n[2] M. Bibes et al., Nat. Mater. 7, 425 (2008);  M. Bibes, Nat. Mater. 11, 354−357 (2012).  \n[3] N.A. Hill, J. Phys. Chem. B 104, 6694 (2000).  \n[4] A.P. Pyatakov et al., Phys. Usp.  55, 557 (2012 ). \n[5] M. Gich, I. Fina, A. Morelli,F. Sánchez, M. Alexe,J.Gàzquez, J. 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Rev. \nLett. 110, 067202 (2013).  \n \nFigure captions  \nFig. 1 Profile fitting of XRD pattern (left hand side) and Micro -Raman spectra (righ hand  \nside) confirm rhombohedral structure in Ga doped hematite system.  \nFig. 2 (a) M(H) loops of hematite sample measured at different electric voltages, magnetic \nrelaxa tion at 5 kOe for 0 V (b), for ON -OFF modes of +100 V (c), for ON -OFF modes of      \n-100 V.  22 \n Fig.3 M(H) loops at different measurement voltages (a), magn ified M(H) loops at 0V,  200 V \n(b), variation of magnetic exchange bi as field with applied v oltages (c). Time dependence of \nmagnetic moment at a magnetic field of 5 kOe and different voltage conditions 0V and \n+100V (d), 0V and -100V (e), and 0V -  ± 500V in steps of 100V (f).  \nFig. 4(a) Room temperature M(H) loops at different measurement voltages,  (b) magnified  \nM(H) loops, (c) M(H) loops at 0 V and  ± 200 V for Ga04 MA100V 8 sample, (d) variation of \nexchange bias with diff erent positive applied voltages . \nFig. 5 Magnified form of room temperature M(H) loops of Ga04MA50V8  sample, measured \nat different electric voltages (a), shown for +200 V and -200 V with respect to 0 V loop (b).  \nInsets show the variation of magnetic exchange bias field for different electric voltage (a) and \nfor electric bias voltages at ±100 V and ± 200 V (b).  \nFig. 6 Room temperature time dependent magnetic moment at a magnetic field of 5 kOe  \nand different voltage conditions (a) 0V, (b) 0V and 100V, (c) 0V and +100V, and (d) 0V  \nand -100V, (e) 0V and +200V, (f) 0V and -200V  for Ga04MA100V8 sample . \nFig. 7 Time dependent magnetic moment at a magnetic field of 5 kOe and voltage  \nconditions (a) 0V and +100V, (b) 0V and -100V, (c) 0V and 100 V, (d) branch of (a),  \n(e) branch of (b), (f) magnetization at voltage 0 -500 V, (g) change of magnetization.  \nFig. 8  Ro om temperature M(H) loops at different positive voltages (a) and a comparative  \nM(H) loops at  ± 200 V and at 0 V (b). The Inset shows variation of exchange bias field  \nand coercivity at different voltages for Ga08MA25V8 sample.  \nFig.9 In field magnetic relaxation at magnetic field of 5 kOe for applied electric voltage at  \n0 V (a). The in -field magnetic relaxation at ON -OFF mode of voltage for +200 V (b), for  \n-200V (c), for  ±200V (d), and increment of voltage up to ± 500V in steps of 100V (e).  \nThe charge of switched magnetization at peak values with applied electric field (f) is  \nshown for Ga08MA25V8 sample.  23 \n Fig. 10 M(H) loops for Ga10MM50V7 sample measured at different electric voltages  \n(a) and loop shift is shown in magnified plot (b). Variation of the magnetic coercivity  \nand exchange  bias shift at 200 V with Ga content in the Ga doped hematite system (c).  \nFig. 11 A schematic diagram of the spin order between two planes in -Fe2O3 and Ga doped \n-Fe2O3 (a), Core -shell spin structure in  a grain of Ga doped -Fe2O3 (b), in -field magnetic \nrelaxation  at V = 0 (c) and in the presence of constant voltage (d),  response of spin vectors \nduring M(H) measurement in the presence of  constant +ve (e) and –ve (f)  voltage.  \n \n \nTable 1. Structural information of the samples used in the present study  \nSample code   Structural \nformula  Preparation condition  Cell parameters  \na(±0.005Å), c( ±0.002Å),       \nV (±0.05(Å)3 Grain \nsize \n(nm)  \na (Å) c (Å) V(Å3) \nGa02MA100 V8 α-Fe1.8Ga0.2O3 Mechanical alloyed for 100 \nh and annealed at 800 °C \nunder vacuum for 6 h  5.033  13.738  301.41  53 \nGa04MA50 V8 α-Fe1.6Ga0.4O3  Mechanical alloyed for 50 \nh and annealed at 800 °C \nunder vacuum for 2 h  5.036 13.744  301.99  37 \nGa04MA100 V8 α-Fe1.6Ga0.4O3  Mechanical alloyed for 100 \nh and annealed at 800 °C \nunder vacuum for 2 h  5.035  13.715  301.15  26 \nGa08MA25 V8 α-Fe1.2Ga0.8O3 Mechanical alloyed for 25 \nh and annealed at 800 °C \nunder vacuum for 2 h  5.025  13.703  299.70  50 \nGa10MM50 V7 α-FeGaO 3 Mechanical alloyed for 100 \nh, followed by heating at \n1250 °C in air. This is \nfollowing mechanical \nmilling for 50 h and \nsubsequent vacuum \nannealing at 700 0C (2 h ) 5.024  13.611  297.56  19 \n \n \n \n 24 \n \n200400600 1200 1400 1600 20 30 40 50 60 70 80 Ga04MA50V8 \n \n Ga04MA100V8\n  \n Ga08MA25V8 \n \nWave number (cm-1)\nFig. 1 Profile fitting of XRD pattern (left hand side) and Micro-Raman spectra (righ hand \nside) confirm rhombohedral structure in Ga doped hematite system.2 (deg.)Ga02MA100V8 \n  \n  (012)\n(104)\n(110)\n(113)(202)\n(024)\n(116)\n(122)\n(214)(300)\n(208)\n(109)\n(217)\nGa08MA25V8 \n  \n Ga04MA50V8 \n Ga04MA100V8Ga02MA100V8 \nmagn/phonA1g(2)A1g(1)\nEg(4)Eg(1)\nEg(3) Experiment data\n fitted data\n  Eg(5)\nEu(LO)\n Ga10MM50V7\n Ga10MM50V7 \n   \n \n \n \n 25 \n \n-15 -10 -5 0 5 10 15-0.4-0.20.00.20.4\n0100 200 300 400 500 6000.1420.1430.1440.145\n0 100 200 300 400 500 6000.1440.1450.1460100 200 300 400 500 6000.14440.14450.14460.1447\n-2-1012-0.20.00.2 \n  -Fe2O3\n 0 V\n 50 V\n 100 V\n 300 V\n  \n1.00% \n \n  \n  (c)  H=5 kOe\n+100 V ON-OFF\n-100V-100V\n-100V+100V\n+100V+100V\n0V\n0V0V\n0V0V\n0V(a)\n  0.87%\n \n (d)  H=5 kOe\n-100 V ON-OFF\n  \n (b) magnetic relaxation\nat magnetic field 5 kOe\nand electric voltage   0 V \n \nTime (s) Time (s)Time (sec.)M (emu /g)H (KOe)\nFig. 2 (a) M(H) loops of hematite sample measured at different electric voltages, \nmagnetic relaxation at 5 kOe for 0 V (b), for ON-OFF modes of +100 V (c), for ON-OFF \nmodes of -100 V. \n \n \n \n 26 \n \n0.1230.1260.129\n0.124\n0 200 400 6000.1220.1240.1260.128-15 -10 -5 0 510 15-0.40.00.4\n0 100 20005001000-4 -2 0 2 4-0.10.00.1\n+100 V+100 V\n(d)\n+400 V+300 V+200 V\nV=0\n+100 VH=5kOe\n0 V+100 V\n0 V\n  \n0 VGa02MA100V8\n0 V0 V\n0 V\n-100 V-100 V(e)\n-100 V\n  \n-500 V-400 V-300 V(f)\n-200 V\n+500 V(a)\nTime (s)-100V\n  \n0 V reference\n0.96%M (emu /g)M (emu /g) \n \nFig.3 M(H) loops at different measurement voltages (a), magnified M(H) loops at 0V, \n200V (b), variation of magnetic exchange bias field with applied voltages (c). Time \ndependence of magnetic moment at a magnetic field of 5 kOe and different voltage \nconditions 0V and +100V (d), 0V and 100V (e), and 0V - \n  500V in steps of 100V (f).M (emu /g)\nH (KOe)\n  \n V0\n V50\n V100\n V200\n V200Nincrease of \nvoltage\n(c)\n  \nApplied Voltage (V)Hexb (Oe)(b)  0V\n 200 V\n -200 V \n \n \n \n 27 \n \n-15 -10 -5 0 5 10 15-0.6-0.4-0.20.00.20.40.6\n-1.0 -0.5 0.0 0.5 1.0-0.040.000.04-2 -1 0 1 2-0.20.00.2\n0 100 2000200400M (emu /g) M (emu /g)   0 V\n 10 V\n 20 V\n 50 V\n 100 V\n 200 VM (emu /g)\nH (KOe)\n  \nincrease \nof positive  \nvoltage(a) applied electric voltage\nincrease of \npositive voltage\nH (KOe)\n  \n(b)(c) 0 V\n +200 V\n -200 V\nH (KOe)\n   \nFig. 4(a) Room temperature M(H) loops at different measurement voltages, (b) magnified \nM(H) loops, (c) M(H) loops at 0 V and \n  200 V for Ga04MA100V8 sample, (d) variation of  \nexchange bias with different positive applied voltages.V (Volts)Hexb (Oe)HC line\n(d) \n \n \n \n 28 \n \n-0.8 -0.4 0.0 0.4 0.8-0.04-0.020.000.020.04\n-2 -1 0 1 2-0.10-0.050.000.050.100 100 2000200400\n100 150 200320340360380   Electric voltage\n   0 V\n 10 V\n 20 V\n 30 V\n 50 V\n 100 V\n 200 V\n  \n0 V 200 VHexb (Oe)(a)\n  \n\n Electric Voltage (V)Hexb (Oe)\n  \nH (kOe)M (emu/g)(b)\n0 V+200 V-200 V Hexb\n HC \n Electric Voltage (V)\n-200 V+200 V\n-100 V\nFig. 5 Magnified form of room temperature M(H) loops of Ga04MA50V8 sample, \nmeasured at different electric voltages (a), shown for +200 V and -200 V with respect to \n0 V loop (b). Insets show the variation of magnetic exchange bias field for different electric \nvoltage (a) and for electric bias voltages at      100 V and       200 V (b). \n  \n+100 V\nreference line \n \n \n \n 29 \n \n0100 200 300 400 500 6000.20400.20450.2050\n0100 200 300 400 500 6000.20880.20970.21060.21150100 200 300 400 500 6000.20880.20960.21040.2112\n0100 200 300 400 500 6000.2060.2080.2100.2120.214\n0100 200 300 400 500 6000.2080.2100.2120.214\n0100 200 300 400 500 6000.20880.20960.2104M (emu/g)\n  \n(a)\nV=+100V=+100V=+100\nV=0V=0\nV=0V=0\n  \n(b)-100V -100V+100V+100V\n0V 0V0V\n0V\n0+100V\n  \n(d)\n+200V +200V\n0V0V0V+200V\nTime (s)\n  \n(e)\n0V 0V\n0V\n-200V\n-200V-200V\n  \n(f)\nFig. 6 Room temperature time dependent magnetic moment at a magnetic field of 5 kOe \nand different voltage conditions (a) 0V, (b) 0V and \n 100V, (c) 0V and 100V, and (d) 0V \nand -100V, (e) 0V and +200V, (f) 0V and -200V for Ga04MA100V8 sample.0V0V0V 0V\nV=0\n-100V-100V\n-100V-100V\n  \n(c) \n \n \n \n 30 \n \n0100 200 300 400 500 6000.2000.2020.204\n0100 200 300 400 500 6000.2020.204\n0100 200 300 400 500 6000.2020.2040.206\n0100 200 300 400 500 6000.2040.2060.208\n0 200 400 600012400 450 500 550 600\n400 450 500 550 6000.13%1.03% \n +100V+100V\n0 V 0 V0 V+100V\n  \n1.36%\n0.26%1.14% \n M (emu/g)\n-100 V -100 V0 V\n0 V 0 V0 V\nTime (s)-100 V\n  (a)\n(b)\n(c) \n -100V0 V+100V\n-100V0 V0 V\n0 V+100V\n  \n2.51%1.23%\n  (f) H=5 kOe\nTime (s)M (emu/g)\n  \n500 V450 V400 V350 v300 V250 V200 V150 V100 V50 VV=0\nFig. 7 Time dependent magnetic moment at a magnetic field of 5 kOe and voltage \nconditions (a) 0V and +100V, (b) 0V and 100V, (c) 0V and \n 100 V, (d) branch of (a), \n(e) branch of (b), (f) magnetization at voltage 0-500 V, (g) change of magnetization. (g)\n% of change of peak value \n M (%)\nApplied voltage (V)% of change of base valueGa04MA50V8\n(d) \n +100V\n0 V\n  \n(e)\n  \n-100 V0 V 0 V\nTime (sec)\n   \n \n \n \n 31 \n \n-1.0 -0.5 0.0 0.5 1.0-0.10-0.050.000.050.10\n-1.0 -0.5 0.0 0.5 1.0-0.10-0.050.000.050.10\n0100 200 3000100200300 \n M (emu /g)\nH (kOe)\n  \n 0V\n +100V\n +200V\n +300V(a) \n (b)\n 0 V\n + 200 V\n -200 VElectric Voltage (V)\nFig. 8  Room temperature M(H) loops at different positive voltages (a) and a comparative \nM(H) loops at     200 V and at 0 V (b). The Inset shows variation of exchange bias field\nand coercivity at different voltages for Ga08MA25V8 sample. \nM (emu /g)\nH (kOe)\n  \n\n Hexb (Oe)HC line\n  \n \n \n \n 32 \n \n0.32100.32160.3222\n0.3150.3200.325\n0100 200 300 400 500 6000.3150.3200.3250.3150.3200.325\n0100 200 300 400 500 6000.3150.3200.3250.330\n0100 200 300 400 500-2-1012M (emu /g)M (emu /g)\n  \n(a)\n5 kOe (0 V)\n(b)\n+200V+200V+200V\n0V0V0VH=5 kOe (0 V)\n  \n(c)\n0V\n-200V -200V\n-200V0V0V\n  (d)\n0V0V\n0V0V\n-200V -200V+200V +200V\n-200V0V+200V\n0V\n   (e)\nTime (s) Applied Voltage (V)-500V -400V-300V-200V-100V+500V +400V+300V\n+100V+200V\nTime (s)\n  \no V reference\nFig.9 In field magnetic relaxation at magnetic field of 5 kOe for applied electric voltage at \n0 V (a). The in-field magnetic relaxation at ON-OFF mode of voltage for +200 V (b), for \n00V (c), for \n  200V (d), and increment of voltage up to \n  500V in steps of 100V (e). \nThe charge of switched magnetization at peak values with applied electric field (f) is \nshown for Ga08MA25V8 sample. (f)\n  negative voltagepositive voltageM (%) for peak value\n   \n \n \n \n 33 \n \n \n \n \n \n 34 \n \n(c) Hext<HAFM, V = 0Hext\n<> =0\nMagnetization (arb. unit)\nTime (s)Hext = 5 kOe, \nV = 0M(t)<> =0\nHextup spin\ndown spin (normal)\n(d) Hext<HAFM, V 0\nA\nB\nbc\na\nA\nB\nbca\nFe (  ), Ga (  ), Oxygen atom (  ) (a)\nCore ( ), Shell ( )(b)H\nV = 0\nM0\n(f)M0-Mv-H\nV 0, (-V)\nM0\n(e)M0-Mv+H\nV 0, (+V)\nM0\nFig. 11A schematic diagram of the spin order between two planes in -Fe2O3and Gadoped -Fe2O3(a),  Core -shell spin \nstructure in  a grain of Gadoped -Fe2O3(b),  in -field magnetic relaxation  at V = 0 (c) and in the presence of constant \nvoltage (d),  response of spin vectors during M(H) measurement in the presence of  constant + ve(e) and –ve(f)  voltage. "    },    {        "title": "1708.02948v2.Weyl_magnons_in_pyrochlore_antiferromagnets_with_all_in_all_out_orders.pdf",        "content": "Weyl magnons in pyrochlore antiferromagnets with all-in-all-out orders\nShao-Kai Jian1and Wenxing Nie2,\u0003\n1Institute for Advanced Study, Tsinghua University, Beijing 100084, China\n2Center for Theoretical Physics, College of Physical Science and Technology, Sichuan University, Chengdu 610064, China\n(Dated: December 25, 2017)\nWe investigate novel topological magnon band crossings of pyrochlore antiferromagnets with all-\nin-all-out (AIAO) magnetic order. By general symmetry analysis and spin-wave theory, we show\nthat pyrochlore materials with AIAO orders can host Weyl magnons under external magnetic \felds\nor uniaxial strains. Under a small magnetic \feld, the magnon bands of the pyrochlore with AIAO\nbackground can feature two opposite-charged Weyl points, which is the minimal number of Weyl\npoints realizable in quantum materials and has not be experimentally observed so far. We further\nshow that breathing pyrochlores with AIAO orders can exhibit Weyl magnons upon uniaxial strains.\nThese \fndings apply to any pyrochlore material supporting AIAO orders, irrespective of the forms\nof interactions. Speci\fcally, we show that the Weyl magnons are robust against direct (positive)\nDzyaloshinskii-Moriya interactions. Because of the ubiquitous AIAO orders in pyrochlore magnets\nincluding R 2Ir2O7, and experimentally achievable external strain and magnetic \feld, our predictions\nprovide promising arena to witness the Weyl magnons in quantum magnets.\nIntroduction.| Weyl fermions, which are originally\nproposed as solutions of massless Dirac equation [1],\nemerge as linear crossing points [2{8] of two elec-\ntronic bands in solid-state materials, the so-called Weyl\nsemimetals. The Weyl points emit Berry \rux and be-\nhave like monopoles in momentum space, making Weyl\npoints robust again perturbations and leading to many\nexotic properties. For instance, there are fermi arcs con-\nnecting projections of two opposite-charged Weyl points\non the surface of Weyl semimetals. It also leads to in-\nteresting transport phenomena [9{14], such as negative\nmagnetoresistivity [9], quantum anomalous Hall e\u000bects\n[10], and chiral magnetic e\u000bects [10] etc.\nThese robust and interesting topological properties\ntrigger the search for topological magnon modes [15, 16],\nespecially the bosonic analog of Weyl points. Weyl (or\nDirac) points are discovered in systems with phonons\n[17], photons [18, 19] as well as magnons [20{27].\nLike Weyl semimetals, Weyl magnon semimetals exhibit\nmagnon arcs on the surface connecting the projections of\nopposite-charged Weyl points. Unlike other categories,\nthe Weyl magnon semimetals generically break time-\nreversal symmetry owing to the presence of magnetic or-\nders. Recently, people have also proposed detections of\nchiral anomaly of Weyl magnon semimetal by Aharonov{\nCasher e\u000bect [28].\nSince there exists plenty of distinct magnetic orders\nin three-dimensional materials, one would naturally ask\nwhich material provides the promising platform to real-\nize Weyl magnons. In this letter, by symmetry analy-\nsis and model calculations, we show that under external\nmagnetic \felds, pyrochlore antiferromagnets with AIAO\norders can host Weyl points. We also show that the\nbreathing pyrochlore [29{31] with AIAO order exhibits\nWeyl magnons upon uniaxial strains. It is the \frst time\nthat Weyl magnons are shown to exist in materials with\nAIAO magnetic orders. We emphasize that our symme-try analysis applies to the materials that share the same\nsymmetry and representation with the e\u000bective Hamilto-\nnian constructed near \u0000 point, irrespective of the speci\fc\nforms of complicated interactions, like Dzyaloshinskii-\nMoriya (DM) interaction [32, 33] as we show later. Since\nthe AIAO orders exist in many heavy transition metal\ncompounds with 4d and 5d elements [34, 35], such as\nNd2Ir2O7[36], Eu 2Ir2O7[37, 38], Cd 2Os2O7[39], and\nSm2Ir2O7[40] etc, and Weyl points can be easily manip-\nulated by external magnetic \felds [21], our proposals call\nfor experimental detections of Weyl points in pyrochlore\nantiferromagnets with AIAO magnetic orders.\nModel and classical phases.| The point group of py-\nrochlore lattice is Oh, while it is lowered to Thwith\nthe presence of AIAO orders. To model the AIAO mag-\nnetic orders, we consider following Hamiltonian [20] on\n(breathing) pyrochlore lattice,\nH=JX\nhiji2u~Si\u0001~Sj+J0X\nhiji2d~Si\u0001~Sj+DX\ni(~Si\u0001^zi)2;(1)\nwhereJ >0 andJ0>0 are antiferromagnetic couplings\nbetween nearest neighbors in up-pointing and down-\npointing tetrahedra, respectively. As shown in Fig. 1(a),\nwe takeJ=J0for pyrochlores, and J6=J0for breathing\npyrochlores. The last term in Eq. (1) describes the local\nspin anisotropy, where ^ ziis the local anisotropic direction\npointing to the center of each tetrahedra as indicated by\nFig. 1(b). For D > 0, spin tends to lie in the plane\nperpendicular to local ^ zi. Thus there is an accidental\nU(1) degeneracy of classical orders, which is found to be\nbroken by quantum disorder, leading to the existence of\nWeyl magnons [20].\nHere we consider an easy-axis spin anisotropy D < 0.\nIt is easy to show that the classical ground state is AIAO\nordered. For D< 0, the third term is maximally satis\fed\nif~Sialigns or anti-aligns in local ^ zi-directions. There are\ntotally sixteen con\fgurations with two AIAO con\fgura-arXiv:1708.02948v2  [cond-mat.str-el]  22 Dec 20172\n(a)\n(b)\n(c)\nΓXW L Γ0.00.10.20.30.40.50.6ω/J (d)\nFIG. 1. (a) A schematic plot of breathing pyrochlore lattice.\nThe tetrahedron in blue color is up pointing while the tetra-\nhedron in red color is down pointing. The uniaxial strains in\n(001) direction is shown. (b) A tetrahedra and AIAO order\nin one cubic, with ^ zdirections of local frames. The magnetic\n\felds applied in (100) direction is indicated by the orange ar-\nrow. (c) The \frst Brillouin zone of fcclattice. The basis of\nreciprocal lattice are given by ~b1=1\n2(0;1;1),~b2=1\n2(1;0;1),\n~b3=1\n2(1;1;0), where the lattice constant is set to be unit. (d)\nThe magnon bands in pyrochlore lattice with AIAO magnetic\norder. Note that the four bands split into a triple degeneracy\nand a singlet at \u0000 point.\ntions among them. Only AIAO con\fguration satis\fes the\nlocal constraintP\ni2u(d)~Si= 0 from the \frst two terms\nto optimize the exchange interactions simultaneously.\nSymmetry analysis.| A key observation of the magnon\nbands is that the states at \u0000 point are triply degenerate\nas shown in Fig. 1(d), a spin-wave spectrum of Eq. (1)\nwithD< 0. Thus these degenerate states form a Tgrep-\nresentations of Thgroup in the presence of AIAO order,\ndistinct from other magnetic orders in pyrochlore, e.g.\nthe easy-plane orders [20], where the \u0000 point is at most\ndoubly degenerate.\nWe \frst consider pyrochlore lattice with AIAO mag-\nnetic order, which preserves Thsymmetry, and will show\nthe presence of two Weyl points under external mag-\nnetic \felds. Given that the states at \u0000 point form three-\ndimensional Tgrepresentations of Thgroup, we can con-\nstruct an e\u000bective k\u0001ptheory near the \u0000 point. The\nHamiltonian up to quadratic order in momentum space\nis constructed in Ref. [1, 42] and is given by\nHT(~ p) =\u000b1j~ pj2+\u000b2X\nip2\niL2\ni\n+[pxpy(\u000b3fLx;Lyg+\u000b4Lz) +c:p:];(2)\nwhereLi(i= 1;2;3) is a 3\u00023 matrix [42] and \u000bj(j=1;:::;4) is a constant which characterizes the dispersion\naround \u0000 point. fgdenotes anticommutator and c:p:\nmeans cyclic permutations. Note that \u000b4term breaks\ntime-reversal symmetry. The subscript TindicatesTh\ngroup.\nThere is a double degeneracy along each axes, namely,\n(100), (010) and (001), as shown in Fig. 1(d), protected\nbyC2rotational and \u001bhhorizontal re\rection symmetries.\nA simple strategy to get Weyl magnons is to break these\nsymmetries, especially to split these two-fold degenerate\nbands. An experimentally accessible way to lower the\nsymmetry is applying an external magnetic \feld. Ac-\ncordingly, we introduce a Zeeman term HZ=~\fZ\u0001~L\ninto Eq. (2), where ~\fZis proportional to applied mag-\nnetic \feld. The subscript Zindicates Zeeman e\u000bect. For\nsimplicity, we assume the magnetic \feld is along (100)\ndirection as indicated by orange arrow in Fig. 1(b). The\nZeeman term is simpli\fed as HZ=\fx\nZLx. Then the sym-\nmetry group is lowered from ThtoC2h. The double de-\ngeneracy along (010) and (001) directions is split entirely;\nwhile the degeneracy along (100) direction is split ex-\ncept two crossings at ( \u0006Q1;0;0), whereQ1=p\nj\fx\nZ=\u000b2j.\nThus we get a minimal Weyl magnon band structure from\nHT+HZwith only two Weyl points as shown in Fig. 2(a).\nNote that the Weyl points are locked at pxaxis owing to\nC2rotational symmetry. In general, the magnetic \feld\ncan be applied in arbitrary direction. Two Weyl points\nstill survive but will be shifted away from px= 0 axis as\nlong as the magnetic \feld is small enough. Note that the\nWeyl magnons also appear in breathing pyrochlore with\nAIAO orders when magnetic \felds are applied.\nAnother convenient way to lower the symmetry is ap-\nplying strains upon the materials [5, 6, 23]. For instance,\nuniaxial strain along (001) direction lowers the symmetry\ntoD2hgroup, giving rise the following term,\nHstrain =\fstrain(2L2\nz\u0000L2\nx\u0000L2\ny); (3)\nwhere\fstrain is a real constant corresponding to the\nstrength of the strain. In the presence of Hstrain, there\nemerges a nodal line in pz= 0 plane, if \u000b2\u0001\fstrain>0, as\nshown in Fig. 2(c) (also see Supplemental Materials [42]\nfor details). This nodal line is protected by \u001bhhorizon-\ntal re\rection symmetry [42]. It may lead to \rat magnon\nsurface states [44, 45].\nNow, we consider the breathing pyrochlore lattice, in\nwhichOhgroup of pyrochlore lattice is lowered into Td\ngroup. (Note that AIAO order in breathing pyrochlore\npreservesTgroup). Follow the same strategy, we add\n(001)-directional strains to lower the symmetry from T\ngroup down to D2group. Accordingly, we \fnd besides\nEq. (3), the following term is also allowed by symmetry,\nHD(~ p) =\fD(pxLx\u0000pyLy); (4)\nwhere\fDis a real constant corresponding to the strength\nof strains. The subscript DindicatesD2group. After\naddingHstrain +HDtoHT, the nodal line splits into3\npxpy\nΓ XWY\nB\n(a)\nΓXW ΓY0.30.40.50.6ω/J (b)\npxpy\nΓ XWY\n(c)\nΓXW ΓY0.30.40.50.6ω/J (d)\nFIG. 2. (a) A schematic plot of two Weyl points at pxaxis.\nRed and blue colors denote opposite monopole charges. The\napplied magnetic \feld is shown. (b) The magnon bands along\nthe path \u0000- X-W-\u0000-Ywhen magnetic \felds are applied, with\nS= 1=2;J0=J;D =\u00000:2J;B = 0:05J. The band-crossing\npoint is along \u0000- X. (c) A schematic plot to show a nodal\nline emerges in the pyrochlores upon a uniaxial strain along\n(001) direction. (d) The magnon bands along \u0000- X-W-\u0000-Y\npath withS= 1=2;J0=J;D =\u00000:2J;\r= 2%;Bz= 0:05J.\nIt clearly shows the nodal-line crossings.\nfour Weyl points located at the points ( \u0006Q2;0;0) and\n(0;\u0006Q2;0), since the the horizontal re\rection symmetry\n\u001bhis broken, where Q2is given in [42]. A schematic plot\nof these Weyl points is shown in Fig. 3(a).\nWeyl magnons in pyrochlores under a magnetic \feld.|\nBy symmetry analysis, with the background of AIAO\nmagnetic order, the Weyl magnons and nodal-line\nmagnons [44, 45] appear under external magnetic \felds\nor uniaxial strains. Now we use linear spin-wave the-\nory to show the emergence of Weyl magnons explic-\nitly. The spin operator can be expressed in terms of the\nHolstein{Primako\u000b bosons ~S\u0016\u0001^z\u0016=S\u0000ay\n\u0016a\u0016,~S\u0016\u0001^x\u0016=p\n2S(a\u0016+ay\n\u0016)=2, and~S\u0016\u0001^y\u0016=p\n2S(a\u0016\u0000ay\n\u0016)=(2i), where\na\u0016(ay\n\u0016) is annihilation (creation) operator of Holstein{\nPrimako\u000b boson at \u0016th sublattice, and the local frames\nat each sublattice are listed in Appendix. The spin-wave\nHamiltonian up to quadratic order is given by\nH=X\n~ pX\n\u0016\u0017[ay\n~ p;\u0016A\u0016\u0017(~ p)a~ p;\u0017+ (a~ p;\u0016B\u0016\u0017(~ p)a\u0000~ p;\u0017+H:c:)];\n(5)\nwhereA\u0016\u0017(~ p) =S[\u000e\u0016\u0017(J+J0\u00002D)\u00001\n3(1\u0000\u000e\u0016\u0017)(J+\nJ0ei(p\u0016\u0000p\u0017))], andB\u0016\u0017(~ p) =S(1\u0000\u000e\u0016\u0017)ei\u001e\u0016\u0017(J+\nJ0e\u0000i(p\u0016\u0000p\u0017)) with\u001e01=\u001e23=\u0000\u0019\n3;\u001e02=\u001e13=\u0019\n3and\u001e03=\u001e12=\u0019, are 4\u00024 matrices in sublattice space.\nAndp\u0016\u0011~ p\u0001~b\u0016, with~b0= (0;0;0) and~b1;2;3de\fned in the\ncaption of Fig. 1(c). The spin-wave spectrum is shown\nin Fig. 1(d), where the coupling constants are chosen to\nbeJ=J0;D=\u00000:2J;S= 1=2. A gap between the \frst\nand the upper three bands at \u0000 is obvious.\nConsider applying a magnetic \feld, i.e., \u000eHZ=\n~B\u0001P\ni~Si, where~Bis the magnetic \feld (we have\nabsorbed the coupling into magnetic \feld, thus ~B\nhas the dimension of energy), in the linear spin-wave\nregime (i.e., the magnetic \feld are small enough),\nwe get\u000eAZ\n\u0016\u0017=P\nj=x;y;zBj\u000eAZ\nj;\u0016\u0017, where\u000eAZ\nx=\ndiag(\u00001;\u00001;1;1)=p\n3,\u000eAZ\ny= diag(\u00001;1;\u00001;1)=p\n3 and\n\u000eAZ\nz= diag(\u00001;1;1;\u00001)=p\n3. To be speci\fc, we con-\nsider a small magnetic \feld in (100) direction, i.e., ~B=\n(B;0;0). The spectrum of H+\u000eHZis plotted in Fig.\n2(b), showing a Weyl point along \u0000- Xline. The two\nWeyl points are restricted in pxaxis due to C2rotational\nsymmetry along (100) direction. The magnetic \feld can\nbe along any direction and it shift the positions of the two\nWeyl points, realizing a tunable Weyl points by external\nmagnetic \felds [21].\nNodal-line magnons in strained pyrochlores.| Upon\n(001) uniaxial strains, the antiferromagnetic exchange in-\nteractions within an up-pointing (down-pointing) tetra-\nhedra become inequivalent, i.e., the two bonds lying in xy\nplane are distinct from the rest four bonds. The coupling\nstrengths become J!(1\u0006\r)J, where\rcharacterizes\nthe e\u000bect of the strain. The perturbation introduced by\nstains is captured by \u000eHstrain =P\n~ r[\rJ(~S~ r;0\u0001~S~ r;3+~S~ r;1\u0001\n~S~ r;2)\u0000\rJ(~S~ r;0\u0001~S~ r;1+~S~ r;0\u0001~S~ r;2+~S~ r;3\u0001~S~ r;1+~S~ r;3\u0001~S~ r;2),\nwhere the summation is over all unit cells. It leads to\n\u000eAstrain\n\u0016\u0016 =\rA\u0016\u0016=3, and\u000eXstrain\n\u0016\u0017 =\rX\u0016\u0017for\u0016+\u0017= 3,\n\u000eXstrain\n\u0016\u0017 =\u0000\rX\u0016\u0017for\u00166=\u0017;\u0016+\u00176= 3, where X=A;B.\nThe spectrum of magnons with a small tensile strain\n\r= 2% is shown in Fig. 2(d). The crossing points in\n\u0000-X, \u0000-Yand \u0000-Wlines show the evidence of the pre-\ndicted nodal line in pz= 0 plane, which is protected by\n\u001bhsymmetry.\nThe applied strains will generically modify the back-\nground AIAO order, since the constraintP\ni2u(d)~Si= 0\ncan no longer be satis\fed. However, if the strain is much\nsmaller than J,J0andD(\r= 2% in our case), we can\nneglect the titling of the background from AIAO order.\nNote that the local frames are also adjusted correspond-\ning to the strains, but we can neglect this e\u000bect since it\nonly shifts the positions of the nodal lines.\nWeyl magnons in strained breathing pyrochlore.| The\npresence of two distinct tetrahedra in breathing py-\nrochlore, where J6=J0, lowers the symmetry from Oh\ngroup toTdgroup. The symmetry is further lowered\nby the applied strain in (001) direction to D2group.\nThe e\u000bect of stains is captured by replacing J;J0by\n(1\u0006\r)Jand (1\u0006\r0)J0, respectively, leading to \u000eH0\nstrain =P\n~ r2u[\rJ(~S~ r;0\u0001~S~ r;3+~S~ r;1\u0001~S~ r;2)\u0000\rJ(~S~ r;0\u0001~S~ r;1+~S~ r;0\u00014\npxpy\nΓ XWY\n(a)\nΓXW ΓY0.20.30.40.5ω/J (b)\n-1.5 1.5-1.51.5\npxpy\n(c)\n-1.5 1.5-1.51.5\npxpy (d)\nFIG. 3. Four Weyl points emerge in breathing pyrochlore\nupon uniaxial strains along (001) direction. (a) A schematic\nplot of four Weyl points. Di\u000berent colors denote opposite\nmonopole charges in pz= 0 plane. (b) The magnon bands\nalong the path \u0000- X-W-\u0000-Y, withS= 1=2;J0= 0:6J;D =\n\u00000:2J;\r= 2%, showing the Weyl points are located at px\nandpyaxis. Berry curvatures of the four Weyl points within\nthe orange frame in (a) are plotted in (c) and (d). (c) The\nprojections of the directions of Berry curvature to pz= 0\nplane forpx;y2(\u00001:5;1:5). It shows the four Weyl points\nwith monopole charge 1 (blue color) and \u00001 (red color). (d)\nThe projections of directions of Berry curvature to pz= 0\nplane with applied magnetic \feld Bx= 0:005J.\n~S~ r;2+~S~ r;3\u0001~S~ r;1+~S~ r;3\u0001~S~ r;2)+P\n~ r2d[\r0J0(~S~ r;0\u0001~S~ r;3+~S~ r;1\u0001\n~S~ r;2)\u0000\r0J0(~S~ r;0\u0001~S~ r;1+~S~ r;0\u0001~S~ r;2+~S~ r;3\u0001~S~ r;1+~S~ r;3\u0001~S~ r;2).\nAs predicted by the symmetry analysis, four Weyl points\nemerge in the pz= 0 plane in the spectrum of H+\u000eH0\nstrain\nas shown in Fig. 3(b). Actually they are located at px\nandpyaxes due to C2rotational symmetry. The projec-\ntions of Berry curvature to pz= 0 plane are plotted in\nFig. 3(c), where four Weyl points with monopole charge\n1 (blue) and\u00001 (red) are clearly shown.\nOne can again apply magnetic \felds to the system, and\nconsequently tune the positions of Weyl points in the re-\nciprocal space [21]. To explicitly show an example, we\napply a small magnetic \feld B= 0:005Jin (100) direc-\ntion. The shifts of Weyl points are shown in Fig. 3(d)\nindicated by the Berry curvatures. Owing to the C2sym-\nmetry along (100) direction, two Weyl points (red) are\nstill locked in pxaxis. Keeping increasing the magnetic\n\felds, two Weyl points with monopole charge 1 (blue)\nwill hit one of the Weyl points with monopole charge \u00001\n(red) and then these three Weyl points merge as a sin-\ngle Weyl point with monopole charge \u00001. But there will\nemerge another four Weyl points [42].\nWeyl magnons in the presence of DM interactions.|\nHere, we emphasize that the results from the symmetry\nΓ X W L Γ0.30.40.50.60.7ω/J(a)\n●\nΓ X W L Γ0.30.40.50.60.7ω/J (b)\nFIG. 4. Spin-wav spectra with DM interaction. (a) The\nmagnon bands along the path \u0000- X-W-L-\u0000 withS= 1=2;D=\n\u00000:2J;DM = 0:18J. Note that the four bands split into a\ntriple degeneracy and a singlet at \u0000 point. (b) The magnon\nbands with applied magnetic \feld B= 0:05Jalong (100) di-\nrection. The band-crossing point is indicated by a red point\nalong \u0000-Xline.\nanalysis is applicable not only to the representative model\nHamiltonian in Eq. (1), but also to all the magnetic\nmaterials that satisfy Tdsymmetry and whose magnon\nbands ful\fll three-dimensional Tgrepresentation near \u0000\npoint. Especially, the Weyl magnon can be robust in the\npresence of the DM interactions which are important in\nthose strongly correlated materials with spin-orbit cou-\npling. To demonstrate that, we add DM interaction as\nan example,\nH=X\nhiji(J~Si\u0001~Sj+~Dij\u0001~Si\u0002~Sj) +DX\ni(~Si\u0001^zi)2;(6)\nwhere DM vector ~Dijis dictated by Tdgroup [42] whose\nmagnitude is given by j~Dj=DM.\nThe \\direct\" (positive) DM interaction is found to fa-\nvor the AIAO ground state [2, 40, 42]. Our spin-wave\nanalysis for Eq. (6) shows that the magnon bands at \u0000\npoint also form a singlet and a three-dimensional Tgrep-\nresentation as shown in Fig. 4(a), where DM = 0:18J\n[40]. Thus all the symmetry analysis can apply to such\na case, namely, there will emerge two Weyl points in\npyrochlore under magnetic \feld, and there will emerge\nfour Weyl points in breathing pyrochlore under strains.\nHere, we explicitly show the emergent Weyl points in py-\nrochlore under magnetic \feld which is indicated by red\npoint in Fig. 4(b), where the magnetic \feld B= 0:05J\nis along (100) direction.\nConclusions.| With the aid of symmetry analysis,\nwe show that under magnetic \felds pyrochlores with\nAIAO orders can host Weyl magnons. Moreover, the\npyrochlores exhibit nodal line upon a uniaxial strain.\nWe also predict that four Weyl points would emerge in\nbreathing pyrochlore antiferromagnets with AIAO orders\nupon uniaxial strains. To con\frm the predictions, we\nperform a linear spin-wave calculation to a simple mi-\ncroscopic spin model. Owing to the ubiquitous existence\nof AIAO orders in nature and achievable technique in\nexperiments, our \fndings shed light on experimental re-5\nalization of Weyl magnons.\nWe remark on the possible experimental detections of\nthese topological Weyl magnons. The bulk Weyl points\ncan be detected by inelastic neutron scattering. While\nfor the magnon arcs, for instance, there is a magnon\narc connecting the projections of opposite-charged Weyl\npoints in (001) surface Brillouin zone of the pyrochlore\nunder magnetic \felds, it should be possible to detect\nthem by using surface-sensitive probes, such as high-\nresolution electron energy loss spectroscopy, or helium\natom energy loss spectroscopy. Besides the spectroscopic\nexperiments, the Weyl magnon semimetals would also\nlead to anomalous thermal Hall e\u000bects [46{51], just like\nin the Weyl semimetals [52, 53]. In the materials showing\nthe AIAO orders, such as the pyrochlore oxides Eu 2Ir2O7\n[37, 38], and Cd 2Os2O7[39], the breathing pyrochlores\nBa3Yb2Zn5O11[31], our predictions can be veri\fed by\nabove techniques experimentally.\nWe would like to thank Hong Yao and Luyang Wang\nfor helpful discussions. S.-K. Jian is supported in part\nby the NSFC under Grant No. 11474175 and by the\nMOST of China under Grant No. 2016YFA0301001.\nW. Nie is supported by the NSFC under Grant No.\n11704267 and by the MOST of China under Grant No.\n2017YFB0405700. W. Nie acknowledge the hospitality\nof Renmin University of China and Tsinghua University,\nwhere part of the work was initiated.\n\u0003wenxing.nie@gmail.com\n[1] H. Weyl, Z. Phys. 56, 330 (1929).\n[2] S. Murakami, New J. Phys. 9, 356 (2007).\n[3] X. Wan, A. M. Turner, A. Vishwanath, and S. Y.\nSavrasov, Phys. Rev. B 83, 205101 (2011).\n[4] A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205\n(2011).\n[5] J. Ruan, S.-K. Jian, H. Yao, H. 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McMorrow, Phys. Rev. Lett. 117,\n037201 (2016).\n[41] J. M. Luttinger, Phys. Rev. 102, 1030 (1956).\n[42] See the Supplemental Materials for details.\n[43] M. Elhajal, B. Canals, R. Sunyer, and C. Lacroix, Phys.\nRev. B 71, 094420 (2005)\n[44] A. Mook, J. Henk, and I. Mertig, Phys. Rev. B 95,\n014418 (2017).\n[45] K. Li and J. Hu, Chin. Phys. Lett. 34, 077501 (2017).\n[46] L. Zhang, J. Ren, J.-S. Wang, and B. Li, Phys. Rev. B\n87, 144101 (2013).\n[47] R. Shindou, R. Matsumoto, S. Murakami, and J. I. Ohe,\nPhys. Rev. B 87, 174427 (2013).\n[48] A. Mook, J. Henk, and I. Mertig, Phys. Rev. B 89,\n134409 (2014).\n[49] A. Mook, J. Henk, and I. Mertig, Phys. Rev. B 90,\n024412 (2014).\n[50] S. A. Owerre, J. Phys.: Condens. Matter 28, 386001\n(2016).\n[51] S. A. Owerre, Phys. Rev. B 94, 094405 (2016).\n[52] K.-Y. Yang, Y.-M. Lu, and Y. Ran, Phys. Rev. B 84,\n075129 (2011).6\n[53] A. A. Zyuzin, S. Wu, and A. A. Burkov, Phys. Rev. B\n85, 165110 (2012).\nSUPPLEMENTAL MATERIAL\nA. The~k\u0001~ ptheory\nThe point group of pyrochlore lattice is Oh. The AIAO oder preserves Thgroup. The magnon bands have a triple\ndegeneracy at \u0000 point, which forms Tgrepresentation. Now we construct the ~k\u0001~ ptheory near the \u0000 point. Three\nspin-one matrices serve as Tgrepresentations [1],\nLx=0\n@0 0 0\n0 0i\n0\u0000i01\nA;Ly=0\n@0 0\u0000i\n0 0 0\ni0 01\nA;Lz=0\n@0i0\n\u0000i0 0\n0 0 01\nA: (S1)\nUsing these matrices, one can construct all nine 3 \u00023 hermitian matrices: I3;Lx;c:p:;fLx;Lyg;c:p:; 2L2\nz\u0000L2\nx\u0000L2\ny;L2\nx\u0000\nL2\ny, whereI3is identity matrix and c:p:means cyclic permutations. On the other hand, the momentum piis aTu\nrepresentation. We can construct the Hamiltonian by using these two representations, as shown in Table. S1. Then\nthe most general Hamiltonian up to quadratic order in momentum is given by\nHT(~ p) =\u000b1j~ pj2+\u000b2X\nip2\niL2\ni+ [pxpy(\u000b3fLx;Lyg+\u000b4Lz) +c:p:]; (S2)\nwhere\u000biare real parameters. Note that \u000b4term breaks time reversal symmetry. By \ftting to the spin wave dispersion\nforS= 1=2;J=J0= 1;D=\u00000:2 in 10% Brillouin zone, these parameters are given by \u000b1=\u00002:75\u000210\u00003;\u000b2=\n30:15\u000210\u00003;\u000b3= 26:03\u000210\u00003;\u000b4= 18:80\u000210\u00003.\nTABLE S1. Representations of Thgroup constructed from ~Land~ p.\nReps. ~L ~ p\nAg 1 p2\nx+p2\ny+p2\nz\nEg (2L2\nz\u0000L2\nx\u0000L2\ny;L2\nx\u0000L2\ny) (2p2\nz\u0000p2\nx\u0000p2\ny;p2\nx\u0000p2\ny)\nTg (Lx;Ly;Lz) and (fLx;Lyg;fLy;Lzg;fLz;Lxg) (pxpy;pypz;pzpx)\nTu (px;py;pz)\nB. Nodal-line magnons in strained pyrochlores\nTo lower the symmetry of the system, one adds uniaxial strains along (001) direction. The symmetry group is\nlowered toD2h. To capture the strain, we add Hstrain =\fstrain(2L2\nz\u0000L2\nx\u0000L2\ny) into the Hamiltonian. There is a band\ncrossing along (100) [or (010)] direction though the double degeneracy is split. The crossing point is at ( \u0006Q;0;0) and\n(0;\u0006Q;0), whereQ=p\n3\fstrain=\u000b2. Note that \fstrain=\u000b2must be positive to have a band crossing. Actually, there\nemerges a nodal line at pz= 0 plane, given by the function\n6\f2\nstrain + 4(\u000b2\n3+\u000b2\n4)p2\nxp2\ny+\u000b2\n2(p2\nx\u0000p2\ny)2\u0000\u000b2(p2\nx+p2\ny) = 0: (S3)\nThe nodal-line is protected by \u001bhhorizontal re\rection symmetry.\nC. Weyl magnons in strained breathing pyrochlores\nAs shown in the main text, applying uniaxial strains to breathing pyrochlore with AIAO order leads to four Weyl\npoints. Diagonalizing HT+Hstrain +HDleads to four Weyl points located at ( \u0006Q2;0;0) and (0;\u0006Q2;0), where\nQ2=p\n(3\u000b2\fstrain +\f2\nD)=\u000b2\n2.7\n(a)\n-1.5 1.5-1.51.5\npypz (b)\nFIG. S1. (a) A schematic plot of six Weyl points in Brillouin zone. Two of them are restricted in pxaxis due to C2symmetry\nalong (100) symmetry. (b) The projections of the directions of Berry curvatures in px= 0 plane.\nD. Spin wave analysis\nSince there is not U(1) symmetry of Holstein{Primako\u000b bosons, we can write the Hamiltonian as H=P\n~ p\by\n~ pH(~ p)\b~ p\nin spinor space \b ~ p= (a~ p;\u0016;ay\n\u0000~ p;\u0016), whereH(~ p) =\u0010A(~ p)=2By(~ p)\nB(~ p)AT(\u0000~ p)=2\u0011\n:The local frame for the four sublattices of\nthe pyrochlore lattices is shown in Table S2. The expressions of the matrix elements of A;B are given in the main\ntext. The dispersions given in the main text are obtained by diagonalizing gH, whereg=diag(I4\u00024;\u0000I4\u00024), including\nproperly\u000eH0\nstrain and (or)\u000eHZ.\nIn the strained breathing pyrochlore, applying an external magnetic \felds shift the positions of two Weyl points as\nshown in the main text. Keep increasing the magnetic \felds, there emerge four extra Weyl points which are shown in\nFig. S1: a schematic plot of total six Weyl points in Fig. S1(a) with the projections of directions of Berry curvatures\nin Fig. S1(b). Note that these points in Fig. S1(b) do not indicate the Weyl points locate at px= 0 plane since there\nis no symmetry to \fx it.\nE. Spin-wave analysis with Dzyaloshinskii-Moriya interaction\nThe Dzyaloshinskii-Moriya (DM) vectors are given by [2, 3]\n~D03=DMp\n2(\u00001;1;0);~D12=DMp\n2(\u00001;\u00001;0);~D01=DMp\n2(0;\u00001;1); (S4)\n~D23=DMp\n2(0;\u00001;\u00001);~D13=DMp\n2(1;0;1);~D02=DMp\n2(1;0;\u00001): (S5)\nSince AIAO is already the classical ground state for Eq.(1) in the main text, we only need to check AIAO is also a\nground state for DM term. Assuming translation symmetry, the problem reduces to the con\fguration inside a unit\ncell which has only eight degrees of freedom corresponding to four spins. Expressed in local frame in Table S2, one\ncan do a variation with respect to these eight parameters. When DM > 0, it turns out that the lowest energy is\ngiven by the con\fguration where each spin is parallel to the ^ z\u0016direction in local frames, namely, the con\fguration\ncorresponds to AIAO order.\nTABLE S2. Local frame in each sublattice for AIAO orders, indicated in Fig. 1(b).\n\u0016 ^x\u0016 ^y\u0016 ^z\u0016\n01p\n2(\u00001;1;0)1p\n6(\u00001;\u00001;2)1p\n3(1;1;1)\n11p\n2(\u00001;\u00001;0)1p\n6(\u00001;1;\u00002)1p\n3(1;\u00001;\u00001)\n21p\n2(1;1;0)1p\n6(1;\u00001;\u00002)1p\n3(\u00001;1;\u00001)\n31p\n2(1;\u00001;0)1p\n6(1;1;2)1p\n3(\u00001;\u00001;1)8\n\u0003wenxing.nie@gmail.com\n[1] J. M. Luttinger, Phys. Rev. 102, 1030 (1956).\n[2] M. Elhajal, B. Canals, R. Sunyer, and C. Lacroix, Phys. Rev. B 71, 094420 (2005)\n[3] Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa and Y. Tokura, Science 329297 (2010)"    },    {        "title": "1708.05100v1.Copper_Tellurium_Oxides___A_Playground_for_Magnetism.pdf",        "content": "Copper Tellurium Oxides - A Playground for Magnetism\nM. R. Norman1\n1Materials Science Division, Argonne National Laboratory, Argonne, IL 60439\n(Dated: August 18, 2017)\nA variety of copper tellurium oxide minerals are known, and many of them exhibit either unusual\nforms of magnetism, or potentially novel spin liquid behavior. Here, I review a number of the\nmore interesting materials with a focus on their crystalline symmetry and, if known, the nature\nof their magnetism. Many of these exist (so far) in mineral form only, and most have yet to have\ntheir magnetic properties studied. This means a largely unexplored space of materials awaits our\nexploration.\nIn 2005, Dan Nocera's group reported the synthesis of\nthe copper hydroxychloride mineral, herbertsmithite [1].\nA number of years later, they were able to report the\ngrowth of large single crystals [2]. Since then, a num-\nber of relatives of this mineral have been discovered and\ncharacterized [3]. Despite the existence of a large Curie-\nWeiss temperature of order 300 K, herbertsmithite does\nnot order down to 20 mK [4].\nThe reason these events have signi\fcance is that these\nminerals could be a realization of an idea proposed by\nPhil Anderson back in 1973 [5] that was based on an early\ndebate in the \feld of magnetism between Louis N\u0013 eel and\nLev Landau. N\u0013 eel had proposed the existence of anti-\nferromagnetism, where there are two sub lattices of fer-\nromagnetic moments oppositely aligned. This state was\nsubsequently seen by neutron scattering (which resulted\nin a Nobel prize for N\u0013 eel, and later for the neutron scat-\nterer, Cli\u000bord Shull). But at the time, there was great\nskepticism about the existence of this state. The rea-\nson is that it is not an eigenstate of the spin operator\n(unlike ferromagnetism). There was suspicion that the\ntrue ground state would be a singlet. We now know that\nthe origin of the N\u0013 eel state is broken symmetry [6], and\nthat \ructuations are usually not enough to destabilize\nlong range order. But Phil realized that if the spins sat\non a non-bipartite lattice, matters could change. Imag-\nine a triangle with Ising spins. Then if two spins are\nanti-aligned, the direction of the third spin is undeter-\nmined. Phil speculated that instead of N\u0013 eel order, the\nspins instead paired up to form singlets, and this would\nbe preferred in two dimensions (where thermal \ructua-\ntions have a tendency to suppress order) and for low spin\n(where quantum \ructuations are more important). This\nis particularly obvious for S=1/2, where a singlet bond\nhas an energy of -3 J/4 compared to - J/4 for an antifer-\nromagnetic bond. But to avoid the energy loss from the\nunpaired spin, these singlets should \ructuate from bond\nto bond, much like Pauling's model for how double car-\nbon bonds in benzene rings resonate from one link to the\nnext (hence the name, resonating valence bonds).\nMost attention has been given to the Heisenberg\nmodel, given the more important role of \ructuations in\nthis case. But we now know that the near neighborHeisenberg model on a triangular lattice does order, with\nthe spins rotating by 120\u000efrom one sub lattice to the\nnext [7]. This is most clear from exact diagonalization\nstudies, where precursors of the broken symmetry state,\nand associated magnon excitations, are evident in the\neigenvalue spectrum [8]. But for the kagome case, the\nspectrum is qualitatively di\u000berent, with no signature of\nthese e\u000bects [9]. Over the years, a number of numeri-\ncal studies have been done, either purporting a valence\nbond solid (an ordered array of singlets), or various types\nof quantum spin liquid states (gapped Z 2, gapless U(1),\nchiral, etc., where the group corresponds to an emergent\ngauge group associated with the symmetry of the spin\nliquid). The uncertainty is connected to the fact that all\nof these states have energies comparable to one another.\nThe real interest, though, is that these solutions are char-\nacterized by fractionalized excitations (typically free spin\n1/2 neutral fermions known as spinons, or gauge \rux ex-\ncitations known as visons). Proving the existence of such\nexcitations is a major challenge in physics [10, 11].\nThis brings us to real materials. Many of them\neither have contributions over and beyond that of\na near neighbor Heisenberg model (longer range ex-\nchange, anisotropic exchange, Dzyaloshinskii-Moriya in-\nteractions, etc.) or distorted lattices, all of which can in\nprinciple either change the nature of the ground state, or\npromote or destabilize order. Hence the interest in \fnd-\ning as many materials as possible that have frustrated\nlattices for their magnetic ions, and then characteriz-\ning these materials. In that context, we can often allow\nMother Nature to do the hard work for us. Many miner-\nals are known which have the appropriate magnetic lat-\ntices, making them ideal sources for \fnding desired ma-\nterials. In fact, Dan Nocera's group had \frst synthesized\nthe iron jarosite minerals (where the iron ions sit on a\nkagome lattice), but realized that the same would not\napply to a copper (S=1/2) version, since Cu2+would\nnot go onto the Fe3+site. Hence the turn to herbert-\nsmithite once they had seen that structure in the miner-\nalogical literature. In that context, there is one mineral\nwhere copper goes into a jarosite-type structure, osariza-\nwaite [12], but in this case, the copper kagome lattice is\nstrongly diluted by Al3+ions. The magnetic propertiesarXiv:1708.05100v1  [cond-mat.str-el]  16 Aug 20172\nFormula unit SG Lattice Ref\nCu6IO3(OH) 10Cl R3 maple leaf [14]\nCu6TeO 4(OH) 9Cl R3 maple leaf [14]\nPb3Cu6TeO 6(OH) 7Cl5 R\u00163 maple leaf [15]\nCu2ZnAsO 4(OH) 3 P\u00163 maple leaf [16]\nZn6Cu3(TeO 6)2(OH) 6Y P \u001631m kagome [17]\nMgCu 2TeO 6(H2O)6 P\u001631m honeycomb [18]\nCu3TeO 6(H2O)2 P21/n honeycomb/dimer [19]\nCu3TeO 6 Ia\u00163 hexagons [20]\nPb3(Cu 5Sb) 1=3(TeO 3)6Cl P4 132 hyperkagome [21]\nPbCuTe 2O6 P4132 hyperkagome [22]\nCuTeO 4 P21/n square lattice [23]\nSr2CuTeO 6 I4/m square lattice [24]\nSrCuTe 2O7 Pbcm orthorhombic [25]\nCu3BiTe 2O8Cl Pcmn kagome staircase [26]\nTABLE I: Table of various copper tellurium oxides. SG is the\nspace group, Lattice is the arrangement of copper ions, and\nRef is the associated reference to the literature.\nof this mineral, along with a host of others, are unknown\nat this time.\nLast year, in searching the mineralogical literature, I\nbecame aware of a review article on tellurium oxides [13].\nCopper (desired since it is an S=1/2 ion) has a tendency\nto be associated with tellurium, and indeed there are\nmany interesting copper tellurium oxides (and related hy-\ndroxides and hydroxyhalides) that can be found in this\narticle. Based on this, a search was done for interesting\nones where the copper ions sat on a frustrated lattice. A\npartial list of these can be found in Table I. The intent\nof this paper is to go through this table and point out in-\nteresting materials that might be worth synthesizing and\nstudying for their magnetic properties.\nBluebellite (Cu 6IO3(OH) 10Cl) and mojaveite\n(Cu 6TeO 4(OH) 9Cl) are recently discovered miner-\nals found in the Mojave Desert [14]. Though the \frst\nhas I5+as opposed to Te6+for the second, their crystal\nstructures are similar, and also similar to another mineral\ndiscovered there, fuettererite (Pb 3Cu6TeO 6(OH) 7Cl5)\n[15], as well as the mineral sabelliite (Cu 2ZnAsO 4(OH) 3)\n[16] discovered in Sardinia. In these four examples,\nthe copper ions sit on a so-called maple leaf lattice\n(1/7-depleted triangular lattice). This lattice (with a\ncoordination number of z=5) is intermediate from a\nfrustration viewpoint between a triangular lattice (z=6)\nand a kagome (1/4-depleted triangular lattice with\nz=4), and is thought to be (barely) on the ordered side\n[27]. There has been one copper mineral with such a\nlattice that has had its magnetic properties investigated,\nspangolite (Cu 6Al(SO 4)(OH) 12Cl(H 2O)3), whose sus-\nceptibility resembles that expected for a singlet ground\nstate [28], with a small upturn at low temperatures dueFormula unit X S L L/S\nCu6IO3(OH) 10Cl I 2.899 3.900 1.345\nCu6TeO 4(OH) 9Cl Te 2.999 3.572 1.191\nPb3Cu6TeO 6(OH) 7Cl5 Te 3.033 3.322 1.095\nCu2ZnAsO 4(OH) 3 Zn 3.028 3.166 1.046\nCu6Al(SO 4)(OH) 12Cl(H 2O)3Al 3.004 3.214 1.070\nTABLE II: Table of copper maple leaf lattices, 1/7-depleted\ntriangular lattices with a sub formula unit of Cu 6X, with X\nsitting in the middle of the hexagonal hole. S is the smallest\nCu-Cu near neighbor distance (in \u0017A), L the largest, with L/S\ntheir ratio.\nFIG. 1: Crystal structure of mojaveite [14]. View is along c,\nwith zranging from 0.2 to 0.5. The two di\u000berent crystallo-\ngraphic Cu sites are shown as blue and cyan, with Cl as green,\nTe as gold and oxygen as red.\nto about a 7.5% concentration of orphan spins (similar\nto what is observed in herbertsmithite). In all cases,\nthough, the maple leaf lattice is distorted (Fig. 1). In\nTable II, these distortions are tabulated, with sabelliite\nthe least distorted, and bluebellite the most. But for\nsabelliite, even though there is only one crystallographic\nCu site (as compared to two for the others), signi\fcant\nsite disorder exists in this material, with Zn on the Cu\nsites, and Sb on the As sites, making a synthetic variant\na desirable goal.\nWe next come to a more promising min-\neral, quetzalcoatlite (Zn 6Cu3(TeO 6)2(OH) 6Y, with\nY=Ag xPbyClx+2ya neutral unit) [17], found in the3\nFIG. 2: Crystal structure of quetzalcoatlite [17]. View is along\nc, with zranging from 0.25 to 0.75. Cu is blue, Cl green, Te\ngold and oxygen red.\nsame Blue Bell claims as bluebellite. Interestingly,\nthis was the \frst mineral whose crystal structure was\ndetermined at the Advanced Photon Source at Argonne\n(studied there because of the small size of the crystals).\nIt exhibits a perfect kagome net of copper ions (Fig. 2),\nwith Cl ions sitting at the center of the hexagonal holes.\nThe stacking of the kagome planes is AA (as opposed\nto ABC stacking in herbertsmithite), thus similar to\nkapellasite, which is a polymorph of herbertsmithite\n(both have the same space group). Like herbertsmithite,\nthese layers are separated by Zn ions (Fig. 3), but here,\nthe Zn ions have tetrahedral coordination (ZnO 2(OH) 2).\nBecause of this, the disorder seen in herbertsmithite\n(where Cu can sit on the Zn sites) should be absent in\nthis mineral. Moreover, the Cu-O-Zn-O-Cu pathway\nconnecting successive kagome layers is quite tortuous,\nimplying weak coupling between the layers. But there\nis still some disorder because of the variability of the\nY unit (ideally one would have AgCl dimers along\nthecaxis). Turning to the planar properties (Fig. 2),\nunlike herbertsmithite where one has Cu-O-Cu superex-\nchange pathways, here Te intervenes, leading to weaker\ncouplings of the form Cu-O-Te-O-Cu or Cu-O-O-Cu\n(super-superexchange). Because of this, the anticipated\nCurie-Weiss temperature should be signi\fcantly smaller\nthan herbertsmithite. Still, as one of the few known\nmaterials where Cu ions fall on a perfect kagome lattice,\nFIG. 3: Crystal structure of quetzalcoatlite [17]. Shown is a\nside view (vertical axis along c). Cu is blue, Cl green, Te gold,\noxygen red, Zn light gray, Pb dark gray, and OH groups pink\n(the positions of the hydrogens are not known).\nFIG. 4: Crystal structure of leisingite [18]. View is along c,\nwithzranging from 0.4 to 0.6. Cu is blue, Te gold and oxygen\nred.\none would hope that the available crystals could be\nstudied for their magnetic properties, and an attempt at\nsynthesis would be a desirable goal as well.\nThe next mineral in Table I is leisingite\n(MgCu 2TeO 6(H2O)6). Here, the copper ions form\na perfect honeycomb lattice (z=3), with Te ions sitting\nin the hexagonal holes (Fig. 4). The Cu ions are con-\nnected by a superexchange pathway, but the Cu-O-Cu\nbond angle is 93.5\u000e, which is near the crossover from\nF to AFM behavior. The layers are AA stacked, being4\nconneced by Mg(H 2O)6octahedra with Mg ions sitting\nbelow the Te ions. But again, disorder is present, with\nFe sitting on the Mg sites, and some Mg sitting on the\nCu sites. Again, a synthetic variant is highly desirable.\nJensenite (Cu 3TeO 6(H2O)2) is mentioned in passing.\nLike leisingite, it is composed of layers of Te and Cu\nforming a honeycomb lattice (but in this case, it is dis-\ntorted) separated by other layers which contain isolated\ncopper dimers. Mcalpenite has the same formula unit as\njensenite (except for the waters), but has a cubic space\ngroup instead. Although known in mineral form [29], it\nhas been synthesized as well by a variety of techniques\n[20, 30{32]. It is composed of a lattice of corner-sharing\ncopper hexagons whose normals point in di\u000berent direc-\ntions (a di\u000berent lattice of hexagons has been seen in frus-\ntrated spinels like ZnCr 2O4[33]). This material, which\nhas been studied by a number of groups, has been called\na `spin web' compound [31] with a N\u0013 eel temperature of\n61 K [30].\nIf one takes a kagome lattice and then stretches it along\nthecaxis, one gets a hyperkagome structure, with corner\nsharing triangles arrayed in a cubic space group. The\nmineral choloalite (Pb 3(Cu 5Sb) 1=3(TeO 3)6Cl) has this\nlattice [21], with the same P4 132 space group as the well\nknown spin liquid iridate, Na 4Ir3O8[34]. A simpler syn-\nthetic material, PbCuTe 2O6, has the same hyperkagome\nlattice and space group [22]. No magnetic ordering has\nbeen seen by NMR and \u0016SR down to 20 mK [35], though\nthermodynamic data indicate some type of transition oc-\ncuring at 0.87 K [22], perhaps related to impurities [35].\nBecause of the distoted nature of the lattice, modeling\nthis material is somewhat challenging, but exchange cou-\nplings have been proposed based on electronic structure\ncalculations [22]. A Sr variant is known that orders at\n5.5 K [36].\nThe \frst attempt to make Cu 3TeO 6was by high tem-\nperature hydrothermal synthesis [20]. CuTeO 4has been\nmade under similar conditions [23], though Cu 3TeO 6is\nthermodynamically more stable. The interest in CuTeO 4\nis that it exhibits a square lattice net for the copper ions\n(Fig. 5). But unlike a typical cuprate, the Cu-O-Cu bond\nangles are either 122.5\u000eof 126.1\u000e, more simlar to her-\nbertsmithite than other cuprates. This buckling of the\nCuO 2planes is due to an attempt to lattice match with\na TeO 2layer. Interestingly, the copper ions are nearly\noctahedrally coordinated, with short, medium and long\nCu-O bonds which at most are 14% di\u000berent in length for\nCu2 ions, and 18% di\u000berent for Cu1 ions. Despite these\ndi\u000berences from other cuprates, the electronic structure\nis remarkably similar, with a predicted magnetic ground\nstate which is a quasi-2D N\u0013 eel state [37]. It has been\nproposed that replacing Te6+by Sb5+could hole dope\nthis material, potentinally leading to a superconducting\nphase [37]. Experimenally, only the structure is known,\nand beyond the original synthesis paper [23], the only re-\nports in the literature is \fnding it as a secondary phase.\nFIG. 5: Crystal structure of CuTeO 4[23]. View is along the\nbaxis, with yranging from 0.4 to 0.6. Cu is blue and oxygen\nred, with short Cu-O bonds in black, medium bonds in green\nand long bonds in pink.\nFIG. 6: Crystal structure of Sr 2CuTeO 6[24]. View is along\nc, with zranging from 0.4 to 0.6. Cu is blue, Te gold and\noxygen red.\nIn the material Sr 2CuTeO 6[24], the Te ions move down\ninto the plane (Fig. 6). So, though the copper ions again\nform a square net, the exchange pathway is Cu-O-Te-O-\nCu (or Cu-O-O-Cu) as in quetzalcoatlite. This material\nis rather straightforward to synthesize, and a number of\nvariants are known (with Ba instead of Sr, or W instead\nof Te). The Sr variant orders at \u001880 K [24], but the Ba\nanalgoue does not appear to order (for the W version,\nthe Ba analogue orders at \u001830 K, but the Sr one does\nnot). These maetrials have been extensively studied, as\nwell as modeled by DFT calculations [38]. Crystals have\nbeen large enough to do both elastic [39] and inelastic [40]\nneutron scattering. One \fnds a simple N\u0013 eel lattice, but\nwith interactions signi\fcantly reduced from the layered\ncuprates due to the super-superexchange nature of the\nmagnetic coupling.\nSrCuTe 2O7has two types of Te sites (one 4+, one 6+)5\nFIG. 7: Crystal structure of SrCuTe 2O7[25]. View is along\nthebaxis, with yranging from 0 to 0.2. Cu is blue, Te gold\nand oxygen red, with Sr as green.\n[25]. The Cu ions sit on an orthorhombic lattice, either\nbeing connected by a Cu-O-Sr-O-Cu pathway (long direc-\ntion) or an orthogonal Cu-O-Te-O-Cu pathway (Fig. 7).\nThere are, though, four di\u000berent layers containing copper\n(Z=4). No evidence for magnetic order has been found\n[25], but typically, an upturn is seen in the susceptibility\nat low temperatures, indicating the presence of orphan\nspins. Similar behavior is seen in Pb and Ba variants.\nThe last material we mention is Cu 3BiTe 2O8Cl [26],\nsimilar to the mineral francisite, Cu 3BiSe 2O6Cl. The lat-\ntice formed is a distorted version of a so-called kagome\nstaricase, with the copper kagome layers strongly buck-\nled. Francisite itself has been synthesized and studied\nquite a bit [41{43], and orders at 27.4 K, but the order\nis complicated due to the low symmetry of the lattice.\nThere are many other copper tellurium oxides that are\nnot mentioned here, perhaps the best known being the\nperovskite CuTeO 3, the Se variant of which is a ferromag-\nnet [44]. Interestingly, it has been recently modeled by\nByung Il Min's group [45], he being a former student of\nArt's that I had the pleasure of working with when I was\na postdoc of Art's. Many materials not mentioned here\nexhibit chains instead, or more complex lattices. Vari-\nants are also known where Sb replaces Te. Certainly,\nMother Nature has been kind to us in providing a won-\nderful playground of unexplored materials. It is up to us\nto do the exploring.\nI know that if Art were still with us, he and his group\nwould likely take on the challenge of trying to better un-\nderstand this fascinating class of materials. Ironically,\nthe motivation behind the study of spin liquids is the\noriginal work of Phil Anderson as outlined at the begin-\nning of this article, and the two of them were well known\nfor not seeing eye to eye (Art being a student of Slater's,\nAnd so it happened that the world was asunder\ninto two camps, the Van-Vleckians and Slaterians.\nAt present, the Van-Vleckians are led by Admiral\nP. W. Anderson who is sometimes seen swimming,\nand the Sl aterians by General A. J. Freeman who\nstands on the firm  ground of LSD.LSDRVBFIG. 8: Cartoon presented by Mike Norman to Art Freeman\non his 80thbirthday. The quote is from Martin Peter [46].\nand Phil of van Vleck's). In that context, I cannot resist\nshowing a cartoon I presented to Art on his 80thbirthday\n(Fig. 8) motivated by a quote from Martin Peter that suc-\ncinctly illustrates the complex relation between Art and\nPhil [46]. I should end by saying that the physicist I\ncame to be was shaped by the melding of the in\ruences\nthat these two individuals had on me, both of which I\nam eternally grateful to.\nWork supported by the Materials Sciences and Engi-\nneering Division, Basic Energy Sciences, O\u000ece of Sci-\nence, US DOE. 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Peter, Physica B 172, viii (1991)."    },    {        "title": "1708.07974v1.Scaling_of_Memories_and_Crossover_in_Glassy_Magnets.pdf",        "content": "Scaling of Memories and Crossover in Glassy Magnets \n \nA. M. Samarakoon1,2+, M. Takahashi3,+, D. Zhang1, J. Yang1, N. Katayama1,4, R. Sinclair5, \nH. D. Zhou5, S. O. Diallo2, G. Ehlers2, D. A. Tennant2, S. Wakimoto6, K. Yamada7, G-W. \nChern1, T. J. Sato3*, S.-H. Lee1* \n \n1 Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA \n2 Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA \n3 Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, \nKatahira, Sendai 980-857, Japan \n4 Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan \n5 Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee \n37996, USA \n6 Materials Sciences Research Center, Japan Atomic Energy Agency, 2-4 Shirakata \nShirane, Tokai, Naka, Ibaraki 319-1195, Japan \n7 Institute of Materials Structure Science, High Energy Accelerator Research \nOrganization, Oho, Tsukuba 305-0801, Japan \n \n+ AMS and MT made equal contributions to this work. \n* To whom correspondence should be addressed. E-mail: shlee@virginia.edu  or \ntaku@tagen.tohoku.ac.jp \n \n \nAbstract: Glassiness is ubiquitous and diverse in characteristics in nature. \nUnderstanding their differences and classification remains a major scientific \nchallenge. Here, we show that scaling of magnetic memories with time can be used to \nclassify magnetic glassy materials into two distinct classes. The systems studied are high temperature superconductor-related materials, spin-orbit Mott insulators, \nfrustrated magnets, and dilute magnetic alloys. Our bulk magnetization \nmeasurements reveal that most densely populated magnets exhibit similar memory \nbehavior characterized by a relaxation exponent of −≈.(). This exponent \nis different from −≈/ of dilute magnetic alloys that was ascribed to their \nhierarchical and fractal energy landscape, and is also different from −= of \nthe conventional Debye relaxation expected for a spin solid, a state with long range \norder. Furthermore, our systematic study on dilute magnetic alloys with varying \nmagnetic concentration exhibits crossovers among the two glassy states and spin solid.  \n \nMagnetic glassy systems present a unique opportunity for searching possible universal \nphenomena associated with glassy behaviors. This is because glass phase exists in a wide \nrange of magnetic materials that are described by seemingly very different spin interactions. \nThe most well-known common features of the magnetic glassy behaviors are the lack of \nlong range magnetic order and the field-cooled (FC) and zero-field-cooled (ZFC) hysteresis \nfound in the bulk susceptibility1,2. The term spin glass was coined in 1970s to describe the \nlow temperature behaviors of dilute magnetic alloys that are made of nonmagnetic metals \nwith low concentrations of magnetic impurities1,3. The canonical glassy behaviors are \nmanifested in intriguing phenomena called aging, rejuvenation, and memory effects4. \nWhile aging simply refers to the time-span dependence of relaxation phenomena in the \nglassy state, rejuvenation describes the re-thermalization whenever the system is further \ncooled after waiting at some temperature. The states accessed while aging can be retrieved \nupon re-heating, which is called memory effect.  Several theories have been proposed to \nunderstand the physics of the spin glass.  \nVarious systems other than the dilute magnetic alloys also exhibit the aforementioned \ncharacteristic glassy behaviors at low temperatures, even when the magnetic moments are \ndensely populated. For example, glassy behaviors have been observed in the phase diagrams of high temperature superconducting materials, cuprates5,6 and iron-based \nsuperconductors7. Another example is the so-called spin-orbit Mott insulators, Li 2RhO38,9 \nand Na 2Ir1-xTixO310, which exhibit anisotropic Kitaev-type exchange interactions. Yet \nanother is a set of geometrically frustrated magnets, pyrochlores such as Y 2Mo2O711, \nspinels such as ZnFe 2O412, and the quasi-two-dimensional bi-pyramid compounds \nSrCr9pGa12-9pO19 (SCGO)13-17 and BaCr 9pGa12-9pO19 (BCGO)18. We emphasize that the \nmagnetic interactions of these systems seem to be quite different in nature. For instance, \nthe parent compound of high-T c superconductors La 2-xSrxCuO4 (LSCO) is a Mott insulator \nwith a conventional Neel spin order19. The entire magnetic excitation spectrum of La 2CuO4 \ncan be understood by an effective spin Hamiltonian with dominant nearest neighbor \nantiferromagnetic coupling constant ܬ= 104 meV20. The iron chalcogenide Fe 1+yTe \ndisplays a bi-collinear antiferromagnetic stripe order21,22. Magnetic interactions in the two \nspin-orbit Mott insulators, Li 2RhO3 and Na 2Ir1-xTixO3 are dominated by highly anisotropic \nKitaev exchange couplings23,24. Remarkably, despite their different nature of magnetic \ninteractions, all the systems show the same FC-ZFC hysteresis at low temperatures. A \nnatural question to ask is whether or not there is a unifying concept that can unite and also \nclassify these various glassy magnets.  \nHere, we address this issue by investigating memory effects of several of the \naforementioned exemplary systems using the bulk magnetization measurements. We \nperformed thermo-remanent magnetization (TRM)25-27 measurements on five different \ncompounds, which can be divided into three categories: (1) the high temperature \nsuperconducting materials, cuprates and Fe-chalcogenides, (2) Kitaev-model-related \nsystems Li 2RhO3 and Na 2Ir1-xTixO3, and (3) a semi-conducting pyrochlore Y 2Mo2O7. \nIntriguingly, despite their distinct microscopic Hamiltonians, all of them exhibit \nunconventional glassy behaviors in the TRM measurements, that are weak and broad \nshoulder-like memory effects as in the prototype spin jam compounds SCGO/BCGO, \nstarkly contrasting the strong and dip-like memory effects observed in the canonical spin glass such as CuMn2%. Interestingly, all the data can be well reproduced by a modified \nstretched exponential function of ൜1−exp൬−ቀ௧ೢ\nఛቁଵି\n൰ൠ. More importantly, all the \ndensely populated magnets except Y 2Mo2O7 yield an exponent of 1−݊≈ 0.6(1). This \nvalue is different from 1−݊≈ 1/3 of dilute magnetic alloys28-30 that was ascribed to \ntheir hierarchical and fractal energy landscape31-34, and is also different from 1−݊= 1 \nof the conventional Debye relaxation expected for a crystal. Based on these results, we \nargue that the glass magnets can be categorized into two distinct classes with different \nrelaxation behaviors characterized by the exponent: 1−݊≈ 1/3 for glassy magnets with \nhierarchical energy landscape and 1−݊≈ 0.6(1)  for the ones with non-hierarchical \nenergy landscape.  \nThe TRM measurement is the most effective way to probe the memory effects in detail \nas explained in Supplementary Information and as shown most recently in the comparative \nstudy35 of SCGO/BCGO and the canonical spin glass CuMn2%. While a dip-like memory \neffect with clear rejuvenation was observed, as expected, in CuMn2%, a shoulder-like \nmemory effect seen in SCGO/BCGO implies lack of rejuvenation. Figure 1 shows the \nTRM data obtained from five different compounds: (a) Fe 1.02Se0.15Te0.85, (b) \nLa1.96Sr0.04CuO4 (LSCO(x=0.04)), (c) Li 2RhO3, (d) Na 2Ir0.89Ti0.11O3, and (e) Y 2Mo2O7. \nThese TRM data were taken after waiting at the waiting temperature ܶ௪~ 0.7 ܶ for \nseveral different waiting times ranging from 1.5(5) min to maximally 100 hours. For all \nsystems aging and memory effect appears, i.e., the magnetization decreases near ܶ௪ when \nthe measurements were performed after waiting. The memory effect gets enhanced as the \nwaiting time, ݐ௪, increases. Surprisingly, Fe 1.02Se0.15Te0.85 and LSCO(x=0.04) whose \nparent compounds, FeTe and La 2CuO4, respectively, are long-range ordered state, i.e., spin \nsolid, exhibit very weak memory effects. The memory effects in both systems are \nnegligible for short waiting time ݐ௪≤ 6 min. For ݐ௪≳ 1 hr, both systems show a very \nweak and broad shoulder appearing around ܶ௪ (see Figs. 1a and 1b), regardless of how \nlarge ݐ௪ is. For Fe 1.02Se0.15Te0.85, the memory effect even seems to saturate for ݐ௪≳  30 hrs (Fig. 1a). Similar weak shoulder-like memory effects were also observed in the spin-\norbit Mott insulators, Li 2RhO3 and Na 2Ir0.89Ti0.11O3 (see Fig. 1c and 1d, respectively). Note \nthat, similarly to the two superconductivity-related systems, the two Kitaev-model-related \nsystems also exhibit negligible memory effects for short waiting time ݐ௪≤ 6 min.  \nThe weak shoulder-like memory effects have been recently observed in frustrated \nmagnets, SCGO and BCGO, that are in vicinity of spin liquid, and here we show that \nanother frustrated magnet, Y 2Mo2O7, also exhibits similar features (Fig. 1e). These data \nclearly show that the weak shoulder-like memory effect is universal in these densely \npopulated magnets, regardless of their magnetic interactions. It is in stark contrast to the \ncanonical spin glass such as CuMn2%, where the memory effects in the magnetization \ncurve were readily seen even for such short waiting times as ݐ௪= 1.5(5) min (see Fig. \n3a, and Fig. S2 in the Supplementary Information), and the effects become sharp and strong, \nappearing as a large dip at ܶ௪ for ݐ௪≥ 3 hrs35.  \nFigure 2a summarizes the ݐ௪ dependence of the memory effect for the densely \npopulated magnets along with the canonical spin glass CuMn2%. The relative change of \nthe magnetization ∆ܯ=൫ܯ−ܯ൯ܯൗ induced by the aging, in which ܯ and \nܯ are the magnetizations with and without waiting, respectively, is plotted. Overall, it \nis clear that the memory effect is much weaker in densely populated magnets than in the \ncanonical spin glass. Firstly, ∆ܯ of ݐ௪≥ 30 hrs for all the densely populated magnets \nexcept Y 2Mo2O7 is smaller than ∆ܯ of ݐ௪= 6 min for CuMn2%. Secondly, for ݐ௪≤\n6 min, most of them show negligible memory effects as shown in Fig. 2a and Fig. 1. \nThirdly, the memory effect of the spin jams except Y 2Mo2O7 seems to saturate for ݐ௪≥\n30 hrs, while for CuMn2% it seems to keep increasing with increasing ݐ௪ over the time \nperiod. It is interesting that the densely populated Y 2Mo2O7 exhibits both spin glass and \nspin jam behaviors. This is probably due to the fact that Y 2Mo2O7 is a semi-conductor \nevidenced by its resistivity of ߩ~10ିଶ Ω∙cm at 300 K, and has an unquenched orbital \ndegree of freedom36,37. As a result, Y 2Mo2O7 is not a typical frustrated antiferromagnet, which is manifested in the relatively small frustration index ݂= Θௐ/ܶ≅ 2.3 that is \ntwo orders of magnitude smaller than that of SCGO. \nIn search of possible underlying scaling behavior, we have fitted the ݐ௪ dependence \nof ∆ܯ to the following phenomenological function  \n    ∆ܯ(ݐ௪)=ܣ൜1−exp൬−ቀ௧ೢ\nఛቁଵି\n൰ൠ,  (1) \nwhich is modified from the stretched exponential function that was proposed to describe \nrelaxation phenomena in glassy systems28-34. The modification made here is to take into \naccount the experimental observation that ∆ܯ seems to saturate for long waiting times. \nHere ܣ= ∆ܯ(ݐ௪→ ∞) is a measure of degree of aging, ߬ is a microscopic time scale \nfor relaxation dynamics. A positive non-zero exponent ݊ would tell us how much the \nrelaxation deviates from the conventional Debye behavior ( ݊= 0). The exponent 1−݊ \ncan be related to critical exponents for the spin glass transition within the framework of a \nrandom cluster model32,33. For example, assuming that the growth of clusters involves no \nconserved mode, the droplet model predicts an exponent 1−݊= 1/238. The dashed lines \nin Figure 2a are the fits of the experimental data to Eq. (1) for all the materials. It is \nremarkable that the same phenomenological function, albeit with different parameters, \nreproduces all the data of both spin jams and spin glass over the wide range of the waiting \ntime. This indicates that a universal scaling may be in play in the aging or relaxation \nphenomena of all glassy magnets, as shown in Fig. 2c.  \n The difference between the spin glass and spin jam is clearly manifested in different \nparameters in Fig. 2d. For spin glass CuMn2%, the exponent 1−݊≈ 1/3 that deviates \nsignificantly from the conventional Debye behavior of 1−݊= 1. This is consistent with \nthe previous studies on several other dilute magnetic alloys such as CuMn1% and \nAgMn2.6% 28, NiMn23.5% 29, Au90Fe10 30. This deviation observed in the spin glasses was \nascribed to the underlying hierarchically constrained dynamics31-34. On the other hand, the \ndensely populated glassy magnets, SCGO, Fe 1.02Se0.15Te0.85, LSCO, and the two spin-orbit \nMott insulators, yield the exponent of 1−݊≈ 0.6(1), indicating a smaller deviation from the conventional Debye relaxation. This implies that their energy landscapes are not \nhierarchical as in the canonical spin glass. These are summarized in Fig. 2d in which the \nexponent 1−݊ is plotted as a function of the degree of aging, ܣ .We note that there is a \npositive correlation between the deviation from the Debye limit and the degree of aging.  \n To further support the aforementioned scenario, we have performed the TRM \nmeasurements on Cu 1-xMnx as a function of the Mn concentration, x. This series of \ncompounds provides an excellent platform also to investigate how the spin glass is \nconnected with the spin jam, and eventually magnetic ordered states. On one hand, Cu 1-\nxMnx is a canonical spin glass for small x. On the other hand, pure Mn exhibits a long-\nrange spin-density wave (SDW) order at low temperatures. The magnetic ground state of \nsamples with large x thus can be viewed as large domains of SDW order disrupted by non-\nmagnetic Cu atoms, similar to that observed in the densely populated magnets such as \nFe1.02Se0.15Te0.85 and La 1.96Sr0.04CuO4.  \nAs shown in Fig. 3a, 3b, 3c and 3d, for dilute alloys with small values of ݔ≲ 0.45, \nthe data exhibits prominent dip behaviors, i.e., the presence of rejuvenation. As x increases \nfurther, the dip behavior is gradually replaced with the shoulder behaviors, i.e., lack of \nrejuvenation, similar to spin jam (see Fig. 3e and 3f). The crossover seems to occur at x ~ \n0.45 that is close to the percolation threshold for a three-dimensional system39. Note the \nnon-monotonic behavior of the degree of aging ܣ= ∆ܯ(ݐ௪→ ∞) that maximizes at \nݔ∼ 0.15. The initial growth of A for small x is related to the increasing number of \nmagnetic impurities, giving rise to a stronger magnetic signal. For very large x where the \nsystem is in the spin jam regime, the degree of aging is expected to decrease as observed \nfor x = 0.75 and 0.85 shown in Fig. 3e and 3f  respectively. Thus, even though the exact \nvalue of x for the maximal A is determined by the balancing between the exact nature of \nthe magnetic interactions and the magnetic concentration, the maximum of A should occur \nmost likely somewhere close to the middle of x = 0 and the percolation threshold, which is \nqualitatively consistent with the observed value of ݔ∼ 0.15.   Surprisingly, regardless of x, ∆ܯ of Cu 1-xMnx follows the same stretched \nexponential relaxation function, as shown in Fig. 2b, but with varying values of the \nexponent, 1−݊ ,from 0.34(1) for x=0.02 to 0.66(9) for x=0.85 (see Fig. 2d). And thus, all \ntheir ∆ܯ can be collapsed into a same function, once the waiting time is properly scaled, \nand it is so even with those of the densely populated glassy systems, as shown in Fig. 2c. \nThe change in the exponent, 1−݊ ,as a function of x clearly shows that the glassy state \nof the dilute magnetic alloy (for small x) is replaced by a glassy state for large x similar to \nthe one observed in the densely populated magnets (see Fig. 2d) Interesting, the crossover \noccurs as the magnetic concentration go beyond the percolation threshold39. This clear \ncrossover phenomenon strongly indicates that there are two distinct glassy states: spin glass \nand spin jam. \nWhy do the densely populated systems exhibit the large exponent 1−݊≈ 0.6(1) \nsimilar to the quantum-fluctuation-induced spin jam SCGO, compared to the canonical \nspin glass state of dilute magnetic alloys? A clue comes from neutron scattering studies; \nthe magnetic structure factor, ܫ(ܳ) ,of all the densely populated magnets studied here \nexhibit prominent peaks that are centered at a non-zero momentum ( ܳ )corresponding to \nshort-range spin correlations, as those of the frustrated magnets SCGO17,40 and BCGO18. \nThis indicates that those systems have dominant antiferromagnetic interactions between \nlocalized spins and short-range spin correlations. For example, the cuprate41,42 and iron \nchalcogenide43,44 exhibit strong incommensurate peaks near the antiferromagnetic ordering \nwave vector of their parent compounds. As shown in Fig. 4b, the spin-orbit Mott insulator \nNa2Ir0.89Ti0.11O3 exhibit a prominent peak centered at ܳ= 0.87(2) Åିଵ. The common \ncharacteristics of the antiferromagnetic and short-range magnetic structure factor starkly \ncontrast with the nearly featureless magnetic structure factor of the spin glass CuMn2%, as \nshown in Fig. 4d. In the dilute magnetic alloys such as CuMn2%, magnetic impurities \ninteract among themselves through the Ruderman-Kittel-Kasuya-Yosida (RKKY) \ninteractions that are mediated by the itinerant electrons. The RKKY interactions are long-ranged, and oscillate from ferromagnetic to antiferromagnetic as a function of the distance. \nAs a result, the random distances among the magnetic moments lead to their random \ninteractions that even change the sign, resulting in the featureless magnetic structure factor.  \nThe featureless ܫ(ܳ) of CuMn2% is consistent with the real-space droplet model for \nspin glass38,45 in which low-energy excitations are dominated by connected spin clusters of \narbitrary length scales. The real-space clusters or droplets correspond to the meta-stable \nground states or local minima in the energy landscape. Their arbitrary length scales and \nrandom RKKY interactions yield a multitude of energy scales, resulting in the complex \nhierarchical fractal energy landscape35,46-48. As a consequence, the spin glass exhibits the \nobserved strong dip-like memory effect. In contrast to the droplet model for spin glass, the \nclusters in spin jams are more uniform in size, as evidenced by the prominent peak of ܫ(ܳ) .\nThis feature, combined with the short-range exchange spin Hamiltonian, leads to a \nnarrowly distributed energy scale, and the weak memory effect as observed in our \nsusceptibility measurements.  \nThe distinct nature of the two magnetic glass phases, spin glass and spin jam, also \nmanifests in their characteristically different low energy excitations. The thermodynamic \nbehavior of canonical spin glass at low temperatures is dominated by thermally active \nclusters or droplets, particularly those with a free energy less than or of the same order of \n݇ܶ where ݇ is the Boltzmann constant. The fact that there is a finite density of clusters \nwith limiting zero free energy naturally leads to the linear- T specific heat38,49, which is a \nsignature of canonical spin glass. On the other hand, the low-energy excitations in spin jam \nare the Halperin-Saslow (HS) spin waves with finite spin stiffness over large length scales \n(often larger than the typical cluster sizes)50-52. These gapless HS modes exhibit a linear \ndispersion relation and are the source of a ܶଶ dependence of the specific heat for a two-\ndimensional system. Indeed, such ܶଶ behavior has been observed in the glass phase of \nSCGO14, Li2RhO38,9, and doped Na 2IrO310.  \n The memory effect measurements provide crucial information about the nature of relaxation dynamics in different magnetic states, which allows us to classify the semi-\nclassical magnetic glassy materials as shown in Fig. 5. At the lower left corner of the \ntriangle lies the spin solid that is realized in densely populated semi-classical magnetic \nmaterials with small disorder and weak frustration that order long-range at low \ntemperatures with Debye relaxation. The typical energy landscape associated with spin \nsolid is a smooth vase with a global minimum. At the lower right corner of the triangle lies \nthe spin glass that is realized in dilute magnetic alloys with random magnetic interactions. \nIts typical energy landscape is dominated by hierarchical meta-stable states that correspond \nto spin clusters of arbitrary length scales in real space, exhibiting hierarchical rugged \nfunnels and fractal geometry, and the observed strong deviation from the conventional \nDebye relaxation. Finally, at the top corner is the new magnetic state dubbed spin jam that \nencompasses many densely populated compounds with short-range exchange magnetic \ninteractions, disorder and frustration. Disorder can be either extrinsic as in LSCO, FeTeSe \nand Na 2Ir1-xTixO3, or intrinsic due to quantum fluctuations as in SCGO and BCGO 15,16. \nOne salient feature of the spin jam, represented by a nonhierarchical energy landscape with \na wide and nearly flat but rough bottom, is the lack of widely distributed energy and time \nscales. This in turn leads to a significantly weaker memory effect and the relaxation \nexponent that is closer to the Debye exponent than that of the spin glass, as observed in our \nexperiments. Remarkably, the canonical spin glass Cu 1-xMnx with small x crosses over to \nthe spin jam state when the magnetic concentration x increases beyond the percolation \nthreshold.  \n Our classification of a wide range of semi-classical glassy magnets based on \nnonequilibrium relaxation dynamics to two distinct states has implication to other non-\nmagnetic structural glasses. Indeed, recent studies have found two distinct low frequency \nmodes in structural glass: one related to a hierarchical energy landscape and the other \nrelated to jamming53-55. The rather distinct aging and memory behaviors observed in the \nspin glass and jam might also shed light on the relationship between nonequilibrium dynamics and connectivity among elementary interacting agents in networks and socio-\neconomic systems56. \n \nAcknowledgments \nWork at University of Virginia by S.H.L. and A.S. was supported by US National Science \nFoundation (NSF) Grant DMR-1404994 and Oak Ridge National Laboratory, respectively. \nThe work at Tohoku University was partly supported by Grants-in-Aid for Scientific \nResearch (24224009, 23244068, and 15H05883) from MEXT of Japan. The work at \nUniversity of Tennessee was supported by the US National Science Foundation (NSF) \ngrant DMR-1350002.  \n \nAuthor contributions \nS-H. L. designed the research. J.Y., N. K., R. S., H. D. Z., S. W. and K. 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The jamming transition and the marginally jammed solid, \nAnnu Rev Condens Matter Phys  1(1) 347-369 (2010).  \n56. Albert, R. & Barabasi, A. -L. Statistical mechanics of complex networks, Rev Mod Phys  \n74:47-97 (2002). \n \n \n \nFigure captions  \nFig. 1. Memory Effect as a function of waiting time.  Bulk susceptibility, ߯=  ܯ/ܪ ,\nwhere ܯ and ܪ are magnetization and applied magnetic field strength, respectively, \nobtained from (a) Fe 1.02Se0.15Te0.85 (b) La 1.96Sr0.04CuO4, (c) Li 2RhO3 (d) Na 2Ir0.89Ti0.11O3 \nand (e) Y 2Mo2O7, with H = 3 Oe. Symbols and lines with different colors indicate the data taken with different waiting times, ݐ௪, ranging from zero to 100hrs, at \nܶ௪/ܶ ~ 0.7 where ܶ௪ and ܶ are the waiting and the freezing temperature, \nrespectively. For Fe 1.02Se0.15Te0.85, the Curie-Weiss Temperature ߠ௪ was estimated by \nfitting its high-T susceptibility data as shown in Fig. S1A in the Supplementary \nInformation. For La 1.96Sr0.04CuO4, the high-T susceptibility does not follow the simple \nCurie-Weiss law (see Fig. S1b in Supplementary Information). In order to show how \nstrong the magnetic interactions are in LSCO, we quote the coupling constants of the \nparent compound La 2CuO4 that were experimentally determined by inelastic neutron \nscattering (ref. 20); the antiferromagnetic nearest-neighbor ܬ≈ 104 ܸ݁݉ and the \nferromagnetic next-nearest-neighbor ܬᇱ≈ −18 ܸ݁݉ .ߠ௪ for Li 2RhO3, \nNa2Ir0.89Ti0.11O3 and Y 2Mo2O7 were taken from ref. 9, 24 and 36, respectively. \nFig. 2. Summarizing the memory effect.  From the data shown in (a) Fig. 1 and (b) Fig.3, \nthe aging effect was quantified for the eleven systems by plotting the relative change of \nthe magnetization ∆ܯ=൫ܯ−ܯ൯ܯൗ where ܯ is the magnetization \nwithout waiting, and it was plotted as a function of ݐ௪ in a log scale. The aging effects \nof a spin jam prototype, SrCr 9pGa12-9pO19 (SCGO(p=0.97)), and a spin glass prototype \nCuMn2% were taken from Ref. 35, except the ݐ௪= 1.5(5) ݊݅݉ data are new (see Fig. \nS2 in the Supplementary Information), and are also plotted here for comparison. Each \nset of ∆ܯ(ݐ௪) for each sample shown in panels (a) and (b) was fitted to the modified \nstretched exponential function, Eq. (1). After the fitting, in (c) −log(1−∆ܯܣ⁄) \nwas plotted as a function of (ݐ௪߬⁄)ଵି in a log-log scale. (d) The degree of aging, ܣ ,\nand the inverse exponent, 1/(1−݊) ,obtained for all the samples are plotted against \neach other. \n Fig. 3. Memory Effect of Cu -x at. % Mn samples as a function of waiting time.  Bulk \nsusceptibility, ߯=  ܯ/ܪ , where ܯ and ܪ are magnetization and applied \nmagnetic field strength, respectively, obtained from Cu 1-xMnx with (a) x=0.02, (b) \nx=0.15, (c) x=0.30, (d) x=0.45, (e) x=0.75 and (f) x=0.85, with H = 3 Oe. Symbols and \nlines with different colors indicate the data taken with different waiting times, ݐ௪, \nranging from zero to 100hrs, at ܶ௪/ܶ ~ 0.7 where ܶ௪ and ܶ are the waiting and the \nfreezing temperature, respectively.  \nFig. 4. Neutron scattering measurements.  (a) T-dependence, ܫ௦ (ܶ) ,and (b) Q-\ndependence, ܫ௦ (ܳ) ,of elastic magnetic neutron scattering intensity obtained from \nNa2Ir0.89Ti0.11O3. The measurements were done at the Cold Neutron Chopper \nSpectrometer (CNCS) at the Spallation Neutron Source (SNS). (c) T-dependence, \nܫ௦ (ܶ) ,and (d) Q-dependence, ܫ௦ (ܳ) ,of elastic magnetic neutron scattering \nintensity obtained from the magnetic alloy CuMn2%. The measurements were done at \nthe Backscattering Spectrometer (BASIS) at SNS. For both ܫ௦ (ܳ) in (b) and (d), the \nnon-magnetic background was determined from the data above the freezing temperature \nand subtracted from the base temperature data. The black solid line in (b) is the fit of the \nmagnetic peak centered at ܳ = 0.87 Åିଵ to a simple Gaussian, while the line in (d) is \na guide to eyes. The red horizontal bar at the center of the peak in (b) represents the instrument Q-resolution, ܳ݀≈ 0.06 Åିଵ, that was determined by fitting a nearby Bragg \npeak centered at 1.2 Åିଵ. \nFig. 5. Schematic phase diagram. Classification of semi-classical magnetic states into \nthree distinct phases, spin solid, spin jam, and spin glass, was made based on the memory \neffect. \n \n  Fig. 1 \n \n \n  \nFig. 2 \n \n  \nFig. 3 \n \n \n \n  \nFig. 4 \n \n \n  \nFig. 5 \n \n \n  \nSupplementary Information \nScaling of Memories in Glassy Magnets \nA. M. Samarakoon, M. Takahashi, D. Zhang, J. Yang, N. Katayama, R. Sinclair, H. D. \nZhou, S. O. Diallo, G. Ehler, D. A. Tennant, S. Wakimoto, K. Yamada, G-W. Chern, T. J. \nSato, S.-H. Lee \n \nThis PDF file includes: \nI. Experimental procedure of the Thermo-Remanent Magnetization (TRM) \nII. Neutron Scattering Methods \nFigures S1-S3 \n  I. Experimental procedure of the Thermo-Remanent Magnetization (TRM) \nThe Figure S1 shows the DC susceptibility data obtained from Fe 1.02Se0.15Te0.85 and \nLa1.96Sr0.04CuO4 that exhibit their glassy transitions at low temperatures. \nThe Thermo-Remanent Magnetization (TRM) data, shown in Figure 1 and Figure 3 of the \nmain text, and Figures S2 and S3 of the Supplementary Information, were collected using \nthe following procedure. First, each sample was cooled down from well above the freezing \ntemperature, ܶ, to base temperature with a single stop for a period of time, ݐ௪, at an \nintermediate temperature ܶ௪  below ܶ  under zero field. Once cooled down to base \ntemperature, the thermo-remanent magnetization is measured by applying a small field of \na few gauss upon heating at a constant rate. For all the measurements reported in this paper, \nwe used a Superconducting Quantum Inference Device (SQUID) magnetometer, Quantum \nDesign MPMS-XL5 equipped with the ultra-low-field option together with the \nenvironmental magnetic shield. Since it is necessary to have zero-field conditions at the \nsample during the cooling process including the waiting at an intermediate temperature, \nthe remanent magnetic field at the sample position was measured by the instrument’s \nfluxgate, and has been eliminated by introducing a compensating field using non-\nsuperconducting DC coil to get the remaining uncompensated magnetic field less than \n0.005 G at the sample position. After that, a small DC magnetic field of 3 G was generated \nby the DC non-superconducting coil and applied to the sample during the TRM \nmeasurements. \nFigure S3 shows that for the spin jam systems the memory effect with ݐ௪ =  10 hrs \nis maximal when the waiting temperature ܶ௪ ~ 0.7 ܶ and it becomes weaker for other \nvalues of ܶ௪ over a wide range of ܶ௪.  \n \nII. Neutron Scattering Methods \nFor the neutron scattering study of CuMn2%, the Backscattering Spectrometer (BASIS) at \nSpallation Neutron Source (SNS) was used. A 10 g polycrystalline sample of CuMn2% was sealed in a standard aluminum (Al) and was cooled in a standard liquid He-4 cryostat. \nDuring the measurements, the wavelength of scattered neutrons was fixed to be 6.2 Å by \nsilicon analyzer crystals, yielding an elastic energy resolution of ~4 ܸ݁ߤ .For the neutron \nscattering study of Na 2Ir0.89Ti0.11O3, the Cold Neutron Chopper Spectrometer (CNCS) at \nSNS. A 2.3 g polycrystalline sample of Na 2Ir0.89Ti0.11O3 was sealed in an Al annular can \nwith thickness of 1 mm to reduce the neutron absorption by Ir, and was placed inside a \nstandard liquid He-4 cryostat that can go down to 1.4 K. The wavelength of incident \nneutrons was fixed to be ߣ= 5 Å , yielding an elastic energy resolution of ~ 70 ܸ݁ߤ . \nElastic magnetic Neutron scattering intensity ܫ௦(ܳ,ܶ) =ܫ(߱,ܳ,ܶ)߱݀ఠబ\nିఠబ , where \n߱  is the instrument’s elastic energy resolution has been determined by subtracting  \nmeasurements done well above the freezing temperature  ܶ.  \n \nFigure Captions \n \nFigure S1:  High-Temperature bulk susceptibility (black) and inverse susceptibility (red) \nrespectively, obtained from (a) Fe 1.02Se0.15Te0.85 and (b) La 1.96Sr0.04CuO4. The data above \n120 K of Fe 1.02Se0.15Te0.85 has been fitted to the Curie-Weiss law (red dash line) and the \nestimated Curie-Weiss temperature is -265.5(8) K. The measurements have done under \nmagnetic fields of 0.01 T and 0.1 T respectively. \nFigure S2:  Bulk susceptibility, ߯= ܪܯ ⁄ , where ܯ  and ܪ  are magnetization and \napplied magnetic field strength, obtained from (a) CuMn2% and (b) SrCr 9pGa12-9pO19 \n(p=0.97) with ܪ= 3 ܱ݁ .The ݐ௪= 1.5(5) min data is new while all other data for ݐ௪≥\n6 min are taken from Ref. 35.  \nFigure S3.  Temperature Dependence of memory effect. ߯  and ൫ܯ−ܯ൯ܯൗ \nmeasured for (a) Fe 1.02Se0.15Te0.85 (b) La 1.96Sr0.04CuO4, (c) Li 2RhO3 (d) Na 2Ir0.89Ti0.11O3 and \n(e) Y2Mo2O7, with ݐ௪= 10 ℎݏݎ ,at various waiting temperatures. \n Fig. S1 \n \nFig. S2 \n \nFig. S3 \n \n \n \n \n \n \n"    },    {        "title": "1709.03323v1.Magnetic_order_and_phase_transition_in_the_iron_oxysulfide_La2O2Fe2OS2.pdf",        "content": "1 \n Magnetic order and phase transition in the iron oxy sulfide La 2O2Fe 2OS 2 \nReeya K. Oogarah 1, Emmanuelle Suard 2 and Emma E. McCabe 1* \n1 School of Physical Sciences, Ingram Building, Univ ersity of Kent, Canterbury, Kent, CT2 7NH, U.K. \n2 Intitut Laue Langevin, 71 avenue des Martyrs - CS 20156 - 38042 GRENOBLE CEDEX 9, France \n* corresponding author: e.e.mccabe@kent.ac.uk  \n \nSubmitted for publication: 16 th  June 2017; accepted for publication 11 th  September 2017 \n \nKeywords: iron oxychalcogenides, magnetic structure, neutron diffraction, magnetic phase transition  \n \nAbstract \nThe Mott-insulating iron oxychalcogenides exhibit c omplex magnetic behaviour and we report here a neut ron diffraction investigation \ninto the magnetic ordering in La 2O2Fe 2OS 2. This quaternary oxysulfide adopts the anti-Sr 2MnO 2Mn 2Sb 2-type structure (described by \nspace group I4/ mmm ) and orders antiferromagnetically below TN = 105 K. We consider both its long-range magnetic structure and its \nmagnetic microstructure, and the onset of magnetic order. It adopts the multi-k vector “2 k” magnetic structure ( k = (½ 0 ½) and k = (0 ½ \n½) and has similarities with related iron oxychalco genides, illustrating the robust nature of the “2 k” magnetic structure. \n1.  Introduction \nMixed-anion systems, containing more than one kind of anion, often adopt anion-ordered structures with  transition metal cations in \nunusual oxidation states and environments.1-2 They therefore have the potential to exhibit inter esting properties, including iron-based \nsuperconductivity in Ln FeAsO-related materials ( Ln  = lanthanide), 3-4 thermoelectricity in BiCuOSe 5 and wide-band gap semiconductivity \nin Ln CuO Q (Q = S, Se). 6-8 The anion ordering in these materials, resulting f rom the different sizes and characters of the oxide  and \npnictide or chalcogenide ions, often gives layered crystal structures with the oxide anions usually co ordinating the “harder” cations in \nfluorite-like layers, whilst the “softer” transitio n metal cations are coordinated by the more covalen t pnictide or chalcogenide anion. \nThe ZrCuSiAs (or “1111”) structure 9 (Figure 1a) adopted by Ln FeAsO, BiCuOSe and Ln CuO Q listed above is relatively simple but its \nbuilding blocks, e.g. the fluorite-like oxide layer s, or anti-fluorite-like transition metal layers, c an be incorporated into more complex \nmixed anion materials to give new functional materi als 10-18  and it’s interesting to study the influence of the  anion Q on the resulting \nproperties and electronic structures. 2 \n  \nThe iron oxychalcogenides La 2O2Fe 2OQ2 (Q = S, Se) were first reported in 1992 19  and adopt a body-centred tetragonal crystal struct ure \n(I4/ mmm ) consisting of fluorite-like [La 2O2]2+  layers and [Fe 2O] 2+  layers separated by Q2– anions (Figure 1b). (This structure can be \ndescribed as an anti-Sr 2MnO 2Mn 2Sb 2-type structure 20  with the cation and anion sites swapped.) 1 Substitutions on the transition metal \nsite ( M = Mn, Fe, Co) and in the fluorite-like layers can be carried out and a number of members of this fami ly have been reported. 2, 21-32  \nTheir magnetism has been the focus of several studi es because their magnetic order results from three competing exchange \ninteractions: nearest-neighbour (nn) J1 exchange interactions (either direct, or via 60 – 70° M – Se – M exchange interactions); next-\nnearest-neighbour (nnn) J2 ~100° M – Q – M exchange and nnn J2’  180° M – O – M exchange (Figure 1b). The relative strengths of th ese \nexchange interactions changes with transition metal , with nn J1 exchange dominating for the phases with the less e lectronegative Mn \ncation,21, 25, 33  whilst AFM nnn J2’  dominates for the dominates for the analogues with  the more electronegative Co cation;26, 28  the sign of \nthe nnn J2 M – Q – M exchange also changes with transition metal (ferro magnetic (FM) for M = Fe, Co 23, 29, 34-36  and antiferromagnetic \n(AFM) for M = Mn 33, 37 ). For iron analogues (with iron of intermediate el ectronegativity), an AFM two k-vector magnetic structure \n(referred to here as the “2 k” structure, described by the two perpendicular k-vectors k1 = (½ 0 ½) and k2 = (0 ½ ½) 30 ) first proposed by \nFuwa et al 27  has been reported for Sr 2F2Fe 2OS 229  and for La 2O2Fe 2OSe 2 (Figure 1c). 30  This non-collinear magnetic structure, with Fe 2+  \nmoments directed along the Fe – O bonds 27, 38  (consistent with 2D-Ising-like character) 29-30  allows AFM nnn J2’  Fe – O – Fe exchange and \nFM nnn J2 Fe – Q – Fe exchange, while nn Fe 2+  moments are perpendicular to one another. The elec tronic structure of the iron-based \n“Fe 2O” systems has attracted much interest, with theore tical studies highlighting their Mott-insulating na ture 34, 39-40  and possible \nrelationship to the parent phases of the iron-based  superconductors. Inelastic neutron scattering on t he oxyselenide La 2O2Fe 2OSe 2 \nsuggests that the exchange interactions are weaker than previously thought, which may suggest some add itional electron localisation \nwhich has not been fully explored theoretically.30  \nAlthough La 2O2Fe 2OS 2 was first reported in 1992, 19  its magnetic structure has not been investigated b y neutron powder diffraction \n(NPD) experiments. Liu et al recently investigated the role of the oxychalcogenide anion Q in the seri es Nd 2O2Fe 2OSe 2-xSx and suggested \nthat introducing S 2- onto the chalcogenide site induced an enhanced FM component. 41  This prompted us to investigate the magnetic \nordering and structure of closely-related La 2O2Fe 2OS 2; we report here a structural and magnetic study us ing variable temperature NPD \ndata that allows us to confirm its magnetic structu re, and by comparison with related “Fe 2O” materials, to highlight the robust nature of \nthis “2 k” magnetic structure, and its role in giving a magne tic microstructure common to all “Fe 2O” materials studied. \n \n \n3 \n 2. Experimental \n4.44 g of La 2O2Fe 2OS 2 were prepared by the solid state reaction of La 2O3 (Sigma-Aldrich, 99.99%), Fe (Alfa-Aesar, 99+%) and  Se (Alfa-\nAesar, 99+%). Stoichiometric quantities of these re agents were intimately ground together by hand usin g an agate pestle and mortar. \nThe resulting grey powder was pressed into several 5 mm diameter pellets using a uniaxial press. These  pellets were slowly heated in an \nevacuated, sealed silica tube to 400°C and held at this temperature for 12 hours, and then heated to 6 00°C and then 850°C held at this \nreaction temperature for 12 hours. The sample was t hen cooled to room temperature in the furnace. Prel iminary structural \ncharacterisation was carried out using powder X-ray  diffraction data collected on a Panalytical Empyre an diffractometer from 10° - 90° \n2θ. The diffractometer was fitted with a germanium monochromator, an X’Celerator detector and an Oxfor d Cryosystems Phenix \ncryostat.  \nNeutron powder diffraction data were collected on t he high flux diffractometer D20 at the ILL with neu tron wavelength 2.41 °A. The \npowder was placed in an 10 mm diameter cylindrical vanadium can (to a height of 2.5 cm) and data were collected from 5-130° 2θ. A 40 \nminute scan was collected at 1.8 K and 10 minute sc ans were collected on warming at 2 K min -1 to 168 K. Rietveld refinements were \nperformed using TopasAcademic software.42-43  The diffractometer zero point and neutron waveleng th were refined using data collected \nat 160 K for which lattice parameters were known fr om XRPD analysis and were then fixed for subsequent  refinements. A background \nwas refined for each refinement, as well as unit ce ll parameters, atomic positions and a pseudo-Voight  peak shape. TopasAcademic \npermits nuclear-only and magnetic-only phases to be  included in refinements and the unit cell paramete rs of the magnetic phase were \nconstrained to be integer multiples of those of the  nuclear phase. The scale factor scales with the sq uare of the unit cell volume; the \nscale factor for the nuclear phase was refined and that for the “2 k” magnetic phase (with cell volume 8 times that of t he nuclear phase) \nwas constrained to be 0.015625 × that of the nuclea r phase. The web-based ISODISTORT software was used  to obtain a magnetic \nsymmetry mode description of the magnetic structure ; 44  magnetic symmetry modes were then refined correspo nding to either the \ncollinear magnetic structure, or the “2 k” magnetic structure (see Section 3.2 below). Magne tic susceptibility data were measured using \na Magnetic Properties Measurement System (MPMS, Qua ntum Design). Field-cooled and zero-field-cooled da ta were collected on \nwarming from 2 K to 300 K at 5 K min -1 in an applied magnetic field of 1000 Oe. \n \n3.  Results \n3.1 Structural characterisation \nPreliminary structural characterisation by Rietveld  refinement using room temperature XRPD data indica ted that our La 2O2Fe 2OS 2 \nsample was of high purity, with room temperature la ttice parameters a = 4.04431(5) Å and c = 17.8786(3) Å (see supplementary \nmaterial), in good agreement with the crystal struc ture reported by Mayer et al. 19  NPD data were collected on a 4.44 g sample of \nLa 2O2Fe 2OS 2 on warming from ~1.8 K to ~168 K. NPD data collect ed at 168 K (above TN) were consistent with the crystal structure \nreported by Mayer et al 19  and allowing Fe, S and O site occupancies to refin e (whilst that of the La site was fixed at unity) g ave site \noccupancies close to unity (Fe: 0.985(3); S: 0.985( 8); O(1): 1.001(4) and O(2): 0.995(5)) (see supplem entary material). The slight iron \ndeficiency may indicate a very small amount of oxid ation of Fe 2+  ions, but this stoichiometry is very close to idea l and the room \ntemperature lattice parameters and property measure ments (e.g. magnetic susceptibility measurements, s ee supplementary material) \nare consistent with those reported in the literatur e. Sequential Rietveld refinements using data colle cted on warming revealed a smooth \nincrease in unit cell volume with temperature with expansion of ~0.08% along [100] and ~0.16% along [0 01] (Figure 2 and \nsupplementary material). \n 4 \n  \nFigure 2  Unit cell parameters as a function of tem perature for La 2O2Fe 2OS 2 determined from sequential Rietveld refinements us ing NPD data. \n \n3.2 Magnetic ordering \nA broad asymmetric Warren-like peak 45  centred at ~39° 2θ was observed in data collected immediately above TN (from 106.5 K) which \ndecreased in intensity rapidly and could not be det ected above 116 K. This peak, centred around the po sition of the most intense 4.032 4.034 4.036 4.038 4.040 \n0 30 60 90 120 150 180 a(Å) \nTemperature (K) \n17.820 17.825 17.830 17.835 17.840 17.845 17.850 17.855 \n0 30 60 90 120 150 180 c(Å) \nTemperature (K) \n290.0 290.2 290.4 290.6 290.8 291.0 291.2 291.4 \n0 30 60 90 120 150 180 volume (Å 3)\nTemperature (K) 4.416 4.417 4.418 4.419 4.420 4.421 \n0 30 60 90 120 150 180 c/a\nTemperature (K) 5 \n magnetic Bragg reflection (which appears below TN) is characteristic of two-dimensional short-ranged  magnetic order, 45  similar to the \nmagnetic Warren peak observed for other Ln 2O2Fe 2OSe 2 analogues ( Ln  = La, Ce, Pr). 30-32  Fitting with a Warren function gave a correlation \nlength of ~100 Å (~25 times in the in-plane lattice  parameter) at 106.5 K (immediately above TN) which decreases very rapidly (Figure 3 \nand supplementary material). \n \n6 \n Figure 3 [colour online] (a) NPD data collected for  La 2O2Fe 2OS 2 close to TN showing (above) 10 – 70° 2θ data and (below, enlar ged) the narrow 2θ \nregion in which the Warren peak is observed and (b)  shows the fit to this Warren peak in 106.5 K data to a model suggesting an in-plane \nmagnetic correlation length of ~100 Å (observed dat a points in black; calculated Warren profile + back ground shown by red line; peak \npositions for the nuclear phase are shown by black tick marks); (c) shows the evolution of Fe2+ magnet ic moment on cooling with data \npoints in blue and solid red showing as a guide to the eye showing critical behaviour for a 2D-Ising s ystem (with critical exponent β = \n0.132(1), TN = 105.48(3) K and M0 = 3.508(7) µ B); (d) shows nuclear and magnetic structure at 1.8 K for La 2O2Fe 2OS 2 with La, Fe, O and S \nions shown in green, blue, red and yellow, respecti vely, and Fe 2+  moments shown by blue arrows.  \n \nAdditional Bragg reflections are observed in data c ollected for T ≤ 104.5 K which increase smoothly in intensity wit h decreasing \ntemperature. The web-based ISODISTORT software 44  was used, in conjunction with TopasAcademic refine ment software 42-43  to explore \npossible magnetic structures. \nIndexing these additional reflections suggested a m agnetic unit cell described by k vector k = (0 ½ ½) as proposed for La 2O2Fe 2OSe 2 and \ncollinear magnetic order with a high degree of frus tration.24  However, similar magnetic Bragg scattering has sin ce been observed for \nrelated iron oxychalcogenides and found to be fitte d equally well by a non-collinear two- k-vector magnetic structure (the “2 k” magnetic \nstructure).29-30   \nThe magnetic Bragg reflections observed for La 2O2Fe 2OS 2 were found to be consistent with the “2 k” magnetic structure proposed by \nFuwa et al 27  for Nd 2O2Fe 2OSe 2 and by Zhao et al for Sr 2F2Fe 2OS 2.29  (Equally good fits could be obtained using a colli near model (as \ndescribed by Free et al 24 ), but given the narrow temperature-range for which  the Warren-peak is observed (suggesting low levels  of \nfrustration) and the observed 2D-Ising-like onset o f Fe 2+  magnetic moment on cooling (Figure 3c), the “2 k” magnetic ordering seems \nmore likely for this oxysulfide, consistent with cl osely-related materials. 29-32  ) \nInitially, our Rietveld model (including the nuclea r phase and a magnetic-only phase describing “2 k” magnetic order on the Fe 2+  \nsublattice) gave a poor fit to the data (Figure 4a)  due to significant anisotropic broadening of the m agnetic Bragg reflections. This \nbroadening was fitted with a model describing antip hase boundaries (e.g. stacking faults) 46  perpendicular to the c axis in the magnetic \nstructure which gave a good fit to the data (Figure s 4b, c). The magnetic correlation length along c, ξc is ~51(2) Å at 2 K and changes little \nwith temperature ( ξc = 55(6) at 103 K). We note at two additional weak reflections (at 24° 2θ and at 52° 2θ) are observed in our NPD \ndata at all temperatures (Figures 3, 4 and suppleme ntary materials) and are thought to be due to a sma ll amount of an unidentified \nimpurity phase, and are not thought to relate to th e magnetic behaviour of La 2O2Fe 2OS 2, the focus of this study. 7 \n  \nFigure 4 Rietveld refinement profiles with “2 k” magnetic model at 1.8 K showing (a) wide 2θ range  with reflections for nuclear and magnetic phases \nshown by blue and black ticks, respectively, (b) na rrow 2θ range highlighting fit to magnetic reflecti ons with the same peak shape for both \nnuclear and magnetic phases and (c) showing improve d fit to magnetic reflections by the “2 k” magnetic model with stacking faults. Magnetic \nscattering is highlighted by solid black line in pa nels (b) and (c), whilst the observed, calculated a nd difference lines are shown in blue, red \nand grey, respectively. \n \nSequential NPD Rietveld refinements indicate a smoo th increase in Fe 2+  moment on cooling (Figure 3c and supplementary mat erial). \nFitting this magnetic order parameter by models for  critical behaviour suggests similar 2D-Ising-like character around TN before long-\nranged three-dimensional order develops, as observe d for other “Fe 2O” materials 30-32  and consistent with “2 k” magnetic ordering with \nFe 2+  moments directed along the Fe – O bonds. The low t emperature (1.8 K data) were fitted well by a model  containing the I4/ mmm  \nnuclear phase and a magnetic-only phase describing “2k” magnetic order on the Fe 2+  sublattice (with stacking faults in the magnetic \nstructure as described above) and an ordered moment  of 3.30(4) µ B on Fe 2+  sites. This is in good agreement with similar mode ls for \nother “Fe 2O” materials including Sr 2F2Fe 2OS 2 (Fe 2+  moment = 3.3(1) µ B)29  and La 2O2Fe 2OSe 2 (Fe 2+  moment = 3.50(5) µ B). 30  Full details \nfrom the 1.8 K refinement and selected bond lengths  are given in Tables 1 and 2 and the final structur e (nuclear and magnetic) is \nillustrated in Figure 3d. \n \nTable 1 Details from Rietveld refinement using 1.8 K NPD data for La 2O2Fe 2OS 2. The refinement was carried out with the nuclear s tructure \ndescribed by space group I4/mmm  with a = 4.0353(1) Å and c = 17.8237(9) Å, and “2 k” magnetic ordering on the Fe 2+  sublattice (as \ndescribed above) with magnetic correlation length ξc = 51(2) Å; R wp  = 5.680% and R p = 3.961%, RB = 0.89% (nuclear phase) and RB  = 1.80% \n(magnetic phase).  \nAtom Site x y z U iso  × 100 (Å 2) Fe 2+  moment (µ B) \nLa 4 e 0.5 0.5 0.1803(1) 0.9(2)  \nFe 4 c 0.5 0 0 0.9(2) 3.30(4) \nS 4 e 0 0 0.0934(5) 0.9(2)  \nO(1) 4 d 0.5 0 0.25 0.9(2)  \nO(2) 2 b 0.5 0.5 0 0.9(2)  \n 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 300,000 \n250,000 \n200,000 \n150,000 \n100,000 \n50,000 \n0\n-50,000 \n-100,000 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 300,000 \n250,000 \n200,000 \n150,000 \n100,000 \n50,000 \n0\n-50,000 \n-100,000 70 65 60 55 50 45 40 35 30 25 20 15 10 800,000 \n600,000 \n400,000 \n200,000 \n0\n-200,000 \n2θ (°) Intensity  (a)  \n(b)  Intensity  \n2θ (°)  \n(c)  Intensity  \n2θ (°)  8 \n Table 2 Selected bond lengths and angles from Rietv eld refinement using 1.8 K NPD data for La 2O2Fe 2OS 2.  \nBond lengths (Å) \nLa – O(1) 4 × 2.370(1) \nLa – S  4 × 3.214(2) \nFe – Fe  4 × 2.85341(9) \nFe – O(2) 2 × 2.01766(6) \nFe – S  4 × 2.616(6) \nBond angles (°)  \nFe – O – Fe  180 \nFe – S – Fe(1) 66.1(1) \nFe – S – Fe(2) 100.9(3) \n \n \n4.  Discussion \nThe crystal structure of La 2O2Fe 2OS 2 is very similar to that of the oxide-fluoride-sulf ide Sr 2F2Fe 2OS 2.23  The slightly larger Sr 2+  cation (eight-\ncoordinate ionic radii are 1.26 Å for Sr 2+  and 1.160 Å for La 3+ )47  in the fluorite-like [ A2X2]2+  layers of the latter give a slightly larger \nseparation of the “Fe 2O” layers, but this has very little effect on the A FM ordering temperature ( TN = 106(2) K for Sr 2F2Fe 2OS 2;23, 29  105(1) \nK for La 2O2Fe 2OS 2 here). \nWe note that the change in c lattice parameter with temperature for this oxysul fide La 2O2Fe 2OS 2 does not show as marked a \ndiscontinuity at TN as the analogous oxyselenide La 2O2Fe 2OSe 2 (see supplementary material for comparison). 24  The more rapid decrease \nin c lattice parameter below TN for oxyselenides Ln 2O2Fe 2OSe 2 was ascribed to magnetostrictive effects 24, 31-32  and it is noticeably less \npronounced in the oxysulfide La 2O2Fe 2OS 2 with its shorter c lattice parameter. This is consistent with the tre nd observed across \nLn 2O2M2OQ2 systems ( Ln  = lanthanide ion; M = Mn, Fe, Co; Q = S, Se) so far, in which the more rapid decrease in c below TN occurs for \nsystems with larger separation between subsequent m agnetic “ M2O” layers. 25, 31-32  \nThe observation of a Warren peak immediately above TN suggests some two-dimensional short-ranged magneti c order before three-\ndimensional magnetic order develops below TN. This is consistent with the broad maximum in magn etic susceptibility measurements \n(see Mayer et al 19  and supplementary material). The observation of th e Warren peak over such a narrow temperature range above TN \n(~ 10 K here for La 2O2Fe 2OS 2; ~14 K for La 2O2Fe 2OSe 2;30  ~140 K for La 2O2Mn 2OSe 221 ) reflects the low degree of magnetic frustration \nexpected for the “2 k” magnetic structure, 30  with both FM J2 (Fe – Se – Fe) and AFM J2’  (Fe – O – Fe) nnn interactions satisfied. We note \nthat the occurrence of this multi- k non-collinear magnetic structure rather than a sin gle-k structure implies the presence of higher-order \nterms which couple the two orthogonal k-vectors and maintain the C4 symmetry of the nuclear crystal structure, 30  whilst the collinear \nstructure 24  breaks this tetragonal symmetry.  \nNPD data for several ”Fe 2O” materials suggest stacking faults in the magneti c structure and it’s interesting that this “magneti c \nmicrostructure” observation is common to all “Fe 2O” materials studied, but hasn’t been reported for manganese or cobalt analogues \nwhich have sharp magnetic Bragg reflections. The st acking faults in the Fe 2+  magnetic structure are likely to result from the b ody-\ncentred crystal structure, which means that consecu tive “Fe 2O” layers are offset from one another by (½ ½ 0) an d therefore the O(2) \nsite alternates between (0 0 z) and (½ ½ z) in successive layers. Although the Fe 2+  sites are coincident in consecutive layers, Fe 2+  \nmoments (directed along Fe – O bonds) in consecutiv e layers are perpendicular to one another. Flipping  the direction of moments in \none layer (by 180°) relative to the layer below is therefore likely to be a relatively low energy defe ct. This would explain the presence of \nstacking faults in “Fe 2O” materials and their absence in manganese analogu es with their collinear magnetic structure. We note  that the \nmagnetic correlation length along c, ξc, is slightly longer in this oxysulfide (~51(2) Å a t 2 K with only 8.9119(9) Å between magnetic \n“Fe 2O” layers) than in the analogous oxyselenide La 2O2Fe 2OSe 2 (45(3) Å at 2 K with 9.258(5) Å between layers), p resumably due to the \nslightly closer “Fe 2O” layers in the oxysulfide. \nThe magnetic behaviour of La 2O2Fe 2OS 2 is extremely similar to that of Sr 2F2Fe 2OS 229  as well as to that of the analogous oxyselenide \nLa 2O2Fe 2OSe 2, although the exchange interactions in the oxysulf ides are slightly stronger resulting in a slightly higher TN than for the \noxyselenides (presumably due to the shorter bond di stances and better overlap in the oxysulfides). Thi s suggests that the relative \nstrengths of these exchange interactions, and the F e 2+  anisotropy are unchanged. Liu et al suggest that a s selenide ions are replaced by \nsmaller sulfide ions in Nd 2O2Fe 2OSe 2-xSx, a FM component is induced as the relative strengt hs of the exchange interactions are modified \nby chemical pressure. 41  Comparing room temperature crystal structures for various “Fe 2O” materials indicates that the Q – Fe – Q angles 9 \n are similar for both Nd 2O2Fe 2OSe 1.6 S0.4  and for Nd 2O2Fe 2OSe 2 (α 1 is 95.88(2)° for Nd 2O2Fe 2OSe 1.6 S0.4 41  and 95.72(2)° for Nd 2O2Fe 2OSe 231 ). \nWhile Fe – O and Fe – Fe distances are comparably s horter for Nd 2O2Fe 2OSe 1.6 S0.4  than for Nd 2O2Fe 2OSe 2,31  La 2O2Fe 2OS 2 (here) and \nSr 2F2Fe 2OS 2,23  it’s interesting that the room-temperature Fe – Q bond length for Nd 2O2Fe 2OSe 1.6 S0.4  (2.7026(3) Å)41  is intermediate \nbetween those reported for Nd 2O2Fe 2OSe 2 (2.7154(4) Å) 31  and for Sr 2F2Fe 2OS 2 (2.633 Å). 23  The similar magnetic behaviour observed for \nLa 2O2Fe 2OS 2 (here), Sr 2F2Fe 2OS 229  and Ln 2O2Fe 2OSe 230-31  (including critical behaviour, long-range magnetic  ordering, degree of frustration \nand magnetic microstructure) illustrates the robust  nature of the “2 k” magnetic order in “Fe 2O” materials, little changed with anion Q or \nsmall variations in crystal structure. However, it raises the question as to whether the ~300 K FM com ponent observed in magnetic \nsusceptibility data for Nd 2O2Fe 2OSe 2-xSx41  is intrinsic to the material or might arise from t race amounts of a ferromagnetic impurity. \nInelastic neutron scattering studies to determine t he strengths of the exchange interactions in these oxysulfides would be interesting \nand allow comparison with the Mott-insulating oxyse lenides. \n \n5.  Conclusions \nThe magnetic behaviour in La 2O2Fe 2OS 2 reported here is extremely similar to that of othe r “Fe 2O” materials which illustrates the robust \nnature of the “2 k” magnetic structure. This magnetic ordering is dom inated by nnn J2 and J2’  interactions and the Fe 2+  magnetic \nanisotropy and is relatively independent of variati ons in unit cell size, the chalcogenide Q2- and structural distortions. 32  The onset of \nmagnetic order in this oxysulfide is very sudden wi th two-dimensional short-ranged order developing in  a narrow temperature range \nimmediately above TN, in contrast to the more three-dimensional like ch aracter of the manganese analogue. 21  We’ve shown that the \nstacking faults in the magnetic structure of these “Fe 2O” materials are a consequence of the 2D-Ising natu re of the two- k vector \nmagnetic order.  \n \nAcknowledgements \nWe are grateful to Mr Ben Coles for assistance coll ecting NPD data and to Dr Andrew Wills for helpful discussions. \n \nReferences \n1. Clarke, S. J.; Adamson, P.; Herkelrath, S. J. C. ; Rutt, O. J.; Parker, D. R.; Pitcher, M. J.; Smura , C. F., Inorg. Chem. 2008,  47 , 8473-8486. \n2. Stock, C.; McCabe, E. E., J. 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Topas Academic: General profile and structure analy sis software for powder diffraction data , Bruker AXS: Karlsruhe, Germany, \n2012. \n44. Campbell, B. J.; Stokes, H. T.; Tanner, D. E.; Hatch, D. M., J. Appl. Cryst. 2006,  39 , 607-614. \n45. Warren, B. E., Phys. Rev.  1941,  59  (9), 693-698. \n46. Her, J.-H.; Stephens, P. W.; Gao, Y.; Soloveich ik, G. L.; Rijssenbeek, J.; Andrus, M.; Zhao, J.-C. , Acta Cryst 2007,  B63 , 561-568. \n47. Shannon, R. D., Acta Cryst. 1976,  A32 , 751.  \n \nGraphical Abstract \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nJ1 \nJ2 \nJ2’  \nO \nS \nFe  \na \nb \n-1 01234\n0 30 60 90 120 Fe moment (μB)\nTemperature (K) 12000 16000 20000 24000 \n34 38 42 46 50 54 58 Intensity (arbitrary units) \n2θ (°)\n104.6 106.5 K 108.5 K 11 \n Supplementary material \nSM1 Rietveld refinement profiles and details using room temperature XRPD data. \n \nFigure SM1 Rietveld refinement profiles for La 2O2Fe 2OS 2 using room temperature XRPD data showing observed (black), calculated (red) and \ndifference (grey) profiles with tick marks showing the positions of the main phase (upper blue; 98.9%)  and LaFeO 3 impurity (lower \nblack, 1.1%). R wp  = 8.81% and R p = 6.98%. \n \nSM2 168 K NPD refinement \n \nFigure SM2 Rietveld refinement profiles for La 2O2Fe 2OS 2 using 168 K NPD data showing observed (blue), calc ulated (red) and difference (grey) \nprofiles with tick marks showing the peak positions  for the tetragonal nuclear structure. \n \nTable1 SM2 Details from Rietveld refinement using 1 68 K NPD data for La 2O2Fe 2OS 2. The refinement was carried out with the nuclear s tructure \ndescribed by space group I4/mmm  with a = 4.0387(1) Å and c = 17.8522(9) Å; R wp  = 5.367% and R p = 3.534%, RB = 0.99% (nuclear \nphase).  \nAtom Site x y z Uiso  × 100 (Å 2) \nLa 4 e 0.5 0.5 0.1805(1) 1.2(2) \nFe 4 c 0.5 0 0 1.2(2) \nS 4 e 0 0 0.0932(5) 1.2(2) \nO(1) 4 d 0.5 0 0.25 1.2(2) \nO(2) 2 b 0.5 0.5 0 1.2(2) \n \n \n \n \n \n \n \n \n \n \n \n \n 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 2,000 \n1,500 \n1,000 \n500 \n0\n2θ (°)  Intensity  \n70 65 60 55 50 45 40 35 30 25 20 15 10 200,000 \n150,000 \n100,000 \n50,000 \n0\n2θ (°)  Intensity  12 \n SM3 Magnetic susceptibility measurements for La 2O2Fe 2OS 2 \n \n \nSM4 NPD data collected on warming from 1.8 K to 168  K for La 2O2Fe 2OS 2; ticks for the nuclear and magnetic phases are sho wn in \nblack and blue, respectively. \n \n \n \n 0.0E+00 2.0E-08 4.0E-08 6.0E-08 8.0E-08 1.0E-07 1.2E-07 \n0 50 100 150 200 250 300 χm(m 3mol -1)\nTemperature (K) ZFC FC \n1.8 K  \n40.8 K  \n83.2 K  \n124.2 K  \n167.8 K  \n10  25  \n 40  55  70  \n2θ(°)  13 \n  \nSM5 Results from sequential Rietveld refinements us ing NPD data collected for La 2O2Fe 2OS 2 on warming. \n \nFigure SM5 Selected bond lengths and angles for La 2O2Fe 2OS 2 as a function of temperature, determined from sequ ential Rietveld refinements \nusing NPD data. We notes that the relatively large esds on some bond lengths reflect the very small ch ange in these values in the \ntemperature range studied, and the low Q-range of o ur data (making it relatively insensitive to subtle  structural changes).  \n \n \n \n \n \n \n \n \n \n \n \n \n \nFe – S – Fe(2) \nFe – S – Fe(1) 2.8530 2.8535 2.8540 2.8545 2.8550 2.8555 2.8560 2.8565 \n0 30 60 90 120 150 180 Fe - Fe (Å) \nTemperature (K) 2.605 2.610 2.615 2.620 2.625 2.630 \n0 30 60 90 120 150 180 Fe - S (Å) \nTemperature (K) \n2.0170 2.0175 2.0180 2.0185 2.0190 2.0195 2.0200 \n0 30 60 90 120 150 180 Fe - O (Å) \nTemperature (K) 65.8 65.9 66.0 66.1 66.2 66.3 66.4 66.5 \n0 30 60 90 120 150 180 Fe -S -Fe(1) ( °)\nTemperature (K) \n100.4 100.6 100.8 101.0 101.2 101.4 101.6 \n0 30 60 90 120 150 180 Fe -S -Fe(2) ( °)\nTemperature (K) 14 \n SM6 Comparison of unit cell parameters on cooling f or La 2O2Fe 2OS 2 and La 2O2Fe 2OSe 2. \n \nFigure SM6 Comparison of normalised unit cell param eters for La 2O2Fe 2OS 2 (Q = S, results reported here) and La 2O2Fe 2OSe 2 (Q = Se, from \nreference 30). Unit cell parameters on cooling were  normalised to their values at 168 K (above TN) for both materials to aid \ncomparison.  \n \nSM7 Comparison of evolution of Fe 2+  moment on cooling for La 2O2Fe 2OQ2 (Q = S, Se)  \n \nFigure SM7 Comparison of evolution of Fe 2+  moment on cooling for La 2O2Fe 2OS 2 (Q = S; blue; results reported here) and La 2O2Fe 2OSe 2 (Q = Se; \nred; reference 30). Data points are shown with erro r bars, and solid lines are fits to critical behavi our with parameters: \n Q = S: M0 = 3.508(7) µ B, TN = 105.48(3) K, β = 0.132 (1). \n Q = Se:  M0 = 3.701(8) µ B, TN = 89.50(3) K, β = 0.122(1). \n \n \n \n \n \n \n \n 0.9985 0.9990 0.9995 1.0000 1.0005 1.0010 \n0 30 60 90 120 150 180 a/a 168 \nTemperature (K) Q = S Q = Se \n0.9980 0.9985 0.9990 0.9995 1.0000 1.0005 1.0010 \n0 30 60 90 120 150 180 c/c 168 \nTemperature (K) Q = S Q = Se \n-0.2 0.0 0.2 0.4 0.6 0.8 1.0 \n0.5 0.7 0.9 1.1 1.3 Fe 2+ MT/ M0\nT/ TNQ = S \nQ = Se "    },    {        "title": "1709.03764v1.Noise_reduction_in_heat_assisted_magnetic_recording_by_optimizing_a_high_low_Tc_bilayer_structure.pdf",        "content": "Noise reduction in heat-assisted magnetic recording by optimizing a high/low\nTc bilayer structure\nO. Muthsam,1,a)C. Vogler,1and D. Suess1\nUniversity of Vienna, Physics of Functional Materials\n(Dated: 9 October 2018)\nIt is assumed that heat-assisted magnetic recording (HAMR) is the recording technique of\nthe future. For pure hard magnetic grains in high density media with an average diameter\nof 5 nm and a height of 10 nm the switching probability is not su\u000eciently high for the use in\nbit-patterned media. Using a bilayer structure with 50% hard magnetic material with low\nCurie temperature and 50% soft magnetic material with high Curie temperature to obtain\nmore than 99.2% switching probability, leads to very large jitter. We propose an optimized\nmaterial composition to reach a switching probability of Pswitch>99:2% and simultaneously\nachieve the narrow transition jitter of pure hard magnetic material. Simulations with a con-\ntinuous laser spot were performed with the atomistic simulation program VAMPIRE for a\nsingle cylindrical recording grain with a diameter of 5 nm and a height of 10 nm. Di\u000berent\ncon\fgurations of soft magnetic material and di\u000berent amounts of hard and soft magnetic\nmaterial were tested and discussed. Within our analysis, a composition with 20% soft mag-\nnetic and 80% hard magnetic material reaches the best results with a switching probability\nPswitch>99:2%, an o\u000b-track jitter parameter \u001bo\u000b;80=20= 14:2 K and a down-track jitter\nparameter\u001bdown;80=20= 0:49 nm.\nI. INTRODUCTION\nHeat-assisted magnetic recording (HAMR)1{4is con-\nsidered to be a promising approach to increase the areal\nstorage density of recording media in the future. High\nareal density means small recording grains which require\nhigh anisotropy to be thermally stable. The available\n\feld of the write head limits the anisotropy of the grain.\nHAMR overcomes this so-called recording trilemma by\nusing a local heat pulse to heat the material near or\nabove the Curie temperature. By doing this, the coer-\ncivity of the material is reduced such that the available\nhead \feld is su\u000ecient to switch the grain. However, ther-\nmally written-in errors are a serious problem of HAMR.\nIt has been shown that for pure FePt-like hard-magnetic\ngrains with a height of 10 nm and a diameter of 5 nm, the\nswitching probability of one grain is clearly below 99%5\nwhich is too low for practical use in bit-patterned media.\nThe idea to overcome this problem is to use a bilayer\nstructure with graded Curie temperature which consists\nof a hard magnetic layer with low Curie temperature and\na soft magnetic layer with high Curie temperature6. Sim-\nilar to this, a thermal spring magnetic medium was also\nproposed by Co\u000bey et al7. Nevertheless, it was shown8\nthat using a 50/50 low/high Tc bilayer structure leads\nto a signi\fcant increase of both the down-track and the\no\u000b-track jitter parameter. In this paper, we optimize an\nexchange coupled grain to obtain a switching probability\nofPswitch>99:2% while maintaining the low down-track\nand o\u000b-track jitter of a pure FePt grain with the same\ndimensions. This is achieved by varying the composition\nof the soft magnetic layer as well as the ratio between\nthe soft and the hard magnetic layer. The simulations\nwere performed with the atomistic simulation program\na)Electronic mail: olivia.muthsam@univie.ac.atVAMPIRE, which solves the stochastic Landau-Lifshitz-\nGilbert equation9.\nThe structure of this work is as follows: In Section II, the\nHAMR models that are used in the simulations as well\nas the composition of the materials and the optimization\nparameters are explained. Section III summarizes the re-\nsults and shows which material composition works best.\nIn Section IV, the results are discussed.\nII. MODELING HAMR\nFor the simulations a cylindrical recording grain is con-\nsidered with a height of 10 nm and a diameter of 5 nm.\nIt can be interpreted as one island of a patterned media\ndesign with ultra high density. A simple cubic crystal\nstructure is used. In the atomistic simulations, only near-\nest neighbor exchange interactions between the atoms are\nincluded. A continuous laser pulse with Gaussian shape\nand full width at half maximum (FWHM) of 20 nm is\nassumed in all simulations. The temperature pro\fle of\nthe heat pulse is given by\nT(x;y;t ) = (Twrite\u0000Tmin)e\u0000(x\u0000vt)2+y2\n2\u001b2 +Tmin (1)\nwith\n\u001b=FWHMp\n8 ln(2)(2)\nand\nTpeak= (Twrite\u0000Tmin)e\u0000y2\n2\u001b2+Tmin: (3)\nThe speedvof the write head is assumed to be 20 m/s.\nx0=vtdenotes the down-track position of the write headarXiv:1709.03764v1  [cond-mat.mtrl-sci]  12 Sep 20172\nFIG. 1. Schematic representation of the atoms within the\ncylindrical recording grain with a bilayer structure composed\nof soft magnetic material (red) and hard magnetic material\n(blue).\nwith respect to the center of the bit. xandylabel the\ndown-track and the o\u000b-track position of the grain, respec-\ntively. In our simulations both the down-track position\nxand the o\u000b-track position yare variable. The \fnal\ntemperature of all simulations is Tmin= 270 K. Since\nthe laser pulse is continuously switched on, the correct\ntiming of the \feld pulse is very important. The applied\n\feld is modeled as a trapezoidal \feld with a write fre-\nquency of 1 Ghz and a \feld rise and decay time of 0.1 ns.\nThe \feld strength is assumed to be +0.8 T and -0.8 T\ninz-direction. Initially, the magnetization of each grain\npoints in + z-direction. The trapezoidal \feld tries to\nswitch the magnetization of the grain from + z-direction\nto\u0000z-direction. At the end of every simulation, it is\nevaluated if the bit has switched or not.\nA. HARD MAGNETIC LAYER\nThe composition of the FePt like hard magnetic mate-\nrial is the same in all simulations. Only the amount of the\nhard magnetic material is optimized later. The parame-\nters are chosen as follows: For the damping constant we\nuse\u000bHM= 0:1:The atomistic spin moment in \u0016HM=\n1:7\u0016Bwhich corresponds to a saturation polarization\nJHM= 1:42 T. The exchange energy within the hard\nmagnetic material is Jij;HM= 5:18\u000210\u000021J/link. We use\nuniaxial anisotropy in z-direction with an anisotropy con-\nstant 9:12\u000210\u000023J/link which corresponds to an uniaxial\nanisotropy K1;HM= 6:6 MJ/m3. This material composi-\ntion was also used in former simulations by Suess et al5\nand Vogler et al8.\nB. OPTIONAL PARAMETERS\nIn order to \fnd the material composition which maxi-\nmizes the switching probability and simultaneously min-imizes the transition jitter, di\u000berent parameters of the\nsoft magnetic material are varied and simulations for the\ndi\u000berent material compositions are performed. However,\nsome parameters of the soft magnetic material are specif-\nically chosen to model the material as realistic as possi-\nble. The \fxed parameters, on one hand, are the damping\nconstant\u000bSM= 0:1 and the anisotropy constant K1;SM\nwhich is set to zero. On the other hand, the exchange\nenergy within the layer, the exchange energy between the\nlayers and the atomistic spin moment are variable. The\namount of hard magnetic material in these simulations is\nalways 50%.\nFirst, the optimal parameters for the soft magnetic ma-\nterial are determined, then the amounts of soft and hard\nmagnetic material are additionally varied to further opti-\nmize the recording grains. Figure 1 shows such a bilayer\ncomposition with 20% soft magnetic material and 80%\nhard magnetic material.\nIII. RESULTS\nA. HARD MAGNETIC LAYER\nFirst, a phase diagram, where the switching probabil-\nity depending on the the down-track position xand the\no\u000b-track position yis computed for a pure FePt like grain,\nsee Figure 2. If the write temperature Twrite of the heat\nspot is \fxed, every peak temperature Tpeakcorresponds\nto an unique o\u000b-track position y(see eq. (3)). Therefore,\nin the phase diagram, the switching probability is shown\nas a function of the down-track position xand the, to y\ncorresponding, peak temperature Tpeak.\nThe schematic position between the heat pulse and the\ntrapezoidal \feld at down-track position x= 0 nm can be\nseen in Figure 3. In the simulations, only the cooling of\nthe heat pulse is considered, i.e. the simulations do not\nstart before the peak of the heat pulse.\nThe resolution in the down-track direction is \u0001 x= 2 nm\nand the resolution in the peak temperature direction is\n\u0001Tpeak= 25 K. The velocity of the write head is assumed\nto bevh= 20 m/s. In each phase point, 128 trajectories\nare simulated. Thus, the switching probability phase dia-\ngram contains almost 60.000 switching trajectories with\na length of 2 ns. The areas with less than 1% switch-\ning and the areas with more than 99 :2% switching are\nmarked by the contour lines. The phase diagram of the\npure hard magnetic grain shows only a few areas with\ncomplete switching. In particular, no complete switch-\ning occurs for high peak temperatures larger than 650 K.\nThis shows the high DC noise of pure hard magnetic\ngrains.\nBoth the o\u000b-track and the down-track jitter can be\nextracted from the switching probability phase diagram.\nThe transition in o\u000b-track direction at a speci\fc down-\ntrack position can be obtained by making a vertical cut\nin the phase diagram. For example, the o\u000b-track transi-\ntion at down-track position x= 0 nm is marked by the\ndashed vertical line at x= 0 nm in Figure 2. On the other\nhand, the down-track transition is marked by a cut in the\nhorizontal direction at a speci\fc o\u000b-track position which3\n−20 −10 0 10 20500\n600\n700\ndown-track [nm]Tpeak[K]\n0 0.2 0.4 0.6 0.8 1\nFIG. 2. Switching probability phase diagram of a pure FePt\nlike hard magnetic grain. The contour lines indicate the\ntransition between areas with switching probability less than\n1% (red) and areas with switching probability higher than\n99.2% (blue). The dashed lines mark the switching probabil-\nity curves of Figure 4.\n0 0.5 1 1.5 2\n·10−9−0.8−0.400.40.8\ntime [s]field strength [T] Heat pulse\nApplied field\n300400500600700\ntemperature [K]\nFIG. 3. Schematic representation of a trapezoidal \feld and a\nGaussian heat pulse for the HAMR simulations at down-track\nposition x= 0 nm.\nis depicted by the corresponding peak temperature. For\na peak temperature Tpeak = 700 K, the switching prob-\nability curve at down-track position x= 0 nm and the\ntransition curve in down-track direction at o\u000b-track po-\nsitiony= 0 nm (Tpeak=Twrite) can be seen in Figure 4\n(a) and (b).\nB. SOFT MAGNETIC LAYER\nAn exchange coupled bilayer structure is considered.\nThe parameters for a suitable soft magnetic composi-\n400 500 600 70000.20.40.60.81\nTpeak[K]switching probabilityFePt(a)\n−20 −10 0 10 2000.20.40.60.81\ndown-track [nm]switching probabilityFePt(b)FIG. 4. Switching probability curves of a FePt like hard mag-\nnetic grain. (a) P(Tpeak) which corresponds to o\u000b-track jitter\nfor a \fxed down-track position x= 0 nm. (b) Down-track jit-\nterP(x) for a \fxed o\u000b-track position y= 0 nm and the peak\ntemperature Tpeak= 700 K.\ntion are sought. Since the atomistic simulations are very\ntime consuming, it is not possible to calculate a switch-\ning probability phase diagram for every material con\fg-\nuration. For this reason, only the switching probability\ncurve along the o\u000b-track direction at x= 0 nm is calcu-\nlated, again as a function of the peak temperature that\ncorresponds to the respective o\u000b-track position. The re-\nsult of these simulations is a switching probability curve\nP(Tpeak) for 400 K\u0014Tpeak\u0014700 K which gives both,\nthe maximum switching probability in the center of the\ngrain and the o\u000b-track jitter. In a \frst optimization step,\nsoft magnetic materials reaching complete switching in\nthe center of the grains are pre-selected. To do this, the\nswitching probability curves are \ftted with a Gaussian4\ncumulative distribution function\n\b\u0016;\u001b2=1\n2(1 + erf(x\u0000\u0016p\n2\u001b2))\u0001P (4)\nwith\nerf(x) =2p\u0019Zx\n0e\u0000\u001c2d\u001c; (5)\nwhere the mean value \u0016, the standard deviation \u001band\nthe mean maximum switching probability P2[0;1] are\nthe \ftting parameters. The standard deviation \u001bde-\ntermines the steepness of the transition function and\nis a measure for the transition jitter and thus for the\nachievable maximum areal grain density of a recording\nmedium. The \ftting parameter Pis a measure for the\naverage switching probability for su\u000eciently high tem-\nperatures. In Figure 5, one can see the \ftting parameter\nPfor a recording grain as a function of di\u000berent atom-\nistic spin moments \u0016SMand di\u000berent exchange energies\nJij;SMwithin the soft layer. Materials with Pless or\nequal 0.992 are not further considered. The phase dia-\ngram shows, that there are a few material compositions\nwith su\u000eciently high P. For these materials, the \ftting\nparameter \u001bis additionally compared for the di\u000berent\ncon\fgurations and the material with the lowest \u001bis cho-\nsen. Two materials, namely that with \u0016SM= 1:7\u0016Band\nJij;SM= 7:25\u000210\u000021J/link and that with \u0016SM= 2:0\u0016B\nandJij;SM= 7:25\u000210\u000021J/link nearly have the same\nand the lowest \u001b:Since the atomistic spin moment for\nthe hard magnetic material is \u0016HM= 1:7\u0016Bthe same\nvalue is chosen for the soft magnetic material.\nIn summary, the following parameters for the soft mag-\nnetic composition are chosen for further simulations: For\nthe exchange constant within the soft magnetic mate-\nrialJij;SM= 7:25\u000210\u000021J/link is chosen. The ex-\nchange energy between the materials is set to Jij=p\nJij;HM\u0001Jij;SM= 6:13\u000110\u000021J/link. The atomistic spin\nmoment of the soft magnetic material is equal to that of\nthe hard magnetic material namely \u0016SM= 1:7\u0016B.\nC. BILAYER COMPOSITION\nWith the chosen parameters, the amount of soft and hard\nmagnetic material is optimized. The idea is to use as\nlittle soft magnetic material as possible to get narrow\ntransitions, but as much soft magnetic material as neces-\nsary to get 100% switching probability in the simulations.\nPhase diagrams are only computed for the most promis-\ning materials. To \fnd contemplable materials, again an\no\u000b-track transition with switching probabilities at down-\ntrack position x= 0 nm and for temperatures in a range\nfrom 400 K to 700 K in steps of 25 K is calculated for each\nmaterial.\nIn Table I the resulting maximum switching probabili-\nties for di\u000berent material compositions with Tpeakslightly\nabove the Curie temperature of the material can be seen.\nOne observes that materials with an amount of soft mag-\nnetic material of 20% or more have a switching proba-\nbility of 100% for temperatures around the Curie tem-\nperature. Thus, the transition curves of these materials\n0.5 1 1.5 25\n6\n7\n8\n9\nµSM[µB]Jij,SM ·10−21[J/link]\n0.92 0.94 0.96 0.98FIG. 5. Phase diagram of a recording grain with 50% hard\nmagnetic material and 50% soft magnetic material where the\n\ftting parameter Pof eq. (4) can be seen for di\u000berent con\fg-\nurations of the soft magnetic layer. The contour lines mark\nthe areas with P > 0:992:\nThickness HM =SMTpeakPswitch\nFePt 654K 98:4%\n90=10 660K 98:4%\n89=11 660K 98:4%\n88=12 666K 99:2%\n87=13 666K 99:2%\n86=14 672K 100%\n85=15 672K 100%\n84=16 678K 97:6%\n83=17 678K 97:6%\n82=18 678K 97:6%\n81=19 684K 100%\n80=20 690K 100%\n79=21 696K 100%\nTABLE I. Results for material compositions with di\u000berent\namounts of hard magnetic (HM) and soft magnetic (SM)\nshare. The maximum switching probability is calculated at\ndown-track postition x= 0 nm and for a peak temperature\nTpeakwhich is 20% higher than the Curie temperature of the\nmaterial composition.\nare compared to that of FePt. This is done by \ftting the\ntransitions with a Gaussian cumulative distribution func-\ntion as in eq. (4). The important \ftting parameter is the\nstandard deviation \u001bwhich is a measure for the transition\njitter and thus for the achievable maximum areal grain\ndensity of a recording medium. The transition curves and\nthe corresponding \ftting curves of the di\u000berent materials\ncan be seen in Figure 6 for peak temperatures between\n400 K and 700 K. Note, the \ftting curve of FePt with\n\u001bo\u000b;FePt = 9:7 K shows that the switching probability\ndoes not reach 100% in the simulations.\nFor the other material con\fgurations, full switching5\n400 500 600 70000.20.40.60.81\nTpeak[K]switching probabilityFePt\nFit FePt\n80/20\nFit 80/20\n75/25\nFit 75/25\n50/50\nFit 50/50(a)\n600 7000.90.920.940.960.981\nTpeak[K]switching probabilityFePt\nFit FePt\n80/20\nFit 80/20\n75/25\nFit 75/25\n50/50\nFit 50/50(b)\nFIG. 6. (a) O\u000b-track jitter ( P(Tpeak) curves) and correspond-\ning \fts of grains with di\u000berent amounts of soft magnetic share\nat down-track position x= 0 nm for di\u000berent peak tempera-\ntures Tpeak. (b) Zoomed transition and \ftting curves for peak\ntemperatures in a range from 550 K to 700 K.\ncan be seen for su\u000eciently high temperatures. Further,\nin Figure 6, one can see that the o\u000b-track transition jit-\nter gets larger for a higher amount of soft magnetic ma-\nterial. For 20% soft magnetic material, the transition\nis the steepest one among the bilayer compositions with\n\u001bo\u000b;80=20= 14:2 K. Thus, it is much steeper than that\nof a material with 50% soft magnetic share for which\n\u001bo\u000b;50=50= 27:87 K is almost twice as large as for 80/20.\nActually, the transition of the composition with 20% soft\nmagnetic material is the best compared to that of FePt\nalthough even here \u001bo\u000b;80=20is 46% larger than \u001bo\u000b;FePt.\nSince a grain with 80% hard magnetic material and 20%\nsoft magnetic material (80/20) is the most promising\nmaterial to ful\fll our purpose, a switching probability\nphase diagram is calculated for 80/20. In Figure 7, this\nswitching probability phase diagram is illustrated. In\ncontrast to the phase diagram of FePt (see Figure 2), the\n−20 −10 0 10 20500\n600\n700\ndown-track [nm]Tpeak[K]\n0 0.2 0.4 0.6 0.8 1FIG. 7. Switching probability phase diagram of recording\ngrain consisting of a composition of 80% FePt like hard\nmagnetic material and 20% Fe like soft magnetic material.\nThe contour lines indicate the transition between areas with\nswitching probability less than 1% (red) and areas with\nswitching probability higher than 99.2% (blue). The dashed\nlines mark the switching probability curves in Figure 6 and\nFigure 8.\n80/20 phase diagram shows complete switching also for\nhigher peak temperatures. Indeed, the bilayer structure\nshows 100% switching probability for peak temperatures\nhigher than 550 K in a range from down-track position\nx=\u000010 nm tox= 6 nm. It can also be seen that the\njitter in o\u000b-track and down-track direction of both ma-\nterials does not di\u000ber much from the one of FePt.\nTo compare the jitter of the materials more accu-\nrately, the jitter in down-track direction for one peak\ntemperature and all down-track positions is calculated\nwith higher resolution for both materials. Since writing\nof the grain starts at Tpeak = 475 K for both materi-\nals, the jitter is considered at the same peak tempera-\nture,Tpeak= 700 K, for both materials. The simulations\nare done for down-track positions xbetween -20 nm and\n22 nm with a resolution of \u0001 x=2 nm. Again, in the\narea of the transitions the resolution is \fner, namely\n\u0001x= 0:16_6 nm.\nIn Figure 8, the results of these simulations can be seen.\nForTpeak = 700 K, the 80/20 material reaches 100%\nswitching probability whereas the switching probability\nof the pure FePt is clearly below 100%. In contrast to\nthe distinct switching probabilities of the materials, the\ntransitions of FePt and 80/20 are almost the same. How-\never, to compare the down-track jitter in more detail,\nthe transitions are again \ftted with a Gaussian cumula-\ntive distribution function like in eq. (4) and eq. (5). For\nboth temperatures the steepness of the transitions dif-\nfers marginally. This can seen by the \ftting values which\nare\u001bdown;FePt = 0:45 nm and \u001bdown;80=20= 0:49 nm.\nFor comparison, a composition with 50% hard and 50%6\n−20 −10 0 10 2000.20.40.60.81\ndown-track [nm]switching probabilityFePt\n80/20(a)\n−10 −8 −6 −400.20.40.60.81\ndown-track [nm]switching probabilityFePt\nFit FePt\n80/20\nFit 80/20\n50/50\nFit 50/50(b)\nFIG. 8. Comparison of down-track jitter at a peak tempera-\ntureTpeak= 700 K for a pure hard magnetic grain and a 80/20\nhard/soft layer composition (a) for all down-track positions x\nfrom - 20 nm to 22 nm. In (b) the jitter in down-track direc-\ntion is shown more closely for down-track positions between\nx=\u000010 nm and x=\u00004 nm.\nsoft magnetic material has a \ftting value \u001bdown;50=50=\n0:64 nm.\nIV. DISCUSSION\nTo conclude, we simulated HAMR for a cylindrical\nrecording grain ( d= 5 nm,h= 10 nm) with an exchange\ncoupled bilayer structure with graded Curie temperature.\nHere, the hard magnetic layer has a low Curie temper-\nature and the soft magnetic layer a high Curie temper-\nature. Our goal was to vary the composition and the\namount of soft magnetic material such that at the same\ntime both the AC noise and the DC noise are minimized.AC noise determines the distance between neighboring\nbits in bit-patterned media. DC noise on the other hand\nlimits the switching probability of a bit in bit-patterned\nmedia. Pure hard magnetic material shares high DC\nnoise for a head velocity vh= 20 m/s. In contrast, for\na bilayer structure with too high soft magnetic fraction,\nthe DC noise is signi\fcantly reduced but unfortunately\nthe AC noise is signi\fcantly higher than for pure hard\nmagnetic material.\nVarying the soft magnetic composition showed that the\natomistic spin moment does not in\ruence the switching\nprobability as much as the exchange interactions. Thus, a\nsoft magnetic material with the same atomistic spin mo-\nment as the hard magnetic material and a higher Curie\ntemperature was chosen. The used composition is simi-\nlar to the one used by Vogler et al8. Further simulations\nto vary the amount of hard and soft magnetic material\nshowed that more soft magnetic material leads to higher\nswitching probabilities, whereas less soft magnetic ma-\nterial leads to narrower transitions. These results are\nnot surprising. Because of the low coercivity of the soft\nmagnetic layer and the exchange spring e\u000bect, the soft\nmagnetic layer helps to switch the magnetization of the\nhard magnetic layer more reliably. Thus, the switching\nprobability increases for a thicker soft magnetic layer. On\nthe other hand, this increase of the switching probability\nis also visible for the jitter. Temperatures for which the\ngrain does not switch for pure hard magnetic media start\nto switch for a bilayer composition. This explains the\nincrease of the jitter in o\u000b-track and down-track direc-\ntion for the bilayer structure. We showed that a material\ncomposition consisting of 20% soft magnetic material and\n80% hard magnetic material reduces both the AC and the\nDC noise.\nThe 80/20 composition shows full switching in a wide\nrange with a maximum switching probability Pswitch>\n99:2%. The transition jitter is comparable to that\nof FePt with the jitter parameters \u001bo\u000b;80=20= 14:2 K\nand\u001bdown;80=20= 0:49 nm. These are only marginally\ndi\u000berent to those of FePt, i.e. \u001bo\u000b;FePt = 9:7 K\nand\u001bdown;FePt = 0:45 nm. The 80/20 composition\nis much better than that of a 50/50 bilayer structure\nwhich has the jitter parameters \u001bo\u000b;50=50= 27:87 K\nand\u001bdown;50=50= 0:64 nm. Indeed, a 2 nm thick soft\nmagnetic layer is in the expected range for optimal\nswitching10,11. The optimal thickness of the soft mag-\nnetic layer to reduce the switching \feld of the hard mag-\nnetic layer is around the exchange length between the\nsoft and the hard magnetic layer12. Since this exchange\nlength is around 2 nm, our material ful\flls this require-\nment perfectly.\nV. ACKNOWLEDGEMENTS\nThe authors would like to thank the Vienna Sci-\nence and Technology Fund (WWTF) under grant No.\nMA14-044, the Advanced Storage Technology Consor-\ntium (ASTC), and the Austrian Science Fund (FWF)\nunder grant No. I2214-N20 for \fnancial support. The\ncomputational results presented have been achieved us-7\ning the Vienna Scienti\fc Cluster (VSC).\n1Hiroshi Kobayashi, Motoharu Tanaka, Hajime Machida, Takashi\nYano, and Uee Myong Hwang. Thermomagnetic recording .\nGoogle Patents, August 1984.\n2C. Mee and G. Fan. A proposed beam-addressable memory. IEEE\nTransactions on Magnetics , 3(1):72{76, 1967.\n3Robert E. Rottmayer, Sharat Batra, Dorothea Buechel,\nWilliam A. Challener, Julius Hohlfeld, Yukiko Kubota, Lei Li,\nBin Lu, Christophe Mihalcea, Keith Mount\feld, and others.\nHeat-assisted magnetic recording. IEEE Transactions on Mag-\nnetics , 42(10):2417{2421, 2006.\n4Mark H. Kryder, Edward C. Gage, Terry W. McDaniel,\nWilliam A. Challener, Robert E. Rottmayer, Ganping Ju, Yiao-\nTee Hsia, and M. Fatih Erden. Heat assisted magnetic recording.\nProceedings of the IEEE , 96(11):1810{1835, 2008.\n5Dieter Suess, Christoph Vogler, Claas Abert, Florian Bruckner,\nRoman Windl, Leoni Breth, and J. Fidler. Fundamental limits\nin heat-assisted magnetic recording and methods to overcome it\nwith exchange spring structures. Journal of Applied Physics ,\n117(16):163913, 2015.\n6Dieter Suess and Thomas Schre\r. Breaking the thermally in-\nduced write error in heat assisted recording by using low and high\nTc materials. Applied Physics Letters , 102(16):162405, 2013.7Kevin Robert Co\u000bey, Jan-Ulrich Thiele, and Dieter Klaus Weller.\nThermal springmagnetic recording media for writing using mag-\nnetic and thermal gradients . Google Patents, April 2005.\n8Christoph Vogler, Claas Abert, Florian Bruckner, Dieter Suess,\nand Dirk Praetorius. Areal density optimizations for heat-\nassisted magnetic recording of high-density media. Journal of\nApplied Physics , 119(22):223903, 2016.\n9Richard FL Evans, Weijia J. Fan, Phanwadee Chureemart,\nThomas A. Ostler, Matthew OA Ellis, and Roy W. Chantrell.\nAtomistic spin model simulations of magnetic nanomaterials.\nJournal of Physics: Condensed Matter , 26(10):103202, 2014.\n10Thomas Schre\r, Josef Fidler, and H. Kronmller. Remanence\nand coercivity in isotropic nanocrystalline permanent magnets.\nPhysical Review B , 49(9):6100, 1994.\n11Dieter Suess. Micromagnetics of exchange spring media: Opti-\nmization and limits. Journal of magnetism and magnetic mate-\nrials, 308(2):183{197, 2007.\n12L. S. Huang, J. F. Hu, and J. S. Chen. Critical Fe thickness for ef-\nfective coercivity reduction in FePt/Fe exchange-coupled bilayer.\nJournal of Magnetism and Magnetic Materials , 324(6):1242{\n1247, 2012."    },    {        "title": "1710.07707v1.Anisotropic_magnetic_properties_of_the_triangular_plane_lattice_material_TmMgGaO4.pdf",        "content": "Anisotropic magnetic properties of the triangular plane lattice material TmMgGaO 4\nF. Alex Cevallos, Karoline Stolze, Tai Kong, and R.J. Cava\nDepartment of Chemistry, Princeton University, Princeton NJ 08544 USA\n*Corresponding authors: fac2@princeton.edu (F. Alex Cevallos), rcava@princeton.edu (R.J. Cava)\nAbstract\nThe crystal growth, structure, and basic magnetic properties of TmMgGaO 4 are reported. The \nTm ions are located in a planar triangular lattice consisting of distorted TmO 6 octahedra, while the Mg \nand Ga atoms randomly occupy intermediary bilayers of M-O triangular bipyramids. The Tm ions are \npositionally disordered. The material displays an antiferromagnetic Curie Weiss theta of ~ -20 -25 K, \nwith no clear ordering visible in the magnetic susceptibility down to 1.8 K; the structure and magnetic \nproperties suggest that ordering of the magnetic moments is frustrated by both structural disorder and \nthe triangular magnetic motif. Single crystal magnetization measurements indicate that the magnetic \nproperties are highly anisotropic, with large moments measured perpendicular to the triangular planes. \nAt 2 K, a broad step-like feature is seen in the field-dependent magnetization perpendicular to the plane\non applied field near 2 Tesla.\n1\nIntroduction \nGeometrically frustrated magnetism has been an actively studied property of materials since at \nleast the 1970s1-2. In a geometrically frustrated system, the geometry of the crystal lattice inhibits the \nlong-range ordering of the magnetic moments; the simplest examples are systems with \nantiferromagnetic nearest neighbor interactions on a triangular lattice. As there is no simple way to \norder the spins in such a system, the magnetic ordering transition will typically be below the Weiss \ntemperature. Geometrically frustrated systems can be used to study a variety of unusual ground state \nconditions that are difficult to achieve in more conventional materials1-7. The influence of structural \ndisorder, especially in geometrically frustrated rare-earth based systems, is expected to be significant4.\nThe YbFe2O4 structure type has been observed to demonstrate a wide variety of electronic and \nmagnetic properties, including geometric magnetic frustration, spin-glass behavior7-11, charge density \nwaves12-13, and ferroelectricity14-16. This structure crystallizes in the R-3m space group, and is defined by\ntriangular planes of metal-oxygen octahedra separated by a bilayer of metal-oxygen triangular \nbipyramids17. Recently, the YbFe 2O4-type compound YbMgGaO 4 has drawn interest as a potential \ncandidate for exhibiting quantum spin liquid (QSL) behavior18-21. YbMgGaO4 places the magnetic Yb3+ \nions on the triangular planes, with non-magnetic Mg2+ and Ga3+ ions randomly mixed in the triangular \nbipyramidal sites (but with no mixing between magnetic and nonmagnetic ions). The Mg-Ga mixing, \noff the magnetic plane, causes a considerable variation in local magnetic interactions that is reflected in\nthe compound's properties20, 22-23. Here we report the structure and elementary magnetic properties of \nthe isostructural and closely related compound TmMgGaO 4, in which similarly complex magnetic \nbehavior is possible. Our results suggest that further, more detailed study may be of future interest.\nExperimental\nSynthesis\nPolycrystalline samples of TmMgGaO 4 were synthesized via solid state reaction. \nStoichiometric quantities of Tm 2O3 (99.99%, Alfa Aesar), Ga 2O3 (99.999%, Alfa Aesar) and MgO \n2\n(99.95%, Alfa Aesar) were ground with an agate mortar and pestle. Pellets of the starting composition \nwere heated in ceramic crucibles in air at 1450°C for four days, with intermediate grinding. A small \namount of impurity, believed to be Tm 3Ga5O12, was observed in some samples and attributed to an \ninsufficient starting quantity of MgO due to partial hydration of the starting material. The addition of \nMgO to the impure sample at the 5% level, followed by reheating at 1450°C for 24 hours resulted in \nthe reduction or elimination of this impurity in all cases.\nSingle crystals in the mm size range were grown by the floating zone method. Single phase \npolycrystalline powder was loaded into rubber tubes and hydrostatically compressed at 40 MPa, \nresulting in polycrystalline rods typically measuring 6 mm in diameter and 5-8 cm in length. The rods \nwere then sintered at 1450°C for 3 hours in air before being transferred to a four-mirror optical floating \nzone (FZ) furnace (Crystal Systems, Inc. Model No. FZ-T-10000-HVP-II-P) with 4 x 1000 W lamps. \nThe rods were then further sintered in the FZ furnace at 65.0% output power, rotating at 15 RPM, with \na rate of travel of 1.5 mm/hr. Crystal growth from the feed rods was performed at 68.0% output power, \nwith rods rotating in opposite directions at 20 RPM and an upwards travel rate of 0.5 mm/hr. Growth of\nfaceted millimeter-scale crystals of TmMgGaO 4 was observed within a few mm of growth initiation, \nbut large single crystals were not obtained for growths up to 5 cm in length, even when using previous \ngrowths as a seed.\nCharacterization\nRoom-temperature powder X-ray diffraction (PXRD) measurements were performed with a \nBruker D8 Advance Eco diffractometer with Cu Kα radiation (λ = 1.5418 Å) and a LynxEye-XE \ndetector. Phase identification was performed using the Bruker EVA program. Powder Rietveld \nrefinements were performed using Fullprof Suite. Magnetic measurements were taken using a Quantum\nDesign Physical Property Measurement System (PPMS) Dynacool with a vibrating sample mount. DC \nmagnetic susceptibility, defined as the measured magnetization M divided by the applied magnetic \nfield, was measured between 1.8 K and 300 K in an applied field of 1000 Oe, and the resulting values \n3\nwere divided by the number of moles of Tm3+ present to obtain the magnetization values per formula \nunit. Field-dependent magnetization was measured at 2 K. Powder samples for magnetic \ncharacterization were obtained by grinding small single crystals. Anisotropic magnetization \nmeasurements were taken on a 1 x 0.2 mm plate-like single crystal (large face perpendicular to the c-\naxis), either mounted on a silica sample holder with GE varnish or placed in a plastic sample holder, \nand oriented along the axis of study.\nResults and Discussion\nStructure\nTmMgGaO4, as has been previously reported24, is isostructural to YbFe 2O4, and therefore to a \nlarge family of related isostructural Ln+2M2+M3+O4 compounds8, 25-26. In agreement with previous \nreports, our colorless, transparent TmMgGaO 4 crystallizes in the space group R-3m (166), with lattice \nparameters of a = 3.4195(3) Å and c = 25.1231(1) Å. The structure is composed of triangular layers of \ndistorted TmO6 octahedra, separated by bilayers of mixed occupancy Mg- and Ga-O triangular \nbipyramids (Figure 2). The Tm atoms sit displaced, slightly off the ideal [0 0 0] position along c, with z\n= 0.0078(2). This distortion has been observed in other materials in this family and is attributed to the \nrandom distribution of non-lanthanide metals in the neighboring bilayers8. As the Tm atom can only \noccupy one of the resulting displaced sites, they are both determined to have 1/2 occupancy (the sites \nare too close together to be simultaneously occupied). Although the lattice parameters and overall \nstructure are in good agreement with previously published results24-25, the disordered slightly off-ideal \nlocation of the Tm3+ ion did not appear to be noted in the only previous study reporting refined atomic \npositions25. However, this positional disorder of the rare earth atom has been observed in a variety of \nrelated systems, including LuCuGaO 4, LuCoGaO4, YbCuGaO4, LuCuFeO4, LuZnFeO48, and \nYbMgGaO421, suggesting that TmMgGaO 4 behaves similarly. Our single-crystal XRD refinement of the\nstructure (see Supplementary Information) showed that the statistic positional disorder of the Tm \npositions can also be described by an approximately equivalent model wherein the Tm atom sits \n4\ndirectly on the high-symmetric Wyckoff site 3a [0 0 0] site with a very large anisotropic displacement \nparameter along the c-axis. The results of this refinement can be seen in the supplementary \ninformation, along with a map of the measured atomic scattering density showing the distribution of \nthulium positions in a section of the unit cell, derived from the observed structure factors, Fobs.\nMagnetism\nThe temperature-dependent magnetic susceptibility of a powder sample of TmMgGaO 4 \n(sample produced by grinding a collection of small single crystals) is shown in Figure 3. The \nsusceptibility was fit to the Curie-Weiss law χ – χ 0 = C / (T – θW), where χ is the magnetic susceptibility,\nχ0 is a temperature-independent contribution, C is the Curie constant, and θW is the Weiss temperature. \nThe inverse susceptibility, 1/(χ – χ 0), was found to be almost linear for a χ 0 value of 0.0010 emu mol-1. \nThe χ0 value for the polycrystalline sample is consistent with the single crystal susceptibility obtained \nfor fields in the plane of the triangular lattice, as described below.  The inverse susceptibility was fit at \nhigh and low temperature. At high temperature (150 K – 290 K), C was found to be 7.26 and θW was \nfound to be -25.6 K. The effective magnetic moment per ion, μ eff, was determined by the relationship \nμeff =√8C, yielding a moment of 7.62 μB/Tm, in good agreement with the ideal value for a free \nTm3+ ion, 7.57 μB. At low temperatures, a linear fit yielded values of C = 6.68, θW = -19.7, and an \neffective magnetic moment of 7.31 μB/Tm. The negative Weiss temperatures determined by these fits \nsuggest that TmMgGaO 4 has dominantly antiferromagnetic coupling, but no magnetic ordering is \nobserved down to 1.8 K. At the lowest temperatures studied here (1.8-10 K) there is a deviation from \nthe paramagnetic behavior predicted by the higher temperature Curie Weiss fits (Figure 3, inset). \nTmMgGaO4 therefore exhibits a frustration index f (θW /TM) of 10 or more, suggesting that it is a \nstrongly frustrated magnet4.\nA magnetization vs. applied field measurement was taken on the same powder sample at 2 K. \nThe material exhibited a nonlinear magnetization response with an apparent saturation of \napproximately 5 μB/Tm by applied fields of 8 Tesla. Repeating this measurement on a small single \n5\ncrystal of TmMgGaO 4 perpendicular to the triangular plane (i.e. parallel to the c-axis) resulted in a \nmuch higher saturation magnetization, of approximately 7 μB/Tm (Figure 4). Both the powder and \nsingle-crystal magnetizations demonstrate a slope anomaly with an onset field of approximately μ0H ~ \n2 Tesla. The origin of this anomaly is currently unknown, but its appearance is reminiscent of \nmetamagnetism and its presence is consistent across all samples. In contrast, magnetization vs. applied \nfield measurements condu cted parallel to the triangular planes (i.e. perpendicular to c) yielded very \ndifferent results: a much smaller magnetization with a very slight curvature and no apparent saturation. \nFrom this it can be safely concluded that TmMgGaO 4 has a high degree of magnetic anisotropy, which \nwe attribute to crystal electric field (CEF) effects originating from its layered crystal structure and \ndistorted TmO6 octahedra. This distortion of the octahedra results from the previously-described off-\nsite nature of the central Tm atoms, which is itself attributed to the local distortions caused by disorder \non the Mg/Ga sites. We note that strong CEF effects have been found in the isostructural compound \nYbMgGaO420, lending credence to our hypothesis concerning the origin of the magnetic anisotropy.\nIn order to observe this anisotropy more clearly, magnetization vs temperature measurements \nwere taken on a single crystal (pictured in Figure 1), both parallel and perpendicular to the triangular \nlayers. As can be seen in Figure 5, the anisotropy is clearly displayed in this measurement as well. \nMeasurements parallel to the perpendicular to the triangular planes (i.e. along the c-axis) yielded a \nsimilar curve to the powder sample, but much larger values of molar magnetic susceptibility. \nMeasurements parallel to the planes show a much weaker signal, with a broad region spanning from 50 \nK to 300 K that appears to be nearly temperature-independent. The purple curve in Figure 5 is a \nweighted average of the temperature-dependent susceptibilities of the single crystal (1/3 of the value \nparallel to the c-axis plus 2/3 of the value perpendicular to the c-axis). This calculated value overlaps \nalmost exactly with the measured susceptibility of the powder. In the right panel of Figure 5, the \ninverse is shown of all four susceptibilities. The weighted average of the anisotropic susceptibilities in \nthis case is more linear than the unmodified inverse susceptibility of the powder, but when the term χ 0 \n6\n(which we attribute to the container for the powder sample) is applied to the powder measurement, the \ntwo once again overlap almost exactly. The observation that the magnetic moment as measured in a \npolycrystalline powder is effectively a simple average of the moments along the spatial directions is \nconsistent with the hypothesis that this anisotropy arises as a result of CEF effects within the crystal \nlattice27.\nConclusion\nSingle crystals of TmMgGaO 4 have been synthesized in an optical floating zone furnace, and \nthe structure has been refined. The resulting values for lattice parameters and atomic positions are in \ngood agreement with previously published results for this and other isostructural compounds. \nMagnetization measurements were performed, finding the material to display dominantly \nantiferromagnetic interactions, with no ordering above 1.8 K. Thus the magnetic characterization \nsuggests a magnetic frustration index of 10 or more. Magnetization measurements on a single crystal \nshow that the magnetic properties are highly anisotropic, and suggest that this anisotropy may arise \nfrom CEF effects. Further characterization of the low temperature properties of this material is \nwarranted; the uncommon interplay of an isolated, geometrically-frustrated magnetic 2D lattice, and \nthe subtle structural distortions induced on the magnetic interactions due to the off-plane disorder of \nnon-magnetic ions may give rise to unusual electronic or magnetic ground states, and similar to the \ndisordered transition metal pyrochlores, may provide interesting system for testing the effect of \ndisorder in geometrically frustrated magnets28-33. Further efforts to grow larger high-quality single \ncrystals for complimentary measurement purposes, such as neutron scattering, may also be of interest.\nAcknowledgments\nThis research was supported by the US Department of Energy, Division of Basic Energy Sciences, \nGrant No. DE-FG02-08ER46544, and was performed under the auspices of the Institute for Quantum \nMatter. \n7\nReferences\n1. 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Jpn., 59, 4443-4448 (1990).\n12. Hearmon, A. J., Prabhakaran, D., Nowell, H., Fabrizi, F., Gutmann, M. J., Radaelli, P. G., Phys. \nRev. B, 85, 014115 (2012).\n13. Zhang, Y ., Yang, H. X., Guo, Y. Q., Ma, C., Tian, H. F., Luo, J. L., Li, J. Q., Phys. Rev. B, 76, \n184105 (2007).\n14. Liu, Y., Zou, T., Wang, F., Zhang, X., Cheng, Z., Sun, Y . Physica B. Cond. Mat. , 405 (16), 3391-\n3394 (2010).\n15. Ikeda, N. J. of Phys. Cond. Mat.  20 (43), (2008).\n16. Zhang, Y ., Yang, H. X., Ma, C., Tian, H. F., Li, J. Q. Phys. Rev. Lett., 98, 247602 (2007).\n17. Kato, K., Kawada, I., Kimizuka, N., Katsura, T., Zeit. Für Krist., 141, 314-320 (1975).\n18. Li, Y., Chen, G., Tong, W., Pi, L., Liu, J., Yang, Z., Wang, X., Zhang, Q. Phys. Rev. Lett. 115, \n167203 (2015).\n19. Paddison, J. A. M., Daum, M., Dun, Z. L., Ehlers, G., Liu, Y ., Stone, M. B., Zhou, H. D., Mourigal, \nM. Nature Physics 13, 177-122 (2017).\n20. Li, Y., Adroja, D., Bewley, R., Voneshen, D., Tsirlin, A., Gegenwart, P., Zhang, Q. Phys. Rev. Lett. \n118, 107202 (2017).\n8\n21. Li, Y., Liao, H., Zhang, Z., Li, S., Jin, F., Ling, L., Zhang, L., Zou, Y., Pi, L., Yang, Z., Wang, J., \nWu, Z., Zhang, Q. Sci. Reports, 5, 16419 (2015).\n22. Shen, Y., Li, Y .-D., Wo, H., Li, Y., Shen, S., Pan, B., Wang, Q., Walker, H. C., Steffens, P., Boehm, \nM., Hao, Y ., Quintero-Castro, D. L., Harriger, L. W., Frontzek, M. D., Hao, L., Meng, S., Chang, Q., \nChen, G., Zhao, J., Nature, 540, 559-562 (2016).\n23. Zhu, Z., Maksimov, P. A., White, S. R., Chernyshev, A. L., Phys. Rev. Let., 119, 157201 (2017).\n24. Kimizuka N., and Takayama, E. J. Solid State Chem. 41, 166-173 (1982).\n25. Beznosikov, B. V. and Aleksandrov, K. S. Krasnoyarsk. Pre-print (2006).\n26. Kimizuka, N., and Mohri, T. J. Solid State Chem.  60 (3), 382-384 (1985).\n27. Dunlap, B. D., J. Mag. and Mag. Mat. , 37, 211-214 (1983).\n28. Sanders, M. B., Krizan, J. W., Plumb, K. W., McQueen, T. M., Cava, R. J., J. Phys. Condes Matter , \n29, 045801 (2017).\n29. Krizan, J. W., Cava, R. J., Phys. Rev. B, 92, 014406 (2015).\n30. Krizan, J. W., Cava R. J., Phys. Rev. B, 89, 214401 (2014).\n31. Saunders, T. E., Chalker, J. T., Phys. Rev. Lett., 98, 157201 (2007).\n32. Andreanov, A., Chalker, J. T., Saunders, T. E., Sherrington, D., Phys. Rev. B, 81, 014406 (2010).\n33. Silverstein, H. J., Fritsch, K., Flicker, F., Hallas, A. M., Gardner, J. S., Qiu, Y., Ehlers, G., Savici, A.\nT., Yamani, Z., Ross, K. A., Gaulin, B. D., Gingras, M. J. P., Paddison, J. A. M., Foyetsova, K., Valenti, \nR., Hawthorne, F., Wiebe, C. R., Zhou, H. D., Phys. Rev. B, 89, 054433 (2014).\n9\nTable 1. Refined crystal structure of TmGaMgO 4 at ambient temperature, space group R-3m (no. \n166), unit cell parameters a = 3.4195(3) Å and c = 25.1231(12) Å, from the ambient temperature \npowder X-ray diffraction data. Rf factor = 17.1, Rp = 10.1, Rwp = 14.7,  χ2 = 13.7\nAtomWyckoff site xyzOcc.\nTm6c000.0078(2) 0.5\nMg6c000.2147(2) 0.5\nGa6c000.2147(2) 0.5\nO16c000.2836(4) 1\nO26c000.1368(4) 1\n10\nFigure Captions\nFigure 1. Powder Rietveld refinement of TmMgGaO 4. Inset: A small, colorless transparent single \ncrystal of TmMgGaO 4 on a 1-mm grid.\nFigure 2. The crystal structure of TmMgGaO 4 showing the coordination polyhedra. TmO 6 octahedra \nare blue, GaO5 or MgO5 triangular bipyramids are orange.  On the right is the triangular magnetic lattice\nformed by the Tm layers, as viewed along the c-axis. In the bottom right is a closer look at the distorted\nnature of the TmO6 octahedra.\nFigure 3. Left panel: The temperature-dependent DC magnetic susceptibility and reciprocal \nsusceptibility of a polycrystalline powder of TmMgGaO 4 in an applied field of 1000 Oe. Curie-Weiss \nfits are shown in black. Right panel: The field-dependent magnetization at 2 K. Inset: Magnified view \nof the low-temperature region of the inverse susceptibility.\nFigure 4.  Magnetization vs applied field measurements on a single crystal of TmMgGaO4 parallel to \nthe c-axis, perpendicular to the c-axis, and on a powder sample of the same material. Inset: Derivative \nof magnetization parallel to the c-axis with respect to applied field.\nFigure 5. Left panel: The temperature-dependent DC susceptibility of a single crystal of TmMgGaO 4 \nas measured parallel to the c-axis (red) and perpendicular to the c-axis (blue). In purple is a weighted \naverage of the directional susceptibilities. The measured susceptibility of the polycrystalline powder is \nsuperimposed in green. Right panel: The reciprocal susceptibilities of the single crystal and \npolycrystalline powder.\n11\nFigure 1.\n12\nFigure 2.\n13\nFigure 3\n14\nFigure 4. \n15\nFigure 5\n16\nSupplementary Information\nExperimental\nThe  crystal  structure  of  TmMgGaO4  was  further  determined  by  single-crystal  X-ray  diffraction\n(SXRD), especially to analyze the positional disorder of Tm and to evaluate the Ga/Mg split position.\nThe SXRD data was collected at 296 K with a Kappa APEX DUO diffractometer equipped with a CCD\ndetector (Bruker) using graphite-monochromatized Mo- Kα radiation (λ = 0.71073 ) Å. The raw data was\ncorrected for background, polarization, and the Lorentz factor using APEX2 softwarei, and multi-scan\nabsorption correction was applied.ii The structure was solved with the charge flipping methodiii and\nsubsequent difference Fourier analyses with Jana2006. iv,  v,  vi Structure refinement against  Fo2 was\nperformed with Shelxl-2017/1.vii, viii \nThe initial structure model with anisotropic  refined thermal  parameters  for Tm revealed a broad\nelectron distribution on the Tm1 position (Wyckoff site 3 a, [0 0 0]) along the z-axis. Thus, a new site\n(Wyckoff site 6c) was introduced to describe the statistic positional disorder of the rare earth atom,\nwhich could be refined to [0, 0, 0.00528(3)] using an isotropic thermal parameter and 50 % occupancy\nfor Tm. The new coordinates were taken as start position for the powder Rietveld refinement . As the\nrefinement of the anisotropic thermal parameters Uij[Tm1] results in a shift of the coordinates back to\n[0 0 0], the high symmetric site 3 a with a large displacement parameter along z was used to describe\nthe positional disorder of Tm in the final single-crystal structure model.  Ga and Mg share a split\nposition with 50% occupancy each. Due to correlation of parameter values  the displacement parameters\nand coordinates for Ga and Mg were restricted be the same.  \nThe examination of the Fourier ( Fobs) map (Figure S1) revealed significant elongated electron density\nmaxima on the Tm1 position along z, clearly indicating the positional disorder of the rare earth metal,\nand a less distinct but still prolate electron distribution on the Ga1/Mg1 split position along z as well,\nwhich also points towards a positional disorder in addition to the mixed occupancy. Crystallographic\ndata are summarized in Table S1, final atomic parameters are listed in Table S2 and S3.\n17\nTable S1. Crystallographic data and details of the structure determination of TmMgGaO 4 derived from\nsingle-crystal experiments measured at 296(1) K .\nSum Formula TmMgGaO4\nFormula weight / (g · mol–1) 326.96\nCrystal System trigonal\nSpace group R´3m (no. 166)\nFormula units per cell, Z 3\nLattice parameter a / Å 3.4250(6)\nc / Å 25.169(4)\nCell volume / ( Å3) 255.7(1)\nCalculated density / (g · cm–3) 6.370\nRadiation  (Mo-Kα) λ= 0.71073 Å\nData range2θ ≤ 82.23°\n–6 ≤ h ≤ 6\n–6 ≤ k ≤ 6\n–45 ≤ l ≤ 44\nAbsorption coefficient / mm–133.79\nMeasured reflections 3119\nIndependent reflections 262\nReflections with I > 2σ(I) 249\nR(int) 0.024\nR(Fo2) 0.012\nNo. of parameters 10\nR1(obs) 0.010\nR1(all Fo) 0.012\nwR2(all Fo) 0.022\nResidual electron density / (e · Å –3) 0.67 to –0.74\nTable  S2.  Wyckoff  positions,  coordinates,  occupancies ,  equivalent  and  isotropic  displacement\nparameters respectively for  TmMgGaO 4 single-crystal measured at 296(1) K. The coordinates of G1\nand Mg1 were equalised;  Ueq is one third of the trace of the orthogonalized Uij tensor. \nAtom Wyck. Site x y z Occupancy Ueq/Uiso\nTm1 3a 0 0 0 1 0.01154(5)\nGa1 6c 0 0 0.21437(2) 0.5 0.00580(6)\nMg1 6c 0 0 0.21437(2) 0.5 0.00580(6)\n18\nO16c000.29101(7) 10.0085(3)\nO26c000.12881(8) 10.0144(3)\nTable S3. Anisotropic displacement parameters for TmMgGaO 4  single-crystal  measured at 296(1) K.\nThe coefficients Uij (/Å2) of the tensor of the anisotropic temperature factor of atoms are defined by\nexp{–2π2[U11h2a*2 + … + 2U23klb*c*]}; Uij[Ga1] and Uij[Mg1] were equalised .\nAtom U11 U22 U33 U12\nTm1 0.00417(4) 0.00417(4) 0.02627(9) 0.00208(2)\nGa1/Mg1 0.00505(8) 0.00505(8) 0.00730(15) 0.00253(4)\nFigure S1. Fourier map (Fobs) for TmMgGaO 4 based on room-temperature data calculated in \nspace group R´3m, map summed up between –0.10 < y < 0.35, contour lines correspond to 10 \ne/Å3. \n19\ni APEX2, Version v2013.10, Bruker AXS Inc., Madison, Wisconsin, USA, 2013.\nii M. Sheldrick, Sadabs: Area-Detector Absorption Correction , Version 2014/5, Bruker AXS Inc., \nMadison, Wisconsin, USA, 2014.\niii G. Oszlányi, A. Sütő, Ab initio structure solution by charge flipping,  Acta Crystallogr. A  60, 134 \n(2004).\niv G. Oszlányi, A. Sütő, The charge flipping algorithm , Acta Crystallogr. A  64, 123 (2008).\nv V . Petřiček, M. Dušek, L. Palatinus, Jana2006, The crystallographic computing system, Institute of \nPhysics, Praha, Czech Republic, (2014).\nvi V . Petricek, M. Dusek, L. Palatinus, Crystallographic Computing System JANA2006: General \nfeatures, Z. Kristallogr. 229, 345 (2014).\nvii G. M. Sheldrick, SHELX2017, Programs for crystal structure determination , Universität Göttingen, \nGermany, 2017.\nviii G. M. Sheldrick, A short history of SHELX, Acta Crystallogr. A  64, 112 (2008).\n"    },    {        "title": "1712.03501v1.Switching_of_magnons_by_electric_and_magnetic_fields_in_multiferroic_borates.pdf",        "content": "arXiv:1712.03501v1  [cond-mat.mtrl-sci]  10 Dec 2017Switching of magnons by electric and magnetic fields in multi ferroic borates\nA. M. Kuzmenko,1D. Szaller,2Th. Kain,2V. Dziom,2L. Weymann,2A. Shuvaev,2Anna Pimenov,2\nA. A. Mukhin,1V. Yu. Ivanov,1I. A. Gudim,3L. N. Bezmaternykh,3,∗and A. Pimenov2\n1Prokhorov General Physics Institute, Russian Academy of Sc iences, 119991 Moscow, Russia\n2Institute of Solid State Physics, Vienna University of Tech nology, 1040 Vienna, Austria\n3L. V. Kirensky Institute of Physics Siberian Branch of RAS, 6 60036 Krasnoyarsk, Russia\n(Dated: July 13, 2018)\nElectric manipulation of magnetic properties is a key probl em of materials research. To fulfil\nthe requirements of modern electronics, these processes mu st be shifted to high frequencies. In\nmultiferroic materials this may be achieved by electric and magnetic control of their fundamental\nexcitations. Here we identify magnetic vibrations in multi ferroic iron-borates which are simultane-\nously sensitive to external electric and magnetic fields. Ne arly 100% modulation of the terahertz\nradiation in an external field is demonstrated for SmFe 3(BO3)4. High sensitivity can be explained\nby a modification of the spin orientation which controls the e xcitation conditions in multiferroic\nborates. These experiments demonstrate the possibility to alter terahertz magnetic properties of\nmaterials independently by external electric and magnetic fields.\nPACS numbers: 75.85.+t, 78.20.Ls, 78.20.Ek, 75.30.Ds\nThe continuous development of electronic devices\ndrives the necessity to obtain an electric control of mag-\nnetic effects [1, 2]. Compared to external magnetic field,\nelectric voltage may be applied to smaller spatial area\nand with much less switching power, thereby improving\nthe performance and increasing the density of integrated\ncomponents. Inrecentyearsasubstantialcontributionto\nachievingelectricmanipulationofmagneticproperties[2]\nhas been realized through the application ofmultiferroics\n(i.e., materials with simultaneous electric and magnetic\nordering)[3–7]. In several multiferroics, the coupling be-\ntweenelectricityandmagnetismisstrongenoughtoallow\na mutual influence of both properties. This magnetoelec-\ntric coupling has been demonstrated to lead to manipu-\nlation of magnetic moments [8–14] and magnetic struc-\nture [15–18] by external electric field. These effects have\nbeen shown to survive up to room temperature [19, 20].\nRecentreviewsofthetopiccanbefoundinRefs.[1,2,21].\nHaving in mind possible applications, the time scale of\nswitching is an important issue. For example, in typical\nferroelectric devices, this time is limited by the speed of\ndomain wall propagation which sensitively depends upon\nthe amplitude of electric field [22] and may be as short\nas few tenths of nanoseconds [23–26]. In multiferroics\nthe problem of fast switching is not fully settled. Due\nto low static electric polarization in spin-driven multi-\nferroics [6, 27] substantial degradation of the switching\ntime has been reported [28]. Extremely short switch-\ning times of electric polarization and of magnetization\ncan be reached using pulsed laser light. Depending on\nthe specific mechanism of the interaction of the light\npulse and the spins, the switching rate may be as short\nas 40fs [29]. Several interesting recent developments in\nthe field of light-matter interaction include spin modula-\ntion via thermalisation processes [30], pumping the en-\nergy into the electronic transitions [31], using magneticcomponent of a terahertz pulse [32], or directly exciting\nthe magneto-electric excitation in a multiferroic mate-\nrial [33]. Detailed discussion of the experiment and the-\nory of the short-time optical manipulation of magnetism\nis given in Refs. [29, 34, 35].\nBesides the electric modification of static magnetic\nstructures, a control of the high-frequency properties is\nof substantial interest [36]. To accomplish this control\nin the practice, the dynamic processes, which are sensi-\ntive to the influence of the static electric field, have to\nbe identified. Especially for terahertz light, the multifer-\nroics are promising as they possess magnetoelectric exci-\ntationsallowingthe combinationofelectric and magnetic\nfields. These excitations are called electromagnons [37–\n39] and an external magnetic field may easily control\nthem. However, until now, only a few experiments could\ndemonstrate the electric control of excitations in multi-\nferroics[40,41]. Similartostaticexperiments,thecontrol\nhere is achieved by modifying the electric domain struc-\nture with the gate voltage. In addition, ferromagnetic\nresonance in ferromagnetic thin films has been demon-\nstrated to be sensitive to static voltage [42–45]. The\nmechanism of the last effect is generally attributed to\nthe voltage control of the magnetic anisotropy. In this\nwork, we utilize another route to electric control of dy-\nnamic magnetic properties based on an influence of elec-\ntric and magnetic fields on the spin orientation which de-\ntermines the excitation conditions of fundamental mag-\nnetic modes.\nRare-earth iron-borates represent one exotic class of\nmultiferroics [46–48]. At high temperatures, all rare-\nearth borates reveal a non-centrosymmetric trigonal\nstructurebelongingto the spacegroupR32[49–52] which\npersistdowntolowesttemperaturesforSm-andNd-iron-\nborates[53]. The connection between magnetic and elec-\ntric orderingin iron boratesis realizedviathe couplingof2\nelectric polarization to the antiferromagnetically ordered\nspin lattice[54–58].\nWithout losing any generality, we consider\nSmFe3(BO3)4below. As an approximation, the\nmagnetoelectric coupling in iron-borates with easy-\nplane antiferromagnetic order may be written in the\nsymmetry-dictated form [54–56]\nPx∼L2\nx−L2\ny. (1)\nHere,Pxis the electric polarization along the crystallo-\ngraphica-axisand Lx=M1x−M2xandLy=M1y−M2y\narethex,ycomponentsoftheantiferromagneticvectorof\nthe ordered iron moments. Here, the magnetic structure\nis modeled by two antiferromagnetically coupled sublat-\ntices,M1andM2,respectively(boldsymbolsdenotevec-\ntorial quantities). A peculiarity of Eq. (1) is due to the\nfactthat SmFe 3(BO3)4isaneasyplaneantiferromagnet.\nWe note that in high enough magnetic fields the antifer-\nromagnetic vector realigns perpendicular to the field (i.e.\nL⊥H). In agreement with Eq. (1), for H/bardblb-axis one\nobtains [55] Lx/negationslash= 0,Ly= 0,Px>0, and for H/bardbla-axis,\nLx= 0,Ly/negationslash= 0,Px<0. That is, the electric polariza-\ntion rotates by 180◦after a 90◦rotation of the external\nmagnetic field.\nIn zero-field, the magnetic moments of different do-\nmains or regions are distributed approximately homoge-\nneously, as illustrated schematically in Fig. 1(a), thus\naveraging the electric polarization to zero. The mag-\nnetic fields as weak as 0.5 T are enough to break the\nhomogeneous distribution, which leads to a nonzero elec-\ntric polarization [47, 54–56] according to Eq. (1) and\nFigs. 1(c,e). This effect is quadratic in small magnetic\nfields and may be described as a first order magnetoelec-\ntric effect. Due to the symmetry of the magnetoelec-\ntric coupling [59], the opposite effect must be possible\nas well: the magnetization must be sensitive to an ex-\nternal electric field. Indeed, such sensitivity has been\nrecently demonstrated [60, 61] in static experiments for\nSmFe3(BO3)4and for NdFe 3(BO3)4.\nMultiferroic iron borates present a rich collection of\nexcitations in the terahertz range [62–65]. According\nthe optical experiments [66, 67], in the iron borates the\nsplitting of the ground rare-earth doublets are close to\nthe magnon frequencies of the magnetic Fe-subsystem.\nTherefore, not only the static properties of the iron bo-\nratesarestronglyinfluencedbytherare-earth[54–56,58],\nbutalsothemagneticmodesinthesesystemsarestrongly\ncoupled. The last effect is seen experimentally as, e.g.,\na redistribution of the mode intensities and shifts of the\nresonance frequencies [62, 63].\nOur experiments revealed that only coupled Fe-rare-\nearthmodesshowmeasurablesensitivitytostaticelectric\nfields. The strongest effect has been detected for the Sm-\nFe mode around10 cm−1. In case ofSmFe 3(BO3)4other\nmodes [62] may be also expected to reveal voltage sen-\nsitivity. For the low-frequency electromagnon [64, 65]\nFIG. 1:Electric and magnetic ordering in rare-earth iron bo-\nrates.(a) Homogeneous distribution of Fe spins (blue arrows)\nin theab-plane and in the absence of static magnetic ( H) and\nelectric ( E) fields. Different arrows refer to different domains\nin the sample. Both, antiferromagnetic vector Land static\npolarization Pequal zero in this case. (c,d) Either magnetic\nfieldH/bardblb(c) or electric field E⇈a(d)induce P⇈aandL/bardbla.\n(e,f) Rotation of the magnetic field to H/bardbla(e) or inversion\nof the electric field to E↑↓a(f) leads to the inversion of the\nstatic polarization and the rotation of the antiferromagne tic\nvector. Panel(b)showsthecrystalstructureofSmFe 3(BO3)4.\nstrong static magnetic field must be applied to raise\nthe resonance frequency up to the millimeter frequency\nrange. Magnetic field thus would align the Fe moments\n(see Fig. 1) suppressing the voltage effect. The mode\naround 14 cm−1is too weak to reveal observable modu-\nlation. The high-frequency mode of Sm around 16 cm−1\nhas wrong excitation conditions ( h/bardblc-axis) for which it is\nnot sensitive to a rotation of spins in the ab-plane. In\ncase of NdFe 3(BO3)4for the Fe mode around 4 cm−1\nno effect could be observed due to the weakness of this\nexcitation.\nTerahertz transmission experiments were carried out\nusing quasi-optical terahertz spectroscopy [53, 68, 69].\nSingle crystals of SmFe 3(BO3)4and NdFe 3(BO3)4with\ntypical dimensions of ∼1 cm, were grown by crystalliza-\ntion from the melt on seed as described in Ref. [70].\nIn SmFe 3(BO3)4, the coupledFe-Sm antiferromagnetic\nmode around 10 cm−1is of purely magnetic character\nand it may be excited by an acmagnetic field perpen-3\n/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s50/s52/s54\n/s57/s46/s48 /s57/s46/s53/s49/s48/s45/s53/s49/s48/s45/s51/s49/s48/s45/s49/s48/s46/s53/s84\n/s48/s84\n/s32/s48/s84\n/s32/s48/s46/s49/s84\n/s32/s48/s46/s50/s84\n/s32/s48/s46/s51/s84\n/s32/s48/s46/s52/s84\n/s32/s48/s46/s53/s84\n/s32/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s99/s109/s45/s49\n/s41/s72 /s124/s124/s97\n/s32/s84/s114/s97/s110/s115/s109/s105/s116/s116/s97/s110/s99/s101/s32/s97/s109/s112/s108/s105/s116/s117/s100/s101/s97\n/s57/s46/s48 /s57/s46/s53/s48/s46/s52/s84/s32/s48/s84\n/s32/s48/s46/s49/s84\n/s32/s48/s46/s50/s84\n/s32/s48/s46/s51/s84\n/s32/s48/s46/s52/s84/s72 /s124/s124/s98\n/s83/s109/s70/s101\n/s51/s40/s66/s79\n/s51/s41\n/s52\n/s104 /s124/s124/s98/s44/s32 /s101 /s124/s124/s99\n/s49/s49/s46/s55/s75\n/s32/s48/s84\n/s98/s72 /s124/s124/s98/s32\n/s72 /s124/s124/s97/s32/s49/s48/s51\n/s98\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s99\nFIG. 2:Manipulation of magnon excitation by magnetic field\n(a)Suppressionof themagnon at 9cm−1in SmFe 3(BO3)4by\nexternal magnetic fields along the a-axis. (b) Increasing of\nthe mode intensity in magnetic fields H/bardblb-axis. (c) Magnetic\nfield-dependence of the mode intensity from fitting the spec-\ntra in (a) and (b) by a Lorentzian function (blue and orange\nspheres, respectively) and from model calculation based on\nEq. (2) (solid gray lines).\ndicular to the antiferromagnetic L-vector [62, 63]. In the\nnotations of Fig. 1(a) and without external fields the lo-\ncal magnetic moments are homogeneously distributed in\ntheab-plane. This means that an average of 50 % of\nmagnetic moments is excited for any orientation of the\nacmagnetic field in the ab-plane. The situation changes\ndrastically if external magnetic or electric fields within\ntheab-plane are present. As demonstrated in Figs. 1(c-\nf), external fields destroy the homogeneous distribution\nof the magnetic moments in the ab-plane. In the ex-\nperiment, this breaks the balance between the excitation\nconditions with h/bardblaandh/bardblb, respectively, thus shifting\nthe mode intensity to one or the other direction ( hande\nrefer to the oscillating magnetic and electric field of light,\nrespectively).\nThe control of the observed mode intensity by an ex-\nternal magnetic fields is shown in Fig. 2 where panels\n(a,b) demonstrate that the mode strength may be either\nsuppressedor increaseddepending on the direction ofthe\nexternal magnetic field. As the fields above 0.5 Tesla are\nsufficient toorientthe magneticmomentsfully, theinten-\nsity of the mode is either saturated at the doubled value\ncompared to H= 0 case (Fig.2(b)) or it is suppressed to\nzero (Fig.2(a)). As follows from the scheme of Fig. 1 and\nas demonstrated experimentally [47, 55], in both cases\neither positive or negative static electric polarization is\nobserved along the crystallographic a-axis.\nThe coupling of electric polarization with an external\nmagnetic field in multiferroic iron borates provides the\nmain idea how to control the magnetic excitations by\nan electric voltage. By different configurations shown in\nFig. 1, the application of a static voltage along the a-\naxis would favor one of the two possible orientations of\nthe electric polarization. Simultaneously with the static\nmagnetic configurations the excitation conditions for the\nselected coupled Sm-Fe mode are changed which may be\nemployed for electric field control of the dynamic mag-\nnetic properties./s49/s48 /s50/s48 /s51/s48/s48/s53/s49/s48\n/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49\n/s45/s50/s48/s48 /s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48/s48/s50/s52/s32/s32/s32/s40/s99/s109/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s49/s48/s48\n/s84/s32/s61/s32/s49/s49/s32/s75\n/s104 /s73/s73/s98/s44/s32 /s101 /s73/s73/s99\n/s69 /s124/s124/s97/s44/s32 /s72 /s124/s124/s98\n/s32/s69/s61/s48/s44/s32\n/s48/s72/s61/s48\n/s32/s69/s61/s45/s50/s53/s48/s86/s47/s109/s109/s44/s32\n/s48/s72/s61/s48\n/s32/s69/s61/s43/s50/s53/s48/s86/s47/s109/s109/s44\n/s48/s72/s61/s48\n/s32/s69/s61/s43/s50/s53/s48/s86/s47/s109/s109/s44 /s72/s61/s48/s46/s50/s84/s83/s109/s70/s101\n/s51/s40/s66/s79\n/s51/s41\n/s52/s32\n/s32/s84/s114/s97/s110/s115/s109/s105/s116/s116/s97/s110/s99/s101/s32/s97/s109/s112/s108/s105/s116/s117/s100/s101/s97\n/s57/s46/s48 /s57/s46/s53 /s49/s48/s46/s48/s56/s46/s53/s57/s46/s48\n/s32/s80/s104/s97/s115/s101/s32/s115/s104/s105/s102/s116\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s99/s109/s45/s49\n/s41/s98\n/s84/s114/s40 /s69 /s41/s32/s47/s32/s84/s114/s40/s48/s41/s99\n/s84/s32/s61/s32/s49/s49/s32/s75\n/s48/s72/s61/s48\n/s48/s72\n/s97/s61/s48/s46/s50/s32/s84/s32\n/s49/s48/s51\n/s98/s32\n/s69/s108/s101/s99/s116/s114/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s86/s47/s109/s109/s41/s100\n/s48/s72\n/s98/s61/s48/s46/s50/s32/s84/s32\nFIG. 3: Manipulation of the magnon in SmFe 3(BO3)4by\nstatic electric field. (a) Transmittance amplitude and (b)\nphase shift spectra in electric field. Symbols: green - ini-\ntial state, blue - negative electric field, and red - positive\nelectric field. The black line demonstrates that an external\nmagnetic field can approximately compensate the effect of the\nelectric field. (c) Direct modulation of the transmittance a m-\nplitude signal by a static electric field at different frequen cies.\nThe inset shows the temperature dependence of the resonance\nfrequency as observed (dark gray line) and calculated (ligh t\ngray line) in Ref. [62]. The temperatures and frequencies of\nthe transmittance amplitude measurements are marked in the\ninset by circles. (d) Changes in magnon contribution in the\nelectric field. Symbols are experimental results while the s olid\nlines come from model calculation based on Eq. 2. The orange\nand blue symbols correspond to a simultaneous application o f\nelectric and magnetic fields.\nThe basic results on electric field control of the mag-\nnetic excitation in SmFe 3(BO3)4are shown in Fig. 3. In\naddition to the magnetic field dependence presented in\nFig. 2, close to the resonance position of about 9.5 cm−1\nwe observe strong dependence both of the transmittance\namplitude and of the phase shift in the electric fields of\n∼2.5 kV/cm. Particularly in the case of transmittance\namplitude we observe more than one order of magnitude\nchanges in the terahertz signal as influenced by the elec-\ntric field.\nIn spite of the large spectral changes close to the res-\nonance frequency, far from the resonance we observe\nno measurable changes in the signal. This is due to\nthe fact, that the contribution of the present magnetic\nmode, shown in Fig. 3(d), is small as compared to unity,\nthe relative magnetic permeability of vacuum. In the\nscale of Fig. 3(b), the changes of the optical length of\nthe sample far below the resonance can be estimated\nas ∆l∼1·10−2mm, which is below the sensitivity\nof the setup. On the other hand, the ∆ µ= 3.4·10−3\ncontribution of the resonance under study and the in-\ncrease of ∆ µin higher fields agrees with the behaviour\nof the static susceptibility [71]. The electric field modu-4\n/s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s46/s48/s49/s48/s46/s49/s49\n/s49/s56 /s50/s50/s49/s46/s48/s49/s46/s49\n/s45/s50/s48/s48 /s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48/s48/s46/s57/s57/s49/s46/s48/s48/s49/s46/s48/s49\n/s48 /s49/s48 /s50/s48 /s51/s48/s48/s53/s49/s48\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s49/s48/s46/s56/s32/s99/s109/s45/s49/s49/s48/s32/s99/s109/s45/s49/s56/s46/s57/s32/s99/s109/s45/s49\n/s57/s46/s51/s32/s99/s109/s45/s49/s56/s46/s49/s32/s99/s109/s45/s49/s55/s46/s51/s32/s99/s109/s45/s49\n/s32/s84/s114/s97/s110/s115/s109/s105/s116/s116/s97/s110/s99/s101/s32/s97/s109/s112/s108/s105/s116/s117/s100/s101/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s78/s100/s70/s101\n/s51/s40/s66/s79\n/s52/s41\n/s51\n/s104 /s73/s73/s98/s44/s32 /s101 /s73/s73/s99/s44/s32 /s69 /s73/s73/s97/s32\n/s97/s84/s32/s40/s75/s41/s56/s46/s57/s32/s99/s109/s45/s49\n/s56/s46/s49/s32/s99/s109/s45/s49\n/s55/s46/s51/s32/s99/s109/s45/s49\n/s84/s114/s40/s84/s41/s47/s84/s114/s40/s109/s105/s110/s41/s98\n/s84/s32/s61/s32/s49/s51/s75\n/s32/s84/s114/s40/s85/s41/s47/s84/s114/s40/s48/s41\n/s69/s108/s101/s99/s116/s114/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s86/s47/s109/s109/s41/s56/s46/s49/s32/s99/s109/s45/s49/s56/s46/s57/s32/s99/s109/s45/s49\n/s55/s46/s55/s32/s99/s109/s45/s49/s99/s32/s32/s32/s40/s99/s109/s45/s49\n/s41\n/s84/s32/s40/s75/s41\nFIG. 4: Electric field-effect in NdFe 3(BO3)4.(a) Tempera-\nture dependence of the relative transmittance amplitude si g-\nnal for different frequencies. Weak saw-tooth modulation of\nthe curves is due to ±500 V sweeping of electric voltage dur-\ning the cooling process. The inset shows the temperature de-\npendence of the resonance frequency as observed (dark gray\nline) and calculated (light gray line) in Ref. [62]. The tem-\nperatures and frequencies of the resonances seen in (a) are\nmarked in the inset by circles. (b) An example of a detailed\nview of the data in (a). Here the spectra are normalized by\nthe minimal transmitted intensity. (c) Direct modulation o f\nthe terahertz transmittance amplitude at selected frequen cies\nin NdFe 3(BO3)4.\nlation of magnetic susceptibility in the dynamic regime\nd(∆µ)/dE≈4·10−7cm/V is directly connected to the\nstatic magnetic susceptibility via d(∆µ)/dE=dχy/dEx.\nThe static values in SmFe 3(BO3)4were recently mea-\nsured [60] giving dχy/dEx= 2.5·10−8cm/V, which is\nabout an order of magnitude lower in value. The sim-\nplest explanation of this deviation would be to attribute\nthe static result to the sample twinning, which leads to\nthe suppression of the magnetoelectric signal. However,\nother mechanisms, such as domain wall motion, cannot\nbe excluded.\nThe influence of electric and magnetic fields on the\nmagnetic mode contribution ∆ µy∼∆µ0/angbracketleftLx/angbracketright2is deter-\nmined by the square of the x-component of the antifer-\nromagnetic moment /angbracketleftLx/angbracketright2, averaged over the sample. To\nclarify this effect in more detail, we analyzed the actual\npart of the Landau free energy corresponding to the vec-\ntorL= (cosϕ,sinϕ,0) in the xy-plane[56] (vector com-\nponents are considered in the x,y,zCartesian basis):\nΦ(ϕ,E,H) =1\n6K6cos6ϕ−1\n2K1ucos2ϕ−1\n2K2usin2ϕ\n−1\n2χ⊥H2sin2(ϕ−ϕH)−P⊥(Excos2ϕ−Eysin2ϕ).(2)\nHere the first term representsthe crystallographichexag-\nonal anisotropy energy, while the second and third terms\nstand for the magnetoelasticanisotropy K1u∼σxx−σyy,K2u∼σxy, which are induced by the internal elas-\ntic stress of compression/elongation ( σxx−σyy) and\nσxyin theab-plane of a real crystal. The fourth term\ndetermines the Zeeman energy due to the canting of\nthe antiferromagnetic structure in the magnetic field\nH=H(cosϕH,sinϕH,0) and results in ϕ=ϕH±π/2\nwhen the magnitude of magnetic field dominates the ab-\nplane anisotropy and the effect of electric field. The last\nterm of Eq. (2) accounts for the magnetoelectric cou-\npling, i.e., the interaction of the spontaneous polariza-\ntionP= (P⊥cos2ϕ,−P⊥sin2ϕ,0) with external electric\nfields. The amplitude of the spontaneous polarization is\ndeterminedbythemagnetoelectriccouplingconstantand\nthe electricsusceptibility as describedin Ref. [53, 56]. By\nminimizing the free energy and taking into account that\nthe crystallographic hexagonal anisotropy is small[56]\ncompared to other contributions in Eq. (2), one can find\nthe local orientation of the vector Lin theab-plane as a\nfunction of electric and magnetic fields:\ntan2ϕ=2K2u−χ⊥H2sin(2ϕH)−4P⊥Ey\n2K1u−χ⊥H2cos(2ϕH)+4P⊥Ex.(3)\nAssuming random distribution of the magnetoelastic\nanisotropies K1uandK2uobeying a two-dimensional\nGaussian curve we have simulated the behavior of ∆ µy\nin magnetic and electric fields. These results are shown\nin Figs. 2(c) and 3(d) which demonstrate a good de-\nscription of the experiment. The main parameters of\nthe model were taken from Ref. [56] (mean square de-\nviation of the anisotropy ∆ K1u= ∆K2u≈5.5×\n103erg/cm3and the transverse magnetic susceptibil-\nityχ⊥= 1.2×10−4cm3/g), while the maximal value\nof the spontaneous electric polarization was taken as\nP⊥≈ −240µC/m2. Thisvalueisslightlylowerthanthat\nobserved in Refs. [55, 56] likely due to a larger amount\nof crystallographic inversion twins in the enantiomorph\ncrystal. Remarkably, according to Eq. (3), the simulta-\nneous application of both EandHcould lead to a com-\npensation of their action as a result of an interrelation\nbetweenthem. Forexample,for E/bardblaandH/bardblbthecom-\npensation effect occurs according to χ⊥H2+4P⊥Ex= 0,\nwhich is in a good agreement with our measurements for\nE= +250 V/mm and µ0Hb= 0.2 T (Figs. 3(a,b,d)).\nFigure 4 shows typical results of electric field experi-\nments in NdFe 3(BO3)4. Panel (a) demonstrates the tem-\nperature dependence of the transmittance amplitude at\nselected frequencies. Characteristicminima in these data\ncorrespond to a crossing of the temperature-dependent\nresonance frequency of the mode and the frequency of\nthe experiment, as shown in the inset. These mea-\nsurements were obtained with cooling at 1 K/min and\nthe simultaneous sweeping of the gate voltage between\n-500 V and +500 V at a rate of ∼0.1 Hz. The char-\nacteristic saw-tooth profile of these curves demonstrate\nthe nonzero effect of the electric field on this mag-\nnetic mode in NdFe 3(BO3)4. From the slopes shown5\nin Fig. 4(c) the field-dependent susceptibility may be\nestimated as dχy/dEx= 2.4·10−8cm/V, which is an\norder of magnitude smaller than the same values from\nSmFe3(BO3)4. This difference is due to a small value\nof the spontaneous electric polarization and to larger\nthreshold magnetic field to suppress the spiral magnetic\nstructure in NdFe 3(BO3)4(∼1 T compared to ∼0.3 T\nfor SmFe 3(BO3)4)[55, 72].\nThe reaction time of the present experimental setup\ncan be estimated as ∼45 ms. Within this time scale an\ninstantaneous response ofthe the magnetic system to the\nchangesofelectricfieldhavebeen observed. Basedonthe\nacresultsgivenin Ref. [60] the switchingtimes ofat most\n1 ms may be expected. As mentioned in the introduc-\ntion, in case of domain wall motion the switching time\nof the devices are limited by tenths of nanoseconds. In\nmagnetoelectric ferroborates the process includes both,\nrotation of the magnetic moments and switching of the\nelectric polarization. The characteristic time scale for\nthe magnetic part is determined by the in-plane antifer-\nromagnetic resonance frequency ( ∼5GHz at H= 0)[64],\nwhich will probably determine the switching rate. Fi-\nnally, for short pulses, electric and magnetic fields are\npresent simultaneously. This mixing may influence the\nswitching on the short time scales.\nIn conclusion, magnetic modes in multiferroic ferrob-\norates are shown to be sensitive to both, external mag-\nnetic field and static voltage. Nearly 100% modulation\nof the terahertz radiation in an external electric field is\ndemonstrated for SmFe 3(BO3)4. The experimental re-\nsults can be well explained using a theoretical model\nwhich includes the magnetoelectric coupling in multifer-\nroic borates. High sensitivity to electric voltage is due\nto a strong effect of both magnetic and electric fields on\nthe spin orientation in an easy plane antiferromagnetic\nstructure and significant coupling of the rare-earth and\nthe iron magnetic subsystems.\nAcknowledgements\nThis workwassupported by the Russian Science Foun-\ndation (16-12-10531: AMK, VYuI and AAM), by the\nRussianFoundationforBasicResearch17-52-45091IND-\na : IAG and LNB), and by the Austrian Science Funds\n(W1243, I 2816-N27, I 1648-N27).\nSupplementary Information\nMultiferroic borates\nRare-earth iron-borates represent one exotic class of\nmultiferroics [46–48]. At high temperatures, all rare-\nearth borates reveal a non-centrosymmetric trigonal\nstructurebelongingtothe spacegroupR32[49, 50]whichpersist down to lowest temperatures for compounds with\nlarge ionic radius of the rare-earth (La-Sm). In iron bo-\nrates with smaller ionic radius (Eu-Er, Y) a phase tran-\nsition to the structure within P3 121 space group takes\nplace for lower temperatures [51, 52].\nIn multiferroic iron borates the coupling of the param-\nagnetic rare earth (R) and antiferromagnetically ordered\nFe-moments arises due to the exchange interaction and it\nresults in an induced antiferromagnetic order in the Sm\nsubsystem. For the easy-plane ground state of Fe-spins\nin SmFe 3(BO3)4the orientation of the Sm magnetic mo-\nments also occurs in the easy ab-plane and the Sm-order\ntakes place at the N´ eel temperature of the Fe-subsystem\n∼30 K. Strictly speaking, this is the ordering temper-\nature of both, Fe- and Sm- subsystems. However, since\nthe Fe-Fe exchange interaction is much stronger than the\nSm-Fe, it is reasonable to consider the Fe ordering as a\nprimary order parameter.\nThe role of anisotropy of the R-subsystem is very im-\nportant in determining the orientation of the iron spins.\nIn the case of SmFe 3(BO3)4the ground doublet of Sm3+\nissplitbytheSm-Feexchangeinteractionthusstabilizing\nthe easy-plane state [62]. In case of NdFe 3(BO3)4the\ncollinear easy-plane state below the N´ eel temperature\nis transformed into the spiral easy-plane state around\n13-15K. The origin of this transition still remains un-\nclear. In several other iron borates like GdFe 3(BO3)4\nand HoFe 3(BO3)4the spin reorientation from the easy\nplane to the easy axis state exists due to competitions of\nthe Fe- and R-subsystem anisotropies [46–48].\nTerahertz spectroscopy\nTerahertz transmission experiments were carried out\nusing quasi-optical terahertz spectroscopy [68]. This\ntechnique utilizes linearly polarized monochromatic radi-\nation provided by backward-wave-oscillators. He-cooled\nbolometerswereused asdetectorsofthe radiation. Using\nwire grid polarizers, the complex transmission coefficient\ncan be obtained both in parallel and crossed polarizers\ngeometry. The phase information was obtained by com-\nparing the mirror positions necessary to reach an inter-\nference minimum between the two arms of our Mach-\nZender interferometer for the sample and for the refer-\nence aperture. Static magnetic fields, up to ±7 Tesla,\nhave been applied to the sample using a split-coil super-\nconducting magnet. Frequency dependent transmission\nspectra were analyzed using the Fresnel optical formu-\nlas for the transmittance of a plane-parallel sample [69]\nassuming a Lorentzian form of the magnetic excitations\nµ(ω) = 1+∆ µω2\n0/(ω2\n0−ω2−iωg). Here ∆ µis the mode\ncontribution, ω0isthe resonancefrequency, gisthe mode\nwidth, and ωis the angular frequency of the experiment.6\nMagnetic interactions\nThe magnetic part of the thermodynamic potential in\nEq.(2)ofthemaintextisderivedfromEq.(2)ofRef.[56]\nfor the special case of magnetic moments restricted to\nthexymagnetic easy-plane. Considering the antiferro-\nmagnetic vector as L= (cosϕ,sinϕ,0), and the external\nmagnetic field as H=H(cosϕH,sinϕH,0), both lying\nin thexy-plane, Eq. (2) of Ref. [56] takes the form\nΦm(ϕ,H) =1\n6K6cos6ϕ−1\n2K1ucos2ϕ−1\n2K2usin2ϕ\n−1\n2χ⊥H2sin2(ϕ−ϕH), (4)\nwhich is identical to the magnetic part of Eq. (2) of the\nmain text.\nMagnetoelectric coupling\nThe magnetoelectric term of Eq. (2) in the main text\nis a simplified form of the more general expressions of\nRef. [56]. In Ref. [56] the Eqs. (3) and (4) give the mag-\nnetoelectric and electric part of the thermodynamic po-\ntential, respectively, as\nΦme(ϕ,P)+Φe(P,E) =−c2(Pxcos2ϕ−Pysin2ϕ)\n+P2\nx+P2\ny\n2χe\n⊥−PE. (5)\nThe first term of the right-hand side represents the mag-\nnetoelectric coupling between the magnetic order, i.e. ϕ,\nand thePelectric polarization. Here c2is the magneto-\nelectric coupling constant, while in the electric term χe\n⊥\ndenotes the electric susceptibility of the crystal in the\nxy-plane and Eis the external electric field. For a given\nmagnetic orderin the xy-plane, i.e. for a given ϕ, the po-\nlarizationminimizing the thermodynamicpotential reads\nas [56]\nPx=P⊥cos2ϕ+χe\n⊥Ex (6)\nPy=−P⊥sin2ϕ+χe\n⊥Ey, (7)\nwhere the amplitude of the spontaneous polarization is\nP⊥=c2χe\n⊥. Omitting the terms from Eq. (5) which are\nindependent of the spin configuration, i.e. independent\nofϕ, we arrive at the −P⊥(Excos2ϕ−Eysin2ϕ) mag-\nnetoelectric term of Eq. (2).\n∗deceased\n[1] C. A. F. Vaz, J. 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Mukherjee * \nSchool of Basic Science s, Indian Institute of Technology Mandi, Mand i Himachal Pradesh \nIndia 175005  \n*Email:kaustav@iitmandi.ac.in  \n \nABSTRACT  \nWe report  the effect  of partial su bstitution of Dy -site by rare-earths  (R=Gd, Er and \nLa)on the magnetic and magnetocaloric behavior of a mixed metal oxide  \nDyFe 0.5Cr0.5O3.Structural studies reveal  that substitution of Dy by R has a minimal influence \non the crystal structure . Magnetic and heat capacity studies show that the magnetic transition \naround 121 K observed for DyFe 0.5Cr0.5O3 remains unchanged with rare -earth substitution, \nwhereas the lower magnetic transition temperature is suppressed/enhanced by magnetic/non -\nmagnetic substitution. In all these compounds , the second order nature of magnetic transition \nis confirmed by Arrot t plots.  As compared to DyFe 0.5Cr0.5O3, the values of magnetic entropy \nchange  and relative cooling power  are increased  with magnetic rare-earth substitution  while it \ndecreases with non -magnetic rare -earth substitution . In all these compounds, m agnetic \nentropy change follows the power law dependence of magnetic field and the value of the \nexponent n indic ate the presence of ferromagnetic correlation in an antiferromagnetic  state. A \nphenomenological universal master curve is also constructed for all the compounds by  \nnormalizing the entropy change with rescaled temperature  using a single reference \ntemperatur e. This master  curve also reiterate s the second order nature of the magnetic phase \ntransition in such mixed metal oxides .  \n  2 \n 1. Introduction  \n         Investigation of materials showing large magnetocaloric  effect (MCE) has been an \nimportant area of research for its potential application in magnetic refrigeration technology \n[1-4]. Magnetic refrigeration provides a highly efficient and environment friendly cooling in \ncomparison to conventional gas compression /expansion techniques [ 3, 4]. Generally , in two \ntemperature  (T) regime s this technology is important: near the room T where  it can be used \nfor domestic and industrial refrigeration, and also, in the low T where it can be useful for \nsome specific technological applications in liquefaction of hydrogen in fuel industry and \nspace science [ 1-3, 5 and 6 ]. For a material to be a good solid refrigerant it should have a  \nhigh density of magnetic moments and a strong T and field dependence of the magnetization \nfor the occurrence of a magnetic  phase transformation around the working T. Additionally, \nthe material should have insignificant m agnetic hysteresis to  avoid energy losses during the \nmagnetization/demagnet ization cycles  [3, 4].Also the material should have large magnetic \nentropy change (∆ SM) and lar ge relative cooling power (RCP) . Apart for technological \napplications, from the viewpoint of basic physics, investigation of magnetocaloric  parameters \nof magnetic materials is interesting and important as one can acquire insight about complex \nmagnetic phases present in the system, which may not be possible by just studying \nmagnetization. For example, a detailed analysis of the field dependenc e of magnetocaloric \neffect can provide useful information about the performance of a refrigerant for magnetic \nfield ranges used in actual refrigeration cycles. Beside this, such a study can also be helpful to \nget deeper understanding of the nature of magne tic phase transitions and phase coexistence in \nthe material.  \n          Many oxides containing rare -earths and transition metals have been found to be \nshowing  good MCE [ 4]. In this context rare -earth transition metal perovskites  of the form \nABO 3 (where A is the rare -earth ion and B is the transition metal ion) which show magnetic \ntransition at low T are interesting for investigation of low T MCE [ 7-12]. In last few  years , a \nnew family of mixed metal oxides (combining orthoferrites and orthochromi tes) of the form \nRFe 0.5Cr0.5O3, is discovered in perovskites  and are being extensively investigated  [13-18]. \nSuch compounds are important as combining the two transition metals within the perovskite  \nstructure can be an effective approach to enhance the magnetic properties and at the same \ntime tune/induce functional properties as compared to their parent compounds.  For example, \na single compound, DyFe 0.5Cr0.5O3, exhibits both magnetoelectric (ME) coup ling as well as \nMCE [ 13 and16], whereas DyCrO 3 shows only MCE [ 9] while DyFeO 3 shows magnetic field 3 \n induced multiferroicity [ 19]. DyFe 0.5Cr0.5O3 show s complex magnetic nature due to co -\nexistence of several types of order parameters. In such compounds magnetic transition  T and \nhence the MCE can be tuned by substitution at the Dy -site. \n         Hence , in this paper we report an extensive investigation magnetic and magnetocaloric \nproperties of a series of compound s Dy0.8R0.2Fe0.5Cr0.5O3 (R = Gd, Er  and La). DyFe 0.5Cr0.5O3 \nundergoes in three antiferromagnetic  (AFM)  ordering around T~261 K, T1 ~ 121 K and T2 \n~13 K [ 13 and 16]. It is observed that  T1is unaffected, but T2 is changed due to partial \nreplacement of Dy. Due to weakness of magnetic features around 261 K, the effect of partial \nsubstitution on this transition could not be tracked. Hence , in this manuscript we have \nrestricted our study in the T range of 2 to 150 K. Interestingly  a significant MCE is observed \nonly around the T2 in all these compounds.  The MCE and RCP values are \nincreased/decreased with magnetic/nonmagnetic substitution with respect to that observed in \nDyFe 0.5Cr0.5O3. Arrott  plot confirms the second order nature of the magnetic transition in all \nthe compounds and magnetic entropy change follows the power law of the dependence of \nmagnetic field of the form ∆ SM  ~ Hn. The obtained value of  exponent n indicate s the \npresence of ferromagnetic  (FM) correlation in an AFM  state in all these compound s. A \nphenomenological universal curve of all the compounds is created  by normalizing the entropy \nchange with rescaled T. This master  curve also restate s the second order n ature of the \nmagnetic phase transition in such materials.  \n2. Experimental  \n Polycrystalline samples of Dy 0.8Gd0.2Fe0.5Cr0.5O3 (DGFCO), Dy 0.8Er0.2Fe0.5Cr0.5O3 \n(DEFCO) and Dy 0.8La0.2Fe0.5Cr0.5O3 (DLFCO) were prepared by solid state reaction method \nunder the similar c onditions as reported in Ref [ 16]. The DyFe 0.5Cr0.5O3 (DFCO) sa mple is \nthe same as used Ref [ 16]. The structural analysis is carried out by x -ray diffraction (XRD, \nCu Kα) using Rigaku  Smart Lab instruments. The Rietveld refinement of the powder \ndiffraction data is  performed using FullProf Suite software. Temperature  (2 - 300 K) and \nmagnetic field ( H) (up to 50 kOe) dependent magnetization were performed using Magnetic \nProperty Measurement System (MPMS) from Quantum design, USA.  Heat capacity ( C) \nmeasurements in the T range 2 -150 K were performed using the Physical Property \nMeasurements System (P PMS).  \n \n 4 \n 3. Results and discussion  \n Fig. 1 show the XRD patterns of all the compounds recorded at room T.All these \ncompounds  crystallize  in orthorhombic perovskite structure with Pbnm space group . The \ncrystallographic parameters obtained from Rietveld analysis are listed in T able 1. A small \nshifting in peak w ith respect to parent compound is observed  (inset Fig . 1), which confirms \nthe expansion and contraction  of lattice  with R substitution ; implying that the dopant goes to \nthe respective sites .  \n Fig. 2  (a)-(c) shows t he T response of dc magnetic susceptibility ( χ) data taken i n zero \nfield cooling (ZFC) and  field cooled (FC) condition  in the T range from 2 to 150 K  at 100 Oe  \nfor DGFCO, DEFCO and DLFCO compounds.  All these compounds undergo  two distinct \nAFM  ordering as co nfirmed by dχ/dT vs T plots . For all the compounds this curve shows a \nslope change around T1 ~ 121 K(not shown) and a peak at T2 ~ 9, 11 and 18 K  respectively  \n(inset of Fig. 2(b ); curve for DFCO is added from Ref [ 16] for comparison ). DFCO  is \nreported to  undergo  two AFM  ordering around 121and 13 K  [13, 16 ]. It is observed that \npartial replacement of Dy by R ions result i n shifting of the T2 while T1 remains unchanged. It \nhas been reported in DFCO that the magnetic ordering a t T1 is caused  due to the Cr -O-Cr \nordering , while the ordering at T2 is caused due to Dy -O-Fe/Cr magnetic interactions  [13]. To \nconfirm magnetic ordering behaviour of these compounds, heat -capacity is measured as a \nfunction of T (2-150 K)  shown in Fig 2 (d) -(e). For all the compounds, a weak, but a distinct \npeak is observed around T2, but such a peak around T1is not clear in the raw data. In order to \nsee the features more clearly, the T response of d C/dT, is plotted in the inset s of Fig. 2 (e) and \n(f). The derivative curves show a minima  and peak around T2 and T1 respectively. These \nfeatures in C measurements confirm the magnetic ordering temperatures  of the respective \ncompounds. Hence from the above observations , it can be said that  partial replacement of Dy \nby R ions does not affects the magnetic ordering due to Cr.  However , with magnetic dopants \nT1 is suppressed while with non -magnetic dopant it is enhanced. In these compounds , there is \na competition between transition metal sub -lattice moments  (at B-site) and rare earth sub -\nlattice moments  (at A -site). Due to the dominance of rare earth field the ordering of moments \nis observed  at low T. In fact that Gd and Er might be on the threshold of magnetic ordering as \nthe T is lowered to 2 K and the internal magnetic field arising because of this, however small \nit may be, is such that it suppresses AFM  ordering due to Dy. However, when Dy  site is \ndiluted by nonmagnetic La ions, transition metal sub -lattice moments dominate  resulting in \nthe observation  magnetic ordering at higher T.  5 \n To get a better understanding about the magnetic behavior and effect of substitution, \nisothermal M is measured as a function of H (in the range ±50 kOe) at different T (2 to 150 \nK). Fig.  3 (a-c) exhibits the  representative M (H) curves at  selected temperatures  of 2, 40 and \n150 K for DGFCO, DEFCO and DLFCO compounds. For all  these  compounds , a weak \nmagnetic hysteresis is observed below 5 kOe (shown in inset s) and magnetic saturation is \nabsent at high fields . Such behaviour indicates presence of weak FM correlation along with \nAFM coupling in these compounds.  The origin of s uch FM correlation  is not  due to the \nmagnetic field induced metamagnetic transition , as such effect is prevalent only at low \ntemperatures, w hereas FM correlation persist upto room temperature . In fact  the magnetic \nbehavior of such mixed metal oxide is sensitive to the nature of rare earth ions, as observed \nfrom the temperature response of coercive field , which is found to be different for different \ncomposition [16].  With reference to Fig . 2 (b) of Ref [ 16], it is observed that, i n case of \nmagnetic doping (G d and Er), there is an enhancement experienced by the magnetization \nvalue while there is a decrement in magnetization value with the nonmagnetic doping.  \n To identify the nature of magnetic transition , H/M vs.M2 plots were done using the \nvirgin curves of isothermal magnetization.  According to Ba nerjee ’s criterion , this Arrott  plot \nexhibits a  negative and positive slope for first and second order  nature of magnetic  transition  \nRef [ 20].Fig. 4 (a)-(d) shows the H/M vs.M2 plots for all the compounds  at selected \ntemperatures . A positive sl ope is observed around T2 and T1 (insets of Fig. 4 (a)) indicating \nsecond order natu re of the magnetic transition s. \nIn order to see the effect magnetic and non -magnetic R substitution on the Dy site of \nDFCO, magnetocaloric effect  is calculated from the virgin curves of M (H) isotherms  (in the \nT range of 2 -150 K) .  MCE is generally measured in terms of the change in isothermal \nmagnetic entropy ( ∆SM) produced by changes in applied magnetic field. It is to be noted \nhere, that each isotherm is measured after cooling the respective compound  from room  T to \nthe measurement T and ∆SMis calculated using the following expression  [21] \n∆SM=Σ [( Mn- Mn+1)/(Tn+1 -Tn)] ∆Hn  (1) \nwhere  Mn and Mn+1 are the magnetization values measured at field Hn and Hn+1at temperature  \nTn and Tn+1 respectively. Fig.  5 (a) shows the T-dependent ∆SM at 50 kOe  applied field change \nfrom zero field .  An inverse MCE is observed for all the compounds in the T2 region  while  \nMCE is negligible around T1. Hence it can be said that in these compounds the observed \nMCE is due to the magnetic entropy variation arising from strong rare earth and transition 6 \n metal sub -lattice interactions. Also the broadness of the ∆SM peak indicate the second order \nnature of this transition [ 22] which is in analogy with the Arrott plots. Interestingly , it is \nobserved that ∆ SM increases upto ~ 14.6 J/kg-K at 50 kOe for DGFCO as compared to 10.8 \nJ/kg-K for DFCO. Er substitution results in insignificant changes to ∆ SM value  (~10.9 J/kg -K) \nwhile La substitution results in  slightly smaller value ~ 9.3  J/kg-K with respect to DFCO.  \nThese compounds  have a higher  ∆SM at 50 kOe in comparison to other transition metal oxides \nlike TbMnO 3, DyMnO 3 [7], DyCrO3 [ 9, 10] and pure and Fe doped HoCrO 3 [23]. \n          In the magnetocaloric  material research another parameter, namely relative cooling \npower (RCP)  is required to further evaluate a material for the ir suitability in magnetic \nrefrigeration device.   RCP is the measure of the amount of heat transfer between cold and \nhot reservoirs in an ideal refrigeration cycle. The RCP is defined as the product of maximum \n∆SM (∆SMMax) and full width of half maximum  of the peak in  ∆SM (∆TFWHM ), i.e.  \n                                                RCP = ∆ SMmax× ∆TFWHM        ....... (3)   \nHowever, in these mixed metal oxides, ∆ SM vs. T plot is not found symmetric in \nnature. Thus, RCP is calculated by numerically integrated the area under the ∆ SM vs T curve, \nwith T limit Tc (cold end) and T h (hot end) of thermodynamic cycle [ 10]. It is expressed as  \n( , )c\nhT\nM\nTRCP S T H dT  \n   (4) \nThe T response of RCP value for the studied compound s is display ed in Fig 5  (b). The \nobserved RCP values are comparable with another manganite  and chromite system  [7, 8, 10 \nand23]. The RCP values are increased/decreased with magnetic/nonmagnetic substitution . \nTherefore,  mixed metal oxides show good magnetocaloric properties in the cryogenic T \nregion . \nIn the field of magnetocaloric material research  it is essential  to compare  the \nexperimental data for different materials because  all MCE parameters exhibit the applied \nfield dependent behavior and th is parameter  varies  for compounds  to compounds . Therefore,  \nmagnetic field dependent study of ∆SM is necessa ry for a better understanding of  intrinsic  \nnature of MCE  [24]. The field dependent ∆SM is described in terms of power law behavior  \n(∆SM ~Hn), [12, 15, 25 and 26 ] where n is exponent directly related to magnetic state of the \nmaterials . Magnetic field response of ∆ SM fitted well with power law behavior above the \npeak T. For all studied compound, the representative curve of ∆ SM vs H, at selected T, is 7 \n plotted in Figure 6  (a)-(d). Applied field dependent  ∆SM follows the power law behavior \n(∆SM ~Hn) and observed value of the exponent ( n) lies between 1.4  to 1.6 .But for  ideal AFM \nsystem , the value of  n should be 2 [ 26]. This lower value of n <2 observed  in these \ncompounds is due to the presence  of FM correlations in AFM  state and this observation is in \nanalogy to the inference drawn from the M(H) isotherms of these compounds .  \nPower laws and universal scaling have been extensively used to investigate  MCE  in \nmagnetic materials  [15, 25-30]. They are key tools which allow us to compare the \nperforming properties of the materials  regardless of their nature, processing or experimental \nconditions during measurements.  A phenomenological universal curve for the field \ndependence of Δ SM has been pro posed in Ref  [29]. Generally, if a universal curve exists, \nthen the equivalent points of the Δ SM(T) curves measured at differen t applied fields should \nmerge into the single  universal curve.  Universal curve are also plotted for different \ncomposition s of material [ 30]. The MCE data of different materials of same universality \nclass should fall onto the same curve, irrespective of the applied magnetic field.  This curve \nalso identifies the nature of magnetic transition in materials [ 31]. Franco  et al.,  and Bi swas et \nal., have shown  the universal behavior for conventional  [27-29], and inverse MCE [ 26].For \nIMCE, in compounds where symmetrical peak is observed,  rescaling in T axis is not \nrequired . But in these mixed metal oxides  symmetrical peak is not observed and a universal \ncurve is not obtained. However , after rescaling the T axis, a  universal curve is obtained for \nall these compounds . It is to be noted that in these compounds, the observed MCE is \nobserved near the magnetic transition which is o f second order. Generally , in such cases the \nT axis is rescaled to obtain the universal curve  using single reference temperature  [30]. The \nrescaled temperature (θ)  is defined as  [30]: \nθ= (T-Tpk)/(Tr-Tpk)  (2) \nwhere Tpk is peak T at ∆SMmax and Tr is reference T. Tr is selected as the T according to the \nrelation ∆ S(Tr) = ∆ Spkmax/2.Normalize d ∆SM as function of θ at selected fields for all the \ncompound s are shown in Fig.  7 (a)-(d).As observed from the figure, t he ∆ SM curves of the \ncompounds  merges  to a single phenomenological curve which is independent of field. This \nbehavior is distinctly different than that observed for typical  first order phase transition [3 1] \nand reaffirms the  second order nature magnetic phase transition.  Also as observed , ∆SM vs T \nplot display a shifting in peak T with rare earth substitution with changes in ∆SM value s. \nHence ,we tr ied to construct a universal curve for different composition at same field. In 8 \n order to check whether this universal curve  holds for different composition of series  the \nrescaled curve s of all the compounds is plotted  for maximum applied field  (50 kOe) . Inset  of \nFig. 7(a)  show that the curves for all the compound s merge  on single master curve of the \nmagnetic entropy change . Thus, this series of compound follows the universal curve of ∆ SM \nwhich is calculated using a single reference temperature.  \n4. Conclusion  \nIn summary, a detailed investigation of magnetic and magnetocaloric properties of \nrare-earth substituted  mixed metal based oxide, Dy 0.8R0.2Fe0.5Cr0.5O3(R = Dy, Er and La) is \ncarried out . Our results reveal that the magnetic transition around 121 K is unaffected by rare -\nearth substitution, whereas the lower magnetic transition T is suppressed/enhanced by \nmagn etic/non -magnetic substitution due to  modification of Dy/R -O-Fe/Cr magnetic exchange \ninteraction. Arrot t plot confirm s the second order nature of the  magnetic transition in all these \ncompounds . The ∆SM and RCP values are increased/decreased  with magnetic/nonmagnetic \nsubstitution.  Magnetic entropy change follows the power law dependence of magnetic field \nand the value of the exponent n indicate the presence of ferromagnetic correlation in an AFM \nstate in all these compound. A phenomenological universal master curve of all the \ncompounds is constructed by normalizing the entropy change with θ. This universal curve \nalso reaffirm s the secon d order nature of the magnetic phase transition in such materials. \nThus based on the above observations , it can be said that the mixed metal oxides show good \nmagnetocaloric properties in the cryogenic T region.  \nAcknowledgement  \n        The authors acknowledge IIT, Mandi for financial support. The experimental facilities of \nAdvan ced Materials Research Center (AMRC), IIT Mandi are also being acknowledged.  \nReferences  \n[1] V. K. Pecharsky  and K. A. Gschneidner Jr, J. Magn. Magn. Mater. 200, (1999) 44-56. \n[2] V. Provenzano, J. Li, T. King, E. Canavan, P. Shirron, M. DiPirro, and R.D. Shull, J. \nMagn. Magn. Mater. 266, (2003) 185.  \n[3] K. A. Gschneidner Jr, V. K. Pecharsky and A. O. Tsokol, Rep. Prog. Phys. 68, (2005) \n1479  and references therein . \n[4] M.-H. Phan  and S. -C. Yu, J. Magn. Magn. Mater. 308, (2007) 325.  9 \n [5] B. F. Yu, Q. Gao, B. Zhang, X. Z. Meng, and Z. Chen, Int. J. 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Phys.  115, (2014 ) \n17D728 . \n[15] Mohit K. Sharma, Karan Singh and K. Mukherjee, J. Magn. Magn. Mater.  414, (2016) \n116. \n[16] Mohit K. Sharma, Tathamay Basu, K. Mukherjee and E. V. Sampathkumaran, J. Phys.: \nCondens. Matter  28, (2016) 426003.  \n[17] Mohit K. Sharma, Tathamay Basu, K. Mukherjee  and E. V. Sampathkumaran, J. Phys.: \nCondens. Matter  29, (2017) 085801 . \n[18] Fernando Pomiro, Rodolfo D. S´ anchez, Gabriel Cuello, Antoine Maignan, Christine \nMartin, and Raul E. Carbonio Phys. Rev B  94, (2016) 134402.  \n[19] Y. Tokunaga, S. Iguchi, T. Arima, and Y. Tokura, Phys. Rev. Lett.  101, (2008) 097205.  \n[20] S. K. Banerjee, Phys. Lett.  12, (1964) 16.  \n[21] M. Foldeaki, R. Chahine and T . K. Bose J. Appl. Phys.  77, (1995)  3528.  \n[22] J. Lyubina, J. Phys. D: Appl. Phys.  50, (2017) 053002.  \n[23] Shiqi Yin, Vinit Sharma, Austin McDannald , Fernando A. Reboredob and Menka Jain \nRSC Adv.  6, (2016) 9475.  \n[24] S. Fahler, U. K. Rossler, O. Kastner, J. Eckert, G. Eggeler, E. Quandt and K Albe, Adv. \nEng. Mater.  14, (2012) 10.  \n[25] R. M’nassri, Bull. Mater. Sci  39, 2, (2016) 551 -557pp.  10 \n [26] A. Bis was, S. Chandra,  T. Samanta, M. H. Phan, I. Das, and H. Srikanth, J. Appl. Phys.  \n113, (2013) 17A902.  \n[27] S. Chandra, A. Bis was, S. Datta, B. Ghosh, V. Siriguri, A. K. Raychaudhuri, M. H. \nPhan, and H. Srikanth, J. Phys.: Condens. Matter  24, (2012) 366004.  \n[28] V. Franco and A. Conde, Int. J. Refrigeration 33, (2010) 465.  \n[29] V. Franco, J. S. Blazquez, and A. Conde, Appl. Phys. Lett.  89, (2006) 222512.  \n[30] V. Franco, J. S. Blázquez, and A. Conde J. Appl. Phys.  103, (2008) 07B316.  \n[31] C. M. Bonila, J. H. Albillos, F. Bartolome, L. M. Garcia, M. P. Borderias, V. Franco, \nPhys. Rev. B  81, (2010) 224424.  \n  11 \n Table 1: Lattice parameters for the compounds obtained from Rietveld  refinem ent of XRD \ndata. The parameter for DFCO is added fr om Ref [16 ] for comparison.  \nParameters  DFCO  DGFCO  DEFCO  DLFCO  \na (Å ) 5.2860 (1)  5.2946 (1)  5.2781 (0)  5.3369 (1)  \nb (Å) 5.5606 (1)  5.5594 (1)  5.5573 (1)  5.5539 (1)  \nc (Å) 7.5902 (2)  7.5982 (1)  7.5835 (1)  7.6395  (2) \nV (Å3) 223.02 (8)  223.65(8)  222.44 (9)  226.44 (1)  \nχ2 1.88 2.46 1.987  2.6 \n \n \n  12 \n Figures  \n \n \nFig. 1: Room temperature  of X -ray diffraction pat terns for DGFCO, DEFCO and DLFCO \ncompounds . Inset: s hows the expanded XRD patte rn of all the compounds.  Inset shows the \npattern in an expanded form for one  peak, to bring out that the peaks shift with  substitution.  \nThe black curve is for DFCO from [ 16]. \n \n \n \n \n \n \n \n \n13 \n  \nFig. 2: Left Panel  1: Temperature  (T) response of dc magnetic susceptibility ( χ = M/H) \nobtained  under zero -field-cooled (ZFC) and  field cooled (FC) condition at 100 Oe  for (a) \nDGFCO,  (b) DEFCO and (c) DLFCO . Inset of  Fig. (b) : dχ/dT plotted as a function of Tin the \nT range 2 to 30 K.  The black curve is for DFCO from [ 16]. Right panel: T response of heat \ncapacity (C vs T) for (a) DGFCO,  (b) DEFCO and (c) DLFCO . Inset of  Fig. ( e): dC/dT vs T \nplot in the T range 6 - 22 K. Inset of  Fig. ( f):dC/dT vs T plot in the T range 115 - 130 K.  The \nblack curve s in both the inset is for DFCO from [1 6]. \n \n14 \n  \nFig. 3: Isothermal m agnetization ( M) plot as function of applied field ( H) at 2, 40 and 150 K, \nfor (a) DGFCO,  (b) DEFCO and (c) DLFCO . Inset s: M (H) curves expanded at the low field  \nregion  for the respective compounds.  \n \n \n \n \n \n \n15 \n  \nFig. 4: Arrot t plot (H/Mvs.M2) for (a) DFCO, (b) DGFCO,  (c) DEFCO and (d) DLFCO  in the \ntemperatur e range of 2 -24 K. Inset: same  plot in the range  of 115 – 142 K for DFCO . \n \n \n \n \n \n \n \n \n \n \n16 \n  \nFig. 5: (a) Isothermal magnetic entropy change  (∆SM) plotted as a function of temperature  \n(T) for DGFCO, DEFCO and DLFCO compounds  for a magnetic field  change  50 kOe.  The \nblack curve is for DFCO from [ 16]. (b) T response of RCP for all the compound s. \n \n \n \n \n \n17 \n  \nFig. 6:  Magnetic field  (H) response of isothermal magnetic entropy  (∆SM) change  at select ed \ntemperatures  for (a) DFCO, (b) DGFCO,  (c) DEFCO and (d) DLFCO . The curve through the \npoints is  the fit to the power law behaviour  (as described in text) . \n \n \n \n \n \n18 \n  \nFig. 7:  Normalized magnetic entropy change (∆ SM/∆Smax) as a function of reduced \ntemperature  θ=(T -Tpk)/(T r-Tpk) for (a) DFCO, (b) DGFCO, (c) DEFCO and (d) DLFCO . Inset \nof Fig (a):  Same figure  for all the compounds at 50 kOe . \n"    },    {        "title": "1712.10283v1.Multiphase_Magnetic_Systems__Measurement_and_Simulation.pdf",        "content": "1 \n Multiphase Magnetic Systems: Measurement and Simulation \n \nYue Cao,1 Mostafa Ahmadzadeh ,1 Ke Xu,1 Brad Dodrill,2 John S. McCloy1,3,* \n1. Materials Science & Engineering Program, Wash ington State University, Pullman, WA, 99164, USA \n2. Lakeshore Cryotronics, We sterville, OH, 43082, USA \n3. School of Mechanical & Materials Engineering, Washington State University, Pullman, WA, 99164, USA \n*john.mccloy@wsu.edu ; +1-509-335-7796 \n \nABSTRACT \nMultiphase magnetic systems are common in nature and are increa singly being recognized in \ntechnical applications. One characterization method which has s hown great promise for \ndetermining separate and collective effects of multiphase magne tic systems is first order reversal \ncurves (FORCs). Several examples are given of FORC patterns whi ch provide distinguishing \nevidence of multiple phases. In parallel, a visualization metho d for understanding multiphase \nmagnetic interaction is given, which allocates Preisach magneti c elements as an input ‘Preisach \nhysteron distribution pattern’ (PHDP) to enable simulation of d ifferent ‘wasp-waisted’ magnetic \nbehaviors. These simulated systems allow reproduction of differ ent major hysteresis loop, FORC \npattern, and switching field distributions of real systems and parameterized theoretical systems. \nThe experimental FORC measurements and FORC diagrams of four co mmercially obtained \nmagnetic materials, particularly  those sold as nanopowders, sho ws that these materials are often \nnot phase pure. They exhibit comp lex hysteresis behaviors that are not predictable based on relative \nphase fraction obtained by characterization methods such as dif fraction. These multiphase materials, \nconsisting of various fractions of BaFe 12O19, ε-Fe 2O3, and γ-Fe 2O3, are discussed. \nKeywords: Wasp-waisted hysteresis; FORC; barium hexaferrite; ε- Fe2O3 \n  2 \nⅠ. INTRODUCTION  \nEven though the ideal characteris tics of hysteresis in perfect single magnetic domains is generally \naccepted, it is still a challenge t o interpret the hysteresis o f real materials, as it depends on particle \nsize, shape, and distribution, as well as stress, defects, and impurities. The hysteresis of magnetic \nmixtures is even more complex, as interactions between non-dilu te phases distort the simple loop \nshape. However, the parameters extracted from major hysteresis loop ‒ for instance, coercivity ‒ \nare not very efficient at showi ng the interactions of magnetic phases in a multiphase system.  \nFirst order reversal curve (FO RC) analysis is one technique whi ch has developed rapidly in the last \ntwo decades,1 providing a powerful tool for observing magnetic switching con tributed by different \nmagnetic phases. FORC is now applied to understand various hyst eretic behaviors in metal-\ninsulator transitions,2 ferroelectricity,3 magnetocalorics,4 and perpendicular magnetic recording \nmedia.5 The FORC technique can also be employed to distinguish the dif ferent magnetic signals in \nmultiphase systems4,6 to investigate subtle magnetic features caused by interaction of multiple \nmagnetic phases.  \nIn this paper, a series of multiphase magnetic nanopowders are investigated experimentally using \nmajor hysteresis loops and FORC diagrams and these systems are also modeled using ‘Preisach \nhysteron distribution patterns’ (PHDPs) based on the classical Preisach model (CPM) to aid in \ndescriptive understanding of the reduction of coercivity, or ‘w asp-waistedness’ features, observed \nin major loops of multiphase materials. The FORC diagrams of th em generally indicate the \nexistence of a low coercivity phase, a high coercivity phase, a nd a coupling region, which can all \nbe simulated using PHDPs. Together, the simulations and measure ments provide a coherent picture \nof magnetic interactions in a ‘wasp-waisted’ system,7,8 and they confirm the utility of the CPM for \ndescribing such systems.  3 \nⅡ. SIMULATIONS AND EXPERIMENTAL METHODS \nA. Theory and Simulation \nAs the basis of First Order Reversal Curves (FORC), the Preisac h model9 is a typical model of \nhysteresis, and it is a simple and straightforward description of magnetic switching. It has been \ndeveloped with increasing compl exity to refine its description of magnetic behaviors observed \nunder different conditions,10,11 but the classic Preisach model (CPM) is used here as a useful \nphenomenological approach to unde rstanding highly interacting m ultiphase magnetic materials \n(see Supplementary Material for more details on simulations and relationship to the Preisac h \nmodel). In the proposed embodiment of the CPM, all hysterons ar e distributed onto a two-\ndimensional (2D) coordinate system to build a ‘Preisach hystero n distribution pattern’ (PHDP) ( Fig. \nS-1(b) ). The magnetization of the PHDP at a given external field is c alculated via summing the \nmagnetization of all hysterons, as the overall magnetization of  a real material could be decomposed \ninto a series of these hysterons.  Since each hysteron has magn etization of nominally +1 or -1, the \nnumber of hysterons in a simula tion, while phenomenological on a first order, represents both the \nmole fraction of a phase and the relative magnetization of that  phase. In other words, if a phase has \nhigher molar magnetization, a mole of said phase would be repre sented by more hysterons than a \ndifferent phase with a smaller magnetization.   \nThe FORC technique captures hysteretic features on multiple rev ersal curves, and the \nmagnetization of every data point on each reversal curve is det ermined by the reversal field ( Hr) \nand the magnetic field ( H). Thus, for each data point on a given FORC, its reversal fiel d Hr is \nequivalent to the Preisach switching field a, and its magnetic field H represents the Preisach \nswitching field b. Then a FORC diagram is plotted with contours of FORC distribu tion density ( Hr, \nH) with coercivity Hc(Hr, H) as the x-axis and interaction Hu(Hr, H) as the y-axis. In this paper, the \nFORC distribution density ߩ of PHDP is obtained via least squares fitting a second-order \npolynomial function using Matlab 2014b12,13 and the corresponding FORC diagrams are processed 4 \nin Origin 9.1. More detail regarding the PHDP and FORC techniqu e have been discussed more \nthoroughly in previous publications10,14 and in the Supplementary Material . \nB. Sample selection and FORC measurements \nTo investigate the two-component PHDP of a real sample to compa re to simulation, FORC data \nwas acquired on a series of commercial materials, all purchased  as M-type barium hexaferrite \n(BaFe 12O19). BaFe 12O19 (Ba-M) is a commercial and competitive permanent magnet15,16 as magnetic \nrecording material,17,18 and millimeter wave absorber.19 T h e  m i c r o m e t e r - s i z e d  s i z e d  B a F e 12O19 \npowders  were obtained from advanced fe rrite technology (AFT) GmbH and n anometer-sized \nmaterials were obtained from Aldrich.20 For the purposes of this paper, these samples have been \ndenoted “Micro” (large particle BaFe 12O19), and “Nano-1”, “Nano-2”, and “Nano-3” for the three \nlots of nanopowders purchased as “BaFe 12O19.” The particle size,21 crystal phase identification by \nX-ray diffraction (XRD),20 temperature-dependent magnetization,21 and major hysteresis loops20 \nhave been previously assessed. These materials were assessed to  have multiple crystalline phases, \nmost of which were not BaFe 12O19, and major loops exhibited characteristic shapes indicative of  \ninteracting phases with different  magnetic properties.  The XRD  and major loops up to 50 kOe are \nreviewed and replotted in the Fig S-4  and Fig S-3 . \nMost samples considered had very high coercivity and high satur ation field. Multiple attempts were \nmade to capture the whole hysteresis behavior in FORC (see Supplementary Material ). Ultimately, \nall samples were measured on a Lakeshore 7400 vibrating sample magnetometer (VSM) with \nsaturation magnetic field Hsat of ±33 kOe, which is considerably higher than standard VSMs us ed \nfor FORC measurements. The specific parameters, including the f ield steps, the number of FORCs \nand the smoothing factors, for each sample are listed in Table I . Note that in multiple cases, the \nloops in FORC appeared to saturate in lower field instruments, but previous major loop behavior \nobserved up to 50 kOe could not be reproduced (see Supplementary Material ). This indicates an \nimportant caveat in FORC measurem ents of high concentrations of  very high coercivity materials 5 \nwhich are typically not encountered in geologic contexts. All F ORC diagrams were processed using \nthe VARIFORC function in FORCinel,22,23 which is written in Igor (V2.02 in IGOR Pro7, \nWaveMetrics, Portland, OR). In processing the FORC data, the lo wer edge artifact was removed24 \nprior to plotting the FORC diagram. \nTable I . Final FORC parameters. H u1 and H u2 are the limits for the y-axis (interaction or bias) on \nthe FORC diagram. H c1 and H c2 are the limits for the x-axis (coercivity axis) on the FORC di agram. \nHCal and HSat are the fields used for calibration or saturation, re spectively, applied at the end of \neach FORC. H Ncr is field step between reversal fields and N Forc is the number of FORCs. \nSample H u1 \n(kOe) Hu2 \n(kOe) Hc1 \n(kOe) Hc2 \n(kOe) HCal \n(kOe) HSat \n(kOe) HNcr \n(Oe) NForc \nMicro -1.5 1.5 0 10 11.845 20 89.65 150 \nNano-1, 2, and 3 -1.5 1.5 0 28 29.5 33 208.84 150 \n \nⅢ. RESULTS \nA. FORC diagrams and interpretation \n1. Micro sample \nThe experimentally obtained raw FORCs and calculated FORC diagr am of the ‘Micro’ sample are \nshown in Fig. 1 . It is apparent that a small but more compact peak appears at low coercivity (~100 \nOe) and a widely distributed peak appears at high coercivity (~ 3 kOe). However, a significant \n‘wasp-waistedness’ was not clear in the previously measured maj or hysteresis loop, probably due \nto a large field step in the original major loop measurement, w hich was 0.5 kOe (see Fig. S-3 ). Note \nthat the FORC measurements used significantly smaller step size s which did resolve multiple \nfeatures. The concentric distribution at high coercivity sugges ts single domain (SD) particles,25,26 \nand the large vertical spread of the high coercivity component suggests the interaction between \nthese SD particles.1,27 The bias to negative interaction ( Hu) suggests mean field magnetizing \ninteractions,28 which would be expected from packed hexagonal plate21 crystals. The distribution \nof low coercivity component also showed spread along the H u axis, but the spread gradually \ndecreases with increasing coercivity, which suggests pseudo-sin gle domain (PSD) particles.29 The 6 \ndesignation of PSD is a transition region between SD and multi domain (MD), and the coe rcivity \nof PSD and MD particles decrease with increasing particle size.30,31  \nThough XRD suggested only BaFe 12O19 phase,21 it is possible that peaks of magnetite (Fe 3O4) \nand/or maghemite (γ-Fe 2O3) could be hidden in the pattern (see Supplementary Material ). \nMaghemite is the fully oxidized equivalent of magnetite with th e same crystal structure. Its typical \nreported saturation magnetization is ~74 emu/g, whereas that of  magnetite is ~92 emu/g.31 The SD \ncritical grain size of maghemite is ~60 nm and it is ~50-84 nm for magnetite,32 although these \nvalues can depend on particle shape.33 Almeida et al34 have observed almost identical FORC \nbehavior for magnetite and maghe mite with similar particle size . Therefore, their similar behaviors \nmakes it difficult to exclusively assign the low-H C component to one of these phases. It is also \npossible, though unlikely, that th e low coercivity peak is larg e domain BaFe 12O19 of the PSD size. \nHowever, the coercivity of BaFe 12O19 does not appear to drop for particles at least 1 μm in size,35 \nwhich is about the observed particle size for this material,21 though some larger particles are \npossible. The high coercivity component is most likely single d omain BaFe 12O19 particles, since the \npeak in coercivity is ~3 kOe, which is well in line with previo us reports on the coercivity of this \nmaterial from ~50 nm to >1 μm (2.5 – 6 kOe).20,21,35 \n2. Nano-1 sample \nThe FORC diagram of ‘Nano-1’ ( Fig. 2 ), on the other hand, shows two separated peaks at Hc = \n~0.2 kOe and Hc = ~6.5 kOe and one obscure peak at Hc = ~2 kOe.  Again, the low coercivity \ncomponent behaves as PSD, and the middle and high coercivity co mponents are most likely two \npopulations of SD particles. The coercivity of the middle peak is quite close to that of SD BaFe 12O19 \nparticles in the ‘Micro’ sample, and therefore, the middle coer civity component is probably SD \nBaFe 12O19. The high coercivity component was initially assigned to SD \tߝ-Fe 2O3 nanoparticles; \nhowever, the reasonable coercivity of  SD ߝ-Fe 2O3 nanoparticles within sim ilar particle size has \nbeen reported as ~15-20 kOe, which is much higher than the valu e observed.36 The high coercivity 7 \ncomponent was finally determined as SD BaFe 12O19 (but with a different particle size distribution \nthan the middle peak)  after comparing to the theoretical and experimental results.35,36 We attempted \nto capture the signal of single domain ߝ-Fe 2O3 nanoparticles by changing the intensity of contour, \nas XRD shows considerable amount of this phase, but still did n ot observe any peak beyond 6.5 \nkOe. The disappearance of the low-magnetization, high-coercivit y component has been reported in \nsome magnetic mixtures, especially in multi-component systems w hich show significant \nmagnetization difference between two different magnetic phases.  For example, 95% hematite ( Ms \n~0.4 emu/g) was dominated by 5% magnetite ( Ms ~92 emu/g),37 and the resulting FORC diagram \nonly suggested the existence of magnetite phase at room tempera ture.38,39 By comparison, \nBaFe 12O19 has Ms ~50-70 emu/g21 and ε-Fe 2O3 has Ms ~15 emu/g.36 Note that, the fact that there is \nno offset for the high coercivity component indicates those sin gle domain nanoparticles are \nrandomly oriented.12 Corroborating evidence from the XRD result (see Supplementary Material ), \nthe low coercivity component is likely PSD γ-Fe 2O3 or magnetite. The average particle size of \n‘Nano-1’ is in the PSD transition region for γ-Fe 2O3 and magnetite.31   \n3. Nano-2 sample \nThe major loop of ‘Nano-2’ shows the most significant ‘wasp-wai stedness’ since the two observed \ndistribution peaks ( Hc ~ 0.2 kOe and 24 kOe) are widely separated in its FORC diagram  (Fig. 3 ). \nThe XRD identified  that the major phase in ‘Nano-2’ was ߝ-Fe 2O3 phase, but could not rule out \nthe possibility of small amounts of other Fe oxides, such as Fe 3O4 or γ-Fe 2O3. The FORC diagram \nshows a clear peak located at ~24 kOe, which is close to the pe ak value of coercivity of SD ߝ-Fe 2O3 \nparticles.36 Note that, the primary particles size is 30 to 50 nm,21 which is still in the SD range of \nߝ-Fe 2O3. The low coercivity component, on the other hand, could be larg er PSD\tߝ-Fe 2O3 particles \nor PSD Fe 3O4/ γ-Fe 2O3. Since a small amount of Fe 3O4/ γ-Fe 2O3 can dominate over the FORC signal \nfrom\tߝ-Fe 2O3, due to their magnetization difference, the low coercivity compo nent in the FORC \ndiagram of ‘Nano-2,’ like the other samples, is likely Fe 3O4/ γ-Fe 2O3 particles. 8 \n4. Nano-3 sample \n‘Nano-3’ shows a pinched major loop with low major loop coerciv ity, presumably due to high \nconcentration of PSD γ-Fe 2O3 particles (low coercivity and thin loop) in addition to SD ߝ-Fe 2O3 \nparticles. It is worth pointing out that the XRD patterns of γ- Fe2O3 and Fe 3O4 are very similar, so \nagain magnetite versus maghemite cannot be reliably distinguish ed with the current data. The \nFORC diagram of ‘Nano-3’ ( Fig. 4 ) does not show multiple peaks, unless measured up to very high  \nmagnetic field. A subtle high coercivity peak then emerges at v ery high coercivity (~25 kOe), which \nsuggests that this peak still belongs to single domain ߝ-Fe 2O3 particles, with lower fraction than \nNano-2, while the low coercivity component is assigned to PSD γ -Fe 2O3/Fe 3O4 particles which \ncontribute more magnetization, and hence higher intensity. XRD showed that Nano-3 contains ~50% \nγ-Fe 2O3/Fe 3O4 phase which could contribute mor e magnetization, and eventually  hide the signal of \nSD ߝ-Fe 2O3 particles in its FORC diagram.  \nB. Simulations to match experimental FORC diagrams \nIn order to tailor further the resulting FORC diagrams for comp arison with experimental diagrams, \na phenomenological Preisach model was created using three diffe rent concentric ellipses/circles of \nhysterons to represent the po ssible phases described in Sec. III A . The concentric ellipses are used \nto describe BaFe 12O19 phase(s) and ε-Fe 2O3 phase(s), and the concentric circles are created for γ-\nFe2O3 phase. In addition, to emphasize the difference between the ma gnetization of each phase, the \ntotal number of hysterons in a concentric ellipse/circle for di fferent phases are intentionally \ndesigned. For example, one concentric ellipse for ε-Fe 2O3 contains only 20 hysterons, but one \nconcentric ellipse for BaFe 12O19 consists of 120 hysterons due t o its high magnetization. BaFe 12O19 \nphase and ε-Fe 2O3 phase are distinguishable by the long axis/short axis ratio. S ince BaFe 12O19 \nphases show wider spread along the H u axis in the experimental results, the long axis/short axis \nratio of the BaFe 12O19 phase is designed larger than that of the ε-Fe 2O3 phase. All parameters of \ndifferent phases are listed in Table II . 9 \nTable II. Summary of simulation parameters of BaFe 12O19, ε-Fe 2O3 and γ-Fe 2O3 phase. \nPhase Name # of hysterons in one \nconcentric ellipse/circle Long axis/short \naxis ratio Color \nBaFe 12O19 100 4/3 Green \nε-Fe 2O3 20 2/1 Blue \nγ-Fe 2O3 120 N/A Red \n \nFour different elliptical PHDPs were created to simulate the el ongated distributions along the H c \naxis in the FORC diagram of Micro and Nano samples. All possibl e phases discussed in Sec. III A \nare represented by grouped concentric ellipses or circles of hy sterons. The centers of the elliptical \ndistribution are allocated on the diagonal, and their major axe s are parallel to the diagonal to obtain \nsymmetric distributions. Previous study showed that the center of hysteron distribution determined \nthe coercivity; therefore, the same phase of particles that sho w size-dependent coercivity are \nexhibited by locating the center of ellipses/circles at various  position on the PHDP diagonal. It is \nalso valid that a phase present in a larger amount is modeled b y multiple concentric distributions \nin this simulation. To slow down the magnetization switching at  the remnant state, all concentric \nellipses and circles are distributed on a uniform matrix where the hysterons are placed with distance \nof 0.1 horizontally and vertically.  \nThe simulated FORC dataset of each PHDPs consist of 40 FORCs, a nd the magnetization of each \ndata point on every reversal cur ve is calculated at a defined s tep of 0.05 until the magnetic field H \nreturns to saturation again. It was found that simulation of 40  FORCs was sufficient to present all \nthe major features of typically obtained experimental FORC diag rams, and additional FORCs only \nslightly improved resolution. It should be recognized that eith er increasing the number of FORCs \nor decreasing the step of FORC will change the quality of FORC diagrams,26 resulting in longer \ncollecting time in a FORC meas urement and longer processing tim e to extract the FORC diagram. \nAdditionally, the quality of a FORC diagram is also affected by  the smoothing factor (SF).12 In this \nmodeling, SF=2 is chosen to map FORC diagrams in better resolut ion and higher quality. Smaller 10 \nSF will induce more noise and larger SF will cut off partial co ncentric circles at high coercivity \nregion.  \nThe simulated phases and corresponding simulation parameters in  four elliptical PHDPs are \nsummarized in Table III . Note that the low coercivity phase was deliberately changed f or M1 \n(BaFe 12O19), M2 (γ-Fe 2O3), and M3 (ε-Fe 2O3) to illustrate the fact that the FORC diagram is not \nstrongly affected by the choice of this phase. It has been conf irmed in selected cases that \nreplacement of γ-Fe 2O3 for ε-Fe 2O3 in M3, for instance, gives a FORC pattern and FORC diagram \nif the total number of hysterons  for the group of concentric ci rcles or ellipses is kept constant. \nTable III . Summary of elliptical PHDP for Ba-M; for all simulations, the  matrix step size is 0.1, \nthe number of FORCs is 40, the  step is 0.05, and the SF=2. \nPHDP # Phase 1  Centroid 1 Phase  2 Centroid 2 Phase 3 Centroid 3 Phase 4 Centroid 4 # of concentric \ndistribution of \neach phase Similar \nsample \nM1 BaFe 12O19 [-0.05, 0.0.5] BaFe 12O19 [-0.3, 0.3] N/A N/A N/A N/A 1;5 Micro \nM2 γ-Fe 2O3 [-0.05, 0.05] BaFe 12O19 [-0.2, 0.2] BaFe 12O19 [-0.4, 0.4] ε-Fe 2O3 [-0.2, 0.2] 1;1;5;1 Nano-1 \nM3 ε-Fe 2O3 [-0.05, 0.05] ε-Fe 2O3 [-0.6, 0.6] N/A N/A N/A N/A 4;4 Nnano-2 \nM4 γ-Fe 2O3 [-0.05, 0.05] ε-Fe 2O3 [-0.6, 0.6] N/A N/A N/A N/A 1;1 Nano-3 \n \n \nⅢ. DISCUSSION \nBarium hexaferrite written as BaFe 12O19 can also be written as BaOꞏ6Fe 2O3.  Thus, one can imagine \nthat incomplete reaction with BaO during synthesis could result  in Fe 2O3 phases, and the BaO could \nbe present as an amorphous phase and be undetectable with XRD.  It has been suggested that under \ncertain synthesis conditions, the smallest particles of Fe 2O3 tend toward the γ-Fe 2O3 phase, while \nthe ε-Fe 2O3 phase is stable from ~3-8 to ~30 nm.40,41  Small γ-Fe 2O3 particles <4 nm are \nsuperparamagnetic, while those 4-8 nm have coercivitites less ≤  2.1 kOe.41 While not probably in \nthe size-range of PSD maghemite, these low coercivities could a ccount for the low coercivity \ncomponents observed in all the samples. It has also been report ed that ε-Fe 2O3 can form in the \npresence of Ba2+ if the Fe/Ba ratio is 10-20.42 Hematite, α-Fe 2O3, can also form in this series as 11 \nparticles of ε-Fe 2O3 become larger than ~30 nm.40,41 Apparently, an uncontrolled commercial \nsynthesis process resulted in the nanopowders with one or more Fe2O3 phases rather than BaFe 12O19. \nFORC is readily able to distinguish these different phases, pro vided that the maximum applied field \nin the FORC sequence is high enough. In a number of cases, a lo wer field resulted in an incomplete \ndiagram being collected, and the highest coercivity would have easily been missed if the major \nloops had not already been measured on a high field instrument.20 These FORCs which do not \nachieve high enough field are not representative of the whole m agnetic behavior of the system, but \nrather only the low coercivity components, much like minor loop s.\nUse of phenomenological simulation allowed the recreation of th e expected FORC and FORC \ndiagram behavior in the presence of multiple phases with differ ent magnetizations and coercivities. \nThere are more robust simulations  for FORC diagrams based on mi cromagnetic simulations, for \nexample,24 but are restricted to SD particles. FORC measurements provide a quick way for \nidentifying characteristic phases in a mixed sample. When coerc ivities are much different than one \nanother, especially when the high coercivity phase has low magn etization, the remanence and hence \nFORC diagram can be dominated by the high magnetization phase, such as in the case of \nmagnetite/hematite mixtures.38 While it is tempting to consider a simple superposition of the  \nsignatures separate phases, it is  clear that this does not alwa ys happen and considerable interaction \nbetween the particles takes place, particularly in highly inter acting, non-dilute conditions such as \nconsidered here. \nⅣ. SUMMARY AND CONCLUSION \nFour samples of commercially obtained “barium hexaferrite” were  identified by First Order \nReversal Curves at room temperature and up to high fields. In t wo cases, high coercivity ε-Fe 2O3 \ncould be identified by FORC when it was suggested by XRD, but n ot in the third case where \nroughly half the sample was γ-Fe 2O3 (or Fe 3O4) a high magnetization and low coercivity phase. \nEven in the apparently crystallographically pure micron-sized B aFe 12O19, strong evidence of low 12 \ncoercivity phases, possibly pseu do-single domain structures, we re found at coercivities <2 kOe.  \nHigh coercivity phases BaFe 12O19 (2-5 kOe) and ε-Fe 2O3 ( 2 0 - 2 6  k O e ) ,  h a d  b r o a d  c o e r c i v i t y  \ndistributions shown by elongated positive regions on the FORC d iagram along the Hc axis. Most \nFORC diagrams show evidence of s trong interparticle magnetic bi as by broadening in the Hu \ninteraction axis. In the case of the ‘micro’ sample, the centro id BaFe 12O19 was clearly shifted to \nnegative Hu, suggesting magnetizing mean fie ld interactions. Simple phenom enological Preisach \nmodeling allowed reproduction of t he main features of the FORC diagrams, and showed that simple \nmixing of the phases does not necessarily produce a peak for ea ch phase.     \nSUPPLEMENTARY MATERIAL  \nSee the Supplementary Material for details on simulation theory  and methods, X-ray diffraction \ndata and discussion, complete FO RC and FORC diagrams of all ite rations of the experiments, and \nisothermal remnant magnetiza tion (IRM) data for Nano-1. \nACKNOWLEDGEMENT  \nThis research was funded by the U.S. Department of Energy in su pport of the Nuclear Energy \nEnabling Technologies– Reactor Materials (NEET-3) program. The authors thank the anonymous \nreviewer and Neil Dilley for help improving the manuscript.     13 \nREFERENCES \n1C. R. Pike, A. P. Roberts, and K. L. Verosub, J. Appl. Phys. 85, 6660 (1999). \n2J. G. Ramírez, A. Sharoni, Y. D ubi, M. E. Gómez, and I. K. Schu ller, Phys. Rev. B 79, 235110 \n(2009). \n3A. Stancu, D. 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(a) Experimental FORCs (every 2nd FORC displayed) with major loop and (b) \ncorresponding FORC diagram of ‘Mi cro’ at room temperature with VARIFORC smoothing \nfactor S c0 = S b1 = 5, S c1 = S b1 =7. The unit of the FORC density  intensity scale bar is emu/kg/ Oe2. \nInset shows close-up near Hc~0. \nFig. 2.  (a) Experimental FORCs (every 2nd FORC displayed) with major loop and (b) \ncorresponding FORC diagram of ‘Nano-1’ at room temperature with  VARIFORC smoothing \nfactor S c0 = S b1 = 5, S c1 = S b1 =7. The unit of the FORC density  intensity scale bar is emu/kg/ Oe2. \nInset shows close-up near Hc~0. \nFig. 3.  (a) Experimental FORCs (every 5th FORC displayed) with major loop and (b) \ncorresponding FORC diagram of ‘Nano-3’ at room temperature with  VARIFORC smoothing \nfactor S c0 = S b1 = 5, S c1 = S b1 =7. The unit of the FORC density  intensity scale bar is emu/kg/ Oe2. \nInset shows close-up near Hc~0. \nFig. 4.  (a) Experimental FORCs (every FORC displayed) with major loop and (b) corresponding \nFORC diagram of ‘Nano-3’ at room  temperature with VARIFORC smoo thing factor S c0 = S b1 = 5, \nSc1 = S b1 =7. The unit of the FORC density intensity scale bar is emu/kg/ Oe2. Inset shows close-up \nnear Hc~0 and near max field.  In the main FORC diagram, contour of th e high coercivity \ncomponent is drawn in, while in the inset, the color-sale was a djusted to make this component \nvisible; this inset color scale does not correspond to main fig ure. \nFig. 5.  (a-d) Simulated elliptical PHDPs and (e-h) corresponding FORC and (i-l) FORC diagrams \nfor EP1, EP2, EP3 and EP4 with SF=2. BaFe 12O19, γ-Fe 2O3 and ε-Fe 2O3 phases are represented by \ngreen, red and blue, respectively. Black spots are matrix hyste rons. \n\n\n  17 \nFIGURES \n \n \n \n \n \nFig. 1.  (a) Experimental FORCs (every 2nd FORC displayed) with major loop and (b) \ncorresponding FORC diagram of ‘Mi cro’ at room temperature with VARIFORC smoothing \nfactor S c0 = S b1 = 5, S c1 = S b1 =7. The unit of the FORC density  intensity scale bar is emu/kg/ Oe2. \nInset shows close-up near Hc~0. \n\n18 \n\nFig. 2.  (a) Experimental FORCs (every 2nd FORC displayed) with major loop and (b) \ncorresponding FORC diagram of ‘Nano-1’ at room temperature with  VARIFORC smoothing \nfactor S c0 = S b1 = 5, S c1 = S b1 =7. The unit of the FORC density  intensity scale bar is emu/kg/ Oe2. \nInset shows close-up near Hc~0. \n\n\n\n19 \n\n\n\nFig. 3.  (a) Experimental FORCs (every 5th FORC displayed) with major loop and (b) \ncorresponding FORC diagram of ‘Nano-3’ at room temperature with  VARIFORC smoothing \nfactor S c0 = S b1 = 5, S c1 = S b1 =7. The unit of the FORC density  intensity scale bar is emu/kg/ Oe2. \nInset shows close-up near Hc~0. \n \n20 \n \nFig. 4.  (a) Experimental FORCs (every FORC displayed) with major loop and (b) corresponding \nFORC diagram of ‘Nano-3’ at room  temperature with VARIFORC smoo thing factor S c0 = S b1 = 5, \nSc1 = S b1 =7. The unit of the FORC density intensity scale bar is emu/kg/ Oe2. Inset shows close-up \nnear Hc~0 and near max field.  In the main FORC diagram, contour of th e high coercivity \ncomponent is drawn in, while in the inset, the color-sale was a djusted to make this component \nvisible; this inset color scale does not correspond to main fig ure. \n21 \n\n\n\n\n\nFig. 5.  (a-d) Simulated elliptical PHDPs and (e-h) corresponding FORC and (i-l) FORC diagrams \nfor M1, M2, M3 and M4 with SF=2. BaFe 12O19, γ-Fe 2O3 and ε-Fe 2O3 phases are represented by \ngreen, red and blue, respectively. Black spots are matrix hyste rons. \n\n\n\n\n\n\nS-1 \n  \nSUPPLEMENTARY MATERIAL \n Multiphase Magnetic Systems: Measurement and Simulation \n \nYue Cao,1 Mostafa Ahmadzadeh ,1 Ke Xu,1 Brad Dodrill,2 John S. McCloy1,3 \n1. Materials Science & Engineering Program, Wash ington State University, Pullman, WA, 99164, USA \n2. Lakeshore Cryotronics, We sterville, OH, 43082, USA \n3. School of Mechanical & Materials Engineering, Washington State University, Pullman, WA, 99164, USA \n \n  \n 23 pages  \n 4 tables  \n 16 figures  \n \n  S-2 \n I. DETAILS ON SIMULATIONS \n Theory \nAs the basis of First Order Reversal Curves (FORC), the Preisac h model1 is a typical model of \nhysteresis, and it is a simple and straightforward description of magnetic switching. It has been \ndeveloped with increasing compl exity to refine its description of magnetic behaviors observed \nunder different conditions.2,3 However, despite the growing diversity of the Preisach model \nmanifestations, none satisfactorily applies to all situations, so the utility of the classic Preisach \nmodel (CPM) may still be considerable given its simplicity.  \nIn the CPM, one ‘Preisach hysteron’ is a mathematical magnetic element which has two \nindependent switching fields ( a and b) and two states of magnetiza tion (1 and -1). The shape of \nhysteresis of one hysteron is perfectly rectangular in the CPM.2 The magnetization ( M) of each \nhysteron is determined by switching fields and external magneti c field ( H): \n                                                    ܯሺܪሻൌቐ1\tሺܪ  ܾሻ\t\nെ1\tሺܪ  ܽሻ\n\t\t\t\t\t\t\t\t\t\t\t\t\tݔሺܾ  ܪ  ܽሻ\t                                               (1) \nwhere x is -1 when the hysteresis loop is ascending and +1 when it is descending. The configuration \nfor a hysteresis loop of a single hysteron is shown in Fig. S-1(a).  \nIn the proposed embodiment of the CPM, all hysterons are distri buted onto a two-dimensional (2D) \ncoordinate system to build a ‘Pr eisach hysteron distribution pa ttern’ (PHDP) ( Fig. S-1(b) ). It should \nbe noted that this is a very general embodiment of a distributi on, whereas other more specific ones, \nsuch as lognormal, Lorentzian, o r Gaussian, have been previousl y suggested.3-5 The magnetization \nof the PHDP at a given external field is calculated via summing  the magnetization of all hysterons, \nas the overall magnetization of a real material could be decomp osed into a series of these hysterons.  \nSince each hysteron has magnetization of nominally +1 or -1, th e number of hysterons in a \nsimulation, while phenomenological on a first order, represents  both the mole fraction of a phase \nand the relative magnetization of that phase.  In other words, if a phase has higher molar \nmagnetization, a mole of said phase would represent more hyster ons than a different phase with a \nsmaller magnetization.   \nFor every hysteron in the PHDP, its coercivity ( Hc) and interaction ( Hu) (see Fig. S-1(a) ) are defined \nas:2 \n                                              \t\t\t\t\t\tܪሺܾ,ܽሻൌି\nଶ\t\t\t\t\t\t\t\t\t\t\t\tܪ ௨ሺܾ,ܽሻൌା\nଶ                                         (2) \n   S-3 \n  \nFig. S-1 . (a) Schematic of a single magnetic hysteron element. (b) Simu lated PHDP (middle) and \n5 hysterons with switching fields ( a,b) as (I) (-0.4, 0.8); (II) (-0.8, 0.4); (III) (-0.4, 0); (IV) ( -0.4, \n0.4) and (V) (0, 0.4). \n \n Classic Preisach model of multi-phase materials \nFig. S-2(a-d)  shows selected simulated PHDPs using the CPM, and the simulati on parameters of \nmore PHDPs are summarized in Table S-Ⅰ . Each PHDP contains two parts: 1) a uniform matrix and \n2) series of concentric ellipses (with a circle being the symme tric case) with various centroids and \nradii. The radius of the outside concentric circle is taken as 0.05, 0.1 or 0.25 to obtain various \nhysteretic switching behavior 2. More concentric circles are placed by gradually decreasing th e \nradius with a gap of 0.05 un til the centroid is reached. Each c oncentric circle consists of 80 evenly \ndistributed hysterons, and it is distributed on the uniform mat rix where hysterons are placed with \ndistance of 0.2 horizontally and vertically. In this paper, the  effect from the position and \nconcentration of the concentric circles is primarily discussed,  since it impacts the hysteresis of \nPHDPs more significantly than the matrix.2 Simulations P1 and P2 have only one distribution \ncentroid, but P3, P4, P5, and P6  are designed to contain two ce ntroids, representing either a single \nmagnetic phase with a bimodal size distribution or two differen t magnetic phases.  The impact of \na shift from a circular to elliptical distribution is discussed  in Sec. Ⅲ B  in the main paper.  \nThe simulated FORC dataset of each PHDPs consist of 40 FORCs, a nd the magnetization of each \ndata point on every reversal cur ve is calculated at a defined s tep of 0.05 until the magnetic field H \nreturns to saturation again. It was found that simulation of 40  FORCs was sufficient to present all \nthe major features of typically obtained experimental FORC diag rams, and additional FORCs only \nslightly improved resolution. I t should be recognized that eith er increasing the number of FORCs \nor decreasing the step of FORC will change the quality of FORC diagrams,6 resulting in longer \ncollecting time in a FORC measu rement and longer processing tim e to extract the FORC diagram. \nAdditionally, the quality of a FORC diagram is also affected by  the smoothing factor (SF).7 In this \nmodeling, SF=2 is chosen to map FORC diagrams in better resolut ion and higher quality. Smaller \nSF will induce more noise and larger SF will cut off partial co ncentric circles at high coercivity \nregion. \nS-4 \n The corresponding FORCs of each PHDP are simulated and shown in  Fig. S-2(e-h) . The coercivity \n|Hc| of the major hysteresis loop is identical for P1 and P2 since  they have the same centroid of \nconcentric circles.2 The coercivity of the PHDP is located at an intermediate point  between the \ncentroids for two-component PHDPs. Another feature which can be  readily appreciated is that the \nradius of concentric circles controls the susceptibility χ c of the hysteresis loop at the coercivity. It \nhas been shown that increasing the radius of concentric circles  (P1 versus P2) decreases the \nsharpness of the hysteresis as well as the susceptibility χ c.2 However, it can be observed that adding \nanother series of concentric circles at another centroid can al so decrease the susceptibility (e.g., P1 \nversus P3). Additionally, the  radius (P5 versus P6, in Table S-Ⅰ ) and the centroid (P3 versus P5) of \nthe added concentric circles can lead to either an increase or a decrease of the susceptibility.  \nThe third observation is that moving the two centroids away fro m each other can result in an \nobservable “wasp-waistedness” in the hysteresis (e.g., P3 and P 4). The wasp-waisted hysteresis is \ngenerally caused by: 1) two phases or materials which have diff erent magnetic behaviors8,9 or 2) \nan inhomogeneously distributed single magnetic phase.10 It is worth mentioning that the wasp-\nwaisted hysteresis is not only attributed to multiple magnetic phase coupling. For example, the \nshape asymmetry11 or vortex state magnetization reversal12 can result in wasp-waistedness as well.  \nNote that a pair of tails are observed in simulated FORC diagra ms, which are numerical artifacts \nfrom the data points that cannot be perfectly fitted the polyno mial FORC function. These features \nare also referred to as the ‘lower edge artifact’ that is remov able using commercial FORC \nprocessing software (e.g. FORCinel).13  \nThe FORC diagrams of P3 and P4, representing two-components PHD Ps, (Fig. S-2(c-d) ) contain \ntwo peaks which correspond to the two centroids of PHDPs. One i nteresting feature is that two \npeaks are extracted in the FORC diagrams of P3, while its major  hysteresis loop does not show \nobvious wasp-waistedness. This is because these two centroids i n P3 are very close to each other, \ndemonstrating that a FORC diagra m can be more sensitive than ot her standard magnetic \nmeasurements.  \n    S-5 \n Table S-Ⅰ . Summary of PHDP of CPM; for a ll simulations, the matrix step size is 0.2, the number \nof FORCs is 40, the step is 0.0 5, and the SF=2. Here radius is the radius of the outermost circle, \nχc is the susceptibility (i.e., slope dM/dH) at the coercivity, | Hc| is the absolute value of the \ncoercivity. See text for descrip tion of the distribution densit y. \nPHDP \n# Centroid \n1 Radius \n1 Distribution \ndensity 1 Centroid \n2 Radius \n2 Distribution \ndensity 2 χc |Hc| \nP1 [-0.4, 0.4] 0.1 2.42 N/A N/A N/A 7.39 0.40 \nP2 [-0.4, 0.4] 0.25 6.06 N/A N/A N/A 5.43 0.40 \nP3 [-0.2, 0.2] 0.1 2.42 [-0.4, 0.4] 0.1 2.42 2.85 0.30 P4 [-0.1, 0.1] 0.1 2.42 [-0.5, 0.5] 0.25 6.06 4.06 0.45 \nP5 [-0.2, 0.2] 0.1 2.42 [-0.5, 0.5] 0.1 2.42 1.76 0.35 \nP6 [-0.2, 0.2] 0.1 2.42 [-0.5, 0.5] 0.25 6.06 4.05 0.45 \n \n \n \nFig. S-2.  (a-d) Simulated PHDPs and (e-h) corresponding FORC and (i-l) F ORC diagrams for P1, \nP2, P3 and P4 with SF=2. \n \n  \nS-6 \n The simulated elliptical PHDPs, the resulting FORCs, and the co rresponding FORC diagrams are \nshown in Fig. S-3 and parameters in Table S-II . The FORC diagrams of EP1, EP2 and EP3 show \ntwo peaks spread at the low coercivity region and high coercivi ty region, representing the two \ncentroids in the elliptical PHDPs. Apparently, the distance bet ween the centroids decides the ‘wasp-\nwaistedness’ of the hysteresis loop. For example, the hysteresi s loop of EP1 closely resembles the \nloop of ‘Micro,’ which exhibit insignificant ‘wasp-waistedness. ’  \n \nTable S-II . Summary of elliptical PHDP; for all simulations, the matrix s tep size is 0.1, the \nnumber of FORCs is 40, the s tep is 0.05, and the SF=2. \nPHDP \n# Centroid 1 \n(ellipse) Long \naxis Short \naxis Distribution \ndensity 1 Centroid 2 \n(circle) Radius \n2 Distribution \ndensity 2 χc |Hc| \nEP1 [-0.3, 0.3] 0.3 0.1 3.03 [-0.05, 0.05] 0.05 0.61 4.61 0.3 \nEP2 [-0.4, 0.4] 0.3 0.1 3.03 [-0.05, 0.05] 0.05 0.61 4.86 0.37 \nEP3 [-0.5, 0.5] 0.3 0.1 3.03 [-0.05, 0.05] 0.05 0.61 5.12 0.45 \nEP4 [-0.05, 0.05] 0.1 0.03 3.03 [-0.05, 0.05] 0.05 0.61 12.74 0 .13 \n \n \n \nFig. S-3.  (a-d) Simulated elliptical PHD Ps and (e-h) corresponding FORC and (i-l) FORC \ndiagrams for EP1, EP2, EP3 and EP4 with SF=2. Red circle in (d)  represents the circular \ndistribution, and the elliptical distribution (black) overlays the red circle.  \n \n  \nS-7 \n II. DETAILS ON EXPERIMENTS \n Initial characterization of commercial powders \n  Magnetic data \nPreviously collected major loops are shown in Fig. S-4 . Note the strong wasp-waistedness in some \nsamples, particularly “nano -2” and “nano-3” samples. \nMajor loop saturation magnetization Ms and coercivity Hc are consistent with those measured \npreviously, indicating that the phases had not magnetically cha nged in the period between these \ntwo measurements (~6 years). Experimental FORCs were collected on several different instruments. \nMagnetic measurements for ‘barium hexaferrite’ systems were col lected at room temperature.  \n \n  \nFig. S-4.  Major hysteresis loops from powders of (a) and (b) ‘barium hex aferrite’ systems at \nroom temperature \n \n \n X-ray diffraction data \nA summary of the crystallographic phase determination from X-ra y diffraction (XRD) obtained via  \nRietveld refinement is shown in Table S-IⅡ , based on previously analyzed data.14 Note that some \nsamples of “barium hexaferrite” did not contain any measurable quantity of this phase. \n \nTable S-ⅡI . Details on studied nanoparticle systems \nName Description XRD Phase ID (vol%)14 \nBa-hex-micro ~ 1 μm particles of barium hexaferrite30,31 100% BaFe 12O19 \nBa-hex-nano-1 ~60 nm particles of mixed Fe-oxides14,15 56% BaFe 12O19, 44% ε-Fe 2O3 \nBa-hex-nano-2 ~60 nm particles of mixed Fe-oxides14 100% ε-Fe 2O3 \nBa-hex-nano-3 ~60 nm particles of mixed Fe-oxides14 51% γ-Fe 2O3, 49% ε-Fe 2O3 \n \nS-8 \n Given the first order reversal curve data (FORC) data obtained on these samples, a reanalysis of \nthe XRD data was conducted. \nRaw XRD spectra of the samples, along with the reference patter ns of the identified phases are \nshown in Figure S-4 . Ba-hex (ICSD-980066757) is the only phase that is detected in  the Micro \nsample which shows intense shar p peaks. However, small amount o f magnetite (ICSD-980158745) \ncould be present since the main  magnetite diffraction peak over laps with Ba-hex peaks. Moreover, \none should note that the diffrac tion patterns of magnetite and maghemite (γ-Fe 2O3) are very similar, \nwhich makes it difficult to discriminate between the two phases  in XRD spectra. Therefore, the \npseudo-single domain (PSD) behavior in the corresponding FORC d iagram is likely due to the \npresence of magnetite and/or maghemite with relatively large sp ontaneous magnetization.  \nThe nano-crystalline nature of the Nano samples leads to the pe ak broadening observed in their \ndiffraction patterns. XRD spectru m of Nano-1 powder reveals the  presence of considerable \namounts of other phases in addition to Ba-hex. The XRD analysis  shows the characteristic peaks \nof ε-Fe 2O3 phase (ICSD-980415250), which intrinsically has an extremely l arge coercive field. \nMinor fractions of magnetite/ma ghemite and hematite (ICSD-98016 1292) are likely to be present \nin this sample. The PSD signature in the FORC diagram can be as signed to magnetite/maghemite, \nwhile the single domain (SD) distributions from FORC are likely  due to Ba-hex phase with distinct \nsize distributions. Hematite’s signature (if any) cannot be obs erved in the FORC, because FORC \nmeasurements are not able to reveal signals from hematite (whic h has very small spontaneous \nmagnetization) when other high-M S phases are simultaneously present.9 Similarly, due to relatively \nlow magnetization of the ε-Fe 2O3 phase, no sign of this phase is observed in the FORC diagram of  \nNano-1 sample, even at very high fields. \nHigh-H C ε-Fe 2O3 is the major phase identified in  the XRD pattern of Nano-2 samp le. This phase is \nresponsible for the distribution peak with very high coercivity  (~ 25 kOe) in the FORC diagram. \nThe PSD component in the FORC diagram is likely due to the pres ence of magnetite/maghemite \nwhose main diffraction peaks can  be seen as low-intensity broad  peaks in the XRD spectrum. It is \nworth mentioning that, despite the large fraction of ε-Fe 2O3 phase in this sample, its distribution \npeak shows lower intensities in the FORC diagram than that of m agnetite/maghemite due to its \nlower magnetization. \nThe major phase in the Nano-3 sample is magnetite/maghemite. Ho wever, XRD shows the presence \nof considerable ε-Fe 2O3 as well. Therefore, the corresponding FORC diagram shows an obv ious \nPSD behavior assigned to magnetite/maghemite, along with a subt le significantly high-H C S D  \ncomponent from ε-Fe 2O3.  S-9 \n  \nFig. S-5.  X-ray diffraction results for t he four samples measured, along  with labeled characteristic \npeaks and patterns for the pote ntial phases.  ICSD numbers for the phases are provided in the text. \n \n  \nS-10 \n \n Isothermal Remnant Magneti zation (IRM) measurements \nThere are several different techniques to statistically “unmix”  the measured magnetization curves, \na procedure which is critical f or enhancing the understanding o f individual magnetic component \ncontributions in magnetic assemblages. One of those applicable techniques is isothermal remnant \nmagnetization (IRM) acquisition curves.16 The IRM acquisition curve method has been used in the \nenvironmental and rock magnetis m community to advance the under standing of natural processes \nby studying magnetic mineral assemblages that carry convolved i nformation on environmental \nmechanisms.17 It was first suggested by Robertson and France16 that IRM acquisition curves of a \nmixed magnetic mineral is a composite curve linearly combined f rom cumulative log-Gaussian \n(CLG) functions of individual mine rals. Each CLG distribution o btained from IRM acquisition \ncurves represents one individual magnetic phase and allows esti mation of its contribution to the \nbulk magnetization. The raw IRM acquisition curves are usually plotted in ‘gradient’ obtained by \ncalculating the first derivative of the IRM curve to easily see  the distribution of different magnetic \nphases.18 In processing a gradient of acquisition plot (GAP),18 each magnetic component is \nrepresented as a log-Gaussian pr obability density function with  a given mean coercivity, dispersion, \nand relative proportion. \nIn a typical IRM measurement, t he sample should be demagnetized  first, then the magnetic field \nwill be applied from zero to a maximum field with a certain ste p. In order to obtain the remanent \ndata of each magnetic field, the magnetic field will switch bac k to zero field and the moment at \nzero field will be recorded as the remanence at this magnetic f ield. 100 remanence data points are \ntaken in the IRM measurement fo r Nano-1 at the magnetic field f rom 0 to 12 kOe. \nThe experimental IRM acquisition curve for Nano-1 is shown in Fig. S-6(a) , and then plotted as a \nGAP19 (Fig. S-6(b) ) in a coordinate of logarithm of H [log(H)], versus gradient o f acquisition, \ndM/dlog(H).18 The GAP, which demonstrates the coercivity distribution, revea ls that there are three \nmagnetic components with distinct coercivities. The sharp peak in Fig. S-6(b)  indicating a Gaussian \ndistribution has a mean coercivity of ~6.5 kOe, which carries t he majority of remnant magnetization, \nand this is consistent with the most intense peak located at ~6 .5 kOe in the FORC results (see main \npaper). Similarly, another peak is fitted at the mean coercivit y ~100 Oe, representing the low \ncoercivity phase in the FORC diagram. However, the third fitted  peak (green curve) shows a mean \ncoercivity of ~3.2 kOe, which is considerably greater than the peak (~2 kOe) observed in the FORC \ndiagram. The mismatch is presumably due to the coupling between  magnetic particles, which \naffects the successfulness of CLG analysis and, as a result, ca n suggest misleading \ninterpretations.16-20 For instance, Heslop et al.20 found that magnetostatic interaction resulted in left-\nskewed individual distributions. Since the coercivity distribut ion of individual components does \nnot always follow the CLG distribution, Egli21 proposed that this mismatch on peak can be \novercome by using skewed generalized Gaussian (SGG) distributio ns, which enables a more \nflexible fitting. Therefore, by applying SGG distribution on th e second peak, the mean coercivity \nof it can be modified to a lower magnetic field. This largest g radient obtained at low coercivity \nphase is presumably due to the imperfect demagnetization proces s which starts from a very small \nnegative magnetization rather tha n exactly zero magnetization. This small imperfection is amplified \nin the low coercivity region. However, the middle peak is now l ocated at a mean coercivity ~2 kOe \nin Fig. S-6(c) which corresponds to the middle p eak as it occurred in the FORC  diagram. S-11 \n \nFig. S-6. Isothermal remnant ma gnetization (IRM) analysis.  (a)  IRM acquisition curve of \n‘barium hexaferrite -lot 1’ and the gradient curves in (b) log scale and (c) linear scale. Three \ncumulative log Gaussian distribu tion functions were used to fit  the gradient curve in log scale.    \n\n \n  \nS-12 \n \n FORC data collections \nFour commercial samples purchased as M-type barium hexaferrite (BaFe 12O19) had FORCs \nmeasured multiple times at Washi ngton State University (WSU) or  Lakeshore. All parameters of \neach FORC measurement are listed in Table S-IV .  \n \nTable S-IV . FORC parameters. H u1 and H u2 are the limits for the y-axis (interaction or bias) on the \nFORC diagram. H c1 and H c2 are the limits for the x-axis (coercivity axis) on the FORC di agram. \nHCal and HSat are the fields used for calibration or saturation, re spectively, applied at the end of \neach FORC. H Ncr is field step between reversal fields and N Forc is the number of FORCs. \nSample H u1 \n(kOe) Hu2 \n(kOe) Hc1 \n(kOe) Hc2 \n(kOe) HCal \n(kOe) HSat \n(kOe) HNcr \n(Oe) NForc Date Figure # \nMicro -1.5 1.5 0 10 11.8 20 59.65 150 1/5/2017 i \nNano-1 -1.5 1.5 0 10 11.8 20 208.84 150 1/3/2017 ii \nNano-1 -1.5 1.5 0 10 11.5 32 131.34 100 6/22/2016 iii \nNano-1 -1.5 1.5 0 28 29.5 33 208.84 150 6/22/2016 iv \nNano-2 -1 1 0 1.5 2.5 32 35.15 100 6/23/2016 v \nNano-2 -1.5 1.5 0 28 29.5 33 208.84 150 1/23/2017 vi \nNano-3 -1.5 1.5 0 1 2.5 32 40.08 100 6/22/2016 vii \nNano-3 -1.5 1.5 0 2 3.72 10 52.6 100 1/5/2017 viii \nNano-3 -0.5 0.5 0 10 10.5 28 73.31 150 7/24/2017 ix \nNano-3 -1.5 1.5 0 28 29.5 33 208.84 150 12/11/2017 x \n \nOn the following pages, iterativ e FORC and FORC diagram example s are given using the different \nparameters shown in the Table a bove.  The final FORC pattern in  the main paper are shown for \ncompleteness.  Note that these presented diagrams do not have t he lower edge artifact13 removed \nduring data processing. \n\n\n\n\nS-13 \n i. Sample: Ba-hex-micro \nDate: 1/5/2017 (Lakeshore PMC 3900 VSM) – NOTE:  this is the sa me data presented in the \nmain paper \n \n Measurement parameters: \nHb1                            -1.500000E+03 Hb2                            +1.500000E+03 Hc1                             0.000000E+00 \nHc2                            +10.00000E+03 \nHCal                           +11.84507E+03 \n  \nS-14 \n ii. Sample: Ba-hex-nano-1-WSU \nDate: 1/3/2017 (Lakeshore PMC 3900 VSM, WSU) \n  \n Measurement parameters:  \nHb1                            -1.500000E+03 \nHb2                            +1.500000E+03 \nHc1                             0.000000E+00 Hc2                            +10.00000E+03 \nHCal                           +11.84507E+03 \nHNcr                           +89.65527E+00 \nHSat                           +20.00000E+03 NForc                           150 \n \n \nS-15 \n  \niii. Sample: Ba-hex-nano-1-10 kOe \nDate: 6/22/2016 (Lakeshore PMC 3900 VSM)  \n \n Measurement parameters: \nHb1         -1.50000E3 Hb2         1.50000E3 \nHc2         10.0000E3 \nHCal        11.5000E3 \nHSat        32.0000E3 NForc       100     \n  \nS-16 \n iv. Sample: Ba-hex-nano-1 -28 KOe  \nDate: 12/12/2017 (Lakeshore 7400 VSM) – NOTE:  this is the same  data presented in the main \npaper\n \n Measurement parameters: \nHb1         -1.50000E3 Hb2         1.50000E3 \nHc2         28.0000E3 \nHCal        29.5000E3 HSat        33.0000E3 \nNForc       150     \n  \nS-17 \n v. Sample: Ba-hex-nano-2-1.5 kOe \nDate: 6/23/2016 (Lakeshore PMC 3900 VSM) \n \n \n Measurement parameters:  \nHb1         -1.00000E3 Hb2         1.00000E3 \nHc2         1.50000E3 \nHCal        2.50000E3 \nHSat        32.0000E3 NForc       100 \n  \nS-18 \n vi. Sample: Ba-hex-nano-2-28 kOe \nDate: 1/23/2017 (Lakeshore 7400 VSM) – NOTE:  this is the same data presented in the main \npaper \n\n Measurement parameters: \nHb1         -1.50000E3 \nHb2         1.50000E3 \nHc2         28.0000E3 \nHCal        29.5000E3 \nHSat        33.0000E3 \nNForc       150     \nS-19 \n vii. Sample: Ba-hex-nano-3-Lakeshore-1 kOe \nDate: 6/22/2016 (Lakeshore PMC 3900 VSM) \n  \n \n Measurement parameters: \nHb1         -1.50000E3 Hb2         1.50000E3 \nHc2         1.00000E3 \nHCal        2.50000E3 \nHSat        32.0000E3 NForc       100       \nS-20 \n viii. Sample: Ba-hex-nano-3-2 kOe \nDate: 1/5/2017 (Lakeshore PMC 3900 VSM) \n  \n Measurement parameters:  \nHb1                            -1.500000E+03 Hb2                            +1.500000E+03 \nHc1                             0.000000E+00 \nHc2                            +2.000000E+03 \nHCal                           +3.726484E+03 \nHNcr                           +52.63163E+00 HSat                           +10.00000E+03 \nNForc                           100 \nS-21 \n ix. Sample: Ba-hex-nano-3-10 kOe \nDate: 7/24/2017 (Lakeshore PMC 3900 VSM) \n  \n Measurement parameters: \nH b 1     - 5 0 0  Hb2    500 \nHc2    10000 \nH C a l     1 0 5 0 0  \nH S a t     2 8 0 0 0  NForc    150 \nS-22 \n x. Sample: Ba-hex-nano-3-28 kOe \nDate: 12/11/2017 (Lakeshore 7400 VSM) – NOTE:  this is the same  data presented in the main \npaper \n \n Measurement parameters: \nHb1         -1.50000E3 Hb2         1.50000E3 Hc2         28.0000E3 \nHCal        29.5000E3 \nHSat        33.0000E3 \nNForc       150        \nS-23 \n III. REFERENCES \n1F. Preisach, Z. Physik A 94, 277 (1935). \n2Y. Cao, K. Xu, W. Jiang, T. Dr oubay, P. Ramuhalli, D. Edwards, B. R. Johnson, and J. McCloy, \nJ. Magn. Magn. Mater. 395, 361 (2015). \n3I. Bodale, L. Stoleriu, and A . Stancu, IEEE Trans. Mag. 47, 192 (2011). \n4A. Berger, Y. Xu, B. Lengsfield , Y. Ikeda, and E. Fullerton, IE EE Trans. Magn. 41, 3178 (2005). \n5G. Bertotti, Hysteresis in magnetism: for physicist s, materials scientists, and engineers  \n(Academic press, 1998). \n6A. R. Muxworthy and A. P. Roberts, in Encyclopedia of Geomag netism and Paleomagnetism , \nedited by D. Gubbins and E. Herre ro-Bervera (Springer, Dordrech t, The Netherland, 2007), p. \n266. \n7A. P. Roberts, C. R. Pike, and K. L. Verosub, J. Geophys. Res. 105, 28461 (2000). \n8J. E. Davies, O. Hellwig, E. E. F ullerton, G. Denbeaux, J. B. K ortright, and K. Liu, Phys. Rev. B \n70, 224434 (2004). \n9C. Carvallo, A. R. Muxworthy, a nd D. J. Dunlop, Phys. Earth Pla n. Inter. 154, 308 (2006). \n10L. Tauxe, T. Mullender, and T. Pick, J. Geophys. Res. 101, 571 (1996). \n11R. K. Dumas, K. Liu, C.-P. Li, I . V. Roshchin, and I. K. Schull er, Appl. Phys. Lett. 91, 202501 \n(2007). \n12R. K. Dumas, C.-P. Li, I. V. Ros hchin, I. K. Schuller, and K. L iu, Phys. Rev. B 75, 134405 \n(2007). \n13J. S. McCloy, K. Korolev, J. V. C rum, and M. N. Afsar, IEEE Tra ns. Mag. 49, 546 (2013). \n14J. McCloy, R. Kukkadapu, J. Cru m, B. Johnson, and T. Droubay, J . Appl. Phys. 110, 113912 \n(2011). \n15D. J. Robertson and D. E. France, Phys. Earth Plant. Int. 82, 223 (1994). \n16D. Heslop, Earth Sci. Rev. 150, 256 (2015). \n17P. P. Kruiver, M. J. Dekkers, and D. Heslop, Earth Planet. Sci.  Lett. 189, 269 (2001). \n18D. Heslop, M. Dekkers, P. Kruiver , and I. Van Oorschot, Geophy.  J. Int. 148, 58 (2002). \n19D. Heslop, G. McIntosh, and M. Dekkers, Geophy. J. Int. 157, 55 (2004). \n20R. Egli, J. Geophys. Res. Sol. Earth 108, 2081 (2003). \n21R. J. Harrison and I. Lascu, G eochem. Geophys. Geosys. 15, 4671 (2014). \n "    },    {        "title": "1801.03114v1.Magneto_electric_effect_in_doped_magnetic_ferroelectrics.pdf",        "content": "arXiv:1801.03114v1  [cond-mat.mtrl-sci]  9 Jan 2018Magneto-electric effect in doped magnetic ferroelectrics\nO. G. Udalov1,2and I. S. Beloborodov1\n1Department of Physics and Astronomy, California State Univ ersity Northridge, Northridge, CA 91330, USA\n2Institute for Physics of Microstructures, Russian Academy of Science, Nizhny Novgorod, 603950, Russia\n(Dated: January 11, 2018)\nWe propose a model of magneto-electric effect in doped magnet ic ferroelectrics. This magneto-\nelectric effect does not involve the spin-orbit coupling and is based purely on the Coulomb in-\nteraction. We calculate magnetic phase diagram of doped mag netic ferroelectrics. We show that\nmagneto-electric coupling is pronounced only for ferroele ctrics with low dielectric constant. We\nfind that magneto-electric coupling leads to modification of magnetization temperature dependence\nin the vicinity of ferroelectric phase transition. A peak of magnetization appears. We find that\nmagnetization of doped magnetic ferroelectrics strongly d epends on applied electric field.\nI. INTRODUCTION\nMultiferroic (MF) materials with strongly coupled fer-\nroelectricity and magnetism is an intriguing challenge\nnow days [1–6]. Among various MF materials the doped\nmagnetic ferroelectrics (DMFE) attract a lot of atten-\ntion since these materials demonstrate the existence of\nelectric polarization and magnetization at room temper-\natures [7–15]. DMFEs are fabricated by doping of ferro-\nelecrics (FE) with magnetic impurities. Transition metal\n(TM)-doped BaTiO 3(BTO) is the most studied material\nin this family. While both order parameters are simulta-\nneously non-zero in DMFE, the coupling between them\n(magneto-electric effect) is very weak and not enough\nstudied [16–18]. Mostly the magneto-electric (ME) effect\nin DMFE is related to spin-orbit interaction leading to\ninfluenceofelectricpolarizationonthematerialmagnetic\nproperties.\nIn the caseofBTOthe roomtemperatureferroelectric-\nity is the internal property of the material. Magnetiza-\ntion appears due to artificially introduced magnetic im-\npurities [7–15]. Several mechanisms of coupling between\nmagnetic impurities are known [19]. At high doping the\nadjacent magnetic moments directly interact with each\nother due to electron wave function overlap. This in-\nteraction is usually antiferromagnetic. At low impurities\nconcentration ( <10%) the direct coupling is not possible.\nHowever, the room temperature ferromagnetic (FM) or-\ndering is observed in this limit. The reason for FM in-\nteraction between the impurities in this case is shallow\ndonor electrons which inevitably present due to defects\nsuch as oxygen vacancies. Donor electrons have weakly\nlocalized wave function spanning over several lattice pe-\nriods. Donor electron interacts with impurities forming\nso-called bound magnetic polaron (BMP) in which all\nmagnetic moments are co-directed. The polaron size es-\nsentially exceeds the interatomic distance. Interaction of\nthe polarons leads to the formation of long-range mag-\nnetic order in the system. Due to large BMP size the\ncritical concentration of defects and magnetic impurities\nat which FM ordering appears can be rather low. BMPs\nand their interaction are well understood in doped mag-\nnetic semiconductors [19–23].In the present work we propose a model of magneto-\nelectric (ME) coupling in DMFE. The idea behind this\nmodel is based on the fact that shallow donor electron\ninteracts not only with magnetic impurities but also\nwith phonons forming not just magnetic polaron but the\nelectro-magnetic one. Magnetic and orbital degrees of\nfreedom are strongly coupled in such a polaron. In con-\ntrast to the most magneto-electric effects based on the\nspin-orbit interaction we consider here the ME coupling\noccurring purely due to the Coulomb interaction. Note\nthat the Coulomb based ME effects were considered re-\ncently in a number of other systems [24–28].\nThe size of magnetic polaron is defined by the wave\nfunction of a donor electron. In its turn the size of\nthe donor electron wave function is defined by electron-\nphonon interaction and depends on the dielectric prop-\nerties of the FE matrix [29–32]. Well known that the\ndielectric constant of FEs strongly depends on tempera-\nture and applied electric field. This opens a way to con-\ntrol magnetic polarons with electric field or temperature.\nFinally, the magnetization of the whole sample becomes\ndependent on the external parameters. In the present\nwork we study this mechanism of ME coupling. In par-\nticular, we study magnetic phase diagram of DMFE and\nshow that one can control magnetization with electric\nfield in such a system.\nIn DMFEs based on FEs with high dielectric constant\nthis effect is negligible, which is consistent with observed\nweak ME effect in doped BTO. A good FE matrix would\nbe Hf0.5Zr0.5O2[33–35] which has a low dielectric con-\nstant (ε <50) strongly dependent on applied electric\nfield. Currently there are no data on Hf 0.5Zr0.5O2doped\nwithmagneticimpurities. AnothermagneticFEwithlow\ndielectric constant is (Li,TM) co-doped zinc oxide [36–\n39]. The ME effect in this material can be also strong.\nThe paper is organized af follows. We present the\nmodel in Sec. II. Properties of single magnetic and elec-\ntric polarons are discussed in Sec. III. Mechanisms of\ninteraction of BMPs are considered in Sec. IV. Magnetic\nphase diagram of DMFE and ME effect in a number of\nsystems are presented in Sec. V.2\nII. THE MODEL\nAs we mentioned in the Introduction, the long range\nmagnetic order in DMFE appears due to interaction of\nBMPs formed by shallow donor electrons and magnetic\nimpurities. To understand magnetic properties of DMFE\nwe first study the interaction of two BMPs.\nConsider FE with magnetic impurities localized at\npointsri\ni. Each impurity has a spin S0. The impuri-\nties concentration is low (below 20% [19]) and there is no\ndirect interaction between them. There are also defects\nwith positions rd\niin the system. Their concentration is\nsmaller than concentration of magnetic impurities. Or-\ndinarily, oxygen vacancies serve as such defects. A de-\nfect creates a point charge potential ( ∼e2/|r−rd\ni|). A\ncharge carrier is bound to each of these defects. The car-\nrier spin is s0= 1/2. The carriers (electrons) interact\nwith impurity spins forming bound magnetic polarons.\nConsider two neighbouring defects. They are described\nby the Hamiltonian\nˆH=ˆHe+ˆHph+ˆHe−ph+ˆHe−imp,(1)\nwhere carriers energy is given by\nˆHe=−/planckover2pi12\n2m∗/summationdisplay\ni∆i−e2\n4πε0ε\n/summationdisplay\ni,j1\n|ri−rd\nj|−1\n|r1−r2|\n.\n(2)\nHerem∗is the effective mass of electron in conduction\nband of the material in the model of rigid lattice, εis\nthe static dielectric constant, ε0is the vacuum dielectric\nconstant, riare the carriers coordinates, indexes iandj\ntake two values, 1 or 2.\nThe interaction between the carriers and impurities is\ngiven by the Hamiltonian\nˆHe−imp=J0/summationdisplay\ni=1,2/summationdisplay\nj(ˆsiˆSj)δ(ri\nj−ri),(3)\nwhereˆsiandˆSjaretheelectronand impurityspin opera-\ntors, respectively. The impurity spin, S0is usually much\nlarger than one half. J0is the interaction constant. In-\nteraction with magnetic impurities leads to formation of\nmagnetic polaron.\nTermsˆHphandˆHe−phin Eq. (1) are the Hamilto-\nnians of phonons and electron-phonon interaction, re-\nspectively [29]. We assume that carrier interacts mostly\nwith longitudinal optical phonons. Generally, coupling\nto acoustical phonons and piezoelectric interaction can\nbe taken into account. We neglect them for simplicity,\nsince they are usually weaker than interaction with opti-\ncal phonons and do not lead to any qualitative changes.\nThe electron-phonon coupling leads to formation of elec-\ntricpolaron. Wewilluseresultsofelectricpolarontheory\nto describe the electron wave function [29]. The whole\nsystem of electron, magnetic impurities and phonons is\nan electro-magnetic bound polaron.A. Dielectric properties of FEs\nBelowwewill showthatdielectricpropertiesofFEma-\ntrix play crucial role in formation of the magnetic state\nof DMFE. Therefore, we need to introduce some model\nof dielectric susceptibility for considered FEs. For sim-\nplicity we assume that dielectric properties of FE matrix\nare isotropic. We introduce the dependence of dielec-\ntric permittivity on applied electric field below the Curie\ntemperature\nε±(E) =εE\nmin+∆εE\n1+(E∓Es)2/∆E2s.(4)\nSuperscripts “+” and “ −” correspond to the upper and\nthe lower hysteresis branch, respectively, Esis the elec-\ntric field at which the electrical polarization switching\noccurs, ∆ Esis the width of the switching region. εE\nmin\nand ∆εEdefine the minimum dielectric constant and its\nvariation with electric field. Equation (4) captures the\nbasic features of dielectric constant behavior. The per-\nmittivity has two branches corresponding to two polar-\nization states. In the vicinity of the switching field, Es\nthe dielectric permittivity, εhas a peak.\nNot much data are currently available on voltage de-\npendencies of ε(E) for FEs with low dielectric constants.\nThe dielectric constant of Hf 0.5Zr0.5O2can be described\nusing the following parameters: εE\nmin= 30, ∆εE= 15,\nEs= 0.1 V/nm, ∆ Es= 0.1 V/nm.\nWemodel thetemperaturedependenceofFEdielectric\nconstant using the following formula\nε(T) =εT\nmin+∆εT\n/radicalBig\n(T−TFE\nC)2+∆T2.(5)\nThis function allows to describe the finite height peak\nat FE phase transition temperature T=TFE\nCas well\nas 1/(T−TFE\nC) dependence in the vicinity of T=TFE\nC.\nFor simplicity we neglect different behavior of dielectric\nconstant above and below TFE\nC. This does not lead to\nany qualitative changes in the properties of considered\nsystem.\nIII. SINGLE POLARON PROPERTIES\nFirst, consider a single electron located at a defect and\ninteracting with impurities and phonons. In the models\nof electric and magnetic polarons the electron wave func-\ntion is chosen in the form of spherically symmetric wave\nfunction\nΨ = Ψ 0e−r/aB, (6)\nwhereaBis the decay length, |Ψ0|2= 1/(πa3\nB).3\nA. Bound magnetic polaron\nFirst, consider the interaction of bound electron with\nimpurities leading to formation of magnetic polaron.\nProperties of a single magnetic polaron was investigated\nin the past [19, 20, 22, and40]. Lets calculatethe average\nelectron-impurities interaction energy, /angbracketleftˆHs\ne−imp/angbracketright(super-\nscript s indicates that we consider a single polaron) for\ngivenaB. Following Ref. [23] we average the magnetic\nenergy over the spatial coordinates\nˆHs\ne−imp=−/summationdisplay\niJ0(ˆsˆSi)|Ψ(ri\ni)|2. (7)\nThe strongest interaction between electron and impu-\nrities appears inside the sphere of radius rp\nrp=aB\n2ln/parenleftbiggJ\nkBT/parenrightbigg\n, (8)\nwhereJ=S0J0|Ψ0|2/2,kBis the Boltzman constant,\nandTis the temperature. We assume that rp>aB\n(J >6kBT). Only in this case magnetic percolation\ncan appear prior to electric percolation (insulator metal\ntransition). Number of impurities within the sphere is\nNp= 4πr3\npni/3, where niis the impurities concentra-\ntion.\nWe assume that the number of impurities within the\nradiusrpis big enough. The total spin of impurities re-\nlaxes much slowerthan the spin ofthe chargecarrier. We\ncan introduce the “classical” exchange field (measured in\nunits of energy) acting on the electron magnetic moment\nB=J0/angbracketleft/summationdisplay\ni|Ψ(ri\ni)|2ˆSi/angbracketright ≈J0/angbracketleft/summationdisplay\nri\ni<rp|Ψ(ri\ni)|2ˆSi/angbracketright ≈\n≈2J/angbracketleft/summationdisplay\nri\ni<rpˆSi/angbracketright/S0.(9)\nThe maximum value of the field is Bmax≈2NpJ. Note\nthan in the absence of an electron the impurities spins\nare independent and the average field value is zero. Field\nfluctuations are given by/radicalbig\n/angbracketleftB2/angbracketright ≈Bmax//radicalbig\nNp.\nEquation (7) can be rewritten as follows\nˆHs\ne−imp=−(ˆsB). (10)\nThis Hamiltonian has two non-degenerate eigenstates.\nForJ > k BTthe average magnitude of the impurities\nfield/radicalbig\n/angbracketleftB2/angbracketright ≫kBTmeaning that even in the case of\nindependent impurities the electron spin should be cor-\nrelated with the instant meaning of the average field B,\nand/angbracketleftˆs/angbracketright= (1/2)(B/B). For the energy averaged over the\nelectron spin states we find\nHs\ne−imp=/angbracketleftˆHs\ne−imp/angbracketright=−B/2. (11)\nNow we determine the field Bby taking into account\nthe interaction of electron and impurities. This interac-\ntion does not lead to the appearance of average B. Theaverage absolute value (fluctuations) of B=/radicalbig\n/angbracketleftB2/angbracketrightis\nnon-zero and is defined by the competition of entropy\nand internal energy. To find the average B, consider the\nstates of the system close to the state with full polar-\nization of impurities (within sphere r <rp). The fully\npolarizedstatemeansthat alltheimpurityspinshavethe\nsame and maximum projection on a certain axis. There\nis only one such state, but it has the lowest energy. If\none reduces total impurities spin by 1, the energy in-\ncreases by J/S0. At the same time the number of states\nwith reduced spin is Np. If ln(Np)≫J/(S0kBT), the\nentropy is the stronger factor than internal energy. In\nthis case donor electron can not couple spins of impu-\nrities and they are almost independent. In this limit\nHs\ne−imp≈ −Bmax/(2/radicalbig\nNp). This corresponds to fluc-\ntuation regime of BMP. In the opposite limit, ln( Np)<\nJ/(S0kBT), the internal energy is dominant. In this case\nall impurities spins are correlated due to interaction with\nthe electron and Hs\ne−imp=−Bmax/2. ForJ= 12kBT\n(which is in agreement with our requirement J0>6kBT\nand corresponds to rp≈1.25aB) andS0= 5/2 we find\nln(Np)<4.8 andNp<120. This estimate is reasonable\nand the number of impurities in BMP is always within\nthis range [19]. In our work we consider the case of well\ncorrelated BMP since only in this limit one can expect\nstrong magnetism.\nSinceNp∼a3\nBandJ∼(a3\nB)−1the magnetic energy of\nBMP is independent of the characteristicsize of the wave\nfunction, aB. Therefore, in this regime the interaction\nwith impurities does not influence the electron spatial\ndistribution.\nB. Electric polaron\nIn previous section we have shown that interaction\nwith impurities does not influence the Bohr radius of the\nbound carrier wave function. Therefore, aBis defined\nby the interaction of the electron with defect charge and\nwith phonons. The problem of electric polaron was stud-\nied in the past [29–32]. There are numerous approaches\nto this problem. We will follow a variational approach\nof Ref. [41]. The Hamiltonian of a single electric polaron\nhas the form\nˆHp=−/planckover2pi12\n2m∗∆−e2\n4πε0εr+ˆHph+ˆHe−ph.(12)\nThe electron wave function is given in Eq. (6). Wave\nfunction of phonons is given in Ref. [41]. The radius\nof electric polaron, aBis defined by the minimization of\naverage energy /angbracketleftˆHp/angbracketrightwith respect to aB. In the case of\nstrong coupling between the carrier and phonos the Bohr\nradius is given by [41]\n(aB)−1=m∗e2\n16/planckover2pi12/parenleftbigg11\nε+5\nε∞/parenrightbigg\n, (13)\nwhereε∞is the optical dielectric constant.4\nInFEmaterialsthestaticdielectricconstant εdepends\non temperature Tand external electric field E. There-\nfore, one can control the donor electron wave function\nsizeaBwith external electric field or by varying temper-\nature.\nFor materials with large static dielectric constant ε∼\n1000 (as in BTO, for example) the Bohr radius becomes\nindependent of ε(aB= 16/planckover2pi12ε∞/(5m∗e2)). In this case\nvariation of εwith temperature or electric field does not\ninfluence the polaron size.\nIn a number of FEs the static dielectric constant is\nof the same order as the optical one. For example, in\nHf0.5Zr0.5O2the static dielectric constant is about 30\nwhile optical one is about 4.5 (there is no experimen-\ntal data on ε∞in Hf0.5Zr0.5O2, therefore we use data on\nHfO2and ZrO 2for estimates). Static dielectric constant\nofthis material depends on applied electric field [35]. Ac-\ncording to Eq. (4) the FE dielectric constant has a peak\nin the vicinity of switching field. The polaron radius\ngrows with ε. Thus, the aB(E) has also a peak in the\nvicinity of switching field, Es. Variation of εwith field in\nHf0.5Zr0.5O2is about 50% ( εE\nmin= 30, ∆εE= 15). This\nleads to 10% changes of polaron radius.\nIn Li-doped ZnO oxide the static dielectric constant\nstrongly depends on temperature and is not very large.\nFE properties strongly depend on Li concentration. FE\nphase transition in these materials is usually above the\nroom temperature [42–45]. In the vicinity of the FE\nCurie temperature the static dielectric constant varies\nfrom 5 to 60. Such a strong growth of the dielectric\nconstant can increase the polaron radius twice.\nIV. INTERACTION OF TWO\nELECTRO-MAGNETIC POLARONS\nIn this section we consider magnetic interaction of two\nelectro-magnetic polarons in DMFE. We introduce here\nmagnetic moments of these polarons. They have direc-\ntionsm1,2. Since there is a large number of impurities\nin each polaron we can treat these quantities as classical\nvectors. Thedistance R=|rd\n1−rd\n2|betweenthesetwopo-\nlaronsexceeds2 aBand2rp. In thiscasetheinter-polaron\nmagnetic interaction is weak comparing to magnetic en-\nergy of a single polaron. There are three mechanisms\nof magnetic coupling between polarons: 1) exchange due\nto the Coulomb interaction in the Hamiltonian Eq. (2)\n(Heitler-London interaction); 2) magnetic coupling due\nto kinetic energy term in the Hamiltonian Eq. (2) (su-\nperexchange); and 3) magnetic coupling mediated by im-\npurities, Eq. (3).\nA. Heitler-London interaction between polarons\nConsider the Hamiltonian in Eq. (2). If two defects\nare far away from each other the Hamiltonian can be\nsplit into zero order Hamiltonian of two non-interactingcarriers\nˆH(0)\ne=−/planckover2pi12\n2m∗/summationdisplay\ni∆i−e2\n4πε0ε/summationdisplay\ni1\n|ri−rd\ni|,(14)\nand perturbation term\nˆH(1)\ne=−e2\n4πε0ε/parenleftbigg1\n|r1−rd\n2|+1\n|r2−rd\n1|−1\n|r1−r2|/parenrightbigg\n.\n(15)\nThe wave functions of non-interacting electrons are de-\nnoted as Ψ 1,2. In the first order perturbation theory the\nHamiltonian in Eq. (15) produces the spin-dependent in-\nteraction between carriers\nˆHHL= 4HHL(ˆs1ˆs2) =HHLcos(θ).(16)\nHere we introduce the angle θbetween magnetic mo-\nments of polarons. Since the polaron magnetic moment\nis large we can treat it as classical value. As was shown\nin the previous section the average spin of electron is co-\ndirected with corresponding polaron magnetic moment.\nThe exact formulas for the exchange constant HHLis\ngiven elsewhere [46]. The only important thing for us is\nthat it exponentially decays with the distance between\ndonor centres Ras exp(−2R/aB) and is inversely pro-\nportional to ε. Thus, we can write\nHHL=HHL\n0e−2R/aB\nε. (17)\nGenerally, the constant HHL\n0can be found numerically\nfor wave functions given by Eq. (6).\nB. Superexchange\nMagnetic interaction between two electrons appears\nalso due to virtual hopping of electrons between defect\nsites, so-called superexchange. The coupling appears in\nthe second order perturbation theory with respect to the\nhopping matrix elements, t=/angbracketleftΨ1Ψ1|ˆHe|Ψ1Ψ2/angbracketright. Effec-\ntive Hamiltonian describing the superexchange is given\nby [46]\nˆHse=4t2\nU(ˆs1ˆs2). (18)\nHereUis the onsite repulsion of electrons calculating as\nU=/angbracketleftΨ1Ψ1|ˆHe|Ψ1Ψ1/angbracketright. We assume that Uis mostly de-\ntermined by the Coulomb interaction between two elec-\ntrons situated at the same site. Uis inversely propor-\ntional to the size of the Bohr radius and the system di-\nelectric constant, U∼(1/(εaB)). Hopping matrix ele-\nment decreases with increasing of distance between the\ndefects,t2∼exp(−2R/aB). Finally, we arrive to the\nfollowing expression for the interaction energy\nˆHse= 4Hse\n0aBεe−2R/aB(ˆs1ˆs2) =\n=Hse\n0aBεe−2R/aBcos(θ) =Hsecos(θ).(19)5\n/s112/s111/s108/s97/s114/s111/s110/s32/s50/s97\n/s66/s114\n/s112\n/s115\n/s49/s83\n/s105\n/s115\n/s50/s83\n/s105\n/s82/s97\n/s66/s73/s110/s116/s101/s114/s97/s99/s116/s105/s111/s110/s32/s114/s101/s103/s105/s111/s110\n/s112/s111/s108/s97/s114/s111/s110/s32/s49\n/s50/s40 /s97/s32/s32/s82 /s41\n/s66/s49/s47/s50\n/s105\nFIG. 1. (Color online) Two electro-magnetic polarons in\nDMFE separated by a distance R. Red arrows show direction\nof electrons average magnetic moments. Angle between mag-\nnetic moments of two such electrons is θ. The electrons wave\nfunction characteristic size is aB. Black arrows show mag-\nnetic moments of impurities in DMFE. Within the magnetic\npolaron radius rpthey are co-directed with average impurity\nmagnetic moment. In the central area between the polarons\nthere is a lens shaped interaction region Ω i. Impurities spins\nin this region are not fully polarized by donor electrons but\ncorrelated with them leading to interaction between the car -\nriers. Width of the interaction region is about aB. Lateral\nsize of the region is about 2√RaB.\nC. Impurities mediated interaction\nConsider the situation where the distance between po-\nlaronsRexceeds the single polaron size rp(see Fig. 1).\nBeyond the polaron radius rpthe interaction between\nelectron and impurities much weaker than inside the po-\nlaron. In the central region between two polarons im-\npurities interact with both electrons leading to magnetic\ninteraction between carriers (see Fig. 1). We will fol-\nlow the simplified approach of Ref. [23] to calculate this\ncoupling. According to Ref. [23] the main contribution\nto the inter-polaron interaction is given by lens-shaped\nregionwith lateral size of√RaBand width of aB. We as-\nsume that interaction of electron with impurities in this\nregions is independent of impurity position. Magnetic\nenergy of this region is given by\nˆHp−p\ne−imp= 2Je−R/aB(ˆs1+ˆs2)/summationdisplay\nj∈ΩiˆSj/S0,(20)\nwhere summation is over the region of interaction Ω i.\nNumber ofimpurities inside the interaction regioncan be\nestimated as Ni=πa2\nBRni. Treating the total polarons\nspins as classical magnetic moments we obtain\nˆHp−p\ne−imp= 2Je−R/aBcos(θ/2)/summationdisplay\nj∈ΩiˆS(z)\nj/S0,(21)\nwhereθis the angle between average magnetic moments\nof polarons. We assume that both polarons are similarand have the same magnetic moment. ˆS(z)\njis the pro-\njection of the impurity spin on the direction m1+m2.\nInteraction of donor electron and impurities in the region\nΩiis weak. Therefore, the average magnetic moment\ncreated by this interaction is defined as /angbracketleft/summationtext\nj∈ΩiˆS(z)\nj/angbracketright ≈\n2NiS0Je−R/aBcos(θ/2)/(3kBT). Introducing this result\ninto Eq. (21) we get the averageinteraction energyof two\npolarons\nHp−p\ne−imp=4NiJ2e−2R/aBcos2(θ/2)\n3kBT=Hp−p(cos(θ)+1).\n(22)\nV. MAGNETIC PHASE DIAGRAM OF DMFE\nThe distance at which two polarons can be considered\nas coupled ( rc) is defined by the condition\n|HHL+Hse+Hp−p|=kBT. (23)\nNote that the Heitler-London coupling, HHL>0, and\nthe superexchange, Hse>0, produce antiferromagnetic\n(AFM)couplingwhileimpuritymediatedcouplingisFM,\nHp−p<0. On one hand the first two interactions de-\ncay faster with distance ( e−2R/aB) than the third one\n(e−R/aB). But on the other hand the impurity mediated\ninteraction depends on concentration niand tempera-\nture. It decreases with increasing of temperature and re-\nducing of ni. Experimental results on DMFE show that\nin most cases FM order appears at low magnetic impuri-\nties concentration[9, 11, 13, 16, 17, and 47] meaning that\nimpurity mediated coupling dominates. However, AFM\norderis also reported in DMFEs with low impurities con-\ncentration [7].\nIn the case of HHL\n0=Hse\n0= 0, Eq. (23) for the inter-\naction distance rcat given temperature Tturns into\nkBT=S0J0√nirce−rc/aB\n√\n3πa2\nB. (24)\nApproximately one can write\nrc∼aB/parenleftbigg\nln/bracketleftbiggS0J0\n2πa3\nBkBT/bracketrightbigg\n+1\n2ln/bracketleftbigg\na3\nBniln/parenleftbiggS0J0\n2πa3\nBkBT/parenrightbigg /bracketrightbigg /parenrightbigg\n.\n(25)\nAccording to percolation theory [48] the long range\nmagnetic order in the system of randomly situated po-\nlarons appears approximately at rcn1/3\nd= 0.86, where\nndis the defects concentration. Introducing rcfrom this\nrelation into Eq. (23) one can find the ordering tempera-\nture. Depending on the sign of the total interaction the\nordering can be either FM or AFM (or superspin glass\nstate).\nFirst, consider the case when the polaron-polaron in-\nteraction is the dominant one and we can neglect the\nHeitler-London and superexchange contributions. In this\ncasethereisonlyFMtype interactionbetweenimpurities6\n/s48/s46/s48/s48/s48 /s48/s46/s48/s50/s53 /s48/s46/s48/s53/s48 /s48/s46/s48/s55/s53 /s48/s46/s49/s48/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48/s84 /s32/s40/s75/s41\n/s110 /s32/s32/s40/s49/s47/s110/s109/s32/s32/s41\n/s105/s51/s84 /s32/s40/s75/s41/s77 /s47/s110 /s32\n/s105/s110/s32/s32/s61/s32 /s48/s46/s55/s56/s32/s49/s47/s110/s109\n/s105/s51/s80/s77/s32/s115/s116/s97/s116/s101\n/s70/s77/s32/s115/s116/s97/s116/s101/s84/s70/s69\n/s67\n/s52/s48/s48 /s54/s48/s48 /s56/s48/s48/s48/s49/s50/s66/s84/s79\nFIG. 2. (Color online) Approximate magnetic phase transi-\ntion temperature Eq. (26) as a function of impurities con-\ncentration ni. The system parameters, which correspond to\nBTO doped with Fe, are in the text. The inset is the mag-\nnetic moment per Fe impurity as a function of temperature\nforni= 0.78 1/nm3(5% doping).\nand only the FM/paramagnetic (PM) transition is possi-\nble. The transition temperature is given by the equation\nkBT=S0J0√0.86niexp(−0.86/(aBn1/3\nd))√\n3πn1/6\nda2\nB.(26)\nNote than according to Eqs. (5), (4) and (13) the Bohr\nradius,aB(T,E) depends on temperature and external\nelectric field. This makes the PM/FM transition tem-\nperature more complicated function of niandndand\nmakes it dependent on electric field.\nDimensionless magnetization of the DMFE is given by\nthe following equation [23 and 48]\nM(T) =S0niVinf((rc(T))3nd), (27)\nwhereVinfis the relative volume of infinite cluster (or\nprobability that an impurity belongs to an infinite clus-\nter) in site percolation problem. We found the func-\ntion using Monte-Carlo simulations approach developed\nin Ref. [49].\nA. BaTiO 3based DMFE\nFigure 2 shows the magnetic phase diagram of DMFE\nwith the following parameters. Impurities magnetic mo-\nment is S0= 5/2. High frequency dielectric constant\nisε∞= 5.8 and the static one is ε= 1000. We chose\nsuch a value of m∗thataB= 0.45 nm. This corre-\nsponds to BTO crystal with Fe impurities. Concentra-\ntion of defects (oxygen vacancies) is about 0.043 nm−3/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s50/s48/s48/s52/s48/s48/s84 /s32/s40/s75/s41\n/s110 /s32/s32/s40/s49/s47/s110/s109/s32/s32/s41\n/s105/s51/s84 /s32/s40/s75/s41/s77 /s47/s110 /s32\n/s105\n/s110/s32/s32/s61/s32 /s49/s46/s55/s32/s110/s109\n/s105/s45/s51/s80/s77/s32/s115/s116/s97/s116/s101\n/s70/s77/s32/s115/s116/s97/s116/s101/s110/s32/s32/s61/s32 /s49/s32/s110/s109\n/s105/s45/s51\n/s110/s32/s32/s61/s32 /s48/s46/s55/s32/s110/s109\n/s105/s45/s51/s84\n/s67/s70/s69/s84/s70/s69\n/s67/s40/s76/s105/s44/s84/s77/s41/s32/s99/s111/s45/s100/s111/s112/s101/s100/s32/s90/s110/s79\n/s48 /s50/s48/s48 /s52/s48/s48/s48/s49/s50\nFIG. 3. (Color online) Magnetic phase diagram of (Li,TM)\nco-doped ZnO. The system parameters are provided in the\ntext. The inset is the magnetic moment per TM impurity as\na function of temperature for ni= 1.7 nm−3(solid blue line),\n1 nm−3(dashed green line) and 0 .7 nm−3(dotted black line).\n(0.27%, lattice period in BTO is about 0 .4 nm). Parame-\nterS0J0= 6·104K·nm3. At impurities concentration of\nabout 5% this gives spin splitting of the carrier of about\n2.4 eV. This splitting occurs due to interaction with all\nimpurities within the polaron. We neglect the Heitler-\nLondon and superexchange contributions.\nThe figure shows magnetic state of the system as a\nfunction of impurities concentration and temperature.\nThe system is FM at low temperatures and high impu-\nrities concentration and is PM at high temperatures and\nlow concentration of magnetic impurities. The curve in\nFig. 2 shows approximate boundary between these two\nmagnetic states. For such a high dielectric constant the\nBohr radius aBis independent of εand the temperature\ndependence of the dielectric constant does not play any\nrole in magnetic properties of the material.\nTheinsetshowsmagnetizationasafunction oftemper-\nature for impurities concentration ni= 0.78 1/nm3(5%\nfor BTO crystal). Magnetic phase transition appears at\nT≈650 K. This is in agreement with experiment in\nRef. [9]. Ferroelectric phase transition in BTO appears\naroundTFE\nC=360 K. In this region the dielectric con-\nstant has a strong peak. However, because of very large\nεthe ME effect is weak and no peculiarities appear in\nthe vicinity of TFE\nC.\nB. ZnO based DMFE\n(Li,TM) co-dopedZinc oxideis oneofthe most studied\ndoped magnetic ferroelectrics [50–52]. Ordinarily, both\nferroelectricity and magnetism in these materials appear\ndue to doping. In contrast to “classical” FEs such as7\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48/s114/s32/s32/s44/s114/s32/s32/s47/s50/s44/s32/s97/s32/s32/s32 /s40/s110/s109/s41\n/s84 /s32/s40/s75/s41/s112\n/s66/s99/s114/s32 /s32/s47/s50\n/s99\n/s114\n/s112\n/s97\n/s66/s84\n/s67/s70/s69\nFIG. 4. (Color online) The Bohr radius aB, BMP radius rp\nand BMP interaction distance rcas a function of temperature\nfor DMFE with εdescribed by Eq. (4) with εmin= 25, ∆ ε=\n45,TFE\nC= 370 K and ε∞= 4. Effective mass is chosen such\nthat the Bohr radius at zero temperature is aB= 0.75 nm.\nBTO, the ZnO based multiferroics have relatively low\ndielectric constant. Inevitable defects in doped ZnO ma-\nterials also provides shallow donor states. Due to low di-\nelectric constant of the material the Bohr radius of these\nstates can be temperature dependent.\nMagnetism in TM doped ZnO was theoretically and\nexperimentally studied in numerous works [36–39]. Two\ndistinct cases were recognized when the material is either\ndiluted magnetic semiconductor (DMS) or diluted mag-\nnetic insulator (DMI) [38]. In the first case carriers are\ndelocalizedonthescaleofthewholesampleandmagnetic\norderingappearsduetoRuderman-Kittel-Kasuya-Yosida\ninteraction. In the second case carriers are strongly lo-\ncalized and the coupling is due to magnetic polarons. We\nwill assume the small concentration of defects and BMP\nbased coupling.\nDielectric and magnetic properties in ZnO-based ma-\nterials strongly depend on the dopand type, concentra-\ntion and fabrication procedure. Figure 3 shows magnetic\nphase digram of the DMFE with parameters close to\n(TM,Li) co-doped ZnO. Impurities magnetic moment is\nS0= 5/2andS0J0= 3.3·104K·nm3givingthespinsplit-\nting ofthe electronofabout 1.7eVfor impurities concen-\ntrationni= 1 nm−3. High frequency dielectric constant\nisε∞= 4 [19]. Static dielectric constant strongly de-\npends on temperature with εT\nmin= 25, ∆ εT= 45 and\nthe ferroelectric Curie temperature, TFE\nC= 370 K [53].\nWe chose m∗such that aB= 0.75 nm at zero tempera-\nture [19]. Concentration of defects (oxygen vacancies) is\nabout 0.02 nm−3(∼0.1%). We neglect Heitler-London\nand superexchange contributions.\nMagnetic phase transition curve has a peculiarity in\nthe vicinity of FE phase transition temperature TFE\nC=\n370 K. The peculiarity is related to non-monotonic be-havior of the BMP coupling radius rcin the vicinity of\nTFE\nC(see Fig. 4). Since static dielectric constant is com-\nparable to optical one and it has a peak as a function\nof temperature at T=TFE\nCthe Bohr radius also has a\npeak in this region. Increasing aBleads to the increase of\nBMP interaction distance rcand enhancement of mag-\nnetic properties. Note that while the Bohr radius aBand\ninteraction distance rchave a peak in the vicinity of FE\nphase transition, the BMP radius rphas a deep (at least\nfor given parameters).\nInset in Fig. 3 shows magnetization of DMFE as a\nfunction of temperature for several concentrations of TM\nimpurities. Magnetization also has a peak at T=TFE\nC.\nSuch a peak is the consequence of coupling between elec-\ntric and magnetic subsystems in this material and can be\nconsidered as magneto-electric effect.\nIn Refs. [51 and 53] temperature dependence of\n(Li,TM) co-doped ZnO magnetization were studied in\nthe vicinity of FE phase transition. No peculiarities\nin magnetization in the vicinity of the FE transition\npoint were observed. Two possible reasons for the ab-\nsence of magneto-electric coupling in these particular\nsamples may exist. The first one is that samples stud-\nied in Ref. [51] are nanorods of (Li,Co) co-doped ZnO\nwith very large surface/volume ratio. The origin of mag-\nnetism in such structures is also under question. On one\nhand the conductivity of these samples is small mean-\ning that the material is DMI with possible BMP-based\nmagnetism. On the other hand the magnetism can be\nrelated to surface effects as often happens in nanoscale\nmetal oxides [54 and 55]. The second possible reason is\nthat the model of electric polaron described in Sec. IIIB\nis not applicable to this particular material. Ferroelec-\ntricity in this material is related to Li doping and oxygen\nvacancies [51]. Electric dipoles in this material are in-\nhomogeneously spread across the sample. Therefore, the\nlow-frequency dielectric constant related to these dipoles\nshould be also rather inhomogeneous. Inhomogeneity of\ndielectric constant probably appears at the same spatial\nscale as the distance between magnetic polarons in the\nsystem. Therefore, electric dipoles responsible for ferro-\nelectricity and static dielectric constant do not influence\nthe polaron size.\nC. Hf xZr1−xO2based DMFE\nAnother FEs family with low dielectric constant is ma-\nterials based on HfO 2. Doping of Hf oxide with vari-\nous elements leads to the appearance of FE properties\n(spontaneous electric polarization, hysteresis loop, elec-\ntric field dependent dielectric constant) [33–35]. In the\npresent work we discuss Hf 0.5Zr0.5O2FE [35]. This ma-\nterial is homogeneousin contrast to FEs based on weakly\ndoped zinc oxide. This allows to expect that variation of\ndielectric constant in this material leads to variation of\npolaron size. The source of carriers in this material is\nalso oxygen vacancies. No data is available on magnetic8\n/s45/s51/s48/s48 /s45/s50/s48/s48 /s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54/s50/s46/s48/s77/s47/s110\n/s69 /s32/s32/s40/s77/s86/s47/s109/s41/s110/s32/s32/s61/s32 /s48/s46/s56/s32/s110/s109\n/s105/s45/s51\n/s110/s32/s32/s61/s32 /s48/s46/s54/s32/s110/s109\n/s105/s45/s51\n/s110/s32/s32/s61/s32 /s48/s46/s52/s32/s110/s109\n/s105/s45/s51/s72/s102/s90/s114/s79/s105/s45/s69\n/s115\nFIG. 5. (Color online) Dimensionless magnetic moment per\nmagnetic impurity (maximum moment is S0= 5/2) as a func-\ntion of applied electric field Eforni= 0.8 nm−3(solid lines),\n0.6 nm−3(dashed lines) and 0 .4 nm−3(dotted lines). Arrows\nshow the hysteresis bypass direction.\ndoping of this material.\nDielectric constant of Hf 0.5Zr0.5O2depends on exter-\nnal electric field. Therefore, one can control magnetic\nproperties of Hf 0.5Zr0.5O2doped with magnetic impuri-\nties using electric field. Figure 5 shows the dependence\nof magnetization of DMFE on external electric field at\nroom temperature and for different impurity concentra-\ntions. Other parameters are chosen as follows. The Bohr\nradius at zero electric field is 0.5 nm. Impurities mag-\nnetic moment is S0= 5/2 andS0J0= 3.3·104K·nm3\ngiving the spin splitting of the electron of about 1.1 eV\nfor impurities concentration ni= 1 nm−3. Defects con-\ncentration is nd= 0.05 nm−3(∼0.6% in the case of\nHf0.5Zr0.5O2which has the lattice constant of 0.5 nm).\nOptical dielectric constant ε∞= 4.5. Static dielectric\nconstant as a function of electric field is given by Eq. (4)\nwithεE\nmin= 30, ∆εE= 15,Es= 100 MV/m and switch-\ning regionwidth ∆ Es=Es[35]. The Heitler-London and\nsuperexchange contributions are neglected.\nStatic dielectricconstantdepends on electricfield lead-\ning to electric field dependence of magnetization in the\nsystem (ME effect). εdemonstrates hysteresis behav-\nior causing hysteresis of magnetization as a function of\nelectric field E. Dielectric constant reaches its maximum\nat the switching field ±Es. According to Eq. (25) the\nBMP interaction distance grows with ε. Therefore, the\nmagnetization has peaks at E=±Es. While interaction\ndistance variation is not large (about 10%) the magneti-\nzation variation is significant.D. Influence of Heitler-London and superexchange\ncontributions\nSince the Heitler-Lodon and superexchange interac-\ntions are antiferromagnetic ones, they compete with the\nBMP-based coupling. These interactions decay faster\nwith distance between defects than the impurities medi-\nated magnetic coupling, but they do not depend on con-\ncentration niand temperature. Therefore, at low impu-\nrities concentration and high temperature antiferromag-\nnetic interactions can dominate leading to antiferromag-\nnetic (or spin glass) ordering. At temperature indepen-\ndent dielectric constant the AFM ordering temperature\ncan be found as follows\nTAFM=HHL+Hse±/radicalBig\n(HHL+Hse)2+4˜Hp−p\n2,\n(28)\nwhere˜Hp−p=−Hp−pkBT. Solutions exist only if\nHHL+Hse>2√Hp−pkBT. This condition is alway sat-\nisfied at low enough impurities concentration. Compe-\ntition between AFM and FM interactions in DMS was\nconsidered in Ref. [22].\nFigure6 showsmagnetic phasediagramofDMFE with\nsignificant contribution of the Heitler-London and su-\nperexchange interactions. The following parameters are\nused.S0= 5/2,S0J0= 4·104K/nm3,nd= 0.02\nnm−3,m∗is chosen such that the Bohr radius away from\nTFE\nCis about 0 .75 nm,ε∞= 4,εT\nmin= 25, ∆ εT= 35,\n∆T= 20 K, TFE\nC= 370 K, HHL\n0= 3.5·107K (main\ngraph),Hse\n0= 5·103K/nm (red curves), 10 ·103K/nm\n(green curves), 15 ·103K/nm (blue curves). In the inset\nwe useHHL\n0= 5·106K,Hse\n0= 3·103K/nm.\nIn contrast to the previously considered cases the re-\ngion of AFM ordering appears at finite HHLand/orHse.\nFM/PM boundary also changes. Region of AFM order-\ning exist only at low impurity concentration, since only\nin this case AFM interactions overcome strong impurity\nmediated FM coupling. The main figure shows the case\nwhere direct interactions ( HHLandHse) are strong and\ninduce AFM ordering at high temperatures close to FE\nphase transition. Note that Heitler-London interaction\ndecreases with increasing of dielectric constant while the\nsuperexchange interaction behaves oppositely. There-\nfore, behavior of the phase boundaries strongly depends\non the ratio between these two contributions. Superex-\nchange mostly influences the region in the vicinity of FE\nphase transition. AFM region grows and FM region de-\ncreases with increasing of Hse\n0in the vicinity of TFE\nC.\nHeitler-London interaction influences the phase diagram\naside ofTFE\nC, but this influence is mostly quantitative.\nInset shows the case when HHLand/orHseare rela-\ntively small and do not lead to magnetic ordering in the\nvicinityofFEphasetransition. In thiscasemodifications\nof FM/PM boundary is weak. AFM region exists at low\ntemperatures and low impurities concentration.9\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s50/s48/s48/s52/s48/s48\n/s65/s70/s77/s32/s115/s116/s97/s116/s101/s80/s77/s32/s115/s116/s97/s116/s101\n/s70/s77/s32/s115/s116/s97/s116/s101/s84 /s32/s40/s75/s41\n/s110 /s32/s32/s40/s49/s47/s110/s109/s32/s32/s41/s65/s70/s77/s32/s115/s116/s97/s116/s101/s84 /s32/s40/s75/s41\n/s110 /s32/s32/s40/s49/s47/s110/s109/s32/s32/s41\n/s105/s51/s80/s77/s32/s115/s116/s97/s116/s101\n/s70/s77/s32/s115/s116/s97/s116/s101\n/s105/s51/s72/s32/s32/s32/s32 /s103/s114/s111/s119/s115/s115/s101\n/s48/s46/s48/s48 /s48/s46/s48/s51 /s48/s46/s48/s54/s48/s52/s48/s56/s48\nFIG. 6. (Color online) Magnetic phase diagram of DMFE\nwith strong Heitler-London and superexchange interaction s.\nSolid lines show boundary between PM and AFM states,\ndashed lines demonstrate PM/FM transition. All curves are\nplotted for the same HHL\n0. Red lines correspond to the sys-\ntem with the smallest superexchange contribution, green li nes\nshow magnetic phase diagram for a system with intermediate\nHse\n0and blue curves are for the highest Hse\n0. All system pa-\nrameters are provided in the text. Main figure shows the\nsituation of strong direct coupling leading to AFM ordering\nin the vicinity of FE phase transition. Inset shows the case\nwhen direct coupling ( HHLandHse) is weak andinduce AFM\nstates only at low temperature.\nVI. CONCLUSION\nIn the present work we proposed a coupling mecha-\nnismofmagneticandelectricdegreesoffreedomindoped\nmagnetic ferroelectrics. Magnetic order in DMFE ap-\npears due to formation and interaction of BMPs. There\nare three different contributions into interaction betweenmagnetic polarons. All these contributions depend on di-\nelectric constant of the FE matrix. The most significant\nis the impurities mediated interaction between polarons.\nIt depends on the radius of polaron wave function. Due\nto interaction with phonons this radius linearly depends\non the dielectric constant of FE matrix. Since the dielec-\ntric constant of FEs can be controlled with applied field\nor varying temperature, one can control the interpolaron\ninteraction and magnetic state of the whole system. Pe-\nculiarity of this magneto-electric effect is that it does not\ninvolve the relativistic spin-orbit coupling and relies only\non the Coulomb interaction.\nWe calculated magnetic phase transition temperatures\nasafunctionofimpurities concentrationandshowedthat\nstrong temperature dependence of dielectric permittivity\nin the vicinity of FE phase transition leads to essential\nmodification of magnetic phase diagram. We found mag-\nnetization as a function of temperature and showed that\nit has a peak in the vicinity of FE phase transition. This\npeak is a consequence of ME effect appearing in DMFE.\nWe calculated magnetization as a function of electric\nfield in DMFE and demonstrated that magnetic moment\nof the system can be effectively controlled with applied\nbias. Magnetization shows hysteresis behavior as a func-\ntion of electric field. It has two peaks associated with FE\npolarization switching.\nStrong magneto-electric coupling can appear only in\nDMFE with low dielectric constant such as (Li,TM) co-\ndoped ZnO or Hf 0.5Zr0.5O2. TM-doped BaTiO 3is not\na very promising candidate to observe our effect due to\nvery larger dielectric constant.\nVII. ACKNOWLEDGEMENTS\nThis research was supported by NSF under Cooper-\native Agreement Award EEC-1160504 and NSF PREM\nAward. O.U. was supported by Russian Science Founda-\ntion (Grant 16-12-10340).\n1M. Lilienblum, T. Lottermoser, S. Manz, S. M. Selbach,\nA. Cano, and M. Fiebig, Nature Physics 11, 1070 (2015).\n2W. Eerenstein, N. D. Mathur, and J. F. 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R.Rao, NanoToday 4, 96(2009)."    },    {        "title": "1801.03479v1.The_hidden_magnetization_in_ferromagnetic_material__Miamagnetism.pdf",        "content": "The h idden magnetization  in ferromagnetic material : \nMiamagnetism  \n \nSouri M ohamed  Mimoune1,a, Lotfi Alloui1, Mourad Hamimid2, Mohamed Lotfi  Khene1, \nNesrine Badi1 and Mouloud Feliachi3 \n \n1LMSE, Université de Biskra, BP 145, 07000 Biskra, Algérie  \n2Université de Bordj Bou Arréridj, 34265 Bordj Bou Arréridj, Algérie  \n3IREENA -IUT, CRTT, Boulevard  de l’Université , BP 406, 44602 Saint -Nazaire Cedex, France  \nacorresponding author : s.mimoune@univ -biskra.dz  \n \nAbstract : This paper presents the hidden magnetization feature s of ferromagn etic materials : called mia magnetism. As \nwe know, we have several form s of magnetization:  the diamagnetism, the paramagnetism, the ferromagnetism  etc. The \nmain character of the diamagnetism is that its magnetic susceptibility  is negative ( from -10-9 for gas and -10-6 for \nliquid and solid to -1 for superconducting material s of type I ) and it is not  less th an -1 unless for special materials like \nmetamaterials at high frequencies . The mia magnetism has  the character that the magnetic susceptibility  can reach at low \nfrequencies a negative value of -155 of magnitude  leading to a negative permeability . We can  not see it because it is \nhidden by the ferromagnetic character which has a high positive magnetic susceptibility. We use the  discrete Fourier \ntransform to illustrate this hidden character and the hysteresis model can be represented only by harmonics of (2n+1) f0 of \nmagnitude.  This magnetization follow s a Boltzmann  distribution for the modulus of theses harmonics.  \n  \n1. Introduction  \nIn the literature, different models are used to represent  the \nferromagnetic character of the hysteresis . We can cite some \nmodels : the Jiles Atherton model  [1], the P reisach model  \n[2], the H ausser model [3] and other.  These models are \nsupposed to be phenomenological ones  and are quite good \nto represent the hysteresis . This paper presents  an \nalternative  representation of the hyster esis by using Fourier \nseries of the flux density B and the magnetic field strength \nH. This representation shows a hidden characte r of \nmagnetization which we call  Miamagnetism . The \nMiamagnetism illustrate s a high diamagnetic character with  \nhigh negative magnetic susceptibility and magnetic \npermeability  in low frequencies . The negative permeability \nis also seen in metamaterial [4] and in a composite medium \ninterspaced split ring resonators and wires t hat exhibits a \nfrequency region in the micro wave regime with negative \nvalue of perme ability . This paper explain s also the first \nmagnetization curve  behaviour . \n \n2. Purpose  \nWhen applying a sinusoidal voltage throw a conducting \nwinding to a closed ferromagnetic material, a n hysteretic  \nbehavio ur occurs.  The magnetomot ive force  acting on \nferromagnetic material produces at saturated regime  a non \nsinusoidal flux density B and a distorted magnetic field H. \nFigure 1 gives the experimental result of the voltage in the \nsecondary and the current in the primary  in relative units  at \nthe frequency f0 equal to 50 Hz . Figure 2 shows the \nvoltage -current characteristic. ( Note that in all that follows, \nthe figures are in relative units ).  \nFig. 1. Experimental voltage and curr ent versus time . \n \nFig. 2. Experimental c urrent i versus experimental  voltage  v. \n \n3. Harmonic diagram s \nWe apply the discrete F ourier transform to the voltage and \nto the  current in order to  show  the harmonics and then the \nfrequencies which are important for forming these two \nsignals . We notice that the (2n+1) f0 harmonics are \nimportant for both voltage and the current  (see Fig. 3) .  \n0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04-1-0.8-0.6-0.4-0.200.20.40.60.81\ntime (s)relative voltage and currentcurrent i/imax\nvoltage v/vmax\n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81\ncurrent (i/imax)voltage (v/vmax) \nFig. 3. Modulus diagram of the first thirteen harmonics  \nof the voltage v and the current i \n \nThe flux density B and the magnetic field H are derived \nfrom the voltage v and the current i respectively by the two \nexpressions : \n              \nand                \nwhere S and l are respectively the section and the mean \nlength  of the magnetic core and n1 and n2 are the primary \nand the secondary turns  respectively . \nThe figure 4 shows the flux density and the magnetic field \nversus the time.  Note that the flux density is obtained by  \nnumerical integration.  Figure 5 shows the hysteretic  \ncharacter  of (B,H). We are not here going to redo all that \nhas been done before and well known about hysteresis but \nto show other hidden aspects of this phenomenon.  As we \ncan see in figure 6 for the harmonic diagram, the flux \ndensity B and the magnetic field H revel also the \nimportance of the (2n+1) f0 harmonics.  The 2n f0 harmonics \nexist but they are not fundamental for the derivation of the \nhysteresis . \n \nFig.4. The  flux density B and t he magnetic field H versus time.  \n \nFig.5. The hysteretic characteristic of the flux density B and the magnetic \nfield H.  \nFig.6 . Modulus diagram of the flux density B and the magnetic field H. \n \nTo reconstruct these two signals we can in first \napproximation consider that are the sum of a finite number \nof harmonics having a frequency equal to (2n+1) multiple \nof the principal frequency f0 (where n is an integer number \nvarying from 0  to nmax). \n \n             \n                            \n     \n             \n                            \n     \n \nFigure 7  shows that to have a good approximation of the \nhysteresis, we need to take only the four first harmonic s \n(n=0:3) and by putting the corresponding values of the \nmodulus  and phase  angles of Bn(t) and Hn(t) we can \nconstruct the hysteresis . \n \n            \n    and             \n     (1) \nwith :  \n \n                                \n                                 \n                                \n                                 \n \n                                \n                                \n                                \n                                 \n \nFig.7. Reconstruction of the hysteretic characteristic  \nby the four first harmonics . \n \nWe can see that by using only the first harmonic of the \nmagnetic field H and the flux density B we obtain the \nferromagnetic character of the hysteresis but by using the \n0 100 200 300 400 500 600 700 80000.51voltage (v/vmax)\n0 100 200 300 400 500 600 700 80000.51\nfrequency (Hz)current (i/imax)\n-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015-1-0.8-0.6-0.4-0.200.20.40.60.81\ntime (s)Flux density B and magnetic field HMagnetic field (H/Hmax)\nFlux density (B/Bmax)\n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81\nmagnetic field (H/Hmax)Flux density (B/Bmax)\n0 100 200 300 400 500 600 700 80000.511.5Flux density\n(B/Bmax)\n0 100 200 300 400 500 600 700 80000.20.40.60.8\nfrequency (Hz)Magnetic field\n(H/Hmax)\n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1.5-1-0.500.511.5\nMagnetic fieldFlux density\n1st harmonic only\n1st+2nd harmonics\n1st+2nd+3rd harmonics\nfour harmonicsfirst and the second harmonics we obtain a negative slope  \nof the hysteresis and the relative permeability ca n reach a \nhundred of magnitude.  Let call the hysteresis  obtained with \nthis four harmonics  the ‘harmonic hysteresis ’. Figure 8 \nshows the harmonic hysteresis compared to the \nexperimental one.  \n \nFig. 8 . harmonic hysteresis compared to the experimental one.  \n \nWe can approximate the modulus of the flux density B of \nthe (2n+1) frequencies ( n≤6) by the following empirical \nexpression (Fig. 9 ): \n \n        \n   \n             \n    \n \nand for the magnetic field  H (Fig. 10 ): \n \n \n        \n             \n    \n \nFig. 9 . Modulus of the (2n+1) harmonics  of the flux density . \n \nFig. 10 . Modulus of the (2n+1) harmonics of the magnetic field . This magnetization follows a Boltzmann distribution for \nthe modulus of theses harmonics . \n \n4. Miamagnetism  \nWe will now turn to the study the harmonics of the \nrelations (1) apart . The superposition method implies that \neach harmonic with a frequency different from the other s \nacts independently  on the ferromagnetic material without \nmutual  induction with other harmonics.  \nIf we represent the (Bn,Hn) characteristics  apart in figure \n11, we can see that the principal harmonic characteristic  \n(B0,H0) has a ferromagnetic character (Fig. 11a) but for \nn=1:3 the (Bn,Hn) harmonic characteristic s have a  very high  \ndiamagnetic character  (Fig.11b) . Figure 11b s hows the \nthree harmonics 1, 2 and  3 with negative slope showing \nincreased diamagnetic appearance.  \nThis diamagnetic character has a negative relative magnetic \npermeability and a high negative magnetic susceptibility \n(-1) and can reach the average value of -155. The so -\ncalled ferromagnetic material has a ferromagnetic character \nwith a very high positive permeability  535 which hides a \nvery high diamagnetic character with a high negative \npermea bility . The absolute value of this permeability is \nlower than the ferromagnetic permeability.  \na)  \nb)  \n \nFig.1 1. (Hn,Bn) characteristics : a) first harmonic and b ) 2nd, 3rd and 4th \nharmonics.  \n \nTo explain this negative behavior, we suppose that the \nmagnetic dipoles are not all oriented  following the \ndirection of the flux density B for all domains but a part of \nthem are directed following the opposite direction of B in \nsome domains. The delayed time is important due to their \nhigh frequencies.  \n \n \n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81\nMagnetic field (H/Hmax)Flux density (B/Bmax)Harmonic hysteresis\nExperimental hysteresis\n0 100 200 300 400 500 600 70010-410-310-210-1100101\nfrequency (Hz)modulus of the flux densityExperiment\nTheory\n0 100 200 300 400 500 600 70000.10.20.30.40.50.60.7\nfrequency (Hz)Magnetic fieldExperiment\nTheory\n5. Evaluation of the  magnetic permeability  of \nferromagnetism  and miamagnetism  \nThe harmonic study allows us to determine the \npermeability of each harmonic. The real part of the \nmagnetic field and flux induction signals for each harmonic \nmakes it possible to determine the value of the real \npermeability whereas the imaginary part gives us \ninformation on the other losses, namely th e Joule and \nexcess losses. T able I gives the value o f the real \npermeability for each harmonic.  \nFor the first harmonic, we find a ferromagnetic effect \nwhich is represented by a very large positive value of the \nrelative permeability, 535. For the other harmonics we \nnotice an increased diamagnetic effect which  is represented \nby a very large negative value of the average relative \npermeability : -154 for the second, third and fourth \nharmonics. To this phenomenon of diamagnetism, we \nattribute the name of ‘Miamagnetism’ . This miamagnetic \nbehavior is hidden by the fe rromagnetism which is more \nimportant.  The Miamagnetism is closely related to the \ndomains precession related to the multiple frequency of f0. \n \nTable 1. Relative p ermeability for each harmonic at the fundamental \nfrequency 50 Hz.  \nHarmonics  Permeability  (µ) Relative  \npermeability  (µr) \n1st harmonic  6.7265 10-4 535.2744  \n2nd harmonic  -1.6854 10-4 -134.1173  \n3rd harmonic  -2.1144 10-4 -168.2576  \n4th harmonic  -2.0055 10-4 -159.5888  \n \n6. First magnetiz ation c urve  \nTo obtain the first magnetization curve we can set the \nphase angle of all harmonics to 90° with respect the signs. \nFigure 12 gives the first magnetization curve compared to \nharmonic curve.  \n \n                           \n                           \n                           \n                            \n \n                           \n                           \n                           \n                            \n \nIf we now suppose that the e ddy current and the excess \nloses  create  a phase angle of 5° (i.e. -5°) we obtain an \nharmonic hysteresis  which approach the first one  (Fig. 13) . \n \n                           \n                           \n                           \n                             \nFig. 12 . harmonic hysteresis and first magnetisation curve . \n \n \nFig. 13 . harmonic hysteresis and hysteresis with phase angle of 5° . \n \n7. Conclusion  \nThe M iamagnetism has the character that the \nmagnetic susceptibility can reach at low freque ncy a \nhigh negative value of -154 and greater magnitude \nleading to a negative permeability. We can  not see it \nbecause it is hidden by the ferromagnetic character \nwhich has a higher positive magnetic susceptibility. \nWe have used the discrete Fourier transform tff to \nillustrate this hidden character and the hysteresis \nmodel can only be represented by harmoni cs of \n(2n+ 1)f0 of magnitude.  \n Acknowledgemen ts \n The authors would like to knowledge Pr. Salah Eddine \nZouzou for his helpful in experimental setup.  \n \n8. References  \n[1] D. C. Jiles, D.  L. Atherton, J. Appl. Phys. 55, 2115  \n(1984).  \n[2] I. D. Mayergoyz, Mathematical models of hysteresis , \n(Springer Verlag, New York,  1991 ). \n[3] M. Hamimid, S. M. Mimoune, M. Feliachi, \nInternational Journal of Numerical Modelling , John \nWiley & Sons, Ltd , 30, 6 (2017 ). \n[4] M. S. Wartak , K. L. Tsakmakidis and O. Hess, Physics \nin Canada , 67, 1 (2011 ). \n"    },    {        "title": "1801.09897v1.Open_Material_Property_Library_With_Native_Simulation_Tool_Integrations____MASTO.pdf",        "content": "arXiv:1801.09897v1  [cond-mat.mtrl-sci]  30 Jan 20181\nOpen Material Property Library With Native\nSimulation Tool Integrations – MASTO\nAntti Stenvall and Valtteri Lahtinen\nAbstract —Reliable material property data is crucial for trust-\nworthy simulations throughout different areas of engineer ing.\nSpecial care must be taken when materials at extreme condi-\ntions are under study. Superconductors and devices assembl ed\nfrom superconductors and other materials, like supercondu cting\nmagnets, are often operated at such extreme conditions: at l ow\ntemperatures under high magnetic fields and stresses. Typic ally,\nsome library or database is used for getting the data. We have\nstarted to develop a database for storing all kind of materia l\nproperty data online called Open Material Property Library\nWith Native Simulation Tool Integrations – MASTO. The data\nthat can be imported includes, but is not limited to, anisotr opic\ncritical current surfaces for high temperature supercondu cting\nmaterials, electrical resistivities as a function of tempe rature,\nRRR and magnetic field, general fits for describing material\nbehaviour etc. Data can also depend on other data and it can\nbe versioned to guarantee permanent access. The guiding ide a in\nMASTO is to build easy-to-use integration for various progr am-\nming languages, modelling frameworks and simulation softw are.\nCurrently, a full-fledged integration is built for MATLAB to allow\nusers to fetch and use data with one-liners. In this paper we\nbriefly review some of the material property databases commo nly\nused in superconductor modelling, present a case study show ing\nhow selection of the material property data can influence the\nsimulation results, and introduce the principal ideas behi nd\nMASTO. This work serves as the reference document for citing\nMASTO when it is used in simulations.\nIndex Terms —experimental data, material property database,\nnumerical modelling, simulations\nI. I NTRODUCTION\nIN all simulation work a guiding principle is garbage\nin means garbage out . The inputs of typical simulations\ninclude the device under study, the operation conditions an d\nthe material properties characterizing the components of t he\ndevice. The model of the device under study can typically\nbe taken, and often simplified, from the design drawings, or\nfrom the constructions. Operation conditions may be known\nor certain conditions are sought. It can be difficult to find\nreliable material property data for special materials or ty pical\nmaterials at extreme operation conditions. Characterizat ion is\noften possible but can be very time consuming. Therefore,\nmodellers often rely on known sources – material property\nlibraries or databases. In principle they are the same thing\nwith different names.\nThree larger material property libraries, databases or sou rces\nare well-known in the superconducting magnet community,\nManuscript received August 28, 2017.\nThe authors are with Tampere University of Technology, Labo ratory of\nElectrical Energy Engineering, Group of Modelling and Supe rconductivity,\nP.O. Box 692, FIN-33101 Tampere, Finland (phone: +385-40-8 490403; fax:\n+358-3-31152160; e-mail: antti.stenvall@tut.fi; www: htt p://www.tut.fi).\nThe authors would like to acknowledge the following financia l support:\nAcademy of Finland project number 287027.among others. The data from the Cyogenics Technologies\nGroup at the Material Measurement Laboratory, National\nInstitute of Standards and Technology (USA) is often called\nNIST data in modellers’ jargon [1]. It includes thermal con-\nductivities, electrical resistivities and specific heats, among\nother properties, for materials commonly used in construct ing\nsuperconducting magnets such as different aluminum alloys ,\nstainless steels and copper. Fits as a function of temperatu re,\nand in some cases for different material purities and as a\nfunction of magnetic flux density, are provided with the data .\nThe data are available via web pages of NIST in HTML format\nfrom which one can copy the fits and the coefficients to one’s\nsimulation tool. The data are originally from measurements\nat NIST. The NIST database represents data from a national\nlevel public organization.\nMATPRO – A Computer Library of Material Property at\nCryogenic Temperature represents another such library [2] .\nMATPRO was developed in a collaboration between CERN\nand University of Milano, Italy. Typical materials (insula tors,\nmetals, alloys) utilized in constructing superconducting mag-\nnets are included in MATPRO including the most common low\nand high temperature superconductors. The data in MATPRO\nare restricted to densities, specific heats, electrical res istivi-\nties and thermal conductivities of materials. In the electr ical\nresistivity and thermal conductivity data, the magnetic fie ld\nand RRR dependence are also included in addition to the\ntemperature dependence. MATPRO is written in Fortran77 and\nit also has a Windows compatible executable that one can\nuse via the command prompt. The executable can be used to\nextract numerical data in ASCII format. The data in MATPRO\nare collected from different sources, all documented in [2] .\nCryoComp is a commercial material property library from\nEckels Engineering Inc. [3]. A Windows compatible exe-\ncutable is provided for a license buyer. Data can be accessed\nvia an index and can be represented with tables, which can be\ndirectly saved or copied and pasted into, for example, Excel .\nThis database also includes typical materials for supercon duct-\ning magnet design which are grouped by type in the index. For\nexample, seven different nickel alloys are included. One ca n\nalso add private data into the database and use it via the same\ngraphical user interface as the built-in data. Data in CryoC omp\ncomes without references and without any warranty or even\na guarantee of suitability for the condition for which it is\nprovided. One should, however, note that even though this\nmay sound strange, this is standard license agreement text.\nCommon to the presented three material property libraries\nis that they are standalone ones, not integrated to modellin g\nsoftware, and not extendable by people not working directly\nwith these software. Data from the presented databases, as w ellas others, has been partially collected and implemented int o\nsome special software, like for performing quench analyses in\nROXIE [4]. This integration, however, is not straightforwa rd\nas it is merely an export, and requires implementation of the\nproperties to the software, rather than a dependency which\nupdates itself when the original software is updated.\nThis paper introduces an ongoing effort to build a material\nproperty database with the same principles as social coding\nnetworks like GitHub [5] and tools utilizing shared softwar e\nlike Composer [6] or Bower [7], but with the possibility of\nfeatures following scientific practices. This includes opt ional\ndata review to promote their reliability. The database is ca lled\nThe Open Material Property Library With Native Simulation\nTool Integrations – MASTO. A domain name (masto.eu.com)\nfor MASTO has been reserved and an early prototype is\naccessible there via a web browser. MASTO will not be\nlimited to just hosting for material property data, but also\nfor networking among all the different stakeholders around\nmaterials and their properties. The completely new way of\nacting with material property data that MASTO represents\naims to establish a sustainable, open, extendable and relia ble\nmaterial property network from which one can find materials\nand their properties, find experts to characterize material s and\nwhich one can use as a store window to make materials\navailable for possible customers.\nIn this paper we first compare selected data from the three\nintroduced material property databases – NIST, MATPRO and\nCryoComp – and study how the variation in the data influences\non modelling results. The aim of this comparison is not to\nquestion the reliability of the databases, but to emphasize how\nimportant it is to select the material properties for simula tions\nand how to get error estimates for the results by using differ ent\ndata sources. After this we introduce the fundamental ideas\nin MASTO and present its MATLAB integration. Finally,\nconclusions are drawn.\nII. I NFLUENCE OF MATERIAL PROPERTY DATA ON\nSIMULATION RESULTS –ACASE STUDY\nA simple way to estimate hot spot temperature in a super-\nconducting magnet follows the so-called adiabatic MIITs [8 ]\napproach in which the heat conduction is neglected and the\nhot spot temperature can be directly estimated from the curr ent\ndecay curve – either measured or simulated. The adiabatic he at\nbalance equation:\nc∂T\n∂t=ρ/parenleftbiggI\nA/parenrightbigg2\n, (1)\nwherecis the volumetric effective specific heat, ρis the sta-\nbilizer resistivity, Iis the current and Ais the cross-sectional\narea of the stabilizer, can be separated into material prope rties\nand current-dependent terms, and integrated separately as :\n/integraldisplayTmax\nTopA2c\nρdT=/integraldisplay∞\n0I2dt, (2)\nwhereTopcorresponds to the operation temperature at which\nheat starts to be generated and Tmax corresponds to the upper\nlimit of the hot spot temperature, typically at the origin ofthe quench, at which the integrals on the right and left hand\nsides are equal. For the material properties, those average d\nover the magnet’s unit cell are used. Therefore, by measurin g\nthe current decay curve in a quench experiment, which is\nstraightforward to measure, one can find Tmax. One should\nnote that because the heat conduction, and any cooling, is\nneglected, Tmax does not represent the hot spot temperature,\nbut an upper limit for it – within the error of material proper ty\ndata and assumption of isothermal unit cells. MIITs abbrevi -\nation comes from the scaled units: mega, current, current an d\ntime and (2) is called the MIITs equation.\nThe computation of the material property integral in the\nMIITs equation to different values of Tmax can reveal the\nsensitivity of the estimated hot spot temperature to change s in\nmaterial properties. To study this, we use the three present ed\nmaterial property sources: NIST, MATPRO and CryoComp.\nTo simplify, we use only material properties of copper. We\nconsider RRR of 100 and magnetic flux densities of 0 T and\n12 T to take into account also the magnetoresistivity which a t\nlow temperatures is significant.\nThe databases use different definitions for RRR. Here, we\ndefine RRR as the ratio ρ(273 K)/ρ(4.2 K) at 0 T. The RRR\nvalues for the ratio of 100 defined accordingly correspondin g\nto the RRR values in the different databases are the followin g:\nNIST 99.7, MATPRO 99 and CryoComp 100.\nWe tabulate the material property values from 4 to 300 K\nwith a spacing of 1 K and perform the integration with the\ntrapezoidal method. Our Topis 4.2 K. We normalize our results\nto the NIST data. Because the material properties change\nby orders of magnitude as a function of temperature, the\ndifferences are better visible in the normalized graphs.\nFig. 1 compares the resistivities from the three different\ndatabases. As can be seen, CryoComp data at 0 T is almost\nthe same as NIST data, differing by less than 0.5%. At 12 T the\nlargest difference is still less than 4%. This characterize s the\ndifficulty of describing resisitivity as a function of RRR an d\nmagnetic flux density. NIST and MATPRO data differ substan-\ntially both at low temperatures and intermediate temperatu res.\nAt 0 T field, the MATPRO data plummets to 22% below the\nNIST value around 35 K. At 12 T this peak is reduced to 6%\nat the same temperature. Another local maximum in relative\ndifference between the MATPRO and NIST data can be found\nat 147 K and 142 K for 0 T and 12 T, respectively. In the case\nof 12 T, the difference is 13%.\nFig. 2 displays the difference in the values of specific heat.\nAt low temperatures the relative variation oscillates wher eas\nat higher temperatures it changes less rapidly. Above 160 K\nCryoComp and MATPRO data fit very well together and differ\nat most 2% from the NIST data. One should keep in mind that\na small absolute variation at low temperature results in a la rger\nvariation in relative error than at high temperatures. The r atio\nof copper’s specific heat at 300 K to 4.2 K is more than 4000.\nTo study how these variations in material property values\ninfluence modelling, it is important to consider their inte-\ngrals. For example, the typical quantity of interest in quen ch\nmodelling is the hot spot temperature at the end of quench.\nBecause the ratio of the specific heat and the resistivity pla y\na crucial role in the temperature increase, the integral of t hat0 100 200 300\nTemperature [K]0.80.850.90.9511.051.1Normalized resistivityMATPRO 0 T\nCryoComp 0 T\nMATPRO 12 T\nCryoComp 12 T\nFig. 1. Normalized copper resistivities from the MATPRO and CryoMat\ndatabases for RRR=100 and 0 T and 12 T. Normalization is done a ccording\nto the corresponding data from NIST database.\n0 100 200 300\nTemperature [K]0.850.90.9511.05Normalized specific heatMATPRO\nCryoComp\nFig. 2. Normalized specific heat of copper from MATPRO and Cry oMat\ndatabases. Normalization is done to the NIST data.\nis the most visible parameter showing the difference. Furth er,\nvariation of this at low temperatures can have an influence on\npredicting, for example the quench protection system delay .\nSuch sensitivity analysis has been done in [9].\nFig. 3 displays how different current decay curves, i.e.\nMIITs, predict different hot spot temperatures when materi al\nproperties are taken from different libraries. We do not pre sent\nactual MIITs values but only compare how the predicted Ths\nchanges. At 0 T, MATPRO is more optimistic than NIST\nwhich is still slightly more optimistic than CryoComp. If\nthe MIITs correspond to such a value that NIST data gives\n300 K for Ths, the MATPRO data predicts Thsof only 279 K.\nThe corresponding value for CryoComp data is 304 K. In\nparticular, the very low resistivity of MATPRO below 60 K\ninfluences its considerably lower value. Interestingly, at 12\nT the situation is reversed. With the MIITs corresponding to200 220 240 260 280 300\nThs based on NIST data [K]200220240260280300320340Ths with corresponding MIITs [K]MATPRO 0 T\nCryoComp 0 T\nMATPRO 12 T\nCryoComp 12 T\nFig. 3.Thspredicted by material property data from MATPRO and CryoMat\ndatabases as a function of Thspredicted by the data from NIST database at\n0 T and 12 T, when the operation temperature was 4.2 K.\nNIST data of 300 K, MATPRO predicts 334 K and CryoComp\n296 K. Because as a function of field both of the datasets\nintersect NIST data, a field value which matches at a given\ntemperature for NIST and MATPRO or NIST and CryoComp\ncan be found. However, real situations are seldom like that,\nusually the field changes during the current decay and is\ndifferent in different parts of the system.\nThere exists large variation in the literature data for mate rial\nproperties of even the most common material, and possibly th e\nmost important, from the stability point of view, copper. Th is\nvariation can have a significant influence when deciding if\ndevices designed on their limits are feasible or not. Furthe r, to\ninclude this data into a simple simulation tool, we needed to\nimplement relevant functions to our simulation tool manual ly\nby extracting the data from the databases and compiling to an\nappropriate format. In the case of a database with simulatio n\ntool integrations, it would be easy to change the data source\nand re-run the simulations to get an estimate for the reliabi lity\nof the results.\nIII. P RINCIPAL IDEAS BEHIND MASTO AND ITS\nMATLAB INTEGRATION\nMASTO is a new kind of effort for constructing a central-\nized database for material property data that is also integr ated\nto simulation tools. The level of integration will depend on\nthe particular tool at hand.\nCritical questions in a centralized effort for building a\nmaterial property database are\n•How to ensure the reliability of data?\n•How to ensure that no user is blocking another one from\nentering similar data?\n•How to ensure the persistence of data?\nIn a fully open system aimed at distributing information,\nsuch as arXiv [10], GitHub or any social media platform like\nTwitter and Instagram, in principle anyone can make anythin g\nvisible for all web users. This is contrary to the scientificpeer reviewing practice, in which experts check the submitt ed\nmaterial beforehand and communicate their findings to edito rs\nwho decide if revision is needed, material can be accepted or\nif it must be rejected. Still, arXiv is popular among scienti sts,\npeople upload a lot of software to GitHub and social media\nis used as the main information source by many people. To\nincrease the reliability of data, MASTO features an optiona l\npeer review that one submitting a material can ask for. In\nthis case, the foreseen MASTO editorial and development\noffice finds experts to blind review the data. One can use\nthe advanced search functionality to only search among the\nreviewed materials. Other options for one browsing MASTO\nin estimating the reliability of data include a star ranking\nfrom 1 to 5 (similar to systems for evaluating e.g. movies) or\nsearching for materials marked as confirmed . Naturally, all the\nuploaded data can include citations to literature and descr iption\nof the characterization mentioning the standards that have been\nfollowed.\nStainless steel, for example, means different things to dif -\nferent people; and even a standard such as AISI 316L could\nbe in question. The microstructure and manufacturing histo ry\nhas a non-negligible effect on the yield strength. Therefor e,\na stainless steel expert must be able to identify her steels\nin detail while a typical user can import data that is named\njust as stainless steel. This leads to the requirement of non -\nblocking namespaces for materials. To allow this, material s are\norganized under communities . The name of each community\nis unique in the system, but different communities can have\nmaterials with the same names. A material is an entity in\nwhich a user has composed her data in a way she wants to.\nFor example, a material can be the yield strength of a stainle ss\nsteel having a certain manufacturing history, or it can be a\nlist of densities of all solid materials at standard tempera ture\nand pressure conditions (STP) [11], [12]. The communities a re\nthe data sources that the editorial and development office ca n\ntag as trusted in a similar way as in Twitter one can identify\nthat the account @realDonaldTrump really belongs to the\n45thpresident of the United States and is not one of the\nmany parody accounts. The administration of communities ca n\nbe distributed and therefore many researchers can contribu te\nto the same community. A unique identifier for a material\nproperty, data, fit etc, can be experts/aisi-304 , with\nformatcommunity/material . This community idea is\nalso in use in GitHub for sharing code. Communities can also\nhave private material properties for internal use.\nThe International Digital Object Identifier Foundation [13 ]\nhas developed a wide spread way to identify publications and\ntheir locations in the internet allowing publishers to chan ge the\nactual links in case of content management system upgrades\netc. In the MASTO system there is an additional dimension\nfor this, as material property data can have various versions .\nConsider one entering electrical resistivity of copper as a\nfunction of temperature there. Because some code can depend\non this, this cannot be removed from the system any more, in\nthe same way as one cannot undo a published publication.\nTherefore, when the data owner updates this material to\nalso include magnetic field dependence and the effect of\nRRR, a new version is created from the data. The versioningsystem follows the schema x.y.z , wherezmeans a small\nfix, or a bug fix, e.g. in data or its description; ymeans\nminor improvement, or addition of new simulation software\ndependency, to the data that kees the same interface; and xis\na major upgrade. Because the guiding idea is to make MASTO\ncompatible with various simulation software, one can detai l the\nversion to use. Therefore, persistence means identifying d ata\nlikeexperts/aisi-304/1.0.5 or in a simulation tool\nto have a dependency experts/aisi-304/1.ˆ , which\nmeans the latest with major version 1. This allows one to\nalways find the data from MASTO, no matter if the internal\nlinking system is changed.\nMASTO offers a Representational State Transfer (REST)\narchitecture [14] based application programming interfac e\nto fetch, import and update data via HTTP. Currently,\nan utility package is built to MATLAB to allow full\nMASTO integration to in-house software [15], but the\ndevelopment of other integrations is underway. The\nutility package can be found from MASTO community\nstenvala [16] and package utils . With this utility\npackage (that an initialization script will install), for e xample\ndensities of basic elements can be fetched with command\nmasto.stenvala.utils.latest.require(\n’stenvala’, ’element-densities’); . To get the\ndensity of solid copper at STP conditions, one uses command\nmasto.stenvala.elementDensities.latest(\n’cu’, ’solid’) .\nIV. C ONCLUSIONS\nSeveral material property databases or libraries for mater ials\nutilized in cryogenic environments exist. Some of these are\nopenly available on the internet, some meet the definition of\nproperietary software and some are developed in collaborat ion\nbetween research institutes. Typically, common to all of th ese\nare that when material property data for a given material\nat given conditions are fetched, it may differ, people canno t\ncontribute to the libraries and any software integration mu st\nbe done manually.\nWe studied three different databases called NIST, MATPRO\nand CryoComp and considered the electrical resistivity and\nspecific heat of copper. We considered a simple adiabatic Jou le\nheating case and showed that at 0 T when NIST data predicted\na temperature increase from 4.2 K to 300 K, MATPRO\ndata gave only 279 K. With CryoComp data the result was\n304 K. At 12 T the corresponding numbers were 334 K for\nMATPRO and 296 K for CryoComp. We presented an ongoing\neffort to construct a new online material property database ,\nOpen Material Property Library With Native Simulation Tool\nIntegrations – MASTO, to which anyone can contribute with\none’s own material property data, the credibility of data ca n be\nassessed in multiple ways, where persistence of data as well\nas its versioning is guaranteed, and the material propertie s\ncan be linked as dependencies to external software with no\nprogramming effort. We aim to make MASTO a new research\ninfrastructure connecting different people: modellers, e xper-\nimentalists, material providers etc. around material prop erty\ndata.REFERENCES\n[1] Cryogenics Technologies Group, Material Measurement L aboratory,\nNational Institute of Standards and Technology USA [Online ]. Available:\nhttp://cryogenics.nist.gov/, Accessed on: Aug. 18, 2017.\n[2] G. Manfreda, L. Rossi, and M. Sorbi ”MATPRO upgraded vers ion 2012:\na computer library of material property at cryogenic temper ature” INFN\nTechnical Report INFN-12-04/MI, 2012.\n[3]CryoComp Rapid Cryogenic Design, 88 Materials in the Properties\nDatabase. Thermal Analysis Software. Eckels Engineering I nc. 1993-\n2012 [Online]. Available: http://www.eckelsengineering .com/, Accessed\non: Jan. 22, 2018.\n[4] S. Russenschuck, ”A Computer Program for the Design of Su percon-\nductor Accelerator Magnets” 11th Annual Review of Progress In Applied\nComputational Electromagnetics, 1995, Monterey, CA, USA . LHC-Note\n354, CERN, Geneva.\n[5] GitHub, A Git repository hosting service. [Online]. Ava ilable:\nhttps://github.com/, Accessed on: Jan. 22, 2018.\n[6] Composer, A tool for dependency management in PHP. [Onli ne]. Avail-\nable: https://getcomposer.org/, Accessed on: Jan. 22, 201 8.\n[7] Bower, A package manager for the web. [Online]. Availabl e:\nhttps://bower.io/, Accessed on: Jan. 22, 2018.\n[8] E. Todesco, ”Quench limits in the next generation magnet s”CERN\nYellow Report CERN-2013-006, 10-16, 2013.\n[9] T. Salmi, D. Arbelaez, S. Caspi et al. , ”A Novel Computer Code forModeling Quench Protection Heaters in High-Field Nb 3Sn Accelerator\nMagnets” IEEE Trans. Appl. Supercond. 24(4), 4701810, 2014.\n[10] arXiv, An e-print archive. [Online]. Available: https ://arxiv.org/, Ac-\ncessed on: Jan. 22, 2018.\n[11] IUPAC. Compendium of Chemical Terminology, 2nd ed. (th e ”Gold\nBook”). Compiled by A. D. McNaught and A. Wilkinson. Blackwe ll\nScientific Publications, Oxford (1997).\n[12] M. Nic, J. Jirat, B. Kosata; updates compiled by A. Jenki ns,\n”IUPAC XML on-line corrected version: STP”, 2006-, last\nupdate Feb 24, 2014, version: 2.3.3. [Online]. Available:\nhttps://doi.org/10.1351/goldbook.S06036, Accessed on: Jan. 22,\n2018.\n[13] International DOI Foundation (IDF), A not-for-profit m embership or-\nganization that is the governance and management body for th e fed-\neration of Registration Agencies providing Digital Object Identifier\n(DOI) services and registration, and is the registration au thority for\nthe ISO standard (ISO 26324) for the DOI system. [Online]. Av ailable:\nhttps://www.doi.org/, Accessed on: Aug. 18, 2017.\n[14] R. Fielding, ”Architectural Styles and the Design of Ne twork-based\nSoftware Architectures” University of California, Irvine Doctor of\nPhilosophy Dissertation, 2000.\n[15] MASTO ”Programming language support for Matlab”. [Onl ine]. Avail-\nable: http://dev.masto.eu.com/public/api/languages/l anguage/matlab, Ac-\ncessed on: Dec. 11, 2017.\n[16] stenvala, A MASTO community for A. Stenvall. [Online]. Available:\nhttp://dev.masto.eu.com/materials/stenvala, Accessed on: Dec. 11, 2017."    },    {        "title": "1802.03044v2.Manipulating_Anomalous_Hall_Antiferromagnets_with_Magnetic_Fields.pdf",        "content": "Manipulating Anomalous Hall Antiferromagnets with Magnetic Fields\nHua Chen,1, 2Tzu-Cheng Wang,3Di Xiao,4Guang-Yu Guo,3, 5Qian Niu,6and Allan H. MacDonald6\n1Department of Physics, Colorado State University, Fort Collins, CO 80523, USA\n2School of Advanced Materials Discovery, Colorado State University, Fort Collins, CO 80523, USA\n3Department of Physics and Center for Theoretical Physics,\nNational Taiwan University, Taipei 10617, Taiwan\n4Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA\n5Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan\n6Department of Physics, the University of Texas at Austin, Austin, TX 78712, USA\nThe symmetry considerations that imply a non-zero anomalous Hall e\u000bect (AHE) in certain\nnon-collinear antiferromagnets also imply both non-zero orbital magnetization and a net spin mag-\nnetization. We have explicitly evaluated the orbital magnetizations of several anomalous Hall e\u000bect\nantiferromagnets and \fnd that they tend to dominate over spin magnetizations, especially so when\nspin-orbit interactions are weak. Because of the greater relative importance of orbital magnetization\nthe coupling between magnetic order and an external magnetic \feld is unusual. We explain how\nmagnetic \felds can be used to manipulate magnetic con\fgurations in these systems, pointing in\nparticular to the important role played by the response of orbital magnetization to the Zeeman-like\nspin exchange \felds.\nIntroduction |We have previously [1] pointed out that\nspin-orbit interactions induce an anomalous Hall conduc-\ntivity i.e.an antisymmetric contribution to the conduc-\ntivity tensor \u001b\u000b\f=@j\u000b=@E\f, in some common antiferro-\nmagnets (AFMs) with non-collinear magnetic order. Be-\ncause the anomalous Hall e\u000bect (AHE) is usually associ-\nated with ferromagnetism, we refer to these systems as\nAHE AFMs. One way to understand the \fnite anoma-\nlous Hall conductivity of AHE AFMs is to view it as a\ntime-reversal-odd pseudovector \u001bAH\n\u000b=\u000f\u000b\f\r\u001b\f\r=2 that\nonly vanishes in magnetic systems when required to do\nso by some lattice symmetry. This idea of spatial symme-\ntry controlled AHE has also been extended to collinear\nAFMs [2].\nSince the total magnetization is also a time-reversal-\nodd pseudovector, it must be nonzero in AHE AFMs.\nIndeed, Mn 3Ir, the prototypical AHE AFM identi\fed in\nRef. [1], has a \fnite magnetization [3, 4], as do other AHE\nAFMs such as Mn 3Sn and Mn 3Ge [5{8]. It is precisely\nbecause of the nonzero magnetization, that the sign of\nthe AHE can be \ripped by reversing the magnetic \feld\ndirection in experiments. However, the microscopic pic-\nture of magnetization in AHE AFMs is far from clear. In\nparticular, it is expected that typical AHE AFMs should\nhave vanishingly small total spin magnetization due to\nthe much larger exchange coupling than the magnetic\nanisotropy of sublattice moments. As a result, the or-\nbital contribution to the total magnetization [9{14] is no\nlonger negligible, and could play a key role in determining\nhow AHE AFMs respond to external magnetic \felds.\nOur goal in this Letter is to develop a quantitative\ndescription of manipulating the order parameter direc-\ntion of AHE AFMs coherently using orbital magnetic\n\felds, which is appropriate for those AHE AFMs with\ndominating orbital magnetization over the spin contri-\nbution. To this end, we \frst provide a general criterion,backed by \frst-principles calculations, for searching for\nsuch orbital-magnetization-dominant AHE AFMs. We\nthen point out, in the framework of relativistic spin den-\nsity functional theory (SDFT), that the orbital magnetic\n\feld reorients the order parameter through an unusual\norbital-spin susceptibility, for which we give a convenient\nformula based on linear response theory. With these\npreparations, we \fnally explain our method for investi-\ngating \feld-induced coherent order parameter switching\nin such AHE AFMs, by keeping track of energy extrema\nevolution in the con\fguration space, and illustrate the\nvarious unusual switching behaviors by applying this ap-\nproach to a toy model mimicking Mn 3Ir.\nGround State Orbital and Spin Magnetizations |\nOrbital magnetization arises from circulating electron\ncurrents. In a \fnite system it can be unambiguously\nde\fned as the expectation value of \u00001\n2j\u0002r[10]. In\nan extended system this de\fnition of orbital magneti-\nzation becomes ambiguous because the position operator\nis unbounded. Historically this conundrum posed both\nconceptual and practical challenges, but have been fully\nsolved recently [11{14]. In particular, we now know that\nthere are two gauge-invariant contributions to the total\norbital magnetization of an extended system, due to the\nmagnetic moments of individual Bloch wave packets and\nto the Berry phase modi\fcation of the electron density\nof states in a magnetic \feld, respectively [11, 16].\nTo verify that orbital magnetization has a larger rel-\native importance in AHE AFMs we have calculated\nboth orbital and spin magnetizations in Mn 3Ir, Mn 3Pt,\nMn3Rh, Mn 3Sn, and Mn 3Ge, all AHE AFMs accord-\ning to previous work [1, 5{8], listed in Table I. The\norbital magnetization Morbis calculated with the zero-\ntemperature expression given in e.g. [14] using Wannier\ninterpolation of results from relativistic SDFT [15], which\nadds corrections from spin-orbit coupling to the Kohn-arXiv:1802.03044v2  [cond-mat.mes-hall]  14 May 20192\nTABLE I. Ground state spin and orbital magnetization (in\nm\u0016Bper formula unit) for some common AHE AFMs. The\npartial orbital magnetizations M1\norbandM2\norbare respectively\nthe Bloch state orbital moment and magnetic-\feld-dependent\ndensity-of-states contributions.\nMspinM1\norbM2\norbMtot\norb\nMn3Ir 26.9 -76.7 106.1 29.7\nMn3Pt 11.2 -17.0 29.4 12.2\nMn3Rh 2.4 -24.0 35.0 11.0\nMn3Sn 0.9 40.5 -42.5 -2.0\nMn3Ge 0.9 -17.5 35.2 17.7\nSham single particle equations, but employs exchange-\ncorrelation energy functionals that retain the structure\nof the non-relativistic limit [21]. We \fnd that Morbis at\nleast comparable to the total spin magnetization Mspin\nin size, and that it is much larger than the latter in cer-\ntain materials, e.g.Mn3Rh. This is in sharp contrast to\nconventional metallic ferromagnets such as Fe in which\norbital magnetization is more than one order of magni-\ntude smaller than spin magnetization. We are also aware\nof earlier SDFT calculations showing the importance of\norbital magnetization in Mn 3Sn [17] prior to the estab-\nlishment of a gauge-invariant form of the orbital magne-\ntization in crystalline solids.\nInterestingly, comparing MorbandMspinacross Table\nI, we see that heavier elements have smaller Morb=Mspin\nvalues. This trend can be understood by taking spin-\norbit coupling as a weak perturbation [18{20], as we ex-\nplain below. We consider \frst the atomic limit in which\nspin-orbit coupling can be approximated by \u0015soL\u0001S.\nHereLandSare the orbital and spin angular momen-\ntum operators that are proportional with appropriate g-\nfactors to the local orbital and spin magnetic moments.\nIt follows that magnetic order, which leads to a nonzero\nspin density averaged over an atomic sphere surround-\ning each magnetic atom, results in an e\u000bective magnetic\n\feld that couples directly to the local orbital moment.\nWe write this e\u000bective coupling as \u0000Morb\u0001H, where\nMorb=\u0000go\u0016BL=~andH=~\u0015soS^\n=go\u0016B, withSand\n^\n the magnitude and the direction of the local spin den-\nsity, andgothe appropriate g-factor. The orbital mag-\nnetization is then the orbital-orbital susceptibility !\u001fo,\na rank-2 tensor that is non-zero even in the absence of\nspin-orbit coupling, times this e\u000bective magnetic \feld. It\nfollows that the orbital magnetization is linear in spin-\norbit coupling strength in the perturbative limit. In non-\ncollinear antiferromagnets, the total orbital magnetiza-\ntion is a sum over sublattices of orbital-orbital suscepti-\nbilities times local orbital magnetic \felds [15].\nIn a similar way, the spin-canting that produces a non-\nzero total spin magnetization in these AHE AFMs can\nbe viewed as the net spin density induced by the orbital\n0.0 0.2 0.4 0.6 0.8 1.0\n/so\n0.0000.0050.0100.0150.0200.0250.030Magnetization (B)\nOrbital\nSpinFIG. 1. Dependence of net MspinandMorbon spin-orbit\ncoupling strength in Mn 3Ir.\u0015=\u0015 sois the ratio of spin-orbit\ncoupling strength to its realistic value.\nmagnetic \feld through a spin-orbital susceptibility !\u001fso\nthat connects spins and orbital magnetic \felds. Since !\u001fsois clearly zero in the absence of spin-orbit coupling,\nit must be at least linear in \u0015so, and the spin-canting must\ntherefore be at least of 2nd order. The same conclusion\ncan be reached by relating !\u001fsoto magnetocrystallline\nanisotropy [15]. It will be shown below that !\u001fsoplays a\ncentral role in the reorientation of the noncollinear mag-\nnetic order parameters by external magnetic \felds.\nAlthough these atomic limit considerations do not\nstrictly apply to metallic AFMs, we expect that the gen-\neral trend should still hold. As an explicit check, we\ncalculated the total orbital and spin magnetizations of\nMn3Irvs. spin-orbit coupling strength by arti\fcially\nvarying the speed of light when generating the fully-\nrelativistic pseudopotentials. The results shown in Fig. 1\nagree well with the qualitative picture explained above.\nIt follows that in an AHE AFM family of given sym-\nmetries, larger Morb=Mspinvalues should be expected in\nmaterials with weaker, not stronger, atomic spin-orbit\ncoupling.\nManipulating AHE AFM Order with an Orbital Mag-\nnetic Field |Having established the importance of or-\nbital magnetization in AHE AFMs, below we discuss the\norder parameter reorientation induced by orbital mag-\nnetic \felds within the relativistic SDFT formalism. Im-\nportant di\u000berences between the present formulation and\nthe conventional approach of solving the Landau-Lifshitz-\nGilbert (LLG) equation for a classical spin model with\nlocal Zeeman coupling to external \felds will be discussed\nat the end.\nWe \frst consider the simpler case of a ferromagnet in\nwhich the order parameter is a vector that speci\fes the\nspin-orientation ^\n. Because the energy scales associated\nwith external magnetic \felds are small, it is su\u000ecient\nto account only for the contribution to energy that is\nof \frst order in H, namely the coupling of Hto total\nmagnetization. Minimizing total energy in the presence3\nof a \feld then yields\n0 =\u000eEani(\u000e^\n)\u0000\u000eM(\u000e^\n)\u0001H; (1)\nwhereEaniis the dependence of energy on order param-\neter direction in the absence of a \feld. When Mis\npurely due to spin its magnitude is essentially \fxed at\nthe saturation magnetization Ms. Eq. (1) then simply\nimplies that the magnetization direction adjusts so that\nthe anisotropy \feld Hani\u0011\u0000\u000eEani=(Ms\u000e^\n) cancels the\nexternal magnetic \feld. When Mis dominated by the\norbital contribution, on the other hand, Eq. (1) must be\ngeneralized to\nHani+\u000eMorb\nMs\u000e^\n\u0001H= 0: (2)\nTo go further, we discuss the meaning of Eq. (2) within\nthe framework of relativistic SDFT. For magnetic sys-\ntems SDFT has the convenience of explicitly accounting\nfor the Zeeman-like exchange coupling between the mag-\nnetic condensate and the Kohn-Sham quasiparticle spins\nin the exchange-correlation potential. Although the rel-\nativistic SDFT has some subtle disadvantages [22], no-\ntably a failure [23] to capture the interaction physics\nresponsible for Hund's second rule, it is regularly and\nsuccessfully applied and is built into common electronic\nstructure software packages. Its practical success is likely\ndue to the fact that the degree to which local spin align-\nment reduces interaction energies is not strongly altered\nby relativistic corrections.\nIn this formalism ^\n enters the exchange-correlation po-\ntential in the form of \u0000\u0001ex^\n\u0001S\u0011\u0000g\u0016BHspin\u0001S=~, where\n\u0001exis the exchange \feld strength. Using a simpli\fed no-\ntation in which the variation of \u0001 exwithin an atomic cell\nis left implicit, we have\n\u000eMorb\nMs\u000e^\n=~\u0001ex\ng\u0016BMs\u000eMorb\n\u000eHspin=~\u0001ex\ng\u0016BMs !\u001fos; (3)\nwhereg\u0019\u00002 is the Lande g-factor, and !\u001fos= !\u001fT\nso\nis the orbital-spin susceptibility discussed further below.\nWith this notation Eq. (1) becomes\nHani=\u0000~\u0001ex\ng\u0016BMs !\u001fos\u0001H: (4)\nIt follows that when the magnetization is orbitally dom-\ninated, the anisotropy \feld must be balanced by an ad-\njustment in Morbproduced by the orbital-spin suscep-\ntibility !\u001foswhich, among the various magnetic sus-\nceptibility contributions identi\fed in solid state sys-\ntems [24{27], is the one seldom addressed in the liter-\nature [24, 28, 29]. !\u001fosin crystals can be calculated using the standard\nlinear response theory which we sketch below. More de-\ntails can be found in [15]. To apply a uniform orbital\nmagnetic \feld to we consider a periodic vector potential\nA(r) =Borb\u0002q\nq2sin(q\u0001r), then take the q!0 limit [14, 26]withq\u0001Borb= 0 [30]. One can then obtain for a grand\ncanonical ensemble [15]\n\u001f\u000b\f\nos=\u0000e~g\u0016B\n4kBT\u000f\u000b\r\u000e (5)\n\u0002ImX\nnZ\n[dk] tr\u0000\nG0v\rG0v\u000eG0\u001b\f\u0001\n;\nwhere Greek letters label Cartesian coordinates x;y;z ,\nG0is the Kohn-Sham thermal Green's function, vis the\nvelocity operator, \u001bis the spin-space Pauli matrix vector,\nnis a fermionic Matsubara frequency label. Completing\nthe Matsubara sum then yields a variety of terms that\ncan be grouped as either Fermi-surface or Fermi-sea con-\ntributions [15], which was not done in [24]. The Green's\nfunction formalism is convenient when generalizing the\ntheory to cover disorder and interaction e\u000bects [15, 26].\nWe now turn to the speci\fc case of AHE AFMs, in\nwhich it is convenient to view the magnetic sublattice de-\npendent spin-density directions ^\ni(ilabels the total N\nmagnetic sublattices) as the order parameter. Because\nthe exchange coupling between local moments is strong,\nthe relative orientations between local moments on di\u000ber-\nent sublattices are normally nearly \fxed. Then, as in the\ncase of a classical rigid body, the number of parameters\ncan be reduced to three for any N[31{34]. Generalization\nto include non-rigid rotation, which is necessary in, e.g.,\nthe case of Mn 3Sn, is discussed in [15]. The counterpart\nof Eq. (1) for the noncollinear case is\n0 =\u000eEani(\u000e!)\u0000\u000eM(\u000e!)\u0001H; (6)\nwhere!represents the three variables parameterizing the\nthree-dimensional rotation group SO(3). For in\fnitesi-\nmal rotations the three components of \u000e!commute, and\ncan be chosen as in\fnitesimal rotation angles around the\nthree Cartesian axes \u000e!\u000b. It follows that\n\u000eEani\n\u000e!\u000b=\u000eMorb\n\u000e!\u000b\u0001H=H\u0001NX\ni=1\u000eMorb\n\u000e^\ni\u0001\u000e^\ni\n\u000e!\u000b(7)\n=~\u0001ex\ng\u0016BH\u0015NX\ni=1(\u001fi\nos)\u0015\r\u000f\r\u000b\f\ni\n\f:\nwhere Greek letters label x;y;z , !\u001fi\nosis the total orbital\nresponse to a local Zeeman \feld on sublattice i, which\ncan be evaluated by using Eq. (5) and projecting the\nspin operator onto site i. The Levi-Civita symbol comes\nfrom the antisymmetric in\fnitesimal rotation matrix in\nCartesian coordinates.\nWith above preparations we propose the following\nstrategy for studying coherent magnetic switching in\nAHE AFMs. Switching through domain nucleation and\ngrowth will be discussed elsewhere. With a microscopic\nHamiltonian we can identify energy extrema that sat-\nisfy Eq. (7). These correspond to local minima, maxima,\nand saddle points in the SO(3) parameter space. Both\nthe positions in SO(3) space and the energies of these4\nFIG. 2. (a) Structure of a s-dmodel resembling Mn 3Ir with\nits bands shown in (b). The smaller arrows in (a) represent\ntheC2axes ^\u0011mnin Eq. (8).\nextrema change smoothly with increasing external mag-\nnetic \feld. Whenever a minimum is converted to a saddle\npoint, magnetic switching to a new minimum can pro-\nceed. For numerical implementation one can discretize\nthe SO(3) space, calculate Eani,Morb,\u000eEani\n\u000e!and !\u001fi\nosat\neach grid point, and search for the H-dependent energy\nextrema.\nApplication to a Model AHE AFM |We now give an\nexample of the procedure proposed above using a toy\nmodel that mimics the magnetic structure of Mn 3Ir.\nWe consider a 1 =4-depleted fcc lattice (Fig. 2), with\nans-orbital on each site, nearest-neighbor hopping, and\nsublattice-dependent exchange \felds whose directions\nreplicate the triangular antiferromagnetic order of Mn 3Ir.\nWe add spin-orbit coupling Hso, being careful to respect\ntheC2symmetry axis ^ \u0011along bond-dependent lines (see\nFig. 2) that pass through the center of each nearest neigh-\nbor bond:\nHso=X\nhim;jni\u000b\fitso(^dim;jn\u0002^\u0011mn)\u0001\u001b\u000b\fcy\nim\u000bcjn\f:(8)\nHereijlabel unit cells, mnlabel sublattices, \u000b\fla-\nbel spin components, ^dim;jn is a unit vector pointing\nfrom siteimto sitejn, and\u001bis the vector formed\nby three Pauli matrices. As discussed above the spin-\norbit coupling vector ^ \u0011mnis chosen to be parallel to\nrc\nmn\u0000(rm+rn)=2, where rc\nnmis the mean of all neigh-\nbors of the bond mn. The band structure of this model is\nillustrated in Fig. 2 (b). This s-dmodel allows us to cal-\nculate Morband !\u001fi\nos, but not the full Eanithat should\ncome from a microscopic Hamiltonian of the delectrons.\nWe thus supplement the model with a phenomenological\nsite-dependent uniaxial anisotropy of the exchange \felds\n[4] consistent with the crystal symmetry.\nConsider the starting ground state con\fguration with\nMorbalong the (111) direction, and site-dependent ex-\nchange \felds with 120\u000erelative orientations in a perpen-\ndicular plane. The eight equivalent (111) directions have\nidentical energy minima in the absence of a magnetic\n\feld. We apply a \feld Halong the (1 \u001611), with the ex-pectation that with increasing Hthe system will even-\ntually switch to a con\fguration with a parallel Morb.\nBased on symmetry considerations we focus on the path\nin SO(3) de\fned by rotation around the ( \u0016101) direc-\ntion with angle \u0012. If the order parameter were that\nof an ordinary ferromagnet, Eaniwould, in the absence\nof a magnetic \feld, have four equivalent minima along\nthis path at \u0012= 0;arccos(\u00001=3)\u0019109:47\u000e;180\u000eand\narccos(\u00001=3) + 180\u000ecorresponding to four of the eight\n(111) directions. However, plotting our Eanivs.\u0012in\nFig. 3 (b) shows only two energy minima located at the\n\frst two rotation angles. The other two orientations dif-\nfer in the chirality of the three exchange \felds and do\nnot have the same energy. Among the two remaining\nminima,\u0012= arccos(\u00001=3) rotates the (111) plane nor-\nmal to the ( \u001611\u00161) direction. However, Morbis surprisingly\nrotated oppositely to the (1 \u001611) direction. (Similar behav-\niors exist in Mn 3Sn and Mn 3Ge [35]). Thus the magnetic\nswitching induced by a \feld along (1 \u001611) corresponds to\nreaching the minimum at \u0012= arccos(\u00001=3) through the\nsaddle point initially at \u0012\u001955\u000e.\nFig. 3 (c) shows the energies of these three extrema\nas a function of H. AsHincreases, the energy of the\n\fnal\u0012= arccos(\u00001=3) state moves below that of the\ninitial minimum, and the latter eventually disappears\nafter merging with the saddle point. At this time the\nmagnetization con\fguration will switch to the \fnal state\n\u0012= arccos(\u00001=3). Switching between time-reversed\nstates for noncollinear AHE AFMs is more complicated\nsince it may not be able to be achieved through a single\nrotation around a \fxed axis. One example is shown in\n[15]. In general one needs to consider the 3 degrees of\nfreedom of SO(3), at least locally, in order to determine\nthe smooth switching path connecting two time-reversed\nstates. These points are also relevant to the formation\nand dynamics of magnetic domain walls driven by mag-\nnetic \felds.\nDiscussion |We have been ignoring spin contributions\nto the coupling with H. For AHE AFMs with orbitally-\ndominant magnetization in the absence of a magnetic\n\feld, the orbital magnetization can still be shown to dom-\ninate in \fnite \felds, by treating spin-orbit coupling as\na perturbation. More generally, spin coupling to mag-\nnetic \felds can be included in our formalism in a similar\nway as the orbital coupling, but through the spin-spin\nsusceptibility that can also be obtained microscopically.\nCompared to the conventional method of using classical\nspin models combined with LLG equation in studying\n\feld-induced switching, our formalism has much fewer\nassumptions as all the essential quantities can be given by\nmicroscopic calculations. In particular, since !\u001fosis pro-\nportional to spin-orbit coupling, our work indicates that\nif one were going to describe orbitally-dominated AHE\nAFMs with phenomenological Heisenberg-type models,\nthe e\u000bective g-tensor will be strongly dependent on or-\nder parameter direction.5\n(111)\n(1-11)\n(-11-1)\n(-1-1-1)\n(a)\n(b)\n(c)(-101)\nSaddle poin t\nInitial stateFinal state\nFIG. 3. (a) Rigid counterclockwise rotation of the non-\ncollinear order parameter with respect to the ( \u0016101) direc-\ntion. The unrotated structure has orbital magnetization along\n(111). (b) Anisotropy energy vs. rotation angle. (c) Total\nenergy at three lowest extrema along the rotation path vs.\nstrength of an external magnetic \feld along (1 \u001611).\nHC and AHM were supported by SHINES, an En-\nergy Frontier Research Center funded by DOE BES un-\nder Grant No. SC0012670 and by the Welch Founda-\ntion Research Grant No. TBF1473. HC was also sup-\nported by the start-up funding from CSU. TCW and\nGYG were supported by Ministry of Science and Tech-\nnology of the Republic of China (MOST 104-2112-M-002-\n002-MY3). DX was supported by DOE BES Grant No.\nde-sc0012509. QN was supported by DOE (DE-FG03-\n02ER45958, Division of Materials Science and Engineer-\ning), NSF (EFMA- 1641101) and Welch Foundation (F-\n1255). The authors are grateful to Satoru Nakatsuji,\nMasaki Oshikawa, Yasuhiro Tada, and Junren Shi for\nhelpful discussions.[1] H. Chen, Q. Niu, and A. H. MacDonald, Phys. Rev. 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Shi, G. Vignale, D. Xiao, and Q. Niu, Phys. Rev. Lett.\n99, 197202 (2007).\n[15] Supplementary Information\n[16] I. Souza and D. Vanderbilt, Phys. Rev. B 77, 054438\n(2008).\n[17] L. M. Sandratskii and J. K ubler, Phys. Rev. Lett. 76,\n4963 (1996).\n[18] J. G. Gay and R. Richter, Phys. Rev. Lett. 56, 2728\n(1986).\n[19] Patrick Bruno, Phys. Rev. B, 39, 865 (1989).\n[20] P. Bruno, Physical origins and theoretical models of mag-\nnetic anisotropy . Ferienkurse des Forschungszentrums\nJ ulich, J ulich, 1993.\n[21] See for example A.H. MacDonald and S.H. Vosko, J.\nPhys. C 12, 2977 (1979).\n[22] A. Cr\u0013 epieux and P. Bruno, Phys. Rev. B 64, 094434\n(2001).\n[23] I. Galanakis, Phys. Rev. B 71, 012413 (2005).\n[24] P. K. Misra and L. Kleinman, Phys. Rev. B 5, 4581\n(1972).\n[25] Y. Gao, S. A. Yang, and Q. Niu, Phys. Rev. B 91, 214405\n(2015).\n[26] H. Fukuyama, Prog. Theor. Phys. 42, 1284 (1969).\n[27] M. Ogata and H. Fukuyama, J. Phys. Soc. Jap. 84,\n124708 (2015).\n[28] L. M. Roth, J. Phys. Chem. Solids 23, 433 (1962).\n[29] P. K. Misra and L. M. Roth, Phys. Rev. 177, 1089 (1969).\n[30] We assume Bto be proportional to Hand use them\ninterchangeably, although it should be kept in mind that\na linear proportionality is not guaranteed in magnetically\nordered materials.\n[31] A. F. Andreev and V. I. Marchenko, Phys. Usp. 23, 21\n(1980).\n[32] T. Dombre and N. Read, Phys. Rev. B 39, 6797 (1989).\n[33] O. Gomonay, Phys. Rev. B 91, 144421 (2015).\n[34] C. Ulloa and A. S. Nunez, Phys. Rev. B 93, 134429\n(2016).6\n[35] T. Nagamiya, S. Tomiyoshi, and Y. Yamaguchi, Solid\nState Commun. 42, 385 (1982)."    },    {        "title": "1802.07355v1.Controlling_magnetism_in_2D_CrI3_by_electrostatic_doping.pdf",        "content": "\t1 Controlling magnetism in 2D CrI3 by electrostatic doping  Shengwei Jiang 1,2, Lizhong Li 1,2, Zefang Wang 1,2, Kin Fai Mak 1,2,3*, and Jie Shan 1,2,3*  1 School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA 2 Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA 3 Kavli Institute at Cornell for Nanoscale Science, Ithaca, New York 14853, USA  *E-mails: kinfai.mak@cornell.edu; jie.shan@cornell.edu  The atomic thickness of two-dimensional (2D) materials provides a unique opportunity to control material properties and engineer new functionalities by electrostatic doping. Electrostatic doping has been demonstrated to tune the electrical 1 and optical 2 properties of 2D materials in a wide range, as well as to drive the electronic phase transitions 3. The recent discovery of atomically thin magnetic insulators 4, 5 has opened up the prospect of electrical control of magnetism and new devices with unprecedented performance 6-8. Here we demonstrate control of the magnetic properties of monolayer and bilayer CrI3 by electrostatic doping using a dual-gate field-effect device structure. In monolayer CrI3, doping significantly modifies the saturation magnetization, coercive force and Curie temperature, showing strengthened (weakened) magnetic order with hole (electron) doping. Remarkably, in bilayer CrI3 doping drastically changes the interlayer magnetic order, causing a transition from an antiferromagnetic ground state in the pristine form to a ferromagnetic ground state above a critical electron density. The result reveals a strongly doping-dependent interlayer exchange coupling, which enables robust switching of magnetization in bilayer CrI3 by small gate voltages.  Control of magnetism by electrical means is an attractive approach for magnetic switching applications because of its low-power consumption, high speed and good compatibility with conventional semiconductor industry 6-8. Electrical control of magnetism has been explored in a variety of magnetic materials including dilute magnetic semiconductors 9, 10, ferromagnetic metal thin films 11, 12, magneto-electrics 13 and multiferroics 14-16. The emergence of atomically thin magnetic insulators/semiconductors 4, 5 that can be readily integrated into van der Walls heterostructures 17, 18 to form field-effect devices has presented a unique and promising system for electrical control of magnetism.  A recent experiment has shown electrical control of the magnetic order in bilayer CrI3 based on a linear magneto-electric effect 19 (Supplementary Sect. 3). Such a strategy, however, is limited to non-centrosymmetric materials 6, 13-16 magnetically biased near the antiferromagnet - ferromagnet transition 19. Here we demonstrate effective tuning of the magnetic properties of both monolayer and bilayer CrI3 by a more general approach based on electrostatic doping. Our results provide the basis for future voltage-controlled spintronic and memory devices based on 2D magnetic materials, as well as stimuli for future theoretical and experimental investigations of the microscopic mechanisms of the observed physical phenomenon.     \t2 Pristine monolayer CrI3 has been shown a model Ising ferromagnet below ~ 50 K 4.  The magnetic moment carried by the Cr3+ ions, which are arranged in a honeycomb lattice structure and octahedrally coordinated by nonmagnetic I- ions, is aligned in the out-of-plane direction by anisotropic exchange interaction mediated by the I- ions 20-22. On the other hand, pristine bilayer CrI3 has been shown an antiferromagnet with antiparallel magnetization from two ferromagnetic (FM) monolayers below a critical temperature of ~ 58 K 4.  The antiferromagnet - ferromagnet transition occurs at a low critical field of ~ 0.6 – 0.7 T at low temperature 4, reflecting the interlayer exchange interaction being weak in the system 23.  The weak interlayer exchange interaction is expected to be susceptible to external perturbations, such as doping, thus providing a unique route for nonmagnetic control of the antiferromagnet - ferromagnet transition as we demonstrate below.  To electrically gate atomically thin CrI3, we fabricated dual-gate field-effect devices using the van der Waals assembly method 18, 24. Figure 1a shows the schematic side view of our devices, and figure 1b, optical micrograph of two sample devices. CrI3 is encapsulated in either hexagonal boron nitride (hBN) or graphene thin layers to minimize environmental effects.  Graphene is also used as contact and gate electrodes, and hBN as gate dielectric. The dual-gate structure allows independent control of the electric field and the doping level, as well as achieving high doping levels in the channel. We will focus on the effect of doping in this study. For nearly symmetric top and back gates the doping level is controlled by the sum of the two gate voltages, which we refer to simply as gate voltage below. The gate-induced doping density in CrI3 was calculated from the gate voltage using a parallel-plate capacitor model with the dielectric constant (≈ 3 25, 26) and thickness of the hBN gate dielectric measured independently. We estimate the uncertainty for the doping density to be on the order of 10 % due to uncertainties in the hBN thickness and dielectric constant. Because the electronic density of states in CrI3 is substantially higher than that in graphene due to the flat Cr d-bands 22, 27-29, the presence of graphene contacts embedding the CrI3 channel has little influence on the efficiency of doping the magnetic material. Details on the device fabrication and characterization are provided in Methods and Supplementary Sect. 2. The magnetization of CrI3 was characterized by the magnetic circular dichroism (MCD) at 633 nm using a confocal microscope. The MCD, linearly proportional to sheet magnetization M, can be calibrated by assuming that under saturation each Cr3+ ion carries a magnetic moment of 3𝜇! (𝜇! is the Bohr magneton) in pristine samples 20, 21. Figure 1c and 1d are the magnetic-field 𝜇!𝐻 dependence of the MCD for a monolayer and bilayer device, respectively, at 4 K near zero doping (𝜇! is the vacuum permeability). The results are fully consistent with an earlier report 4. Namely, monolayer CrI3 is a ferromagnet with a coercive force 𝜇!𝐻! of ~ 0.13 T, and bilayer CrI3 is an antiferromagnet which turns into a ferromagnet at a spin-flip transition field 𝜇!𝐻!\" of ~ 0.6 T. A clear hysteresis is seen for the antiferromagnet - ferromagnet phase transition, indicating its first-order nature. To accurately determine the Curie temperature 𝑇! of monolayer CrI3, we measured the temperature dependence of its magnetic sheet susceptibility 𝜒(𝑇)=!\"(!)!\" (Fig. 2b). To this end, a conducting ring was fabricated around the sample (left, Fig. 1b) and an ac current at 30 kHz was applied to the ring to generate a small ac magnetic field of ~ 6 Oe in amplitude at the sample. The modulated \t3 MCD was measured by a lock-in amplifier under zero dc magnetic field to yield the sheet susceptibility 𝜒. The value of 𝑇! was extracted by analyzing the behavior of 𝜒(𝑇) when 𝑇! is approached from above.  The solid lines in Fig. 2b are fits to the data using the result for a 2D Ising model: 𝜒∝𝑇−𝑇!!! with a critical component γ = 1.75 30. See Methods and Supplementary Sect. 1 for more details. We first examine the effect of electrostatic doping on the magnetic properties of monolayer CrI3. The top and bottom panels of Fig. 2a are the magnetic-field dependence of magnetization M(H) at three selected gate voltages at 4 K and 50 K, respectively.  Significant doping-induced changes in the FM hysteresis loop can be observed. In contrast, the effect of pure electric field on M(H) is negligible (Supplementary Sect. 3). At 4 K, both the saturation magnetization 𝑀! and the coercive force 𝐻! increase (decrease) with hole (electron) doping while the shape of the hysteresis loop remains approximately unchanged. At 50 K, the shape of M(H) also changes significantly, indicating a doping-induced change in the Curie temperature. This is fully consistent with the magnetic susceptibility measurement at varying doping levels (Fig. 2b). We summarize the doping effect on magnetic properties of monolayer CrI3 in Fig. 2c. The saturation magnetization, coercive force and Curie temperature all have been normalized to their values at zero gate voltage. For the entire doping range, all three parameters increase (decrease) linearly with hole (electron) doping (solid lines are linear fits to the experimental data), i.e. hole (electron) doping strengthens (weakens) the magnetic order in monolayer CrI3. Significant tuning range up to ~ 75%, 40% and 20% has been achieved for 𝐻!, 𝑀!, and 𝑇!, respectively. The result also shows that among these three parameters, the coercive force can be tuned the most and the Curie temperature the least.  Next we examine the effect of electrostatic doping on bilayer CrI3. Figure 3a shows the magnetic-field dependence of the MCD at several representative gate voltages at 4 K. A small remnant magnetization is present in the antiferromagnetic (AFM) phase due to a built-in electric field in the sample and the corresponding magneto-electric effect. It can be completely removed by applying an opposing electric field through gating (Supplementary Sect. 3). The remnant magnetization has a weak doping dependence (Fig. 4c trace 5). (We therefore did not apply any electric field to cancel it in order to access a wider range of doping densities using the combination of two gates.) This behavior is consistent with the picture of two antiferromagnetically coupled FM monolayers: since the two monolayers are doped nearly equally, the doping-induced change in M from two monolayers largely cancels out. For a similar reason, in the FM phase the saturation magnetization, which is the combined magnetization of the two monolayers, follows a linear doping dependence as in monolayer CrI3 (Fig. 4c, trace 1).  The most remarkable observation, however, is the drastic change with doping in the spin-flip transition field 𝐻!\". The field decreases monotonically with electron density and eventually drops to zero at a critical applied density of ~ 2.6×10!\" cm-2. This indicates that the AFM phase has vanished and the material has turned into a ferromagnet above this critical density! A detailed doping density-magnetic field phase diagram is mapped out in Fig. 3b. The AFM phase shrinks continuously with increasing electron density. The extracted spin-flip field 𝐻!\" (averaged over forward and backward scans) as a function of applied density is shown in Fig. 3c (right axis). It spans a range of ~ 0.6 - 0.7 T.  Doping, therefore, not only changes the properties of constituent monolayers in \t4 bilayer CrI3, but also significantly modifies the interlayer exchange coupling. The interlayer exchange constant 𝐽! can be estimated from the experimental result by noting that 𝐻!\" can be viewed as an exchange bias field produced by one of the layers in bilayer CrI3 acting on the other layer 31. This allows us to write the following expression  2𝐽!=𝜇!𝑀!(𝐻!\"−𝑀!/2𝑡)      (1) by comparing the free energy of the AFM and FM phases at the transition (Supplementary Sect. 4). Here the sign convention is 𝐽! > 0 for AFM coupling, and < 0 for FM coupling; 𝑡≈ 0.7 nm is the interlayer separation 20, 21. The second term on the right hand side takes into account the demagnetization energy in the FM phase when 𝐻>𝐻!\". Using the measured doping dependence for 𝐻!\" and 𝑀!, we obtain the doping dependence of 𝐽! in Fig. 3c (left axis). As expected, it decreases monotonically with electron doping and changes sign (i.e. changes from AFM to FM coupling) at a critical applied density of ~ 2.1×10!\" cm-2. The material, however, remains AFM until the doping density reaches ~ 2.6×10!\" cm-2, at which the demagnetization energy is overcome by the interlayer exchange interaction.  The microscopic mechanisms for the observed doping effects on magnetism in 2D CrI3 remain not understood and present an exciting topic for future theoretical studies.   Our experimental findings suggest the importance of the doping effect on intralayer Cr-Cr exchange interaction, anisotropy of the exchange interaction, and the interlayer Cr-Cr exchange interaction mediated by I- ions in few-layer CrI3. (See Supplementary Sect. 5 for more discussions). For instance, the large tuning in the saturation magnetization cannot be explained alone by the doping effect on the electron occupancy of the Cr3+ ions (which carry all the magnetic moment in CrI3). Due to the octahedral crystal field and the exchange interaction, the Cr d-orbitals are split into spin-polarized t2g and eg levels 22, 27-29. Recent ab initio calculations show that the energy of the fully filled majority-spin t2g levels is significantly below the valence band edge, which is largely composed of spin-unpolarized I p-orbitals 22, 27-29. Our observation of decreasing 𝑀! with electron doping thus suggests that the sample is unintentionally n-doped and the conduction band is composed of minority-spin Cr t2g levels. This is consistent with a recent ab initio study 22 but disagrees with earlier ab initio studies 27-29, highlighting the need for more accurate band structure calculations. Furthermore, based on this picture of band filling, only up to a ~ 3% change in 𝑀! is expected in the monolayer (the maximum doping density range achieved in our experiment ~ 5.3×10!\" cm-2 corresponds to ~ 0.1 electron per Cr3+ ion). This is nearly an order of magnitude smaller than the observed value of 40% (Fig. 2c), suggesting other mechanisms also in play.  Finally, we explore switching of magnetization in 2D CrI3 by electrostatic doping.  This is highly relevant to the technologically important area of voltage control of magnetism 6-8. Figure 4a shows an attempt with a monolayer device. The device was first prepared in the magnetization “up” state by applying a magnetic field at 0.8 T. The gate voltage was then swept while the device was biased right below its coercive force at - 0.12 T. As the gate voltage was scanned from negative to positive values, the device was switched from the “up” to the “down” state.  The switching is enabled by the doping-dependent coercive force (Fig. 2c). It occurs when the coercive force drops below the bias magnetic field. However, it is one-time only switching as the device stayed in the \t5 “down” state afterwards regardless of the gate voltage because the “down” state has a lower energy than the “up” state in monolayer CrI3 under a negative bias magnetic field. The system cannot overcome the energy barrier to go back to the “up” state. Such one-time switching of ferromagnets through tuning 𝐻! electrically has been observed in other magnetic systems 10. In contrast, the magnetization in bilayer CrI3 can be repeatedly switched between the “up” or “down” state (FM state) and the “zero” state (AFM state). As shown in Fig. 4b, a bilayer device was first prepared in the “up” state by applying a magnetic field at 1 T. The gate voltage was then swept while a bias field at 1, 0.6, 0.5, 0.3 and 0 T was applied, corresponding to trace 1 through 5 in Fig. 4c, respectively. Complete and repeatable switching between the “up” and “zero” states has been achieved in this device for the three intermediate bias fields. The switching is enabled by the doping-dependent 𝐽!, which changes sign with doping and shifts the global energy minimum of the system between the AFM and FM states. Compared to earlier studies based on the magneto-electric effect to tune the exchange bias for electrical switching of magnetic moments 13-15, 19, switching of magnetization by electrostatic doping here is more efficient and is applicable for a much wider range of external bias magnetic field because of the much stronger doping dependence of the exchange bias field 𝐻!\" (Fig. 3c). Our findings have thus identified an efficient and versatile approach based on electrostatic doping to modulate magnetization in 2D magnets, and bilayer CrI3 as a particularly promising system for voltage-controlled magnetic switching applications.   Methods Device fabrication.  Van der Waals heterostructures of atomically thin CrI3, hexagonal boron nitride (hBN) and graphene were fabricated by the layer-by-layer dry transfer method 24, 26.  Images of sample devices are shown in Fig. 1b.  First, atomically thin samples were mechanically exfoliated from their bulk crystals (HQ Graphene) onto silicon substrates covered by a 300-nm thermal oxide layer.  The thickness of thin flakes was initially estimated from their optical reflectance contrast on silicon substrates and later verified by the atomic force microscopy measurements.  Monolayer and bilayer CrI3 were further confirmed by the magnetization measurement under a varying out-of-plane magnetic field.  The typical thickness of hBN gate dielectrics was ~ 20 nm, and the typical thickness of graphene was ~ 3 layers.  The selected thin flakes were then picked up one-by-one by a stamp consisting of a thin layer of polycarbonate (PC) on polydimethylsiloxane (PDMS).  The complete stack was then deposited onto substrates with pre-patterned Au electrodes.  The residual PC on the device surface was dissolved in chloroform before measurements.  Since atomically thin CrI3 is extremely air sensitive, it was handled inside a glovebox with less than one-part-per-million oxygen and moisture  and was removed from the glovebox only after being fully encapsulated.  Electrical characterization.  The sample conductance in the out-of-plane direction was measured by biasing CrI3 using the graphene source and drain electrodes.  Due to the high sample resistance/small conductance, the bias voltage was modulated at a low frequency of ~ 17 Hz and the resultant current was measured with a lock-in amplifier.  The sample conductance was recorded as a function of doping density by varying the \t6 (sum) gate voltage while the electric field generated by the top and back gate voltage cancels each other at the sample (Fig. 1c). See Supplementary Sect. 2 for more details. Magnetization measurement.  The sample magnetization was characterized by the magnetic circular dichroism (MCD) microscopy in an Attocube closed-cycle cryostat (attoDry1000) down to 4 K and up to 1.5 Tesla in the out-of-plane direction.  The MCD microscopy was performed using a HeNe laser at 633 nm with an optical power of ~ 5 µW.  An objective of numerical aperture (= 0.8) was used to focus the excitation beam to a sub-micron spot size on the devices.  The reflected light was collected by the same objective and detected by a photodiode. The helicity of the optical excitation was modulated between left and right by a photoelastic modulator (PEM) at 50.1 kHz.  The MCD was determined as the ratio of the ac (measured by a lock-in amplifier) and dc component (measured by a multimeter) of the reflected light intensity. The MCD was converted to sheet magnetization M by assuming that it is linearly proportional to M and the saturation magnetization in pristine samples is 𝑀! = 0.137 mA per layer.  The saturation magnetization was obtained by assuming each Cr3+ cation carry a magnetic moment of 3𝜇! 20, 21 and evaluating the density of Cr3+ ions from the crystallographic data of bulk CrI3 (space group 𝑅3 with unit cell parameters of 𝑎=6.867 Å, 𝑏=6.867 Å, 𝑐=19.807 Å and 𝛽=90°) 20, 21. The coercive force 𝐻! for monolayer CrI3 and the spin-flip transition field 𝐻!\" for bilayer CrI3 are defined as the field corresponding to the steepest change in the sheet magnetization M(H).  They were determined from the peak position of dM/dH, which was numerically calculated from the measured M(H). The spin-flip transition is relatively broad.  We recorded the full-width-at-half-maximum (FWHM) of the peaks as the transition width (vertical bars in Fig. 3c).  Magnetic susceptibility measurement.  The ac magnetic susceptibility 𝜒 of 2D CrI3 was measured using an excitation ring around the sample (Fig. 1b, left).  An ac current of 50 mA at 30 kHz was applied to the ring to generate a small oscillatory magnetic field of ~ 6 Oe in amplitude at the sample.  The MCD under zero dc magnetic field was measured by a lock-in amplifier to yield 𝜒.  For each doping level, the value of 𝜒 was measured as a function of temperature.  To this end, the sample temperature was first raised to 90 K (well above 𝑇!), and the measurement was followed during cooling.  The value of 𝑇! can be obtained by fitting the temperature dependence of 𝜒 above 𝑇! using the result for a 2D Ising model as described in the main text.  The extracted value of 𝑇! (dashed lines in Fig. 2b) also agrees well with the peak position of 𝜒(𝑇).   References 1. Novoselov, K.S. Nobel Lecture: Graphene: Materials in the Flatland. Reviews of Modern Physics 83, 837-849 (2011). 2. Sun, Z., Martinez, A. & Wang, F. Optical modulators with 2D layered materials. Nature Photonics 10, 227 (2016). 3. Saito, Y., Nojima, T. & Iwasa, Y. Highly crystalline 2D superconductors. Nature Reviews Materials 2, 16094 (2016). \t7 4. Huang, B. et al. Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 546, 270 (2017). 5. Gong, C. et al. Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals. Nature 546, 265 (2017). 6. Matsukura, F., Tokura, Y. & Ohno, H. Control of magnetism by electric fields. Nature Nanotechnology 10, 209 (2015). 7. Ralph, D.C. & Stiles, M.D. Spin transfer torques. Journal of Magnetism and Magnetic Materials 320, 1190-1216 (2008). 8. Song, C., Cui, B., Li, F., Zhou, X. & Pan, F. Recent progress in voltage control of magnetism: Materials, mechanisms, and performance. Progress in Materials Science 87, 33-82 (2017). 9. Ohno, H. et al. Electric-field control of ferromagnetism. Nature 408, 944 (2000). 10. Chiba, D., Yamanouchi, M., Matsukura, F. & Ohno, H. Electrical Manipulation of Magnetization Reversal in a Ferromagnetic Semiconductor. Science 301, 943 (2003). 11. Weisheit, M. et al. Electric Field-Induced Modification of Magnetism in Thin-Film Ferromagnets. Science 315, 349 (2007). 12. Maruyama, T. et al. Large voltage-induced magnetic anisotropy change in a few atomic layers of iron. Nat Nano 4, 158-161 (2009). 13. He, X. et al. Robust isothermal electric control of exchange bias at room temperature. Nature Materials 9, 579 (2010). 14. Chu, Y.-H. et al. Electric-field control of local ferromagnetism using a magnetoelectric multiferroic. Nature Materials 7, 478 (2008). 15. Wu, S.M. et al. Reversible electric control of exchange bias in a multiferroic field-effect device. Nat Mater 9, 756-761 (2010). 16. Heron, J.T. et al. Electric-Field-Induced Magnetization Reversal in a Ferromagnet-Multiferroic Heterostructure. Physical Review Letters 107, 217202 (2011). 17. Zhong, D. et al. Van der Waals engineering of ferromagnetic semiconductor heterostructures for spin and valleytronics. Science Advances 3 (2017). 18. Geim, A.K. & Grigorieva, I.V. Van der Waals heterostructures. Nature 499, 419-425 (2013). 19. Shengwei Jiang, Jie Shan & Mak, K.F. Electric-field switching of two-dimensional van der Waals magnets. Under review (2018). 20. McGuire, M.A. Crystal and Magnetic Structures in Layered, Transition Metal Dihalides and Trihalides. Crystals 7, 121 (2017). 21. McGuire, M.A., Dixit, H., Cooper, V.R. & Sales, B.C. Coupling of Crystal Structure and Magnetism in the Layered, Ferromagnetic Insulator CrI3. Chemistry of Materials 27, 612-620 (2015). 22. Lado, J.L. & Fernández-Rossier, J. On the origin of magnetic anisotropy in two dimensional CrI 3. 2D Materials 4, 035002 (2017). 23. Stryjewski, E. & Giordano, N. Metamagnetism. Advances in Physics 26, 487-650 (1977). 24. Wang, L. et al. One-Dimensional Electrical Contact to a Two-Dimensional Material. Science 342, 614 (2013). \t8 25. Fallahazad, B. et al. Shubnikov-de Haas Oscillations of High-Mobility Holes in Monolayer and Bilayer WSe2: Landau Level Degeneracy, Effective Mass, and Negative Compressibility. Physical Review Letters 116, 086601 (2016). 26. Wang, Z., Shan, J. & Mak, K.F. Valley- and spin-polarized Landau levels in monolayer WSe2. Nat Nano 12, 144-149 (2017). 27. Liu, J., Sun, Q., Kawazoe, Y. & Jena, P. Exfoliating biocompatible ferromagnetic Cr-trihalide monolayers. Physical Chemistry Chemical Physics 18, 8777-8784 (2016). 28. Zhang, W.B., Qu, Q., Zhua, P. & Lam, C.H. Robust intrinsic ferromagnetism and half semiconductivity in stable two-dimensional single-layer chromium trihalides. Journal of Materials Chemistry C 3, 12457-12468 (2015). 29. Wang, H., Eyert, V. & Schwingenschlögl, U. Electronic structure and magnetic ordering of the semiconducting chromium trihalides CrCl 3 , CrBr 3 , and CrI 3. Journal of Physics: Condensed Matter 23, 116003 (2011). 30. Blundell, S. Magnetism in Condensed Matter (Oxford University Press, Oxford, 2001). 31. Nogués, J. & Schuller, I.K. Exchange bias. Journal of Magnetism and Magnetic Materials 192, 203-232 (1999).    \t9 Figures and figure captions \n Figure 1 | 2D CrI3 field-effect devices.  a, Schematic side view of a dual-gate bilayer CrI3 field-effect device.  Bilayer CrI3 is encapsulated in few-layer graphene, which also serves as source and drain electrodes for out-of-plane transport measurements.  The top and back gates are made of hBN and graphene.  b, Optical micrograph of two sample devices.  The scale bars are 50 and 20 𝜇m in the left and right panel, respectively.  The metallic ring structure (left panel) was used to create an ac magnetic field for the susceptibility measurement for monolayer CrI3.  The red dashed line marks the boundary of a bilayer sample in the right panel. c, d, MCD versus magnetic field for monolayer (c) and bilayer CrI3 (d) near zero doping at 4 K.  The insets illustrate the magnetic states corresponding to various ranges of magnetic field. \n\t10  Figure 2 | Controlling ferromagnetism in monolayer CrI3 by doping. a, MCD signal versus magnetic field at three representative doping levels at 4 K (top panel) and 50 K (bottom panel).  b, AC susceptibility (symbols) as a function of temperature measured at the same three doping levels as in a.  Solid lines are fits to the experimental data above the Curie temperature using the critical dependence of a 2D Ising model as described in the text.  The dashed lines indicate the extracted Curie temperatures.  c, Coercive force (purple), saturation magnetization (pink) (both at 4 K), and Curie temperature (orange) normalized by their values at zero gate voltage as a function of gate voltage (bottom axis) and applied doping density n (top axis) with positive (negative) values for applied electron (hole) density. The lines are linear fits to the data.  \n\t11  Figure 3 | Doping-controlled interlayer magnetism in bilayer CrI3.  a, MCD signal versus magnetic field at 4 K at representative gate voltages (Device #2). The finite remnant magnetization in the AFM phase is caused by a built-in interlayer electric field (see main text for details).  The spin-flip transition field decreases monotonically with electron doping.  b, Doping density - magnetic field phase diagram at 4 K. The gate voltage is given in the left axis and the applied doping density in the right axis. The FM and AFM phase correspond to the region of high and low MCD signal, respectively.  c, Interlayer exchange constant 𝐽! (black, right axis) and spin-flip transition field 𝐻!\" (blue, left axis) as a function of gate voltage (bottom axis) and applied doping density (top axis). The spin-flip transition width determined from the M(H) measurement was shown as error bars for 𝐻!\", and the errors for 𝐽! were propagated from that for 𝐻!\" using Eqn. (1). The material turns into a ferromagnet at a doping density of ~ 2.6×10!\" cm-2. \n\t12  Figure 4 | Switching of magnetism in 2D CrI3 by electrostatic doping. a, Gate-voltage control of MCD of monolayer CrI3 at 4 K (Device #1). The sample was prepared in the “up” state by a magnetic field at 0.8 T and then biased at – 0.12 T during the gate voltage sweeps.  The magnetization is switched from the “up” to the “down” state at ~ 10 V when the gate voltage sweeps from negative to positive values.  It stays in the “down” state in subsequent sweeps of the gate voltage.  b, The MCD of bilayer CrI3 (device #2) versus magnetic field under zero gate voltage at 4 K.  c, Gate-voltage control of MCD of bilayer CrI3 at 4 K.  The sample was prepared in the “up” state by a magnetic field at 1 T and then biased at 1, 0.6, 0.5, 0.3 and 0 T corresponding to trace 1 through 5, respectively.  Black and red curves denote forward and backward sweep directions.  Under bias field of 1 T (trace 1) and 0 T (trace 5), the magnetization stays in the “up” (FM) and “zero” (AFM) state, respectively.  Under intermediate bias fields (trace 2, 3 and 4), repeatable gate-voltage switching between the “up” and “zero” state has been achieved.  \n"    },    {        "title": "1802.09821v2.Computational_study_on_microstructure_evolution_and_magnetic_property_of_laser_additively_manufactured_magnetic_materials.pdf",        "content": "Noname manuscript No.\n(will be inserted by the editor)\nComputational study on microstructure evolution and\nmagnetic property of laser additively manufactured magnetic\nmaterials\nMin Yi \u0001Bai-Xiang Xu \u0001Oliver Gut\reisch\nReceived: July 10, 2018 / Accepted:\nAbstract IAdditive manufacturing (AM) o\u000bers an un-\nprecedented opportunity for the quick production of\ncomplex shaped parts directly from a powder precursor.\nBut its application to functional materials in general\nand magnetic materials in particular is still at the very\nbeginning. Here we present the \frst attempt to compu-\ntationally study the microstructure evolution and mag-\nnetic properties of magnetic materials (e.g. Fe-Ni al-\nloys) processed by selective laser melting (SLM). SLM\nprocess induced thermal history and thus the residual\nstress distribution in Fe-Ni alloys are calculated by \f-\nnite element analysis (FEA). The evolution and distri-\nbution of the \r-Fe-Ni and FeNi 3phase fractions were\npredicted by using the temperature information from\nFEA and the output from CALculation of PHAse Di-\nagrams (CALPHAD). Based on the relation between\nresidual stress and magnetoelastic energy, magnetic prop-\nerties of SLM processed Fe-Ni alloys (magnetic coer-\ncivity, remanent magnetization, and magnetic domain\nstructure) are examined by micromagnetic simulations.\nThe calculated coercivity is found to be in line with\nthe experimentally measured values of SLM-processed\nFe-Ni alloys. This computation study demonstrates a\nfeasible approach for the simulation of additively manu-\nfactured magnetic materials by integrating FEA, CAL-\nPHAD, and micromagnetics.\nKeywords Additive manufacturing \u0001Magnetic\nmaterials\u0001Selective laser melting \u0001Microstructure\nevolution\u0001Micromagnetic simulation\nM. Yi , B.-X. Xu, O. Gut\reisch\nInstitute of Materials Science, Technische Universit at Darm-\nstadt, Darmstadt 64287, Germany\nTel.: +49 6151 16-22922\nFax: +49 6151 16-21034\nE-mail: yi@mfm.tu-darmstadt.de; xu@mfm.tu-darmstadt.de1 Introduction\nFe-Ni permalloys are typical soft magnetic materials\nwith extraordinary magnetic, mechanical, and electrical\nproperties [1]. Due to their low coercivity, high magne-\ntoconductivity, high permeability, and moderate satu-\nration magnetization, they are of great interests for ap-\nplications in electromagnetic devices, including trans-\nformers, sensors, and electric motors [2{4]. In order to\nrealize these applications, suitable manufacturing tech-\nniques have to be identi\fed since they signi\fcantly af-\nfect the magnetic properties. In the past, numerous\nconventional manufacturing methods such as sintering,\nthermal spraying, ball milling, and magnetron sput-\ntering have been used to obtain the desirable perfor-\nmance of Fe-Ni alloys. Nevertheless, within the scope\nof these methods, the direct consolidation of di\u000berent\ntypes of powders into bulk magnetic components with\nmagnetism preserved is always challenging. Moreover,\nthese conventional methods may lead to the decrease of\nmagnetic properties due to the excessive grain growth\nunder a low-speed heating and cooling. They are also\nweak in producing precise magnetic components with\ncomplex shape and geometry.\nSelective laser melting (SLM), as a typical addi-\ntive manufacturing (AM) technique, enables the quick\nproduction of complex shaped three-dimensional (3D)\nparts directly from metal powders. Up to now, a large\nnumber of studies about SLM-AM or electron-beam-\nAM have been focused on structural materials with me-\nchanical properties as the focus, such as aluminium al-\nloys [5], Ti-Al-V alloys [6{8], Ni-based superalloys [9],\nstainless steel [10, 11], etc. In contrast, the application\nof SLM-AM to functional materials is still in its infancy.\nNonetheless, SLM-AM undeniably provides a promis-\ning route for breaking the bottlenecks of traditionalarXiv:1802.09821v2  [cond-mat.mtrl-sci]  13 Feb 20192 M. Yi et al.\ntechniques to fabricate complex shaped functional and\nminiaturized magnetic devices or systems directly from\nmetal powders. Ongoing e\u000borts have been devoted to\nthe production of magnetic materials by SLM-AM. The\ninitial work was carried out on the SLM processing of\nmagnetic Fe-Ni alloy by Zhang et al. [12{14]. Depend-\ning on the composition and processing, Fe-Ni alloy can\nbe either a soft magnetic material or in the L1 0phase\nas a rare-earth-free alternative for permanent magnets\n[15, 16], thus making Fe-Ni alloy a very interesting ma-\nterial. Later, Moore et al. fabricated magnetocaloric\nLa(Fe,Co,Si) 13geometries by SLM [17]. However, after\nthe early work [12{14, 17] in 2012 and 2013, it is found\nfrom the literature survey that few studies followed.\nOnly lately in 2016 \u00002018, studies continue with focus\non SLM processed magnetic materials such as Fe-Si al-\nloy [18], Fe-Si-Cr alloy [19], Fe-80%Ni permalloy [20],\nFe-30%Ni alloy [21], Ni-Fe-V and Ni-Fe-Mo permalloys\n[22, 23], Fe-Co-1.5V soft magnetic alloy [24], perma-\nnent magnets including NdFeB [25] and AlNiCo [26],\nect. Electron beam melting (EBM) is also tried to pro-\nduce MnAl(C) magnets [27]. Apart from the SLM and\nEBM based AM technique, other 3D printing technolo-\ngies without high energy input and high temperature,\nsuch as binder jetting and material extrusion, are re-\ncently applied to the production of polymer-bonded\nmagnets [28{32]. These experimental studies reveal the\nnotable e\u000bect of AM process on the microstructure and\nmagnetic properties of magnetic materials, and provide\ninsight into the challenges for the design and control of\nmagnetic properties by AM.\nDespite of these recent experimental e\u000borts, no liter-\nature is found about the modeling and simulation of the\nfabrication of magnetic alloys by SLM-AM. Almost all\ncomputational studies are dedicated to the structural\nmaterials by SLM-AM with a focus on the tempera-\nture, microstructure, residual stress, strength and duc-\ntility [33{36], possibly driven by the related experimen-\ntal contributions which are continuously \rourishing. As\nfor fabricating magnetic alloys by SLM-AM, numeri-\ncal simulations are also essential for the optimization\nof SLM-AM processes without intensive and expensive\ntrial-and-error experimental iterations, as well as for\nthe understanding of underlying physical phenomena\nwhich are di\u000ecult to observe experimentally.\nIn this work, taking magnetic Fe-Ni alloy as a model\nmaterial, we attempt to computationally predict the\nmicrostructure evolution and coercivity of SLM pro-\ncessed magnetic materials through the integration of\n\fnite element analysis (FEA), CALculation of PHAse\nDiagrams (CALPHAD), and micromagnetic simulations.\nTemperature history and distribution were calculated\nby FEA within the framework of heat transfer. By us-ing the temperature information as the input, ther-\nmomechanical simulation by FEA were performed to\nget the residual stress distribution. Furthermore, in-\ntegrating temperature results with CALPHAD output\nresulted in the temporal evolution of liquid, \r-Fe-Ni\nphase, and FeNi 3phase. Finally, by incorporating the\nresidual stress into the magnetoelastic energy of micro-\nmagnetics, the magnetic hysteresis and coercivity were\ncalculated. It is anticipated the computational study\ncould provide a possible general routine or procedure\nfor enlarging the process understanding of the under-\nlying physical mechanisms in the SLM processed mag-\nnetic materials.\n2 Thermal analysis\nThe fabrication of magnetic Fe-Ni alloy by direct SLM\nprocessing of powders is illustrated in Fig. 1. An Fe-\nNi alloy substrate with a dimension of 360 \u0016m\u0002300\n\u0016m\u0002250\u0016m is chosen for the additional layer-by-\nlayer growth of new Fe-Ni alloy layers. One of the most\nimportant features of SLM-AM is the complex temper-\nature history generated by the laser irradiation. Pre-\ndicting the temperature history forms the foundation\nfor the subsequent simulation of residual stress, mi-\ncrostructure, and magnetic property. Using the com-\nmercial FEA code ABAQUS [37], here we design and\nimplement a non-linear transient thermal 3D model to\nobtain the laser induced global temperature history.\nThe governing equation for the energy balance of\nheat transfer in the SLM process is given as\nk(\u001e;T)T;ii=\u001a(\u001e;T)Cp(\u001e;T)dT\ndt; (1)\nFe-Ni powder\nFe-Ni alloy substratelaser 𝒗𝑷\n250μm\nFig. 1. Schematics of fabricating magnetic Fe-Ni alloys\nby SLM process.Computational study on additive manufacturing of magnetic materials 3\n00.511.52\n0 1000 2000 3000Thermal expansion coefficient (1/K)  \nTemperature (K)  400050006000700080009000\n0 1000 2000 3000Density (kg/m3) \nTemperature (K)  -30-101030507090\n0 1000 2000 3000Thermal conductivity (W/m/K)  \nTemperature (K)     0 0.9 \n050100150200250\n0 1000 2000 3000Elastic modulus (GPa)  \nTemperature (K)  0100200300400\n0 1000 2000 3000Yield stress (MPa)  \nTemperature (K)  ×10-5 𝜙=1 \n𝜙=0 𝜙: 0→1 𝜙=1 \n𝜙=0 𝜙: 0→1 𝜙=0, 1  \n𝜙=1 \n𝜙=0 𝜙=1 \n𝜙=0 (b) (a) (c) \n(e) (d) 0\n1\n2\n3\n4\n50 1000 2000 3000\nHardening coefficient (GPa ) \n𝜙=0 𝜙=1 (f) \n𝜙=1 𝜙=0 \n(0.1 GPa) 02004006008001000\n0 1000 2000 3000Specific heat (J/kg/K)  \nTemperature (K)  \nFig. 2. Material parameters as a function of temperature Tand material state \u001e. The blue dashed vertical line\nrepresents the temperature Ts\u0019Tl= 1709 K.\nin whichTis the temperature, \u001ais the material density,\nCpis the speci\fc heat capacity, tis the time, and kis\nthe thermal conductivity. The initial condition for Eq.\n(1) is\nT(xi;t0) =T0; (2)\nin whichT0is the ambient temperature 300 K. The\ntemperature of the substrate bottom surface is set as\nconstantT0= 300 K. On other surfaces, the thermal\n\rux includes convection part qconand radiation part\nqradwhich can be given as\nqcon=hc(T)(T\u0000T0) (3)\nand\nqrad=\u001bsb\u000fsb(T4\u0000T4\n0); (4)\nrespectively. In Eqs. (3) and (4), hcis the tempera-\nture dependent convective heat transfer coe\u000ecient, T\nis the temperature of the corresponding surface, \u001bsbis\nthe Stefan\u0000Boltzmann constant, and \u000fsbis the surface\nemissivity.\nIn Eq. 1,\u001eis a \feld variable to indicate the material\nstate, i.e., whether the material has ever gone beyond\nthe liquidus temperature Tl. Each element stores its\ntemperature Tand\u001e. We set\u001e= 0 for a powder state\nand\u001e= 1 for a bulk state. \u001eis designed to realize theirreversible melting process from powder to bulk state\nby the subroutine USDFLD of ABAQUS. The powder\nelements are initialized with \u001e= 0.\u001eis changed from 0\nto 1 upon melting and will retain 1 afterwards, i.e. the\nfused material can never go back to powder. \u001eof the\nsubstrate elements is initialized and always remains as\n1.\nThe material parameters \u001aandCpare determined\nby the CALPHAD approach which is capable of pre-\ndicting thermodynamically consistent properties. In the\nCALPHAD model, the Gibbs free energy per gram of\none phase in a multicomponent system can be expressed\nas\nG1g(P;T) =X\niciG0\ni+RTX\nicilnci+Gexcess; (5)\nin whichciis the composition of element iin the mul-\nticomponent system, G0\nithe Gibbs free energy of pure\nelementi,Rthe gas constant, and Gexcessthe excess\nGibbs energy of mixing. Once the Gibbs energy is ob-\ntained from CALPHAD data, all other thermodynamic\nproperties can be derived. For example, \u001aandCpcan\nbe calculated as\n\u001a(T) =1 g\nV1g(P;T)=1 g\n(@G1g=@P)T(6)\nand\nCp(T) =\u0000T(@G2\n1g=@T2)P; (7)4 M. Yi et al.\nas shown in Fig. 2(b) and (c). The enthalpy per gram\ncan be derived as\nH1g(T) =G1g\u0000T(@G1g=@T)P: (8)\nThe latent heat Ldue to the change in enthalpy \u0001H 1g\nof the system during the entire solid-liquid phase change\ncan be calculated from Eq. (8) as 291.14 J/g. It should\nbe mentioned that according to the phase diagram of\nFe-Ni alloy [38], the solidus temperature Tsof permalloy\nwith the composition around Fe 20Ni80is about 1,709 K,\nwhich is only 0.2 K lower than Tl. Therefore, a constant\nLis taken here. For the system with a wide tempera-\nture region of solid-liquid mixture, \u0001H 1gis a function\ntemperature and the latent heat e\u000bects are often in-\ncluded in the temperature dependent e\u000bective speci\fc\nheat [9, 39].\nThe thermal conductivity of powder ( kp) is usually\nvery small and here is assumed to be 1% of that of\nbull material ( kb) and slightly increase with tempera-\nture before melting. The powder density is assumed to\nbe half of the bulk density. The powder speci\fc heat is\nset the same as the bulk one. The temperature depen-\ndent material parameters used for thermal analysis are\nshown in Fig. 2(a){(c).\nThe interaction between the top surface and laser\nbeam is simulated by a moving surface heat \rux with\na Gaussian distribution, i.e.\nqa=2\u0011Pa\n\u0019R2aexp\n\u00002kr\u0000r0(va;t)k2\nR2a\n; (9)\nwhere\u0011is the powder bed absorption coe\u000ecient with\nan assumed value of 0.5, Pais the laser power, Rais\nthe laser beam radius, ris the coordinate of the point\nin the material, and r0is a function of laser moving\nspeed, and vais the coordinate of laser beam center.\nThe moving laser heat \rux is dependent on the scanning\nstrategy and can be realized by the subroutine DFLUX\nof ABAQUS. If not speci\fed in the following, the laser\nbeam parameters are chosen as Pa= 100 W,Ra= 50\n\u0016m, andva= 0:4 m/s, according to the experimental\nwork [14]. This laser parameter may be di\u000ecult to be\nrealized in industrial applications. Here we limit our-\nselves to the feasibility of the proposed computational\nscheme and will not focus on its application to the real\nindustrial AM at the current stage.\nFig. 3 shows the thermal history results for the single-\ntrack scan along the middle line perpendicular to y\naxis. The powder layer is 50 \u0016m thick and the sub-\nstrate/powder model is discretized by hexahedral FE\nmeshes, as displayed in Fig. 3(a). Fig. 3(b) presents the\ntemperature pro\fle around the laser center at t= 0:6\nms, as well as the temporal evolution of temperature at\nthree FE nodes, e.g. one on the powder surface, one in\nSubstrate:  \n250 mm Powder:  \n50 mm t = 0.6 ms \ninterface  (b) \n5071  \n3878  \n2686  \n1493  \n300 T (K)  (a) \n020004000\n0 1 2 3 4 5T (K) \nt (ms) node I\nnode II\nnode III\nTl\nTl  \n5071  \n3878  \n2686  \n1493  \n300 T (K)  54 mm 226 mm \n160 mm (c) Fig. 3. Thermal analysis results for the single-track\nSLM scan along the middle line perpendicular to yaxis.\n(a) Model geometry and FE mesh, with a 50 \u0016m thick\npowder layer. (b) Temporal evolution of temperature\nin three FE nodes. (c) Temperature distribution on the\ncross-section of the molten pool at t= 0:6 ms.\nthe powder interior, and one in the substrate/deposit\ninterface. It can be found that the powder at all these\nthree nodes is quickly heated up to temperature above\nthe melting point and then gradually cools down to the\nroom temperature. The melting of node III in Fig. 3(b)\nensures the good connection between the substrate and\nthe deposited layer. By measuring the slope of the line\nconnecting the maximum temperature of the peak to\nthe temperature at 5 ms, an average cooling rate in the\norder of 105K/s can be obtained. Analogously, the av-\nerage heating rate can be estimated to be in the order\nof 106K/s. These estimated rates indicate the feature\nof fast heating and cooling during SLM.\nBy examining the temperature pro\fle at a certain\ntime (e.g.t=0.6 ms), the molten pool geometry can be\nobtained, as shown in Fig. 3(c). The length and depth of\nthe molten pool is estimated as 226 and 54 \u0016m, respec-\ntively. The molten pool appears like a comet tail, whose\nasymmetry could be attributed to the laser movement,\nas well as the temperature and material state dependent\nthermal conductivity. In front of the laser, the mate-\nrial is in powder state with low thermal conductivity,\nthus leading to slow heat transfer and high temperature\ngradient. On the contrary, in rear of laser the bulk ma-\nterial state possesses higher thermal conductivity and\nwide temperature distribution. A similar melt pool ge-Computational study on additive manufacturing of magnetic materials 5\nt = 0.5 ms \n18.2 ms 0100020003000\n0 10 20T (K) \nt (ms) node\n3198\n1st 2nd 3rd 4th 1st scan \n2nd scan \n3rd scan 4th scan node 3198  \nnode 7844  node 3198  \nnode 7844  \nTl \n01000200030004000\n0 10 20 30 40 50(a) (b) (c) \n5383  \n \n2842  \n \n300 T (K) \n4236  \n \n2268  \n \n300 T (K) \nt (ms) T (K) 764 459 432 315 288 169 142 251 \nnode 764  \nnode 459  \nnode 432 \nnode 315 \nnode 288 \nnode 169 \nnode 142 \nnode 251 \nTl (d) \n1st layer  2nd layer  3rd layer  4th layer  5th layer  6th layer  7th layer  t = 35.9 ms \nFig. 4. Thermal analysis results for the multi-track SLM scan. In-plane four-track scan along xaxis: (a) temper-\nature distribution at t= 0:5 ms and (b) temporal evolution of temperature in the two FE nodes labeled in (a).\nOut-of-plane layer-by-layer scan: (c) temperature distribution at t= 35:9 ms when the seventh layer is being built\n(d) temporal evolution of temperature in the FE nodes on the surface of each layer.\nometry is also reported in literature on nonmagnetic\nmaterials [9].\nIn contrast to the single-track scan for a strip-like\nmaterial, the in-plane and out-of-plane multi-track scans\nare simulated to build a one-layer and multi-layer bulk\nmaterial, respectively. The associated thermal results\nare presented in Fig. 4. During the multi-track scan-\nning process, an idle time of 5 ms between the comple-\ntion of one track and the beginning of the subsequent\ntrack is assumed. The idle time is demonstrated to be\nimportant [40], but its optimization is out of the scope\nhere. Cyclic heating and cooling is remarkable during\nthe multi-track scanning process, as shown in Fig. 4(b)\nand (d). The temporal evolution of temperature at the\ninterfacial nodes between the \frst and second scanning\ntrack (nodes 3198 and 7844 marked in Fig.4(a)) indi-\ncates four heating-cooling cycles. Especially, these two\nmaterial nodes experience notable melting in the \frst\nscan and remelting in the second scan. The remelting\nmeans that during the second scan the molten pool can\nextend to the previously deposited track, resulting in\ngood inter-track bonding. During the third and fourth\nscans, although these two nodes do not melt again,\nheating and cooling with a temperature change around\n500 K still occurs and may raise debonding and thermal\nfatigue issues. For the out-of-plane layer-by-layer multi-\ntrack scan, the continuous addition of powder is consid-ered by using the element deactivation and activation,\ni.e. successive discrete addition of new elements into the\nnew scanning track at the beginning of each time step.\nBy using the heat accumulation e\u000bect in SLM process\n[41] in which the heat stored in the previous layer af-\nfects the next processing layer and induces overheating,\nthe laser power can be varied layer by layer, i.e. large\npowder for the initial layers and small power for the\nsubsequent layers. Fig. 4(c) and (d) presents the typ-\nical thermal results in the case of 100 W for the \frst\nlayer, 60 W for the second layer, and 50 W for the\nother \fve layers. The temperature evolution of nodes\n(marked in Fig. 4(c)) at the surface of each layer in\nFig. 4(d) shows melting and remelting process, indicat-\ning the possibility of inter-layer bonding and the inte-\ngration of deposited layers into a bulk material.\n3 Mechanical analysis\nThe mechanical analysis is subsequently performed in-\ndependently, since it is reasonable that the mechanical\nresponse has a neglectable e\u000bect on the thermal history,\nand the thermal and mechanical analyses are weakly\ncoupled. The analysis is based on the thermal history\ndependent quasi-static mechanical model, which takes\nthe above thermal results as thermal loads. The gov-6 M. Yi et al.\nerning equation for the stress equilibrium is\n\u001bij;j= 0; (10)\nin which\u001bijis the stress. For the mechanical bound-\nary condition, the rigid body motion is restricted and\nthe substrate bottom surface is free to deform. The me-\nchanical constitutive law can be given as\n\u001bij=Cijkl(\u001e;T)\u000fe\nkl; (11)\nwhere the elastic tensor Cijklcan be expressed by tem-\nperature dependent elastic modulus (Fig. 2(e)) and a\nPoisson ratio of 0.33 for the isotropic material behav-\nior considered here. The total strain is decomposed into\nelastic strain \u000fe\nij, plastic strain \u000fp\nij, and thermal strain\n\u000fT\nij, i.e.\n\u000fij=\u000fe\nkl+\u000fp\nkl+\u000fT\nkl: (12)\nThermal strain is given by \u000fT\nij=\u000b(\u001e;T)(T\u0000T0)\u000eijin\nwhich\u000bis the thermal expansion coe\u000ecient, T0is the\ninitial temperature, and \u000eijis the Kronecker delta. \u000b\nis calculated by the CALPHAD approach through the\nGibbs free energy in Eq. (5), i.e.\n\u000b(T) =1\n31\nV@V\n@T=1\n31\n(@G1g=@P)T@2G1g\n@P@T; (13)\nas shown in Fig. 2(d). For the computation of plastic\nstrain, the linear isotropic hardening model and von\nMises yield criterion are used. The yield function is\ncomputed as\nf(\u001bij;\u001b0\nY;\u001bh\nY) =\u001bmises\u0000(\u001b0\nY+\u001bh\nY): (14)\nIn Eq. 14,\u001bmises is the von Mises stress calculated from\nthe stress tensor \u001bij.\u001b0\nY(\u001e;T) is the initial yield stress\nwithout equivalent plastic strain, as shown by the left\ncurve in Fig. 2(f). \u001bh\nYrepresents the hardening and lin-\nearly correlates with the equivalent plastic strain \u000fp\ne\nthrough the hardening coe\u000ecient Eh(\u001e;T) (right curve\nin Fig. 2(f)), i.e. \u001bh\nY=Eh\u000fp\ne. The plastic strain is com-\nputed by combing the yield criterion in Eq. 14 and the\nPrandtl{Reuss \row rule.\nFig. 5 gives the results of residual stress in the case\nof single-track scan. The distribution of residual stress\n\u001b11along the scanning direction is shown in Fig. 5(a).\nIt is apparent from the contour plot that compressive\n\u001b11appears in the deposited layer and tensile \u001b11in\nthe substrate region close to the deposited layer. Fig.\n5(b) shows the distribution of through-thickness resid-\nual stresses and plastic strain \u000fp\n11(along the line 0 marked\nin Fig. 5(a)). It can be seen that both the stress com-\nponents\u001b11and\u001b22are compressive in the deposited\nlayer, but change from tensile to compressive in the sub-\nstrate. The stress component \u001b33through the thicknessis relatively small. Furthermore, the residual stress dis-\ntribution along xdirection in the midplane is examined\nin terms of the 4 lines de\fned in Fig. 5(a). It can be\nseen from Fig. 5(c) that \u001b11gradually changes from\ncompressive along line 1 to tensile along line 4. The\ncompressive \u001b11on the free surface of the deposit are\ncaused by the steep temperature gradient, i.e. the ex-\npansion of the hotter top-layer material is prohibited\nby the underlying material with much lower tempera-\nture. In addition, the thermally induced plastic strain\nshould be responsible for the residual stress; because\npure elasticity with homogeneous material parameters\nunder no external constraint will not generate residual\nstress after cooling down to a uniform temperature. The\ndistribution of plastic strain \u000fp\n11in the deposited layer,\nas shown in Fig. 5(b), also favors the compressive \u001b11\nafter cooling down to the room temperature. The ten-\nsile stresses in the substrate/deposit interface can be\nattributed to the cooling down of the molten material\n[42] and the self-balance of the whole structure. Gen-\nerally, compressive residual stresses in the top part of\nthe deposit are favorable for increasing the load resis-\ntance and preventing crack growth. But tensile residual\nstresses in the bottom part of the deposit are disadvan-\n(a) \nline 1  𝑥 line 2  \nline 3  \nline 4  \n20 μm \n30 μm \n20 μm \nline 0  \n-400-2000200400\n0 100 200 300 Stress ( MPa ) \nDepth from free surface ( mm) \n-400-2000200400\n0 100 200 300 𝜎11 (MPa ) \nPosition along 𝑥 (mm) (c) 𝜎11 \n𝜎22 \n𝜎33 \nline 1  \nline 2  line 3 \nline 4 347 \n-210 \n-389 s11 (MPa ) \n-1.5-1-0.50𝜖11p (%) \n-200-1000100200\n0 100 200 300s33 (MPa ) \nPosition along x (mm) line 1  \nline 2  line 3 \nline 4 (d) (b) \nFig. 5. Calculated residual stress when the single-track\nlaser beam has been switched o\u000b and the temperature\nhas equilibrated to 300 K. (a) Contour (a vertical y-\nmidplane cutting through the model) of stress \u001b11dis-\ntribution. (b) Plastic strain \u000fp\n11and stress component\ndistribution along line 0 displayed in (a). Distribution\nof stress (c) \u001b11and (d)\u001b33along the four lines marked\nin (a).Computational study on additive manufacturing of magnetic materials 7\ntageous since they could reduce the load resistance and\naccelerate crack growth.\nThe residual stress distribution in the multi-track\nscan is in Figs. 6 and 7. For the in-plane four-track\nscan in Fig. 6, the stress distribution in Fig. 6(b) and\n(c) is similar to that in the single-track case in Fig. 5.\nBut the tensile stress \u001b11in the substrate is lower. For\nthe out-of-plane layer-by-layer multi-track scan in Fig.\n7, the residual stress is even much lower. The through-\nthickness stress distribution in Fig. 7(c) shows a average\nresidual stress around 50 MPa, much smaller than that\nin the single-track and in-plane multi-track scan. The\nreason could be related to the partial relief of stress\nunder reheating and cooling during by the subsequent\nlaser scanning for depositing the adjacent layers. The\ncyclic heating and cooling during the multi-track scan\nalso result in cyclic stress history, as depicted by Fig.\n6(d) and Fig. 7(c). It can be seen that all stress com-\nponents are almost zero when the liquidus temperature\nis reached. Most importantly, \u001b11and\u001b22evolution in\nFig. 6(d) and Fig. 7(c) manifest that the marked nodes\nexperience somewhat cyclic tension and compression\nalongxandydirections. Since the marked nodes are\nin the interface of adjacent layers, the cyclic tension\n288 \n110 \n-69 \n-247 \n-435 s11 (MPa ) (a) \nline 1  𝑥 \nline 2  \nline 3  \nline 4  \n20 μm \n30 μm \n20 μm \nline 0  \n-400-300-200-1000100\n0 100 200 300 𝜎11 (MPa ) \nPosition along 𝑥 (mm) -400-2000200400\n0 100 200 300 Stress ( MPa ) \nDepth from free surface ( mm) \n-300-100100300500\n0 10 20 30 40 50\nt (ms) Stress ( MPa ) \nT (K)  T \n Tl (b) \n(d) node 3336  𝜎11 \n𝜎22 \n𝜎33 \n𝜎11 \n𝜎22 \n𝜎33 288 \n-69 \n-435 s11 (MPa ) \n line 1  \nline 2  line 3 \nline 4 (c) \nFig. 6. Calculated residual stress when the in-plane\nmulti-track laser beam has been switched o\u000b and the\ntemperature has equilibrated 300 K. (a) Contour (a ver-\nticaly-midplane cutting through the model) of stress\n\u001b11distribution. (b) Stress component distribution\nalong line 0 marked in (a). (c) Distribution of stress \u001b11\nalong the four lines marked in (a). (d) Temporal evo-\nlution of stress and temperature at surface node 3336\nmarked in (a).and compression could weaken the interface bonding,\nor even lead to interface failure.\nIt should be mentioned that in the above thermal\nand mechanical analysis, the calculation methodology\nfor the magnetic FeNi material is similar to that for the\nconventional alloys. No special treatment is proposed\nto deal with the magnetic contribution to the tempera-\nture and stress/strain. This is an approximation which\nis reasonable due to the following two aspects. Firstly,\nthe in\ruence of magnetic properties of FeNi on the\nheat-transfer thermal analysis is negligible. Secondly,\nthe magnetostrictive coe\u000ecient of FeNi is in the order\nof 10\u00007to 10\u00006, which is so small that the e\u000bect of mag-\nnetization on the stress/strain can be neglected when\ncompared to the e\u000bect of thermal expansion [43]. So the\nmechanical analysis can be performed by using the simi-\nlar method for conventional alloys. However, if one deals\nwith giant magnetostrictive materials (e.g. Terfenol-D\nwith 10\u00003order of magnitude of magnetostriction), the\nmagnetization contribution in the stress/strain calcula-\ntion cannot be ignored [43].\n-200-1000100200\n0 50 100 150 200 250 300 350\n-200-1000100200\n0 10 20 30 40 50 60220 \n87 \n-46 \n-180 \n-313 s11 (MPa ) \n𝜎11 \n𝜎22 \n𝜎33 \n𝜎11 \n𝜎22 \n𝜎33 z \n459 \nz (mm) Stress ( MPa ) \nt (ms) Stress ( MPa ) (a) (b) \n(c) \nFig. 7. Calculated residual stress when the out-of-plane\nmulti-track laser beam has been switched o\u000b and the\ntemperature has equilibrated 300 K. (a) Contour of\nstress\u001b11distribution. (b) Stress component distribu-\ntion along the line marked in (a). (c) Temporal evo-\nlution of stress at an inter-layer node 459 marked in\n(a).8 M. Yi et al.\n4 Microstructure evolution\nMicrostructure plays a critical role in the property of\nthe products processed by SLM-AM and is required to\nbe predicted if possible. Here we attempt to predict\nthe microstructure evolution in Fe-Ni permalloy dur-\ning SLM process by using the thermal history and the\ntemperature-dependent phase fraction estimated from\nCALPHAD. The CALPHAD method is capable of pre-\ndicting not only thermodynamical properties for mate-\nrial design, but also microstructural evolution through\ncomprehensive physical models of materials process-\ning [44]. For example, recently CALPHAD has been\ncombined with phase-\feld simulation [9, 45] and heat-\ntransfer simulation [33, 39] to predict the process-phase\nrelationships.\nBased on the thermodynamic data of Fe-Ni alloy\nin the CALPHAD software Thermo-Calc [47], we can\npredict the phase at any temperature for a given Fe-\nNi composition. Since in present work we are inter-\nested in the magnetic Fe-Ni permalloy whose compo-\nsition is around Fe 20Ni80, only the Thermo-Calc calcu-\nlated results around Fe 20Ni80are shown in Fig. 8. The\nphase distribution of permalloy region in Fig. 8 is parti-\ntioned into four parts by four curves f1(T),f2(T),f3(T)\nandf4(T), which can be either \ftted by piecewise-\nsmooth functions or directly used as scattered data\nfrom Thermo-Calc output. For numerical study here,\nwe extract scatter data from these four curves and in-\nterpolatefi(T) values at any temperature. The peak\ntemperature is around 787 K and the corresponding\nmole fraction of Ni ( xNi) is around 0.72.\n0.520.570.620.670.720.770.820.870.920.97\n500 600 700 800 900 1000𝑓4 (𝑇) \nTemperature (K)  Mole fraction of Ni  \nFeNi3 \n𝑓3 (𝑇) 𝑓2 (𝑇) 𝑓1 (𝑇) Tc (𝛾-Fe-Ni) \n𝜸-Fe-Ni \nTc (FeNi3) \nFig. 8. CALPHAD informed phase boundary curves\naround a Ni atomic percent of 80%. The shadow region\nindicates the coexistence of \r-Fe-Ni and FeNi 3phases.\nThe Curie temperature cures for \r-Fe-Ni and FeNi 3are\nalso presented [46].As a \frst attempt and for simplicity, here we only\nconsider FeNi 3and\r-Fe-Ni phases. The real experimen-\ntal case should be more complicated, whose comprehen-\nsive modeling cannot be achieved within one step and\nwill be continuously explored in the near future. In this\nway, we can \fnd from Fig. 8 that \r-Fe-Ni phase exists at\ntemperature between 761 K and 1709 K for xNi= 0:8.\nAbove 1709 K, only liquid exists. In the region bounded\nby curvesf2(T) andf3(T), only FeNi 3exists. When\nxNiis 0.72, single phase appears at any temperature\nand the stoichiometric is the same as the powder. In\nthe shadow region bounded by curves f1(T) andf2(T),\nand curves f3(T) andf4(T), the coexistence of FeNi 3\nand\r-Fe-Ni phases occurs. For the phase-coexistence\nregion with 0 :72<x Ni<0:90, by using the lever rule,\nwe can calculate the FeNi 3phase mole fraction \t FeNi 3\nand the\r-Fe-Ni phase mole fraction \t \r-Fe-Ni by apply-\ning functions f1(T) andf2(T). Similarly, for the case\n0:52< x Ni<0:72, \t FeNi 3and \t\r-Fe-Ni can be calcu-\nlated from functions f3(T) andf4(T). Then ifxNiof\nthe initial powder is given, at any time tthe level rule\nreads\n\tFeNi 3[T(t)] =8\n>><\n>>:f1[T(t)]\u0000xNi\nf1[T(t)]\u0000f2[T(t)];0:72<xNi<0:90;\nxNi\u0000f4[T(t)]\nf3(T)\u0000f4[T(t)];0:52<xNi<0:72;\n(15)\nand\n\t\r-Fe-Ni [T(t)] =8\n>><\n>>:xNi\u0000f2[T(t)]\nf1[T(t)]\u0000f2[T(t)];0:72<xNi<0:90;\nf3[T(t)]\u0000xNi\nf3[T(t)]\u0000f4[T(t)];0:52<xNi<0:72:\n(16)\nBy using Fig. 8, Eqs. 15 and 16, and the tempera-\nture evolution available at each FE node from the ther-\nmal analysis, the concurrent and spatially varying mi-\ncrostructure evolution could be predicted. Taking the\ninitial powder with composition xNi= 0:8 as an ex-\nample, Fig. 9 shows the temporal evolution of powder,\nliquid phase, and the phase fraction of \r-Fe-Ni and\nFeNi 3in the case of single-track scan. At t= 0:6 ms\n(Fig. 9(a)), the laser beam is on and the temperature\ngradient is very large. In front of liquid phase (black\ncolor), powder is still there and no phase forms. Due\nto the high temperature around the liquid phase, the\nalready deposited layer behind the liquid phase and the\nsubstrate region close to the liquid phase possess only\n\r-Fe-Ni phase. Because of the neglectable di\u000berence be-\ntween the liquidus and solidus temperatures, it can beComputational study on additive manufacturing of magnetic materials 9\n-0.200.20.40.60.811.2\n1 ms\n2 ms\n3 ms\n4 ms\n5 ms\n5071  \n3878  \n2686  \n1493  \n300 T (K) \n(a) (b) \nt = 3.85 ms t = 0.6 ms 00.20.40.60.811.2\n0 100 200 300 400 500t = \nPosition along 𝑥 (𝜇m) liquid  \nt = \n7.9 ms \n8.4 ms \n8.9 ms \n9.4 ms \n9.9 ms \n10.4 ms Ψg-Fe-Ni (d) Ψg-Fe-Ni \nΨg-Fe-Ni \n0 1 \nliquid  powder  powder  \n-0.200.20.40.60.811.2\n0 100 200 300\n00.20.40.60.81\n0 100 200 300ΨFeNi 3 \n0 1 \nliquid  powder  793 \n670 \n546 \n423 \n300 T (K) \nΨg-Fe-Ni \n0 1 \nliquid  powder  \nΨFeNi 3 \n0 1 \nliquid  powder  powder  \nliquid  \n0.2 ms \n0.6 ms \n0.9 ms \n1.3 ms Ψg-Fe-Ni 3.77 ms \n3.85 ms \n4.01 ms \n4.17 ms \n4.33 ms t= \nPosition along 𝑥 (mm) (c) \n(d) \nFig. 9. Phase evolution during the single-track SLM scan, with the powder composition xNi= 0:8. Snapshots\n(a vertical y-midplane cutting through the model) of the predicted distribution of temperature, liquid, powder,\n\r-Fe-Ni phase fraction \t \r-Fe-Ni , and FeNi 3phase fraction \t FeNi 3at time (a) 0.6 ms and (b) 3.85 ms. (c) and (d)\n\t\r-Fe-Ni distribution along the node path marked in (b).\nseen from Fig. 9(a) that the region with liquid-solid\ncoexistence is almost unobservable and the interface\nbetween liquid and solid is sharp. This result is dif-\nferent from the previous work on stainless steel 316L\nwith a solidus-liquidus temperature di\u000berence around\n120 K calculated from CALPHAD [39]. Meanwhile, the\nregion with coexistence of \r-Fe-Ni and FeNi 3phases is\nalso very narrow and only exists in substrate. When\nthe time goes to 0.9 ms, the laser beam is o\u000b. As pre-\nsented in Fig. 9(b) at 3.85 ms, the temperature is lower\nthan 793 K, no liquid phase remains, and the deposited\nlayer completely covers xdirection. In the deposited\nlayer, there are wide regions where \r-Fe-Ni and FeNi 3\nphases coexist. More speci\fcally, Fig. 9(c) and (d) de-\npicts the phase fraction of \r-Fe-Ni (\t \r-Fe-Ni ) along the\nsurface node path marked in Fig. 9(b) at various times.\nIt can be found from Fig. 9(c) that within 1.3 ms, \r-\nFe-Ni (\t\r-Fe-Ni = 1), liquid, and powder exist along the\nnode path, but the interface between them is extremely\nnarrow. During the cooling down process in Fig. 9(d),\n\t\r-Fe-Ni is found to be between 0 and 1 in a wide region\nalong the node path, indicating the obvious coexistence\nof\r-Fe-Ni and FeNi 3phases in the deposited layer. In\norder to clearly show the mixture of FeNi 3and\r-Fe-Ni phases, the time in Fig. 9(d) is selected so that the\ncorresponding temperature falls into the region of phase\ncoexistence. Fig. 9 also indicate that after cooling down\nto 300 K, the \fnal state is the ordered intermetallic\ncompound FeNi 3phase. For the composition Fe 20Ni80,\nthe additional Ni element is present with the formation\nof solid solution in FeNi 3phase. The conclusion with \f-\nnal FeNi 3phase also agrees with the experimental X-ray\ndi\u000braction results from the literature [14], in which it is\ndemonstrated that FeNi 3phase is readily identi\fed and\nthe impurity like other Fe-Ni intermetallic compounds\nor Fe-Ni simple substance cannot be found in samples.\nFig. 10 shows the phase evolution during the in-\nplane multi-track scan. At t= 6:5 ms in Fig. 10(a), the\nlaser beam is around the center of the second track and\nthe powder state remains in front of the melt pool. A\nnarrow interface between \r-Fe-Ni and FeNi 3is found\nin both the substrate and the \frst-track deposit. Dur-\ning the cooling down of the whole sample at t= 23:3\nms in Fig. 10(b), \r-Fe-Ni/FeNi 3interface is signi\fcantly\nbroadened and a large region in the deposited layer con-\ntains two phases. It is obvious that in the contour plots\nin Fig. 10(a) and (b), the subsequent laser scan induces\nphase changes in the previously deposited layer. More10 M. Yi et al.\n(a) \n4973  \n3805  \n2637  \n1468  \n300 T (K) \n Ψg-Fe-Ni \n0 1 \nliquid  powder  ΨFeNi 3 \n0 1 \nliquid  powder  \n755 \n641 \n528 \n414 \n300 T (K) \n Ψg-Fe-Ni \n0 1 \nliquid  powder  ΨFeNi 3 \n0 1 \nliquid  powder  (b) \n(c) \n00.20.40.60.811.2\n0 5 10 15 20 25t = 6.5 ms \nt = 23.3 ms \npowder  \nliquid  g-Fe-Ni \nFeNi3 Phase fraction  \nt (ms) node 344 \nFig. 10. Phase evolution during the in-plane multi-track SLM scan along xaxis, with the powder composition\nxNi= 0:8. Snapshots of the predicted distribution of temperature, liquid, powder, \r-Fe-Ni phase fraction \t \r-Fe-Ni ,\nand FeNi 3phase fraction \t Fe-Ni 3at time (a) 6.5 ms and (b) 23.3 ms. (d) Temporal evolution of phase fraction of\nliquid,\r-Fe-Ni, and FeNi 3at the node 344 marked in (a).\nspeci\fcally, Fig. 10(c) depicts the temporal evolution of\nphase fraction at the edge node marked in Fig. 10(a).\nIt can be seen that this material node changes from\npowder to liquid during the \frst-track scan, and then\nexperiences cyclic phase changes between \r-Fe-Ni and\nFeNi 3during the subsequent scans. The cyclic phase\nchange is originated from the cyclic temperature his-\ntory as discussed in Fig. 4(b).\nSimilarly, Fig. 11 presents the phase evolution dur-\ning the out-of-plane multi-track scan. As shown in Fig.\n11(a), during the deposition of the \frst layer at t= 0:6\nms, liquid, powder, \r-Fe-Ni and FeNi 3all exist and the\ninterfaces among them are very sharp. At the beginning\nof depositing the third layer ( t= 17:8 ms), the previ-\nously deposited two layers are \r-Fe-Ni and a wide region\nwith the coexistence of \r-Fe-Ni and FeNi 3appears in\nthe substrate. During the cooling process ( t= 55 ms)\nafter the completion of seven-layer deposition, a lay-\nered region possessing both \r-Fe-Ni and FeNi 3emerges\nin the built vertical structure. The temporal evolution\nof phase fraction at the edge node (as marked in Fig.11(a)) in Fig. 11(b) shows that the material node ex-\nperiences cyclic phase changes.\nIt should be noted that in the above analysis, the\nliquid-solid coexistence is ignored and the solidi\fcation\nbe assumed to occur instantly due to the neglectable\ndi\u000berence between liquidus and solidus temperatures\nfrom the equilibrium phase diagram. However, nonequi-\nlibrium solidi\fcation process could happen in the real\ncase. In order to deal with the nonequilibrium solid-\ni\fcation, we utilize the Scheil-Gulliver model [48, 49]\nwhich has been implemented in the Thermo-Calc pack-\nage. The model assumes perfect mixing in the liquid\nand no di\u000busion in the solid phase. As introduced by\nScheil in 1942 [49], the partition coe\u000ecient ( \u0014) during\nsolidi\fcation can be de\fned as the ratio of the local\ncomposition of the solid phase CSto that of the liquid\nphaseCL,CS=CLas determined from the phase dia-\ngram. IfC0is the starting composition and fSis the\nfraction of solid, CScan be obtained through the fol-\nlowing equation\n(CL\u0000CS)dfS= (1\u0000fS)dCL: (17)Computational study on additive manufacturing of magnetic materials 11\n(a) \n(b) \n459 Ψg-Fe-Ni \n0 1 \nliquid  powder  \nt = 0.6 ms t = 17.8 ms t = 55 ms \n00.20.40.60.811.2\n0 2 4 6 8 10 12 14 16\nliquid  Phase fraction  \nt (ms) 40 44 48 52 56 60powder  \nliquid  g-Fe-Ni \nFeNi3 \nFig. 11. Phase evolution during the out-of-plane multi-track SLM scan, with the powder composition xNi= 0:8.\n(a) Snapshots of the predicted \r-Fe-Ni phase fraction \t \r-Fe-Ni at di\u000berent times. (d) Temporal evolution of phase\nfraction of liquid, \r-Fe-Ni, and FeNi 3at the node 459 marked in (a).\nThe solution of Eq. (17) is CS=\u0014C0(1\u0000fS)\u0014\u00001. The\ncalculated fraction of solid ( fS) in both equilibrium\nand nonequilibrium solidi\fcation for FeNi powder with\ncomposition xNi= 0:8 is shown in Fig. 12. It can be\nfound from Fig. 12(a) that within a temperature range\nof 300{2000 K, the di\u000berence between the equilibrium\nresult and the nonequilibrium result is hardly observ-\nable. Minor di\u000berence only appears around the liquidus\nand solidus temperatures. As shown in the inset of Fig.\n12(a), the liquid-solid coexistence region is about 0.2\nK in the equilibrium result, and is about 0.9 K in the\nnonequilibrium result. The nonequilibrium result gives\nmore accurate prediction on the fraction of solid within\nthe temperature range of 0.9 K (from 1709.36 to 1708.46\nK in Fig. 12(a)). However, the deviation of the above\nequilibrium results from the nonequilibrium result only\nexists in this small temperature range of 0.9 K. In ad-\ndition, Fig. 12(b) indicates that during the nonequilib-\nrium solidi\fcation, microsegregation including Ni de-\npletion and Fe enrichment in \r-Fe-Ni occurs, but it is\nextremely weak.\nSolid phase transformation during the SLM process-\ning induced heating and cooling is also important, i.e.\nthe transformation between \r-Fe-Ni phase and FeNi 3phase for the FeNi alloy with a composition xNiaround\n0.8. It should be mentioned that the results on the\ntransformation between \r-Fe-Ni and FeNi 3at low tem-\n00.20.40.60.81\n300 800 1300 1800Mass fraction of solid  \nTemperature (K)  nonequilibrium  \nequilibrium  00.20.40.60.81\n1708.2 1708.6 1709 1709.4\n00.20.40.60.81\n1708.4 1708.8 1709.2Mass fraction in g-Fe-Ni \nTemperature (K)  Fe\nNi(a) (b) \nFig. 12. (a) Fraction of solid as a function of temper-\nature: equilibrium result from the phase diagram and\nnonequilibrium result from the Scheil-Gulliver solidi-\n\fcation model through the CALPHAD approach. (b)\nMass fraction of Fe and Ni in \r-Fe-Ni phase during\nScheil-Gulliver solidi\fcation.12 M. Yi et al.\nperatures in Fig. 9(d), Fig. 10(c), and Fig. 11(b) are\nobtained by using the phase diagram and lever prin-\nciple without the consideration of nonequilibrium ef-\nfect. This is an extremely rough approximation. More\naccurate or quantitative modeling and computation of\nheating and cooling induced phase transformations be-\ntween\r-Fe-Ni and FeNi 3phases with nonequilibrium\ne\u000bect are highly recommended, but are still challenging.\nOn the one hand, in contrast to the disordered \r-Fe-Ni\nsolid solution, FeNi 3is a ordered phase whose modeling\nis di\u000ecult due to the complexity of involved sublattices\nand the lack of mobility parameters. On the other hand,\nphase-\feld modeling provides the feasibility of simulat-\ning microstructure evolution during solid phase trans-\nformation [50], but it requires the comprehensive ther-\nmodynamic and kinetic data of both disordered \r-Fe-\nNi and ordered FeNi 3phases which are unfortunately\nnot readily available at the current stage. Future ef-\nforts have to be made towards modeling microstruc-\nture evolution and nonequilibrium e\u000bect in heating and\ncooling induced phase transformations between \r-Fe-Ni\nand FeNi 3.\n5 Magnetic properties\nAs a functional magnetic material, Fe-Ni permalloy is\nfeatured by its magnetic properties, in contrast to the\nstructure materials with the merit of strength and duc-\ntility. Therefore, the focus of property prediction here\nis di\u000berent from that in SLM processed structural ma-\nterials. Generally speaking, the phase transformation\nduring laser scanning process will in\ruence the intrin-\nsic magnetic properties of the formed phases, such as\nCurie temperature Tc, saturation magnetization Ms,\nmagnetocrystalline anisotropy Ka, and magnetostric-\ntion\u0015s. Since magnetic properties are concerned only\nwhen the temperature is below Tc(above which a para-\nmagnetic state exits), the Curie temperature curves\nare important and thus are plotted in the phase dia-\ngram, as shown in Fig. 8. It can be found that FeNi 3\npossesses higher Tcthan\r-Fe-Ni. It means that if \r-\nFe-Ni is transformed into FeNi 3in the phase coexis-\ntence region, Tcof the system will be increased. But\nthe quantitative calculation of Tcin this phase coex-\nistence region is still challenging. Taking the composi-\ntionxNi= 0:85 as an example, the formed phases will\nbe magnetic only below 815 K. In addition, Msand\u0015s\nof both\r-Fe-Ni and FeNi 3with the same composition\nxNiare similar [46]. As typical soft magnetic materials,\nboth\r-Fe-Ni and FeNi 3have extremely small magne-\ntocrystalline anisotropy and thus negligible Kavalues.\nTherefore, under the \fxed composition xNiaround 0.8,\nthe phase transformation below the Curie temperaturecurves in Fig. 8 will not signi\fcantly in\ruence the in-\ntrinsic magnetic properties Ms,Ka, and\u0015s. In contrast,\nmagnetic coercivity is an extrinsic magnetic property,\nwhich is a very important indicator for the assessment\nof magnetic materials and is the manifestation of these\nintrinsic magnetic properties combined with extrinsic\nfactors such as shape, stress/strain, etc. To this end,\nwe attempt to calculate the magnetic properties in the\nSLM processed Fe-Ni alloy by using micromagnetic sim-\nulations, with the consideration of SLM induced resid-\nual stress.\nIn the micromagnetic framework [51{53], the spa-\ntial distribution of the magnetization is described as\nmMsin which mis the unit vector of the magnetiza-\ntion direction and Msis the saturation magnetization.\nMicromagnetic model is in the framework of contin-\nuum theory that handles magnetization processes on a\nlength scale that is small enough to resolve the transi-\ntion of the magnetization within domain walls but large\nenough to replace the atomic magnetic moments by a\ncontinuous function of position. The free energy density\nof a magnetic body is de\fned as\nEtot=Eexch+Eani+Edemag+EZeeman+Em-ela:(18)\nIn Eq. (18), the exchange energy\nEexch=Aekrmk2(19)\nis related to the gradient of mand contributes to the\ndomain wall energy, with Aedenotes the exchange pa-\nrameter which is taken as 13 pJ/m for Fe-Ni Permalloy\n[54]. The magnetocrystalline anisotropy energy\nEani=§Ka(m2\n1m2\n2+m2\n2m2\n3+m2\n3m2\n1);cubic anisotropy\nKa[1\u0000(m\u0001u)2]; uniaxial anisotropy\n(20)\nphysically means that a magnetic material is said to\nhave magnetocrystalline anisotropy if it takes more en-\nergy to magnetize it in certain directions than in others.\nKais the magnetocrystalline anisotropy constant and\nuis the unit vector along the anisotropy direction. For\nFe-Ni Permalloy, Kais usually taken as zero. The de-\nmagnetization energy\nEdemag=\u0000\u00160\n2Msm\u0001Hd (21)\nin which\u00160is the vacuum permeability constant and\nHdis the demagnetization \feld generated by the mag-\nnetic body itself. The Zeeman energy\nEZeeman=\u0000\u00160Msm\u0001Hex (22)\nis the energy of magnetization in an external magnetic\n\feldHex.Computational study on additive manufacturing of magnetic materials 13\nAlgorithm 1 Conjugate gradient method for energy minimization and hysteresis calculation in micromagnetics\n!t\n1:i 0\n2:^m(0) initial magnetization direction\n3:whilei\u00142Ndo\n4: ^H(i)\nex 2^Hmax\nexki=N\u00001k\u0000^Hmax\nex\n5: Minimize Etot(^m;^H(i)\nex) =1\n2^mTC^m\u00001\n2^HT\nd^m\u0000^H(i)T\nex^m:\n6:j 0\n7: Initial magnetization direction at ^H(i)\nex:^m0;mI\n0 ^m(i)\n8: E\u000bective magnetic \feld at cell I:HI\n0 \u00001\n\u00160MI\nsrmI\n0Etot(^m0;^H(i)\nex)\n9: Initial search direction: d0 bHHH0= [(m1\n0\u0002H1\n0\u0002m1\n0)T;:::; (mK\n0\u0002HK\n0\u0002mK\n0)T]T\n10: whilekbHHHjk>}do\n11:\u000bj minimizing Etot(^mj+\u000bjdj;^H(i)\nex) w.r.t.\u000bj(line search)\n12: ^mj+1 ^mj+\u000bjdj\n13: ^mj+1;mI\nj+1 ^mj+1(renormalized by Eq. (26))\n14: HI\nj+1 \u00001\n\u00160MI\nsrmI\nj+1Etot(^mj+1;^H(i)\nex)\n15:bHHHj+1 [(m1\nj+1\u0002H1\nj+1\u0002m1\nj+1)T;:::; (mK\nj+1\u0002HK\nj+1\u0002mK\nj+1)T]T\n16:\f ¨\nbHHHT\nj+1bHHHj+1=(bHHHT\njbHHHj) (Fletcher{Reeves method [55])\nbHHHT\nj+1(bHHHj+1\u0000bHHHj)=(bHHHT\njbHHHj) (Polak{Ribiere method [56])\n17: dj+1 bHHHj+1+\fdj\n18:j j+ 1\n19: end while\n20: Final magnetization direction at ^H(i)\nex:^m(i) ^mj\n21:end while\nAll these above energy terms are only related to\nthe magnetization. The most important term consid-\nered here is Em-ela, i.e. the magnetoelastic energy orig-\ninated from the coupling between magnetization and\nstress/strain, which for a polycrystalline with isotropic\nmagnetostriction can be given as [52]\nEm-ela=\u00003\n2\u0015s\n3X\ni=1\u001biim2\ni+3X\ni6=j\u001bijmimj\n; (23)\nwhere\u0015sis the magnetostriction coe\u000ecient. It can be\nseen from Eq. (23) that the residual stress \u001bijinduced\nby the SLM process contributes to the total free en-\nergy and thus will a\u000bect the magnetic hysteresis and\ncoercivity.\nThe minimization of total energy Etotwith respect\ntomcan be realized by the conjugate gradient method.\nThe conjugate gradient method for the hysteresis cal-\nculation is given in Algorithm 1, which is implemented\nin OOMMF code [57] in the framework of \fnite dif-\nference method (FDM). The details of the numerical\nimplementation are concisely described here. Through\nthe discretization by FDM, the energy in Eq. (18) at\nan external magnetic \feld Hexcan be rewritten as\nEtot(^m;^Hex) =1\n2^mTC^m\u00001\n2^HT\ndM^m\u0000^HT\nexM^m:(24)\nIn Eq. (24), the magnetization direction vectors mIat\neach cellIof a FDM mesh are gathered into a vector^m2R3K(Kis the total number of FDM cells), i.e.\n^m=\u0002m1\nx;m1\ny;m1\nz;:::;mK\nx;mK\ny;mK\nz\u0003T: (25)\nIn the similar way, the external magnetic \feld and the\ndemagnetizing \feld at the nodes or cells are gathered\ninto ^Hexand ^Hd, respectively. Since the magnetization\nmagnitude does not change and only the magnetization\ndirection varies (i.e. kmIk= 1), a renormalized vector\n^mhas to be used during the numerical calculations and\nis correspondingly de\fned as\n^m=\nm1\nx\nkm1k;m1\ny\nkm1k;m1\nz\nkm1k;:::;mK\nx\nkmKk;mK\ny\nkmKk;mK\nz\nkmKkT\n:\n(26)\nThe sparse matrix Ccontains grid information associ-\nated with the exchange, anisotropy, and magnetoelastic\nenergies. The matrix Maccounts for the local variation\nof the saturation magnetization Mswithin the magnet.\nThe e\u000bective magnetic \feld at each cell Iis calculated\nas\nHI=\u00001\n\u00160MIsrmIEtot(^m;^Hex): (27)\nIt should be mentioned that in contrast to the normal\nconjugate gradient method which directly includes the\nenergy gradient (e.g. HIin Eq. (27)) in the calcula-\ntion of search direction, here the cross product HHHI=14 M. Yi et al.\nmI\u0002HI\u0002mIis used, as shown in Algorithm 1. By\nusing the constraint kmIk= 1, HHHIis simpli\fed as\nHI\u0000(HI\u0001mI)mI, i.e. it only represents the e\u000bective\nmagnetic \feld component perpendicular to mI. The ap-\nplication of this cross product is physically reasonable,\nsince the \feld parallel to mIcannot induce the mag-\nnetization change according to the Landau{Lifshitz{\nGilbert equation [53, 58{60]. Following the similar no-\ntation for ^m, this cross product vectors HHHIat each cell\nIof a FDM mesh are gathered into a vector bHHH2R3K,\ni.e.\nbHHH= [(m1\u0002H1\u0002m1)T;:::; (mK\u0002HK\u0002mK)T]T:(28)\nIn order to compute the value of \fin Algorithm 1 for\nthe update of the search direction in the conjugate gra-dient method, both the Fletcher{Reeves method [55]\nand Polak{Ribiere method [56] are numerically imple-\nmented. It should be mentioned that for the Polak{\nRibiere method, the vector bHHHjis needed. While for the\nFletcher{Reeves method, only the scalar values bHHHT\njbHHHj\nneeds to be saved between iterations, which reduces\nthe memory requirement and thus is chosen for the mi-\ncromagnetic energy minimization in this work. Repeat-\ning the minimization under di\u000berent external magnetic\n\felds ^H(i)\nexto give the corresponding ^m(i)(Algorithm\n1) will result in the magnetic hysteresis from which the\nmagnetic properties can be calculated.\nThe magnetic parameters of Fe-Ni alloy with three\ndi\u000berent compositions used for micromagnetic simula-\ntions are listed in Table 1 [61]. The nominal magne-\n-1\n-0.5\n0\n0.5\n1\n-400\n-200\n0\n200\n400\n-1\n0\n1\n-1\n-0.5\n0\n0.5\n1\n�0Mz(T)Fe20Ni80\nP= 0 W\nλs= 1× 10-7\n-1\n-0.5\n0\n0.5\n1\n-400\n-200\n0\n200\n400\n-1\n0\n1\n-1\n-0.5\n0\n0.5\n1\n�0Hex(mT)�0Mz(T)Fe20Ni80\nP= 100 W\nλs= 1× 10-7\n×1E-3\n�0Hex(mT)\n×1E-3\n-1\n-0.5\n0\n0.5\n1\n-400\n-200\n0\n200\n400\n-6\n0\n6\n-1.8\n-0.9\n0\n0.9\n1.8\n�0Hex(mT)�0Mz(T)Fe15Ni85\nP= 100 W\nλs= -5.8× 10-6\n×1E-3\n-1\n-0.5\n0\n0.5\n1\n-400\n-200\n0\n200\n400\n-1.5\n0\n1.5\n-1\n-0.5\n0\n0.5\n1\n�0Hex(mT)�0Mz(T)Fe15Ni85\nP= 0 W\nλs= -5.8× 10-6\n×1E-3\n(a) (b)\n(c) (d)\n(e) (f)\nMz(MA/m)\n0.78\n-0.78c-i d-i\nd-i\n c-i\nFig. 13. Magnetic hysteresis predicted by micromagnetic simulations. (a) Non-processed and (b) SLM processed\nFe20Ni80. (c) Non-processed and (d) SLM processed Fe 15Ni85. Remanent magnetic con\fguration: (e) point c-i in\n(c) and (f) point d-i in (d).Computational study on additive manufacturing of magnetic materials 15\n00.20.40.60.81\n90 100 110 120 130Hc (mT)  \nP (W) lamda=-1E-7\nlamda=0\nlamda=1E-7\n00.511.522.53\n90 100 110 120 130Hc (mT)  \nP (W) FeNi75\nFeNi80\nFeNi85\n𝜆s=−1×10−7 \n𝜆s=0 \n𝜆s=1×10−7 \n0 0 Fe25Ni75 \nFe20Ni80 \nFe15Ni85 4 \nExperiment: \n0.06−4 mT  3.5 (a) (b) \nFig. 14. Calculated magnetic coercivity of SLM processed Fe-Ni alloy under di\u000berent laser power. (a) Coercivity\nof SLM processed Fe 20Ni80with di\u000berent magnetoelastic coe\u000ecient. (b) Coercivity of SLM processed Fe 25Ni75,\nFe20Ni80, and Fe 15Ni85. The arrow in (b) indicates the experimentally measured coercivity ranging form 0.06 to 4\nmT [12{14].\nTable 1. Magnetic parameters [61] of Fe-Ni alloy for\nmicromagnetic simulations\ncomposition \u0015sMs(MA/m) Ae(pJ/m)\nFe25Ni75 7:3\u000210\u000060.89 13\nFe20Ni801:0\u000210\u00007\n0\n\u00001:0\u000210\u000070.83 13\nFe15Ni85\u00005:8\u000210\u000060.78 13\ntostriction of Fe 20Ni80is zero. In order to investigate\nthe possible e\u000bect of tiny deviation from the nomi-\nnally zero magnetostriction, small \u0015s(\u00061:0\u000210\u00007) is\nalso considered for the case of Fe 20Ni80. Fe 15Ni85and\nFe25Ni75possess negative and positive magnetostric-\ntion, respectively. The micromagnetic model is meshed\nby cubic cells with a size of 10 \u0016m\u000210\u0016m\u00025\u0016m.\nSince the mesh for stress calculation and micromagnetic\nsimulation is di\u000berent, the stress distribution in the mi-\ncromagnetic model is obtained by performing interpo-\nlation of the nodal stress from the previous mechanical\nanalysis through a Delaunay triangulation of the scat-\ntered data [62].\nThe magnetic hysteresis is calculated by applying\nexternal magnetic \feld along the zdirection. Fig. 13(a)-\n(d) shows typical hysteresis curves for the case of Fe 20Ni80\nand Fe 15Ni85with and without SLM processing. Zoom-\ning in around the original point leads to the coerciv-\nity observable. When there is no SLM processing, both\nFe20Ni80and Fe 15Ni85possess very small coercivity around\n0.5 mT, as depicted in Fig. 13(a) and (c). This is ex-\npected for the magnetically soft Fe-Ni alloys. For Fe 20Ni80,we check the in\ruence of small magnetostriction coe\u000e-\ncient on its coercivity, as shown in Fig. 13(b) and Fig.\n14(a). It can be seen that if \u0015sis set as small values\n\u00061:0\u000210\u00007, SLM processing with di\u000berent laser power\ncan only lead to a coercivity change within 0.05 mT.\nOn the contrary, for Fe 15Ni85which has a negative \u0015s\nof\u00005:8\u000210\u00006, SLM processing with 100 W laser can\nincrease the coercivity to \u00181:5 mT, as presented in\nFig. 13(d). Fig. 14(b) shows the calculated coercivity\nfor di\u000berent alloy compositions and laser power. It can\nbe found that the coercivity of SLM processed Fe 15Ni85\nand Fe 25Ni75is increased to\u00181.7 and\u00180.8 mT, respec-\ntively. Within the laser power P= 100\u0000130 W, the\ncoercivity is found to only slightly increase with P. The\nreason may be that the stress magnitude and distribu-\ntion are not signi\fcantly changed within P= 100\u0000120\nW. It should be mentioned that the experimentally\nmeasured coercivity of SLM processed Fe-Ni alloys is\naround 0:06\u00004 mT [12{14]. Our simulation results\non the coercivity are in accordance with these exper-\nimental measurement. Apart from the coercivity val-\nues, the remanent magnetization ( \u00160Mr) and magnetic\ndomain structure are also a\u000bected by the SLM pro-\ncess. Fig. 13(e) and (f) shows the magnetic con\fgu-\nration at the remanent state of the sample in (c) and\n(d), respectively. For Fe 20Ni80(Fig. 13(a) and (b)) and\nFe15Ni85without SLM processing (Fig. 13(c)), \u00160Mr\nis as low as\u00181 mT. Accordingly, the remanent mag-\nnetic con\fguration is composed of large-area magnetic\ndomains perpendicular to zdirection or along the neg-\nativezdirection, as shown in Fig. 13(e). However, after\nFe15Ni85is processed by SLM, \u00160Mris enhanced to\u0018516 M. Yi et al.\nmT (Fig. 13(d)), and magnetic domains along the pos-\nitivezdirection occupy larger areas (Fig. Fig. 13(f)).\nThese results computationally con\frm that SLM pro-\ncessing could a\u000bect the coercivity, remanent magneti-\nzation, and magnetic domain structure in Fe-Ni alloys.\n6 Summary and outlook\nIn conclusion, we have integrated FEA, CALPHAD out-\nput, and micromagnetics to demonstrate the \frst at-\ntempt for the computational evaluation of the microstruc-\nture evolution and magnetic coercivity of SLM pro-\ncessed magnetic Fe-Ni alloys. The \rowchart and cou-\npling schemes of the integrated framework are sum-\nmarized in Fig. 15, including heat-transfer model, me-\nchanical model, CALPHAD, and micromagnetic model.\nBoth the heat-transfer and mechanical models are nu-\nmerically solved by \fnite element method. The micro-\nmagnetic model and thus the magnetic hysteresis are\ncalculated through the energy minimization by the con-\njugate gradient method within the \fnite di\u000berence frame-\nwork. These models are coupled through the informa-\ntion transfer among them. For example, the tempera-\nture history T(x;t) from the heat transfer model can\nbe used as input for the CALPHAD and mechanical\nmodel. In detail, \fnite element (FE) nodal values of\n\feld variable \u001e(x;t) and temperature T(x;t) from the\n\fnite element simulation of heat transfer model are\nmapped to the CALPHAD model by a Python script to\nobtain the temporal and spatial phase fraction. Mean-\nwhile, FE nodal values of \u001e(x;t) andT(x;t) are also\nimported to the mechanical model for the calculationof transient stress \u001b\u001b\u001b(x;t) and strain \u000f\u000f\u000f(x;t) by \fnite ele-\nment method. In turn, CALPHAD can provide thermo-\ndynamically consistent parameters for the Gauss quadra-\nture points in \fnite element simulations, such as spe-\nci\fc heatCp(T), density\u001a(T), and latent heat Lfor the\nheat transfer model and thermal expansion coe\u000ecient\n\u000b(T) for the mechanical model. Furthermore, the stress\n\felds from the mechanical model can be input to the\nmicromagnetic model for the calculation of magnetic\nproperties, including coercivity, remanent magnetiza-\ntion, and magnetic domain structure. For transferring\nthe stress \felds from the \fnite element method to the\nmicromagnetic model which is solved by \fnite di\u000ber-\nence method, we use a Delaunay triangulation of the\nscattered FE nodal values to perform the linear inter-\npolation. Overall, by using the temperature-dependent\nmaterial states/parameters and the step-by-step ele-\nment activation of powder mesh, \fnite element simula-\ntion of the heat-transfer model is performed to calculate\nthe temperature distribution and evolution during the\nSLM processing of Fe-Ni alloys. With the thermal his-\ntory from \fnite element simulations as input, thermo-\nmechanical analysis is performed by using the elastic-\nplastic material constitutive model. By integrating the\nthermal history and CALPHAD output, the evolution\nand distribution of liquid, powder, FeNi 3and\r-Fe-Ni\nduring the SLM process are calculated. Finally, micro-\nmagnetic simulations which treats the residual stress as\nthe magnetoelastic energy are carried out to calculate\nthe magnetic property of SLM processed Fe-Ni alloy.\nBy using this computational framework, the melting\npool geometry and the cyclic thermal history are iden-\nti\fed. The cyclic tension and compression are con\frmed\nHeat transfer model  \n(finite element method)  CALPHAD  \n(CALculation  of PHAse  Diagrams)  \nMechanical model  \n(finite element method ) Micromagnetic model  \n(energy minimization by \nconjugate -gradient method)  𝝈𝒙,𝑡 Thermal history: 𝑇(𝒙,𝑡) Phase fraction evolution:  \nYpowder𝒙,𝑡, Yliquid𝒙,𝑡, YFeNi3𝒙,𝑡, Y𝛾-Fe-Ni𝒙,𝑡 \nThermodynamic properties:  \n𝐶p𝑇,𝜌𝑇,𝐿,𝛼(𝑇) \nStress/Strain: 𝝈𝒙,𝑡,𝝐𝒙,𝑡 Magnetic properties:  \nCoercivity , remanent  magnetization, \nmagnetic domain structure  FE nodal values  \n𝜙𝒙,𝑡,𝑇(𝒙,𝑡) \nFE GQP  values: 𝐶p𝑇,𝜌𝑇,𝐿 \nFE nodal values  \n𝜙𝒙,𝑡,𝑇(𝒙,𝑡) \nLinear interpolation through Delaunay \ntriangulation of FE nodal values  \nFig. 15. Flowchart and coupling schemes of the integrated framework for the computational study of laser addi-\ntively manufactured magnetic materials. FE: \fnite element; GQP: Gaussian quadrature point.Computational study on additive manufacturing of magnetic materials 17\nin the interface of two neighboring tracks, which could\ndegrade the interface bonding. It is found that the ma-\nterial \frstly changes from powder to liquid and then\nexperiences cyclic phase changes between \r-Fe-Ni and\nFeNi 3. SLM process is found to obviously enhance the\nremanent magnetization and increase the coercivity of\nFe15Ni85and Fe 25Ni75to 0:8\u00001:7 mT. The calculated\ncoercivity is shown to agree with the experimental val-\nues.\nWhile we have shown a promising method for the\nsimulation of additively manufactured magnetic mate-\nrials by integrating FEA, CALPHAD, and micromag-\nnetics, the work here is a \frst attempt and lots of issues\nhave to be thoroughly considered in the near further.\nAs an initial work, here we limit ourselves to the work-\ning principle and potential feasibility of the proposed\ncomputational scheme, and do not focus on the accu-\nrate prediction of industrial or real AM process at the\ncurrent stage. Several issues in this work related to the\nreal AM process have to be deliberated and resolved in\nthe next step, as listed in the following:\n(1) The AM processability of FeNi alloys by SLM\nis an open question. Here we hypothesize defect-free\nprocessing in our simulation. The low processability re-\nlated e\u000bects from \ruid dynamics, surface tension and\ngas \row have to be considered, with more experimental\ninformation on the processability.\n(2) The temperature-dependent material parame-\nters in Fig. 2, which are important for the thermal\nand stress analysis, are not accurate, due to the lack\nof su\u000ecient experimental data. It will be good to make\nmore e\u000borts on experimental measurements and obtain\nthermodynamic property information of magnetic al-\nloys form CALPHAD database. Including the possible\nvaporization may be also necessary for the accurate pre-\ndiction of the peak temperature. The phase change and\ncomposition variation induced stress should also play a\nrole.\n(3) Apart from the weak microsegregation of ele-\nments by using the Scheil-Gulliver model, the phase\nfraction is taken as the main indicator for the microstruc-\nture evolution in this work. However, in the real ex-\nperimental case, microstructure would be more com-\nplicated. Other phases except for \r-Fe-Ni and FeNi 3\nmay also exist and spatial element microsegregation\nand composition inhomogeneity can occur. There are\nalso other general issues in the prediction of microstruc-\nture evolution during SLM process, such as the non-\nequilibrium state, fast cooling induced solute trapping,\ne\u000bect of free energy from magnetic contribution, etc.\nPhase-\feld simulation using CALPHAD information of\nmagnetic materials [63] could be a viable methodologyfor predicting the microstructure evolution during the\nSLM processing, and deserves our future e\u000borts.\n(4) The dependence of magnetic properties on the\nphase fraction and microstructure is very complicated\nand not involved in this work. As for the prediction\nof magnetic properties of SLM-processed magnetic ma-\nterials in the real case, more microstructure informa-\ntion (e.g. phase distribution, grain boundaries, grain\norientation, porosity, surface roughness, etc.) except for\nresidual stress should be considered.\n(5) Even though the calculated coercivity is shown\nto be in line with the experimental one for the laser ad-\nditively manufactured magnetic FeNi, the simulation\nwork here lacks the experimental validation on sev-\neral points such melt pool size, temperature, stress, mi-\ncrostructure, etc. Collaborative experimental work has\nto be carried out in order to make the computational\nframework fully convinced.\nAcknowledgements The support from the German\nScience Foundation (DFG YI 165/1-1 and DFG XU\n121/7-1), the Pro\fle Area From Material to Product\nInnovation { PMP (TU Darmstadt), the European Re-\nsearch Council (ERC) under the European Unions Hori-\nzon 2020 research and innovation programme (grant\nagreement No 743116), and the LOEWE research clus-\nter RESPONSE (Hessen, Germany) is acknowledged.\nThe authors also greatly appreciate their access to the\nLichtenberg High Performance Computer of Technische\nUniversit at Darmstadt.\nReferences\n1. 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Acta\nMater 59(2):521{530. https://doi.org/10.1016/\nj.actamat.2010.09.055"    },    {        "title": "1803.01179v1.Long_Range_Magnetic_order_stabilized_by_acceptors.pdf",        "content": "Long Range Magnetic order stabilized by acceptors\nXiaodong Zhang,1Jingzhao Zhang,1Shengbai Zhang,2and Junyi Zhu1,\u0003\n1Department of Physics, Chinese University of Hong Kong, Hong Kong SAR, China\n2Department of Physics, Applied Physics, and Astronomy,\nRensselaer Polytechnic Institute, Troy, New York 12180, United States\nTuning magnetic order in magnetic semiconductors is a long sought goal. A proper concentration\nof acceptors can dramatically suppress local magnetic order in favor of the long one. Using Mn\nand an acceptor codoped LiZnAs as an example, we demonstrate, by \frst-principles calculation, the\nemergence of a long-range magnetic order. This intriguing phenomenon can be understood from\nan interplay between an acceptor-free magnetism and a band coupling magnetism. Our observation\nthus lays the ground for a precise control of the magnetic order in future spintronic devices.\nFor decades, diluted magnetic semiconductors (DMS)\nhave attracted considerably interests for their unique\nmagnetic properties and \rexible tunabilities used in the\ninformation devices [1{13]. Among the examples, a gi-\nant and tunneling magnetoresistance device was realized\nin trilayer structures of Mn doped GaAs [14]. Magnetic\nmemory cells using the Mn-doped GaAs have also been\nconstructed, in which bit-writing can be processed by\napplying a magnetic \feld or spin-polarized electric \feld\n[15{17]. However, the formation of inhomogeneous mag-\nnetic domains, originated from a short-range magnetic\norder, is a critical bottleneck to further developing the\nDMS technology [18, 19].\nThese domains are formed by an aggregation of the\nmagnetic dopants [9, 20, 21], as a result of a short-range\nattractive interaction among them [9]. It happens that a\ncarrier doping can have a considerable e\u000bect on the ag-\ngregation. For example, Kuroda et al. showed that at\ncertain concentrations of electron donors the aggregation\nof Cr in (Zn 1\u0000xCrx)Te is enhanced. This phenomenon\ncan be explained by an energy gain due to an electron-\nenhanced short-range magnetic interaction among the Cr\natoms [22]. In contrast, a long-range magnetic interac-\ntion can provide a thermodynamic driving force to reduce\nthe undesirable phase separation [23, 24]. Therefore, to\novercome the bottleneck, an approach that can largely\nsuppress the short-range magnetic order, while simulta-\nneously enhancing the long-range one, is highly desirable,\nbut unfortunately lacking.\nThe reason is because such an approach would be\nagainst the common belief that the long-range order is\nalways weaker than the short-range one, because of a\nrapid decrease of the magnetic coupling strength with\nthe distance between magnetic dopants. Such a be-\nlief has led to an impasse in advancing the study of\nDMS, which, in our view, is not \rawless. For example,\nit is known that an acceptor-mediated magnetic order\ncan work against the short-range magnetic order that\ndominates at the acceptor-free condition [25]. In other\nwords, intentionally-introduced acceptors could reduce\nthe short-range magnetic coupling strength, whereby\nmitigating the undesirable magnetic phase separation.At the same time, the long-range magnetic order may\nstay or even be enhanced, as its response to the pres-\nence of the acceptors can be di\u000berent from that of the\nshort-range order. This reasoning raises the hope that\nat a proper magnetic dopant concentration and hole con-\ncentration, the system may suppress the phases that are\nin favor of short-range order by signi\fcantly increasing\ntheir formation energies.\nIn this Letter, we present a theory that reveals the in-\nterplay between di\u000berent magnetic orders in the presence\nof hole doping. As an example, we consider magnetic\norders in a Mn-doped LiZnAs, where Li Znand/or V Li\nserve as the acceptors, to contrast with the ones in the\nabsence of the acceptors. Using density functional theory\ncalculations, we discover a long-range AFM order, which\nmay be attributed to the stepping stone mechanism me-\ndiated by the magnetized Zn dand Aspstates. We also\n\fnd, in line with the above discussions, that at a proper\nacceptor concentration, the short-range AFM order can\nbe removed, while the long-range AFM order gives way\nto an FM order. The net e\u000bect is the stabilization of a\nlong-range FM con\fguration. Our \fndings thus point to\na new direction in rationally designing the DMS.\nAll calculations are performed using projected\naugmented-wave method [26] and density functional\ntheory within generalized gradient approximation of\nPerdew-Burke-Ernzerhof [27] as implemented in VASP\ncode [28]. The Mn doped LiZnAs is simulated by 2 \u00022\u00022\nsupercell (96 atoms). All atoms are relaxed with force\ntolerance of 0 :01 eV=\u0017A. A plane-wave cut-o\u000b energy of\n500 eV was used in all calculations. For Brillouin-zone\nintegration, Monkhorst-Pack k-points grid of 4 \u00024\u00024\nwas employed. In order to consider strong correlation\ne\u000bect of transition metals, the LDA+U method [29] is\nused. The Hubbard parameter U=3:5 eV and Hund rule\nexchange parameter J=0:6 eV are taken as suggested by\nRef. [30]. HSE06 calculations are also performed to check\nthe accuracy of LDA+U method and found that results\nof LDA+U method are qualitatively consistent with re-\nsults of HSE calculations. Convergence tests in respect\nto cell size, energy cuto\u000bs, k-points, U, and force cuto\u000bs\nhave been performed.arXiv:1803.01179v1  [cond-mat.mtrl-sci]  3 Mar 20182\nFIG. 1. Simulation cells with the dopant separation from\nthe \frst nearest neighboring con\fguration to the 5th nearest\nneighboring one. The Li atoms are deleted for clarity.\nTo simulate the short range and long range magnetism,\nwe use 2 \u00022\u00022 supercell with two Mn atoms substituting\ntwo Zn atoms, which corresponding to Mn concentration\nof 6.25%. Considering the symmetry of LiZnAs, there\nare 5 types of in-equivalent con\fgurations, corresponding\nto \frst nearest neighbor (1st-NN) con\fguration to \ffth\nnearest neighbor (5th-NN) con\fguration shown in Fig.1.\nThe formation energy of Mn doped LiZnAs is de\fned\nas\nEf=E(doped )\u0000E(undoped )+nZn\u0016Zn\u0000nMn\u0016Mn(1)\nwhere\u0016Znand\u0016Mnare chemical potential of Zn and\nMn, respectively. And nZnandnMnare number of Zn\nand Mn, respectively. In the comparison among di\u000berent\nmagnetic con\fgurations, these two chemical potentials\nwill be cancelled.\nFirstly, we calculated the relative formation energy\nwithout acceptors as a function of di\u000berent neighbor-\ning con\fgurations, as shown in Fig.2(a). Here, the rel-\native formation energy is calculated in reference to an\nAFM state of the \frst nearest neighboring (1st-NN) sites,\nwhich is the most stable con\fguration.\nNext, we introduced acceptors by replacing Zn atoms\nwith Li atoms. Various Zn sites have been checked and\nwe found that the most stable con\fguration is obtained\nby replacing Zn sites that are the nearest neighbors to\nthe magnetic dopants. Still, the total energies are simi-\nlar for di\u000berent replacement sites. Also, we calculated the\nformation energy of the Mn pairs as a function of di\u000ber-\nent doping con\fguration. To our surprise, the most sta-\nFIG. 2. (a) Relative formation energy of Mn dopants at di\u000ber-\nent nearest neighboring sites for acceptor free case; (b) that\nenergy for the acceptor doping case; (c) magnetic coupling\nstrength of di\u000berent con\fgurations.\nble con\fguration is the FM state with magnetic dopant\natoms occupying the \ffth nearest neighboring (5th-NN)\nsites and the 1st-NN sites becomes the most unstable\nFig.2(b). This discovery strongly suggests that with the\nintroduction of acceptors, the local magnetic dopants\nclustering is largely hindered and the long range order\nof magnetic dopants emerges.\nTo further understand this dramatic change in the rel-\native stabilities of di\u000berent con\fgurations, we calculated\nthe magnetic coupling strength of each con\fguration, as\nshown in Fig.2(c). The magnetic coupling strength is de-\n\fned as a half of the di\u000berence between the AFM and\nFM state. The calculated results demonstrate that the\n\frst nearest neighboring sites prefer a AFM state for ac-\nceptor free case. However, when acceptor are introduced,\nthe magnetic coupling strength decreases to almost zero.\nWhen the distance between the dopant pairs becomes\nlonger, the magnetic order changed from AFM or non-\nmagnetic to FM. This discovery is di\u000berent from the\n\\common sense\" belief that magnetic coupling strength\nis the strongest among the neighboring sites and decays\nvery fast when the distance between dopant atoms in-3\ncreases.\nIn order to understand how acceptor doping changes\nthe magnetic interaction and further in\ruence the rela-\ntive formation energy, we \frstly need to understand the\nmagnetic interaction in acceptor free case.\nIn acceptor free case, the magnetic interaction of 1st-\nNN con\fguration can be explained by the superexchange\ntheory, which suggests that for the half occupied dstate\nof Mn, the electron hopping from the As pstate strongly\nfavors AFM coupling [31{33]. Still, the magnetic cou-\npling between the third or fourth nearest neighbors are\nnonzero. A similar magnetic order has been discovered in\nCr doped Bi 2Te3and Sb 2Te3system, where an antibond-\ning state derived from the s lone pair on stepping stone\nBi atoms plays a critical role for the long range magnetic\norder. However, in this system, there lacks such slone\npair state.\nTo understand the long range magnetic coupling mech-\nanism, we calculated the spin texture, as shown in\nFig.3(a). We found that spin density exists near the cen-\nter of As and Zn bond, demonstrating a covalent nature.\nThe coupling between the As- pstate and Zn- dstate is\nclearly demonstrated in projected density of states, as\nshown in Fig.3(b). The electron hopping among the spin\npolarized covalent states near the center of the two As\nand Zn bonds on the chain lowers the total energy and\nenhances the long range correlation between the magnetic\ndopants.\nFurther, we substitute Zn site in the Mn-As-Zn-As-Mn\nchain by Ca or Cd atoms in order to investigate the role\nof Zn for such long range magnetic interaction. We found\nthat the spin density becomes localized around As atoms.\nThis is due to the high ionicity of the Ca and As bond, as\nshown in Fig.3(c). Note that, there is no dorbital in Ca2+\nand theselectron are mostly transferred to the orbitals\nnear the As atom. The electron hopping between the\norbitals near di\u000berent As atoms on the chain is largely\nhindered because the overlap of orbitals almost disap-\npeared (refer to supplementary materials for details). As\nexpected, we found that the magnetic coupling strength\nis almost zero when Zn sites in the four Mn-As-Zn-As-\nMn chains surrounding one Mn atom are substituted by\nfour Ca atoms (Fig.4). Further, we also substitute four\nZn atoms by Cd atoms. We found that the magnetic\ncoupling preserves because the coupling between the d\norbitals of Cd and the p orbitals of As is covalent and\nnear the center of the As and Cd bond. Therefore, the\nelectron hopping among these states can lower the total\nenergy and enhance the long range correlation between\nMn dopants. The large coupling strength in Cd doping\ncase is due to the enhanced p-dcoupling strength be-\ntween Cd and As, since the d orbital of Cd is higher than\nthat of Zn.\nThe above analysis demonstrates that the covalent na-\nture ofp-dhybridized orbital is the direct reason for\nsuch long range magnetic interactions. However, such\nFIG. 3. (a) Local spin density near dopant in Mn doped\nLiZnAs (b) Projected density of states (PDOS) (c) Local spin\ndensity near dopant in Mn,Ca codoped LiZnAs.\nenhancement in the magnetic coupling is not limited to\nthep-dcoupling mechanism.\nNext, we'll investigate the interplay between acceptor\nmediated magnetism and the intrinsic long range mag-\nnetism. We \frst studied acceptor doping. Band coupling\nmodel [25] suggests that the exchange coupling strength\nis sensitive to the position of dlevel, relative to the VBM.\nUnder the crystal \feld of Tdsymmetry, the Mn dorbitals\nare split into lower egstates and higher t2gstates. All the\n\fvedelectrons of Mn will occupy these states, which are\nthe majority spin states below the VBM. The minority\nspin states are all empty and above the VBM. The orig-\ninal band coupling model [25] proposed that the energy4\nFIG. 4. Changes of the coupling strength with respect to the\nreplaced atoms.\ndi\u000berence between FM and AFM phase is\n\u0001EFM\u0000AFM =\u0000\u000bnh(\u00011\npd+ \u00012\npd) + 6\u00011;2\ndd(2)\nwhere\u000bis a parameter related to the localization of hole\n(acceptor) states and Mn-Mn distance; nhis the hole\n(acceptor) density; the \u00011\npdand \u00012\npdare contributions\nfrom acceptors and described by Zener's model [1, 34, 35];\nthe \u00011;2\nddis coming from intrinsic magnetic interaction.\nMore details about band coupling model and parameters\ncan be found in Ref.[25]. However, this energy di\u000berence\nis not a function of the distance between the dopants.\nTo include the distance as an important variable in the\nlong range magnetic investigation, we slightly modi\fed\nthe above equation, as listed below,\n\u0001EFM\u0000AFM =\u0000\u000b(R)nh(\u00011\npd+ \u00012\npd) + 6\f(R)\u00011;2\ndd(nh)\n=nhJacceptors (R) +Jintrinsic (R;nh)(3)\nwhere R is the distance between the two magnetic\natoms;nh, \u00011\npd, \u00012\npd, and \u00011;2\nddare the same as Eq.2.\n\u000b(R) and\f(R) are magnetic interaction parameters as\nfunctions of the distances, based on the acceptor me-\ndiated mechanism, and the intrinsic long range mecha-\nnism, respectively. Usually, the decay of \f(R) is much\nfaster than \u000b(R) because acceptor state is very delocal-\nized and the interaction range can be very long [36].\nThe decay of \f(R) is sensitive to symmetry of bond-\ning, because the intrinsic coupling is mediated by the\nelectron hopping among the magnetized orbitals of the\nhost material. acceptor mediated magnetic interaction\nJacceptor =\u0000\u000b(R)(\u00011\npd+ \u00012\npd) is usually negative while\nintrinsic magnetic interaction Jintrinsic = 6\f\u00011;2\ndd(nh) is\nusually positive. The Jintrinsic (R;nh) is not only de-\npendent on distance between the two magnetic atoms,\nbut also in\ruenced by the acceptor density. We further\nchecked the Jintrinsic at di\u000berent dopant sites and found\nthat the interaction still exists for \frst nearest neighbors\nupon acceptor doping. However, the long range intrinsic\ninteraction is destroyed by acceptors. This is probably\ndue to the change of electron occupation in the steppingstone state, which is more sensitive to the acceptor dop-\ning than the As- pstate that mediates the super exchange\nmechanism. This di\u000berence directly leads to the signif-\nicant di\u000berent magnetic order upon acceptor incorpora-\ntion at di\u000berent magnetic doping sites. A more complete\npicture can be achieved by strict analysis of many body\ne\u000bects in the future, which is out of the scope of this\npaper [refer to Supplementary materials for details].\nSince the \frst term in Eq.3 is dependent upon the ac-\nceptor density, it is possible to tune the magnetic cou-\npling by changing the acceptor concentration. If the sign\nofJacceptor and that of Jintrinsic are opposite, a proper\nacceptor density will result in zero magnetic interaction\nfor short range con\fgurations (1st-NN).\nIn our simulation cell, when we introduce one acceptor\nby removing one electron from the system, the magnetic\ncoupling becomes almost zero on the \frst nearest neigh-\nboring site. When we introduce a Li substitutional de-\nfect on a Zn atom that is close to the Mn atom, or when\nwe remove one Li atom that is close to the Mn atom,\nthe magnetism on the nearest neighboring site con\fgu-\nration also disappeared. Therefore, these three calcula-\ntions con\frmed that the acceptor cancels the magnetism\non nearest neighboring con\fgurations.\nFurther, we checked the magnetic order on other neigh-\nboring sites. For acceptor free case, the 2nd-NN and\nthe 5th-NN con\fguration yield almost zero magnetic cou-\npling. Despite the large di\u000berence of Mn-Mn distance be-\ntween 2nd-NN (5 :94\u0017A) and 5th-NN (10 :28\u0017A), the mag-\nnetic coupling strength of 2nd-NN (13 :5 meV) and that\nof 5th-NN (15 :1 meV) are similar. These results suggest\nthat acceptor induced magnetic interaction is almost a\nconstant shift in di\u000berent neighboring sites, di\u000berent from\nthe fast decay nature under acceptor free condition, as\nshown in Fig.2(c). This di\u000berence largely cancels the\nlocal magnetic interaction and a long range magnetic in-\nteraction emerges.\nBased on all these calculations and analysis, a strat-\negy on tuning the magnetism of di\u000berent sites can be\nproposed. If the short range magnetic interaction under\nacceptor free condition is di\u000berent from the acceptor me-\ndiated magnetic interaction, it's possible to incorporate\na proper amount of acceptors to largely destroy the short\nrange magnetism. As a result, the short range magnetic\ncon\fguration become unstable and long range magnetism\nemerges. Such long range magnetic interaction stabilizes\nlong range con\fgurations and results in a long range mag-\nnetic ordering phase, which can be the global minimum.\nAs shown in Fig.2(b), the formation energies of the con-\n\fgurations with three or more atoms that separate the\ndopant atoms (from 2nd-NN to 5th-NN) are all lower\nthan that of the 1st-NN con\fguration when a acceptor is\ndoped.\nThis is the very \frst time, a sensitive relationship be-\ntween the stability of short range vs.long range magnetic\norder and the concentration of the acceptors is discov-5\nered. With a proper amount of co-dopants are incorpo-\nrated, the long range magnetic order can suppress the\nshort range one and become stable. Such stability is\nvery important during the growth of DMS, be-\ncause once the spinodal decomposition is formed\ndue to the strong short range magnetic coupling,\nit's often kinetically forbidden to change the mag-\nnetic coupling into long range ones via post an-\nnealing techniques. These results are also consistent\nwith early experimental discoveries, which suggest that\nthe formation of nanocrystal (results of spinodal decom-\nposition) can be tuned by acceptors or donors in Cr\ndoped ZnTe [21]. Therefore, we expect our strategy\nshould be general in various transition metals doped in\nDMS and may lead to discoveries of class of magnetic\nmaterials. Our discovery also strongly suggests that it's\nusually naive to use short range magnetic order to rep-\nresent the long range one. To achieve a complete picture\nof magnetic order upon di\u000berent dopant to dopant sepa-\nrations, various doping con\fgurations have to be tested\nto guarantee the correct results.\nIn summary, we observe, based on density functional\ntheory calculations, that the long-range AFM order in\nan acceptor-free Mn-doped LiZnAs is mediated by mag-\nnetizedpdhybridized states. While here the short-range\nmagnetic order can be largely suppressed by a proper ac-\nceptor codoping with Mn, it happens that the long-range\nAFM order also simultaneously ceases, giving way to a\nlong-range FM order. Such a long-range order is the long\nsought goal in the DMS study. Our codoping strategy\nshould be general, whose applications to other DMS may\nlead to a discovery of materials with stable long-range\nmagnetism. 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McCulloch1, Lan Wang1,* & Changgu Lee4,* \n \n1School of Science, RMIT University, Melbourne, VIC 3001, Australia.  \n2School of Mechanical Engineering, Sungkyunkwan University, Suwon, Republic of Korea.  \n3Center for Quantum Materials and Superconductivity (CQMS) and Department of Physics, \nSungkyunkwan University, Suwon, Republic of Korea.  \n4School of Mechanical Engineering and SKKU Advanced Institute of Nanotechnology \n(SAINT), Sungkyunkwan University, Suwon, Republic of Korea.  \n# Equally contribute to the paper  \n* Corresponding authors. Correspondence and requests for materials should be addressed to \nL.W. (email: lan.wang@rmit.edu.au) and  C. L. (email: peterlee@skku.edu.kr ) \n \nABSTRACT:  Two dimensional (2D) van der Waals (vdW ) materials have demonstrated \nfascinating optical, electrical and thickness -dependent characteristics. These have been \nexplored by numerous authors but reports on magnetic pro perties and spintronic applications \nof 2D vdW  materials are scarce by comparison. By performing anomalous Hall effect \ntransport measurements, we have characterised the thickness dependent magnetic properties \nof single crystalline vdW Fe 3GeTe 2. The nanoflakes of this vdW metallic material exhibit a \nsingle hard magnetic phase with a near square -shaped magnetic loop, large coercivity (up to \n550 mT at 2 K), a Curie temperature near 200 K and strong perpendicular magnetic \nanisotropy. Using criticality analysis, we confirmed  the existence of magnetic coupli ng \nbetween vdW atomic layers and obtained a n estimated  coupling length of ~ 5 vdW layers  in \nFe3GeTe 2. Furthermore, the hard magnetic behaviour of Fe 3GeTe 2 can be well described by a \nproposed model. The  magnetic properties of Fe 3GeTe 2 highligh t its potential for integration into vdW magnetic heterostructures, paving the way for spintronic research and applications \nbased on these devices.  \n \n Two dimensional ( 2D) van der Waals (vdW)  materials  have received considerable attention \nsince the successful isolation of graphene\n1, 2. Studies  on these materials have revealed novel \noptical3, 4 and electronic5, 6 properties . Moreover, employing heterostructures based on these \n2D vdW materials has revealed further interesting properties  and suggested applicatio ns 7, 8, 9. \nvdW magnets were known more than 50 years ago10, 11, 12, but interest has  been renewed with \nthe emergence of 2D materials. In the last few years, Raman spectroscopy13, 14, 15, 16, 17 and \nelectron transport measurements18, 19 have been performed on 2D magnets. Importantly , 2D \nferromagnetism has been discovered very recently in two insulating vdW materials, \nCr2Ge2Te620\n and CrI 321 and novel devices based on vdW ferromagne tic heterostructure s have \nbeen demonstrated18, 22, 23. The opportunity exists to design and fabricate many devices based \non vdW magnets . For example, vdW magnetic insulator can magnetize 2D topological \ninsulators by the magnetic proximity effect and thereby generate the quantum anomalous Hall effect in these materials\n24, 25, 26, 27. vdW ferromagnetic metals can be employed in spin orbit \ntorque devices when stacked w ith vdW metals with strong spi n orbit interactions28, 29, 30, 31, 32, \n33. However, in order to exploit  ferromagnetic vdW materials as building blocks for vdW \nheterostructure based spintronics, a ferromagnetic vdW metal with a hard magnetic phase and  \na large magnetic remanence to sat urated magnetization (M R/MS) ratio is essential. T his kind \nof vdW ferromagnetic metal is scarce.  \n Among all the predicted and experimentally observed vdW ferromagnetic materials\n19, 20, 21, 23, \n34, 35, 36, 37, 38, 39, 40, 41, 42, 43, a very promising ferromagnetic metal is Fe 3GeTe 2 (FGT) , which \nexhibits a Curie temperature (T C) near 2 20 K in its bulk state42. Previous e xperimental work has shown that bulk single crystalline  FGT has a ferromagnetic state with a very small \nMR/MS ratio and coercivity at all temperatures42, 43, 44, 45, suggesting  limited potential as a \nbuilding block for  vdW magnetic heterostructures. However, the M R/MS ratio and coercivity \nof a magnetic material strongly depend on its domain structure46, 47, 48, which is thickness  \ndependent . Furthermore , recent research49 shows that MBE -grown wafer -scale FGT thin \nfilms have improved magnetic properties . These findings motivate us to investigate the \nmagnetic properties of exfolia ted FGT nanoflakes of various  thicknesses using anomalous \nHall effect measurements.    \n \nHere, we report anomalous Hall effect measurements on single crystalline FGT nanoflakes \nand show that their magnetic properties are highly dependent on thickness. Import antly, by \nreducing the thickness to less than 200 nm, a hard magnetic phase with large coercivity and \nnear square -shaped hysteresis loop occurs. These characteristics are accompanied by  strong \nperpendicular magnetic anisotropy , making  vdW FGT a ferromagnetic metal suitable for \nvdW heterostructure -based spintronics. By employing  criticality  analysis, the existence of \nmagnetic coupling with a coupling length of ~ 5  vdW layers  between vdW atomic layers is \nestimated  in FGT . Finally , we propose a model to describe the hard magnet ic behaviour of \nFGT  thin flakes. This model is suitable for other vdW ferromagnetic thin films or nanoflakes  \nwith strong perpendicular anisotropy  and square -shaped magnetic loops.  \n In a ferromagnetic material , the relationship between the Hall resistance and the  applied \nmagnetic field is given by E q. 1.   \nZ S Z xy MR BR R + =0                   (1).  \nHere, R xy is the Hall resistance, which  is composed of a normal Hall resistance (the first term \nin Eq. 1) a nd an anomalous Hall resistance (the second term in Eq. 1). B Z and M Z are the applied magnetic field and the sample magnetic moment perpendicular to the sample surface, \nrespectively.  The anomalous Hall resistance is proportional to M Z. As FGT is a metallic \nferromagnetic material, the normal Hall resistance is negligible compared with the anomalous \nHall resistance in the magnetic field range  of interest . Hence, the shape of the R xy vs B loop is \nactually the same as that of the M Z vs B  loop. The coercivity and M R/MS ratio can be \nobtained from the R xy (B) curve.  \n We measured the longitudinal resistance R\nxx and transversal resistance R xy of 11 FGT  \nnanoflake  devices with thickness from 5.8 nm to 329 nm and a bulk FGT crystal . The R xy(B) \nof selected FGT nanoflake devices at 2 K  are shown in Fig. 1  (b-f) with the R xy(B) of the  bulk  \nFGT crystal (Fig. 1a). The applied magnetic field was perpendicular to the surfaces of the samples. The coercivity and M\nR/MS ratio of the bulk crystal sample at 2 K are only  21.6 mT \nand 0.07, respectively. These characteristics agree well with those measured using a \nmagnetometer42. However, the exfoliated nanoflakes display ed different magnetic pr operties. \nThe nanoflakes with thickness es of 329 nm and 191 nm display ed magnetic loops resembling  \ntwo mag netic phases  with different coercivities.  As shown in Fig. 1b and 1c, when the \nmagnetic field sweeps from the positive saturation field to the negative saturation field, the R\nxy values decrease sharply at a lower negative magnetic field and then decrease a gain at a \nhigher negative field, which is similar to the behaviour of the coexistence of two phases. When the thickness es of the FGT nanoflakes decrease further, the “two phase” behaviour \ndisappears. As shown in Fig. 1( d-f), the R\nxy (B) loo ps of the three samples (with thickness es \nof 82 nm, 49 nm and  10.4 nm , respectively) display a near square shape, indicative of a single \nhard magnetic phase. The coercivities of these three samples are much larger than thos e of \nthe samples with 329 nm and 191 nm  thickness and exceed 400 mT at 2 K. As FGT gradually \nevolves from a soft phase (bulk) to a single hard phase  (82 nm, 49 nm and 10.4 nm ), we speculate that the “two phase” behaviour in t he nanoflakes with thicknesses of 329 nm and \n191 nm is due to the gradual evolution of the domain structure. The M R/MS ratios of the 191 \nnm, 82 nm, 49 nm and 10.4 nm thickness nanoflakes are near 1 at 2 K, demonstrating that all \ntheir magnetic moments remain aligned perpendicular to the sample surfaces  at the \nremanence point.  The magnetic moments flip abruptly to the opposite direction at the \ncoercive field. In the magnetic field regime away from the coercivity , the four nanoflakes \nbehave like a single magnetic domain with a strong perpendicular anisotropy. The magnetic domains only appear and flip to the opposite direction near the coercivity. As bulk single crystalline FGT also shows a st rong perpendicular anisotropy, the anisotropy should be \ninduced by the crystalline field. With increasing temperature,  FGT  gradually evolves from \nferromagnetic to paramagnetic state. The Curie temperature (T\nC) of each FGT nanoflake \ndevice can be determined from the tem perature point whe re the remanence value goes to \nzero20, as shown in the supplementary materials. The dependence of T C on thickness (from  \n0.3 mm to 49 nm) is shown in Fig. 2, from which we conclude that the T C of the FGT \nnanoflakes is almost independent of thickness in this range. As shown in the inset of Fig. 2, \nthe T C decreases sharply as the thickness decreases  from 25 nm to 5.8  nm. This behaviour is \nsimilar to that in Cr 2Ge2Te620, but is different from the behaviour in CrI 321. It should  be \nemphasized that nine of the eleven  device s were fabricated in ambient condition with an air \nexposure of ~7 mins. The other two ultra -clean devices were made in a glove box (O 2 < 0.1 \nppm, H 2O < 0.1 ppm) with h- BN and PMMA covering. The two batches  of devices show the \nsame magnetic characteristics (details in supplementary materials). The theory of critical  \nbehaviour50 reveals that the finite thickness of flakes limits the divergence of the spin -spin \ncorrelation length at the T C. As FGT is an itinerant metallic system, its spin -spin coupling \nshould extend for many atomic layers, even along the out of plane directions. The spin- spin coupling range along the out of plane direction can be fitted as shown in the inset of Fig. 2 \nusing the  formula51                 \n1−𝑇𝐶(𝑛)/𝑇𝐶(∞)=[(𝑁0+1)/2𝑛]𝜆      (2), \nwhere T C is the Curie temperature, n is the number of atomic layers of a flake, N 0 is the spin -\nspin coupling range, and λ is a universal critical exponent. A best fitting to the data requires λ \n= 1.66 ± 0.18 and N 0 = 4.96 ± 0.72 monolayers. The fitted λ  = 1.66 is near the value of a \nstandard 3D Heisenberg ferromagnetism52. The correspondence achieved  using a single \nfitting curve also indicates that FGT nanoflakes with a thickness of more than 5 vdW layers \nare still 3D ferromagnets. If FGT  nanoflakes evolve from 3D ferromagnetism to 2D \nferromagnetism from 25 nm to 5.8 nm , we should be able to obtain two fitting curves with \ndifferent critical exponents. However, the data does not show this behaviour, which further confirms 3D magnetism when FGT thickness > 5.8 nm. Scaling behaviour\n40, 41 near the T Cs \nof samples with thicknesses from monolayer to >  10 nm should reveal the evolution of the \nmagnetism from 3D to 2D with decreasing thickness in FGT. As t he focus of this paper is \nrevealing the hard magnetic properties of FGT nanoflakes  and their suitability for future \nspintronic applications,  we propose th is scaling analysis as future work.  \n                                                                                      \nMore detailed measurements were performed on the 10.4 nm  thickness  nanoflake. In Fig. 3a \nand Fig. 3b, the R xy (B) loops from this sample, measured with perpendicular applied \nmagnetic field, are plotted at various tempe ratures. There is  a clear evolution with increasing \ntemperature. At 2  K, the R xy (B) loop is nearly square -shaped with a large coercivity of 552.1 \nmT and M R/MS ~ 1, revealing alignment of spins due to strong perpendicular anisotropy. The \nRxy (B) loops remain approximately square up to 155 K. Figure 3 d displays the temperature \ndependence of coercivity in this temperature regime.  When the temperature ex ceeds T C (~19 1 \nK), the nanoflake becomes paramagnetic.   \nWe also measured the temperature dependence of the R xy at the remanence point of the  \nsample ( the m easurement and data process details  of R xy(T) are shown in the  method). The  \nRxy (T) of the 10.4 nm nanoflake is shown in Fig. 3c. R xy(0) is an extrapolation to T = 0 K \nfrom region ІІ  based on the spin wave model as discussed later . Results from other samples \nare shown in the supplementary materials. Below 155 K, the FGT nanoflake exhibits a \nferromagnetic phase with near square- shaped magnetic loop. The abrupt decrease of the \nmagnetic moment around 155 K in Fig. 3c indicates  a first order magnetic phase transition \nnot yet known, but likely  related to the competition between the perpendicular  anisotropic \nenergy and the thermal  agitation energy . As the bulk FGT single crystal also shows a phase \ntransition with gradually changed magnetic moment around 155 K45 ,we speculate that the \nsharp er phase transition in the FGT nanoflake is due to its decreased thickness, which induces \nsingle domain behaviour at the remanence. In the temperature regime from 155 K  to T C, the \nFGT nanoflake displays a ferromagnetic phase with very small coercivity and remanence. The R\nxy (T) reveals another  phase transition near 1 0 K, where the remanence increases \nsharply with decreasing temperature, indicat ive of the formation of new spins  and magnetic \nmoments. Further understanding of the phases present in this temperature regime would \nrequire neutron scattering measurements, which are beyond the scope of this article . Fig . 3c \nalso shows the temperature dependence of R xy (T) fitted from 2 K to 15 0 K using the mean \nfield theory (the Brillouin function) and the spin wave theory. The R xy (T) behaviour of the \n10.4 nm thickness sample cannot be fitted using mean field theory, but agreement with the spin wave theory for a three- dimensional ferrom agnet is good. This provides further evidence \nthat a FGT nanoflake remains a 3D magnetic system when its thickness exceeds 5 layers. Our experimental results contain information required to construct a correct model for FGT. First, \nas shown in R\nxy (B) meas urements (Fig. 1, Fig. 3 and Fig.4), FGT has a very strong perpendicular anisotropy due to the crystalline field. Second, the thickness dependence of T C \nas shown in Fig. 2 indicates the existence of magnetic coupling between atomic layers in \nFGT with a  coupling range of  about 5 vdW layers. Therefore, a correct Hamiltonian \ndescribing FGT should include a perpe ndicular anisotropic energy, an  in-plane spin- spin \ninteraction energy, an out of plane spin- spin interaction energy and a Zeeman energy induced \nby the applied magnetic field.  \n \nThe evolution of R xy hysteresis loops with the angle θ  between the applied magnetic field and \nthe direction perpendicular to the sample surface (i.e. the direction of magnetic anisotropy) \nfor the 10.4 nm flake at 2 K is shown in Fig. 4a and 4b. As mentioned prior, the spins in the \nFGT nanoflakes align to one direction due to the strong perpendicular anisotropy when the temperature is below 15 5 K. Magnetic domains only appear near the coercive field. This \nsimple magnetic structure makes it possible to construct a model to describe the behaviour of the coercivity. When a magnetic field is applied to a single domain ferromagnetic material with uniaxial anisotropy, the energy of the magnetic system is composed of the magnet ic \nanisotropic energy and the Zeeman energy,  \nφ θφ cos )( ) (sin)( )(2\nS S S A BVTM VTK TE −− =     (3), \nwhere K A is the magnetic anisotropic energy  per volume , VS is the volume of the sample, φ is \nthe angle between the magnetic field and the magnetic moment, θ  is the angle between the \nmagnetic field and the direction of magnetic anisotropy (Fig. 4f ), M S (T) is the magnetic \nmoment of a unit volume FGT at temperature T, and B is the applied magnetic field. With an \napplied magnetic field B and the angle θ, we can calculate φ from Eq. 4, the well -known \nStoner -Wohlfarth model53  \n0 sin )( ) cos() sin()( 2)(= + − − =∂∂φ θφ θφφS S S A BVTM VTKTE\n   (4), For thin FGT nanoflakes with perpendicular anisotropy, the demagnetization effect46 should \nbe considered. Consequently, t he applied magnetic field B and angle θ  in Eq. 3 and 4 should \nbe modified to the effective magnetic B eff and angle θ eff. Further  detail is shown in \nsupplementary materials. Using this  model , we fitted the R xy(B) curve with θ =  85o to obtain \nthe unit magnetic anisotropic energy K A at 2 K, 25 K, 50 K, 80 K, 100 K, and 120 K , as \nshown in Fig. 4 e (additional detail s provided in supplementary materials). Fig . 4c shows the \nfitting curve for the 2 K data. It should be emphasized here that all the R xy (B) with various θ  \nat different temperatures in the magnetic regime away from the coercivity can be well fitted \nby the Stoner -Wohlfarth model. The reason f or using  θ = 85o loops for the K A fitting is that a  \nmagnetic loop of small θ value is nearly a straight line without cu rvature in the magnetic \nregime away from the coercivity, which is not suitable for obtaining a reliable K A.  \n \nAs magnetic domains appear in the magnetic field regime near coercivity, to describe the angular dependence of coercivity , the flip of magnetic domains near coercivity should be \nincluded in model. By considering the thermal agitation energy and utilizing the fitted K\nA \nvalues, a modified Stoner -Wohlfarth model (details  in supplementary materials) can  be used \nto describe the angular dependence of coercivity. As shown in Fig. 4g , if the system can be \nthermally excited from a  meta -stable state (state 1) to an  unstable state (state 2), the magnetic \nmoment can then be flipped to a final stable state (state 3). The energy difference  between \nstate 1 and state 2 is ∆ E. With increasing magnetic field B (more negative B), the energy \ndifference ∆E between the stable state 1 and the unstable state 2 decreases.  In the modified \nStoner -Wohlfarth model, we make two assumptions:  \n1. At a certain B  field, the thermal agitation energy is large enough to overcome the ∆ E in a \nstandard experimental time (100 seconds) causing the magnetic moment to flip. As the \nFGT shows a nearly square -shaped R xy loop (magneti zation  loop), we can assume that this B field is the coercive field , which is a reasonable approximation due to the sharp \ntransition of the magnetic moments.  \n2. When the first domain flips under an applied magnetic field, other un -flipped magnetic \nmoments will generate an effective field on the mag netic moment in the first flipped \ndomain. The processes of domain flip, expansion, and interaction are complex. Micro -\nmagnetic simulation is required to provide a detailed description, which is beyond the \nscope of this  paper. Here, a parameter 𝑎𝑎(T) is use d to describe the mean field interaction \nbetween the flipped and un- flipped magnetic moments.  \nBased on the proposal of Neel  and Brown54,55, we use ∆E = 25k BT as the barrier height where \nthe magnetic moment starts to flip (details in supplementary materials). We thus obtain  \nTk BTMTVTa TVTKBTMTVTa TVTK\nB S AS A\n25] cos)()()]( 1[) (sin)()([] cos)()()]( 1[) (sin)()([\n1 122 22\n= −−−− −−−\nφ θφφ θφ\n   (5), \nwhere V (T) is the volume of the first flipped domain at T,  𝑎𝑎(T) is the parameter describing \nthe effective field due to the coupling between the first flipped domain and the un- flipped \nmagnetic moments at temperature T, which affects the Zeeman energy. The value of 𝑎𝑎(T) lies \nbetween 0 and 1. φ 1 and φ2 are the angles between the applied magnetic field and the \nmagnetic moment for state 1 and state 2, respectively. These angles are calculated using Eq.4. \nDue to the demagnetization effect46, B and θ  here should also be modified to B eff and   θeff \n(Supplementary materials).  \n \nUsing V(T) and 𝑎𝑎(T)  as fitting parameters in Eq.5 in conjunction with the modified Stoner -\nWohlfarth model  provides  excellent agreement with the experimental angular dependence of \ncoercivity at various temperatures  (Fig. 4 d), which further confirms the single domain \nbehaviour induced by the strong perpendicular anisotropic energy  in FGT in the field regime \naway from coercivity. T he volume of the first flipped magnetic domain near the coercivity  V(T) is important for understanding the magnetic dynamics of a ferromagnetic materials . \nThis volume V and the perpendicular anisotropic energy K A at different temperature s are \nshown in Fig. 4e. The modified Stoner -Wohlfarth model proposed here is suitable for \ndescribing the magnetic behaviour of 2D vdW ferromagnetic materials with strong \nperpendicular anisotropy and near square -shaped loop. \n \nTo conclude, FGT nanoflakes are  vdW 2D metallic ferromagnets  with large coercivity , \nMR/MS ratio of 1 , relatively high T C and strong perpendicular anisotropy . Exploitation of this \nnew material in various vdW magnetic heterostructures  with properties  such as giant \nmagnetoresistance and tunnelli ng magnetoresistance, as well as vdW spin- orbit torque \nheterostructures  is expected to yield exciting results . This discovery paves t he way for a new \nresearch field,  namely, vdW heterostructure -based spintronics. \n \nMethod  \nSingle Crystal Growth  \nSingle crystal Fe 3GeTe 2 was grown by the chemical vapor transport (CVT) method. High-\npurity Fe, Ge and Te were blended in powder form with molar proportions of 3:1: 5 \n(Fe:Ge:Te). Iodine (5 mg/cm2) was added as a transport agent and the mixed constituents \nwere seal ed into an evacuated quartz glass ampoule. This ampoule was placed in a tubular \nfurnace, which has a temperature gradient between 700~650 ºC. The furnace center \ntemperature was ramped up to 700 ºC with a heating rate of 1 ºC per minute and was maintained a t the set point for 96 hours. To improve crystallinity, the ampoule was slowly \ncooled down to 450 ºC for over 250 hours. Below 450 ºC, the furnace was cooled more rapidly to room temperature.  \n Device Fabrication and Measurement  \nFirst, the single crystalline Fe 3GeTe 2 was mechanically exfoliated and placed on a Si \nsubstrate with a 280 nm thickness SiO 2 layer. Then, Cr/Au (5 nm/100  nm) contacts were \npatterned by photolithography and e -beam lithography. During intervals between processing, \nthe sample was covered by a P DMS  film and stored in an evacuated glass tube (~10-6 Torr). \nThe sample was exposed to ambient for no more than 7 minutes throughout the fabrication \nprocedure. The transport measurements were performed in a Quantum Design PPMS with 9 \nT ma gnetic field.  \n \nHall effect  measurement  and data processing  \nBecause of the non -symmetry in our nanoflake devices, the measured Hall resistance was \nmixed with the longitudinal magnetoresistance. We processed the data by using (R xyA(+B) - \nRxyB(-B)) / 2 to eliminate the contribution from  the longitudinal magnetoresistance, where \nRxyA is the half loop sweeping from the positive field to the negative field, R xyB is the half \nloop sweeping from the negative field to the positive field, and B is the applied magne tic \nfield.  We also measured the R xy (T) at the remanence point  for most of the  samples. To \nmeasure the R xy (T) at the rem anence point, the magnetic moment of sample s was first \nsaturated by a 1 T magnetic field  and then the magnetic field  was decreased to zero (the \nremanence point). Finally, the temperature dependence of the R xy at remanence was measured \nwhen the temperature was increased from 2 K to 300 K. In order to eliminate the non-\nsymmetry effect of the device, we measured R xy (reman ence) with both 1 T and -1 T \nsaturation. The real R xy (T) at remanence without R xx mixing was calculated using (R xyA(T) - \nRxyB(T)) / 2. Here R xyA and R xyB are the rem anence with 1 T and - 1 T saturation, respectively.  \n \nAssociated content  \nSupplementary Material s, including  the characterization of FGT crystals, detailed \nmeasurement data for other samples, definition of the Curie temperature , anomalous Hall \neffect and the modified Stoner -Wohlfarth model , mean field fitting and spin wave fitting , the \neffect  of the surface amorphous oxide  layer and the confirmation of ohmic contact.  \n Acknowledgement  \n \nThis research was supported by the Australian Research Council Centre of Excellence in \nFuture Low -Energy Electronics Technologies ( Project No.  CE170100039), the  Institute for  \nInformation & Communications Technology  Promotion (IITP) grant (Project No . B0117- 16-\n1003, Fundamental  technologies of 2D materials and devices for the  platform of new -\nfunctional smart devices), and the Basic  Science Research Program ( Project  No. \n2016R1A2B4012931), and the National Research Foundation (NRF) of Korea by a grant \nfunded by the Korean Ministry of Science, ICT and Planning ( Project No . \n2012R1A3A2048816) . \n \nAuthor Information \nAuthor notes  \nCheng Tan  and Jinhwan Lee  contributed equally to this work.  \nAffiliations  \nSchool of Science, RMIT University, Melbourne, VIC 3000, Australia.  \nCheng Tan, Sultan Albarakati,  James Partridge, Matthew R. Field, Dougal G. McCulloch & \nLan Wang  \nSchool of Mechanical Engineering, Sungkyunkwan University, Suwon, Republic of \nKorea  \nJinhwan Lee  \nSchool of Mechanical Engineering and SKKU Advanced Institute of Nanotechnology (SAINT), Sungkyunkwan University, Suwon, Republic of Korea  \nChanggu Lee  \nCenter for Quantum Materials and Superconduc tivity (CQMS) and Department of \nPhysics, Sungkyunkwan University, Suwon, Republic of Korea.  \nSoon- Gil Jung \n& Tuson Park  Contributions  \nC.L. and L.W. conceived and designed the research. J.L. synthesized the material, J.L., S.J. \nand T.P. performed the materia l characterization. M.R.F. and D.G.M. performed the TEM \nscan for the cross -section of nanoflakes. C.T. 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Lett.  80, 4783 (1998).  \n \n53. Robert, C. & Handley, O. Modern Magnetic Materials: Principles and Applications \n(Wiley -Interscience, New York, 2000).  \n \n54. Kneller, E. & Luborsky, F. Particle size dependence of coercivity and remanence of single ‐domain particles. J. Appl. Phys. 34, 656- 658 (1963).  \n 55. Brown Jr, W. F. Relaxational behavior of fine magnetic particles.  J. Appl. Phys. 30, \nS130- S132 (1959). \n \n   \n \n       \n Figure legends :  \nFigure 1 . Rxy (B) for FGT nanoflakes of various thicknesses at 2 K.  Each red scale bars represents 10 \nµm. a Bulk device, L × W × T= 2 mm × 0.8 mm × 0.3 mm.  M R/MS = 0.0715. b A device with a \nthickness of 329 nm, L × W × T = 44.4  µm× 49.4 µm× 329 nm. MR/MS = 0.0807. c  A device with a \nthickness of 191 nm, L × W × T = 14. 7 µm× 9.25 µm × 191 nm. MR/MS = 0.9757. d  A device with a \nthickness of 82 nm, L × W × T=  10.3 µm× 19.5 µm× 82 nm. MR/MS = 0.9839. e  A device with a \nthickness of 49 nm, L × W × T=  12.6 µm× 13.1 µm× 49 nm. MR/MS = 0.9980. f A device with a \nthickness of 10.4 nm, L × W × T= 12.7 µm × 8.79 µm × 10.4 nm.  MR/MS = 0.9973.  \n \nFigure 2 . Thickness dependence of the Curie temperature  (TC). Except ing the device with a thickness \nof 329 nm, the T Cs of the devices were d etermined by the temperature point when the remanence \nbecomes zero. The TC of the device with a thickness of 329 nm was determined from its ρ xx(T) curve. \nInset shows the thickness dependence of the TCs in nanoflakes with thicknesses from 5.8 nm to 25 nm  \nwith fitting curve . Blue dots are the T Cs for the two ultra -clean devices that were covered by h- BN \nand PMMA in a glove box (O 2 < 0.1 ppm and H 2O < 0.1 ppm). \n \nFigure 3 . Anomalous Hall effect measurement s performed on 10.4 nm thickness FGT device.  a Rxy \n(B) loops in the temperature regime from 2 K to 140 K, in which the FGT nanoflake shows \nferromagnetic properties. b Rxy (B) curves in the temperature regi me from 150 K to 185 K. The FGT \nnanoflake shows zero coercivity and remanence  when T > 150 K . c Normalized R xy (T) curve and \nthree fitting curve s based on the mean field theory J = 1, J = ∞ , and the spin wave theory, \nrespectively. Three magnetic regimes exist from 2 K to  TC = 191 K. Regime I (2~ 10K) is an unknown \nphase requiring further investigation. Regime II  (10~155 K)  is a hard ferromagnetic phase.  Regime III  \n(155~191 K)  is a phase with zero coercivity and remanence.  d The temperature dependence of \ncoercivity  from 2 K to 150 K . \n \nFigure 4.  Angular dependent Hall -effect measurements and relevant fitting s using a modified Stoner –\nWohlfarth model . a, b Rxy (B) loops at different angles between the applied magnetic field and the \ndirection perpendicular to the surface of the nanofla ke with a thickness of 10.4 nm. At 0 °, the surface \nof the nanoflake is perpendicular to the magnetic field.  c Normalized R xy (B) curve at 2 K measured at \n85° from 6 T to - 6 T. The fitting curve is based on the Stoner -Wohlfarth model. d  The effective \nangular  dependence of effective coercivities  at different temperatures. The solid curves ar e the fitting \ncurves based on a modified Stoner -Wohlfarth model. From  b we can tell that the remanences of the Rxy loops at angles > 85o show pronounced decreases and a divergent coercivity value was obtained at \n90o, an angle beyond the range included by the modified Stoner -Wohlfarth model. Based on this \nexperimental result, the theoretical fittings were only performed to 85°.  e The temperature \ndependence of K A and V. K A was fitted by the Stoner –Wohlfarth model, while V was fitted based on \nthe fitted K A and the modified Stoner -Wohlfarth model (more details in supplementary information).  f \nIllustration of the variables used in the Stoner –Wohlfarth model. The dashed line is the easy axis of \nthe magnetic anisotropy in FGT nanoflakes.  g Schematic diagram of a magnetic system chang ing \nfrom a stable to an unstable state.  \n \n \n        \n \n   \n \n \n \n \n  \n            \n   \n \n      \n \nFigure 1  0 100 200 300300,000190200210\n5 10 15 20 25150160170180190200210\n Thickness (nm)TC (K)\n Fitting curve\n Experimental points\n Experimental points \n         (h-BN covered) \n Curie Temperature(K)\nThickness(nm)\n  \n \n \n    \nFigure 2  -800 -400 0400 800-20-1001020\n-400 -200 0 200 400-10-50510\n02550751001251500200400600\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.01.2c dbRxy(Ω)\nMagnetic field (mT) 140 K\n 120 K\n 100 K\n   80 K\n   60 K\n   40 K\n   20 K\n   10 K\n     2 Ka\n  \nRxy(Ω)\nMagnetic field (mT) 150 K   155 K\n 160 K   165 K\n 170 K   175 K\n 180 K   185 K\n  Coercivity(mT)\nT (K)\n  ΙΙΙ ΙΙRxy(T) /Rxy(0)\nT/Tc Experimental\n  J=1\n  J=∞\n Spinwave fittingΙTc=191 K\n  \n \n \n \n   \n \n \n \nFigure 3  -6 -3 0 3 6-20-1001020\n0 20 40 60 800.00.51.01.52.0\n-6-4-20246-1.0-0.50.00.51.0-1200 -600 0 600 1200-20-1001020\nRxy(Ω)\nMagnetic field (T) 80°\n 83°\n 85°\n 87°\n 89°\n 90°\n  Effective coercivity (T)Effective tilt angle θ (°)    2 K Exp.      2 K Fitting\n  25 K Exp.    25 K Fitting\n  50 K Exp.    50 K Fitting\n  80 K Exp.    80 K Fitting\n100 K Exp.  100 K Fitting\n120 K Exp.  120 K Fitting\n  e d cbRxy/Rxy(B=0)\nMagnetic field (T)  2K, 85 °\n  Fitting a\n  \n0204060801001200246810121416 \nT (K)V(102 nm3)\n2345678KA(106 erg/cc)\ng f\nRxy(Ω)\nMagnetic field (mT)   0°\n 50°\n 60°\n 70°\n  \n \n \n  \n \n  \n \n    \n \n \nFigure 4  Supplementary Materials  \n \n \nSection 1: The characterization of FGT crystals  \nSection 2: Detailed measurement data for other samples  \nSection 3: Definition of the Curie temperature  \nSection 4: Anomalous Hall effect and the modified Stoner -\nWohlfarth mod el \nSection 5: Mean field fitting and spin wave fitting  \nSection 6: The effect of the surface amorphous oxide  layer  \nSection 7: The confirmation of ohmic contact  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Section 1: The material characterization of FGT crystal  \n \nSection 1: The characterization of FGT crystals  \nChemical composition  \nTo analyse the chemical composition of synthesized Fe 3GeTe 2 (FGT) flakes, energy -\ndispersive X -ray spectroscopy (EDS) was carried out. As shown in Fig. S1(a), we used bulk \nsingle crystalline FGT flakes after peeling off surface layers. EDS (Fig. S1(b)) revealed the \nFGT to be ~ 2.88:1:2.05  Fe:Ge:Te.  \nBasic magnetic properties  \nA magnetic property measurement system (MPMS ), which includes a superconducting \nquantum interference device (SQUID), was employed to characterize the magnetic properties \nof FGT. The volume of sample was 0.00012329 cm3. Figure S1(d) shows the temperature \ndependence of the inverse of magnetization. The data was collected in a magnetic field of 100 Oe, applied along the c -axis direction. We extracted the Curie temperature (T\nc) from fitting \naccording to the Curie -Weiss law in a magnetization graph, as shown in Fig. S1(d).    \n \nSupplementary Fig. S1.  Characterization of the FGT single crystal.  (a) Optical image of a \nsynthesized FGT single crystal. (b) EDS spectrum of synthesized FGT single crystal. The \nstoichiometric composition of three elements Fe, Ge, and Te is 48.59: 16.83: 34.58 for the synthesize d \ncrystal. (c) Field dependence of magnetization  at 2 K. The calculated Ms is 374.48 emu/cm3. (d) \nTemperature dependence of inverse magnetization and fitted line for 100 Oe. The fitted line indicates \nthe T C = 205 K.  \n  \n \n \n  \nSection 2: Detailed measurement data for all the samples  \n-800 -400 0 400 800-0.50-0.250.000.250.50\n-800 -400 0 400 800-0.8-0.40.00.40.8\n050100 150 200 250 3000.70.80.91.01.11.2\n-800 -400 0 400 800-0.9-0.6-0.30.00.30.60.9\n-8-6-4-202468-0.45-0.30-0.150.000.150.300.45\n050100 150 200 250 3000.00.20.40.60.8Rxy(mΩ)\nMagnetic Field (mT) 210 K  207 K\n 205 K  203 K\n 200 K  195 K\n 190 K  180 K\n  \ne fd cbRxy(mΩ)\nMagnetic Field (mT) 180 K\n 170 K\n 150 K\n 120 K\n 100 Ka\n  ρxx(Ω •cm × 10-4)\nT (K) 0 T\n 9 T\n  Rxy(Ω)\nMagnetic Field (mT) 80 K\n 50 K\n 20 K\n 10 K\n   2 K\n  Rxy(mΩ)\nMagnetic Field (mT) 75°\n 90° \n \nRxy(Ω)\nT (K)\n  \n \n \n   \n \n \n    \n \n \nSupplementary Fig. S2. R xy and R xx measurements for the bulk FGT single crystal. The contacts \nfor the bulk device were made of Indium. (a) Rxy (B) curve s show zero hysteresis in the temperature \nregime from 210 K to 180 K . The curve becomes linear at ~207 K. (b)  Rxy (B) loops show no \nhysteresis from 100 K to 180 K. (c)  Rxy (B) loops from 80 to 2 K. In this  regime , the saturated \nmagnetization  decreases with decreasing temperature. At 2 K, the coercivity is ~21.6 mT. (d) \nTemperature dependence of ρ xx at 0 T and 9 T. (e) Angular dependent Hall effect at 50 K with θ  = \n75° and 90.3°, respectively. W hen a high in -plane magnetic field  (θ = 90ο) is applied , the electron \nspins with perpendicular anisotropy will be forced  pa rallel to the sample plane and then point to \nrandom direction when the magnetic field is swept back to zero , therefore  the R xy should be 0 in this \ncondition. (f)  Rxy (T) curve under 1 T field.  -600 -300 0 300 600-150-100-50050100150\n0501001502002503000.480.510.550.59bRxy(mΩ)\nMagnetic Field (mT) 180 K\n 160 K\n 140 K\n 120 K\n 100 K\n   80 K\n   60 K\n   20 K\n     2 Ka\n  \nρxx(Ω • cm × 10- 4)\nT (K)\n  \n \n \n                        \n \n \n       \nSupplementary Fig. S3. R xy (B) and R xx (T) curves for the FGT nanoflake  with a thickness of  \n329 nm. (a) Rxy (B) curve at different temperatures. The hysteresis loops are not square. However, \npronounced hysteresis loops with a non- zero coercive field are displayed. (b)  ρxx (T) curve at zero  \nmagnetic field, an obvious magnetic transition due to spin- flip scattering  is shown at about 205 K, \nfrom which the Curie temperature is determined.  -500 -250 0 250 500-1.0-0.50.00.51.0\n-500 -250 0 250 500-0.2-0.10.00.10.2\n050100 150 200 250 3000.850.900.951.001.05\n-800 -400 0 400 800-1.0-0.50.00.51.0\n050100 150 200 250 3000.00.20.40.60.81.0\n-500 -250 0 250 500-0.8-0.40.00.40.8Rxy(Ω)\nMagnetic Field(mT) 190 K\n 180 K\n 160 K\n 140 K\n 120 K\n 100 K\n   60 K\n   20 K\n     2 K\n  \nf ed cbRxy(Ω)\nMagnetic Field(mT) 210 K\n 209 K\n 207 K\n 205 K\n 203 K\n 200 Ka\n  ρxx(Ω •cm × 10- 4)\nT (K) 0 T\n 9 T\n  Rxy(Ω)\nMagnetic Field(mT)   0°\n 45°\n 60°\n 75°\n 90° \n Rxy(Ω)\nT(K) \n \nRxy(Ω)\nMagnetic Field(mT) 160 K\n 120 K \n \n \n \n      \nSupplementary Fig. S4. R xy and R xx measurements for the nanoflake  with a thickness of \n191 nm. (a) R xy (B) loops from 2 K to 190 K. The coercivity  increases with decreasing  \ntemperature. (b) Rxy(B) curve from 200 K to 210 K. The curve becomes linear at ~209 K . All \nthe curves shows no hysteresis. (c) Angular dependent Hall effect at 50 K, the R xy at 90°  is \nnearly 0, which means the sample is nearly parallel to the magnetic field. At 75 ° an \nasymmetry loop is observed, which may indicate exchange coupling induced uni -directional \nmagnetic anisotropy . (d) Temperature dependence of ρxx at 0 T and 9 T. Although the data is \nnoisy at lower temperature, the spin -flip scattering is still clear. (e) Temperature dependence \nof the remanence. The sudden decrease of remanence indicates that the thermo agitation \nenergy is higher than the perpendicular anisotropic energy, which can also be clearly seen \nfrom the evolut ion of the R xy (B) loop from 160 K to 210 K.  (f) Rxy(B) curve at 120 K and \n160 K. Compared with the 120 K curve, the 160 K curve is indicative of two magnetic phases \nwith increasing temperature.  -800 -400 0 400 800-3-2-10123\n-500 -250 0 250 500-1.0-0.50.00.51.0\n050100 150 200 250 3000123\n-700 -350 0 350 700-3-2-10123\n-6-4-20246-3-2-10123050100 150 200 250 3002.052.102.152.202.252.30Rxy(Ω)\nMagnetic Field (mT) 180 K \n 150 K\n 120 K\n 100 K\n   50 K\n   10 K\n     2 K\n  \nRxy(Ω)\nMagnetic Field (mT) 210 K\n 205 K\n 200 K\n 190 K\n  Rxy(Ω)\nT (K)\n  Rxy(Ω)\nMagnetic Field (mT)   0°\n 45°\n 60°\n 75°\n  f ed cb\nRxy(Ω)\nMagnetic Field (T) 2 T\n 9 Ta\n85° \n  ρxx(Ω •cm × 10-4)\nT (K) 0 T\n 9 T\n  \n \n \n    \n \n \n   \n \n \n \nSupplementary Fig.  S5. R xy and R xx measurements for the nanoflake  with a thickness of  82 nm. (a) \nRxy (B) loops from 2 K to 180 K . The 150 K and 180 K loops  show behaviour indicative of two magnetic \nphases. (b) Rxy (B) curves from 190 K to 210 K. A ll curves do not show any hysteresis. (c) Temperature \ndependence of ρ xx at 0 T and 9 T. Spin- flip scattering happens at ~205 K. (d) Rxy (T) curve  under 1 T \napplied field. (e)(f) Angular dependent Hall effect at various angles θ  at 50 K. (f) The angular dependent \nHall effect at 85o. At first, the magnetic field was swept within 2 T, In this range, the curve did not show \nsaturated behaviour. The R xy (B) shows saturated behaviour when the applied magnetic field exceeds 5 \nT, indicating a strong perpendicular anisotropy. -800 -400 0 400 800-1.8-1.2-0.60.00.61.21.8\n-500 -250 0 250 500-0.6-0.4-0.20.00.20.40.6\n-6 -4 -2 0 2 4 6-1.5-1.0-0.50.00.51.01.5\n-700 -350 0 350 700-1.5-1.0-0.50.00.51.01.5\n050100 150 200 250 3000.00.30.60.91.21.5\n050100 150 200 250 3000.930.940.950.96f ed cRxy(Ω)\nMagnetic Field (mT) 160 K\n 140 K\n 120 K\n 100 K\n   80 K\n   60 K\n   40 K\n   20 K\n   10 K\n     2 K \n \nRxy(Ω)\nMagnetic Field (mT) 180 K\n 190 K\n 195 K\n 200 K\n 205 K\n 210 K\n  Rxy(Ω)\nMagnetic Field (T) 80°  83°\n 85°  87°\n 89°  90°\n  Rxy(Ω)\nMagnetic Field (mT)   0°\n 50°\n 60°\n 70°\n 80°\n  Rxy(Ω)\nT (K)\n  b ρxx(Ω • cm × 10- 4)\nT (K) 0 T\n 9 T\n  a\n \n \n  \n \n \n \n  \n \n \n \n  \n \nSupplementary Fig. S6. R xy and R xx measurements for the nanoflake with a thickness of 49 nm. \n(a) Rxy (B) loops from 2 K to 160 K . (b) Rxy (B) loops from 180 K to 210 K. The curve becomes \nlinear at ~210 K . (c-d) Angular dependent Hall effect at 50 K. Loops taken at angles 0 ° ˗ 50°are not \nshown, because they nearly overlap. Actually , the coercivity slightly  decreases with increasing \nangles from 10o to 40o, which can still be well fitted using the modified Stoner -Wohlfarth model, as \ndescribed in part 4 of the supplementary materials. (e) Temperature dependence of remanence. An \nobvious phase transition occurs  at ~ 155 K. When T < ~15 K , the R xy shows an abrupt  rise. A phase \ntransition may occur  in this temperature regime . (f) The ρxx(T) curve  under zero magnetic field \ndisplays  a peak at ~194 K, attributed to the spin- flip scattering near T C. It can be supressed by a 9 T \nmagnetic field as shown in the figure.  \n  \n-6 -4 -2 0 2 4 6-1.0-0.50.00.51.0\n-6 -4 -2 0 2 4 6-1.0-0.50.00.51.0\n-4 -2 0 2 4-1.0-0.50.00.51.0\n-4 -2 0 2 4-1.0-0.50.00.51.0\n-3 -2 -1 0 1 2 3-1.0-0.50.00.51.00 50 100 150 200 250 3002.52.73.03.2\n0 50 100 150 200 250 3000510152025\n-6 -4 -2 0 2 4 6-1.0-0.50.00.51.0Rxy/Rxy(B=0)\nMagnetic Field(T) 25 K, 85 °\n Fitting\n  \nRxy/Rxy(B=0)\nMagnetic Field (T) 50 K, 85 °\n Fitting\n  \nRxy/Rxy(B=0)\nMagnetic Field (T) 80 K, 85 °\n Fitting\n  Rxy/Rxy(B=0)\nMagnetic Field (T)  100 K, 85 °\n  Fitting\n  \nRxy/Rxy(B=0)\nMagnetic Field (T) 120 K, 85 °\n  Fitting \n ρxx(Ω • cm × 10−4)\nT(K) 0 T\n 9 T\n  c bRxy(Ω)\nT(K) Saturated at 1T\n Remanencea\n  \nh gf e d\nRxy/Rxy(B=0)\nMagnetic Field (T)   25 K\n   50 K\n   80 K\n 100 K\n 120 K\n  \n \n \n \n \n        \n \n \n   \n \n \n   \n \n \n \nSupplementary Fig. S7. Temperature dependent R xx and  Rxy loops from the nanoflake with a \nthickness of 10 .4 nm. (a) Temperature dependence of the ρ xx at 0 T and 9 T. Spin- flip scattering is \nevident near T C. The enhanced resistance with decreasing temperature may originat e from the \ndisorder at the sample surface induced by exfoliation  or oxidization. (b) Temperature dependence of \nthe remanence R xy (T) and the saturated at 1 T R xy (T) curves. (c -h) Normalized a ngular depe nden t \nHall resistance measured with the magnetic field swept from the positive saturated magnetic field to \nthe negative saturated magnetic field at various temperatures. The red solid lines (d -h) are fitting \ncurves based on the Stoner -Wohlfarth model.  Section 3: Definition of the Curie temperature  \n  The definition of Curie temperature is a vital part of magnetic material measurements. \nGenerally, based on Curie -Weiss law, we can determine the Curie temperature of a sample \nthrough a linear fit to the temperature dependence of inverse magnetization above T C. \nHowever, this method is only accurate when the critical exponent is 1 for the temperature-\ndependent susceptibility, which is unknown yet for FGT nanoflakes. Here we define the \ntemperature at which the remanence R xy goes to zero as T C [Nature 546, 265–269 (2017) ], \nwhich can give us more accurate T C values.  \n \n In our experiments, the sampl e was firstly cooled down to 2 K under a magnetic field of 1 \nT(-1T).  Then the magnetic field was slowly (5 Oe/s) decreased to 0 Oe at 2 K. Finally we \nscanned temperature from 2 K to 300 K at 3 K/min and get the R xy vs T curve with 1 T ( -1 T) \nremanence. By  telling the junction of remanence vs T curve of 1 T and - 1 T(Fig. S8b), where \nRxy (T) goes to zero, we can get the value of T C.   \n050100150200250300-2.0-1.5-1.0-0.50.00.51.0\n180 200 220 240-0.34-0.32-0.30-0.28-0.26-0.24-0.22-0.20Rxy(Ω)\nTemperature(K) -1 T Remanence\n  1 T Remanence\nCurie Temperature~204 K\n  Rxy(Ω)\nTemperature(K) -1 T remanence\n  1 T remanence\n  \n \n \n \n \n \nSupplementary Fig. S8. Definition of  the Curie temperature of  one FGT sample. \nThe remanence curves of 1 T and -1 T merge to one curve  at ~204 K, which we \ndefine as the Curie temperature. At this temperature, the remanence (R xy (T)) \nbecomes zero.  Section 4: Anomalous Hall effect and the Stoner -\nWohlfarth model  \n \nAnomalous Hall Effect: Basic for mula and data process  \nIn a ferromagnetic material, the relationship between the Hall resistance and applied \nmagnetic field shows in Eq. S1.   \nZ S Z xy MR BR R + =0          (S1),  \nHere, R xy is the Hall resistance, which is composed of a normal Hall resistance (the first term \nin Eq. S1) and an anomalous Hall resistance (the second term in Eq. S1). B Z and M Z are the \napplied magnetic field and the sample magnetic moment perpendicular to the sam ple surface, \nrespectively. The anomalous Hall effect is proportional to M Z. As FGT is a metallic \nferromagnetic material, the normal Hall resistance is very small compared with the anomalous Hall resistance in the magnetic field range of interest. Hence, th e shape of the R\nxy \nvs B loop is actually the same as that of the M Z vs B loop when the applied magnetic field is \nperpendicular to the sample surface. The coercivity and M R/MS ratio can be obtained from the \nRxy (B) curve.  \n Because of the non -symmetry in ou r nanoflake devices, the measured Hall resistance was \nmixed with the longitudinal magnetoresistance. We processed the data by using (R\nxyA(+B) - \nRxyB(-B)) / 2 to eliminate the contribution from the longitudinal magnetoresistance, where \nRxyA is the half loop  sweeping from the positive field to the negative field, R xyB is the half \nloop sweeping from the negative field to the positive field, and B is the applied magnetic field.  We also measured the R\nxy (T) at the remanence point for most of the samples. To \nmeas ure the R xy (T) at the remanence point, the magnetic moment of samples was first \nsaturated by a 1 T magnetic field and then the magnetic field was decreased to zero (the remanence point). Finally, the temperature dependence of the R\nxy at remanence was meas ured when the temperature was increased from 2 K to 300 K. In order to eliminate the non-\nsymmetry effect of the device, we measured R xy (remanence) with both 1 T and -1 T \nsaturation. The real R xy (T) at remanence without R xx mixing was calculated using (R xyA(T) - \nRxyB(T)) / 2. Here R xyA and R xyB are the remanence with 1 T and - 1 T saturation, respectively.  \n \nDemagnetization effect  \nThe demagnetization effect is significant for FGT nanoflakes with perpendicular magnetic \nanisotropy. The following sentence is f rom Physical Review B vol 58, 3223, which shows the \ndemagnetization factors for thin film with perpendicular anisotropy. “We approximate the \nthin film by a homogeneously magnetized ellipsoid of revolution of volume V  whose radius \nRx = R y = R is much larger  than the ‘film thickness’ 2R z. The magnetostatic self -interaction \nenergy is then given by Dµ0M2V/2, where D ≈ 1 and D ≈ 0 are the demagnetizing factors for \nin-plane and perpendicular magnetization orientations, respectively.”  \nBased on the results above, we can obtain the effective field B eff (magenitude and angle) from \nthe applied magnetic field.  \n\n\n\n − −=− − + =\nθθφ θθθφ θ θ\ncos) sin( sin)) sin( sin() cos(2 2\nBM BarctgM B B B\neffeff\n     (S2) \nThe schematic diagram of the angular relationship of the applied field, magnetization, and \nperpendic ular anisotropy is shown Fig S9a.  \n \nTheoretical fitting by modified Stoner –Wohlfarth model  \nThree different energies, the magnetic anisotropic energy, the Zeeman energy due to the interaction between the applied magnetic field and magnetic moments, and the thermal agitation energy determine the magnetic behavior of  the FGT nanoflakes.  \n 1. Fittin g for the R xy hysteresis loops at different temperatures  \nFrom the M R/MS ratio (~ 1), we know that all the spins in FGT nanoflakes align \nperpendicular to the sample surface at the remanence. Except the regime near the coercive \nfield, an FGT nanoflake behave s like a single domain particle. Therefore, we can use the \nStoner -Wohlfarth model to describe the magnetic behavior of a FGT nanoflake in the \nmagnetic field regime away from the coercivity. As shown in Fig. S9a, the angles between \nthe applied magnetic fiel d  and the direction of the perpendicular anisotropy and the magnetic \nmoment are  θ and φ, respectively. The direction of the magnetic field means the direction of \nthe POSITIVE magnetic field.  \nBased on the Stoner -Wohlfarth model, the energy of a  FGT nanofla ke at temperature T can \nbe written as  \nφ θφ cos )( ) (sin)( )(2\nS eff S eff S A VBTM VTK TE − − =                           (S3), \nwhere K A is the magnetic anisotropic energy, V S is the volume of the sample, M S(T) is the \nmagnetic moment of a unit volume FGT  at temperature T, and B is the applied magnetic field. \nConsidering the demagnetization effect as aforementioned, B and θ here should also be \nmodified to B eff and θeff. With an applied magnetic field B at known θ , we can easily \ncalculate the φ value using the equation  \n0 sin )( ) cos() sin()(2)(= + − − =∂∂φ θφ θφφS eff S eff eff S A VBTM VTKTE\n   (S4), \nIn anomalous Hall measurements, the R xy is proportional to the value of M Z (the \nmagnetization perpendicular to the sample surface). Therefore  \n) cos()( )(eff S xy TM TR θφ− ∝         (S5).  \nBy fitting the measured R xy (T) loops at temperature T based on Eq. S4, and Eq. S5, the \nmagnetic anisotropic energy K A(T) can be obtained. The most accurate K A(T) value can be \nfitted based on the loops with θ = 85o as shown in Fig. 4c and Fig. S7 (d- h). After the K A(T) value is obta ined, the angular dependent coercivity can then be fitted based on a modified \nStoner -Wohlfarth model at temperature T.  \n \n2. Fitting for the angular dependence of coercivity based on a modified Stoner -Wohlfarth \nmodel  \nFGT nanoflakes show multi -domain behaviour near the coercivity. In this field regime, the \ntraditional Stoner -Wohlfarth model cannot be utilized. In this case, we modify the Stoner -\nWohlfarth model and successfully explain the multi -domain behaviour near the coercivity of \nFGT nanoflakes.  \n \nWhen the magnetic field is swept to a negative field, the B field in Eq. S3 and S4 is negative.  \n To fit the angular dependence of coercivity, we make two assumptions , \n1. Eq. S4 can have two kinds of solutions, the stable state (low energy) and the unstable state (high energy state), as shown in Fig. S9b. With an increasing magnetic field B (more \nnegative B value), the energy difference ∆ E between the stable state 1 and the unstable \nstate 2 decreases.  At a certain B field, the thermal agitation energy is large enough to \novercome the ∆E in a standard experimental time and the magnetic moment will flip to a \nstable state in the opposite direction. As the FGT nanoflake shows a nearly square -shaped \nR\nxy loop (magnetic loop), we can assume that this B field is the coer cive field.  \n2. When the first domain flips to the opposite direction under an applied magnetic field, \nother un- flipped magnetic moments will generate an effective field on the magnetic \nmoment in the first flipped domain.  \nAn important issue for the assumption 1 is to determine the ratio ∆ E/k BT to realize the \nexperimentally observable flipping of domains. Neel and Brown proposed that the relaxation \ntime τ for the system to reach thermodynamic equilibrium from the saturated state can be written as  \n) / (01TkE ExpfB ∆− =−τ         (S6),  \nwhere f0 is a slowly variable frequency factor of the order 10-9 sec-1. Assuming τ = 100 Sec, \nEq. S6 yields the condition \nTk EB25=∆           (S7),  \nBased on Eq. S3, Eq. S4, and Eq. S7, we can write the equations to fit the angular \ndependence of coercivity at the temperature T.  \n0 sin)( ) cos() sin()(22 2 2 = + − − φ θφ θφeff S eff eff A BTM TK    (S8a), \n0 sin)( ) cos() sin()(21 1 1 = + − − φ θφ θφeff S eff eff A BTM TK     (S8b),  \nTk BTMTVTa TVTKBTMTVTa TVTK\nB eff S eff Aeff S eff A\n25] cos )()()() (sin)()([] cos )()()() (sin)()([\n1 122 22\n= − −− − −\nφ θφφ θφ\n   (S8c),  \nwhere V(T) is the volume of the first flipped domain at T, 𝑎𝑎 (T) is the parameter describing \nthe effective field due to the coupling between the first flipped domain and the un- flipped \nmagnetic moments at temperature T, which affects the Zeeman energy. The val ue of 𝑎𝑎 (T) \nshould be between 0 and 1. φ1 and φ2 are the angles between the applied magnetic field and \nthe magnetic moment for the stable state 1 and unstable state 2, respectively, which are NOT \nfitting parameters. They were calculated using the Eq. S8a and S8b. The domain dynamics in \nthe system is actually very complex. We should not think that the first flipping domain flips \nto φ2, while the magnetic moments of other parts of the FGT flake still points to the φ1 \ndirection. As the magnetic loop of FGT is  nearly square- shaped, all the magnetic moments \nflip and overcome the barrier ( φ2) in a very narrow range of magnetic field. In this situation, \nusing the calculated φ1 and φ2 based on Eq. S4 is a pretty good approximation. \n V(T) and 𝑎𝑎 (T) are the only two  fitting parameters of the three equations in Eq. S8, by which \nthe experimental angular dependence of coercivity at various temperatures can be well fitted , \nas shown in Fig. 4d. The fitted V(T) and 𝑎𝑎 (T) are also very reasonable. All the 𝑎𝑎 (T) are \nbetween 0 and 1, which means the effective field decreases the effect of the Zeeman energy (𝑎𝑎 < 1), while it cannot totally eliminate the effect of Zeeman energy ( 𝑎𝑎 > 0). The \ntemperature dependence of 𝑎𝑎 (T) shown in Fig. S9d can be explained as below. With \nincreasing temperature, the coercive field decreases and therefore the φ − θ  decreases. The \nratio between the energy due to the coupling of the flipped domain and the un- flipped \nmagnetic moment to the Zeeman energy increases. Hence, 𝑎𝑎 (T) decreases with an  increasing \ntemperature.  \n  It should be emphasized that the fitted V(T) and 𝑎𝑎 (T) values are the only possible values to \nfit the experimental data. Small deviations from the fitting results generate a large difference \nbetween the fitting curve and the exp erimental data(Fig. S9c), which clearly demonstrates \nthat our model provides a reasonable description of the magnetic phenomena in FGT \nnanoflakes.  \n          \n            \n     \nSection 5: Mean Field fitting and Spin Wave fitting \n \n \n \n \n \n \n \n \n \nSupplementary Fig. S9. a , Schematic of the Stoner –Wohlfarth model. b, Schematic diagram of a \nmagnetic system change from stable to unstable state. c , Experimental and fitting curves with \ndifferent values of V and a ( KA=4.53 x 106 erg/cc ). Fitting:  V = 481 nm3, a = 0.407.  Fitting A: V = \n550 nm3, a = 0.407. Fitting B: V = 4 30 nm3, a = 0.407. Fitting C: V = 481 nm3, a = 0.46. Fitting D: V \n= 481 nm3, a = 0.36.  d, The fitted parameter describing the effective field generated by the interaction \nbetween the first flipped domain and the un- flipped magnetic moments.   \nSection 5: Mean Field fitting and Spin Wave fitting \nTo fit the temperature dependent remane  nce in Fig. 3c, we tried the mean field theory (the \nBrillouin function) and the spin wave theory.  \n \nThe Curie -Weiss mean field theory  \nDue to the interaction between magnetic moments, an internal field can be written as \n𝐵𝑖=𝑛𝑤𝑀𝑠𝑝       (S9) \nwhere n w is a parameter describing the strength of the internal field and M sp is the \nspontaneous magnetization. \n𝑀𝑠𝑝=𝑀0ℑ𝐽(x)       (S10)   \n𝑥=𝑀0𝐵𝑖\n𝑘𝐵𝑇∙𝑁       (S11)  \nWhere M 0 is the magnetic moment at zero K, ℑis Brillouin function, k B is Boltzman  \nconstant, N the total number of unit magnetic moments. From the above equations, we can \nobtain \n𝑀𝑠𝑝\n𝑀0=ℑ𝐽(𝑥)       (S12)   \n𝑀𝑠𝑝\n𝑀0= 𝑥⋅𝑁∙𝑘𝐵𝑇)\n𝑛𝑤𝑀02       (S13)  \n Now we calculate the value of T\nC \n  𝑛𝑤𝑀𝑠𝑝∙𝑀0≈ 𝑘𝐵𝑇𝐶     (S14)  \n   \nTherefore,  we obtain    \n𝑛𝑤𝑀0∙ℑ(𝑥)∙𝑀0=𝑘𝐵𝑇𝐶     (S15)  \nWe obtain  𝑛 𝑤𝑀02(𝐽+1)\n3𝐽≈𝑘𝐵𝑇𝐶     (S16)   \nCombine Eqs S11, S12 and S16, we can calculate the M sp vs (T/T C) curves for different J \nvalues.  \n \nSpin wave model  \nThe temperature dependence of magnetic moments of a three dimensional spin wave is \n𝑀∝𝑎𝑎+𝑏(𝑘𝐵𝑇\n2𝐽𝑠)32�, which has been discussed in many text books.  Section 6: The effect of the surface amorphous oxide  layer  \n \nAlthough we tried to minimize the exposure  of samples to ambient conditions, but a \nsignificant oxide layer could still forms quickly on the top of the nanoflakes, which has been \nconfirmed by cross -sectional electron microscopy images shown in Fig. S2a. \nWe choose the same time of ambient exposure as that throughout our device fabrication and transport measurements (~7 mins). The image shows that there is indeed an amorphous oxide \nlayer of ~1.2 nm thick on the sample surface.  \nTo check whether FGT flakes without oxide layer s till show hard magnetic p hase with a near \nsquare -shaped loop, we fabricated ultraclean devices using our new vdW fabrication system \nin a glove box with both O\n2 and H 2O < 0.1 ppm. The image of one of the devices is shown in \nFig. S10c. To fabricate this device, we utilized the method in [Nature Physics 13, 677–682 (2017)]. Firstly, 5 nm thick Pt contacts were fabricated on Si/SiOx substrate in ambient \ncondition. In our glove box, an exfoliated FGT flake was then dry transferred onto the \ncontacts to form very good ohm contacts. Therea fter, a large hBN flake was dry transferred to \ncover the FGT flake to prevent oxidizations in measurements. Finally, the sample was covered by PMMA to prevent any possible oxidization. The R\nxy vs H loop is shown in Fig. \nS10d. It is very clear that ultra -clean FGT flakes still show hard magnetic property with \nsquare shaped loop. The R xy vs T curve also shows the same propert ies as the device \ndescribed in the main text. Thus, the thin oxide layer on the sample surface does not affect the main conclusions (har d magnetic properties with a near square- shaped loop) of this paper. \nMoreover, we can see that FGT is a promising material whose magnetism can survive in ambient environment for a certain time.   \nFrom our experiments, we conclude that the effect of oxide l ayer includes:  \n1. The switch of magnetic moment in the square shape loop of FGT with oxide layer is \nnot as sharp as that in ultra clean FGT flakes, which is due to the pinning effect of the oxide \nlayer.  \n2. The coercivity of FGT  slightly increases after the oxidization, which is also due to the \ndomain wall pinning effect. \n  \n \n \nSupplementary  Fig. S10 (a) Cross -sectional TEM image of an FGT nanoflake on substrate. The top \nlayer is 5 nm Pt layer. The oxide layer is about 1.2 nm. The  thickness of monolayer is about 0.8 nm. \nThe scale bar is 5 nm. (b) Diffraction pattern of the FGT nanoflake. (c) A 5.8 nm FGT device covered \nby h-BN, the bottom contact is 5 nm Pt. The red dashed line is FGT region, yellow dashed line is h-\nBN region. The scale bar is 10 µm. (d) Anomalous Hall effect at 2 K for this device. Magnetic field is \nperpendicular to the sample surface.  \n \n \n \n \n \n \n \n \n \n \n \n \nSection 7: The confirmation of ohmic contact  \nWe fabricated an FGT  device using the same recipe as all the other samples and confirm that \nour procedure of device fabrication produces good ohmic contacts.  \n \n \nSupplementary References  \nS1. Brown Jr, W. F. Relaxational behavior of fine magnetic particles.  J. Appl. Phys. 30, S130- S132 \n(1959).  \nS2. Kneller, E. & Luborsky, F. Particle size dependence of coercivity and remanence of single‐\ndomain particles. J. Appl. Phys. 34, 656- 658 (1963).  \nS3. Skomski, R., Oepen, H -P. & Kirschner. J. Micromagnetics of ultrathin films with perpendicular \nmagnetic anisotropy.  Phys. Rev. B  58, 3223 (1998).   \nS4. Gong, C. et a l. Discovery of intrinsic ferromagnetism in two -dimensional van der                                    \nWaals crystals. Nature 546, 265–269 (2017).  \nS5. Fei, Z.  et al.  Edge conduction in monolayer WTe 2. Nature Phys. 13, 677–682 (2017)  \n \n \nFig. S11 R xx vs Current curve at 2 K for an FGT  sample. (b) Corresponding I -V curve derived \nfrom (a).  "    },    {        "title": "1803.08009v1.Magnetic_illusion__transforming_a_magnetic_object_into_another_object_by_negative_permeability.pdf",        "content": "Magnetic illusion: transforming a magnetic object into another\nobject by negative permeability\nRosa Mach-Batlle, Albert Parra, Sergi Laut, Nuria\nDel-Valle, Carles Navau, and Alvaro Sanchez\u0003\nDepartament de F\u0013 \u0010sica, Universitat Aut\u0012 onoma de Barcelona,\n08193 Bellaterra, Barcelona, Catalonia, Spain\nAbstract\nWe theoretically predict and experimentally verify the illusion of transforming the magnetic\nsignature of a 3D object into that of another arbitrary object. This is done by employing negative-\npermeability materials, which we demonstrate that can be emulated by tailored sets of currents.\nThe experimental transformation of the magnetic response of a ferromagnet into that of its antag-\nonistic material, a superconductor, is presented to con\frm the theory. The emulation of negative-\npermeability materials by currents provides a new pathway for designing devices for controlling\nmagnetic \felds in unprecedented ways.\n1arXiv:1803.08009v1  [cond-mat.mtrl-sci]  21 Mar 2018I. INTRODUCTION\nThe physical appearance of objects is essential for our apprehension of reality and also for\nthe technologies relying on the identi\fcation of objects based on their signature. Illusion,\nthe transformation of the appearance of an object into that of another one, is an active\nresearch \feld [1{6]. The illusion of transforming the response of an object to impinging\nelectromagnetic waves (Fig. 1) has been theoretically studied using the tools of transfor-\nmation optics [1] and experimentally demonstrated in reduced schemes [7{10]. Such optical\nillusions typically require media with negative values of the refraction index. These media\nwere \frst introduced in seminal works by Veselago [11] and Pendry et al. [12] and gave rise\nto a host of new ways of controlling and manipulating light propagation. The theoretical\npredictions were experimentally demonstrated with the help of metamaterials, arti\fcial ma-\nterials with unusual electromagnetic properties that are not found in naturally occurring\nmaterials [13{16].\nMagnetic metamaterials [17{19] have recently been developed, enabling the realization of\nnovel devices for controlling magnetic \felds [19{28]. They include cloaks that make objects\nmagnetically undetectable [19{23] and shells that concentrate magnetic energy in a given\nspatial region [24, 25]. However, illusion has not been realized for static magnetic \felds\nuntil now. The main obstacle for this realization has been the absence of passive materials\nwith negative static permeability [29].If such negative- \u0016materials were available, they would\nallow the realization of novel tools for controlling magnetic \felds, including the illusion of\ntransforming the magnetic signature of an object into other ones. Such magnetic illusions\ncould enable the transformation of an original object into: i) a di\u000berent magnetic object\n(e. g. disguising a ferromagnet as a superconductor), ii) the same material with di\u000berent\ndimensions (enlarging or shrinking the original object [30, 31]), iii) air (making the original\nobject magnetically invisible [19{23]).\nIn this work we theoretically and experimentally demonstrate illusion for magnetic \felds\nby transforming the magnetic signature of a 3D magnetic object into that of another object\nof di\u000berent magnetic nature. The magnetic illusion is achieved by assuming a hypothetical\nnegative-\u0016material; this material can be e\u000bectively emulated in practice by a suitably tai-\nlored set of currents [32]. We experimentally con\frm the theoretical ideas by demonstrating\nthe transformation of extreme antagonistic materials: the response of an ideal ferromagnetic\n2Figure 1: Sketch of the illusions for light (upper row) and magnetic \felds (lower row). In the\nupper row, the image of an object (an apple) illuminated by light is transformed into that of\nanother object (a banana) by surrounding the former with an illusion device (in grey color), as in\n[1]. Analogously, in the lower row, the magnetic response of an object (a ferromagnetic material,\nattracting \felds lines) is transformed by using a magnetic illusion device (in grey color) into that\nof another magnetic object (a perfectly diamagnetic one, repelling \feld lines).\n(\u0016!1 ) sphere is transformed into the response of a perfect diamagnetic ( \u0016!0) one. A\nsketch of this magnetic illusion is shown in Fig. 1, together with its optical illusion counter-\npart [1]. We also discuss how other illusions, such as magni\fcation and cloak can similarly\nbe achieved assuming negative- \u0016materials.\nII. MAGNETIC ILLUSION\nConsider a sphere of radius R1and relative magnetic permeability \u00161in a uniform mag-\nnetic induction Baapplied along the z-direction. We aim at transforming its magnetic\nresponse into that of another sphere of radius Rand permeability \u0016. To this purpose,\nwe surround the original sphere with a spherical shell of inner and outer radii R1andR2,\nrespectively, and permeability \u00162, which acts as the illusion device. We consider that all\ninvolved materials are linear, homogeneous, and isotropic; interestingly, illusion solutions\n3are achieved even within these assumptions. Analytic expressions for the magnetic \feld in\nall the space are obtained by solving magnetostatic Maxwell equations (see Supplemental\nMaterial [33] for the derivations).\nTo achieve the illusion, the magnetic signature of the original object surrounded by the\nillusion device has to be indistinguishable from that of the target object. Since we are\nconsidering spherical objects, the magnetic \feld they create as a response to a uniform\napplied \feld is equivalent to that of a point dipole. By equating the dipolar magnetic\nmoment of the original sphere plus the illusion device to that of the target sphere, we obtain\nthe general condition\n(\u00161\u0000\u00162)(1 + 2\u00162) + (\u00001 +\u00162)(\u00161+ 2\u00162) (R2=R1)3\n2(\u00161\u0000\u00162)(\u00001 +\u00162) + (2 +\u00162)(\u00161+ 2\u00162) (R2=R1)3R3\n2=\u0016\u00001\n\u0016+ 2R3: (1)\nThis relation implies that given an original object (with R1and\u00161) and a target one (with R\nand\u0016), one has the freedom of \fxing R2or\u00162for the illusion device. In the case R=R2, a\nsimilar relation as Eq. (1) can also be obtained from a generalized e\u000bective-medium theory\nfor electromagnetic waves in the limit of long wavelengths [34].\nSeveral interesting cases of magnetic illusion directly result from Eq. (1); the analytic\n\feld distributions for some examples are shown in Fig. 2. A \frst one is transforming the\nobject into another of di\u000berent magnetic nature, \u00166=\u00161. An original ferromagnetic sphere,\nFig. 2(a), surrounded by the illusion shell, Fig. 2(b), has the same magnetic response as\na superconducting sphere, Fig. 2(c). A second example is cloaking, that is, transforming\nthe magnetic signature of the original object into that of empty space, \u0016= 1, rendering the\nobject magnetically undetectable. In Fig. 2(d) the original ferromagnetic sphere is made\nmagnetically invisible by a cloaking shell. A third example is the magni\fcation, R > R 1,\nor shrinking, R<R 1, of the original object, \u0016=\u00161. The ferromagnetic sphere surrounded\nby a magnifying shell, Fig. 2(e), appears as an enlarged object, Fig. 2(f). Similar e\u000bects\nhave been demonstrated for microwaves [10, 35] and dc electric \felds [36], and theoretically\npredicted for electromagnetic waves [5] and magnets [31].\nIII. NEGATIVE MAGNETOSTATIC PERMEABILTY\nOn closer inspection, Eq. (1) reveals why illusion has never been realized in magneto-\nstatics until now; the illusion device requires in general negative values of the permeability\n4Figure 2: Magnetic illusions: transformation, cloaking and magni\fcation. Magnetic induction\n\feld lines and its vertical component, Bz, resulting from a vertically applied uniform magnetic\ninductionBa, normalized to Ba(in color scale), for the cases of (a)an ideal ferromagnetic sphere\n(\u00161= 1000, radius R1),(b)the sphere in (a)surrounded by a spherical-shell illusion device\n(\u00162=\u00000:0942, extending from R1toR2= 1:25R1),(c)an ideal diamagnetic sphere ( \u0016= 1=1000,\nradiusR= 1:5R1),(d)the sphere in (a)surrounded by a spherical-shell cloaking device ( \u00162=\n\u00002071, extending from R1toR2= 1:25R1),(e)the sphere in (a)surrounded by a spherical-shell\nmagnifying device ( \u00162=\u00001:47, extending from R1toR2= 1:25R1) and (f)an ideal ferromagnetic\nsphere (\u00161= 1000, radius 1 :5R1). The magnetic signatures in (b)and(e)are indistinguishable\nfrom those in (c)and(f), respectively, in the region of illusion, the inner border of which is the\ndashed line in (b)and(e).\n\u00162. Whereas e\u000bective negative permeabilities have been obtained for electromagnetic waves\nat several frequency ranges, typically by ressonant elements [13, 37], negative permeability\nhas not been realized by passive materials in magnetostatics [29]. However, we circumvent\nthis limitation here by using the property that in magnetostatics the magnetic response of\na material can be emulated by substituting the material with its magnetization currents,\nJM=r\u0002Min the bulk and KM=M\u0002non the surfaces, where Mis the magnetization\nandnis the vector perpendicular to the surface [32]. Therefore, by externally supplying the\n5adequate set of currents one can exactly emulate the behavior of a material with negative \u0016\nwithout actually having it.\nIV. TRANSFORMING A FERROMAGNET INTO A SUPERCONDUCTOR\nTo demonstrate the potential of negative- \u0016materials, we experimentally realize the il-\nlusion of transforming the magnetic signature of a ferromagnetic sphere into that of its\nantagonistic material, a perfect diamagnetic (e.g. superconducting) one.\nWe choose the radii relations R2= 1:25R1andR= 1:5R1. In general, when \fxing the\nradii, Eq. (1) has two solutions for \u00162. Only when the original sphere has extreme values\n(\u00161!1 or\u00161!0) a single solution exists. In our case, Eq. (1) leads to \u00162=\u00000:0942,\nwhich means that the desired illusion device has negative permeability. This illusion is\ndisplayed in Fig. 2(a)-(c).\nWe next \fnd the currents that emulate the required illusion device. Because \u00161and\n\u00162are homogeneous and isotropic, the volume magnetization currents JMare zero. The\nsurface magnetization current densities KM1andKM2, \rowing at the surfaces r=R1\nandr=R2, respectively, would give the same response as the system constituted by the\ninner ferromagnetic sphere plus the illusion device. Since our setup includes an actual\nferromagnetic sphere, the currents KM1\row on its surface. Therefore, to emulate the\nillusion device there is no need to supply KM1but only the current density\nKM2(\u0012) =\u00003\n2\u00122R3\n1+R3\n2R3\n1+R3\n2\u0013Ba\n\u00160sin\u0012e': (2)\nThese currents create the desired illusion taking into consideration that the ferromagnet\nresponds not only to the applied \feld but also to the \feld created by the currents themselves.\nIn the experimental realization, the surface current distribution of Eq. (2) is substituted\nby a discrete set of 6 loops of current [see Fig. 3(c)]. Finite-element simulations demonstrate\nthat the response of this discretized system [Fig. 3(b)] is very similar to that of the ideal\nsystem [Fig. 3(a)].\nV. EXPERIMENTAL REALIZATION\nTo realize the experiments, we surround a solid steel sphere of radius R1= 20mm with a\nnon-magnetic spherical 3D-printed former of radius R2= 25mm. The radius of the target\n6Figure 3: Magnetic induction \feld lines and its vertical component, Bz, resulting from a vertically\napplied uniform magnetic induction Ba, normalized to Ba(in color scale), for the cases of (a)an\nideal ferromagnetic sphere ( \u00161= 1000, radius R1) surrounded by a spherical-shell illusion device\n(\u00162=\u00000:0942, extending from R1toR2= 1:25R1), and (b)same as (a)with a discretized\ncurrent distribution of the illusion device. (c)Sketch of the ferromagnetic sphere surrounded by\nthe current loops composing the illusion device used in the experiments.\ndiamagnetic object is R= 30mm. The 6 current loops of the discretized illusion device\nare placed in grooves on the former, as shown in Fig. 3(c) and Fig. 4(a). The loops\nare connected to the same power supply, in parallel to each other. The required current\nat each loop is fed by connecting resistors in series. This setup is placed at the central\nregion of a pair of Helmholtz coils [see Fig. 4(a)] that provides a constant applied magnetic\ninduction along the z-direction, Ba;z= 0:353mT, taking into account the contribution of\nthe Earth magnetic \feld. The z-component of Bis measured with a Hall probe, starting\nfromr=Ralong both the z-axis and the x-axis. Measurements are performed for both\nthe original ferromagnetic sphere and the complete illusion scheme, as shown in Fig. 4(b).\nVery good agreement between experimental and theoretical results is obtained for both the\ncomplete setup and the ferromagnet solely. Results demonstrate that we have successfully\ntransformed the magnetic response of a ferromagnet into that of a superconductor, even\nusing a discretized illusion device. In this way, an e\u000bective superconducting response has\nbeen obtained, without actually employing superconducting materials, in the presence of\nother magnetic objects. Further experimental con\frmation for another illusion case with\ndi\u000berent sphere dimensions is shown in the Supplemental Material [33].\n7Figure 4: (a)Experimental setup with a picture of the discretized illusion device with the current\nloops on a non-magnetic 3D-printed holder surrounding the ferromagnetic sphere. (b)Measured\nmagnetic induction Bzalong thez-axis. Open and solid circle symbols correspond to the complete\nillusion setup (SC) and to the ferromagnetic-only (FM) cases, respectively. Solid lines are the\nanalytic results for the target superconducting sphere (magenta) and the original ferromagnetic\none (orange). Finite-elements calculations for the discretized device are shown with triangular\nsymbols. Dashed line indicates the applied Ba;zvalue.\nVI. DISCUSSION\nThe general framework we have constructed based on emulating a hypothetical negative-\n\u0016material with currents allows the design of novel devices for controlling magnetic \felds.\nOne interesting application is the illusion of magnifying a magnetic object, as shown in\nFig. 2(e)-(f). Using negative values of permeability in the illusion device, the apparent\n8radius of the object can be made much larger than the external radius of the illusion device,\nsomething unachievable with positive permeability values. The consideration of negative\npermeability introduces a new solution to cloak the response of a magnetic object, extend-\ning the known solutions with positive permeability [21, 23, 28]. An example of the new\nnegative-permeability cloaking solutions is shown in Fig. 2(d). Only when the permeability\nof the original object, \u00161, has an extreme value (either \u00161!0 or\u00161!1 ) there is a single\ncloaking solution, the positive-permeability one; this kind of solutions has been experimen-\ntally demonstrated for a superconductor-ferromagnetic bilayer magnetic cloak in [21, 23, 28]\nand in other areas involving static \felds, like di\u000busive media [38] or thermal e\u000bects [39, 40].\nSince negative-permeability media can be emulated by supplying energy in the form of\ncurrents - which are simply proportional to the applied \feld [Eq. (2)], these materials can\nbe understood as active [6, 10, 41{44]. This fact makes our illusion device fully controllable\n- e. g. it can be easily switched on and o\u000b at will. By a dynamic control of the currents\none could even transform the magnetic signature of an initial object successively into that\nof di\u000berent ones, or, for example, turn a cloak on and o\u000b. In the present realization, our\nillusion is predetermined [45] because the currents are speci\fc for the applied \feld value.\nOne could make a magnetostatic illusion without prior knowledge of the applied \feld by\nusing a set of sensors and a feedback loop as in [32, 45].\nIt is interesting to compare our magnetostatic illusions to those studied for the full elec-\ntromagnetic wave case. One strategy for illusion for waves typically involves two di\u000berent\nmedia, a complementary media and a restoring one, both involving complicated distribu-\ntions of permittivity and permeability, often obtained by transformation optics [1, 46]. Here\nwe have shown how magnetostatic illusions can be realized in a much simpler way, surround-\ning the original object by a single homogenous and isotropic medium whose response can\nbe emulated by currents. Our strategy is reminiscent to the active electromagnetic cloaks\nand illusions achieved using the equivalence principle [6, 10]. In these works, superimposing\nmagnetic and electric surface current densities at the boundary of an object enables the\ncontrol of scattered \felds. However, if this procedure was applied to our magnetostatic case,\nthe interaction of the material with the magnetic \feld created by the electric currents would\nspoil the desired e\u000bect - e.g. illusion or cloak. In our illusion device, thanks to the intro-\nduction of negative permeability, the current distribution is obtained taking into account its\ninteraction with the material.\n9The results con\frm the static limit as a very promising case for applying the ideas of\ntransforming or disguising materials into di\u000berent ones; a recent example is the transfor-\nmation of the sign of the Hall e\u000bect in semiconductors by metamaterials [47]. Another\nadvantage of the static case is that, since there is no characteristic wavelength involved, the\nillusions can be made at any scale. Our results can also be valid in the quasistatic limit, for\nlow frequency electromagnetic waves such that the wavelength is far larger than the device\ndimensions. In this limit the negative permeability illusion device could be emulated by ac\ncurrent loops.\nIn this work we have limited our study to the illusion of spherical objects. For long\ncylinders in transverse \felds, which also generate a dipolar response, analytic expressions\nanalogous to Eq. (1) can be found. Our method for illusion could also be applied to\ngeometries for which the original and/or the target magnetic object generate a multipolar\nresponse (e. g. a cube). In these cases each term of the multipolar response of the original\nobject would have to be transformed into that of the target object. This could require\nin general an inhomogeneous permeability in the illusion device, which could always be\nconverted into equivalent sets of magnetization currents.\nVII. CONCLUSION\nIn summary, we have shown that negative magnetostatic permeability is a general tool\nfor designing novel magnetic \feld con\fgurations. We have demonstrated that the e\u000bective\nproperties of negative- \u0016materials can be achieved by suitably tailored sets of currents.\nAs a relevant example, we have experimentally realized the illusion of transforming the\nmagnetic signature of a ferromagnet into that of a superconductor. Other phenomena such as\nmagni\fcation and cloaking have been shown to be realizable using the same ideas. Because\nmagnetic signatures are essential in many applications, from the reading and writing of\nmagnetic bits in computer memories or credit cards to the magnetic traces left by hidden\nobjects in security detectors, the transformation of a given magnetic signature into a di\u000berent\none may be relevant in current and future technologies.\n10Acknowledgements\nWe thank Jordi Prat-Camps for discussions and help in the experiments. 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Lett. 118, 016601 (2017).\n14"    },    {        "title": "1803.08013v1.Negative_permeability_in_magnetostatics_and_its_experimental_demonstration.pdf",        "content": "Negative permeability in magnetostatics and its experimental\ndemonstration\nRosa Mach-Batlle1, Albert Parra1, Jordi Prat-Camps2,\nSergi Laut1, Carles Navau1, and Alvaro Sanchez1;\u0003\n1Departament de F\u0013 \u0010sica,\nUniversitat Aut\u0012 onoma de Barcelona,\n08193 Bellaterra,\nBarcelona, Catalonia, Spain\n2Institute for Quantum Optics and Quantum\nInformation of the Austrian Academy of Sciences,\nA-6020 Innsbruck, Austria\nInstitute for Theoretical Physics,\nUniversity of Innsbruck,\nA-6020 Innsbruck, Austria\n\u0003alvar.sanchez@uab.cat\nAbstract\nThe control of magnetic \felds, essential for our science and technology, is currently achieved\nby magnetic materials with positive permeability, including ferromagnetic, paramagnetic, and dia-\nmagnetic types. Here we introduce materials with negative static permeability as a new paradigm\nfor manipulating magnetic \felds. As a \frst step, we extend the solutions of Maxwell magnetostatic\nequations to include negative-permeability values. The understanding of these new solutions allow\nus to devise a negative-permeability material as a suitably tailored set of currents arranged in\nspace, overcoming the fact that passive materials with negative permeability do no exist in mag-\nnetostatics. We con\frm the theory by experimentally creating a spherical shell that emulates a\nnegative-permeability material in a uniform magnetic \feld. Our results open new possibilities for\ncreating and manipulating magnetic \felds, which can be useful for practical applications.\n1arXiv:1803.08013v1  [cond-mat.mtrl-sci]  21 Mar 2018I. INTRODUCTION\nControlling magnetic \felds is fundamental in both science and technology. Magnetic\nmemories for computers, turbines for energy generation, motors for delivering power, medical\ntechniques based on magnetic \felds for treatment and diagnosis, they are all based on\ndetailed spatial distributions of magnetic \felds. Magnetic materials are the conventional\ntools to shape magnetic \felds. This shaping of steady \felds is governed by the laws of\nmagnetostatics. One of the most important consequences of such laws is that \felds decay\nfrom the sources, typically as dipoles, in contrast to the long-distance propagation of time-\ndependent electromagnetic waves.\nThe progress in the science of magnetic materials has been enormous in the last decades.\nNon-linear magnetic materials have been developed to attain complex behaviors including\nhysteresis and history e\u000bects, which are exploited in many actual technologies, like magnetic\nmemories based on remanent magnetization states. In this work we focus on linear magnetic\nmaterials, those that have a magnetization directly proportional to the \feld. They have also\nexperienced very important recent developments. One particularly active line of research is\nthe application of linear magnetic materials as building blocks to construct magnetic meta-\nmaterials. Following the discovery of transformation optics technique and the development\nof metamaterials for the control of electromagnetic waves [1{3], magnetic metamaterials\nhave recently been introduced. They have led to interesting new properties and devices for\ncontrolling magnetostatic \felds [4, 5], including magnetic cloaks [4, 6{13], magnetic concen-\ntrators [14{19], and other novel magnetic phenomena [5, 20{22]. The use of metamaterials\nis particularly attractive in magnetostatics, because of at least two properties. First, in the\nstatic case electric and magnetic \felds decouple, so controlling magnetic \felds requires only\ndealing with permeabilities [4]. Second, natural materials exist with extreme permeability\nvalues, such as \u0016!0 and\u0016!1 . In contrast, for the full electromagnetic case it is\nvery di\u000ecult to fabricate materials with zero permittivity \", for example; only approximate\nresults can be achieved with lossy materials based on resonances [23].\nHowever, some advantages of the full electromagnetic case have not yet a counterpart\nin magnetostatics. One of these is the possibility of having negative- \u0016materials; whereas\nresonances in di\u000berent kinds of natural and arti\fcal substances can yield negative values\nof\u0016and\"at non-zero frequencies [24{28], no such negative- \u0016materials exist in magneto-\n2statics [29]. Negative values of \u0016and\"have enabled very interesting novel phenomena for\nelectromagnetic waves, like perfect lenses [30{32]. Some interesting physical devices such as\n'illusions', in which the waves re\rected by an object are made to resemble those arising from\na di\u000berent one [33, 34] or exterior cloaks, in which an object is cloaked at a distance [35{37],\nhave never been experimentally realized in magnetostatics, because they require materials\nwith negative values of \u0016. Devising ways to create the e\u000bective response of negative- \u0016ma-\nterials would thus pave the way towards the realization of these properties also for static\nmagnetic \felds.\nIn this work we introduce negative static permeability as a new tool for manipulat-\ning magnetic \felds. By solving Maxwell magnetostatic equations for negative-permeability\nvalues, we \fnd that a negative-permeability material can be e\u000bectively realized by a suit-\nably tailored set of currents arranged in space. The theory is con\frmed by experimentally\nconstructing a spherical shell that emulates a negative-permeability material in a uniform\nmagnetic \feld. In this way the e\u000bective properties of three-dimensional negative- \u0016materi-\nals can be produced in practice, overcoming the fact that passive materials with negative\npermeability do no exist in magnetostatics [29].\nThe paper is structured as follows. We start in section II by studying the general mag-\nnetic response of a solid ellipsoid of isotropic permeability \u0016, either positive or negative, to\na uniform applied \feld. In section III we analyze hollow isotropic ellipsoids, focusing on\ncylindrical and spherical shells. In section IV we extend the study to the case of cylindrical\nand spherical shells made of anisotropic materials. In section V we discuss how magneti-\nzation currents can be used to emulate negative-permeability materials. The experimental\ndemonstration of a negative- \u0016spherical shell is presented in section VI. Finally, in sections\nVII and VIII the obtained results and their implications are discussed, and some conclusions\nare extracted.\nII. SOLID ISOTROPIC ELLIPSOIDS WITH NEGATIVE PERMEABILITY\nWe start our study of magnetic materials with negative permeability with the case of solid\nbodies. For the sake of generality, we consider an ellipsoid, with semiaxes a,bandc. The\nellipsoid geometry encompasses two geometries of practical interest, the sphere ( a=b=c)\nand the long cylinder in perpendicular applied \feld ( b=canda!1 ), as well as the two\n3other conceptually interesting cases of a long thin strip (\fnite b;a!1 andc!0), and an\nin\fnite slab (\fnite banda=c!1 ). The four geometries are shown in the insets of Fig.\n1.\nA. Dependence of the \felds on the permeability\nConsider an ellipsoid made of a homogeneous and isotropic material with relative mag-\nnetic permeability \u0016, in a uniform \feld H0applied along the ellipsoid caxis. The magnetic\nmoment mof an ellipsoid in magnetostatics is analogous to the polarizability of a dielectric\nellipsoid in a uniform applied electrostatic \feld. By taking into account that the magneti-\nzation is M= dm=dVand the solutions in [38], [39] and [40], one obtains\nM=\u0016\u00001\n1 +N(\u0016\u00001)H0; (1)\nwhereNis the demagnetizing factor of the ellipsoid, which ranges from 0 (long slab) to 1\n(thin \flm). For a sphere N= 1=3 and for a long cylinder in transversal \feld N= 1=2.\nBy using that the total magnetic \feld is equal to the applied one plus the demagnetizing\n\feld, H=H0+Hd, and that the demagnetizing \feld is related to the magnetization\nthrough Hd=\u0000NM, we \fnd that Hin the ellipsoid volume can be written as H=\nH0=[1 +N(\u0016\u00001)]. The magnetic induction is B=\u00160(H+M) =\u0016\u00160H, where\u00160is\nthe vacuum permeability. These equations for M,HandBshow that all the \felds for\nlinear, isotropic and homogeneous ellipsoids are uniform in the material and only depend on\ntwo parameters: \u0016andN[40, 41]. These equations have been considered until now only for\npositive values of \u0016. However, there is in principle no physical argument against generalizing\nthem to include negative values of \u0016.\nWe now describe how M,H, and Bin the ellipsoid volume depend on \u0016. In Fig. 1 these\n\felds together with the energy density E=B\u0001H=2 are represented for the sphere, cylinder,\nthin \flm, and slab geometries. We start from the positive \u0016region, for which BandHhave\nboth the same direction as H0. The material is paramagnetic when Mis parallel to H0\n(\u0016 >1) and diamagnetic when they are antiparallel ( \u0016 <1). When\u0016= 1,M= 0; this is\nthe only situation in which a solid ellipsoid does not distort an applied magnetic \feld. The\ncase of\u0016!0 corresponds to a perfect diamagnet (e.g. an ideal superconductor), for which\nMexactly cancels H, resulting in B(\u0016= 0) = 0. The hitherto unexplored regime of \u0016<0\n4Figure 1: Normalized magnetization (in red), magnetic \feld (in green), magnetic induction (in\nblue) and energy density (in black) as a function of the permeability \u0016for (a) a sphere, (b) an\nin\fnite cylinder in perpendicular \feld, (c) an in\fnite thin \flm and (d) an in\fnite slab.\ncan be understood as a natural extension of the \u0016>0 behavior. Decreasing \u0016from\u0016= 0\none sees that, while Mis still negative (diamagnetic response) and Hstill positive, the sign\nofBbecomes negative (except for the thin \flm case, for which strong demagnetizing e\u000bects\nyieldBindependent of \u0016). The absolute value of Binside the ellipsoid continuously builds\nup as\u0016decreases from 0 to \u0016!(N\u00001)=N, where all \felds diverge. These asymptotes\nappear at\u0016=\u00002, -1, 0, and\u00001 for the sphere, cylinder, thin \flm, and slab, respectively.\nThe process of increasing B(in the opposite direction to H0) with decreasing \u0016towards\nnegative values is somehow symmetric to the increase of B(in the same direction as H0)\nobserved when \u0016!1 . However, the latter has a bound, B(\u0016!1 ) =\u00160H0=N, whereas\nthe building up of negative Beventually diverges.\nWhen crossing the divergence, with further decrease of \u0016,B,HandMchange their sign,\n5and the ellipsoid becomes paramagnetic. Decreasing \u0016to more negative values results in a\ndecrease of the absolute value of B,HandM. Interestingly, the limit \u0016!\u00001 corresponds\nto the ideal soft ferromagnetic limit, \u0016!1 .\nBased on these results, the conventional concept of diamagnetic and paramagnetic re-\nsponses (negative and positive M, respectively) acquires a new more general meaning. For\nnegative\u0016, the diamagnetic and paramagnetic responses are not longer bounded, as happens\nfor magnetic materials with positive \u0016[H0=(N\u00001)<M <H 0=Nfor\u0016>0]. Instead, the\ndiamagnetic or paramagnetic responses can now take arbitrarily large values, until eventu-\nally diverging at some particular (negative) \u0016value. Interestingly, the giant diamagnetic\nand paramagnetic responses can also be interpreted as the response of a superconducting or\na soft ferromagnetic ellipsoid, respectively, with larger volume than that of the actual body.\nB. Energy analysis\nIn general, magnetostatic phenomena can be regarded as a spatial reorganization of mag-\nnetic energy. The two typical magnetic materials with more extreme values of \u0016, soft fer-\nromagnets ( \u0016!1 ) and perfect diamagnets such as superconductors ( \u0016!0) expel the\nmagnetic energy from their interior (superconductors because B=0 and ferromagnets be-\ncause H=0). Therefore, when a uniform magnetic \feld is applied to a soft ferromagnetic\nor superconducting material, the energy is excluded from the materials volume and redis-\ntributed into the rest of space [Fig. 2(a) and (b)]. The same occurs for materials with\nintermediate \u0016>0, in this case with only partial expulsion of energy. Materials with nega-\ntive\u0016[Fig. 2(c) and (d)] expel even more energy than the \u0016!1 and\u0016!0 cases. Energy\nbalance is preserved in negative- \u0016materials because the extra expelled energy is compen-\nsated by negative energy in the materials. The energy density E=B\u0001H=2 is negative in\nall the volume of a negative- \u0016material since BandHhave opposite signs.\nTo further understand negative- \u0016materials, it is useful to analyze the behavior of mag-\nnetic materials in terms of magnetic \feld lines. When \u0016 > 1, lines are attracted towards\nthe material, being \u0016!1 the case of maximum attraction [Fig. 2(a)]. When 0 <\u0016< 1,\nthe e\u000bect is the opposite, \feld lines tend to avoid the material volume, until the perfect\ndiamagnetic case \u0016!0 is reached and all the \feld lines skip the cylinder [Fig. 2(b)]. With\nfurther reducing \u0016to negative values a double e\u000bect starts to build up: the lines from the\n6applied \feld are expelled farther, as if the material was more diamagnetic than a supercon-\nductor, and at the same time some closed \feld lines appear [Fig. 2(c)]. The closed \feld lines\ngenerated by the material become larger as \u0016approaches the asymptote at \u0016!(N\u00001)=N\n[\u0016=\u00001 for a cylinder, as seen in Fig. 1]. When continuing towards more negative values\nof\u0016[Fig. 2(d)], the closed \feld loops diminish until they disappear in the limit \u0016!\u00001 .\nIn magnetostatics, when dealing with linear materials as in our case, closed \feld lines can\nonly arise from currents, because of Ampere's law. In Section V we discuss how to \fnd the\nrequired current distributions to emulate a negative- \u0016material. Because these currents will\nneed to be readjusted when changing the applied \feld, negative- \u0016materials devised in this\nway can be classi\fed as active [42{45].\nFigure 2: Magnetic induction \feld lines and normalized energy density E=(\u00160H2\n0) (in colors) for\nthe magnetic response of cylinders of (a) \u0016= 104, (b)\u0016= 10\u00004, (c)\u0016=\u00001=2 and (d)\u0016=\u00002 to\na vertically applied magnetic \feld H0.\n7C. Conjugate relations\nA general property of the magnetization of the ellipsoids emerges when negative values of\nthe permeability are considered. For any ellipsoid of permeability \u0016there exists a conjugate\nellipsoid of permeability \u00160, which has exactly the same magnetization with opposite sign,\nM(\u00160) =\u0000M(\u0016). By using Eq. (1), we \fnd that the conjugate permeability of \u0016is\n\u00160= 1\u0000\u0016\u00001\n1 + 2N(\u0016\u00001); (2)\nwhich only depends on the geometry of the ellipsoid.\nConsidering a solid cylinder, N= 1=2, Eq. (2) yields \u00160= 1=\u0016. Then\u00160has the same sign\nas\u0016. This conjugate relation was obtained for positive \u0016values for long rectangular bars\nin transversal \feld in [47] and also appeared in [48] for long hollow cylinders in transversal\n\feld.\nIn the case of a solid sphere, conjugate relations have not been explored before. Using\nN= 1=3, Eq. (2) leads to a conjugate permeability \u00160= (\u0000\u0016+ 4)=(2\u0016+ 1), which shows\nthat the sign of \u00160is not always the same as that of \u0016. When\u0016is larger than 4, the\nconjugate sphere does not have a positive value of \u00160. For this reason conjugate relations for\nspheres could not be obtained without taking into account that \u0016can take negative values.\nThe conjugate \u00160of a soft ferromagnetic sphere ( \u0016!1 ), for example, is a sphere with\n\u00160=\u00001=2, instead of a superconducting sphere ( \u00160=0) as for a cylinder.\nIII. HOLLOW ISOTROPIC ELLIPSOIDS WITH NEGATIVE PERMEABILITY\nWe continue our study of negative- \u0016materials by considering the case of hollow bodies.\nNovel features such as magnetic \feld concentration in the hole of the bodies [14] appear in\nthis geometry.\nConsider an ellipsoidal homogeneous and isotropic material with relative magnetic per-\nmeability\u0016and semi axes a2,b2andc2with a centered hole of semi axes a1,b1andc1; we\nrestrict our study to the case of both the hole and the outer surface having the same shape.\nA uniform magnetic \feld, H0, is applied along a principal axis of the ellipsoid.\nDi\u000berent from solid bodies, the magnetic response of a hollow ellipsoid to a uniform\nmagnetic \feld has in general not only a dipolar term but higher orders as well. Only\nthe cases of hollow spheres and cylinders, because of their high symmetry, have a dipolar\n8response. Analytic expressions for the magnetic \felds for hollow spherical and cylindrical\nshells can be found in the Appendix.\nThe dipolar magnetic moment mof a general hollow ellipsoid is analogous to the polariz-\nability resulting from the application of an electric \feld to a hollow dielectric ellipsoid [38].\nFrom this, one can calculate the averaged magnetization on the whole ellipsoid volume, V,\nincluding the hole volume, as M\u0003= (R\nVM(r;\u0012)dV)=V. Its expression is\nM\u0003=(f\u00001) (N(\u0016\u00001)\u0000\u0016) (\u0016\u00001)\n(f\u00001)(N\u00001)N(\u0016\u00001)2+\u0016H0; (3)\nwhich is uniform and in the direction of the applied magnetic \feld, as in the case of a solid el-\nlipsoid.fis the fraction of the external ellipsoid occupied by the hole, f= (a1b1c1)=(a2b2c2).\nIn the limits \u0016!\u00061 ,M\u0003tends to 1=N, as for solid bodies. When f!0 we recover Eq.\n(1).\nEq. (3) shows that there are two values of \u0016that result in a divergence of the magneti-\nzation,\n\u00161;2= 1 +\u00001\u0006p\n1 + 4N(f+N\u0000fN\u00001)\n2(f\u00001)(N\u00001)N; (4)\nBearing in mind that 0 \u0014f<1 and 0\u0014N\u00141 it is seen that these two values of \u0016are\nnegative for any ellipsoid.\nWhenM\u0003=0 the shell does not create a dipolar response; in the case of a hollow sphere or\ncylinder this makes the object magnetically undetectable because the applied magnetic \feld\nis not distorted. Whereas for solid ellipsoids the magnetization is zero only in the trivial\ncase of no material, \u0016ND= 1, for isotropic hollow ellipsoids there is an extra solution for\nM\u0003(\u0016ND)=0. By using Eq. (3) we \fnd that\n\u0016ND=N\nN\u00001; (5)\nwhich does not depend on f, but only on the geometry of the ellipsoid through its demag-\nnetizing factor. In the particular cases of spherical and cylindrical shells, the non-distortion\npermeabilities are \u0016ND=\u00001=2 and\u0016ND=\u00001, respectively.\nWe show in Fig. 3 the dependence of M\u0003upon\u0016for a hollow sphere ( N= 1=3) and\na hollow long cylinder ( N= 1=2) forf= 1=2, where the two divergences and the two\nnon-distortion permeabilites can be seen, for each geometry.\n9Figure 3: Normalized averaged magnetization M\u0003as a function of the permeability \u0016for a spherical\nshell (blue dashed line) and a cylindrical shell (black solid line), for f= 1=2.\nA. Conjugate relations\nThe consideration of negative values of the permeability leads to conjugate relations for\nhollow ellipsoids, as for solid ones. For any hollow ellipsoid of permeability \u0016two conjugate\nellipsoids of permeabilities \u00160\n1, and\u00160\n2, exist which have exactly the same magnetization with\nopposite sign, M\u0003(\u00160\n1;2) =\u0000M\u0003(\u0016).\nFor a general hollow ellipsoid, the conjugate relations are found using Eq. (3). In the\nparticular case of a hollow cylinder the result \u00160\n1= 1=\u0016, obtained in [48] is recovered. This is\nthe same conjugate relation that appeared for a solid cylinder. Interestingly, a new solution\nappears as\n\u00160\n2=(\u0016\u00001)f+ (\u0016+ 1)\n(\u0016\u00001)f\u0000(\u0016+ 1): (6)\nConjugate relations for a hollow sphere can also be analytically obtained through cum-\nbersome expressions (not shown). None of them corresponds to the case of a solid sphere.\n10IV. HOLLOW ANISOTROPIC CYLINDRICAL AND SPHERICAL SHELLS\nWe now continue our study of hollow bodies by considering shells with homogeneous\nanisotropic permeabilities, extending the results studied above for isotropic materials. We\nrestrict the results to the two more relevant geometries, spherical and cylinidrical shells.\nConsider homogeneous and anisotropic spherical and cylindrical shells of external radius\nR2and internal radius R1, characterized by their angular and radial relative permeabilities,\n\u0016\u0012=\u0016'and\u0016r, and\u0016\u0012and\u0016\u001a, respectively. A uniform magnetic \feld H0is applied in the z\ndirection. Magnetostatic Maxwell equations can be analytically solved (see Appendix for the\nfull derivation), providing the solutions for the magnetic \feld in the three di\u000berent regions:\ninside the hole, in the shell and in the external region. For positive \u0016, these solutions were\nstudied in [14, 18, 46].\nThe solutions show two important properties. First, the magnetic \feld inside the hole of\nthe shells is always uniform and has the direction of the applied one, HINT\ns=\u0000asH0and\nHINT\nc=\u0000acH0, for the spherical and the cylindrical shell, respectively. The expressions for\nthe coe\u000ecients asandacare shown in Eqs. A7 and A11. Second, the magnetic \feld in the\nexternal region is, in general, modi\fed with respect to the applied \feld due to the presence\nof the shell. The \feld created by the shell corresponds to the \feld created by a centered\ndipole with magnetic moment pointing in the applied \feld direction, ms= 4\u0019bsH0for a\nspherical shell and mc= 2\u0019bcH0for a cylindrical one. The expressions for the coe\u000ecients\nbsandbcare shown in Eqs. A10 and A14. A positive (negative) value of bsorbcindicates\nthat the shell is paramagnetic (diamagnetic).\nIn the following we analyze the anisotropic shells that do not distort a uniform applied\nmagnetic \feld as well as those that involve divergent magnetic \felds. The overall results can\nbe seen in Fig. 4, where the permeability relations resulting in a non-distorting shell and\nthose leading to divergent \felds are plotted. It is seen that these two cases alternate, so that\nthere is always a line of no distortion between two consecutive lines of \feld divergence. Also,\nthese lines constitute the borders between diamagnetic and paramagnetic regions. In this\nway, the concept of paramagnetic and diamagnetic materials is enriched. For conventional\nmaterials with positive \u0016(right upper quadrant in Fig. 4) there is a single frontier line\nseparating the two regions. In the general picture that negative \u0016is bringing, the border lines\nand the paramagnetic and diamagnetic regions increase until reaching an in\fnite number of\n11them.\nFigure 4: Relations of non-distortion (red lines) and divergent \felds (blue lines) between the\npermeabilities (a) \u0016\u0012and\u0016rfor a spherical shell and (b) \u0016\u0012and\u0016\u001afor a cylindrical one. R2=R1= 2\nfor both cases. The regions \flled in orange correspond to paramagnetic shells, while the white\nregions correspond to diamagnetic shells.\nA. Non-distortion shells\nIt can be obtained from Eqs. (A10) and (A14) that for a given radial permeability there\nare in\fnite values of the angular permeability for which the coe\u000ecients bsandbcbecome\nzero, and thus the shells do not distort the external magnetic \feld. They can be grouped\n12into two types of solutions. For a spherical shell,\n\u0016\u0012=1 +\u0016r\n2\u0016r; (7)\n\u0016\u0012=\u0000\u0016r\n8\"\u00122\u0019n\nln(R2=R1)\u00132\n+ 1#\n; n = 1;2;3:::; (8)\nand for a cylindrical shell,\n\u0016\u0012=1\n\u0016\u001a; (9)\n\u0016\u0012=\u0000\u0016\u001a\u0012\u0019n\nln(R2=R1)\u00132\n; n = 1;2;3:::: (10)\nThe \frst type of solutions [Eqs. (7) and (9)] corresponds to the red curves in Fig.\n4, extending mainly in the \frst and third quadrant. These solutions were explored for\npositive\u0016in [14, 46] The non-distorting isotropic shells studied above, \u0016r=\u0016\u0012=\u00001=2 and\n\u0016\u001a=\u0016\u0012=\u00001 for a spherical and a cylindrical shell, respectively, are particular cases of\nthese solutions. The second type [Eqs. (8) and (10)] corresponds to the red straight lines in\nFig. 4, extending in the second and fourth quadrants. There is an in\fnite number of these\nlines, and their slope depends upon a single parameter, n.\nB. Magnetic \feld concentration inside the hole of a non-distorting shell\nThe two types of non-distorting solutions of Eqs. (7)-(10) di\u000ber in the \feld concentrated\ninside their hole.\nWe start studying the \frst type of solutions [Eqs. (7) and (9)]. The \feld in the hole for\na spherical and a cylindrical shell is, respectively,\nHINT\ns=H0(R2=R1)1\u00001=\u0016r; (11)\nHINT\nc=H0(R2=R1)1\u00001=\u0016\u001a; (12)\nwhere we have used Eqs. (A7) and (A11).\nThe magnetic \feld concentration can be interpreted in terms of energy reorganization.\nSince we are considering shells that do not distort the external \feld, the energy density\nin the external region is the same as if there was no shell. When the permeabilites are\npositive, the concentration of energy inside the hole [ EINT=\u00160(HINT)2=2] can be simply\n13understood considering that part of the energy that was in the space occupied by the shell\nhas been redistributed and placed inside the hole. When permeabilities are negative, the\nminimum concentration occurs for an in\fnitely large negative radial permeability and is\nHINT\nmin= (R2=R1)H0, independently of the shell geometry. Interestingly, this corresponds\nto the maximum concentration that can be achieved with positive permeabilities, occurring\nwhen the radial permeability tends to + 1. When the radial permeability approaches 0\u0000\nthe \feld concentration increases, and diverges in this limit.\nTo explain how this large magnetic \feld concentration is achieved we compare the be-\nhaviour of two non-distorting shells ful\flling the non-distortion relation of Eq. (9), one with\npositive\u0016and the other one with negative \u0016, for the cylindrical geometry (Fig. 5). When\n\u0016>0, the energy density inside the hole, EINT, is maximum when the energy density in the\nshell is zero. This happens when \u0016\u001a!1 and\u0016\u0012!0 [in Fig. 5(a) this is approximated by\n\u0016\u001a= 100 and\u0016\u0012= 0:01]. In this situation, all the energy that was in the space occupied by\nthe shell has been redistributed and placed inside the shell hole. When considering a shell\nwith negative \u0016[Fig. 5(b)] the energy inside the hole EINTis larger than that for positive\n\u0016. Since the energy in the external region is the same for both cases, energy conservation\nrequires that the energy in a negative- \u0016shell volume is negative, as shown in Fig. 5(b).\nFigure 5: Magnetic induction \feld lines and normalized energy density E=(\u00160H2\n0) in color scale\nfor two cylindrical shells with radii ratio R2=R1= 2 and magnetic permeabilities (a) \u0016\u001a= 100 and\n\u0016\u0012= 0:01 and (b) \u0016\u001a=\u00001=2 and\u0016\u0012=\u00002.\nNow we analyze the \feld concentration corresponding to the second type of non-distortion\nsolutions, resulting from Eqs. (8) and (10). Interestingly, for all shells ful\flling these\nequations the \feld inside the hole is HINT\ns=\u0006H0(R2=R1)3=2for a spherical shell and\n14HINT\nc=\u0006H0(R2=R1) for a cylindrical shell, according to Eq. (A7) and (A11), respec-\ntively; the sign is positive when nis even and negative when nis odd. Therefore, the energy\ndensity inside the hole is the same for all the solutions of this type. This is illustrated in the\nexamples of Fig. 6, where the energy density and the magnetic \feld lines are represented\nfor two cylindrical shells. It is seen that nindicates the number of regions inside the shell\nthat are surrounded by closed magnetic \feld lines.\nFigure 6: Magnetic induction \feld lines and normalized energy density E=(\u00160H2\n0) in color scale for\ntwo cylindrical shells with radii ratio R2=R1= 2. Both have \u0016\u001a= 1 and their corresponding \u0016\u0012is\nobtained from Eq. (10) for (a) n=1 and (b) n=2.\nC. Divergences of \felds\nThe permeability relations yielding divergent \felds can be found from the zeroes in the\ndenominators of Eqs. (A7) and (A11). It is interesting that when \u000b2andk2are negative,\nthere are an in\fnite number of such relations. The divergences occur, for spherical and\ncylindrical shells, respectively, when\np\n\u0000\u000b2ln(R2=R1) = 2arctan\u00123\u0016rp\n\u0000\u000b2\n\f\u0013\n+ 2\u0019n; (13)\np\n\u0000k2ln(R2=R1) = arctan\u0012\u00002\u0016\u001ap\n\u0000k2\n\u0016\u001a\u0016\u0012+ 1\u0013\n+\u0019n; (14)\nwheren=0,1,2.... These expressions are represented as blue lines in Fig. 4 for a particular\nshell withR2=R1= 2.\n15V. EMULATING NEGATIVE-PERMEABILITY MATERIALS\nMedia with negative magnetic permeability do not exist in magnetostatics, as demon-\nstrated in [29]. However, we next show how negative- \u0016media can be e\u000bectively emulated\nby replacing them with a set of currents. In order to \fnd these currents we use the general\nproperty that in magnetostatics the magnetic response of a material can be obtained by\nsubstituting it with its magnetization currents. Given an arbitrary magnetic material in an\napplied magnetic \feld, H0, the corresponding surface and volume magnetization currents\ncan be calculated from the magnetization of the material, M, respectively, as\nKM=M\u0002n; (15)\nJM=r\u0002M; (16)\nwhere nis a unitary vector perpendicular to the material surface.\nThe total magnetic induction in all the space (even at points inside the material), B,\ncan be simply calculated as the applied magnetic induction, B0=\u00160H0, plus the magnetic\ninduction created by these magnetization currents, Bc. Therefore, by externally supplying\nthe adequate set of currents the total distribution of Bwill be exactly the same as if that\nmaterial was present. This allows to emulate any magnetic material, even materials with\nnegative permeabilities.\nWe next \fnd the currents emulating a negative- \u0016material in the case of a spherical\nshell, which is the one we will experimentally demonstrate below. Consider a spherical shell\nwith inner and outer radii R1andR2, respectively, and homogeneous relative magnetic\npermeabilities \u0016r,\u0016\u0012, and\u0016'. Its response to a uniform magnetic \feld H0applied in\nthezdirection is analytically obtained (see Appendix). Restricting to isotropic materials\n\u0016r=\u0016\u0012\u0011\u0016(\u0016'is irrelevant due to the symmetry of the applied \feld), the corresponding\nmagnetization currents are calculated from Eqs. (15) and (16) taking into account that, by\nde\fnition, M=(\u0016\u00001)HandHin the material region can be obtained from Eq. (A2), as\nKM(r=R1) =\u000018\u0016(\u0016\u00001) (R2=R1)3H0sin\u0012\n\u00004(\u0016\u00001)2+ (4\u00162+ 10\u0016+ 4) (R2=R1)3e'; (17)\nKM(r=R2) =6(\u0016\u00001)\u0002\n(\u0016\u00001) + (2\u0016+ 1) (R2=R1)3\u0003\nH0sin\u0012\n\u00004(\u0016\u00001)2+ (4\u00162+ 10\u0016+ 4) (R2=R1)3e'; (18)\nJM= 0: (19)\n16Since we consider homogeneous and isotropic materials, no volume magnetization currents\nappear.\nVI. EXPERIMENTAL DEMONSTRATION OF A NEGATIVE-PERMEABILITY\nMATERIAL\nWe now experimentally demonstrate our theoretical ideas and the plausibility of emulat-\ning magnetic materials with negative \u0016. We consider a homogeneous and isotropic spherical\nshell with inner and outer radii R1andR2, respectively. We choose a permeability \u0016=\u00000:5;\nthis shell does not distort the applied \feld and concentrates the \feld in the hole by a factor\n(R2=R1)3[Eq. (11)]. These properties cannot be simultaneously obtained by conventional\nmaterials with positive \u0016.\nA. Emulation of a negative-permeability material by a \fnite set of currents\nTo construct an actual spherical shell with e\u000bective negative permeability \u0016=\u00000:5, the\nsurface currents given by Eqs. (17) and (18) have to be externally supplied at the inner and\nouter surfaces of the shell, respectively. These continuous current distributions are converted\ninto discrete sets of current loops in our practical realization. Numerical simulations (by\nthe AC/DC module of the Comsol Multiphysics software) indicate that the discretization\ninto 6 current loops at each of the surfaces [Fig. 7(b)] approximates reasonably well the\n\feld created by the theoretical continuous current distribution [Fig. 7(a)]. The current\ncorresponding to each loop is calculated as the integral of the surface current,\nI(Ra;\u0012i) =Z\u0012i+\u0019=12\n\u0012i\u0000\u0019=12KM(r=Ra)Rad\u0012; (20)\nwhere\u0012iis the angular position of each current loop and a= 1;2.\nB. Experimental setup and feedback loop\nIn our experiments, the 6+6 current loops, each consisting of 3 turns of copper wire, are\nplaced onto two specially designed 3D-printed spherical formers, with radii R1=25mm and\nR2=50mm, respectively [Figs. 7(c) and 7(d)]. The spherical shell is placed in between a pair\n17Figure 7: (a) Finite-element simulation of the z-component of B, normalized to B0, when a \feld\nB0is applied in the z-direction to a spherical shell with \u0016=\u00000:5 and radii ratio R2=R1= 2. (b)\nSame for the discretized 6+6 current loops. (c) 3D sketch of the experimental negative- \u0016material,\nconsisting of two sets of 6 circular current loops placed on a 3D-printed plastic former. (d) Picture\nof the actual experimental negative- \u0016material.\nof Helmholtz coils, which create a uniform magnetic \feld in the zdirection in the sphere\nregion, as shown in Fig. 8(a).\nFor a given applied \feld value, the required currents at each loop can be obtained from\nEqs. (17), (18) and (20). They are fed in the 12 loops using a common voltage source from\na Agilent 6671A power supply; each loop is connected in series with a load resistor, whose\nvalue is calculated to provide the required current.\nIf the applied \feld is changed, the value of the current in the loops needs to be readjusted\nin order to keep emulating the same negative- \u0016material. For this purpose we setup a\nfeedback loop that automatically adjusts the currents to the applied \feld value [Fig. 8(b)].\nFor the feedback loop we use a LabView Virtual Instrument as a Control Software, with\na process described as follows. First, the applied \feld is measured with a Hall probe. Then,\n18Figure 8: (a)Picture of the experimental setup with the spherical negative- \u0016material in the middle\nof two Helmholtz coils that create a uniform \feld in the zdirection; the tip of the Hall probe is\nshown on the left of the sphere. (b) Scheme of the feedback loop circuit.\nthe currents corresponding to the reading of the \feld value are calculated according to Eqs.\n(17), (18) and (20). Finally, these currents are fed into the loops by using the same resistors\nand the Control Software automatically readjusts the input voltage. Thanks to the linear\ndependence between the current and the \feld and the simplicity of the experimental compo-\nnents, the feedback loop is very robust against possible instabilities arising from \ructuations\nof the measured applied \feld.\nIn this way, we achieve an e\u000bective negative- \u0016material. Even though the feedback\nloop mechanism is theoretically valid for any applied \feld value, in practice the range of\n19applicability is limited by the power dissipation of the resistors and the overall available input\npower. The discretization we have used is adequate for uniform magnetic \felds applied\nperpendicular to the loops, but the general procedure could be adapted to di\u000berent \feld\ndistributions using other discretization schemes.\nFigure 9: (a) Experimental measurements (black squares), \fnite-element calculation for the dis-\ncretized spherical shell with \u0016=\u00000:5 (red line), and analytic results for the ideal material (blue\nline) for the z-component of Balong thex-axis. (b) Same as (a) for the z-component of Balong\nthez-axis. The shell region is painted in grey color.\n20C. Field measurements\nIn order to verify that the actual device acts as a material with \u0016=\u00000:5, we apply a\nmagnetic induction B0=\u00160H0=0.0543mT and compare the measured \feld pro\fles with the\ntheoretical results. The zcomponent of the magnetic induction is measured with a Hall\nprobe along the x[Fig. 9(a)] and z[Fig. 9(b)] directions. The experimental results show\nthat the \feld inside the hole is uniform and that the external \feld is not modi\fed by the\npresence of the shell, verifying the theory. The \feld in all regions coincides very well with\nthe numerical simulations of the discretized device. Only close to the surfaces there is a\nsmall discrepancy between the ideal and the discretized cases because of the discretization.\nVII. DISCUSSION\nNegative properties of materials are an intense recent topic of research in physics, includ-\ning negative acoustic [49, 50], negative mechanical properties [51{53] and negative capaci-\ntance [54, 55]. Most of these systems are very complicate to realize in practice. In contrast,\nthe negative- \u0016materials we introduce in this work can be simply realized by a set of suit-\nably tailored electrical currents whose analytic expressions are found. These currents are\nproportional to the uniform applied magnetic \feld. Therefore, to emulate the response of\na particular negative- \u0016material, one \frst has to sense the applied magnetic \feld and then\nset the required currents. Because of this sensing-setting requirement we can regard the\nproposed negative- \u0016magnetic materials as active. The feedback loop presented in Section\nVI B automatically adapts the currents to the magnetic \feld, allowing the emulation of a\nnegative-\u0016material even when the applied \feld is changed.\nHaving negative- \u0016magnetostatic materials may enable a whole new set of possibilities for\ncontrolling magnetic \felds, analogous to those proposed or demonstrated for the full elec-\ntromagnetic case. One of the most dramatic properties enabled by materials with negative\nrefraction index is achieving illusions, that is, objects that appear as di\u000berent objects when\nilluminated by light [33]. In [56] we demonstrate how to obtain illusion in magnetostatics\nusing negative- \u0016materials. The magnetic signature of a magnetic material (a soft ferromag-\nnet in [56]) is transformed into that of a di\u000berent one (a perfect diamagnet) by enclosing\nthe former in a shell emulating a negative- \u0016behavior. Other illusions such as magnifying\n21or shrinking materials, cloaks, and anticloaks [57{61] can also be realized using the same\nscheme [56]. Another intriguing possibility that may eventually become possible based on\nour results may be the realization of exterior cloaks [35{37]. As stated by Wegener in [36]\nconventional metamaterial cloak needs to be wrapped around the object, so it would be yet\nmore stunning and useful if it could rather be spatially separated from the object. Such\nexterior cloaking has been demonstrated experimentally in dc electrical conduction using\ne\u000bectively negative electric conductivities in a plane [62] of active metamaterials [63]. Our\nresults open the door to construct a magnetic cloak that can act at a distance in a full 3D\nscheme, something which may have applications in many areas involving magnetic \felds,\nsuch as medical imaging techniques.\nVIII. CONCLUSIONS\nWe have introduced materials with negative static permeability as a new tool for ma-\nnipulating magnetic \felds. We have explored solutions of Maxwell magnetostatic equations\nconsidering negative- \u0016materials. A whole new set of solutions have emerged, extending\nthose previously known for the conventional case of positive \u0016materials. For solid ellipsoid\nbodies, which include the physically interesting cases of a sphere and a cylinder in perpen-\ndicular \feld, the consideration of negative \u0016brings the existence of a divergence of magnetic\n\felds at a particular negative- \u0016value, which only depends on the body demagnetizing fac-\ntor. For hollow isotropic cylinders and spheres with negative \u0016, there are two values of \u0016\nfor which \felds diverge, and also an extra solution for cloaking magnetic \felds, apart from\nthe trivial solution of no material, \u0016= 1. Some conjugate relations between the magnetic\nresponses of bodies of di\u000berent permeabilities have been found, bringing to light some hidden\nsymmetries that become apparent when considering the case of negative \u0016. For cylindrical\nand spherical shells with anisotropic permeability new families of solutions arise, including\nan in\fnite number of cloaking situations, of divergent magnetic \felds, and also of in\fnite\nborders between paramagnetic and diamagnetic regions (at which the magnetization of the\nbody changes from positive to negative, respectively). For all studied cases, magnetization\ncurrents can be obtained from the analytic expressions of the \feld distributions. We have\ndemonstrated that negative-permeability materials can be realized in practice by replacing\nthe material with these magnetization currents. We have experimentally con\frmed these\n22ideas by constructing a set of current loops that emulates the properties of a spherical shell\nwith\u0016=\u00000:5. Our theoretical results and the emulation of negative- \u0016materials by currents\nmay create new ways of controlling magnetic \felds.\nACKNOWLEDGEMENTS\nWe thank European Union Horizon 2020 Project FET-OPEN MaQSens (grant agreement\n736943), and projects MAT2016-79426-P (Agencia Estatal de Investigaci\u0013 on / Fondo Europeo\nde Desarrollo Regional) and 2014-SGR-150 for \fnancial support. A. S. acknowledges a grant\nfrom ICREA Academia, funded by the Generalitat de Catalunya.\nAPPENDIX: ANALYTIC EXPRESSIONS FOR SPHERICAL AND CYLINDRI-\nCAL SHELLS\nConsider homogeneous, linear, and anisotropic spherical and cylindrical shells of external\nradiusR2and internal radius R1, with an applied magnetic \feld H0in the z direction. The\nangular and radial relative permeabilities are \u0016\u0012=\u0016'and\u0016rfor the spherical shell and \u0016\u0012\nand\u0016\u001afor the cylindrical shell. Since there are no free currents in the system, r\u0002H=0,\nand the magnetic \feld can be written in terms of a magnetic scalar potential \u001e,H= -r\u001e,\nin all the space. Using this equation and knowing that r\u0001B= 0, the magnetic \feld in the\nthree di\u000berent regions: inside the hole (INT), in the shell (SHE) and in the external region\n(EXT) can be obtained. For a spherical shell,\nHINT\ns(r;\u0012) =H0[\u0000ascos\u0012er+assin\u0012e\u0012]; (A1)\nHSHE\ns(r;\u0012) =H0\u0014\u0012(1\u0000\u000b)cs\n2r(3\u0000\u000b)=2+ds(1 +\u000b)\n2r(3+\u000b)=2\u0013\ncos\u0012er+\u0012cs\nr(3\u0000\u000b)=2+ds\nr(3+\u000b)=2\u0013\nsin\u0012e\u0012\u0015\n;(A2)\nHEXT\ns(r;\u0012) =H0\u0014\u00122bs\nr3+ 1\u0013\ncos\u0012er+\u0012bs\nr3\u00001\u0013\nsin\u0012e\u0012\u0015\n; (A3)\nand for a cylindrical shell,\n23HINT\nc(\u001a;\u0012) =H0[\u0000accos\u0012e\u001a+acsin\u0012e\u0012]; (A4)\nHSHE\nc(\u001a;\u0012) =H0\u0014\u0012\n\u0000cck\u001ak\u00001+dck\n\u001ak+1\u0013\ncos\u0012e\u001a+\u0012\ncc\u001ak\u00001+dc\n\u001ak+1\u0013\nsin\u0012e\u0012\u0015\n; (A5)\nHEXT\nc(\u001a;\u0012) =H0\u0014\u0012bc\n\u001a2+ 1\u0013\ncos\u0012e\u001a+\u0012bc\n\u001a2\u00001\u0013\nsin\u0012e\u0012\u0015\n: (A6)\nwhere e have used \u000b2= 8\u0016\u0012=\u0016r+ 1 andk2=\u0016\u0012=\u0016\u001a.\nThe coe\u000ecients of the magnetic \feld can be obtained by applying the boundary conditions\n(continuity of radial component of Band tangencial component of Hat both surfaces R1\nandR2). For a spherical shell,\nas=6\u0016r\u000b(R2=R1)(3+\u000b)=2\n\f\u00003\u0016r\u000b\u0000(\f+ 3\u0016r\u000b) (R2=R1)\u000b; (A7)\ncs=3(\u0016r\u000b+\u0016r+ 2)R(3+\u000b)=2\n2R\u0000\u000b\n1\n\f\u00003\u0016r\u000b\u0000(\f+ 3\u0016r\u000b) (R2=R1)\u000b; (A8)\nds=3(\u0016r\u000b\u0000\u0016r\u00002)R(3+\u000b)=2\n2\n\f\u00003\u0016r\u000b\u0000(\f+ 3\u0016r\u000b) (R2=R1)\u000b; (A9)\nbs=\u00002(2\u0016r\u0016\u0012\u0000\u0016r\u00001) [(R2=R1)\u000b\u00001]R3\n2\n\f\u00003\u0016r\u000b\u0000(\f+ 3\u0016r\u000b) (R2=R1)\u000b; (A10)\nwhere\f= 4\u0016r\u0016\u0012+\u0016r+ 4. For a cylindrical shell,\nac=4\u0016\u001ak(R2=R1)1+k\n(\u0016\u001ak\u00001)2\u0000(\u0016\u001ak+ 1)2(R2=R1)2k; (A11)\ncc=2(\u0016\u001ak+ 1)R1\u0000k\n2(R2=R1)2k\n(\u0016\u001ak\u00001)2\u0000(\u0016\u001ak+ 1)2(R2=R1)2k; (A12)\ndc=2(\u0016\u001ak\u00001)R21+k\n(\u0016\u001ak\u00001)2\u0000(\u0016\u001ak+ 1)2(R2=R1)2k; (A13)\nbc=\u0000(\u0016\u001a\u0016\u0012\u00001)R2\n2h\n(R2=R1)2k\u00001i\n(\u0016\u001ak\u00001)2\u0000(\u0016\u001ak+ 1)2(R2=R1)2k: (A14)\nFrom Eqs. (A3) and (A6) the magnetic \feld in the exterior region is, in general, modi\fed\nwith respect to the applied \feld due to the presence of the shell. The \feld created by the\nshell corresponds to the \feld created by a dipole with magnetic moment ms= 4\u0019bsor\nmc= 2\u0019bc, for a spherical and a cylindrical shell, respectively. Eqs. 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Weiss * \n \nInstitute of Experimental and Applied Phys ics, University of Regensburg, D -93040 Regensburg, Germany  \n \nWe report on a comprehensive study  of the magnetic coupling  between soft magnetic Fe layers and hard magnetic \nDysprosium (Dy) layers at low temperatures (4.2 - 120 K). For our experiments we prepa red thin films of Fe  and Dy and \nmultilayers of Fe/Dy by ultra -high vacuum sputtering. The magnetic properties of each material were determined with a super \nconducting quantum interference device. Furthermore, we performed magnetoresistance measurements wit h similarly grown, \nmicrostructured devices, where the anisotropic magnetoresistance (AMR) effect was used to identify the magnetization state \nof the samples.  By analyzing and comparing the corresponding data of Fe and Dy , we show that the presence of a Dy layer on \ntop of the Fe layer significantly influences its magnetic properties and makes it magnetically harder . We perform a systematic \nevaluation of this effect and its dependence on temperature and on the thickness of the soft magnetic layer. All experim ental \nresults can consistently be explained w ith exchange coupling at the interface between the Fe and the Dy layer . Our \nexperiments also yield a negative sign of the AMR effect of thin Dy f ilms, and an increase of the Dy film s’ Curie \ntemperature , which is  due to growth conditions.  \n \nI. INTRODUCTION  \nSemiconductor spintronic device s often engage  \nferromagnetic (FM) electrodes on top of a semiconductor \nchannel [ 1-3]. By using the FM electrodes as spin -sensitive \nprobes, a  non-equilibrium spin accumulation can b e \ngenerated and detect ed in the semiconductor channel . This \nbecomes especially challenging for measurement setups \nwhere the magnetization M of the FM detector is required to \nstay aligned along a pre -defined axis , while an external \nmagnetic field B is appli ed transverse to the  magnetization \naxis. For instance, t his approach is  used in a geometry for \nthe detection of the spin Hall effect (SHE) [ 4-6]. In these  \nexperiments  the spin probes are initially magnetized parallel \nto their axis . After that , a magnetic f ield, which is transverse \nto the FM magnetization axis and lies in the plane of the \nsample,  is applied . This  induce s a Hanle spin precession of \nthe S HE generated spin accumulation which can now be \ndetected  with the FM electrodes. The FM magnetization, \nhowe ver, stays aligned along the pre -defined direction only \nfor small values of the transverse field. For larger magnetic \nfields, it reorients parallel to the B-field, leading  to a rapid \ndecay of the SHE induced signal.  To prevent such signal \ndecay , it is crucial to tailor and to control the magnetic \nproperties of the employed thin FM layer.  \nThe goal of the present  work is to improve the magnetic \nproperties  of soft FM Fe layers, which are commonly used as spin detector or injector [ 1, 2, 7 ], by a simple and wid ely \napplicable technique which makes use of exchange coupling \nbetween soft and hard magnetic materials. The first \nexperimental observation of exchange  (bias) coupling was \nreported for the ferromagnetic/antiferromagnetic (FM/AFM) \nbilayer Co/CoO by Meiklejoh n and Bean in 1956 [ 8, 9]. In \nFM/AFM systems , the antiferromagnetic material acts as a \npinning layer for the FM layer . This gives rise to an \nunidirectional magnetic anisotropy and a shift of the \nhysteresis loop of the FM, when the material is field cooled \nthrough the Néel temperature TN of the AFM (assuming that \nthe Curie temperature TC of the FM is higher  than TN) [10]. \nResearch in the past decades focused on AFM/FM \ninterfaces, including multilayers of AFM/FM and interlayer \nexchange coupling t hrough nonmag netic spacer layer s. \nProgress both in experimental realization as well as in \ntheoretical description has been reviewed by several  authors \n[10-12]. Comparable coupling effects can also be observed \nin bilayer and multilayer systems of soft and hard FMs [ 13-\n15] and even in bilayers of soft and hard ferrimagnets [ 16, \n17]. In soft/hard FM systems , the hard FM material acts  as a \npinning layer for the soft FM.  Based on the  proposals for \nexchange -spring magnets by  Kneller and Hawig [ 18] and \nCoey and Slomski [ 19], exchange -spring coupling has been \nexperimentally observed and theoretically modelled for \nvarious hard/soft FM systems [20-23]. Here we explore the \npotential of Dy/Fe bilayers for hard magnetic contacs.  2 \n The paper is organized as follows. In Sec. II we desc ribe the \ncoupling mechanism between Fe and Dysprosium (Dy), the \nsample growth and the experimental techniques which we \nused in our studies. Data of Dy and Fe single layers is \npresented in Sec. III. and IV. Measurements of the magnetic \ninterplay between bot h materials in a Fe/Dy bilayer are \npresented in Sec. V. The paper conclude s with a summary in \nSec. VI.  \n \nII. EXCHANGE COUPLING IN FE/DY BILAYERS  \nFor our experiments we use the rare -earth element Dy as the \nhard FM material to magnetically pin  the soft FM Fe.  Dy is \nknown to have a large  saturation magnetization up to several \nTesla ( 𝜇0𝑀𝑠=3.75 T) [24] and large coercive fields. Both \nfeatures qualify  Dy to enhance the magnetic stability of soft \nFM spin probes. Detailed studies on the magnetic properties \nof Dy [ 25-28] revealed that its magnetic properties are (like \nmost rare -earth elements) different from those of the FM s of \nthe iron group. With decreasing  temperature , Dy undergoes \ndifferent magnetic phase transitions [ 29-32]. Dy is \nparamagnetic at room -temperature, becomes antiferro -\nmagnetic with a helical phase below the Néel temperature \nTN = 180 K, and finally it turns into an ordinary FM below \nthe Curie temperature TC = 90 K.  Temperatures below 90 K \nare used in most spin injection and detection experime nts. \nIn Fe/Dy bilayer s, the coupling between the rare -earth \nmaterial Dy and the transition metal Fe occurs through \nexchange coupling at the interface of both materials [ 33, \n34]. The magnetic properties of Dy in its FM phase mainly \nderive from its 4f electr ons, whereas those of Fe incorporate \nits 3d  electrons. Since the 3d band of Fe is more than half \nfilled, the 3d spins of Fe couple antiparallel with the 4f spins \nof Dy [ 35, 36 ]. For heavy rare -earth elements like Dy, spin \nand orbital moments are oriented  parallel. Therefore the 3d \nmoments of Fe and the 4f moments of Dy are aligned \nantiparallel , resulting in an antiferromagnetic coupling \nbetween Fe and Dy [ 21]. \nA. SAMPLE GROWTH  \nSince the focus of our work is on the magnetic improvement \nof Fe -based spin probe s, we patterned magnetic stripes (see \nFig. 1) with dimensions comparable to those of typically \nused spin injectors (stripe width W = 1 - 3 µm, length L = \n400 µm) on undoped (001) GaAs substrates  using electron -\nbeam lithography. A fter developing the PMMA re sist, the \nsubstrates were treated with hydrochloric acid (HCl) to \nremove PMMA residues and the GaAs native oxide. \nImmediately after that, w e mounted the sample in an ultra -\nhigh vacuum chamber (base pressure 8 × 10-10 mbar) and \ndeposited thin films of Fe or  Dy and bilayers of Fe/Dy  by \nmagnetron  sputtering . The deposition process took place at \nroom temperature and was controlled by time using \ncomputer controlled shutters.  \n \n FIG. 1. Micrograph  of a microstructured FM stripe (width W). A \nconstant dc current I = 50 µA is flowing through the FM stripe \nbetween two Ohmic contacts. Using the same contacts, the two -\nterminal magnetoresistance is monitored as a function of the \nmagnetic field, which is applied in the xy plane of the  sample.  \nThe Fe single layer was sputte red at low power (5 W) and \nconsists (in order of growth) of 2.9 nm Fe and a cap of 12 \nnm Au , which p revent s the material from oxid ation. The Dy \nsingle layer consists of either 35 nm Dy or 75 nm Dy and \nwas deposited at high sputtering power (50 W).  \nThe Fe/D y bilayer was fabricated by deposition of a thin Fe \nlayer on the GaAs substrate , and subsequent deposition of \nthe Dy layer  without breaking the vacuum. We \nmanufactured different Fe/Dy bilayers where the thickness \nof the Dy layer was fixed at 35 nm and the thickness of the \nFe layer was increased from 2.5 to 15 nm. After deposition \nof the FM layers , Ohmic contacts to the stripes were \nfabricated by electron -beam lithography and th ermal \nevaporation of Ti and Au.  The FM stripes are patterned  \nalong the [ 110] dire ction , however, stripes oriented along \nthe [11̅0] direction yield similar magnetic properties . This  \nsuggests that the deposited  material is polycrystalline.  We \nalso prepared similarly grown Fe, Dy and Fe/Dy full film \nsamples (5 mm x 5 mm) on undoped (001)  GaAs substrate s \nfor measurements in a super conducting quantum \ninterference device ( SQUID ) measurement setup.  \nB. MEASUREMENT TECHNIQUES  \nIn order to characterize the magnetic properties of the \nparticular  FM, we carried out SQUID measurements of full \nfilm s amples and magnetoresistance measurements of \nmicrostructured FM stripes  with a similar  layer sequence.  \nThe hysteresis curves of the full film  samples were obtained \nfrom measurements using  a Quantum Design SQUID  setup , \nwhich provides fields up to 7 T.  The a nisotropic  magneto -\nresistance (AMR) effect [37] was used as a tool to \ndetermine the orientation of the FM stripe’s magnetization \naxis with respect  to the external magnetic field.  The basic \nprinciple  of the AMR effect is that the magnetoresistance of \na FM m aterial depends on the relative angle between the \nmagnetization M and the direction of the electrical current I, \nwhich is flowing  through the FM material. For a FM it is \ncommonly found that 𝑅∥ > 𝑅⊥, i.e. the resistance 𝑅∥ is \nlarger when the mag netization axis M is (anti -) parallel the \ncurrent direction I, compared to the situation when the \n3 \n magnetization of the FM is transverse to the current \ndirection (𝑅⊥) [38]. The AMR value, which is the relative \nchange of the resistance, is commonly defin ed as AMR =\n(𝑅∥−𝑅⊥)/𝑅⊥. \nThe magnetoresistance  measurements were carried out \nbetween 4.2 K and 120 K in a 4He cryostat with Bmax = \n10 T. The samples were mounted in a sample holder which \nallows rotating  the sample in -plane with respect to the \nmagn etic field. We pass a constant dc current I = 50 µA \nthrough a single FM stripe by using its Ohmic contacts (see \nFig. 1) and monitor simultaneously the two -terminal voltage \ndrop between the Ohmic contacts as a function of the \nmagnetic field.  The following measurement routines were \nused to characterize a particular FM layer sequence . \n(1) First, we conduct SQUID measurements with a full film \nsample, from which we obtain the hysteresis curve of the \nFM material . Here, the magnetic field is applied parallel to \nthe in-plane direction of the film.  \n(2) Second, we perform circular magnetic field \nmeasurements  using microstructured FM stripes with the \nsame layer sequence. A constant magnetic field is rotated in \nthe xy plane of the sample (see Fig. 1) by 360°. For FM \nmaterials , the characteristic  AMR dependence of the \nresistance , 𝑅(𝜃)=𝑅⊥+(𝑅∥−𝑅⊥)cos2𝜃, is observed. \nHere, 𝜃 is the angle enclosed by the direction of the current \nI and the magnetization axis M of the FM.  \n(3) By applying a magnetic field in the xy plane of the FM \nstripe , we can determine its coerciv e field. The magnetic \nfield is swept  either (anti -) parallel to stripe (along the x \ndirection, what corresponds to 𝜃=0° and 𝜃=±180° , \nrespectively ) or transverse to the stripe ( y direction, \n𝜃=± 90°). By comparing the magnetoresistance data with \nSQUID dat a of the corresponding full film samples  [see \n(1)], we can determine the coercive field of the FM stripe.  \n(4) Finally, we use a measurement routine which is similar \nto the one  employed  for SHE  experiments d escribed in Sec. \nI. First, we  set the magnetizatio n axis M along the x axis, \ni.e. parallel to the stripe’s axis . After that , a transverse \nmagnetic field By is applied along the y axis. By analyzing \nthe obtained magnetoresistance data , we can determine the \ncritical field Bx,crit where the FM magnetiza tion has mostly \noriented parallel to the applied field, i.e. along the y axis. In \nterms of the previously mentioned SHE experiments , the \ncritical field corresponds the  field where the SHE induced \nsignal, detected by the FM spin probes, becomes zero.  We \nalso p erformed measurements where M was set transverse \nto the stripe ( y axis), and a magnetic field Bx was applied \nalong the stripe’s axis ( x axis).  \nWe first present SQUID and magnetoresistance  data of \nsingle D y and Fe layers, and then of Fe/Dy bilayer s. By \ncompa ring the corresponding results , we are able to distinguish the contributions of the individual  FM layer  to \nthe combined SQUID and magnetoresistance signal of the \nFe/Dy bilayer.  \n \nIII. CHARACTERIZATION OF DY  SINGLE LAYER  \nFirst, we characterize the magnetic pr operties of thin Dy \nfilms. The results presented in the following help to \ninterpret data obtained for Fe/Dy bilayers (see Sec. V.), \nwhere similar Dy films were used.  \nData on experiments with thin (microstructured) Dy films is \nvery rare in literature. Studi es of thin Dy films at low \ntemperatures [ 39] show that large magnetic fields (5 T) are \nnecessary to orient the film ’s magnetization. Since the  \nmaterial employed in our studies is presumably \npolycristalline, even larger fields might be required to fully \nsaturate the material [ 24]. Dy stripes with dimensions \ncomparable to our samples have previously been employed \nto structure magnetic superlattices on 2DEGs [ 40-42]. These \nexperiments also indicate that magnetic fields of 4 -6 T are \nnecessary to orient the magn etization of the Dy superlattice.  \nA. MEASUREMENTS AT HELIUM TEMPERATURE  \nFigure 2(a) shows the SQUID hysteresis loop  of a Dy full \nfilm sample  (tDy = 75 nm) . The curve was recorded at 5 K, \nwhere Dy is in the FM phase. The extracted coercive field \nof the samp le is 1150 mT. This is in good agreement with \nthe coercive field reported in Ref. [ 39] for sputtered, 50 nm \nthick Dy films, which were deposited at room temperature. \nThe magnetization at 7 T , extracted from SQUID data, is \n𝜇0𝑀 = 1.95 T, and the remanent magnetization is 𝜇0𝑀𝑟 = \n0.76 T. Although the maximum applied field (7 T) is in our \nexperiment larger than in Ref. [ 39], the  magnetization curve \ndoes not saturate . \nTherefore  we applied an even larger magnetic fiel d of 10 T \nfor circular magnetic field sweeps . The magnetoresistance \nsignal of a single Dy stripe ( tDy = 75 nm, W = 2 µm) as a \nfunction of the angle 𝜃 (B, I) is shown in Fig. 2(b). At 0°, B \nis parallel to the Dy stripe  (i.e. oriented along the x \ndirection , see Fig. 1), and thus parallel to the current I, \nwhich is flowing  through the FM stripe.  At ±90° the \nmagnetic field is oriented  perpendicular to the stripe  (y \ndirection).  The signal yields a typical AMR signature with a \nsinusoidal shape of 𝑅(𝜃). The res istance 𝑅(𝜃) has a \nminimum for 𝜃=0° (𝐌Dy∥𝐈) and a maximum for \n𝜃=±90° (𝐌Dy⊥𝐈), i.e. 𝑅⊥> 𝑅∥. As a consequence,  Dy \nexhibits a negative AMR effect ( 𝑅∥−𝑅⊥)/𝑅⊥<0 with an \nAMR value of roughly -0.2 %. Previously, a negative sign \nof the AMR effect has only been reported for certain alloys \nor compounds, e.g. for (Ga,Mn)As [ 43, 44 ], Ni - and Mn -\nbased alloys [ 45, 46 ], Fe 4N [ 47] or ha lf-metallic \nferromagnets [48 ], but not for rare -earth elements.  4 \n  \nFIG. 2. (a) SQUID hysteresis loop of a Dy full film sample ( tDy = \n75 nm), obtained at 5 K. (b) Magnetoresistance of a single Dy \nstripe (75 nm Dy, W = 2 µm) at 4.2 K as a function of the angle \nbetween magnetic field B and current I. The constant magnetic \nfield (10 T) was rotated in the xy plane of the sample. B is parallel \nto the stripe and to the current I at 0° ( x direction) and \nperpendicular at ±90° ( y direction). (c) Data of coercive field \nmeasurements at 4.2 K, with Bx oriented parallel to the stripe. \nDashed lines mark the coercive field. (d ) Critical field \nmeasurement of the same stripe at 4.2 K. The magnetization of the \nstripe was set along the x direction (along the stripe’s axis) before a \nperpendicular field By was applied. The critical field is indicated by \ndashed lines.  \nA quantitative ly similar behavior was obtained for Dy \nstripes with different widths ( W = 3 µm and 1 µm). We also \ndeposited Dy stripes with a different layer thickness (35 nm) \non GaAs and on a Si/SiO 2 substrate , and we additionally \ncapped some of the samples with a thin  layer of Au (12 nm) \nto prevent the surface from oxidization. Magnetoresistance  \ndata of these samples also reveal a negative sign of the \nAMR effect, ruling  out that the effect derives either from the \nsubstra te or from surface oxid ation.  For small er magnetic \nfields (i.e. 6 T) the 𝑅(𝜃) shape does not show a regular \nAMR  sin2(𝜃) signature. This suggests, that the \nmagnetization axis does not instantly reorient along B for \nsmall magnetic fields, emphasizing  the hard FM character of \nDy. \nFrom the results desc ribed above we can assume that MDy is \noriented along the B-field vector for B = ±10 T. We now \napply the magnetic field either parallel to the stripe ( Bx) or \ntransverse to the stripe ( By), and sweep it from 10 T to -10 T \nand back . This allows us to investig ate the coercive field of \nthe Dy stripe along the x- and y direction.  The magneto -\nresistance data for the Bx sweep is depicted in Fig. 2 (c). \nWhen the magnetic field is ramped down from 10  T to -10 T \n(red curve) , the slope of the magnetoresistance reverses  its sign at Bx = -1167 mT, when the signal is maximum.  The \ncurve of the upsweep ( -10 T to 10 T , black curve ) is mirror -\nsymmetric, and the signal is maximum at Bx = +116 7 mT. \nThe decrease of the magnetoresistance with increased | Bx| \ncan be explained by the  negative magnetoresistance effect \n(NMR), which is commonly observed for FM materials \n[49]. However, b y comparing the magnetoresistance data \nwith corresponding SQUID data of the full film sample [ see \nFig. 2 (a)] we find that the maximum of the signal / reversal \nof the slope at | Bx| = 1167  mT coincidences with the \ncoercive field , which we  extracted from SQUID data  (1150 \nmT). We assume that the magnetization  reversal leads to an \nincreased  spin disorder  and thus to an increase of the \nmagnetoresistance . The spi n disorder  is maxi mal at the \ncoercive field and decreases, when the Dy stripe reverses its \nmagnetization direction. Thus, t he maximum of the signal / \nreversal of the slope corresponds to the coercive field \n|Bx,coerc | = 1167  mT of the Dy stripe , which  is also marked \nwith dashed lines in Fig. 2(c ). \nThe curve obtained from the magnetic field sweep along the \ny direction exhibit s a similar trace  (not shown here) , the \nreversal of the slope  of the By curve , however,  occurs at a \nlower magnetic field (| By,coerc| = 9 53 mT).  This finding c an \nbe understood in terms of shape anisotropy  [50], by which \nthe long axis of the stripe ( x axis) is magnetically preferred \nover the shorter  axis ( y axis) . \nFinally, we apply the SHE -like measurement routine , which \nwas described in Sec . II., to the Dy stripe , We use a field of \nBx = 10 T to set the magnetization MDy along the stripe ’s \naxis, i.e. along the x direction. When the field is ramped \nback to zero, the magnetization of the stripe will stay mainly  \naligned in this direction, since  the employed Dy film \nexhibits  a large coercive field and a finite  remanence. After \nthat we  sweep the magnetic field By transverse to MDy, i.e. \nalong the y direction, fro m zero field to +10 T or -10 T.  By \nanalyzing the magnetoresistance data , we can determi ne the \ncritical field Bx,crit, where the FM magnetization orients \npreferentially parallel to the applied field . The \ncorresponding data is shown  in Fig. 2(d ). The \nmagnetoresistance exhibits a local minimum at zero field \nand increases monotonously until By ≈ ±500 mT. For | By| > \n500 mT , the slope of the curve reverses its sign, so that the \nmagnetoresistance decreases with increasing | By|. From \nthese data we can extract the critical field. At By = 0 mT , the \nmagnetization of the stripe is oriented along the stripe’s axis \n(x direction) , so that 𝐌Dy∥𝐈. Since Dy exhibits a negative \nAMR effect, the AMR signal is minimum at zero field . \nHowever, w hen By is increased, MDy will start to reorient \nalong the y axis with the applied magnetic field. When MDy \nis largely  oriented parallel to By, i.e. perpe ndicular to the \nstripe,  the AMR induced signal yields a maximum (𝐌Dy⊥\n𝐈). \n5 \n The decrease of the magnetoresistance for  |By| > ±500 mT  is \nagain due to the NMR effect . Overall, the recorded Dy \nmagnetoresistance curve is a superposition of the AMR and \nNMR i nduced signals.  The position of the maximum  of the \nmagnetoresistance curve  is now defined as the  critical field \nBcrit and is determined from the signal’s derivative . Here, we \nextract a critical  field of |Bx,crit| = 445 mT, marked with \ndashed lined in Fig. 2(d). \nWe also performed measurements where the initial \norientation of MDy was set transverse to the stripe ( y \ndirection) and a magnetic field Bx was swept along the ± x \ndirection, i.e. parallel to the stripe. Here , we observe a \ntransition of the signal (not  shown ) from a maximum to a \nrelative minimum of the signal, which is consistent within \nthe AMR effect.  The initial orientation of MDy along the y \ndirection leads to a maximum of the AMR induced signal \n(𝐌Dy⊥𝐈), whereas the final state, when MDy is orie nted \nparallel to the stripe, corresponds to a minimum of the AMR \nsignal ( 𝐌Dy∥𝐈 or 𝐌Dy⥯𝐈). The extracted critical  field \n|By,crit| = 247 mT is lower in this configuration than for the \nprevious setup . Again, th is can be explained by shape \nanisotropy . Both measurements show that the reorientation \nof MDy under the influence of a transverse magnetic field \ncan consistently be explained and traced by means of the \nAMR effect.  \nB. TEMPERATURE DEPENDENT MEASUREMENTS  \nWe also studied the temperature dependence of the coercive \nfield and of the critical  field between 4.2 K and 120 K by \nSQUID and magnetoresistance measurements.  With \nincreasing temperature , the coercive field (Fig. 3, squares  \nand stars ) and the critical  field (Fig. 3, circles)  decrease . \nThis is ascr ibed to the decrease of the  magnetocr ystalline \nanisotropy of the Dy layer  with increasing temperature  [51]. \nAgain, the coercive field extracted from  SQUID \nmeasurements (Fig. 3, stars) and from magnetoresistance \nmeasurements (Fig. 3, squares)  are in good ag reement. At T \n= 15 K, the coercive field has dropped to |Bx,coerc | =  900 mT, \nwhereas |Bx,coerc | = 270 mT at 90 K.   \nSurprisingly we observe a non -zero coercive field ( 70 mT) \nand critical  field ( 36 mT) at T = 120 K where Dy is \nsupposed to be in the antiferr omagnetic phase  (TC = 90 K)  \n[29-32], i.e. the coercive and the critical field  should be \nzero.  Correspondingly , the circular magnetic field \nmeasurement s show  a sinusoidal AMR signature  up to the \nhighest temperature of 120 K. Both findings strongly \nsuggest, that Dy has still a FM component at T = 120 K.  \nTo investigate the FM -AFM phase transition in more detail, \nwe performed two different kinds of SQUID measurements \nwith a Dy full film sample ( tDy = 75 nm Dy).  FIG. 3. Temperature dependence of the coercive fie ld Bx,coerc  \n(square) and of the critical field Bx,crit (circles) of a single Dy stripe \n(75 nm Dy, W = 2 µm). The coercive field extracted from SQUID \ndata (stars) is obtained from a similarly grown full film sample. \nLines are guides for the eye only.  \nFirst,  we recorded the magnetization curve during field \ncooling (FC) from 300 K to 5 K with an applied field of 7 T.  \nThe recorded magnetization of the FC curve, shown in Fig. \n4 (a), becomes non -zero below 160 K. This indicates that \nthe Dy film has a FM component  up to 160 K, i.e. way \nabove the Curie temperature of crystalline Dy.  \nA similar result was also obtained from the second SQUID \nexperiment . Here,  the sample was zero field cooled (ZFC) \nand the magnetization was oriented at 5 K in the plane of \nthe sample wit h a field of 7 T.  Afterwards, the sample was \nwarmed up to 300 K in zero field (ZF) while the sample’s \nremanent magnetization was recorded as a function of \ntemperature.  The warming curve [see Fig. 4(a)] clearly \nshows  that the FM -AFM phase transition is shif ted to \n160 K, since t he recorded magnetization becomes zero only \nfor T > 160 K. Both experim ents confirm  that sputtered Dy \nhas a FM component even above  its nominal Curie \ntemperature and that the AFM -FM magnetic phase \ntransition occurs at a higher temperat ure ( ≈160 K) than \nreported  for single crystalline Dy [29-32]. \nIn addition, we recorded the resistance of a single Dy stripe \n(tDy = 75 nm, W = 2 µm) during ZFC from room temperature \nto 4.2 K  (I = 50 µA) . As expected for metals, a decrease of \nthe resistance is ob served during cooling [see Fig. 4(b)].  At \nT ≈ 169 K, an anomaly of the resistance , marked by dashed \nline in Fig. 4(b),  is visible  and indicates a magnetic phase \ntransition of the Dy stripe about the  same temperature a t \nwhich the magnetization vanishes  in SQUID experiments.  \n6 \n FIG. 4. (a) FC (squares): SQUID m agnetization curve of a full film \nDy sample (75 nm Dy), recorded  during field cooling (applied \nfield = 7 T) from 300 K to 5 K. ZFC/ZF (circles): Zero field \nwarming curve for the same sample. Prior to heati ng, the sample \nwas zero field cooled to 5 K and the magnetization was oriented in \nthe plane of the sample with a field of 7 T. Both FC and ZFC/ZF \ncurves exhibit a magnetic phase transition around 160 K (dashed \nline). (b) Electrical resistance of a single Dy stripe (75 nm Dy, W = \n2 µm) as a function of  temperature. The stripe was zero field \ncooled from room temperature to 1.4 K. The anomaly in the \nresistance ( ≈169 K, marked with a dashed line) indicates a \nmagnetic phase transition.  \nAn absence of the FM/AFM phase transition at T = 90 K has \nalso been reported by Beach et al.  [52, 53 ] for Dy lattices \ngrown on Lutetium layers. They found that the compressive \nepitaxial strain between the Lutetium layers and the Dy \nlattice leads to an enhancement of the Curie temp erature of \nDy up to T = 175 K. A similar finding was also made by \nScheunert et al.  [39] for sputtered Dy films , which were  \ndeposited  at room temperature.  They conclude that the \ndeposition at room temperature induces strain in the hcp \ngrain lattice of Dy , leading to a suppression of the FM-AFM \nphase  transition at 90 K  and a shift of the Curie temperature \nup to 172 K. Since the growth conditions of our samples \n(sputtered at room temperature) are comparable to those of \nScheunert et al.  [39], it is most likely that the shift or \nsuppression of the magnetic phase transition is induced by \nstrain in the Dy lattice.  \nOverall, the hard FM characteristic of the employed thin Dy \nlayers is emphasized  by the large values of the coercive and \nthe critical  field in x as well as in y direction. This shows \nthat Dy can be expected  to establish in -plane magnetization \nstates which are robust against external magnetic field.   \n \nIV. CHARACTERIZATION OF FE SINGLE LAYER  \nWe now characterize the magnetic properties of thin Fe \nfilms . The data helps to evaluate the magnetic interplay \nbetween the Fe  and the Dy layer  in Fe/Dy bilayers, \ndiscussed in Sec. V.  \nFigure 5(a) shows the SQUID hysteresis loop (downsweep) \nof a Fe full film sample (2.9 nm Fe / 12 nm Au), which was \nrecorded at 5 K.  The coe rcive field of the Fe sample is small \n(2.6 mT), characteristic for  the small magnetocry stalline \nanisotropy of the Fe layer . On the other hand , the Fe layer exhibits a large magnetization at 7 T of 𝜇0𝑀 = 2.0 T  and a \nfinite remanent magnetization of 𝜇0𝑀𝑟 = 0.99 T. Both \nfeatures, a small coercive field  and a large saturation \nmagnetization , are typical ly found  for soft FMs  [18]. \nThe microstructured Fe stripes (2.9 nm Fe / 12 nm Au, W = \n3 µm) were first characterized by circular magnetic field \nmeas urements at 4.2 K, as shown in Fig. 5(b). Since the Fe \nlayer’s magnetization saturates  at 40 - 50 mT [see. Fig 5(a)], \na field of B = 1 T was applied. The circular magnetic field \nmeasurements  show that the AMR induced signal  is largest \nat 𝜃=0° when current  and magnetization are parallel \n(𝐌Fe∥𝐈), and is smallest at 𝜃=±90° when 𝐌Fe⊥𝐈. \nTherefore, Fe exhibits (in contrast to Dy)  a positive AMR \neffect with a characteristic cos2(𝜃) dependence and an \nAMR value of roughly 0.1 %. This finding allow s us to \ndistinguish between the contributions of Fe (positive AMR \neffect) and Dy (negative AMR effect) to the \nmagnetoresistance signal of the Fe/Dy bilayer (see Sec. V.)  \nThe coercive field of the Fe stripe is determined by \nsweeping Bx in the range of ±1 T , as shown in Fig. 5(c). \nWhen the magnetic field is ramped down from 1 T  (red \ncurve) , the magnetoresistance increases monotonously  until \nBx ≈+10 mT  is reached . Between +10 mT and -30 mT, a \nchange  of the magnetoresistance from maximum to a \nrelative minimum ( Bx = - 9 mT, marked with red dashed \nline) and back , can be noticed. For Bx > -30 mT, the \nmagnetoresistance decreases monotonously with Bx. The \nmagnetoresistance signal of the upsweep ( -1 T to 1 T, black \ncurve) is mirror -symmetric, and the signal becomes  \nminimum at Bx = +9 mT  (marked with black dashed line) . \nIn the following, we analyze the trace of the downsweep \ncurve (+1 T to -1 T). The decreasing magnetoresistance with \n|Bx| for fields larger than ±30 mT is due to the NMR effect. \nHowever, the change in the magnetoresistance between +10 \nmT and -30 mT cannot be described by the NMR effect, \nsince the NMR effect scales monotonously with the \nmagne tic field. Here, the change in the signal is related to \nthe reorientation of the stripe’s magnetization along the \nreversed magnetic field direction and can be explained by \nmeans of the AMR effect.  \nThe reorientation of the stripe’s magnetization from  +x to –x \ndirection can be mediated by 180° domain wall motion and \nrotation o f the magnetization axis. If  reorientation of the \nmagnetization occurs through 180° domain wall motion \nonly, the AMR signal remains unchanged, since it scales as \ncos2𝜃, i.e. the parallel and antiparallel orientation of the \nmagnetization axis are equivalent. If the reorientation occurs \nthrough rotation of the magnetization axis in the plane of the \nsample, the AMR signal changes with the angle 𝜃 between \ncurrent I and the magnetization axis.  \n7 \n FIG. 5. (a) SQUID hysteresis loop (downsweep) of a 2.9 nm Fe \nfull film sample, obtained at 5 K. (b) Magnetoresistance of a single \nFe stripe (75 nm , W = 3 µm) at 4.2 K as a function of the angle \nbetween the magnetic field B and the  current I. The constant \nmagnetic field (1 T) was rotated in the xy plane of the sample. (c) \nCoercive field measurements at 4.2 K with Bx oriented parallel to \nthe stripe’s axis. Dashed lines mark the coercive fields. (d) Critical \nfield measurements of the same stripe at 4.2 K. Bx,crit is marked \nwith dashed lines.  \nThis explain s consistently  the change of the magneto -\nresistance between +10 mT and -30 mT and determine s the \ncoercive field of the stripe.  \nFor large and positive Bx, the stripe’s magnetization MFe is \noriented parallel to the magnetic field and thus to the stripe . \nWhen the magnetic field reverses sign, MFe starts to realign \nalong the reversed magnetic field.  The magnetization rotates \nin the plane of the sample from its initial orientation (along \nthe +x direction,  𝐌Fe∥𝐈) towards the y axis, and then \ntowards the rever sed magnetic field direction , so that MFe is \nfinally aligned antiparallel to I in –x direction.  \nIn terms of the AMR signal, the rotation of MFe corresponds \nto a change from a maximum (𝐌Fe∥𝐈), to a minimum of \nthe signal ( 𝐌Fe⊥𝐈), and finally back to a maximum ( MFe is \naligned along the -x direction, 𝐌Fe⥯𝐈). With that, we can \ndetermine the coercive field of the Fe stripe , given by the \nAMR minim um. The coercive field extracte d from the \ndown sweep curve is Bx,coerc  = -9 mT [marked with red \ndashed line in Fig. 5 (b)], and the coercive field for the \nupsweep is equally +9 mT (black dashed line).  \nCoercive fields of the same order of magnitude have also \nbeen observed in spin injectio n experiments with Fe spin \nprobes of comparable thickness [ 1, 2, 7]. The coercive field \nof the Fe stripe, which we obtained from magnetoresistance \nmeasurements, is larger than the coercive field obtained \nfrom SQUID data [Fig. 5(a), 2.6 mT]. This is due to shape \nanisotropy (stripe vs. rectangular sample), which is more \ndominant for soft  FMs due to the small magnetocry stalline \nanisotropy.  The magnetoresistance data of the Fe stripe also provide \ninformation on the magnetization reversal  process . By \ncomparing the circular magnetic field measurement [Fig. \n5(b)] and the coercive field measurements , we find  that the \nchange in the AMR signal in the former case is larger \n(Rcircular = 4.2 ) than in the latter case ( Rcoerc = 1.5 ). \nFor the circular magnetic field measurement, the \nmagnetization axis follows instantly the applied field , so \nthat it rotates coherently without domain wall motion . \nTherefore, t he change in the A MR signal is at maximum.  \nOn the other hand, the change of the AMR signal observed \nfor the coercive field measurement  is significantly smaller . \nThis suggests  that the magnetization reversal is not solely \nmediated by rotation  of the magnetization axis, but a lso \nthrough 180° domain wall motion , which does not  \ncontribute to a change of the AMR signa l. The reorientation \nof the Fe stripe ’s magnetization  is therefore mediated by \nincoherent rotation of the magnetization axis and by 180° \ndomain wall motion.  \nFinally,  we apply the SHE -like measurement routine  to the \nFe stripe. This is shown in Fig. 5(d). Here, a field of Bx = \n1 T is sufficient to set the magnetization MFe along the \nstripe ’s axis ( i.e. along the x direction ). When the field is \nramped back to zero, the m agnetization of the stripe will \nmainly stay aligned along its axis. The magnetic field By is \nthen swept transverse to MFe, i.e. along the y direction, from \nzero field to +1 T or -1 T. The trace of the signal is again \nrelated to the rotation of the Fe strip e’s magnetization axis \nwith the applied magnetic field. The recorded curve, shown \nin Fig. 5(d), exhibits a distinct maximum at By = 0 T and \ndrops rapidly between By = ±25 mT. For | By| > 25 mT the \ncurve becomes linear due to the NMR effect.  \nThe trace of the  recorded signal can be explained by means \nof the AMR effect. At zero field , MFe is aligned parallel to \nthe stripe, so that the AMR signal is maximum ( 𝐌Fe∥𝐈). \nWhen By is increased , the magnetization orient s parallel to \nthe applied magnetic field and thus transverse to the stripe. \nThis state corresponds to a minimum of the AMR signal \n(𝐌Fe⊥𝐈). The critical field is now derived from the \nanalysis of  the curve  and its derivative . When  MFe has \noriented mostly parallel to the applied field (i.e. 𝐌Fe⊥𝐈), \nthe AMR signal is minimum and the  slope is zero.  If only \nthe AMR effect would contribute to the total \nmagnetoresistance , the critical field is equa l to the position, \nwhere the derivative of the  signal  becomes zero.  \nNevertheless , we also have to take into account the NMR \nsignal , which is dominant for | By| > 25 mT.  Thus , the critical \nfield is equal to the position, where the derivative \napproaches a fin ite, constant value, i.e. where the AMR \nslope is zero and only the constant NMR slope  contributes \nto the derivative . By analyzing the derivative of the total \nsignal and by fitting the NMR induced background  (not \nshown) ,  \n8 \n FIG. 6. Temperature dependence of the coercive field Bx,coerc  \n(squares) and of the critical field Bx,crit (circles) of a single Fe stripe \n(2.9 nm Fe, W = 3 µm). Both curves were obtained from \nmagnetores istance measurements. Lines are guides for the eye.  \nwe can extract the value of the crit ical field, | Bx,crit| = 25 mT, \nwhich is ma rked with dashed lines  in Fig. 5 (d). \nThe temperature dependence of the coercive field and the \ncritical  field was studied in the same temperature range a s \nfor the Dy single layer (4.2  K - 120 K)  and is shown in \nFig. 6. A decrease of Bcoerc (T) (squares ) and Bcrit (T) \n(circles)  with T can be noticed, whereby the decrease is \nlargest between 4.2 K and 30 K. At T = 15 K, the coercive \nfield and the critical field have dropped to | Bx,coerc | = 6 mT \nand | Bx,crit| = 19 mT . Only small values  for both fields , \n|Bx,coerc | = 0.9 mT and | Bx,crit| = 9 mT, were  recorded at T = \n120 K.  We attribute these findings not only to a decreas ing \nmagnetocr ystalline anisotropy of the Fe film with increasing \ntemperature, but also to domain wall mo tion, which is \nthermally activated . \n \nV. CHARACTERIZATION OF FE/DY BILAYER  \nThe basic motivation of this work is combining Fe with Dy \nin Fe/Dy bilayers to pin the soft FM Fe layer by the hard \nmagnetic Dy film. We first present data at helium \ntemperature for a particular bilayer (2.9 nm Fe / 35 nm Dy, \nT = 4.2 K) and figure out which features arise from the \ninteraction between both materials. Then , we study for the \nsame bilayer the temperature dependence of the magnetic \ninterplay between 4.2 K and 120 K. Finall y, we characterize \ndifferent Fe/Dy bilayers ( T = 4.2 K), where the thickness of \nthe Fe layer tFe was increased from 2.5 nm to 15 nm while  \nthe thickness of the Dy layer was fixed at 35 nm. By varying \ntFe for a given thickness of the hard FM Dy layer , we can  \nsystematically study the magnetic interplay between both \nmaterials. All SQUID measurements are accompanied by \nmagnetoresistance data acquired from similarly grown  full \nfilm samples.  \nA. MEASUREMENTS AT HELIUM TEMPERATURE  \nThe SQUID hysteresis loop of a Fe/D y full film sample (2.9 \nnm Fe / 35 nm Dy)  was recorded at 5 K  and is depicted in \nFigure 7(a).  Although the sample consists of two different magnetic layers , the overall shape of the curve resembles \nthat of a uniform, single phase magnet , thus clearly \nindic ating  exchange coupling between both layers  [18]. The \nmagnetization curve  shows no saturation at 7 T, similar to \nthe one observed for Dy [see Fig. 2(a)].  \nThe degree of coupling between both layers  depends on the \nthickness  ts of the soft FM layer  [18]. If the thickness of the \nsoft magnetic material is below a critical value , bot h FM \nlayers  are rigidly coupled and reverse their magnetization at \nthe same field.  On the other hand, i f ts is larg er than the \ncritical thickness, the soft FM layer  reverses its \nmagne tization at fields smaller than the hard FM layer . \nThe coercive field of the Fe/Dy bilayer (2 25 mT), which we \nextract from the SQUID curve in Fig. 7(a), is lower than that \nof the Dy  layer [1150 mT, see Fig. 2(a)]. This suggests that \nthe latter case applies  for our sample and that both layers are \nnot rigidly coupled. On the other hand, it can also be noted \nthat the coercive field is considerably larger than that of the \nFe sample [2.6 mT, see Fig. 5(a)].  \nFor a soft FM sandwiched between two hard FMs, it was \nfound that the critical thickness is roughly twice the width \nof the domain wall h in the hard FM layer  [20, 54]. For the \nlayer sequence employed in our sample, the critical \nthickness is equal to 𝛿ℎ. Calculations by Egami and Graham \n[55] yield a Dy domain wall thickness of  about 7 atomic \nlayers  (at zero temperature). This value co rresponds to Dy ≈ \n2 nm and is in line with  the experimental observations, \nwhich suggest that the Fe layer thickness ( tFe = 2.9 nm) is \nlarger than the critical thickness.  \nIn the following, we characterize a microstructured Fe/Dy \nbilayer, which has the same  layer sequence as the sample \nused for SQUID measurements (2.9 nm Fe / 35 nm Dy, W = \n2 µm). The magnetoresistance curves yields distinct \nfeatures, which have been recorded neither for the Dy single \nlayer, nor for the Fe single layer. The analysis of the \nobtained magnetoresistance data helps to interpret the \nSQUID magnetization curves and to understand the \ninterplay between the Fe and the Dy layer.  \nFirst, we elucidate how the Fe/Dy bilayer compares \nelectrically to the Fe and the Dy single layers. For that, we \ntreat the Fe/Dy bilayer as a conductor, where the Fe and the \nDy layer form a parallel circuit. The ratio of the currents IDy \nand IFe, which are flowing through the respective layer of \nthe bilayer, is then given by  \n𝐼𝐹𝑒\n𝐼𝐷𝑦=𝑅𝐷𝑦\n𝑅𝐹𝑒 .                                       (1) \nThe resistances RFe and RDy were determined by measuring \nthe four -terminal resistance of the respective Fe and Dy \nsingle layer samples. For the bilayer discussed in the \nfollowing (2.9 nm Fe / 35 nm Dy) we fo und, that the \nresistance of the Dy layer is approximately 2.5 times la rger \nthan that of the Fe layer.  \n9 \n FIG. 7.  (a) SQUID hysteresis loop of a full film Fe/Dy bilayer (2.9 \nnm Fe / 35 nm Dy), obtained at 5 K. (b) Magnetoresistance data of \ncoercive field measu rements at 4.2 K of a single Fe/Dy stripe ( W = \n2 µm) with the same layer sequence. (c) Zoom of the data given in \n(b). Dashed lines mark the coercive field of the exchange coupled \nFe layer. (d) Critical field measurement of the same stripe at 4.2 K. \nThe cri tical field Bx,crit of the exchange coupled Fe layer is marked \nwith dashed lines.  \nIn consequence, the current IFe flowing through the Fe layer \nis 2.5 times larger than the current IDy flowing through the \nDy layer. Thus , ≈71 % of the total current in the F e/Dy \nbilayer is flowing through the Fe layer and ≈29 % through \nthe Dy layer.  \nThe coercive field of the bilayer was determined by \nsweeping Bx in the range of ±10 T.  The obtained curves, \nshown in Fig. 7(b), resemble qualitatively that of the Dy \nsingle layer [see Fig. 2(b)].  When the magnetic field is \nramped down from 10 T to -10 T (red curve), the \nmagnetoresistance first increases monotonously due to the \nNMR effect and the slope of the signal is continuous around \nzero field. The slope reverses its sign at Bx = -1043 mT, \nwhen the signal is maximum.  Then t he magnetoresistance \ndecreases monotonously with Bx for increasing negative \nmagnetic field . The curve of the upsweep ( -10 T to 10 T, \nblack curve) is mirror -symmetric, and the signal is \nmaximum at Bx = +1043  mT. This suggests that the \nmagnetic properties  of the bilayer are mainly determined by \nthe hard FM Dy layer. Therefore we apply the same analysis \nas for the single layer Dy stripe (Sec. III.). With that , we \ndetermine the coercive field of the Dy layer,  |Bx,coerc (Dy)| = \n1043 mT, which is comparable to the value obtained for the \nsingle Dy layer ( 1167 mT). \nHowever, zooming into the curve, shown in Fig. 7(c), \nreveals , e.g.  for the downsweep (red curve) a relative \nminimum  of the signal  at Bx = -217 mT (equally at Bx = \n+217 mT for the upsweep , black curve ). Such feature has \nnot been observed in the Bx data for the Dy single layer . On \nthe other hand, a  similar trace of signal has been recorded for the coercive field measurements of the microstructured \nFe single layer  [see Fig. 5(c)].  \nIf the NMR induced background is fitted and subtracted  (not \nshown) , a transition of the magnetoresistance  from a \nmaximum to a relative minimum is visible . This suggests, \nthat the feature is related to the Fe layer.  Analogous to the \nanalys is of the single Fe layer  (see Sec. IV.) , the trace of the \nsignal can be explained by the AMR effect . \nFor large and positive Bx, the magnetization of the Fe layer \nis aligned parallel to the magnetic field  and thus parallel to \nthe stripe’s axis . When Bx reverses its sign, MFe rotates from \nits initial orientation (along the stripe’s axis, +x direction) \ntowards the y axis ( i.e. perpendicular to the stripe’s axis ) \nand finally towards the rever sed magnetic field direction  (–x \ndirection). In terms of the Fe AMR e ffect, this corresponds \nto a change of the AMR induced signal from a maximum  \n(𝐌Fe∥𝐈) to a minimum ( 𝐌Fe⊥𝐈), and finally back to a \nmaximum  ( 𝐌Fe⥯𝐈). We can exclude that the observed \nchange of the signal stems from the rotation of the Dy  \nlayer’ s magnetization  instead, since this would result in a \nreversed  signal shape due to the nega tive sign of the Dy \nAMR effect.  With that , we extract the enhanced coercive \nfield of the Fe layer , |Bx,coerc  (Fe)| = 217 mT, which is \nmarked with dashed lines in Fig . 7(c) . For magnetic field \nsweep s along the y direction , we could consistently observe \na change in the signal (not shown) from a  minimum to a \nmaximum and back.  \nAnalyzing  the magnetoresistance curve allow s thus \ndistinguishing  between the magneti zation rever sal of the Fe \nand of the Dy layer . Furthermore, the magnetoresistance \ndata also verifies , that the magnetization reversal and the \ncoercive field observed  in the  SQUID curve [Fig. 7(a)] can \nbe attributed to the Fe layer.  Moreover , the coercive fields \nextrac ted from SQUID [ 225 mT in Fig. 7(a )] and \nmagnetoresistance data (|Bx,coerc | = 217 mT) are in good \nagreement.  \nAll in all, we observe that the Fe layer is coupled by the Dy \nlayer and that the Fe layer’s  coercive field is significantly \nenhanced by a factor of  more than 2 4, compared to the Fe \nsingle layer (| Bx,coerc | = 9 mT) . However, t he Fe layer \nreverses its magnetization  at fields smaller than the hard FM \nDy layer, whereas the Dy layer’s coercive field  remains \nnearly unchanged.  This finding supports our assu mption that \nthe Fe thickness is above the critical thickness, so that the \ncoupling between both layers is not perfectly rigid.  \nWe also performed measurements with the Fe/Dy bilayer , \nfrom which we can determine the critical field. In this \nmeasurement routin e we use a field of Bx = 10 T to align the \nbilayer’s magnetization along the  stripe’s axis ( x direction ). \nThe magnetic field is then ramped back to zero . Afterwards, \nthe magnetic field By is swept transverse to the stripe’ s axis \nfrom z ero field to either By = +10 T or to By = -10 T.  \n10 \n Corresponding data is shown in Fig. 7(d), exhibiting  a broad  \nmaximum at By = 0 T, followed by a rapid decrease of the \nmagnetoresistance down to |By| ≈ 350 mT . For | By| > \n350 mT the trace of the curve becomes  roughly  linear . \nComparison of the Fe/Dy bilayer signal with that of the \nsingle Fe layer [see Fig. 5(d)] suggests  that the observed \ntrace is again connected with the rotation of  the Fe layer’s \nmagne tization.  Similar to the Fe layer stripe, the trace of the \nrecorded signal can be explained by m eans of the AMR \neffect.  At zero field , MFe is aligned parallel to the stripe, so \nthat the AMR induced signal shows a maximum ( 𝐌Fe∥𝐈). \nFor large By, the Fe layer’s magnetization has mostly \noriented parallel to the applied magnetic field and thus \ntransverse to the stripe. This state corresponds to a \nminimum of the AMR signal, since 𝐌Fe⊥𝐈. Again, we can \nrule out that the signal stems from the Dy layer’s \nmagnetization instead, because of the negative sign of the \nDy layer’s AMR effect.  \nThe critical field of the Fe layer is derived in th e same \nmanner as before . When MFe has oriented mostly parallel to \nthe applied field (i.e. 𝐌Fe⊥𝐈), the AMR induced signal  is \nminimum , and the signal’s derivative  becomes  zero.  \nNevertheless, we also have to take into account the NMR \ninduced signal, which is dominant for | By| > 350 mT, and \nwhich contributes a s a constant offset to the derivative of \nthe total signal. With that,  we can extract the critical field of \nthe Fe layer, which is marked with dashed lines in Fig. 7(d). \nThe a nalysis of the  total signal’s derivative  yields | Bx,crit (Fe)| \n= 352 mT. \nIn the reversed configuration , the magnetization is set along \nthe y direction  and a transverse field is applied along the \nstripe ’s axis  (x direction).  We observe a transition of the \nsignal from a relative minimum at zero field to a maximum \nof the signal  (not shown here) , whereby the trace of the \nsignal is again in accordance with  the sign of the Fe AMR \neffect.  Compared to the Fe single layer (| Bx,crit (Fe)| = 25 \nmT), the critical field of the Fe layer is significantly \nenlarged by a factor of more than 14.   \nAltogether , SQUID and magnetoresistance measurements \nhave clearly shown that b oth the coercive and the critical  \nfield of the Fe layer are increased by more than one order of  \nmagnitude, when the Fe layer is  brought in contact with the \nhard FM  Dy layer . This strongly suggests that the magnetic \nproperties of the Fe layer  are enhanced through soft/hard \nFM exchange coupling  at the interface between  both layers . \nB. TEMPERATURE DEPENDENCE  \nWe also performed temperature dependent SQUID and \nmagnetoresistance measurements wi th the same bilayer  (2.9 \nnm Fe / 35 nm Dy) . The temperature was varied between \n4.2 K and 120 K.  This allow s us to further characterize the \nexchange coupling  between  both layers , since the magnetic \nproperties  of the Fe and the Dy layer change with \ntemperature [see Figs. 3 and 6].  FIG. 8. Temperature dependent coercive and crit ical fields of the \nexchange coupled Fe layer in a Fe/Dy bilayer (2.9 nm Fe / 35 nm \nDy). Data for Bx,coerc  (squares) and Bx,crit (circles) was obtained \nfrom a microstructured FM stripe ( W = 2 µm). SQUID data of the \ncoercive field (stars) was obtained from a  similarly grown full film \nsample. Lines are guides for the eye.  \n \nThe coercive field of the microstructured Fe/Dy bilayer was \nagain determined by sweeping the magnetic field between  \n±10 T. The shape of the obtained curves  (not shown)  \nresembles for all temp erature s those of single layer  Dy. \nMoreover , at all temperatures the extracted coercive fields  \nof the Dy layer are  similar to those of the single layer Dy. \nThis implies that the Dy layer is also at T = 120 K in the FM \nphase (for the same reason as the Dy s ingle layer) and \nexhibits no FM/AFM transition at 90 K. The interplay \nbetween Fe and Dy is therefore determined by FM/FM \ncoupling also at T = 120 K.  At all temperatures, we record \nfeatures , which have similarly been observed at helium \ntemperature.  This all ows us to clearly identify the rotation  \nof the Fe layer’s magnetization  and thus its coercive field.  \nWe also performed  temperature dependent SQUID \nmeasurements with a similarly grown full film  sample . The \ncoercive fields, which we extracted from the magnet ization \ncurve, are in good agreement with the coercive fields \nobtained from the magnetoresistance data.  \nThe temperature dependence of the critical field was \ndetermined by applying the same measurement procedure as \nfor 4.2 K. At all temperatures , the By curves (pre -\nmagnetization along the x direction, i.e. along the stripe’s \naxis) show a transition of the signal from a maximum at \nzero field to a relative minimum, as it was observed at 4.2 \nK. For the Bx measurements (premagnetization is set along \nthe y axis) we consistently observe a change of the signal \nfrom a maximum to a rel ative minimum (not shown).  At all \ntemperatures, the critical field of the Fe layer can clearly be \nidentified . \n11 \n Figure 8 shows the temperature dependen ce of the coupled \nFe layer’s coercive  field and critical field . The coercive \nfield, which was obtained from SQUID (stars) and from \nmagnetoresistance measurements (squares), decreases \nrapidly between 4.2 K and 30 K, whereas only a modest \ndecrease is visible between 30 K and 120 K . However, a t all \ntemperatures, the Fe layer’s coercive field is enhanced  by a \nfactor of 14 -20, compared to the coercive field of the Fe \nsingle layer (triangles) . The critical field (circles) of the \ncoupled Fe layer also decreases rapidly between 4.2 K and \n30 K , and an  almost linear decay can be noticed for higher \ntemperatures . However, at T = 120 K, the Fe layer’s critical \nfield is still enhanced  by a factor of 8, compared to the Fe \nsingle layer ( see Fig. 6 ). \nThe temperature dependence of the exchange coupling is \nmainl y governed by two competing mechanism. On the one \nhand , the effective exchange length increases with \ntemperature , since the width h of the hard FM domain wall \nincreases [ 56]. This may result in inferior exchange \ncoupling at low temperature ( h < ts), and rigid coupling at \nhigh temperature (h ≥ ts), due to the larger  width of the \ndomain wall  [57, 58 ]. However, the extracted coercive field \nof the coupled Fe layer is smaller than the coercive field of \nthe Dy layer for all considered temperatures. This sugges ts, \nthat the bilayer is in the inferior coupling regime ( tFe > Dy) \nalso at higher temperatures, ruling  out the described \nmechanism  above . \nOn the other hand , the exchange coupling also depends on \nthe anisotropy of the hard FM layer. If the Curie \ntemperatur e of the hard FM layer is small,  the anisotropy of \nthe layer decreases with increasing temperature  relatively \nfast. This leads, in contrast to the above described \nmechanism, to a degradation of the exchange coupling with \nincreased temperature [ 59, 60 ]. \nOur experiments yield a decrease of the exchange coupled \nFe layer’s coercive field and  critical field with increasing \ntemperature . Since the Curie temperature of the employed \nDy layer  is small ( TC ≈ 160 K), we can attribute the \ndecrease of the exchange coupling to the degradation of the \nDy layer‘s anisotropy . This  assumption is also underlined  by \nthe temperature dependence, which we  obtained for  the Dy \nsingle layer (see. Fig. 3).  \nAll in all , we coul d observe exchange coupling between the \nFe and the Dy layer throughout the whole investigated \ntemperature range. We also observed  that t he temperature \ncan be used as a parameter to control the exchange coupling \nin the Fe/Dy bilayer . Since Dy is in the FM p hase at T = \n120 K, the observed coupling is of FM/FM type for all \ntemperatures.  \nC. DEP ENDENCE ON THE THICKNESS  \nOF THE FE LAYER  \nFor AFM/FM bilayers it was found that the strength of the \nexchange field Hex scales inverse ly 𝐻𝑒𝑥∝(𝑡𝐹𝑀)−1 with the  FIG. 9. (a) SQUID hysteresis loops (downsweep) of Fe/Dy full \nfilm samples with different thicknesses tFe of the Fe layer (5 nm, 8 \nnm, 12 nm) and fixed tDy (35 nm), obtained at 5 K. (b) Coercive \nfield measurements at 4.2 K of single Fe/Dy stripes ( W = 3 µm) \nwith the same layer sequence. Dashed lines mark the \ncorresponding coercive field Bx,coerc . (c) Critical field \nmeasurements at 4.2 K for the same Fe/Dy stripes. Dashed lines \nmark the critical field Bx,crit. \nthickness tFM of the FM layer (for tAFM = const.)  [10, 12 ]. \nA qualitative ly similar scaling can also be observed for \nsoft/hard FM bilayer systems. Here, the coercive field  of the \nexchange coupl ed soft FM layer scales (for a given \nthickness of the hard FM layer) inverse ly with its thickness \nts and is given by [ 13, 18] \n                           𝐻𝑐𝑜𝑒𝑟𝑐 = 𝜋2𝐴\n2𝜇o𝑀𝑠⋅(𝑡𝑠)𝑛 (𝑛<0).                (2) \nEquation ( 2) holds if ts is larger than the domain wall \nthickness h of the hard FM layer . Here, A is the exchange \nconstant and  𝜇0𝑀𝑠 is the saturatio n magnetization of the soft \nFM. For ideal  systems, where the soft FM layer has no \nanisotropy  and the hard FM is perfectly rigid, it was found \nthat n = -2 [13, 20]. Micromagnetic calculations f or non -\nideal systems, where the hard FM layer has a finite \nanisotropy, yield n = -1.75 for thick soft FM layers ( ts > 4h) \n[54]. \nIn the previous section, temperature was used to tune the \nexchange coupling.  According to Eq. ( 2), the exchange \n12 \n coupling can al so be controlled by the thickness ts of the \nemployed soft FM layer. Therefore, w e have grown and \ncharacterized Fe/Dy bilayers , where the thickness of the Dy \nlayer was fixed at 35 nm and the thickness of the Fe layer tFe \nwas increase d from 2.5 nm to 15 nm.  This allows us  to \nsystematically study the dependence of the exchange \ncoupling on tFe in the Fe/Dy bilayer system . By evaluating \nBcoerc (tFe) and Bcrit (tFe), we are able to determine  the scaling \nof the coercive  field with tFe. \nFigure 9(a) shows SQUID hys teresis loops (downsweep) of  \nFe/Dy bilayers with different thicknesses of the Fe layer  and \nconstant thickness of the Dy layer ( tDy = 35 nm) . The \ncoercive field  of the curves strongly depends  on the Fe \nlayer ’s thickness and decreases with  tFe, as it is predicted by \nEq. ( 2). The magnetoresistance measurements with \nmicrostructured FM stripes ( W = 3 µm) were carried out at \nT = 4.2 K, and we applied the same methods as in the \nprevious section.  \nThe shape of the curves obtained from the coercive field \nmeasurements  (not shown here) resembles for all bilayers  \nthose of the Dy single layer.  The extracted Dy coercive field \nremains roughly constant for all tFe. This indicates that the \nFe layer does not  noticeably  influence the magnetic \nproperties of the hard FM Dy layer,  even for larg er \nthicknesses of the Fe layer.  Figure  9(b) shows  a zoom of the \nBx curves, which were obtained for different thicknesses of \nthe Fe layer.  All curves  yield a relative minimum / dip in the \nmagnetoresistance  (marked with dashed lines). A similar  \nfeature / trace of the magnetoresistance curve has also been \nrecorded for the bilayer discussed in the previous section \n[2.9 nm  Fe / 35 nm Dy , see Fig. 7(c)] , which allows to \ndetermine  the coercive field of the coupled Fe layer. T he \nextracted coercive fie lds (marked with dashed lines ) decay  \nwith increa sing thickness of the Fe layer, and  are in good \nagreement  for all thicknesses tFe with the SQUID data . \nFigure 9(c) shows magnetoresistance measurements of the \ncritical field for different thicknesses of the F e layer. Here, \nthe magnetization of the bilayer was set along the stripe’s \naxis, before a transverse field By was applied. All curves \nexhibit a transition of the signal from a maximum to a \nrelative minim um, as it has been observed for the bilayer \ndiscussed  in the previous section [2.9 nm Fe / 35 nm Dy, see \nFig. 7(d)].  The observed feature s can clearly be attributed to \nthe critical field of the coupled Fe layer and are in \nagreement with the positive sign of the AMR effect of Fe. \nThe critical field (marked wi th dashed lines) also decreases \nwith the thickness of the Fe layer, as i t has been observed  \nfor the coercive field measurements.  \nWe also note, that t he amplitude of the Fe AMR features \nbecomes smaller for thin ner Fe layers (especially for tFe ≤ \n2.9 nm).  For tFe < 2.9 nm we are no longer able to determine \nthe critical field of the coupled Fe layer, since the amplitude \nof the Fe AMR signal drops and gets superimposed by the \nDy signal.  FIG. 10.  (a) Coercive field Bx,coerc  (tFe) (squares) at 4.2 K for Fe/Dy \nbilayers with different thicknesses tFe of the Fe layer and constant \nthickness of the Dy layer (35 nm). SQUID data of the coercive \nfield (stars) is obtained from similarly grown full film samples. \nThe l ine is a guide for the eye.  (Inset) Log -log plot of Bx,coerc (tFe). \nThe exponent n of the power law 𝐵𝑐𝑜𝑒𝑟𝑐 (𝑡𝐹𝑒)∝(𝑡𝐹𝑒)𝑛 is deduced \nfrom the linear fit of the curve. (b) Critical field Bx,crit (tFe) at 4.2 K \nfor the same Fe/Dy bilayers as presented in (a). The line is a guide \nfor the eye only. (Inset) Log -log plot of Bx,crit (tFe). n is deduced \nfrom the linear fit of the curve.  \nWe can give two possible explanations for this observation.  \nOn the one hand, if the Fe layer thickness is comparable to \nthe Dy domain wall width ( 𝛿𝐷𝑦≈ 2 nm [ 55]), both FM \nlayers are fully coupled and reverse their magnetization at \nthe same field . On the other hand, we can speculate that the \nAMR amplitude of the coupled Fe layer is too small to  be \nobserved  for small tFe. According to E q. (1), the ratio of the \ncurrents IFe (tFe) and IDy flowing through the Fe and Dy  layer \ndepends on the ratio of the individual layer resistances like  \nIFe (tFe) / IDy = RDy / RFe (tFe). Here, t he resistance RDy of the \nDy layer is fixed ( tDy = const.) . If tFe is reduced, the \nresistanc e RFe of the Fe layer increase s and the ratio IFe (tFe) \n/ IDy decreases. Therefore , the current IFe flowing through \nthe Fe layer and the AMR amplitude  decrease s as well, \nwhen the Fe layer becomes thinner.  \nWe now evaluate the dependence of the coupled Fe la yer’s \ncoercive field on tFe. Fig. 10(a) shows the plot of the Bcoerc \n(tFe) curves , which include  data obtained from SQUID \n(stars) and from magnetoresistance measurements  (squares). \nBoth curves show , that the coercive field decreases rapidly \nwith increasing  thickness of the Fe layer, as expected from \n13 \n Eq. ( 2). If plotted against 1/ tFe the curve exhibits  an almost  \nlinear dependence on 1/ tFe (not shown). This suggests that \nthe coupling is an interface effect , as observed for other \nsoft/hard FM bilayer sys tems [ 61-63]. According to Eq. ( 2), \nthe curves can be  well described by a power law \n𝐵𝑐𝑜𝑒𝑟𝑐 (𝑡𝐹𝑒)∝1/(𝑡𝐹𝑒)𝑛 (n > 0). \nThe exponent n is deduced from the log -log plo t of Bx,coerc \n(tFe), shown in the inset of Fig. 10 (a). We extract  n(Bx,coerc ) = \n1.31 ± 0.01  from the magnetoresistance data, and  n(Bcoerc, \nSQUID ) = 1. 40 ± 0.09 from SQUID data . Both values are \nsmaller than the value of n, which is theoretically predicted \nfor non -ideal , exchange coupled systems ( n = -1.75) [ 54]. \nSuch d eviations  have also been reported for MBE grown \nmaterials [ 61]. In this study , the authors as cribe this mainly \nto the surface roughness at the interface , which affects the \nspin pinning.  \nThe dependence of the critical field on tFe, shown in Fig. \n10(b), can also by described with a power law 𝐵𝑐𝑟𝑖𝑡(𝑡𝐹𝑒)∝\n1/(𝑡𝐹𝑒)𝑛. The log -log plot of the critical field curve, from \nwhich we obtain n(Bx,crit) = 0.93 ± 0.02, is shown in the inset \nof Fig. 10(b).  We assume that the deviation between ncoerc \nand ncrit mainly stems from the fact , that both measurement \nroutines are different in terms of magnetization reversal. To \nour knowledge, similar studies have not been reported yet in \nliterature . However, the presence of linear ln( Bcrit) vs ln( tFe)-\ncurves clearly indicates an exchange coupling mechanism \n[64]. \nAll in all, t he experiments presented in this section have \nshown  that exchange coupling in th e Fe/Dy bilayer can be  \ncontrolled  by changin g the thickness of the Fe layer.  This \nallow s improving and engineering  the coercive  field and the \ncritical field of the coupled Fe layer . \n \nVI. SUMMARY  \nWe have performed a comprehensive study  of the magnetic \npropertie s of sput tered Dy layers and their exchange \ncoupling with thin , sputtered  Fe layers at lo w temperatures \n(4.2 K - 120 K).  Magnetoresistance and  SQUID data prove \nthat the deposited Dy single layer is of hard FM nature. \nMoreover magnetoresistance data  exhibit s a negative sign of \nthe AMR  effect of Dy , which has previously not been \nreported . We also observe a shift of the Curie temperature \nfrom 90 K to TC = 16 0 K which is attributed to the growth \nconditions of the material.  Measurements of a Fe/Dy bilayer \n(2.9 nm Fe / 35 nm Dy) yield an enhancement of the \ncoercive field and  of the critical field of the Fe layer by a \nfactor of 1 4 - 22 (compared to the Fe single layer), which is \ndue to exchange coupling between both layers.  Data \ncollected between 4.2 K and 120 K  show s that the exchange \ncoupling depends on the temperature, and persist s even at \n120 K, i.e. close to the C urie temperature of Dy.  We also \nfabricated samples  where the thickness of the Dy layer was \nfixed and the thickness of the  Fe layer was varied between \n2.5 nm and 15  nm. Here, data shows that the coercive field \nand the critical field of the coupled Fe layer scale inverse ly with its thickness, as  predicted by theory. This allows \nengineer ing the coercive field and the critical field of the \nexchange hardened Fe layer by adjusting its thickness.  \n \nOverall, we have demonstrated that the coercive field and \nthe critical field of a microstructured F e layer can be \nenhanced and tailored, if brought in contac t with a hard \nmagnetic Dy layer . Thus, microstructured Fe/Dy bilayer s \ncould be widely  employ ed in the field of spintronics, e.g. for \nSHE experiments , where pre cise control of the FM \nelectrode’s  magn etic properties  is necessary.  \n \nACKNOWDLEDGMENTS  \nWe thank C. H.  Back for fruitful discussions and valuable \ncomments on the manuscript. This work has been supported \nby the Deutsche Forschungsgemein schaft (DFG) via SFB \n689. \n \n* dieter.weiss@physik.uni -regensb urg.de  \n \nREFERENCES  \n[1] X. Lou, C. 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Iihama1,2, Y. Xu1, M. Deb1, G. Malinowski1, M. Hehn1, J. Gorchon1, E. E. Fullerton1, 3, and S. Mangin1*  * stephane.mangin@univ-lorraine.fr  1 Institut Jean Lamour, UMR CNRS 7198, Universite de Lorraine, Nancy, France 2 National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Japan 3 Center for Memory and Recording Research, University of California San Diego USA  All-optical ultrafast magnetization switching in magnetic material thin film without the assistance of an applied external magnetic field is being explored for future ultrafast and energy-efficient magnetic storage and memories. It has been shown that femto-second light pulses induce magnetization reversal in a large variety of magnetic materials. However, so far, only GdFeCo-based ferrimagnetic thin films exhibit magnetization switching via a single optical pulse. Here we demonstrate the single-pulse switching of Co/Pt multilayers within a magnetic spin-valve structure ([Co/Pt] / Cu / GdFeCo) and further show that the four possible magnetic configurations of the spin valve can be accessed using a sequence of single femto-second light pulses. Our experimental study reveals that the magnetization final state of the ferromagnetic [Co/Pt] layer is determined by spin-polarized hot electrons generated by the light pulse interactions with the GdFeCo layer. This work provides a new approach to deterministically switch ferromagnetic layers and a pathway to engineering materials for opto-magnetic multi-bit recording. 2   The possibilities of deterministically manipulating magnetization solely with ultra-short light pulses have attracted a growing attention over the past ten years leading to numerous ultra-fast and low-energy data storage concepts [1-9]. In 2007, all-optical switching (AOS) of magnetization using femtosecond (fs) laser pulses in GdFeCo ferrimagnet was first discovered [1]. Later on, it was shown that the magnetization in GdFeCo can be switched by a single fs-laser pulse independently of the light helicity [2,3] referred to as all-optical helicity-independent switching (AO-HIS) and has only been observed for GdFeCo-based materials. The AO-HIS has been described by a thermal-driven switching mechanism attributed to the transient ferromagnetic like states and the transfer of angular momentum between Gd sub-lattice and FeCo sub-lattice [2,3,17-20]. Very recently, this type of switching has not only been observed in the case of light pulses but also for electron pulses [7-9].  In contrast to AO-HIS, in the case of all-optical helicity-dependent switching (AO-HDS), the final state of magnetization is determined by the circular polarization of the light. AO-HDS has been observed for a large variety of magnetic material such as ferrimagnetic alloy, ferrimagnetic multilayer, ferromagnet thin films and granular recording media [4,5,10-15]. However, so far multiple pulses are necessary to fully deterministically switch the magnetization for AO-HDS [10, 16]. The use of single-pulse switching would be interesting because it is ultra-fast and energy-efficient, however restriction to Gd-based 3  materials limits potential spintronic devices application. Furthermore, in order to move toward ultrafast-spintronic applications, one needs to study and understand the fundamental mechanism not only for single layers as it has been done in most study so far but also in more complex structures like spin valve structures, a key building block of modern spintronics. Selective magnetization switching in spin-valve structures or more complex heterostructures will enable multi-level magnetic storage and memories [21-23].  Here we demonstrate that the four possible magnetic configurations of a magnetic spin-valve structure ([Co/Pt] / Cu / GdFeCo) shown schematically in Fig. 1, where both layers are magnetically decoupled, can be accessed using a sequence of single femto-second light pulses. We show that a single laser pulse is able to switch the magnetization of either the GdFeCo layer alone or the magnetizations of both GdFeCo and [Co/Pt] layers, depending on the optical pulse intensity.  We attribute this magnetic configuration control of the multilayer to, in part, a result of the ultrafast magnetization dynamics in spin-valve structure as well as ultrafast non-local transfer of angular momentum between layers [24-29]. Indeed, ultrafast quenching of magnetization in ferromagnetic or ferrimagnetic layers creates spin-polarized hot electrons that propagate in the metallic spacer layer and transfer the angular momentum to the other magnetic layer. We believe the switching of the [Co/Pt] layer results from a combination of optical excitation and the transfer of spin-polarized hot electron currents generated via the demagnetization of the GdFeCo layers.  4  All Optical Multibit recording in GdFeCo / Cu / [Co/Pt] spin-valve structure Schematic illustration of the Ta(5nm)/Gd23.3(FeCo)76.7(5nm)/Cu(9.3nm)/[Co(0.6nm)/Pt(1.0nm)]4/Ta(5nm)/Glass substrate spin valve structure namely GdFeCo/Cu /[Co/Pt] is shown in Fig. 1a. The GdFeCo and [Co/Pt] layers exhibit perpendicular magnetic anisotropy (PMA) and are magnetically separated by a 9.3-nm continuous Cu layer. The ferrimagnetic GdFeCo layer is FeCo rich at room temperature for this composition as the net magnetization of the alloy is parallel to the magnetization of FeCo sublattice and antiparallel to the magnetization of the Gd sublattice. Using magnetic fields, four remanent magnetic configurations can be reached: Two configurations where the magnetization of the two layers are parallel P+ and P- when the two magnetizations are along the positive and negative field direction, respectively, and two configurations where the two magnetizations are antiparallel AP+ and AP- when the magnetization of the [Co/Pt] is along the positive and negative field direction, respectively. Figure 1b shows the magneto-optic Kerr effect (MOKE) signal (θK) as a function of the applied magnetic field (H) applied perpendicular to film plane. The reversal that appears at low magnetic field is attributed to the GdFeCo reversal. This could be confirmed by the magnetization curves (See supplementary information, sec A). Minor loops in blue and red in Figure 1.b show that the two layers are magnetically decoupled (i.e. the minor loop is centered about zero applied field). 5  In Fig. 2 we show how the four remanent magnetic configurations (P+, AP+, AP- and P-) can be accessed using single 35-fs light pulses under zero applied field. Starting with a saturated sample in the P- state (left most configuration in Fig. 2) one can switch both magnetic layers to the corresponding P+ state by sending a relatively intense pulse of 0.5 µJ where a region of roughly 30 µm in diameter is reversed.  Then exposing the sample to a moderate light pulse of 0.2 µJ only triggers the reversal of the GdFeCo magnetization putting the sample into the AP+ state. A second 0.2-µJ pulse again reverses the GdFeCo layer returning the system to the P+ state. Applying a 0.5-µJ then reverses both magnetic layers back to the original P- state.  Finally, a 0.2-µJ pulse again reverses the GdFeCo layer yielding the AP- state. Somewhat surprisingly if the AP+ or AP- state is excited by a 0.5-µJ pulse only the GdFeCo layer is reversed and the [Co/Pt] layer remains unperturbed. To summarize, 0.5-µJ pulses switch both layers from P+ to P- and back again while 0.2 µJ pulses are sufficient to only switch the GdFeCo layer leaving the [Co/Pt] layer unperturbed. The P+ to P- and P-to-P+ transitions clearly shows that the [Co/Pt] layer can be switched by a single 35 fs pulse.  Energy and radius dependence of the single pulse on the magnetization switching In order to get insights on the energy and configurational changes of the switching of the [Co/Pt]/Cu/GdFeCo structures, we have studied the influence of the pump energy within the beam spot. Indeed, since the beam intensity follows a Gaussian profile, we can explore, 6  with a single pulse, a range of energies. In Fig. 3a starting from an initial state in the P+ configuration, 35-fs single pulses of a given pump energy irradiated the sample consecutively with a typical time between two pulses of 2 seconds. The same experiment has been reproduced for 8 different pump energies ranging from 0.08 µJ to 0.36 µJ (three energies are shown in Fig. 3a). The spatial distribution of the magnetic configuration has been measured after each pulse. From Fig. 3a we can define up to four threshold radii (ri) measured from the center of the optical excitation region whose values depend on the pump energy and allow us to distinguish five different responses to the initial magnetic configuration and light interaction. The radius r1 corresponds to the threshold for the onset of AO-HIS of the GdFeCo layer while leaving the [Co/Pt] layer unperturbed. Thus for r>r1 the magnetization is unperturbed by the light while for r<r1 the GdFeCo layer reverses its magnetization with each pulse from P+ to AP+ or AP+ to P+. This can be seen clearly for 0.08 µJ pulses in Fig. 3a where only within the radius r=r1 there is a switching of the GdFeCo.   For increasing pulse energy of 0.2 µJ pulses (Fig. 3a) two new radii appear. With each pulse the outer ring corresponds to r1 where only the GdFeCo reverses. Near the center of the beam there is a region for r<r3 where both the GdFeCo and [Co/Pt] layers reverse from P+ to P- and then back with each pulse.  There is a second radius, r2, where starting from P+ the GdFeCo magnetization is switched between r1>r>r3 to the AP+ configuration after the 7  first pulse. Then for subsequent pulses there is a narrow region r2>r>r3 where the light has no effect with further pulses. For yet higher pulse energy such as 0.32 µJ as shown in Fig. 3 where for r<r4 the energy density is sufficiently high that both layers demagnetize forming random domain states.   From images like those shown in Fig. 3a and corresponding line scans that quantify the magnetic contrast both the magnetic configurations and corresponding threshold radii can be determined and are shown in Fig. 3b for various pulse energies. The solid lines in Fig. 3d are fits assuming a Gaussian spatial distribution of the pump energy, i.e., 𝐹!!=2𝐸!(𝜋𝑤!!)⋅exp(−2𝑟!!𝑤!!), where 𝐹!!, 𝑟! and 2𝑤! are threshold fluence, threshold radius, and 1/𝑒! spot size, respectively. From the fitting of our experimental data (from Fig. 3b), we are able to extract four (incident) threshold fluences, 𝐹!!= 3.1 mJ/cm2,  𝐹!! = 8.5 mJ/cm2 𝐹!! = 11.8mJ/cm2 and 𝐹!! = 28.4 mJ/cm2 that correspond to radii r1 through r4. We note that the range of fluences for which single shot switching is observed is the highest ever reported. In our films switching fluences cover a whole decade (3.1 – 28.4 mJ/cm2), whereas in GdFeCo single layers doubling the AOS threshold fluence would lead to multidomain creation [30, 31]. We believe this behavior might be related to the higher Curie temperature of the [Co/Pt] layer and the dynamic coupling between the layers. Discussion of the mechanism of single shot switching of different regions For low fluences (r1>r>r2) we observe that only the GdFeCo switches with each 8  pulse as has been observed as in single GdFeCo films. At these fluences the [Co/Pt] layer is not sufficiently excited to be perturbed. This behavior is expected from the extensive literature of AO-HIS on GdFeCo films [32]. The narrow region r2>r>r3 where once the sample is in the antiparallel configuration no changes occur after next pulse indicates some subtle interplay between layers that prefers the antiparallel configuration. However, we do not have a detailed explanation of this interaction at present. We will focus much of the remainder of the discussion on the region r3>r>r4 where we observe the simultaneous reversal of the GdFeCo and [Co/Pt] layers by single pulses. We believe that in this region, GdFeCo is switching by itself via the AO-HIS but in addition there is single shot deterministic reversal of the ferromagnetic layer through a combination of optical excitation and a dynamic coupling mechanism between the layers. We will discuss possible coupling mechanisms below and discuss additional experiments to test the validity of these mechanisms.  The first possible explanation is the presence of a static exchange coupling, which has been reported to be at the origin of the single shot AOS of a [Co/Pt] bilayer coupled to a GdFeCo layer [34]. In this study the final state of the [Co/Pt] layer is determined by the sign of the interlayer coupling. For our samples, the minor loops of the GdFeCo magnetization reversal show no measurable exchange coupling between the two magnetic layers (Fig. 1). The fact that we can independently switch the GdFeCo layer and the energy required for 9  AO-HIS of the GdFeCo layer from P+ to AP+ and from AP+ to P+ are identical, also suggest there is no exchange coupling via the Cu interlayer present in our sample. Moreover we observed this single-shot AO-HIS of the GdFeCo and [Co/Pt] layers for up to a 30 nm-thick Cu spacer layer (See supplementary information, sec C) where no exchange coupling is expected. A second possible explanation could be the presence of some dipolar coupling.  While there are no dipolar fields arising from an ideal uniformly magnetized film, there are dipolar fields generated from domain states and inhomogeneous magnetization.  For circular domains such are shown in Fig. 2 the dipolar fields are strongest at the boundary and may be responsible for the narrow region r2>r>r3.  To explore the role of dipolar coupling we studied samples with a different GdFeCo alloy concentration such that the net magnetization of the ferrimagnetic GdFeCo would be in the direction of the transition-metal sublattice (i.e. “FeCo-rich”) or the rare-earth sublattice (i.e. “Gd-rich”). Figure 4 shows AOS in spin-valve structure of the FeCo-rich sample (Fig. 4a) and the Gd-rich (Fig. 4b) GdFeCo. Since MOKE measurements are mostly sensitive to FeCo sublattice moment (as opposed to the total moment), magnetic contrast values in antiparallel configuration of the magnetization are higher than that in parallel configuration in Gd-rich GdFeCo as shown in Fig. 4b. Single-shot AO-HIS of the GdFeCo and [Co/Pt] layers is observed for both GdFeCo concentrations.  However the final magnetic state reached by the GdFeCo and [Co/Pt] layers corresponds to a 10  parallel alignment of the FeCo sublattice and the Co/Pt magnetizations independently of the GdFeCo concentration and net magnetization as shown schematically in Figs. 4c and d. Thus dipolar interactions (at room temperature) are unlikely to explain the final state of the magnetization of the GdFeCo and [Co/Pt] layers. Having ruled out indirect and dipolar coupling we believe the final state of the ferromagnetic [Co/Pt] layer is determined by spin-polarized hot electrons generated by the light pulse interactions with the GdFeCo layer in addition to local heating of the [Co/Pt] layer. To explain the single pulse AO-HIS of the [Co/Pt] layer, we assumed that the ultrafast laser heating after interacting with the GdFeCo layer is generating a hot electron spin polarized current (i.e. superdiffusive spin currents) parallel to Gd moment which will ultimately transfer its angular momentum to the [Co/Pt] layer and in combination with optical excitation determines the final state of magnetization (schematically shown in Fig. 4c and 4d.). Recent experiments on spin currents generated by the interaction between a light pulse and a GdFeCo layer [29] observed, as a function of time after optical excitation, first positive and then negative spin generations in the conduction band. The authors attribute the positive part of the spin generation to demagnetization of the transition-metal sublattice, and estimate the majority of the negative spin is coming from slower demagnetization of the Gd sublattice. They also suggest a potential contribution to the negative spin current to the spin-dependent Seebeck effect. Both contributions result in the longer time spin currents being parallel to the 11  Gd sublattice, as shown in Figs. 4c and 4d, that will be transferred and absorbed by the [Co/Pt] layer. If the Co/Pt layer is excited by both the optical pulse and the hot electrons [33], the transfer of the hot electron angular momentum could be sufficient to determine the final state of the magnetization of [Co/Pt] as it cools. Thus it is expected that the longer time spin-polarized currents that are parallel to the Gd sublattice will determine the final state.  At the same time, GdFeCo layer is also reversed as a result of the AO-HIS mechanism. Thus even though the [Co/Pt] layer final state magnetization is determined by the Gd sublattice, the FeCo sublattice and the [Co/Pt] magnetization are always parallel in the final state. This mechanism might be also contribute to the recent single shot AOS demonstrated in Co/Gd bilayer system [35], in which non-local transfer of angular momentum might reverse the magnetization of Co since the exchange interaction between Gd and Co exists only at the interface.   To explore the validity of the above spin polarized hot electron transport model to explain the [Co/Pt] magnetization switching, we have grown several spin-valve structures where we have modified the Cu interlayer. First GdFeCo / Cu / [Co/Pt] spin-valve have been grown with different Cu thicknesses ranging from 5 nm to 80 nm. Single-pulse [Co/Pt] switching could be observed up to 30-nm Cu interlayers. The loss of the [Co/Pt] reversal is attributed to the limited spin-diffusion length of the hot electrons in the Cu layer, estimated to 13 nm by Schellekens et al.[26]. On other way to reduce the spin polarization of the hot 12  electrons consists to insert 1 to 5 nm Pt layer in the Cu spacer layer (shown schematically in Fig. 5a). Pt has a significantly shorter spin diffusion length than Cu and is on the order of a few nanometers [26]. While the average magnetic properties of samples with and without Pt layer are very similar (see magnetization curves in supplementary information, sec. E) it is expected that the Pt layers will depolarize the hot electrons before reaching the [Co/Pt] layer. This should limit the ability to deterministically switch the [Co/Pt] layer. Figure 5b shows the MOKE images after the irradiation of fs-laser pulses for the samples with different Pt thicknesses tPt. For tPt = 0 nm we see the P+ to P- and back to P+ switching as described earlier and the [Co/Pt] layer is 100% switched. With increasing Pt thickness, the magnetic contrast around center of the spot gradually decreases indicating the samples are transitioning from AO-HIS to demagnetizing with increasing Pt. By analyzing the magnetic contrast values in details, we plot in Fig. 5c the change in magnetization of the [Co/Pt] layer with increasing Pt thickness. We see that with increasing Pt thickness the samples transition from deterministic switching to demagnetization where the GdFeCo and [Co/Pt] layers break into small domains.  The decay length obtained by exponential fitting was ~ 2 nm as shown in Fig. 5c (See detail of the analysis in supplementary information, sec. F).  This decay length is consistent with a previous report for the spin diffusion length of hot electron in Pt layer (3 nm) [23,26] and is quite a bit shorter than penetration depth of light in Pt layer ~ 13 nm calculated from imaginary part of the refractive index of 2.85+𝑖 4.96 for the wavelength 13  of 800 nm. Thus, the magnetization of [Co/Pt] after irradiation of fs-laser pulse changes with increasing Pt thickness more drastically than the change in light absorption and temperature. Moreover, we also performed light absorption calculation. It was found that total light absorption (i.e., temperature rising) in the sample was not changed significantly with different Pt thickness, (see supplementary information, Sec. G). Based on these studies, we conclude that magnetization switching of the GdFeCo / Cu / [Co/Pt] is mediated by spin-polarized hot electron transport.  Conclusion  In this study, we demonstrated that we could access the four remanent magnetic configurations in GdFeCo / Cu / [Co/Pt] spin-valve structures, without applied field using single femto-second pulses. After studying the effect of the GdFeCo concentration and of the spacer layer we concluded that the final state of the magnetization switching of [Co/Pt] is mediated by spin-polarized hot electron transport. The final state is consistent with the expected spin polarization being parallel to Gd moment in the GdFeCo layer due to the slower demagnetization of Gd compared to the one of the FeCo spins. These hot spin-polarized electrons transfer their angular momenta which, in combination with optical heating, are able to deterministically switch the magnetization of [Co/Pt]. This conclusion is supported by inserting Pt layers inside the Cu spacer layer of the spin-valve to depolarize the 14  optically-induced spin current resulting in the thermal demagnetization of the [Co/Pt] layer. This work provides a new approach to deterministically single-pulse switch ferromagnetic layers and a pathway to engineering materials for opto-magnetic multi-bit recording using spin-valve structures.  Methods 1. Sample preparation All samples were prepared by physical vapor deposition. Base pressure used to deposit multilayer film was about 1×10-7 Torr. Basic stacking structures used in this study for the spin-valve structure are as follows, Glass sub. / Ta (5) / [Pt(1)/Co(0.6)]4 / Cu (tCu) / Gdx(Fe87Co13)1-x (5) / Ta (5), (thickness in nm) Cu thickness was varied from 5 nm to 80 nm. Gd composition was varied from 21.9% to 27.2%. Stacking structures of the reference sample for the Pt layer insertion are as follows, Glass sub. / Ta (5) / [Pt(1)/Co(0.6)]4 / Cu (7) / Pt (tPt) / Cu (7) / Gd23.3Fe66.7Co10 (5) / Ta(5).  2. AOS measurement Ti: sapphire fs laser source and regenerative amplifier are used for the pump laser beam in AOS measurement. Wavelength, pulse duration and repetition rate of the fs laser are 800 nm, 35 fs, and 5 kHz, respectively. The 1/𝑒! spot size 2𝑤! is ~50 µm. No external magnetic field is applied during 15  measurement. Four different magnetic configurations are realized by using permanent magnet before taking images. MOKE images were obtained from the other side of the film. LED light source with the wavelength of 628 nm was used for taking MOKE images.   3. Data analysis   The movies of MOKE images during irradiation of fs-laser pulses were taken by CCD camera. After taking movies, the images of each slice were subtracted from the initial slice of the movies to exclude the background of the images. And then, average contrast values of the initial images were added to each subtracted images. The median and mean filters were applied to reduce noise in the images. The gray scale magnetic contrast values were converted to the color (lower contrast is blue, middle contrast is white, and higher contrast is red).  Reference [1] Stanciu, C. D. et al. All-Optical Magnetic Recording with Circularly Polarized Light. Phys. Rev. Lett. 99, 047601 (2007) [2] Radu, I. et al. Transient ferromagnetic-like state mediating ultrafast reversal of antiferromagnetically coupled spins. Nature 472, 207 (2011) [3] Ostler, T. A. et al. 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Deterministic all-optical switching of synthetic ferrimagnets using single femtosecond laser pulses. Phys. Rev. B 96, 220411(R) (2017)  Acknowledgement S. I. would like to thank the Japan Society for the Promotion of Science (JSPS) for a Grant-in-Aid for JSPS Fellows (28-7881). We would like to thank Crosby Chang and Stephane Suire for their assistance of SQUID-VSM measurement. This work was supported by the ANR-NSF Project, ANR-13-IS04-0008-01, COMAG, ANR- 15-CE24-0009 UMAMI and by the ANR-Labcom Project LSTNM, by the Institut Carnot ICEEL for the project « Optic-switch » and Matelas and by the French PIA project ‘Lorraine Université d’Excellence’, reference ANR-15-IDEX-04-LUE. Experiments were performed using equipment from the TUBE. Davm funded by FEDER (EU), ANR, Région Grand Est and Metropole Grand Nancy.  Author contributions G. M., M. H., E. E. F., and S. M. conceived the research topic. S. I. and Y. X. performed sample fabrication, M. H. helped for sample preparation. M. D. and G. M., developed AOS measurement set-up. S. I. performed AOS experiment and data analysis. J. G. performed light absorption 20  calculation. S. I., J. G., G.M., E.E.F and S. M. wrote manuscript. All authors contributed to discuss the measurement results.  Figures \n  Figure 1 Properties of the GdFeCo / Cu / [Co/Pt] spin valve structure. a) Schematic representation of the Glass sub. / Ta (5 nm) / Gd23.3(FeCo)76.7 (5 nm) / Cu(9.3 nm) / [Co(0.6 nm)/Pt(1.0 nm)]4 / Ta (5 nm) sample where the GdFeCo and [Co/Pt] magnetic layers have perpendicular anisotropy and are separated by a 9.3 nm thick Cu layer. b) Normalized magneto-optic Kerr rotation (θΚ) as a function of the magnetic field (H) applied perpendicularly to the film plane. Red and blue open symbols are minor loops corresponding to the magnetization reversal of GdFeCo which are perfectly centered around the zero field axis.   \n21   \n  Figure 2 Multi-level switching of GdFeCo /Cu/[Co/Pt] spin valve structure using 35-fs single light pulses. a) Experimental demonstration of magnetic configurations obtained consecutively after single optical pulses. Starting from a saturated state P- (resp. P+) a single intense pulse (0.5 µJ) induce a switching into the P+ state (resp. P-). A single moderate light pulse (0.2 µJ) induces a transition from a P- state (resp. P+) to an AP- (resp. AP+) state. All the measurements demonstrate that the GdFeCo layer switching can be obtained using a single moderate light pulse (0.2 µJ) whereas the complete switching of both layer is obtained for single intense pulse (0.5 µJ). b) Normalized averaged magnetic contrast obtained along the black line shown in a), averaged within the width of 5 µm. The four different levels allow to quantitatively define the four magnetic states (P+, P-, AP+, AP-) which can be reached using a sequence of single 35-fs laser pulses. \n22    Figure 2 Single pulse all-optical switching as a function of pump energy for Gd23.3(FeCo)76.7 / Cu(9.3) / [Co/Pt]4 sample. a) MOKE images obtained after four consecutive 35-fs pulses of various pump energy (the time between two pulses is 2 seconds). The same experiments has been repeated with different pump energy ranging from 0.08 to 0.36 µJ. b) Threshold radius (ri) such that for r> r1: there is no light effect, r1>r>r2 AO-HIS single pulse GdFeCo layer reversal. r2>r>r3 Starting from P+ the GdFeCo magnetization is switched once to reach an AP+ configuration and then light pulses have no effect; r3>r>r4: AO-HIS is observed for both layers; r4 >r the energy is too high and the all stack demagnetized.  \n23   \n Figure 3 All-Optical Switching in GdFeCo / Cu / [Co/Pt] spin-valves with different GdFeCo concentration. a) AOS results for FeCo-rich Gd23.3(FeCo)76.7 (5 nm)/ Cu(9.3 nm) / [Co/Pt]4 sample. b) AOS results for Gd-rich Gd26.4(FeCo)73.6 (5 nm) / Cu(10 nm) / [Co/Pt]4 sample. Since MOKE is more sensitive to the FeCo magnetic sublattice, the magnetic contrast signals of AP+ states are larger than that of P+ state in the case of spin-valve sample with Gd-rich GdFeCo. Schematic illustration of spin-polarized hot electron transport induced magnetization switching for c), FeCo-rich and d), Gd-rich GdFeCo / Cu / [Co/Pt] spin valve structures.     \n24  Figure 4 Evolution of the Co/Pt magnetization switching on Pt insertion layers in Cu. a) Schematic illustration of the role of the Pt spin scattering layer inserted in the Cu spacer layer. b) MOKE images after irradiation of consecutive single fs-laser pulses on GdFeCo / Cu (7) / Pt (tPt) / Cu (7) / [Co/Pt] samples and GdFeCo / Cu (15) / [Co/Pt] sample (tPt = 0 nm). c) The changes in magnetization of [Co/Pt] around the center of the spot estimated from magnetic contrast are plotted as a function of Pt thicknesses. 100 % indicates completely switched and 50 % indicates full demagnetization. Solid curve is an exponential fitting and decay length obtained from fitting is ~ 2 nm.   \n"    },    {        "title": "1805.09798v1.Searching_the_weakest_link__Demagnetizing_fields_and_magnetization_reversal_in_permanent_magnets.pdf",        "content": "https://doi.org/10.1016/j .scriptamat.2017.11.020   \n \nSearching the weakest link: Demagnetizing fields and magnetization reversal in \npermanent magnets  \nJ. Fischbacher1, A. Kovacs1, L. Exl2,3, J. Kühnel4, E. Mehofer4, H. Sepehri -Amin5, T. Ohkubo5,  \nK. Hono5, T. Schrefl1 \n1 Center for Integrated Sensor Systems, Danube University Krems, 2700 Wiener Neustadt, Austria  \n2 Faculty of Physics, University of Vienna, 1090 Vienna, Austria  \n3 Faculty of Mathematics, University of Vienna, 1090 Vienna, Austria  \n4 Faculty of Computer Science , University of Vienna, 1090 Vienna, Austria  \n5 Elements Strategy Initiative Center for Magnetic Materials, National Institute for Materials Science, \nTsukuba 305 -0047, Japan  \nAbstract  \nMagnetization reversal in permanent magnets occurs by the nucleation and e xpansion of reversed \ndomains. Micromagnetic theory offers the possibility to localize the spots within the complex structure \nof the magnet where magnetization reversal starts. We compute maps of the local nucleat ion field in a \nNd 2Fe14B permanent magnet using a model order reduction approach.  Considering  thermal fluctuations \nin numerical micromagnetics we can  also quantify the reduc tion of  the coercive field  due to thermal \nactivation . However, the major reduction of the coercive field is caused by the soft magnetic grain \nboundary phases and misorientation  if there  is no surface damage . \n1. Introduction  \nWith the rise of sustainable energy production and eco -friendly transport there is an increasing demand \nfor permanent magnets. The generator of a direct drive wind mill  requires high performance magnets of \n400 kg/MW power; and on average a hybrid and electric vehicle needs 1.25 kg of high end permanent \nmagnets [1]. Modern high -performance magnets are based on Nd 2Fe14B. These magnets have a high \nenergy density product which means that  the magnets can be small and still create a sufficiently large \nmagnetic field. One weak point of Nd 2Fe14B is the relatively low Curie temperature as compared to \nSmCo based magnets. As a consequence, the coercive field of Nd 2Fe14B drops rapidly with increa sing \ntemperature. To enhance the operating temperature of Nd 2Fe14B magnets heavy rare earth elements \nare added. The anisotropy field of (Nd,Dy) 2Fe14B is higher than that of Nd 2Fe14B and therefore the Dy  \ncontaining magnet can be operated at higher temperature. Production techniques that increase the Dy \nconcentration near the grain boundary [2,3]  reduce the share of heavy rare earth s. In these magnets the \nanisotropy field i s locally enhanced near the grain boundary whic h suppresses the formation of reversed \ndomains  [4,5] . Similarly, a n enhancement of the  coercive field has been achieved by Nd -Cu grain \nboundary diffusion [6].  \nThe enhancement of coercivity by modification of the region next to the grain boundaries is a clear \nindication that magnetization reversal in permanent magnets is induced by the nucleation of a reversed domain near the grain surface. In Nd 2Fe14B based permanent magnets weakly ferromagneti c grain \nboundary phases [7] act as nucleation sites [8] if there exist no grains with reversed magnetic domains \nat remanence.  In addition to the presence of the soft magnetic grain boundary phase, magnetization \nreversal is facilitated by the local demagnetizing field. These fields obtain their highest values near the \nedges and corners of the polyhedral grain structure [9]. Traditionally the effect of defec ts where the \nintrinsic properties are different from the bulk and the effect of the demagnetizing field on coercivity is \nexpressed as [10]  \n 𝐻c(𝑇)=𝛼𝐻N(𝑇)−𝑁eff𝑀s(𝑇). (1) \nHere 𝐻N=2𝐾1(𝜇0𝑀s) ⁄  is the ideal nucleation field. The constant µ 0 is the permeability of vacuum. The \nanisotropy constant and the magnetization are denoted by K1 and Ms. HN and Ms are intrinsic magnetic \npropertie s and depend on the composition of the magnetic phase and on the temperature T. On the \nother hand, the coercive field, Hc, strongly changes with the microstructure of the magnet. Therefore, \nmany researcher s refer to  and Neff as microstructural parameters.  The parameter  gives the \nreduction of the coercive field by soft magnetic defects and misorientation of the anisotropy axes with \nrespect to the applied field direction; the parameter Neff describes the reduction of the coercive field \nowing to  the self -demagnetizing fiel d. It can be regarded as a local, effective demagnetization factor.  \nEquation (1) describes the influence of the microstructure on the coercive field. Temperature effects are \nincluded through the temperature dependence of K1(T), Ms(T). A second mechanism ho w temperature \ninfluence s the coercive field are thermal fluctuations on the macroscopic scale. These fluctuations may \ndrive the system over a finite energy barrier. Hysteresis in a non-linear system  like a permanent magnet  \nresults from the path formed by subsequently following local minima  in an energy landscape constantly \nchanged by a varying external field  [11]. With increasing opposing field, the energy barrier that \nseparates the system from the reversed state decreases. The critical field at which the energy barr ier \nvanishes is the switching field of the magnet [12]. Switching at finite temperature can occur at non -zero \nenergy barrier.  If the system can escape over the energy barrier within the measurement time switchi ng \nwill occur. In permanent magnets it is assumed that the system can escape an energy barrier of 25 kBT \nwithin one second [13]. Therefore , to include the reduction of coercivity by thermal activation equation \n(1) can be rewritten as [14,15]  \n 𝐻c(𝑇)=𝛼𝐻N(𝑇)−𝑁eff𝑀s(𝑇)−𝐻f(𝑇). (2) \nThe thermal fluctuation field, Hf, can be expressed in terms of the activation volume v [13] \n 𝐻f(𝑇)=25 𝑘B𝑇\n𝜇0𝑀s𝑣  (3) \nHere kB = 1.38×10−23 J/K is the Boltzmann constant. In this work we will quantify the influence of \ndemagnetizing fields and thermal activation on the reduction of the coercive field with respect to the \nideal nucleation field  using micromagnetic simulations. The paper is organized as follows.  \n1. We will use equation (1) and analyze micromagnetic results for Hc(T) to show how  and Neff \nchanges with the microstructure.  \n2. We will apply  a simplified micromagnetic model based on the local demagnetizing f ield near the \ngrain boundaries  to get a deeper insight on how demagnetizing effects reduce coercivity . \n3. We will calculate the thermally activated switching  to locate the weakest spot where \nmagnetization reversal is initiated within a complex grain structure.  For the computations of  and Neff (see 1 above) we used a finite element micromagnetic solver  [16]. \nThe hysteresis loop is computed by minimization of the Gibbs free energy for decreasing external field. \nAt all surfaces  the mesh size is 2.4 nm. The simplified micromagnetic model (see 2 above) is based on a \nmethod for computing the demagnetizing field f rom surface charges  [17] and a method for computing \nthe switching field as function of field angle in the presence of defects [18]. Thermally activated \nswitching (see 3 above) is computed using a modified string method [19,20] . The finite element meshes \nare created with Tetgen [21]. At all surfaces the mesh size is 2.4 nm.  \n2. Results  \n2.1 Microstructural parameters and the demagnetizing field  \nWe computed the coercive field for a magnet consisting of equi -axed, platelet shaped, or columnar \ngrains. The microstructure used for the simulations is show n in Figure 1.  The grains are perfectly aligned. \nThe edge length of the cube forming the magnet is 200 nm. The volume fraction of the grain boundary is \n26 percent. Its thickness is 3.8 nm. For the grains we used the intrinsic magnetic properties of Nd 2Fe14B \nas function of temperature [22]. We performed two sets of simulations. In set 1  the grain boundary \nphase was non -magnetic; in set 2 the grain  boundary  phase was assumed to be weakly ferromagnetic. \nWe set the magnetization of the grain boundary phase to 1/3 of the value for Nd 2Fe14B, that is  \nMs,GB = Ms/3. The same material as in the grain boundary phase is used as surface layer with a thickness \nof 1.9 nm that covers the magnet . The purpose of this layer is to mimic surface damage.  Please note \nthat we change  the exchange constant of all phases according to A(T) = cM s²(T), where the factor c is \ndetermined from the Nd 2Fe14B values at T = 300 K (µ 0Ms = 1.6 T and A = 8 pJ/m). We apply the finite \nelement method to compute the magnetization as function of the external field, M(Hext). At each step of \nthe external field the micromagnetic energy is minimized using a modified non -linear conjugate gradient \nmethod [16]. To compute the microstructural parameters, we applied the same procedure as usually \ndone in experiments. We  plotted Hc(T)/Ms(T) as function of HN(T)/Ms(T) and fitted a straight line. Table 1 \ngives the microstructural parameter s for the different structures.  At all surface the mesh size was forced \nto be 2.4 nm.  \nTable 1. Microstructural parameter for magnets made of \nequi -axed, platelet shaped, and columnar grains.  \nShape of grains  Grain boundary phase   Neff \ncolumnar  non-magnetic  0.88  0.79  \nequi -axed  non-magnetic  0.88  0.87  \nplatelet  non-magnetic  0.88 0.91  \ncolumnar  weakly ferromagnetic  0.45  0.10  \nequi -axed  weakly ferromagnetic  0.47  0.27  \nplatelet  weakly ferromagnetic  0.51  0.43  \nThe results clearly reflect the shape anisotropy of the grains. For both sets of simulations t he \ndemagnetizing factors increase as we go from columnar, equi -axed to platelet shaped grains. When a \nferromagnetic grain boundary phase is introduced, the parameter  is reduced approximately by a \nfactor of ½. The corresponding reduction in Neff may be understood with reduced surface charge at the \ngrain surfaces . For a uniformly magnetized grain the demagnetizing field arises from magnetic surface \ncharges which are proportional to ( Ms−Ms,GB). So far, the numerical results for  and Neff are consi stent \nwith the conventional interpretation of the role played by soft magnetic defects and shape anisotropy. The value of  is smaller than 1 for perfect grains without any defect and non -magnetic grain boundary. \nThis can be explained by the non -uniform de magnetizing field within the magnet. The finite angle of the \ntotal field, which is the sum of the external field, the demagnetizing field and the exchange field , with \nrespect to the anisotropy axis causes magnetization reversal by nucleation and expansion of reversed \ndomains . \n2.2 Embedded Stoner Wohlfarth model  \nWe now show that we can understand the switching of grains in a permanent magnet using the model \norder reduction approach which we call embedded Stoner Wohlfarth method. By comparing the \nsimplified method with conventional micromagnetic computations, we show that the perpendicular \ncomponent of the demagnetizing field plays a significant role for magnetization reversal.  \nFirst, we compute the demagnetizing field of the magnet. Assuming that each grain is uniformly \nmagnetized we can evaluate the demagnetizing field as a sum of line integrals along the edges of the \ngrains [17]. For field evaluation we developed a highly parallel algorithm to be used on gra phics \nprocessors. The demagnetizing field, Hdemag , is evaluated at several points xi which are located at a \ndistance d from the edges of the grains. Then we loop the external field. For each value of the external \nfield we compute the total field  \n 𝐇tot=𝐇ext+𝐇demag +𝐇x, (4) \nat all points xi. We denote the angle between Htot and the anisotropy axis, with 𝜓. According to the \nStoner -Wohlfarth theory [24], the swi tching field of a small particle with uniform magnetization depends \non the angle between the particle’s easy axis and the field [25] \n 𝐻sw=𝑓(𝜓)𝐻N,   𝑓(𝜓)=1\n(cos2/3𝜓+sin2/3𝜓)3/2. (5) \nThe minimum value of the external field for which | Htot| > Hsw is the switching field of the magnet.  \nEmbedded Stoner Wohlfarth method:  \ncompute the demagnetizing field Hdemag  and set h = |Hext| = 0  \nloop over h: \nevaluate Htot(h) at all points xi \ncompute 𝜓𝑖 and Hsw,i  \nif |Htot| > Hsw,i for any i \nset Hc = h and stop  \nincrease h  \nThe total field given b y (4) is the sum of the external field, Hext, the demagnetizing field, Hdemag , and the \nexchange field, Hx. Near edges the demagnetizing field is tilted and no more antiparallel to the \nmagnetization [9]. Indeed, the perpendicular component of the demagnetizing field will grow to infinity \nas one approaches the edge. This divergence of the magnetostatic  field is compensated by the exchange \nfield leading to a finite total field at any point near the edge [23]. The exchange field is defined as \nfollows: 𝐻x,∥=𝐴/𝑑2 and 𝐻x,⊥=0. The symbols ∥ and ⊥ denote th e component of the exchange field \nparallel and perpendicular to the anisotropy axes.   In other words, we place virtual Stoner -Wohlfarth particles into each grain. These virtual particles are \nlocated at a distance d from the edges.  Then we compute the Ston er-Wohlfarth switching fields of each \nvirtual particle. The lowest value of the external field that leads to switching of at least one virtual \nStoner -Wohlfarth particle gives the coercive field.  Numerical experiments showed that  at 𝑑=\n1.2 √𝐴(𝜇0𝑀s2) ⁄  the results of the embedded Stoner Wohlfarth method coincide with the coercive field \nobtained from conventional micromagnetics [26].  \nFigure 2 compares the grain size dependence of the coercive field computed with the embedded Stoner \nWohlfarth method and conventional micromagnetics.  The switching field of a single, isolated Nd 2Fe14B \ngrain was calculated a s function of grain size.  There is a reasonable good agreement  between the two \nmethods.  The inset shows a map of the local switching field  according to the embedded Stoner \nWohlfarth (ESW) model for h = 0. The lowest values of Hsw are found near the bottom edge of the grain. \nThis is exactly the region where a reversed domain is nucleated in conventional micromagnetics.  \n2.3 Thermal activation  \nWe cho ose a grain boundary diffused magnet for analysis of thermal activation in permanent magnets. \nIn particular we will show that the spot at which magnetization reversal is initiated can be tuned by \ngrain boundary engineering. Figure 3 shows the microstructure used for the simulations. The magnet \nconsists of 64 grains. The average grain size is 57 nm. The grains are separated by a weakly \nferromagnetic grain boundary phase with a thickness of 3 nm. We investigate thermally activated \nmagnetization reversal for different leve ls of grain boundary diffusion as shown on the right -hand side \nof Figure  3. For the intrinsic magnetic properties we used the room temperature values for Nd 2Fe14B and \n(Nd,Dy) 2Fe14B. The material properties used for the Nd 2Fe14B core were K1 = 4.3 MJ/m³, µ0Ms = 1.61 T, A \n= 7.7 pJ/m . For the shell material, (Nd,Dy) 2Fe14B, we assumed K1 = 5.17 MJ/m³, µ 0Ms = 1.15 T, A = 8.7 \npJ/m . The magnetocrystalline anisotropy of the grain boundary phase was zero. Its magnetization and \nexchange constant were µ0Ms = 0.5 T, A = 7.7 pJ/m . The average degree of misalignment of the \nanisotrop y directions was 15 degrees.  \nWe consider three samples: no Dy diffusion, only the surface grains are covered with a dysprosium \ncontaining shell, and all grains are covered with a Dy containing shell. For these three samples we \nquantify the effects that re duce the coercive field with respect to the ideal nucleation field of Nd 2Fe14B \nmain phase. To do so we perform three simulations. We denote the coercive fields obtained by these \nsimulations with Hc,1, Hc,2, and Hc,3, respectively.   \n(1) We compute the demagnetization curve with a conventional micromagnetic finite element solver \n[16] but we switch off the magnetostatic field. We obtain the magnetization reversal proces s without \nany demagnetizing effects. The difference between the ideal nucleation field and the coercive field, \ndefect = HN−Hc,1, has to be attributed to misorientation and soft magnetic defects.  \n(2) We perform a classical micromagnetic simulation of the reversal process. Now magnet ostatic \ninteractions are fully taken into account. The difference demag = Hc,1−Hc,2 is caused by demagnetizing \neffects.  \n(3) We compute the energy barrier for switching as function of the applied field, E(Hext) using a modified \nstring method. The critical field , Hc,3, at which E(Hext) = 25  kBT is the nucleation field at temperature T for a measurement time of one second. The difference thermal  = Hc,2−Hc,3 is caused by thermal \nfluctuations.  \nIn order to address thermal activation in micromagnetics , we follow the procedure outlined by \nFischbacher et al.  [27]. Thermal fluctuations drive the magnetization over an  energy barrier of finite size. \nWe computed the minimum energy path , the most likely path the system takes over an energy barrier,  \nand the associated saddle point using a modified string  metho d [28]. Our implementation of the string \nmethod uses ene rgy minimization [19] and path truncation  [26].  \nWe quantify the effect s that reduce the ideal nucleation field, HN, in the magnets . The results are \npresented in Figure 4. In all three samples, the major reduction of the coercive field is caused by \nmisorientation and the weakly soft magnetic grain boundary.  The reduction by misorientation and the \nsoft-magnetic grain boundary phase  is ab out 50 percent of the ideal nucleation field for the Dy free \nmagnet. It is reduced to  about 40 percent of the ideal nucleation field for the fully Dy diffused sample.  \nThe reduction owing to demagnetizing effects is smaller than 10 percent. For our samples we find \n0.07 HN > demag ≥ 0.03  HN. The reduction of the coercive field by thermal fluctuations is slightly smaller . \n3. Discussion  \nWe showed that the switching field of polyhedral grains can be computed with a simplified model that \ntakes into account the local demagnetization field and applies the Stoner Wohlfarth model locally at \npoints close the edges of the grains. The good agreeme nt of the embedded Stoner Wohlfarth model \nwith conventional micromagnetics suggests that magnetization reversal is governed by strength and \norientation of the demagnetizing field near the edges of the magnet. At the point where magnetization \nreversal start s, the angle between the total field and the anisotropy axis is greater than zero. This \neffective misorientation, which is caused by the demagnetizing field, ex plains why the microstructural \nparameter  is smaller than 1 for perfectly oriented grains witho ut any defects.  \nWe can include a soft magnetic phase at the surface of the grains in the embedded Stoner Woh lfarth \nmethod. We replace (5) with an  analytic formula  that gives the angular dependence of the switching \nfields of particles with a soft magnetic d efect [18]. To test the method, we compare the results of the \nsimplified model with conventional micromagnetics. Figure 5 shows the coercive field as function of the \nwidth of a soft magnetic phase surrounding a Nd 2Fe14B cube with an edge length of 200  nm. Again, there \nis good agreement between the simplified model and conventional micromagnetics.  \nThe embedded Stoner -Wohlfart h model creates a map of the local switching field within a magnet. \nEncouraged by the good agreement between the embedded Stoner Wohlfarth model and conventional \nmicromagnetics we now apply the embedded Stoner Wohlfarth model to a large grained Nd 2Fe14B \nmagnet. The magnet has a size of 8×8×8 (µm)³ and consists of 512 grains. The grains are separated by a \n2 nm thick soft magnetic grain boundary phase  with K1 = 0. For each virtual Stoner -Wohlfarth particle \nthe switching field is computed. The switching field is mapped by color onto the grain surfaces . The \nresulting plot, show n in Figure 6 , gives  the distribution of the local switching field within the grain \nstructure. Magnetization reversal will be initiated where Hsw is lowest.  \nIn the above example the weakes t points of the magnet are next to the grain boundaries. This may \nchange by grain boundary diffusion of Dy. As shown above the (Nd,Dy) 2Fe14B shell around the Nd 2Fe14B \ncore of the grain improves coercivity. This change of coercivity is associated with a shi ft of the spot at which the magnetization reversal starts. The magnetization configuration at the saddle point  shows the \nonset of thermally induced magnetization reversal . At the saddle point the magnetization rotates out of \nthe anisotropy direction by around 90 degrees. Thus , it can be visualized. The spot where magnetiza tion \nstarts is shown in Figure 7  for the three different levels of Dy diffusion. All three samples in Figure 7 have \nthe same grain structure. The only difference is the level of Dy diff usion. The grain structure and the \ndiffusion level s for the 3 simulations are given in Figure 3. The Nd 2Fe14B core has the following material \nparameters K1 = 4.3 MJ/m³, µ0Ms = 1.61 T,  and A = 7.7 pJ/m . The magnetization and exchange constant \nof the 3 nm th ick grain boundary phase are µ0Ms = 0.5 T  and A = 7.7 pJ/m .  The material parameters of \nthe (Nd,Dy) 2Fe14B shell are K1 = 5.17 MJ/m³, µ 0Ms = 1.15 T  and A = 8.7 pJ/m . \nIn the magnet without Dy  magnetization reversal is initiated at a grain boundary at the outer edge of the \nmagnet. The location of the saddle point is shown in the image on the left -hand side of Figure 7.  When \nthe Dy containing shell cover s the outer grains the weakest spot shifts inside. The shell has a thickness of \n5 nm and covers only the grains which are located at the surface of the magnet. Magnetization reversal \nstarts at a grain boundary junction in the middle of the magnet. The image in the center of Figure 7 \nshows a slice through  the sample. In the 3rd simulation the Dy containing shell covers all grains . As a \nresult , the grain surfaces are no more weak points. The Dy containing shell for the grains located at the \nmagnet’s surface has a thickness of 10 nm. For the oth er grain its thickness is 5 nm. Nucleation of \nreversed domain starts inside  a grain.  In order to visualize the nucleus, we again show a slice through \nthe magnet given in the image on the right -hand side of Figure 7. Please note that the grain structure for  \nall three scenarios  is the same . However, in order to capture the initial nucleus, the slice for visualization \nis taken at different positions.  \n4. Conclusion  \nMicromagnetic simulation take into account the detailed shape and morphology of grains and \nintergranular phases as well as the local variation of the intrinsic magnetic properties depending on \nchemical compositions. Using  detailed micromagnetic  simulations  and model order reduction , we can \nquantify the reduction of the coercive field owing to fe rromagnetic boundary phases, demagnetizing \nfields, and thermal fluctuations. Successful hardening of magnets by grain boundary engineering shifts \nthe spot where reversed domains are formed from the grain boundary junction to the center of the \ngrains.  \nAckno wledgement  \nWork supported by the Austrian Science Fund (FWF): F4112 SFB ViCoM and Japan Science and the \nTechnology Agency (JST): CREST.  \nReferences  \n[1] Y. Yang, A. Walton, R. Sheridan, K. Güth, R. Gauß, O. G utfleisch, M. Buchert, B. -M. Steenari, T. Van \nGerven, P.T. Jones, K. Binnemans, J. Sustain. Metall. 3 (2017) 122 –149.  \n[2] M.H. Ghandehari, J. Fidler, MRS Proc. 96 (1987).  \n[3] K. Hirota, H. Nakamura, T. Minowa, M. Honshima, IEEE Trans. Magn. 42 (2006) 2909 –2911.  \n[4] S. Bance, J. Fischbacher, T. Schrefl, J. Appl. Phys. 117 (2015).  \n[5] T. Helbig, K. Loewe, S. Sawatzki, M. Yi, B. -X. Xu, O. Gutfleisch, Acta Mater. 127 (2017) 498 –504.  \n[6] H. Sepehri -Amin, T. Ohkubo, S. Nagashima, M. Yano, T. Shoji, A. Kato, T. Sc hrefl, K. Hono, ACTA \nMater. 61 (2013) 6622 –6634.  [7] Y. Murakami, T. Tanigaki, T.T. Sasaki, Y. Takeno, H.S. Park, T. Matsuda, T. Ohkubo, K. Hono, D. \nShindo, Acta Mater. 71 (2014) 370 –379.  \n[8] G.A. Zickler, J. Fidler, J. Bernardi, T. Schrefl, A. Asali, Adv.  Mater. Sci. Eng. (2017).  \n[9] M. Grönefeld, H. Kronmüller, J. Magn. Magn. Mater. 80 (1989) 223 –228.  \n[10] H. Kronmüller, K. -D. Durst, M. Sagawa, J. Magn. Magn. Mater. 74 (1988) 291 –302.  \n[11] D. Kinderlehrer, Ling Ma, IEEE Trans. Magn. 30 (1994) 4380 –4382.  \n[12] M.E. Schabes, J. Magn. Magn. Mater. 95 (1991) 249 –288.  \n[13] P. Gaunt, Philos. Mag. 34 (1976) 775 –780.  \n[14] D. Givord, P. Tenaud, T. Viadieu, IEEE Trans. Magn. 24 (1988) 1921 –1923.  \n[15] H. Kronmüller, M. Fähnle, Micromagnetism and the Microstructure of  Ferromagnetic Solids, \nCambridge University Press, 2003.  \n[16] J. Fischbacher, A. Kovacs, H. Oezelt, T. Schrefl, L. Exl, J. Fidler, D. Suess, N. Sakuma, M. Yano, A. Kato, \nT. Shoji, A. Manabe, AIP Adv. 7 (2017) 045310.  \n[17] D. Guptasarma, B. Singh, GEOPHYSICS 64 (1999) 70 –74. \n[18] H.J. Richter, J. Appl. Phys. 65 (1989) 3597 –3601.  \n[19] A. Samanta, W. E, Commun. Comput. Phys. 14 (2013) 265 –275.  \n[20] M.F. Carilli, K.T. Delaney, G.H. Fredrickson, J. Chem. Phys. 143 (2015) 054105.  \n[21] H. Si, ACM Trans. M ath. Softw. 41 (2015) 1 –36. \n[22] R. Grössinger, X.K. Sun, R. Eibler, K.H.J. Buschow, H.R. Kirchmayr, J. Phys. Colloq. 46 (1985) C6 -221-\nC6-224.  \n[23] W. Rave, K. Ramstöck, A. Hubert, J. Magn. Magn. Mater. 183 (1998) 329 –333.  \n[24] E.C. Stoner, E. Wohlfarth, P hilos. Trans. R. Soc. Lond. Ser. Math. Phys. Sci. (1948) 599 –642.  \n[25] H. Kronmüller, K. -D. Durst, G. Martinek, J. Magn. Magn. Mater. 69 (1987) 149 –157.  \n[26] S. Bance, B. Seebacher, T. Schrefl, L. Exl, M. Winklhofer, G. Hrkac, G. Zimanyi, T. Shoji, M. Yano , N. \nSakuma, M. Ito, A. Kato, A. Manabe, J. Appl. Phys. 116 (2014).  \n[27] J. Fischbacher, A. Kovacs, H. Oezelt, M. Gusenbauer, T. Schrefl, L. Exl, D. Givord, N.M. Dempsey, G. \nZimanyi, M. Winklhofer, G. Hrkac, R. Chantrell, N. Sakuma, M. Yano, A. Kato, T. Sh oji, A. Manabe, \nAppl. Phys. Lett. 111 (2017) 072404.  \n[28] L. Zhang, W. Ren, A. Samanta, Q. Du, Npj Comput. Mater. 2 (2016).  \n \n  Figure 1: Grain structures used for computing the microstructural parameters as function of the shape of \nthe grains. The volum e fraction of the grain boundary phase is 26 percent.  \nFigure 2: Comparison of conventional micromagnetics and the embedded Stoner Wohlfarth method. \nTop: Magnetization configuration obtained from conventional micromagnetics for perfect \nalignment and a grain  size of 90 nm. Top: µ 0Hext = -5.93 T. Bottom:  µ 0Hext = -5.96 T. Bottom : \nCoercive field as function of the grain size. The dashed  blue  lines are  the results of the \nembedded Stoner -Wohlfarth method; the orange symbols are the results of conventional \nmicrom agnetics. Full symbols refer to zero misalignment; the open symbols give the result with \nthe field rotated 8° off the anisotropy axes. The inset shows a map of the local Stoner -\nWohlfarth switching field.  \nFigure 3 : Model magnet for the analysis of thermal activation. Left:  Granular structure. Right:  Slice \nthrough the finite element mesh.  The grains a re separated by a 3 nm weakly ferromagnetic \ngrain boundary phase. The thickness of the (Nd,Dy) 2Fe14B varies  depending o n the sample.  \nFigure 4: Reduction of coercive field  in Dy diffused permanent magnets. Left: No Dy diffusion, Center: Dy \ncovers only  surface grains. Right: Dy covers all grains . \nFigure 5: Comparison of conventional micromagnetics  (squares)  and a simplified analytic model  \n(triangle)  for the reversal of a Nd 2Fe14B cube with a soft magnetic surface layer of varying \nthickness.  \nFigure 6: Map of the local switching field for a large grained Nd 2Fe14B magnet  for h = 0. The color map \ngives the local switching field c omputed by the embedded Stoner -Wohlfarth model. \nMagnetization reversal will start where the local switching field has its lowest value.  \nFigure 7: The initially reversed nucleus correspond s to the lowest saddle point between the rema nent \nstate and the  reversed state.  At the saddle point the magnetization rotates out of the \nanisotropy axes.  \n   \n \nFigure 1  \n \n \nFigure 2  \n \n \nFigure 3  \n \n \nFigure 4  \n \n \nFigure 5  \n \n \nFigure 6  \n \n \nFigure 7  \n"    },    {        "title": "1805.11831v2.Nonequilibrium_Magnetic_Oscillation_with_Cylindrical_Vector_Beams.pdf",        "content": "Nonequilibrium Magnetic Oscillation with Cylindrical Vector Beams\nHiroyuki Fujita1,\u0003and Masahiro Sato2\n1Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japany\n2Department of Physics, Ibaraki University, Mito, Ibaraki 310-8512, Japanz\n(Dated: March 21, 2022)\nMagnetic oscillation is a generic property of electronic conductors under magnetic \felds and widely\nappreciated as a useful probe of their electronic band structure, i.e., the Fermi surface geometry.\nHowever, the usage of the strong static magnetic \feld makes the measurement insensitive to the\nmagnetic order of the target material. That is, the magnetic order is anyhow turned into a forced\nferromagnetic one. Here we theoretically propose an experimental method of measuring the magnetic\noscillation in a magnetic-order-resolved way by using the azimuthal cylindrical vector (CV) beam, an\nexample of topological lightwaves. The azimuthal CV beam is unique in that when focused tightly,\nit develops a pure longitudinal magnetic \feld. We argue that this characteristic focusing property\nand the discrepancy in the relaxation timescale between conduction electrons and localized magnetic\nmoments allow us to develop the nonequilibrium analog of the magnetic oscillation measurement.\nOur optical method would be also applicable to metals the under ultra-high pressure of diamond\nanvil cells.\nINTRODUCTION\nWhen an electric conductor is under a strong magnetic \feld, the electronic band structure is reconstructed to\nbe Landau levels, and the isoenergy surface of electrons in the momentum space reduce into the so-called Landau\ntubes. For a \fxed Fermi energy, these Landau tubes and their \feld dependence cause oscillations of various electronic\nproperties as a function of the external magnetic \feld [1, 2]. If we focus on the electronic conductivity, the oscillation\nis called the Shubnikov-de Haas e\u000bect and if we on the spin polarization (or magnetic susceptibility), that is called\nthe de Haas-van Alphen e\u000bect. In the latter case, the oscillation frequency is determined by the area of the extremal\norbit on the Fermi surface. Hence, by measuring the oscillation while changing the direction and the strength of the\nmagnetic \feld, we can know the Fermi surface in the full Brillouin zone.\nAlthough developed decades ago, the magnetic oscillation is still an vital tool in condensed matter physics [3{\n12]. For example, recent discovery of the bulk-like quantum oscillation in a Kondo insulator SmB 6[4] triggered\nstudies [13{15] searching for the novel charge-neutral Fermi surface [16{19].\nWhen the target system is a non-magnetic conductor, the magnetic oscillation is indeed a powerful tool. However,\nif we are to apply that to conducting magnets with localized magnetic moments, a problem arises. In this case, the\nstrong static magnetic \feld applied to measure the oscillation itself a\u000bects the magnetic structure. Therefore even\nif we are interested in the electronic structure of antiferromagnetic, ferrimagnetic, or some non-collinear magnetic\nstates, there is a high possibility that a forced ferromagnetic state [Fig. 1( a)] is created by the applied magnetic \feld.\nEven for the collinear magnetic states, the use of the magnetic oscillation is not e\u000bective. Under the strong magnetic\n\feld, the collinear magnetic moments point to the direction of the external \feld. As a result, we cannot obtain the\nfull pro\fle of the Fermi surface because the magnetic oscillation only gives the information about the Fermi surface\ncross section perpendicular to the applied magnetic \feld.\nDue to this, the magnetic oscillation is not suitable for the study of the \feld-induced phase transitions where the\nstrength or direction of the magnetic \feld have drastic impact on the electronic structure, though there are a number\nof such materials of our interest. A notable example is pyrochlore iridates where various electronic structures (metal,\ntopological semimetal, and insulator) appear depending on the magnetic structure determined by the direction of the\nexternal magnetic \feld [20, 21]. Another example of the \feld-induced transition is in thin \flms of chiral magnets where\nwe observe helical magnetic phase, skyrmion crystal phase, and ferromagnetic state as depending on the strength of\nthe external magnetic \feld applied vertically to the \flm [22{24]. In this paper, we propose an extension of the\nmagnetic oscillation measurement applicable to those \feld-sensitive materials.\nThe key idea is to exploit the di\u000berence in the relaxation timescale between the conduction electrons and localized\nspins [25{27]. The energy scale of conduction electrons is of the order of electron volt (eV) while that of spin\nsystem is at most of meV. As a result, even though the electrons reach their thermal equilibrium state within 50-500\nfemtoseconds (fs) after the excitation, it takes more than 1 or 10 picoseconds (ps) for spins to follow that change.\nTherefore, if we apply a magnetic \feld pulse of far-infrared or THz frequency (whose timescale is of the order of\n100 fs to 1 ps), as Fig. 1( b) shows, conduction electrons would follow the change of the magnetic \feld through the\nrapid equilibration to the Landau levels before localized spins respond to that [see also Fig. 1( c)]. Then if we measurearXiv:1805.11831v2  [cond-mat.str-el]  5 Oct 20182\nFIG. 1. (a): Schematics of conventional magnetic oscillation measurement performed to a conducting magnet. Arrows on\nvertices of the lattice represent localized moments and the thick black arrows do the static magnetic \feld perpendicular to the\nlattice. Irrespective to the original spin texture, the static \feld makes the system to be a forced ferromagnet. ( b): Schematics\nof laser-based nonequilibrium measurement proposed in this paper. We apply a cylindrical vector (CV) beam (explained later)\npulse to form the Landau levels to which electrons relax. The latter method has the sensitivity to the magnetic structure of\nthe initial state. Panel ( c) shows the hierarchy of the scattering times in solids [25{27].\nthe conducting-electron spin polarization (or the total magnetization) of this nonequilibrium state, we will see the\nmagnetic oscillation of the electrons under the in\ruence of the initial magnetic structure.\nThe problem is that if we use the conventional Gaussian laser pulses to apply the magnetic \feld, there inevitably\naccompanies the electric \feld which strongly excites and heats the electron system, smearing out the Landau tube\nstructure we are interested in. Therefore, what we need is thus a source of a \\pure magnetic \feld\" without accom-\npanying the electric \feld in the optical frequency regime. Equipments used for NMR or ESR experiments cannot\ngenerate such a high-frequency magnetic \feld. As we see below, what is called the azimuthal cylindrical vector (CV)\nbeam meets the above criterion.\nFIG. 2. Snapshots of the in-plane ( xandy) components of the electric \feld of ( a) linearly polarized Gaussian beam, ( b) linearly\npolarized Optical vortex with unit orbital angular momentum, ( c) azimuthal CV beam, and ( d) radial CV beam. The \feld\namplitude of the Gaussian beam is the strongest at the center while beams ( b)\u0000(d) have vanishing in-plane components there\ndue to their topological nature. The size of the arrows in the \fgure corresponds to the laser amplitude at each point. We show\nthe de\fnition of ( \u001a;\u001e) in the cylindrical coordinate for later use.3\nVECTOR BEAM\nIn this section, we review the properties of the CV beam and how can they be exploited to the observation\nof the magnetic oscillations. In modern optics, a class of laser beams called the topological beams such as optical\nvortices [28, 29] and CV beams [30{32] are intensively studied experimentally. These beams are characterized with the\ntopologically nontrivial spatial pro\fle originating from either the spiral phase structure (optical vortices) or vortices\nin the polarization vector (CV beams) and can be generated by using holograms, structured phase plates, and so\non [29, 32]. Mathematically, CV beam can be seen as a superposition of optical vortices.\nIn Fig. 2, we compare the conventional Gaussian beam with those topological beams. The arrows in the \fgure\ncorrespond to the in-plane components of the electric \feld of those beams propagating perpendicularly to the x-y\nplane. We see that topological beams have vanishing in-plane components at the center re\recting their spatial pro\fle\nwhile those of the Gaussians beam are the strongest there. Optical vortices carry an orbital angular momentum (not\na spin angular momentum) and their applications to condensed matter physics have begun to be actively explored in\nvery recent years [33{39]. Applications of CV beams to solid state physics are, on the other hand, not explored well.\nAccording to Ref. 30, the spatial pro\fles of the electric \feld Eand the magnetic \rux density Bof the azimuthally\npolarized CV beam [Fig. 2( c)] propagating in a vacuum are, in the cylindrical coordinate ( \u001a;\u001e;z ), given by\nE\u001a(\u001a;\u001e;z ) =Ez(\u001a;\u001e;z ) =B\u001e(\u001a;\u001e;z ) = 0 (1)\nE\u001e(\u001a;\u001e;z ) = 2AZ\u000b\n0sin\u0012f(\u001a;\u0012;z; 1)d\u0012;\nB\u001a(\u001a;\u001e;z ) =\u00002\nicAZ\u000b\n0sin\u0012cos\u0012f(\u001a;\u0012;z; 1)d\u0012;\nBz(\u001a;\u001e;z ) =2\nicAZ\u000b\n0sin2\u0012f(\u001a;\u0012;z; 0)d\u0012\nf(\u001a;\u0012;z;n ) = cos1\n2(\u0012)`0(\u0012)Jn(k\u001asin\u0012)eikzcos\u0012\nwherecis the speed of light in a vacuum. The constant \u000bspeci\fes the size of the entrance pupil and Agives the\n\feld amplitude. Here Jn(x) is the Bessel function and kis the wavenumber. We see that at \u001a= 0 \felds are vanishing\nexcept the z-component of magnetic \feld. The details of the apodization function `0(\u0012) and the choice of \u000bdepend\non the pupil but the above property holds irrespective to its choice.\nAs a whole, the \feld distribution of the tightly focused azimuthally polarized CV beam is given as Fig. 3. Following\nRef. 30, we take the pupil apodization function to be: `0(\u0012) = exph\n\u0000\f2\u0000sin\u0012\nsin\u000b\u00012i\nJ1\u0000\n2\fsin\u0012\nsin\u000b\u0001\nand take\f, the ratio of\nthe radius of the pupil and the beam waist to be 1 :5. The parameter \u000bis de\fned as \u000b= sin\u00001(NA=n) wheren= 1:0\nis the refractive index of the vacuum and NA = 0 :95 is the numerical aperture of the lens. There exists a region where\nthe \\longitudinal\" magnetic \feld becomes dominant over the electric \feld (and other components of the magnetic\n\feld) [see Fig. 3 ( b), (c)]. Therefore, if a sample with its size well smaller than the wavelength is placed at the focus,\nwe can virtually apply the desired \\pure magnetic \feld\" for the nonequilibrium magnetic oscillation measurement [see\nFig. 1 (b)]. We stress that as the focusing becomes tighter, the longitudinal \feld does more prominent. The temporal\npro\fle of the electromagnetic \felds is obtained just by multiplying exp( \u0000i!t) to Eq. (1) and taking their real parts.\nThe longitudinal part of the magnetic \rux density Bzoscillates in time.\nMAGNETIC OSCILLATION OF CONDUCTING MAGNETS\nWe have discussed how the CV beam can be utilized to extend the magnetic oscillation measurement. In the\nfollowing, taking a simple concrete model, we see how the magnetic oscillation signal depends on the con\fguration of\nlocalized magnetic moments. We consider the following square lattice tight-binding model under the CV beam whose\nelectrons are coupled with localized moments through the exchange coupling. The Hamiltonian is given as\nH=\u0000tX\nhr;r0i;\u001bcy\nr;\u001bcr0;\u001be\u0000ie\n~Rr0\nrA(x)\u0001dx\n\u00002\u0016BBX\nrsz\nr\u00002JexX\nr;\u000b;\fmr\u0001sr: (2)4\nFIG. 3. (a), (b): Focusing of the Gaussian beam and the azimuthal CV beam with a high numerical aperture lens. The\ncharacteristic spatial pro\fle of the polarization vector of the azimuthal beam results in the vanishing electric \feld and the\ngrowth of the longitudinal magnetic \feld near the focus. ( c) Intensity distribution of the azimuthal beam in the focal plane at\nz= 0. The horizontal axis is the distance from the center of the focal plane.\nThe symbol cr;\u001bis the annihilation operator of a conduction electron at site rwith spin\u001b, and the dimensionless\nvectormris the localized magnetic moment at site r. The \frst term describes the nearest-neighbor hopping with\namplitudetwhich is typically of the order of eV. The magnetic \rux density Bof the beam is introduced by the Peierls\nsubstitution with the vector potential A(r) = (By;0;0). The second term is the Zeeman coupling for the electron\nspins (sr)\u000b;\f=cy\nr;\u000b\u001b\u000b;\fcr;\f. Here the magnetic \rux density is treated to be static because, as we discussed before, a\npulse of the CV beam with su\u000eciently long duration works as a static one for conduction electrons. The third term is\nthe exchange coupling with the coupling constant Jex. Through the exchange coupling, conduction electrons feel the\ne\u000bective magnetic \feld in the direction of the localized magnetic moment mrat siter. The exchange coupling Jex\ntypically takes the value of the order of sub eV to eV. For example, in pyrochlore iridates R 2Ir2O7,Jexis the exchange\ncoupling between fanddorbitals and is considered to be around 5 % of the hopping t[40]. If we regard the Jexas\nthe Hund coupling in transition-metal-compounds like Mn oxides, its energy scale is eV [41, 42]. Below we consider\nthree di\u000berent magnetic structures; ferromagnetic [ mr= (0;0;1) for allr], antiferromagnetic, and ferrimagnetic. For\nthe antiferromagnetic and ferrimagnetic cases, we divide the system into two sublattices AandBand then de\fne\nmr2A= (0;0;1),mr2B= (0;0;\u00001) for antiferromagnetic and mr2A= (0;0;1),mr2B= (0;0;\u00000:5) for ferrimagnetic\ncases. In the following, we take t=\u00003,Jex= 2 in the unit of electron volt and the lattice constant to be a= 5\u0017A. We\nmeasure the magnetic \rux density in the unit of Tesla. We again emphasize that we are assuming that the magnetic\nmomentsmrare independent on the laser magnetic \rux density Bdue to their timescale discrepancy.\nThe conducting-electron spin polarization of the model Eq. (2) at zero temperature is given as:\nhsz\ntotiB=X\nEn<EFhEnjsz\ntotjEni; (3)\nwheresz\ntot=P\nrsz\nris thezcomponent of the total conducting-electron spin, and jEniis the single-electron eigenstate\nwith eigenenergy En. We denote the Fermi energy as EF. Assuming the discrepancy in the relaxation timescale, we\nstudy this electron spin polarization as a function of the applied magnetic \feld of the CV beam.\nFigure 4 summarizes the calculation on a square lattice with linear dimensions Lx=Ly= 30a. Panels (a1-a3)\nand (b1-b3) are the results for the di\u000berent values of the Fermi energy EF=\u00001:5 andEF=\u00002:5 respectively. In\nFig. 4 (a1;b1), we present the Fermi surfaces for each magnetic structure. The magnetic oscillations of the electron spin\npolarization as a function of the inverse magnetic \rux density are shown in Fig. 4( a2;b2) and are Fourier transformed\nas shown in Fig. 4 ( a3;b3). In our setup with t=\u00003 andJex= 2, the band edge is at EF=\u00001 for the ferrimagnetic\nandEF=\u00002 for the antiferromagnetic cases. Therefore, as we see in Fig. 4 ( a1), there is no Fermi surface at\nEF=\u00001:5 for the antiferromagnetic case. The oscillation frequencies indeed depend on the magnetic structure and\nare consistent with the predictions from the Fermi surface cross-section (vertical lines) [1, 2]:\n\u000e\u00121\nB\u0013\n=2\u0019e\n~Sf: (4)\nThe left-hand side is the oscillation period of the magnetic susceptibility as a function of 1 =BandSfis the area of\nthe extremal orbit on the Fermi surface perpendicular to the applied magnetic \feld. Due to the small system size of5\nour calculation, here we are taking an arti\fcially strong magnetic \feld (remember that the Landau level degeneracy\nis determined by the total \rux penetrating the entire system). In realistic situations, as is experimentally veri\fed,\nmagnetic \rux density of O(1-10) Tesla would be su\u000ecient enough to measure magnetic oscillations.\nThe result of Fig. 4 clearly shows that the magnetic oscillation signal measured with the azimuthal CV beam\nwould give the Fermi surface geometry re\recting the magnetic order. Thus, by measuring the spin polarization while\nchanging the peak intensity and direction of the incident CV beam, we can read out the information of the full Fermi\nsurface geometry in a magnetic-order-resolved way. If we use the magnetic oscillation in the conventional setup with\nthe static magnetic \feld, we would obtain the signal in the ferromagnetic case irrespective to the actual magnetic\norder.\nFIG. 4. Calculation results of the magnetic oscillation of the model (2) on a lattice with linear dimensions Lx=Ly= 30a\nfor (a1-a3):EF=\u00001:5 and (b1-b3):EF=\u00002:5. We consider three (ferromagnetic, antiferromagnetic, and ferrimagnetic)\nmagnetic orders. ( a1;b1): Fermi surface structures for each magnetic order. ( a2;b2): \feld dependence of the electron spin\npolarization \u0001hsz\ntotiB=hsz\ntotiB\u0000hsz\ntoti0. The origins are shifted for visibility. These oscillations are Fourier transformed to\nobtain panels ( a3;b3). The vertical lines in these panels correspond to the oscillation frequency calculated from the area of\neach extremal Fermi surface [see Eq. (4)].\nDISCUSSION\nHere we discuss several important issues related to our proposal and calculation. First, we consider the e\u000bect of\nthe exchange coupling on the dynamics of localized magnetic moments. As we mentioned, the exchange coupling\nJexis typically of the order of sub eV to eV. This seems to result in the fast dynamics of localized moments mr\nand break our assumption on the timescale discrepancy. The point is, though the exchange coupling itself has rather\nhigh energy, the laser-induced change in the electron spin polarization \u0001 hstotiB=hstotiB\u0000hstoti0is very small [at\nmost O(10\u00003) of the saturating value even for O(10) Tesla as is read out from Fig. 4 ( a2,b2)]. The laser-induced\ndynamics of localized moments is determined by the e\u000bective magnetic \feld Jexhsri, not the exchange coupling Jex\nitself. Although Jexis large as it is, the change in the local e\u000bective magnetic \feld is reduced by the small factor\n\u0001hsriB=hsriB\u0000hsri0. When the initial state has no net spin polarization hsri0= 0, the e\u000bective \feld determining\nthe the dynamics of the localized moments is given by Jex\u0001hsriBand thus the dynamics turns out to be slow. On\nthe other hand, when there is non-vanishing hsri0, after the laser irradiation the spins mrstart to precess around the\ne\u000bective magnetic \feld determined by hsriB. In this case, the period of the precession is indeed not slow. However,\nbecause \u0001hsriBis very small, namely the directions of the e\u000bective magnetic \felds before and after the application of\nthe CV beam are almost identical, the amplitude of the magnetization precession is quite small. We can then virtually\nneglect the change in the spin structure. Therefore, even though the exchange coupling is strong, the laser-induced\ndynamics is slow, or the magnetization change can be ignored.6\nTo observe the magnetic oscillation, typically we need a magnetic \feld pulse with the magnetic \rux density of\nO(1-10) Tesla. As we mentioned, the frequency of the incident beam should be around THz or far-infrared. In the\nTHz region, the magnetic \feld pulse of O(1) Tesla is just becoming possible, and it would take a while to reach O(10)\nTesla. On the other hand, in the far-infrared region, it is much easier to achieve that. For example, CO 2lasers have\nbeen widely used for industrial purposes requiring the high intensity such as laser ablation or welding. Although CO 2\nlaser itself has the central wavelength of 10 \u0016m and its pulse duration may be too short (approximately 30 fs) for\nelectrons to get relaxed, by using an array of half-cycle pulses we can synthesize a longer duration pulse. Moreover,\nwe can use CO 2lasers to pump other molecular lasers with longer wavelength such as CH 3OH lasers [43].\nThe e\u000bect of heating by the laser irradiation should also be considered. In the magnetic oscillation measurement,\nwhen the (electron) temperature becomes comparable with the bandwidth, the oscillation signal becomes vague. The\nprimary source of the heating under irradiation of lasers is the coupling between the electric \feld and the electrons.\nSince the CV beam has a suppressed electric \feld near the focus, the heating is much weaker compared to that caused\nby the Gaussian beam. Nevertheless, there will exist a small change in the temperature. Therefore, applying our\nmethod to extremely heavy fermion systems with very small electron bandwidth could be problematic.\nOn top of the pump laser pulse, the way of measuring magnetic oscillations is also important for our setup. Since\nour proposal stands on the nonequilibrium property of the electron-spin coupled system, we have to measure the\nnonequilibrium spin polarization in the optical timescale. One way to achieve this is to use the magneto-optical\ne\u000bect such as Kerr or Faraday e\u000bects which is already a common way of measuring the magnetization. To measure\nthe nonequilibrium magnetic oscillation, the temporal resolution of those magneto-optical measurements should be\nshorter than the CV pulse duration, thus we have to use the ultrashort femtosecond pulses to probe the spin density.\nWe can also measure physical quantities other than the spin density. Because the magnetic oscillation originates\nfrom the Landau tube structure, there should appear oscillating behaviors in any physical quantities related to the\nelectronic structure. For example, optical conductivity would be an option.\nIn this paper, we have focused on the magnetic materials and their Fermi surfaces. However, the applicability of\nthe proposed method is not limited to that. A notable example is the Fermi surface measurement of materials under\nultra-high pressure. In producing ultra-high pressure stronger than GPa, commonly the diamond anvil cell [44] is\nused, where the sample is placed in between two polished diamonds and pressed. In such a situation, performing the\nconventional de-Haas type measurement is experimentally di\u000ecult. Hence, Fermi surface of materials under ultra-\nhigh pressure is, even for non-magnetic metals, hard to be probed. On the other hand, since diamonds are highly\ntransparent in the wide range of frequency including THz to far-infrared, our laser-based method is applicable there\nand will provide a new probe of electronic properties of generic metals under ultra-high pressure.\nCONCLUSION\nIn this paper, we propose an extension of the magnetic oscillation measurement for the Fermi surface geometry using\nthe azimuthal cylindrical vector beams. In contrast to the conventional method using the static magnetic \feld, the\nproposed protocol allows us to apply that to conductors with localized magnetic moments with an arbitrary magnetic\norder. Hence, we can study the \feld-induced phase transitions in magnetic materials such as pyrochlore iridates,\nchiral magnets, and so on. 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F. is supported by Advanced Leading Graduate Course\nfor Photon Science (ALPS) of Japan Society for the Promotion of Science (JSPS) and JSPS KAKENHI Grant-in-Aid\nfor JSPS Fellows Grant No. JP16J04752. M. S. was supported by Grant-in-Aid for Scienti\fc Research on Innovative8\nArea, Nano Spin Conversion Science (Grant No.17H05174), and JSPS KAKENHI (Grant No. JP17K05513 and No.\nJP15H02117). A part of the computation in this work has been done using the facilities of the Supercomputer Center,\nthe Institute for Solid State Physics, the University of Tokyo."    },    {        "title": "1806.02576v1.Composite_Skyrmion_bags_in_two_dimensional_materials.pdf",        "content": "Composite Skyrmion bags in two-dimensional materials\nDavid Foster,1Charles Kind,2Paul J. Ackerman,3Jung-Shen B. Tai,3Mark R. Dennis,1, 4and Ivan I. Smalyukh3, 5, 6\n1H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK\n2School of Mathematics, University of Bristol, Bristol BS8 1TW, UK\n3Department of Physics and Soft Materials Research Center,\nUniversity of Colorado, Boulder, Colorado 80309, USA\n4School of Physics and Astronomy, University of Birmingham, Birmingham B12 2TT, UK\n5Department of Electrical, Computer, and Energy Engineering,\nMaterials Science and Engineering Program, University of Colorado, Boulder, Colorado 80309, USA\n6Renewable and Sustainable Energy Institute, National Renewable Energy\nLaboratory and University of Colorado, Boulder, Colorado 80309, USA\n(Dated: June 8, 2018)\nSkyrmions are particle-like topological excitations, studied in various condensed\nmatter systems and models of high-energy physics (HEP). They occur as stable spin\ntextures in certain planar magnetic materials [1{4] and as con\fgurations in chiral\nnematic liquid crystals [5], having been originally proposed as model of atomic nuclei\n[10]. Since magnetic Skyrmions can be accelerated with a current [6{8], they have\nthe potential to encode bits in low-power magnetic storage devices [9]. Drawing on\ntechniques from HEP, we demonstrate that magnetic and liquid crystal Skyrmions\ninteract like orientation dependent, localised particles, explaining previously observed\nSkyrmion behaviour. This interaction motivates the construction of Skyrmion bags :\ntextures of high topological degree which we realise experimentally in liquid crystals, and\nin magnetic materials by computer simulations. These Skyrmion bags con\fgurations\nare nested multiple Skyrmions, which act like single Skyrmions in pairwise interaction,\nand under the in\ruence of a current in magnetic materials. These results emphasize\nequivalent Skyrmion behaviour in di\u000berent physical systems, and suggest new, high-\ndensity magnetic memory storage devices based on Skyrmion bags.\nTopological invariants, such as the number of holes or windings in a texture, can be associated with the quantum\nnumbers in a physical system. This idea led T.H.R. Skyrme, almost 60 years ago, to propose a topological model for\natomic nuclei which has so-called Skyrmions as minimum energy con\fgurations which are localised in space and are\nstabilised by an integer valued topology number called the degree , which corresponds to baryon number. Subsequently,\nWitten [11] derived the Skyrme model as a low energy e\u000bective model of quantum chromodynamics, in the large colour\nlimit. Skyrmions have also been observed in 2D magnetic materials, where n(x) is the orientation of the magnetisation\nat the pointx[1{4]. Trains of magnetic Skyrmions have been proposed as a method of encoding data in racetrack\nmemory technology [8, 12{14] (potentially applicable to biological neural engineering [15]). Skyrmions have also\nbeen experimentally observed in chiral nematic liquid crystal \flms [5], which have a similar theoretical description to\nmagnetic Skyrmions, where the liquid crystal molecules can be either polar vectors [16] (as the magnets) or directors,\nwhich do not distinguish \u0006n. A full theory of Skyrmion-Skyrmion interaction in magnetic materials and liquid\ncrystals, necessary to develop applications, has so far been lacking.\nHere we build on the relationship between Skyrmions in the three di\u000berent areas of physics. Extending a 2D\ncalculation from High Energy Physics (HEP) [17, 18], we show that the interaction between magnetic and liquid\ncrystal Skyrmions is a Yukawa type repulsion, consistent with a previously investigated numerical model [19]. We\nverify this in chiral nematic liquid crystals by experiment, and numerically with simulations of magnetic materials.\nThis interaction turns out to depend on the Skyrmions' relative orientation, explaining several previously observed\nphenomena. This leads us to investigate the viability of Skyrmion con\fgurations with higher topological degree, which\nwe call Skyrmion bags . We develop a common mathematical description for the behaviour of Skyrmion bags, and in\nmagnetic simulation and liquid crystal experiments show that these are stable and interact like single Skyrmions of\nlarge radius. From dynamical micro-magnetic simulations of bags in a racetrack device, we \fnd magnetic Skyrmion\nbags can be driven by a current similar to single magnetic Skyrmions. This suggests the possibility of Skyrmion\nbags racetrack memory devices, which would be more robust and have a higher memory density than conventional\nSkyrmion racetrack memory.\nFor an appropriate parameter regime, both chiral liquid crystals and magnetic con\fgurations can be described by\nthe energy functional\nE=Z\u0012J\n2@in\u0001@in+Dn\u0001(r\u0002n)\u0013\nd2x+Eext; (1)arXiv:1806.02576v1  [cond-mat.mtrl-sci]  7 Jun 20182\nFigure 1: Magnetic and liquid crystal Skyrmions. (a-c) Three conformations of a degree one hedgehog\nSkyrmion (2) with \r= 0;\u0019=4;\u0019=2 respectively. Note that changing the internal orientation rotates the colours in\nthe plane. In each case the set of arrows points in each direction on the 2-sphere once. (d-f) Three conformations of\na hedgehog Skyrmion (2) with \r= 0;\u0019=4;\u0019=2 respectively for the chiral nematic liquid crystal director \feld. Here\nthe directors do not have arrow heads, so opposite points on the 2-sphere are identi\fed. (g)The colour scheme for\nthe vectors in (a-c) , with hues and brightness determined by the position of non the 2-sphere, viewing from the\nside and from above. (h)Theoretical computer-simulated optical micrograph showing a liquid crystal Skyrmion in a\npolarizing optical microscope. (i)Dynamical MuMax3 simulation of two magnetic Skyrmions repelling (points),\nwith the dipole interaction potential \ftted (3). Typical Skyrmion diameter in this simulation regime is 110 nm. (j)\nExperimental measurement of two liquid crystal Skyrmions repelling (points), with the dipole interaction potential\n\ftted (3).\nwhereJandDare coupling constants, and the second term is the Dzyaloshinskii-Moriya interaction (DMI)[20{23]\n(further discussion of the correspondence between magnetic and liquid crystals may be found in the Supplementary\nInformation). For magnetic systems, Eextis the integral over the plane of \u0000B\u0001n, whereB=\u0016eis an external\nmagnetic \feld, normal to the material. In chiral nematic liquid crystals, Eextdepends on the boundary conditions\nof the cell, which causes con\fnement and has a similar e\u000bect to a magnetic \feld. Skyrmions are minimum energy\ncon\fgurations of the energy functional (1). An important tool in understanding Skyrmions is the so-called hedgehog\nansatz [20],\nn(x) = (sinf(r) cos(m\u0012+\r);sinf(r) sin(m\u0012+\r);cosf(r)); (2)\nwherer2=x2+y2,f(r) is some monotonic function satisfying f(0) =\u0019,f(1) = 0,\u0012is the polar angle of nand\nm2Zis the topological degree. \ris the internal orientation of the Skyrmion (sometimes called the chirality). It is\nwell understood [24, 25] that the Skyrmion with m= 1;\r=\u0000\u0019\n2is the minimiser over all homotopy classes. Along\nwith experimental observations [3, 4, 26], this shows that Skyrmions appear in collections of single-degree excitations.\nThus in both magnetic systems and liquid crystals, we only need to consider the asymptotic interaction between two\nwell separated single Skyrmions.\nThe con\fguration of a pair of Skyrmions, each described by the hedgehog ansatz, can be determined using a sum-\nmation approximation. The asymptotics of such an approximation, following techniques from HEP [17] (details given3\nin the Supplementary Information), shows that two Skyrmions with internal orientations \rAand\rBand separation\nR, have the e\u000bective interaction potential\nVint\u0019C2\u0016\n\u0019Jcos(\rA\u0000\rB)e\u0000p\n\u0016=JR\np\nR; (3)\nwhere\u0016=jBjandCis a \ftting parameter. This interaction depends on the Skyrmions' relative internal orientation,\n\rA\u0000\rB, suggesting in fact that Skyrmions attract when j\rA\u0000\rBj<\u0019= 2. Statically, this is counteracted by the DMI\nwhich supports \rA=\rB=\u0000\u0019\n2, hence Skyrmions usually repel. The potential also shows that Skyrmions interact\nless whenjBj=\u0016is large; but this would give rise to sharply located Skyrmion which are potentially unstable.This\ninteraction can be used, in a pointwise manner, to model the interaction between an ensemble of Skyrmions as\npoint-like particles.\nFigure 2: Skyrmion bag con\fgurations. (a-d) Skyrmion bags S(1) toS(4), with magnetic MuMax3 simulations\non the top row, and their experimental liquid crystal counterparts in the middle row. The magnetic and liquid\ncrystal bag con\fgurations are clearly equivalent. (e)the two stable conformations of the S(13) bag, which are\nanalogous to the two best packings of 13 disks [32] (discussed in the Supplementary Information). Examples of more\ncomplex con\fgurations are shown in (f)S(59) bags, and (g)recursiveS(S(3)S(3)S(3)) bags of topological degree 7.\nWe veri\fed this interaction potential using micromagnetic simulations of two Skyrmions, using the MuMax3 \fnite-\ndi\u000berence GPU accelerated code [27]. These simulations, incorporating the energy functional in equation (1) and\nnonlocal demagnetisation e\u000bects, resulted in the repulsive interaction force shown in \fgure 3 a, which agrees with the\nforce derived from the potential energy in equation (3) (\ftted for large separation). Experimental measurements of\nthe repulsion between a pair of interacting Skyrmions in a chiral nematic liquid crystal is shown in \fgure 1 j. Again,\nthere is excellent agreement with a \ft of the interaction Vint.\nThe form of our derived potential Vintaccounts for various previously observed phenomena. Signi\fcantly, the\norientation dependence explains the BiSkyrmion bound state [30] (each Skyrmion has opposite orientation), the\norientations of magnetic bubbles [28], and the relative orientation of Skyrmions in nano discs with respect to the\nboundary [29]. Our analysis also shows that the force is weaker for large separation than previously known [19].4\nFigure 3: Bag repulsion and Racetrack dynamics. (a) Dynamical MuMax3 simulation of two S(3) magnetic\nbags repelling (points), with the \ftted dipole interaction potential (3). Typical S(3) bag diameter in this simulation\nregime is 500 nm. (b)Experimental measurement of two repelling S(3) liquid crystal bags (points), with the \ftted\ndipole interaction potential (3) as in (a). The interaction in these two cases has exactly the same form\nasymptotically as single Skyrmions. (c)Two frames of a dynamical MuMax3 simulation of a collection of bags in a\nracetrack being pushed by a current. (d)A MuMax3 simulation of an encoding of the the word \\SKYRME\" using a\nsimple binary encoding of the alphabet where the outer bag holds the \frst four bits and the inner bag the next four\nbits and each full recursive bag represents a single letter and hence 8 bits of information.\nThe form of the potential in equation (3) suggests that a concentric Skyrmion anti-Skyrmion ( m=\u00001 in equation\n(2)) con\fguration with a relative internal orientation of \u0019, as in \fgure 2(a) might be stable. To create this con\fguration\nwithin MuMax3, we applied, in the region of the Skyrmion, a magnetic \feld opposite the external magnetic \feld,\nwhich caused it to grow. When the Skyrmion was su\u000eciently large, we turned o\u000b the magnetic \feld and then applied\na second seeding magnetic \feld aligned with the external \feld, B. Although this nested Skyrmion con\fguration is\nnot a global minimum energy con\fguration, it is a local minimum and is stable to large perturbations. We denote\nsuch `bag' con\fgurations of nanti-Skyrmions inside of a larger bag Skyrmion as S(n), which has total topological\ndegree 1\u0000n. Similarly, we created Skyrmion bags experimentally in chiral nematic liquid crystals using laser tweezers\nto stretch a Skyrmion to form the bag, and then used laser tweezers to place single Skyrmions inside of the bag. We\nrepeated this process, for both magnetic simulations and experimental chiral nematic liquid crystals (up to S(59)) as\nshown in \fgure 2, we also produced nested bags 2(g). Showing that liquid crystal and magnetic Skyrmion bags have\nsimilar con\fgurations. In the chiral nematic liquid crystal experiments described here, nis a director ( n=\u0000n), so\nthere is no distinction between Skyrmions and anti-Skyrmions.\nSkyrmion bags are the \frst example of stable composite Skyrmions in monochiral 2D materials. Their range of\npossible con\fgurations echoes the plethora of nuclei described by di\u000berent topological degree con\fgurations in the\noriginal Skyrme model. In particular the magnetic and liquid crystal con\fgurations in \fgure 2(a), having total topo-\nlogical degree zero, resemble a sphaleron [31] which plays a crucial role in nonperturbative calculations of electroweak\ntheories.\nSince isolated bags are circularly symmetric, we can apply the previous asymptotic interaction analysis (with\ndi\u000berent constants C). In \fgure 3 the numerical and experimental results, for to interacting S(3) bags, are compared\nwith the previous interaction potential Vint, showing that they interact like large Skyrmions. Since static (anti-\n)Skyrmions repel, the packing of (anti-)Skyrmions inside a static bag can be mapped onto the mathematical circle\npacking problem [32] (discussed in the Supplementary Information). This also explains the packing of Skyrmions in5\nnano discs [29].\nIn the proposed methods of racetrack memory, bits are encoded by Skyrmion separation. The potential Vint\nindicates that in fact Skyrmions in trains repel each other, eventually becoming uniformly distributed along the track,\nleading to data loss. Devices utilising Skyrmion bags, on the other hand, will not have this problem, as 0 ;1 and\nhigher integers can be encoded in di\u000berent integer ninS(n) bags, with no data stored in their separation. A train\nof Skyrmion bags, S(n1);S(n2);:::for di\u000berent integers njstoring bits, could be used as an alternative instead of\na train of single Skyrmions, and the bags can be packed together and driven along by a current. A full MuMax3\nsimulation of Skyrmion bags being accelerated along a race track, by a current, is shown in \fgure 3(c), where it can\nbe seen that the inner anti-Skyrmions force the bags along.\nThere are a variety of ways a Skyrmion bag might be utilised to store data. For instance, a Skyrmion bag could\nstore 4 bits of data, by containing 1 to 16 Skyrmions, with about 220nm of separation, requiring less than 1000\nnm (sizes of bags are considered further in the Supplementary information). To encode 4 bits of data with single\nSkyrmions would need 1320nm, hence bags could have a greater data density, as indicated in \fgure 3(c) to enocde\nthe word `SKYRME'.\nIt is known that near a Skyrmion, the local magnetoresistance varies [33], and this phenomena can be read by\ntunnelling spin-mixing magnetoresistance (TXMR) using a non-magnetic scanning tunnelling microscope tip. This\nhas the potential to `read' a Skyrmion and could be extended to read the contents of a Skyrmion bag. A device\ncould be made from an ultrathin metallic ferromagnetic material [9], where Skyrmions have been previously formed\nand experimentally driven. Skyrmions could be formed using an electric potential from a tip (previously realised\nexperimentally [34]), and the Skyrmions could then be forced to grow using a \fne magnetic \feld. The inner bag\nantiSkyrmions could then be formed, within the larger Skyrmion, by an electric \feld. A train of bags could then be\npushed with a current (as in our MuMax3 simulation in \fgure 3) and read with a TXMR head [33]. But here, due to\nthe di\u000bering bag contents, each site could encode more than a zero or a one, further increasing data density. Such a\nSkyrmion bag racetrack device would further mitigate drawbacks of conventionally proposed racetrack memory, since\nrepulsion or sticking on defects would not lead to data loss.\nIn summary, our theory of interactions demonstrates an analogues interaction between Skyrmions in HEP, magnetic\nsystems and liquid crystal \flms. Not only does the interaction's orientation-dependence describe previous observations\n[4, 19, 28{30], it has also led to the formulation of Skyrmion bags which opens up new possibilities for composite\nSkyrmion excitations in materials. Magnetic Skyrmion bags have the potential to extend Skyrmionics beyond binary\nbits, suggesting Skyrmion racetrack memory devices with a signi\fcantly higher possible storage density. More generally\nour overall approach hints at new physics in magnetic and liquid crystal motivated by theoretical approached in HEP.\nMETHODS\nMicromagnetic Simulations\nThroughout we performed magnetic simulations using the GPU-accelerated micromagnetic simulation program\nMuMax3 [27] with Landau-Lifshitz dynamics in the form,\n@n\n@x= ^\r1\n1 +\u000b2(n\u0002Be\u000b+\u000bn\u0002(n\u0002Be\u000b)); (4)\nwhere ^\ris the gyromagnetic ratio, \u000bthe dimensionless damping parameter, Be\u000bthe e\u000bfective \feld and nthe mag-\nnetisation unit vector. The simulations where performed with free boundary conditions. The e\u000bective \feld energy\nincludes contributions from exchange, anisotropy, DMI and applied \feld.\nThe simulation geometry is typically a 1024 \u00021024 nm2square of 1 nm thickness with a cell size of 2 \u00022\u00021 nm3\nalthough for \fner detail cell sizes of 1 nm3where used. Material parameters are uniformly throughout all simulations:\nsaturation magnetisation Msat= 900 kAm\u00001, exchange A= 10 pJm\u00001, damping\u000b= 0:5, DMID= 0:6mJm\u00002and\nuniaxial anisotropy Ku= 0:56MJm\u00003. The anisotropy is along the + zdirection. The material parameters where\ntaken from Woo et al. [6] and tuned to maximise the stability of the Skyrmion bags.\nLiquid crystal experiment\nMaterials and sample preparation.\nTo assure accessibility and a broad impact of our work, pentylcyanobiphenyl (5CB, from EM chemicals), a commonly6\nused and commercially available nematic liquid-crystal (LC) material, was doped with small amounts of chiral additive,\ncholesterol pelargonate (Sigma-Aldrich), resulting in a chiral nematic LC. The helicoidal pitch pof the LC mixture\nis determined by p= (\u0018\u0001c)\u00001wherecis the weight fraction of the additive and \u0018= 6:25\u0016m\u00001, the helical twisting\npower of the additive in 5CB. Con\fning glass substrates were treated with polyimide SE1211 (obtained from Nissan\nChemicals) to ensure vertical alignment of liquid crystal orientation at the LC-glass interface. Polyimide was applied\nto substrates by spin-coating at 2700 rpm for 30 s and then baked for 5 min at 90oC then 1 h at 180oC. LC cells with\ngap thickness of d= 10\u000020\u0016m were produced by sandwiching glass \fber segments in UV-curable glue. In cells where\nd=p\u00191, spontaneous and controllably generated structures corresponding to minima of free energy were generated\nand manipulated using laser tweezers, as detailed below.\nOptical generation of Skyrmions and Skyrmion bags.\nHolographic laser tweezers capable of producing arbitrary patterns of laser light intensity within the LC sample based\non an Ytterbium-doped \fber laser (YLR-10-1064, IPG Photonics, operating at 1064 nm) and a phase-only spatial light\nmodulator (P512-1064, Boulder Nonlinear Systems) integrated on an inverted microscope (IX81, Olympus) enabled\ngeneration of Skyrmions and Skyrmion bags and subsequent manipulations. Polarizing optical microscopy and video\nmicroscopy observations of LC Skyrmions was achieved with a charge coupled device camera (Grasshopper, PointGrey\nResearch) [5].\nData availability\nThe datasets generated during and/or analysed during the current study are available from the corresponding\nauthor on reasonable request.\n[1] Bogdanov, A. & Yablonskii, D. Thermodynamically stable \\vortices\" in magnetically ordered crystals. The mixed state of\nmagnets. Zh. Eksp. Teor. Fiz 95, 178-182 (1989).\n[2] Cortes-Ortuno, D. et al. Thermal stability and topological protection of skyrmions in nanotracks. Scienti\fc Rep. 7, 4060\n(2017).\n[3] M uhlbauer, S. et al., Skyrmion lattice in a chiral magnet. Science 323, 915-919 (2009).\n[4] Yu, X et al. Real-space observation of a two-dimensional skyrmion crystal. Nature 465, 901 (2010).\n[5] Smalyukh, I. I., Lansac, Y., Clark, N. A. & Trivedi, R. P. Three-dimensional structure and multistable optical switching\nof triple-twisted particle-like excitations in anisotropic \ruids. Nature Mater. 9, 139-145 (2010).\n[6] Woo, S. et al. Spin-orbit torque-driven skyrmion dynamics revealed by time-resolved X-ray microscopy. Nature Commun.\n8, 15573 (2017).\n[7] Iwasaki, J., Mochizuki, M., & Nagaosa, N. Current-induced skyrmion dynamics in constricted geometries. Nature Nanotech.\n8, 742-747 (2013).\n[8] Fert, A., Cros, V., & Sampaio, J., Skyrmions on the track. Nature Nanotech 8, 152{156 (2013).\n[9] Woo, S. et al. Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic\nferromagnets. Nature Mater. 15, 501-506 (2016).\n[10] Skyrme, T. H. R. A non-linear \feld theory. Proc. R. Soc. Lond. A 260, 127-138 (1961).\n[11] Witten, E. Global aspects of current algebra. Nucl. Phys. B 223422-432 (1983).\n[12] Schulz, T. et al. Emergent electrodynamics of skyrmions in a chiral magnet. 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B 87,214419 (2013).\n[20] Nagaosa, N. & Tokura, Y. Topological properties and dynamics of magnetic skyrmions. Nat. nanotech. 8,899{911 (2013).\n[21] Dzyaloshinsky, I. A thermodynamic theory of `weak' ferromagnetism of antiferromagnetics. J. Phys. Chem. Solids. 4,\n241-255 (1958).\n[22] Moriya, T. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120, 91-98 (1960) .\n[23] Fert, A & Levy, P. M. Role of anisotropic exchange interactions in determining the properties of spin-glasses. Phys. Rev.\nLett44, 1538 (1980).7\n[24] Melcher, C. Chiral skyrmions in the plane. Proc. R. Soc. A 470, 20140394 (2014).\n[25] Derrick, G. Comments on nonlinear wave equations as models for elementary particles. J. Math. Phys. 5, 1252-1254 (1964).\n[26] Kiselev, N. S., Bogdanov, A. N., Sch afer, R., & R o\u0019ler, U. K. Chiral skyrmions in thin magnetic \flms: new objects for\nmagnetic storage technologies?. J. Phys. D 44, 392001 (2011).\n[27] Vansteenkiste, A et al. The design and veri\fcation of MuMax3. AIP Advances 4, 107133 (2014).\n[28] Nagaosa N., Yu X. Z. and Tokura Y. Gauge \felds in real and momentum spaces in magnets: monopoles and skyrmions.\nPhil. Trans. R. Soc. A 370, 5806-5819 (2012).\n[29] Zhao, X. et al. Direct imaging of magnetic \feld-driven transitions of skyrmion cluster states in FeGe nanodisks. Proc. Natl.\nAcad. Sci. 113, 4918-4923 (2016).\n[30] Yu,. X. Z. et al, Biskyrmion states and their current-driven motion in a layered manganite. Nature Commun. 5, (2014).\n[31] Manton, N. S. & Klinkhamer, F. R. A saddle-point solution in the Weinberg-Salam theory. Phys. Rev. D ,302212 (1984).\n[32] Fodor, F. The densest packing of 13 congruent circles in a circle. Cont. Algebra and Geom. 431-440 (2003).\n[33] Crum, D. M., et al. Perpendicular reading of single con\fned magnetic skyrmions. Nature Commun. 68541 (2015).\n[34] Hsu, P. J. et al. Electric-\feld-driven switching of individual magnetic skyrmions. Nature Nanotech. 12, 123-126 (2017).\nAcknowledgements\nD.F. and M.R.D. acknowledge the fund by the Leverhulme Trust Research Programme Grant RP2013-K-009, SPOCK:\nScienti\fc Properties Of Complex Knots. DF also thanks P.M. Sutcli\u000be, M. Gradhand, A. Bogdanov and W. Zakrzewski\nfor comments. Experimental research at CU-Boulder (P.J.A., J-S. T. and I.I.S.) was supported by the U.S. Department\nof Energy, O\u000ece of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Award ER46921,\ncontract de-sc0010305.\nAuthor contributions C.K. performed the numerical analysis and D.F. performed the theoretical analysis, with\ninput from C.K. and M.R.D. C.K. and D.F. proposed the investigation. P.J.A. and J-S.T. performed the experiment\nwith suggestions from I.I.S. and D.F. I.I.S. provided experimental techniques and materials. P.J.A., J-S.T. and I.I.S\nanalysed experimental data. D.F., C.K. and M.R.D. prepared the manuscript with input from all authors.8\nSUPPLEMENTARY INFORMATION\nMagnetic and chiral nematic liquid crystals: The link between magnetic and chiral nematic liquid crystal\nSkyrmions can be seen from the Frank free energy,\nE=Z\u0012K11\n2(r\u0001n)2+K22\n2(n\u0001(r\u0002n))2+K33\n2(n\u0002(r\u0002n)) +q0K22n\u0001(r\u0002n)\u0013\nd2x;\nwhere on setting the bend, twist and splay energies to be equal, J=K11=K22=K33,D=K22q0, and the pitch\naxis toq0=2\u0019\np, and including an external magnetic \feld B, we derive the energy functional (1), of the particularly\nsuccessful magnetic Skyrmion model [20] for the spin \feld n(x).\nMinimum energy: The minimum energy Skyrmion con\fguration has been well studied [24], and it is a solution\nof the energy functional's corresponding Euler-Lagrange equations,\nJ@iin\u00002Dr\u0002n+B= 0: (S1)\nThe dot product of (S1) with ncan be used to express the energy functional as the interaction of nwith the external\nmagnetic \feld,\nE=\u00001\n2Z\nB\u0001nd2x: (S2)\nIt must be stressed that this expression can only be used to calculate the energy of a con\fguration, n, which is a\nsolution of (S1).\nMinimiser over all homotopy classes: On substitution the hedgehog ansatz separates angular and radial\ncomponents in the energy functional (1). The angular component can then be integrated out, giving the radially\ndependent energy functional,\nE=2\u0019Z\u00121\n2f02(r) +m2\n2r2sin2f(r)\n+Dsin(m\u0019) sin(\r+m\u0019)\nm\u00001\u0010\nf0+m\n2rsin (2f(r))\u0011\n\u0000\u0016cosf(r))rdr: (S3)\nUsing a scaling argument [25], r!\u0015r, it can be seen that there are no static solutions unlesssin(m\u0019) sin(\r+m\u0019)\nm\u00001is\nnon-zero. Hence, there are only stable solutions for m= 1 Skyrmions, a concise and detailed discussion of magnetic\nSkyrmion stability can be found in [24].\nFinding the minimum energy axial solution: In order to derive the minimum energy solution of (S3), we take\nits functional derivative,\nJrf00(r) +Jf0(r) +Dsin(\r) (1\u0000cos(2f(r))\n\u0000J\n2rsin (2f(r))\u0000\u0016rsin (f(r)) = 0; (S4)\nand then perform a numerical gradient \row on a lattice of 1000 points, with the boundary conditions f(0) =\u0019,\nf(1) = 0, shown in \fgure S2. This is the techniques used to \fnd the one dimensional solutions through the paper.\nApproximating Skyrmions as dipoles. Importantly, asymptotically to \frst order, the Dzyaloshinskii-Moriya\ninteraction becomes negligible for the circularly symmetric Skyrmion. Hence far from the core of the Skyrmion, f(r)\nis a solution of the di\u000berential equation,\nf00(r) +1\nrf0(r)\u0000(1\nr2+\u0016J)f(r) = 0; (S5)\nwhere\u0016J=\u0016=J which is de\fned for convenience.\nThis di\u000berential equation is solved by a modi\fed Bessel function of \frst order, K1, which can readily be approxi-\nmated asymptotically as an exponential. Therefore, as r!1 we \fnd,\nf\u0018Cp\u0016J\n2\u0019K1(p\u0016Jr)\u0018Cep\u0016Jr\npr, (S6)9\nwhereCis a constant which depends on both Jand\u0016, and is found by \ftting to the exponential tail of f(r), as in\n\fgure S2. We further simpli\fed the asymptotic form in Eq. (S6) exploiting the asymptotic behaviour of K1(p\u0016Jr),\nand such an asymptotic tail is also shown in \fgure S2.\nFrom this we can de\fne the asymptotic form of n(x)\u0019e+ (x) as (wheree= (0;0;1)),\n (x) =Cp\u0016J\n2\u0019K1(p\u0016Jr)(cos(\u0012+\r);sin(\u0012+\r);0):\nIf we introduce the orthogonal dipoles,\np1=C(cos\r;\u0000sin\r);p2=C(sin\r;cos\r);\nwe can rewrite the asymptotic form,  (x) = ( 1(x); 2(x);0), as\n a(x) =p\u0016J\n2\u0019pa\u0001^xK1(p\u0016Jr) =\u00001\n2\u0019pa\u0001rK0(p\u0016Jr); (S7)\nwherea= 1;2.\nHence, far from a single Skyrmions the \felds behave as a pair of orthogonal dipoles.\nInteraction potential of two Skyrmions: In order to understand how Skyrmions interact we analytically\nconstruct a two Skyrmion con\fguration. In Ref. [17] it was shown that a con\fguration of two well separated Skyrmions\ncan be approximated by,\nWcon\fg =W1(x) +W2(x):\nwhereW1andW2are the Skyrmion con\fgurations nandmrepresented in stereographic coordinates,\nW1(x) =n1(x) +in2(x)\n1 +n3(x); W 2(x) =m1(x) +im2(x)\n1 +m3(x):\nWe can use this to located one Skyrmion at the origin and a second at R.\nThe vector valued con\fguration, n(x), can then be recovered by inverse stereographic projection and produces the\ncon\fguration shown in \fgure S1. This summation ansatz can be extended to produce a con\fguration of any number\nof Skyrmions by simply adding more con\fgurations together.\nWe will consider a con\fguration of two well separated hedgehog Skyrmions, as in \fgure S1, where one Skyrmion\nnAis in region Aand the second Skyrmion nBis in region B. In region Athe Skyrmion nBis close to the boundary\nvalueeand in region Bthe Skyrmion nAis close to the boundary value e. This means we can approximate the\ncon\fguration nAin regionBand con\fguration nBin regionAas,\nnA=q\n1\u0000 A\u0001 Ae\u0019e+ AandnB=q\n1\u0000 B\u0001 Be\u0019e+ B; (S8)\nwhere A\u0001e= B\u0001e= 0. It can be shown, [17], using the summation ansatz, that a two Skyrmion con\fguration,\nnAB, with a large separation, R=jRj, in regionAcan be approximated as\nnAB\u0019nA+\"B\u0002nA+1\n2\"B\u0002(\"B\u0002nA); (S9)\nand in region Bas\nnAB\u0019nB+\"A\u0002nB+1\n2\"A\u0002(\"A\u0002nB); (S10)\nwhere,\n\"B=1\n2nA\u0002\u0010\n(1 +nA\u0001e) B\u0000(nA\u0001 B)e\u0011\n; (S11)\n\"A=1\n2nB\u0002\u0010\n(1 +nB\u0001e) A\u0000(nB\u0001 A)e\u0011\n: (S12)\nWe can exploit this to approximate the energy of a two Skyrmion con\fguration, nAB, as the sum of the contributions\nover the the two regions AandBas in \fgure S1. This gives the energy functional,10\nFigure S1: A two Skyrmion con\fguration. a, A two Skyrmion con\fguration split into regions. Region A\ncontainsnAand region BcontainsnB.\nE[nAB]\u00192ESkyrmion(S13)\n+Z\nBd2x\u0012J\n2@i A\u0001@i A+1\n4D A\u0001r\u0002nA+1\n2\u0016 A\u0001 A\u0013\n+Z\nAd2x\u0012J\n2@i B\u0001@i B+1\n4D B\u0001r\u0002nB+1\n2\u0016 B\u0001 B\u0013\n+Z\nA\"B\u0001\u0000\nJnA\u0002@2\ninA+ 2DnA\u0002r\u0002nA+B\u0002nA\u0001\n+Z\nB\"A\u0001\u0000\nJnB\u0002@2\ninB+ 2DnB\u0002r\u0002nB+B\u0002nB\u0001\n:\nThe \frst two integrals are the energy of the Skyrmion tails in the complementary empty region, and the second two\nintegrals measure the overlap of one Skyrmion with the other. We can evaluate this integral, for two Skyrmions with\na large separation, by linearising the above integral over the area Aand keeping only \frst order terms in  Aand B.11\nThis gives,\nZ\nA( B\u0001(Jr2\u0000\u0016) A+D B\u0001r\u0002 A)d3x: (S14)\nThe fact that  Aand Bare normal to eand thatr\u0002 Ais parallel with eimplies that B\u0001r\u0002 A\u00190. From\nthe previous asymptotic analysis, we can see that the asymptotic form of ( r2\u0000\u0016J) A= 0. We can then use the\nGreen function, K0(p\u0016Jr), of the 2D Klein-Gordon equation,\n(r2\u0000\u0016J)K0(p\u0016Jr) =\u00002\u0019\u000e(2)(x); (S15)\nto re-express the asymptotic \feld of nain the integral (S14) as a product of orthogonal dipoles,\n\u0000\nr2\u0000\u0016J\u0001\n A\na(x) =pa\u0001r\u000e(2)(x); a= 1;2: (S16)\nFrom equation (S7) we know the asymptotic form of the second Skyrmion, nB, atRis,\n B\na=1\n2\u0019pa(\rB)\u0001rK0(p\u0016Jjx\u0000Rj); a= 1;2: (S17)\nCombining these results we can evaluate the interaction potential de\fned as Vint=E[nAB]\u0000E[nA]\u0000E[nB],\nVint=JZ\nAd2x B\u0001(r2\u0000\u0016J) A\n=C2\u0016\n\u0019cos\rK0(p\u0016JR); (S18)\nwhere\r=\rA\u0000\rB. We \fnd that two well separated Skyrmions interact as a pair of orthogonal dipoles. Using the\nasymptotic form of K0the interaction takes the form\nVint\u0019C2\u0016J\n\u0019cos\re\u0000p\u0016JR\np\nR; (S19)\nwhere\u0016=jBj,Ris the inter-Skyrmion separation and Cis a constant found by \ftting the asymptotic tail of f(r).\nOrientation of Skyrmion inside of the bag. In terms of the hedgehog ansatz, the usual boundary conditions\naref(0) =\u0019;f(1) = 0. For the single Skyrmion bag f(0) = 0;f(1) = 0. As we can see from the images, we can\ndivide the space into two regions, the \frst region r < R 0contains the anti-Skyrmion and the second contains the\nSkyrmion where f(R0) =\u0019. Considering the central region for large R0then we can transform the pro\fle function as\nf!\u0000f+\u0019, this transformation causes the DMI contribution to pickup a minus sign and hence \r=\u0019\n2is the stable\ncon\fguration.\nCircle packing The packing of Skyrmions inside of a bag can be mapped onto the mathematical circle packing\nproblem [32], and is some times referred to as the inimical dictators problem. Circle packing is concerned with the\ncon\fgurations of packing the largest equal size circles into the area of a disk, qualitatively the con\fgurations match\nthe inner Skyrmions of the bags in both magnetic and chiral nematic liquid crystal cases. This is because, as the\ninteraction potential shows, they repel each other exponentially with distance and hence each Skyrmion occupies a\ngiven circular area, similar to the circle packing problem. Importantly, it has been shown that packing thirteen circles\ninto a disk has two optimal con\fgurations, [32]. This corresponds to the S(13) con\fguration, where we also \fnd two\nstable con\fgurations, for both magnetic and chiral nematic liquid crystals, which correspond to the two circle packing\ncon\fgurations and they are shown in \fgure 2.\nThis shows that if a magnetic S(13) bag con\fguration was experimentally produced, data could be stored by the two\ncon\fgurations, where the con\fguration could be switched from one con\frmation to the other. This would alleviate\nthe problem of Skyrmion repulsion a\u000becting Racetrack memory and the possibility of a single Skyrmion `sticking' on\na defect and e\u000becting the encoded data.\nSize of Skyrmion bags. Each additional Skyrmion inside the Skyrmion bag increases the radius of the bag\ndue to the repulsion between internal Skyrmions and between those Skyrmions and the bag boundary. We \fnd this\nrelationship to be very close to linear up to the S(15) bag.12\n 0 1 2 3\n 0 5 10 15 20rf(r)Ce-r/√r \n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8\n 5 10 15 20r\nFigure S2: Minimum energy pro\fle. The minimum energy Skyrmion pro\fle function, f(r), forD=J=\u0016= 1\n(chosen for aesthetics) and \r=\u0000\u0019=2 of the circularly symmetric hedgehog ansatz (2). Also plotted is the\nexponential tail approximation (S6), where C\u00196:95 is found to be optimal. The tail approximation is a good\napproximation for r>2 and is used in the asymptotic interaction potential.13\nFigure S3: Magnetic Skyrmion Bag radius. Mumax3 Skyrmion bag simulations run on a \fxed 1024 nm square\ndomain con\fgured as outlined in the Methods section. The S(n) bags are relaxed and then measured from the\nwidest outer domain wall boundary to show how their radii increase with increasing n.S(0), in this terminology is a\nstandard, charge one, Skyrmion.14\nFigure S4: A magnetic S(S(8)S(8)S(8))bag simulation, of topological degree 22.A stable, recursive\nSkyrmion bag, Mumax3 simulation run on a \fxed 2048 nm square domain con\fgured as outlined in the Methods\nsection."    },    {        "title": "1806.03148v1.Real_Space_Magnetic_Imaging_of_the_Multiferroic_Spinels_MnV2O4_and_Mn3O4.pdf",        "content": "Wolin, et al. 1 \n \nReal -Space Magnetic Imaging of the Multiferroic Spinels MnV 2O4 and Mn 3O4 \n \nB. Wolin,1 X. Wang,1 T. Naibert,1 S. L.  Gleason,1 G. J. MacDougall,1 H. D. Zhou,2,3 \nS. L.  Cooper,1 and R. Budakian1 \n1Department of Physics and Frederick Seitz Materials Research Laboratory,  \nUniversity of Illinois, Urbana, Illinois 61801, USA  \n  \n2Department of Physics and Astronomy , \nUniversity of Tennessee , Knoxville, Tennessee 37996-1200, USA  \n \n3National High Magnetic Field Laboratory , \nFlorida State University, Tallahassee Florida 32310-3706, USA  \n \n \nAbstract \nControlling multiferroic behavior in materials will enable the development of a wide \nvariety of technological applications.  However, the exact mechanisms driving multiferroic \nbehavior are not well  understood in most materials.  Two  such material s are  the spinel s MnV 2O4 \nand Mn 3O4, where mechanical strain is thought to play a role in determining magnetic behavior .   \nBulk studies of MnV 2O4 have  yielded conflict ing and inconclusive results , due in part to the \npresence of mesoscale magnetic inhomogen eity, which complicates  the interpretation of bulk \nmeasurements .  To study the sub- micron -scale magnetic  properties of Mn- based spinel materials , \nwe performed magneti c force microscopy (MFM) on MnV 2O4 samples subject to different levels \nof mechanical strain .  We also used a crystal grain mapping technique to perform spatially \nregistered MFM on Mn 3O4.  These local investigations revealed 100 -nm-scale “stripe” \nmodulations in the magnetic structure of bot h materials.  In MnV 2O4, the magnetization of these \nstripes is estimated to be  Mz ~ 105 A/m, which is on the order of the saturation magnetization \nreported previously.  Cooling in a  strong magnetic field eliminated  the stripe pat terning only in the \nlow-strain  sample of MnV 2O4.  The discovery of nanoscale magnetostructu ral inhomogeneity that \nis highly susceptible to magnetic field control in these materials  necessitates both a revision of \ntheoretical proposals and a reinterpretation of experimental data regarding the low -temperature \nphases and magnetic- field-tunable properties of the se Mn -based  spinels.  \n \n  Wolin, et al. 2 \n \nIntroduction  \n The wide variety of interactions and degrees of freedom in condensed matter systems yield \nsome of t he most complex and challenging problems in physics.  When different types of order \ncompete, materials can exhibit rich phase diagrams with linked structural, magnetic, and orbital \nordering transitions.  Two phenomena of great interest can result from this  competition:  \nmultiferroism —the coexistence and coupling of different types of ferroic order (ferromagnetism, \nferroelectricity, and ferroelasticity) —and magnetoresponsive behavior, i.e., large susceptibilities \nof physical properties to external perturbati ons, such as applied magnetic fields and pressure. \nMagne toresponsive and multiferroic materials  show great promise for practical applications, \nranging from high -frequency actuator s to precision sensors [1]. \nVarious mechanisms can cause a coupling between magnetic and other primar y order \nparameters [ 2-4], including the development of non- collinear spin order that breaks inversion \nsymmetry [ 2,3], and the formation of m ultiferroic domains [4,5] and domain walls [ 6,7].  One of \nthe grand challenges in  the study  of multiferroic and other magnetoresponsive materials has been \nto identify the specific magnetostructural  and magnetoelectric mechanisms responsible for the \ndifferent magnetoresponsive phenomena observed in numerous  complex magnetic materials, \nincluding  ACuO 3 (A=Se,Te) [ 8], Mn- doped BiFeO 3 [9], EuTiO 3 [10], Y2Cu2O5 [11], YbMnO 3 \n[12], and the spinels CoCr 2O4 [5], MnCr 2O4 [13], MnV 2O4 [14,15], and Mn 3O4 [16-19].   \nThe magnetic spinel family of compounds (chemical formula AB 2X4)—which consists of \nan A-site diamond sublattice and a geometrically frustrated B -site pyrochlore sublattice [20]— is a \nparticularly promising class of materials for studying the microscopic origins of  \nmagnet oresponsive behavior in magnetic materials. M agnetic spinels exhibit a range of diverse \nphases and phenomena that can be sensitively tuned using a variety of methods, including A - and/or \nB-site substitution, applied pressure, and/or applied magnetic fi eld [5,13- 19,21].  Due to the  strong \nsensitivity of the ir physical properties to pressure and magnetic field, the magnetic spinels have \nimportant potenti al applications in  catalysis, electrochemistry, and magnetic shape memory [ 22-\n27]. More broadly , magnetic domain formation is known to  play a key role  in raising the \nsusceptibilities  of complex materials  to external perturbations  [6,7,28,29]. H owever, the potential \nrole of this mesoscale inhomogeneity on the magnetoresponsive properties of spinels has not been \nwell investigated , because  most previous rese arch on the spinels has been conducted using bulk \nprobes  focusing on atomic length -scales  such as neutron scattering [ 30-32], SQUID magnetometry \n[33-35], x -ray diffraction [33,36,37], and Raman scattering [ 38,39]. \nIn this report, we explore the role of 0.1 -10 µm  scale magnetic inhomo geneity on the \nmagnetic properties of two specific spinels, MnV 2O4 and Mn 3O4, using magnetic force microscopy \n(MFM).  By using a sub -micron size magnetic probe, MFM can measure magnetic properties that \nare averaged over just tens of unit cells.  Consequently, MFM measurements can reveal small -\nscale (0.1- 100µm) magnetic inhomogeneities that have been overlooked in bulk measurements.  \nWe select the Mn -based magnetic spinels, MnV 2O4 and Mn 3O4, for study, because both materials  \nexhibit similar  magnetostructu ral properties and transiti ons at cryogenic temperatures that depend \nsensitively on the B -site constituent, V or Mn .  For example, MnV 2O4 is a cubic paramagnet at \nroom temperature, and undergoes a magnetic transition to a coll inear ferr imagnetic ( FEM) \nconfiguration below T=57K . A second transition to a Yafet -Kittel (YK) type FEM configuration \naccompanied  by a cubic- to-tetragonal structural transition  occurs  at T=53K  [30,33 ,36,40].  By \ncontrast,  the cubic -to-tetragonal s tructural transition in Mn 3O4 occurs at a significantly higher \ntemperature, T=1440K, and the low -temperature magnetostructural phase behavior is more Wolin, et al. 3 \n \ncomplex:  Mn 3O4 is a tetragonal paramagnet at room temperature and develops a triangular FEM \nconfiguration near T=42K.  Near T=39K, an incommensurate spin ordering develops before \nMn 3O4 finally transitions to a cell -doubled YK -FEM magnetic phase with an orthorhombic crystal \nstructure near T=33K [30- 32,41].    \nIn this study, w e collected MFM images across a wide range of temperatures and magnetic \nfields from two samples of MnV 2O4 with different levels of induced mechanical strain .  We also \nstudied MFM images  from a single sample of Mn 3O4 with inherent strain produced during crystal \ngrowth.  Among a diverse range of magnetic patterns, we observe 100- nm scale “stripe” \nmodulations in the magnetic structure present  in the lowest -field phases of both materials.   These \nstripe modulations are further organized into 1- 10 μm scale domains associated with the local \ncrystal structure.   In Mn 3O4, an observed correlation between stripe width and  encompassing  \ntetragonal domain size ev idences a connection between  mechanical strain and the magnetic \npatter ns.  In MnV 2O4, we observe  100 nm -scale stripe modulations consistent with recent zero -\nmagnetic- field TEM measurements of thin -foil MnV 2O4 [42], and we find different magnetic \nbehaviors in the high - and low -strain MnV 2O4 samples.  We also present a quantitative estimate \nof the local magnetization associated with these stripe domains in MnV 2O4.  We observe that \nmodest applied  magnetic field s (<30 kG) cause dramatic changes to —and the ultimate elimination \nof—the stripe domain patterns in both  Mn 3O4 and low-strain MnV 2O4, but not  in high-strain \nMnV 2O4. These findings are  consistent with theoretical results showing that mesoscale magnetic \ninhomogeneity can significantly lower the energy barrier for strain - and field- dependent phase \nchange s in complex materials [ 28,29], and suggests that magnetic domain formation plays an \nimportant role in the magnetoresponsive behavior of these spinel materials . \n \nMethods  \nSingle crystals of MnV 2O4 were grown at the National High Magnetic Field Laboratory in \nTallahassee  using a traveling -solvent -floating -zone technique. M ixtures of MnO and V 2O3 were \nground, pressed, and calcined to form the seed and feed rods. A greater than stoichiometric amount \nof V 2O3 was used to compensate for evaporation during growth. Details of the growth and \ncharacterization are reported elsewhere [ 33]. Single crystals of Mn 3O4 were grown at the \nUniversity of Illinois using a floating -zone technique. Commercially available Mn 3O4 powder was \npressed and sintered to form the feed and seed rods. The structural and magnetic properties of the \nresulting crystals are also reported elsewhere [ 16,41]. For both materials, crystallographic \norientations were determined via room -temperature x -ray diffraction.  \n After  characterization, the crystal  surface normal to the [001] (cubic) direction was \npolished to <50nm roughness, and sputter coated with  a 5nm layer of  Au-Pd to dissipate static \ncharge .  Two MnV 2O4 samples were prepared from the same growth.  The first sample was a half -\nboule semicylinder  measuring approximately 5mm ×  2.5mm ×  0.5mm .  Epoxy was applied to the \nentire back surface of this sample, whi ch was then attached to a sapphire backing -plate.  The total  \nthermal contraction  occurring between the epoxy curing  temperature and the base temperature \nused in this study (T =4K) is ten times larger for the epoxy than for the MnV 2O4, and therefore  \nsignificant mechanical strain is induced in the sample below T =77K  [42].  A similar order -of-\nmagnitude difference in thermal expansion coefficients between MnV 2O4 foil and the Mo mount \nresulted in an estimated 0.03% compressive strain in MnV 2O4 at 87 K and a <0.1% compressive \nstrain near the cubic- to-tetragonal tran sition at 52K in MnV 2O4 [42].   \nWhile this estimated compressive strain is less than the ~0.15% lattice striction measured in Wolin, et al. 4 \n \nMnV 2O4 at the cubic- to-tetragonal transition [37], it is large enough to influence domain formation \nin MnV 2O4 [42].   The second MnV 2O4 sample was a full -boule cylinder having a 5mm diameter \nand a 2mm length, and was specifically prepared to minimize mechanical strain below T =77K. \nThis sample was attached to a copper backing -plate using a single point of epoxy at one edge, \nallowing  the sample to  thermally contract without  interference from either the epoxy or backing \nplate.  The i ncreased sample thickness and  single epox y point mounting both act to minimize  \nmechanic al strain at the sample surface.  Thermal contact between the sample and backing plate \nwas maintained through the epoxy and physically through the sample- plate interface.  In addition, \nlong soak times (~10 minutes) were used to ensure thermal equilibrium was achieved.  \nSingle crystals of Mn 3O4 were grown at the University of Illinois using a traveling -solvent -\nfloating -zone technique. To prepare the Mn 3O4 sample, the Mn 3O4 rod was  diced into  a rectangular \nblock measuring approximately 1mm ×  2mm × 1mm .  The sample was  polishe d normal to the \n[110] (tetragonal) direction  and sputter coated with 1nm Au -Pd to prevent charging.  T he Mn 3O4 \nsample was  lithographically patterned with  an array of unique location markers to provide spatial \nlocation information.  We performed cryogenic electron backscatter diffraction (EB SD) \nexperiments to determine the tetragonal crystal grain structure for comparison to MFM \nmeasurements.  Using the location mark ers, we were  able to align the magnetic and \ncrystallographic data images with approximately 50nm accuracy , allowing us to  correl ate observed \nmagnetic phenomena with the local crystal domain structure.  \n We performed low -temperature, frequency -modulated  MFM using a 4He bath cryostat that  \nhad a built -in superconducting magnet .  Data was collected in the  temperature rang e from T=4.5K \nto T=80K and the magnetic field rang e from B=0T to B =3T.  In all cases, the magnetic field was \noriented normal to the sample surface, resulting in B  parallel to [001] (cubic) for both MnV 2O4 \nsamples and B  parallel to [110] (tetragonal) for the Mn 3O4 sample.  Commercially available atomic \nforce microscopy cantilevers were evaporatively coated with a 10-nm thick layer of FeCo to \nprovide magnetic sensitivity.   With probe -sample separations of approximately 100 nm and scan \nrates as low as 100 nm/s, we were able to  achieve a spatial resolution of approximately 50  nm for \nmagnetic features.   The cantilevers used in these experiments have resonance frequencies  \napproximately f0~25kHz , spring constants approximately k ~0.3N/m, and quality factors \napproximately  Q~350,000 at T =4K in vacuum.  We measured the cantilever displacement \ninterferometrically using a 1510nm laser  in a fiber- optic Fabry -Pérot configuration [43], and we \nmeasured  the cantilever frequency using a phase- locked loop (see Supplementary Section  [44]).  \n To extract quantitative information from the MnV 2O4 image data, we conducted a \ncalibration experiment  to characterize the magnitude and orientation of the magnetic moment of \nthe MFM probe .  A 70 -nm thick, 70- µm long straight rectangular gold wire was patterned onto a \nSi substrate using electron -beam lithography and thermal evaporation.  The wire measured 4 µm \nwide for half the length and 1µm wide for the other half, with a step -like junction at the center  \n(Figure S2).  We calculated the magnetic field produced by an electric current running through \nthis simple geometry using a finite- element electromagnetic solver.  For areas far f rom the \njunction, the simulation  results showed near- perfect agreement with analytical calculations for an \ninfinite wi re.  To ensure maximum rem nant magnetization, the ambient magnetic field in the \ncryostat was cycled up to B =3T and back to B =0T before any measurements were performed.   With \na constant 5mA current  running through the wire, we recorded MFM  frequency shift  data in the Wolin, et al. 5 \n \narea near the junction.  Comparing this data with the calculated field curvature, we extracted the \npoint spread function (PSF) of the MFM probe.  This function is independent of the sample being \nscanned, and can be used to quantitatively analyze the MnV 2O4 data because it relates the measured \nMFM frequency shift directly to the magnetic field curvature produced by the sample  [45].  See \nthe Supplementary Section [44] for more details. \n \n \n \n \nResults  \nFigure 1(a) shows MFM data collected from a region of the high-strain MnV 2O4 sample after \ncooling from T =70K to T =40K, well into the YK phase [33,35,50], in the presence of a weak \nmagnetic field, B =3kG.  The approximate cubic lattice directions (white arrows and text  in Figure \n1(a)) were determined using room -temperature x -ray diffraction. We observe a space- filling \n \nFigure 1: MFM data of high -strain MnV 2O4 cooled from 70K to 40K in B=0.3T. (a)  When cooled in a weak magnetic \nfield, the magnetic pattern sharpens dramatically.  We observe 20µm -scale domain structure with regular sub-\ndomain stripe modulations.  Regions of overall frequency shift (predominant blue or red color) correspond to areas \nof a single stripe direction.  Approximate cubic lattice axes are indicated in white.  The yellow dashed box highlights \na region where mechanical strain influences the magnetic pattern. (b) Average magnitude of magneti c \ninhomogeneity (characterized by the standard deviation of frequency shift) measured while cooling the high- strain \nMnV 2O4 sample in zero magnetic field. Note the qualitative similarity to measurements of the bulk sample \nmagnetization. (c) Cross sections of the measured point spread function at locations: (top to bottom) 0nm, 250nm, \n500nm, and 750nm away from the PSF center.  (d) Frequency data along the indicated line through the 2 -D image.  \nStripe pitch, frequency offset, and amplitude vary across the domain.  Wolin, et al. 6 \n \nmagnetic patterning with domain and subdomain structures .   Large (µm -scale) do mains of \npredominantly positive (blue) or negative (red) frequency shift  contain and define the boundaries \nof 100- nm scale stripe modulations.  The large domains correspond to a reas of well -defined stripe \ndirection.  Additiona lly, the stripe pitch , amplitude, and offset vary continuously across domains , \nas seen in  Figure 1(d), which shows frequency  shift data along the indicated line- cut (yellow \ndashed line, Figure 1(a)).  The pi tch variation in Figure 1(d) is only approximately 14%, but the \npitch variation between the left -most and right -most domains is as large as 60%.   The stripe pitch \nis anti -correlated between domains : in the boxed region of Figure 1(a), the modulation pitch in the  \nblue domain is highest and the m odulation pitch in the two red domains is lowest, indicating a \nlikely influence of mechanical strain on  the magnetic patterning.   By calculating the standard \ndeviation  (σ) of the frequency shift data from an entire MF M scan, we measure the degree of \nmagnetic inhomogeneity.  Figure 1(b) plots σ versus temperature for data collected during a zero-\nfield cool of the high -strain MnV 2O4 sample.  We observe a sharp onset of magnetic \ninhomogeneity near T= 58K and a peak at T= 54K.  The degree of inhomogeneity distinctly \ndecreases between T= 54K and T=49K,  and at T =49K  the MFM images  show a clear change in the \nmagnetic patterning.  Both the raw MFM data and the derived σ vs. T data clearly indicate two \nmagnetic phase transitions in  MnV 2O4, consistent with previous reports  [30,33,36,40]. \nFurthermore, the results shown in Figure 1(b) are qualitatively similar  to measurements of the bulk \nmagnetization [30,33,36,40].  The correlation between  bulk magnetic behavior and 0.1-10µm scale \nmagnetic inhomogeneity suggests that t he low -temperature magnetic behavior  of MnV 2O4 can be \nwell characterized by  magnetic domain formation  and heterogeneity.   The observed subdomain \nstructure explains the sharp drop in overall inhomogeneity observed below  T=54K.  Without a \nsubdomain structure, we would expect the magnetic inhomogeneity to increase monotonically with \ndecreasing temperature.  These conclusion s will be further explored in the discussion section.  \nTo make a quantitative comparison between the magnitude of magnetic inhomogeneity \nobserved in MFM a nd the bulk magnetic behavior reported for  MnV 2O4, we performed a \ncalibration experiment using previously established techniques [46- 49]. Further details of the \ncalibration experiment are included in the Supplemental Section  [44].  Figure 1(c ) shows the \ninstrument response of the magnetic probe extracted from measurements  of the calibration sample.   \nFrom top to bottom, the traces show cross  sections of the PSF  at locations 0nm, 250nm, 500nm, \nand 750nm away from the  probe apex.  Using this  measured spatial response function of the MFM \nprobe, we  quantitatively model ed the stripe pattern seen in Figure 1(a) to yield an estimate of the \nlocal magnetization associated with the sub -domain stripe features.   We estimate  (to within a \nfactor of 3) the peak -to-peak magnetization associated with the strip e modulations to be  Mpp  ≈ 0.8 \n∙105 A/m.   Because a cantilever -based magnetic probe is sensitive only to the magnetic field \ncurvature , the absolute magnetization of a macroscopic sample cannot be determined using MFM ; \nonly gradients  in the sample magnetization induce a frequency shift .  Thus , our observations are \nconsistent with two extreme possible  interpretations : the stripes define regions with  magnetization \nalternating  either  between  Mz=±Mpp/2 or between  Mz=0 and M z=Mpp.  Magnetometry experiments \non MnV 2O4 at T=40K show that the bulk saturation magnetization is  Mz =0.7 ∙105 A/m [35], so  the \nmagnetization associated with the stripe features is comparable to the overall magnetic behavior \nof the sample in both extreme cases .  From t hese results , we conclude  that the highly \ninhomogeneous nature of the magnetic state of MnV 2O4 represents a dominant contribution to the Wolin, et al. 7 \n \nmagnetization that must be taken into account when  analyzing  the low -temperature magnetic \nbehavior of this material .   \n Figure 2 shows representative MFM  frequency shift data collected after  cooling the low -\nstrain MnV 2O4 sample to T=40K in the presence of different magnetic field strength s.  For fields \nin the range 0kG< B<2.5kG, we observe irregular magnetic patterning with large frequency shifts.  \nRepeated cools with the same parameters yielded qualitatively distinct results, some with no \nregular patterning and others with highly regular strip e patterns.  The observation that different \ncools yield different patter ns indi cates the existence of multiple, nearly degenerate metastable \npattern states  and the absence of signif icant pinning effects.  Figure 2(a) shows an example of \nirregular patternin g observed on cooling in zero applied field.  In the field range 2.5kG< B<7.5kG, \nwe observed 10µm -scale domain features oriented approximately 45º relative to the cubic  crystal \naxes.  We also observed sub -domain stripes that form an interwoven  pattern , as c an be seen in \nFigure 2(b).  Repeated cools  in this field  regime with the same parameters yielded the same domain \nstructure, but  \ndifferent  sub-domain \npattern s.  As the field \nis increased  further, \nthe number of sub-\ndomain stripes \ndecreases  until only \nthe domain features \nremain (Figure 2 (c)).  \nBetween B=15kG \nand B=30kG  (Figure \n2(d)), all magnetic \nfeatures are \neliminated, indicating  \nthat the entire sample \nis a homogeneous \nmagnetic domain.    \nIn the context of \npublished phase \ndiagrams  [35,50], the \ntemperature of  the \nabove measurements \nshould place the \nmaterial well within \nthe tetragonal/YK \nphase for MnV 2O4 for \nthe entire field range investigated. The disappe arance of magnetic features between B=1.5kG and \nB=30kG  is consistent with reports of a weak first -order tra nsition associated with the realignment \nof tetragonal domain structure [33,50], a conclusion supported by x -ray scattering measurements \n[36].   \n \nFigure 2: MFM data of low-strain MnV 2O4 cooled to 40K.  Images are 20x20µm. The  \napproximate cubic axes in (d) apply to all panels. (a) At low fields, the magnetic pattern is \namorphous and causes large frequency shifts.  In addition, we observed several distinct \ntypes of patter ning during different cools at the same field value.  (b) In the intermediate \nfield regime, 10µm -scale magnetic domains were observed.  Irregular sub -domain striping \nwas observed in tweed patterns.  (c) At 10kG, sub- domain striping was eliminated.  (d) By \n30kG, all magnetic contrast was eliminated, indicating a single magnetic domain.  \n Wolin, et al. 8 \n \nConsistent with this interpretation, w e identify t he strong domain features in Figures 2(b) \nand 2(c) as transiti ons between magnetic domains with magnetizations oriente d parallel to  \ndifferent crystal axes (see Supplemental Section  [44]), mirroring previously measured structural \ndomains [36].  As the external magnetic field is increased, tetragonal domains not oriented parallel \nto the external field become energetically unfavorable, resulting in the magnetic uniformity shown \nin Figure 2(d). \nFigure 3 shows representat ive MFM data collected while cooling the high -strain MnV 2O4 \nsample to T=40K in the presence of different magnetic field strengths.   At low fields , we again \nobserve irregular magnetic patterning , as shown in Figure 3(a).  For fields 3kG <B<7.5kG , we \nobserve a less clearly  delineated  domain structure , as well as single direction sub -domain stripes , \nas shown in Figure 3(b).  Finally, for B>7.5kG (F igures 3(c,d)), a somewhat more complex \nmagnetic patterning develops; this patterning  changes as the magnetic field  is increased , and \ninclud es the development of subdomain 100 nm -scale stripe features .  Figure 3(d) shows that \nstrong magnetic inhomogeneity persists up to the highest field measured , B=30kG . Though these \nmeasurements \nnominally explore \nthe same region of \nphase space as \nthose is Figure 2, \nthe current  result s \nreveal  a significant \ndistinction between \nthe high - and low -\nstrain sample \nbehaviors : high \nmechanical strain \nin the crystal lattice \nof MnV 2O4 \nstabilizes magnetic \ninhomogeneity in \nhigher magnetic \nfields.   The distinct \ndifference in \nmagnetic domain \npatterns observed \nin the high -strain \nand low -strain \nsamples also \nindicates a strong \nstructural \ncomponent to the \nmagnetic domain \n \nFigure 3: MFM data of high -strain MnV 2O4 cooled to 40K.  Images are 20x20µm.  The  \napproximate cubic axes in (d) apply to all panels.  (a) At low fields, the magnetic pattern is also \namorphous (similar to the low -strain measurements) and induces large frequency shifts.  (b) In \nthe intermediate fi eld regime, a pattern of domains and sub- domain stripes appeared.  No \ninterwoven striping was observed at any field value.  (c) At 15kG, the domain pattern becomes \nmore segmented, but retains the features seen at lower fields.  Stripe modulations are still  \npresent, but are difficult to observe due to large frequency shifts between domains.  (d) Strong \nmagnetic inhomogeneity remains at B =30kG, in contrast to the low -strain sample.  The stripe \nmodulations also persist up to B =30kG.  Wolin, et al. 9 \n \npattern  in MnV 2O4. This connection could be further explored using a combination of MFM and \nlocal structural measurements , similar to that described below.  \nIn an effort to investigate whether magnetic domain  formation is observed in other Mn-\nbased spinels exhibiting magnetoresponsive properties, we also used MFM to investigate the \nspatial  organization of magnetic patterns in the magnetodielectric spinel, Mn 3O4.  Figure 4 is a \ncomposite MFM image of the  Mn 3O4 sample created by stitching together multiple indivi dual \nMFM scans recorded in succession.  The Mn 3O4 sample was cooled  in the presence of a weak \nmagnetic field, B =2kG from above T =40K  to T=18K; this is well  into the cell-doubled \northorhombic ferrimagnetic  phase , as determined by previous measurements [34, 41, 52].  We \nobserve stripe modulations very similar to those observed in MnV 2O4. In Mn 3O4, the stripes form \na tweed pattern  consisting  of different regions of coordinated stripe direction.  The green dashed \nlines in Figure 4 indicate boundaries between the frozen- in tetragonal crystal grains, as determined \nby electr on backscatter diffraction (EBSD ).  We observe a clear correspondence between the \nlocations of tetragonal domain boundaries and the magnetic stripe region boundaries.  Repeated \ncooling using the same parameters yields an identical set of magnetic domain bo undaries, \nindicating that the magnetic domains are strongly pinned to the tetragonal crystal boundaries , \nsimilar to the behavior observed  in the high -strain MnV 2O4 sample.   Furthermore, the size of the \ntetragonal domain is correlated with  the stripe pitch within the domain  in the Mn 3O4 sample, with \nthe largest tetragonal domains support ing stripes with the lowest pitch.  As the tetragonal domain \nsize shrinks, the stripe pitch increases until  the MFM probe cannot resolve individual stripe \nfeatures.   Similar to our observations in MnV 2O4, the tweed stripe pattern  in Mn 3O4 is eliminated \nby cooling in a sufficiently strong magnetic field ( B=20kG). This is consistent with the observation \nof nearly degenerate orthorhombic phases in Mn 3O4, and the  selection of a universal orthorhombic \ndistortion axis with applied field [34, 52 ].  The relationship between the tetragonal domains and \nthe magnetic pattern is further evidence of the important role that mechanical strain plays in the \nlow-temperature magn etic stripe formation and magnetic properties of these Mn -based spinels .  \nThe presence, magnitude, and similar field -behavior of magnetic inhomogeneities  in both Mn 3O4 \nand MnV 2O4 indicate that such features are likely generic to a wider range of strongly spin -lattice \ncoupled materials, particularly  other magnetic spinels and magnetodielectric materials.  Wolin, et al. 10 \n \n  \n \nDiscussion  \nOur investigations represent the first observations of nanoscale inhomogeneity in the low -\ntemperature magnetic structures of bulk MnV 2O4 and Mn 3O4.  Quantitative estimates of the \nmagnetization associated with these nanoscale magnetic patterns indicate that the magnitude of \nthe magnetic modulations is  large, accounting for much of the bulk magnetic behavior reported in \nthese materials.  Additionally,  our results show for the first time that the magnetic stripe \nmodulations change significantly in modest magnetic field strengths that are comparable to the \nfield strengths at which large magnetodielectric and magnetic- lattice st riction effects are observed \nin MnV 2O4 and Mn 3O4 [30,36,37].  \nThe nanoscale magnetic inhomogeneity we observe in MnV 2O4 and Mn 3O4 raises two \nfundamental questions : (i) what, if any, underlying structural inhomogeneity accompanies the \nmagnetic inhomogeneity; and (ii) to what extent d oes the magnetic inhomogeneity contribute to \nthe magnetoresponsive phenomena observed in MnV 2O4 and Mn 3O4 [16-18]?  \nAddressing the first issue, substantial direct and indirect evidence indicates that the \nnanoscale magnetic inhomogeneity we observe at low t emperatures in MnV 2O4 and Mn 3O4 is \nassociated with an underlying structural modulation.  B ulk x -ray diffraction measurements on \npolycrystalline Mn 3O4 [51] show evidence for a mixture of tetragonal and orthorhombic phases, \nand the coexistence of tetragonal (paramagnetic) and orthorhombic phases at low temperatures in \nMn 3O4 is also supported by recent muon spin resonance measurements of single- crystal Mn 3O4, \nwhich reveal a mixture of magnetically ordered and disordered volumes at low temperatures [ 21].  \nThe p honon and magnon Raman scattering spectra of heavily  twinned samples of Mn 3O4 also show \nevidence for phase coexistence at low temperatures, which may include coexisting orthorhombic \nand tetragonal phases [38].  M ore recent Raman experiments of the phonon and magnon spectra \nof untwinned Mn 3O4 samples show clear evidence for coexisting face- centered orthorhombic and \n \nFigure 4: Composite MFM image of Mn 3O4 at T=18K, B=2kG.  We observe tweed -pattern magnetic stripe features defined by \nthe tetragonal crystal grain pattern (dashed green lines).  The stripe widths are correlated to the domain size, suggesting a \nconnection between the mechanical strain and the associated magnetic pattern. The patchy region in the second subpanel \nfrom the right reveals one of the location markers used to spatially register MFM data with EBSD results.  The non- magnetic \nmarker material does not affect the magnetic behavior of the sample, but appears in the data images because of the \nchanging topography.  Wolin, et al. 11 \n \ncell-doubled orthorhombic phases at low temperatur es [5 2], consistent with the presence of a \nmesoscale structural modulation in this material .  In MnV 2O4, TEM measurements revealed the \ncoexistence of tetragonal twinning domains with different c -axis orientations [42 ], and the \nsensitivity to strain we observe in our measurements of MnV 2O4 support the conclusion that the \nnanoscale magnetic modulatio n we observe in this material is associated with an underlying \nstructural modulation.  Altogether, these results provide strong evidence that the magnetic \nmodulations observed with MFM in both MnV 2O4 and Mn 3O4 are associated with an underlying \nstructural modulation that betrays the strong coupling of spin, orbital, and structural degrees of \nfreedom in these materials [ 36,37]. \nNotably, mesoscale magnetostructural modulations have been observed in other magnetic \nmaterials exhibiting strong spin -lattice coupling, including La1.99Sr0.01CuO 4 [53], \nCo0.5Ni0.205Ga0.295 [54], and the Mn- doped spinel CoFe 2O4 [55].  Mesoscale magnetostructural \npattern formation in materials has been explained using Landau expansions of the elastic energy \nin powers of the strains and the strain gradients [ 54,56- 59], and several key conditions for the \nformation of mesoscale magnetostructural modulations near structural phase transitions of strongly \nspin-lattice coupled materi als have been delineated [ 54,60]: (i) a sensitivity of the system to local \nsymmetry -breaking perturbations, e.g., Jahn-Teller instabilities; (ii) the presence of long -range \ninteractions, such as magnetic interactions, that can stabilize particular structural phases locally; \nand (iii) some local anisotropy, e.g., a surface, defect, or grain boundary, to determine the specific \nmodulation pattern.  All of these essential ingredients for the nucleation of mesoscale \nmagnetostructural domain regions are present in both MnV 2O4 and Mn 3O4.  It is also worth noting \nthat both MnV 2O4 and Mn 3O4 have orbitally active octahedral ( B) sites (V3+ in MnV 2O4 and Mn3+ \nin Mn 3O4), which has been shown to favor an instability toward spinodal decomposition into \ncoex isting structural phases [ 61], consistent with our evidence for coexisting tetragonal and \northorhombic phases in Mn 3O4 and similar to earlier evidence for phase coexistence in the Mn -\ndoped spinel CoFe 2O4 [55].  \nThe newest and most significant demonstration from this MFM study is that the mesoscale \nmagnetic domain patterns observed in MnV 2O4 and Mn 3O4 are readily controlled with modest \nmagnetic fields; indeed, the magnetic field strengths at which we observe the magnetic stripe \nmodulations to change in both in MnV 2O4 and Mn 3O4 correspond c losely to the magnetic field \nvalues at which magnetodielectric effects and magnet -field-tuned lattice striction effects are \nobserved in both MnV 2O4 [30,37] and Mn 3O4 [30,36].  This close correspondence offers strong \nevidence that the magnetically responsive properties of MnV 2O4 and Mn 3O4 are not associated \nwith homogeneous properties of these materials, but are rather associated with the materials’ \nintrinsic magnetic inhomogeneities, which are ultimately driven by the competition between long -\nrange  magnetic interactions and strain energies.  Significantly, the presence of domain walls and \nmesoscale phase separation has been shown to be instrumental in lowering the energy barrier for \nfield-induced phase changes in complex materials [ 28,29], and indee d, we propose that  the \nmesoscale magnetostructural patterns evident in our MFM results —and their strong susceptibility \nto magnetic- field manipulation —are primarily responsible for the large magnetic susceptibilities \nobserved in MnV 2O4 [30,37] and Mn 3O4 [30,36]. \n \nConclusions  Wolin, et al. 12 \n \n We employed cryogenic MFM and room -temperature EBSD to investigate the nanoscale \nmagnetic properties of the two multiferroic spinel materials MnV 2O4 and Mn 3O4.  Our MFM \nmeasurements reveal significant nanoscale magnetic domain formati on that has been overlooked \nby previous bulk probe studies.  The magnitude of the magnetic modulations in these materials are \ncomparable to the bulk magnetizations measured in these materials, and consequently this \nnanoscale magnetic inhomogeneity  cannot be neglected when  considering the overall magnetic \nbehavior of the two materials.  The magnetic patterning cannot be attributed solely to simple \nmagnetic domain formation.  Theoretical proposals and data interpretations  for MnV 2O4 and \nMn 3O4 that rely o n assumption s of magnetic homogeneity must be revisited.   In addition, t he \npresence of nanoscale magnetic inhomogeneity in  these two related compounds suggests this \nphenomenon may be present in other multiferroic spinels.  \n We have established that mech anical strain plays an important role in the phenomenology \nof the low -temperature magnetic patterning.  In Mn 3O4, the tweed stripe pattern is defined by the \ntetragonal crystal grains, and stripe pitch is correlated to grain size. In MnV 2O4, the interwoven \nstripe pattern is also defined by the tet ragonal domain structure.  When the tetragonal domain \nstructure is determined at experimentally accessible temperatures, we can control the magnetic \npatterning through application of an external magnetic field.  Ind ucing mechanical strain in \nMnV 2O4 produces a more complex magnetic pattern at intermediate magnetic fields, and stabilizes \nmagnetic inhomogeneity at higher magnetic fields. These findings are consistent with theoretical \nresults showing that mesoscale magnetic inhomogeneity can significantly lower the energy barrier \nfor strain-  and field -dependent phase changes in complex materials , and offers strong evidence \nthat magnetic domain formation plays an important role in the magnetoresponsive behavior of \nthese sp inel materials.  \n \nAcknowledgments  \nResearch was supported by the U.S. Department of Energy under Award Number DE -FG02 -\n07ER46453 (B.W., X.W., T.N., and R.B),  and by the National Science Foundation under Grant s \nNSF DMR 1464090 and NSF DMR 1800982 (S.L.G. and S.L.C.)  and NSF DMR 1455264 \n(G.J.M).  Research at the National High Magnetic Field Laboratory was supported by the National \nScience Foundation under Grant DMR -1157490, and by the state of Florida. 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Fishman, “Phase  transformations of the  \ndecomposition type in systems with orbital degeneracy,” Low Temperature Physics  28, 613 \n(2002). \n Supplemental  Section  \n \nMethods  \nThe magnetic probes used for the experiment s detailed in this document are derived from \ncommercially available AFM cantilevers  using a unique evaporation process .  Cantilevers h ad \nvarying spring constants and natura l frequencies in the ranges k ~0.01- 0.3 N/m and f 0 ~ 10-30 kHz. \nQuality factors al so varied, centering around Q  ~ 300,000 at T  = 4K in vacuum.  To create magnetic \nprobes suitable for use at high magnetic fields and cryogenic temperatures, we coat only a small \nportion of the cantilever tip, as showed in Figure S1(a).  The cantilever is mounted on a pair of \ntranslation stages, which allow the tip to be positioned directly behind a razor blade.  A trilayer of \nTi-FeCo -Ti is then applied to the exposed portion th rough electron- beam evaporation.  The two Ti \nlayers both promote adhesion and prevent oxidation of the magnetic FeCo layer.  We use a 70%-\n30% mixture of Fe and Co as the magnetic material because it has the maximum saturation \nmagnetization observed in bimetallic alloys  [S1].  The coercive field of the MFM probes is \ntypically around B C=150G as measured using a room -temperature testing apparatus.  \n \nFigure S 1: Magnetic probe preparation. (a) The magnetic probe is created by coating a commercially- available AFM cantilever \nwith magnetic material.  Coating only a portion of the cantilever tip prevents unwanted effects caused by high magnetic fields \nand cryogenic operation. (b) SEM micrograph used as an initial verification that magnetic material has been deposited on the tip.  \nThe material appears as an area of light contrast on the upper -left portion of the cone.   \nThe directional evaporation method we use produc es a half -cone thin -film of magnetic \nmaterial covering approximately half the cantilever height. Figure S1(b) shows an SEM \nmicrograph of the magnetic probe after processing.  The trilayer can be seen as an area of light \ncontrast on the upper- left half of t he cantilever tip.  Our process for creating magnetic probes \nrequires minimal processing and take s good advantage of commercially available products , \nresulting in a fast turn -around time for magnetic probe production and preservation of the integrity \nand m echanical properties of the original cantilever.  \nWe used two different cantilevers for the MnV 2O4 measurements: one for the calibrated \nhigh-strain experiments and one for the low -strain experiments.  The cantilever used for the low -\nstrain experiments had parameters f  = 28.7 kHz, Q ≈ 200,000 and k = 0.23 N/m at room \ntemperature.  The cantilever used fo r the high -strain experiments had f  = 31.1 kHz, Q  ≈ 125,000 \nand k = 0.31 N/m at room temperature.  Both cantilevers were coated with 10 nm of FeCo for \nmagnetic sensitivity.  \nWe prepared the MnV 2O4 and Mn 3O4 samples using the same polishing regimen.  Each \nrough, as -diced sample was mounted on an aluminum block using Crystalbond epoxy. Using a \nrotary polishing machine, we manually polished the samples using increasingly fine (1μm, 0.3μm, \n0.05μm) alumina polishing powder in a water medium.  A final chemi -mechan ical polishing step \nwas done using 0.05μm alumina in a basic (pH = 10) medium.  For the final step, the sample was \nplaced in a vibratory polishing machine for approximately 6 hours. After polishing the samples \nwere thoroughly cleaned in an ultrasonic immer sion cleaner to remove polishing powder and other \ncontaminants.  The Crystalbond epoxy was dissolved using acetone, and we mounted the samples \nas described in the main text.  \n \nCalibration Experiment  \nThe ability to quantitatively interpret MFM measurements i s typically hampered by the \nlack of information about the nm -scale magnetic details of the magnetic probe [ S2,S3].  Without \nquantitative information about the nm -scale magnetization distribution of the probe, the probe-\nsample interaction cannot be accurate ly calculated and modeled. Several effects combine to make \na priori  calculations of the magnetization distribution of the probe difficult, including shape \nanisotropy effects, thin- film effects, and specifics of the magnetic probe material [ S4,S5].  Some \ntechniques exist for measuring the micromagnetic probe structure [ S4,S5 ], but they are time -\nconsuming and still do not provide the necessary accuracy.  The preferred option is to use a model \nmagnetic system, such as a current -carrying wire [ S6-S9] or calibra ted magnetic nanoparticles \n[S10 ], to measure the probe response. \n The quantity of interest for enabling quantitative MFM analysis is the point spread function \n(PSF), which describes the probe response to a point -like feature in the magnetic field curvature.  \nBy using a magnetic system in which the stray magnetic fields of the sample are known, the point \nspread function can be extracted by analyzing the known magnetic field distribution and the \nmeasured MFM frequency shift data.  Following previous work [S2 ], the frequency response of \nthe MFM probe can be expressed in terms of a 2- dimensional convolution between the magnetic \nfield curvature produced by the sample and the PSF of the tip:  \n∆𝑓𝑓(\n𝑟𝑟||→,𝑧𝑧)=𝑓𝑓0\n2𝑘𝑘�𝑀𝑀𝑧𝑧(2)(\n𝑟𝑟||′→−\n𝑟𝑟||→) 𝑑𝑑2𝐵𝐵𝑧𝑧\n𝑑𝑑𝑧𝑧2(\n𝑟𝑟||′→,𝑧𝑧−ℎ) 𝑑𝑑2\n𝑟𝑟||′→ where 𝑓𝑓0is the natural frequency of the cantilever, k is the spring constant, M z(2) is the point spread \nfunction, and h is the height above the sample at which the PSF is calculated.  \n The model magnetic system we used to extract the PSF of our magn etic probe was a 70nm \nthick current -carrying Au wire lithographically patterned onto a silicon substrate, pictured in \nFigure S2.  The wire had a rectangular cross -section and included various features which have \nbeen previously used to calibrate MFM probes  [S6-S9].  We found that the junction feature \noutlined in black (a step -like change in the wire width from 1µm to 4µm) was sufficient to measure \nthe PSF of our magnetic probe.  The small marks surrounding the wire are additional \nlithographically patterned location markers used to locate features on the sample.  At either side \nof the figure, the wire widens where it leaves the measurement region and connects to macroscopic \nelectrodes.  \n  \n \nFigure S2: Composite light micrograph of the MFM calibration sample.  The sample is composed of a current -carrying \nrectangular Au wire.  Constrictions, rings, zig- zags, and step junction features were incorporated to have a diverse set of \ncalibration options.  \n \nFigure S3(a) shows MFM frequency shift data for the area surrounding the junction at \nT=4.5K and B =20kG. The wire carried I=5mA of direct current to produce the known magnetic \nfield distribution. The MFM data show exactly the features we expect for the known magnetic \nfield distribution: opposite -sign frequency shifts ce ntered above the two wire edges and a four-\nfold increase in the frequency shift magnitudes near the narrower section of the wire.  At the inside \ncorner of the junction, there is a peak in the fre quency shift, which corresponds to current crowding \npredicted by the fi nite element analysis.  \nWe found that the magnetic response of the MFM probe was quantitatively captured by \nmodeling the magnetization distribution as a magnetic point -dipole.  This simplified model has \nbeen reported as a good approximation for bulk magnetic MFM probes [S6 -S9], but has never \nbeen investigated for thin -film magnetic probes. Figure S3(b) shows the predicted frequency shift \ndistribution produced by convolving the magnetic field curvature around the junction feature with \nthe point spread function (PSF) of a point -dipole -like magnetic probe (Figure S3(d)).  There is \nexcellent qualitative agreement between the simulation results and the MFM data.  Two line- cuts \nacross the MFM data are shown in Figure S3(c), along with best -fit results o f the point -dipole \nmodel obtained by varying the magnetic moment and position relative to the physical tip apex.  \nFor this particular MFM probe, the best -fit parameters were m z=2.4∙10-15 J/T and h = 400nm.  \nSomewhat surprisingly, these values are comparabl e to those previously reported for bulk \nmagnetic probes, despite the drastic differences in probe materials and geometries [ S6,S9].  This \nsimilarity further emphasizes the short -range nature of the MFM interaction which limits the \neffective volume of magnetic material to the region close to the tip apex, where the half -cone thin -\nfilm geometry of the probe is less apparent.  \n \nFigure S3: Calibration experiment results. (a) MFM data collected at T=4K, B=20kG, and I=5mA. (b) The MFM data shows \nexcellent agreement with simulation results for the magnetic field curvature of the wire (obtained through finite element \nmodeling) is convolved with a point -dipole -like point spread function (d).  (c) Line cuts through the 1 µm and 4 µm sections of the \nwire show quantitative fitting of the model results to the calibration data.  The resulting fitting parameter values were used to \nanalyze the MnV 2O4 data. \nIn order to fully capture the magnetic response of the MFM probe, we recorded data in the \npresence of various exter nal magnetic fields.  The magnetic fields produced by the calibration \nsample do not vary with external magnetic field, but the magnetic structure and overall \nmagnetization of the MFM probe change with increasing external magnetic field.  Between B=0kG \nand B=30kG, the effective dipole moment of the magnetic probe increased by approximately 30%, \nbut there was no discernable change in the overall structure of the PSF.  \n \nModeling of Magnetic Features in MnV 2O4 \n Using the measured PSF, we can determine the relati onship between the magnetic field \ncurvature and the measured frequency shift data, but the ultimate goal is to infer details about the \nmagnetic domain structure of the sample being studied.  Following the same strategy we used to \nfit the point -dipole model  to the measured PSF: we generate a proposed magnetic domain structure, \ncalculate the resulting magnetic field curvature, determine the resulting frequency shift pattern by \nconvolving with the PSF, and compare to the experimental data.  By changing the par ameters of \nthe original proposed domain structure, we can fit the model to the experimental data.  Quantitative \nMFM methods have been previously used to study current crowding effects in artificial systems \n[S2] and domain characteristics in longitudinal magnetic recording media [ S11-S13] but our work \nis the first application of quantitative MFM to a complex, natural material.  \n Due to the extended nature of the tip PSF, a single point in the magnetic field curvature \nproduced by the sample interacts with the tip even when the tip is relatively far away (on the order \nof 1μ m). Consequently, sharp changes in the sample magnetization will be smoothed out in the \nmeasured frequency data.   For the finest stripe features in MnV 2O4 (Figure 1(a)), the frequency \nshift profiles appear approximately sinusoidal, but the underlying sample magnetization is \nprobably much more like a square wave.  Square wave and sinusoidal wave magnetization profiles \nthus provide convenient bounds to our estimates of the sample magnetization.  The sharp \ntransitions of a square -wave -like sample magnetization will induce stronger frequency shifts and \nthe smooth transitions of a sinusoidal magnetization will induce weaker frequency shifts for a \ngiven sample magnetization.   \n \nFigure S4: Modeling magnetic domain.  (a) The depth of a magnetic domain has a measurable effect on the measured frequency \nshift only for depths less than approximately the probe -sample separation (dashed line).  For domains thicker than this distance, \nan infinite -thickness approximation is sufficient. (b) Domain width is a crucial parameter in determining the observed frequency \nshift. For widths less than approximately 900nm, overlap of signals from different features average destructively. (c -e) Different \ndomain wall configurations yield characteristically different frequency shift profiles, which can be used to qualitatively identify \ndomain magnetization direction.  The magnetization vectors (red) on either side of the wall as viewed along the sample surfac e \n(black) are indicated in the upper right of the respective sub- figure.  The three classes of domain wall consist of: (c) 180 ° change \nin the magnetization direction across the wall, where both magnetizations are normal to the sample surface; (d) 180 ° change in \nthe magnetizat ion direction across the wall, where both magnetizations are parallel to the sample surface; and (e) 90 ° change in \nthe magnetization across the wall, where the magnetization changes from normal to parallel to the sample surface.  \n \nFigure S4(a) and (b) show the effects of varying the domain width (in the plane of the \nsample surface) and depth (into the sample from the surface) on the maximum measured frequency \nshift for both sinusoidal and square wave stripe profiles.  The first important conclusion is that for \ndomains with depth comparable to or greater than the tip -sample separation (in this case h = 210nm \nas indicated by the vertical dashed line in S4(a)) the resulting frequency shift is independent of \ndomain depth.  It is highly unlikely that the magnetic domains we observe in MnV 2O4 exist purely \nat the sample surface, so we conclude that domain depth is not an important parameter in \nestimating the sample magnetization.  However, Figure S4(b) shows a non -trivial variation of the \nfrequency shift according to domain width for the range of stripe feature widths we observe in \nFigure 1(a), approximately 200 -400nm.  Note that for features wider than approximately 900nm, \nthe peak frequency decreases slightly.  This is the width at which neighboring domains can be \nresolved by the magnetic probe.  \nWe also used our modeling methods to identify the types of domain walls observed in the \nlow-strain MnV 2O4 sample.  Below the Jahn -Teller transition temperature, the tetragonal  domains \nin MnV 2O4 are oriented parallel to one of the three cubic crystal axes, resulting in different types \nof 90° and 180° domain walls. Furthermore, the MFM probe is sensitive to mainly the z -component \nof the magnetic field curvature, so both the type of domain wall and the relative orientation of the  \ntwo magnetizations to the MFM probe determine the resulting frequency shift profile.  Figure \nS4(c) shows the frequency profile of a single 180° domain wall where the two magnetizations \npoint normal to the sample surface.  In calculating the estimated doma in magnetization in MnV 2O4, \nwe used this type of domain wall.  Figure S4(d) shows the frequency profile of a 180° domain wall \nwhen the two magnetizations lie within the plane of the sample surface.  Due to the anisotropic \nsensitivity of the probe, this typ e of domain wall results in a characteristically different frequency \nprofile.  Finally, Figure S4(e) shows the frequency profile a 90° domain wall where only one \nmagnetization lies within the plane of sample surface.  For certain MFM images of the low -strain \nMnV 2O4 sample, such as Figure 2(c), we identified the types of domain walls and determined the \nmagnetization orientation in the domains defined by those walls.  While beyond the scope of the \ncurrent work, that ana lysis can be found elsewhere [S14]. \n \nConclusion  \nAchieving quantitative magnetic force microscopy results requires an additional step to \ncalibrate the instrument response of the MFM probe.  The easiest and most accurate way to \ncalibrate the probe is to measure a known magnetic system and extract the instrument response by \ncomparing the measured frequency shift data to the known magnetic field curvature.  For our \ncalibration experiment, we used a current -carrying wire with 2 -dimensional features. Additionally, \nwe attempted to exactly reprod uce the experimental conditions of our MnV 2O4 measurements \nduring the calibration experiment, including feature sizes, external magnetic fields, and \ntemperatures.  We found that the point spread function of the magnetic probe is well -characterized \nby a poi nt-dipole model.  We used further modeling of the magnetic domains MnV 2O4 to account \nfor ambiguities in the magnetization profile and identify the key physical parameters. Finally, we \nidentified the frequency shift profiles associated with different types of domain walls for \napplication to low -strain MnV 2O4 data.  \n \n \nReferences  \n[S1] C.W. Chen, Magnetism and Metallurgy of Soft Magnetic Materials  (North-Holland \nPublishing, 1977).   \n[S2] P.J. Rous, R. Yongsunthon, A. Stanishevsky , and E.D. Williams, “Real -space imaging of \ncurrent distributions at the submicron scale using magnetic force microscopy: Inversion \nmethodology,” Journal of Applied Physics  95, 2477 (2004). \n[S3] T. R. Albrecht, P. Grutter, D. Horne, and D. Rugar. “ Frequenc y modulation  detection using \nhigh-Q cantilever for enhanced force micros cope sensitivity”, Journal of Applied Physics  69, \n668 (1991). \n[S4] G. Matteucci, M. Muccini, and U. H artmann, “ Electron holography in the study of the \nleakage fi eld of magneti c force microscope sensor tips ,” Applied Physics Letters  62, 1839 \n(1993). \n[S5] G. Matteucci, M. Muccini, and U.  Hartmann. Flux measurements on  ferromagnetic \nmicroprobes by electron holography,” Physical Review B 50, 6823 (1994).  \n[S6] J. Lohau, S. Kirsch, A. Carl, G. Dumpich, and E.F.Wasserman. Quantitative determination \nof effective dipole and monopole moments of magnetic  force microscopy tips,” Journal of \nApplied Physics  86 (1999).  \n[S7] T. Goddenhenrich, H. Lemke, M. M uck, U. Hartmann, and C. Heiden, “ Probe calibratio n in \nmagnetic force microscopy, ” Applied Physics Letters  57, 2612 (1990). \n[S8] K.L. Babcock, V.B. Elings, J. Shi, D.D. Awshalom, and M. Dugas , “Field -dependence of \nmicroscopic probe s in magnetic force microscopy ,” Applied Physics Letters  69, 705 (1996). \n[S9] L. Kong and S.Y. Chou, “ Quantifi cation of magnetic force microscopy  using a micronscale \ncurrent ring,” Applied Physics Letters  70, 2043 (1997). \n[S10] S. Sievers, K. -F. Braun, D. Eberdeck, S. G ustafsson, E. Olsson, H.W. Schumacher, and U. \nSiegner, “ Quantitat ive measurement of the magnetic moment of individual magnetic \nnanoparticles by magnetic force microscopy, ” Small  8, 2675 (2012). \n[S11] R. Proksch, G. Skidmore, E.D. Dahlberg, S. Foss, J.J. Schmidt, C. Merton, B. Walsh, and \nM. Dugas , “Quantitative magnetic field measurements with the magnetic force microscope,” \nApplied Physics Letters  69, 2599 (1996). \n[S12] E. T. Yen, H. J. Richter, Ga -Lan Chen and G. Rauch, \"Quantitative MFM study on \npercolation mechanisms of longitudinal magnetic r ecording,\"  IEEE Transactions on \nMagnetics  33, 2701- 2703 (1997). \n[S13] T. K. Taguchi, A. Takeo and Y. Tanaka, \"Quantitative MFM study on partial erasure \nbehavio r of longitudinal recording,\"  IEEE Transactions on Magnetics  34,1973- 1975 (1998). \n[S14] B. Wolin, “Real -space magnetic imaging of the spinel MnV 2O4,” Ph.D. Dissertation, \nUniversity of Illinois at Urbana- Champaign (2017). "    },    {        "title": "1806.04177v2.Crystal_fields_and_Kondo_effect__Magnetic_susceptibility_of_Cerium_ions_in_axial_crystal_fields.pdf",        "content": " \n \n \nE-mail address:  H-Ulrich.Desgranges@nexgo.de  \n1 Scientifically unaffiliated  Crystal fields and Kondo effect: Magnetic susceptibility of Cerium  ions in axial  crystal fields  \nH.-U. Desgranges1 \nAlbert -Kusel -Str. 25, Celle, Germany  \n \nThe thermodynamic Bethe ansatz equations of the Coqblin -Schrieffer model have been \nsolved numerically. The full N=6 (J=5/2) degeneracy of the Hund’s rule ionic ground state \nof Ce is taken into account. Results for t he temperature dependent magnetic susceptibility \nparallel and perpendicu lar to the crystal axis are presented.  The deviations, due to the Kondo \neffect, to the non-interacting ion result s are pointed out.  \nKeywords:  \nCoqblin -Schrieffer model, Kondo effect, Thermodynamic Bethe ansatz, crystal fields, magnetic susceptibility  \n \n1. Introduction  \nThe complex behavior of Cerium compounds is \nstill attracting interest  among experimentalist. The \nchange of the effective spin -degeneracy N due to \nthe interplay between Kondo and crystal field ef-\nfects has been studied experimentally by investi-\ngating various Cerium based pseudo -ternary inter-\nmetallic substitution serie s [1]. \nThe single ion Kondo model and its generalization \nto a N -fold degenerate ionic configuration, the \nSU(N) Coqblin -Schrieffer model  [2], has been \nused successfully to describe the thermodynamic \nproperties of dense Kondo systems  [3-5]. How-\never, for fitting the influence of crystal fields on \nthe paramagnetic susceptibility experimentalist s \nhave had to resort to the text-book result for non -\ninteracting ions  [6-9]. \nA broad basis for comparison with experiments on \nthe specific heat in zero magnetic field over the \nwhole temperature range has recently been \nprovided [10] and applied successfully [ 11]. The \nnumerical solution of the  thermodynamic Bethe \nansatz equations for the N = 6 model (Cerium 3+  \nions) with gener al crystal field configurations  was \nachieved by a  new high field / low temperature \nexpansion to calculate the limiting values of the \nunknown functions. In a subsequent paper [12] the \nmethod was extended to the calculation of the \nmagnetic susceptibility. However, this was \nrestricted to t he case of N = 4 that is applicable to \nCerium in a temperature range in which  the highest \nKramers doublet may be neglected.  \nFor the present work the method to calculate the \nmagnetic susceptibility  has been further developed \nto the N = 6 model  with axial  crystal fields that \nsplit the 6 -fold degenerate ground state of the Ce3+ \nions into three doublets that are eigenstates of the total angular momentum operator J z. Here results \nare presented for the case that the energy levels \ncorresponding to the eigenstat es |±1/2>, |±3/2>, \n|±5/2> are sequenced in that order.  \nThe new results allow for a quantitative compari-\nson with experimental data with relevance to e.g. \nCePt 5/Pt(111) [13]. \n \n2. Model and m ethods  \nThe Bethe ansatz solution of the Coqblin -Schrief-\nfer model [14, 15] was used by Schlottmann [16] \nto calculate the anisotropic magnetic susceptibility \nat zero temperature for t he ionic crystal field Ham-\niltonian given by  [16, 17]:  \n00\n2 2 4 4= O   +  \n- H - Hion\nB z B xH b b O\ng S g S\n   (1) \nHere \n0\n2O  and \n4\n2O  denote the usual Stevens opera-\ntors [18], and \nH\n  and \nH  are the parallel and \ntransversal projection of the magnetic field onto \nthe crystal  field axis, respectively.  \nFor a parallel magnetic field the energy levels of \nthe three doublets are given by:  \n1\n1/2 2 4 2\n3\n3/2 2 4 2\n5\n5/2 2 4 2E = 120 H,\nE = 180 H,\nE = 10 60 H.B\nB\nBb b g\nb b g\nb b g\n\n \n \n\n  (2) \nThe calculation for a transversal magnetic field is  \ndone by second order perturbation theory [19] in-\ncluding van Vleck  terms quadratic in H.  \nThe calculation of thermodynamic properties fol-\nlows the lines presented in the preceding publica-\ntions [1 0, 12]. Details will be published elsewhere Crystal fields and Kondo effect : Magnetic susceptibility of Cerium ions in axial crystal fields  2 \n \n \n [20]. The splittings between adjacent ionic energy \nlevels A r ≡ Er+1 – Er, 1 ≤ r ≤ N, serve as generalized \nfields determining the limiting values of the un-\nknown functions of the infinite system of nonlinear \nintegral equations.  The energy levels according to \neq. (2) have to be  put into sequence such that A r ≥ \n0. For reasons of brevity we restrict ourselves to \nthe case E 1/2 < E 3/2 < E 5/2, region II in the notation \nof Schlottmann [16]. The extrapolation of our re-\nsults to zero temperature ha s served as a check on \nour calculation for A 2 = A 4. \nTemperature, magnetic , and crystal fields are \nscaled by the Kondo temperature in the absence of \nall fields TK(N=6). At low temperatures and large \ncrystal fields the influence of the higher doublets \nmay be neglected so that the thermodynamic prop-\nerties are governed by an effective spin -1/2 system \nwith an effective Kondo temperature T K(N=2) \ngiven by the relation  \n\nK²T N=2  = .2  0Bgµ\nT\n   (3) \nFor small temperatures compared with this low \ntemperature scale the susceptibility decreases \nquadratically.  \nIn this case e q. (3) may be used to determine \nTK(N=6) from the scaling relation \n32\nK 4 2 KT (N=6) = C A T (N=2)\n where the numerical \nfactor C 4 has been approximated by \n4 4 2C  = 17.08 + 17.05 A /A\n [10].  \nFor small values of the crystal fields the sixfold de-\ngeneracy at vanishing magnetic field is almost re-\nstored  even at small temperatures . The qualitative \nbehavior of the susceptibility is then the same as \nwithout crystal fields [2 1].  \n \n3. Results  and conclusion  \nThe numerical solution of the thermodynamic Be-\nthe ansatz equations has been achieved for a num-\nber of crystal field splittings. The relative accuracy \nis expected to be better than 1%. All results are de-\npicted in the form of the inverse magnetic suscepti-\nbility χ-1. The symbols indicate the finite tempera-\nture grid. All energies are measured in units of \ntemperature (k B = 1).  \nIn Fig. 1. results for the Coqblin -Schrieffer model \nwith axial crystal fields ( solid  lines) ar e compared \nwith the corresponding curves for non-interacting \nions [22] (dashed lines) . A parallel magnetic field is considered exemplarily  for energy splittings  \nA2 / TK(N=6) = 1 while A 4 / TK(N=6) is varied.  \nThe main point to show here is: While the principal \noutlook of the two sets of curves is sim ilar there is \na consistent quantitative discrepancy. This means \nthat – if the Kondo effect plays a role in the low \ntemperature physics of a certain compound – the \nfitting of the (inverse) susceptibility curves with \nthe model of non -interacting ions cannot reliably \nbe used to determine the crystal field splittings.  \n \nFig. 1. (Color online) Inverse magnetic susceptibility χ-1 \nas function of temperature T scaled by the Kondo \ntemperature in the absence of crystal fields compared \nwith the corresponding non -interacting ion curves \n(dashed lines)  \nTo facilitate a quantitative description of experi-\nmental results the numerical results for  the \nCoqblin -Schrieffer model with axial crystal fields \nare depicted in Figure 2 for a fixed ratio of crystal \nfield splittings A 4 / A 2 =1.0, 2.0, and 4.0, respec-\ntively.  \nHere, the full symbols represent the results for a \nparallel magnetic field while the op en symbols cor-\nrespond to a transversal magnetic field. The dashed \nCrystal fields and Kondo effect : Magnetic susceptibility of Cerium ions in axial crystal fields  3 \n \n \n line shows the Coqblin -Schrieffer  result without \ncrystal fields [2 1]. \n Fig. 2. (Color online) Inverse magnetic  \nsusceptibility χ-1 as function of temperature for parallel \nand transversal magnetic field. The ratio of crystal field \nsplittings is kept fixed to a) 1.0, b) 2.0, and c) 4.0  \n \nUnless the crystal field splitting is too small, the \ninverse susceptibility in a parallel magnetic f ield \nhas its minimum at T = 0 and increases steeply at \nlow temperatures. It then shows a pronounced  peak \nfor smaller values of A4 / A 2. The peak decreases \nin height and turns into a plateau at intermediate \nvalues of A4 / A 2. On increasing A4 / A 2 further the \nplateau turns into a shoulder with an inflection \npoint before the Curie -Weiss like high -temperature \nbehavior (i.e. a straight line) is reached.  \n \nThe inverse susceptibility in a transversal magnetic \nfield in comparison does not display such dis tinct \nfeatures. However, the curves bend to a straight \nline in about the same temperature range as the \ncurves for the parallel magnetic field.  \n \n4. Summary  \nOn the basis of a recent ly found new method the \ninfinite set of coupled, nonlinear integral equations \nCrystal fields and Kondo effect : Magnetic susceptibility of Cerium ions in axial crystal fields  4 \n \n \n describing the thermodynamics of the N=6 \nCoqblin -Schrieffer model has been solved.  \nThereby the full degeneracy of the J=5/2 Hund’s \nrule ionic ground state of Ce is taken into acc ount. \nResults for the (inverse) anisotropic magnetic sus-ceptibility for ions in an axial crystal field are pre-\nsented to provide material for a quantitative analy-\nsis of experimental results  [23]. \nThe deviations, due to the Kondo effect, to the \nnon-interacting ion results are pointed out.  \n \nAcknowledgment  \nI thank Kai Fauth for providing  some  valuable references.  \n \nReferences  \n \n  [1] See for example: C. Gold, L. Peyker, W. Scherer, G. Simeoni, T. Unruh, O. Stockert, H. Michor , \n        and E. -W. Scheidt, J. Phys.: Condens. Matter  24 (2012) 355601 . \n  [2] B. Coqblin and J. R. Schrieffer, Phys. Rev.  185 (1969) 847 . \n  [3] A. C. Hewson, The Kondo Problem to Heavy Fermions  (Cambridge University Press, Cambridge, 1993) . \n  [4] I. Aviani, M. Miljak, V. Zlatić, K. D. Schotte, C. Geibel, F. Steglich, Phys. Rev. B  64 (2001) 184438 . \n  [5] A. P. Pikul, U. Stockert, A. Steppke, T. Cichorek, S. Hartmann, N. Caroca -Canales, N. Oesch ler, \n       M. Brando, C. Geibel , and F. Steglich, Phys. Rev. Lett.  108 (2012) 066405 . \n  [6] G. Motoyama, M. Watanabe, A. Sumiyama, and Y. Oda , J. Phys.: Conf. Ser. 150 (2009) 052173 . \n  [7] H. Mendpara, Devang A. Joshi, A. K. Nigam, A. Thamizhavel, JMMM 377 (2015) 325 . \n  [8] A. Maurya, R. Kulkarni, A. Thamizhavel, S. K. Dhar , and D. Paudyal, .J. Phys. Soc. Jpn. 85  \n        (2016) 034720 . \n[9] V. K. Anand, D. T. Adroja , A. D. Hillier, K. Shigetoh, T. Takabatake, J. -G. Park, K. A. McEwen,  \n        J. H. Pixley, and Q. Si, J. Phys. Soc. Jpn. 87 (2018) 064708 . \n[10] H.-U. Desgranges, Physica B 454 (2014) 135 . \n[11] S. Patil, A. Generalov, M. Güttler, P. Kushwaha, A. Chikina , K. Kummer, T. C. Rödel,   \n        A. F. Santander -Syro, N. Caroca -Canales, C. Geibel, S. Danzenbächer, Y. Kucherenko, C. Laubschat,  \n        J. W. Allen, D. V. Vyalikh, Nature Comm . 7 (2016)  11029.  \n[12] H. -U. Desgranges, Physica B 473 (2015 ) 93. \n[13] C. Praetorius and K. Fauth, Phys. Rev. B  95 (2017) 115113 . \n[14] A. M. Tsvelick and P. B. Wiegmann, J. Phys. C  15 (1982 ) 1707; J. W. Rasul, in Valence Instabilities , \n        edited by P. Wachter and H. Boppart (North -Holland, Amsterdam, 1982), p. 49 . \n[15] For  reviews see N. Andrei, K. Furuya and J. H. Lowenstein, Rev. Mod. Phys.  55 (1983) 331 and  \n        A. M. Tsvelick and P. B. Wiegmann, Adv. Phys.  32 (1983 ) 453. \n[16] P. Schlottmann, JMMM 52 (1985) 211 . \n[17] E. Segal and W. E. Wallace, J. Solid State Chem. 13 (1975) 201 . \n[18] See for example: A. Abragam and B. Bleany, Electron Paramagnetic Resonance of Transition Ions  \n        (Clarendon Press, Oxford 1970) . \n[19] H. Lueken, M. Meier, G. Klessen, W. Bronger, and J. Fleischhauer, J ournal  of the Less-Common Metals  \n        63 (1979) 35 . \n[20] H. -U. Desgranges, in preparation . \n[21] V. T. Rajan, Phys. Rev. Lett. 51 (1983) 308.  \n[22] J. H. Van Vleck, The Theory of Electric and Magnetic Susceptibilities, Oxford University Press, London,  \n        1932.  \n[23] Numerical data, also for different crystal field parameters, can be made available upon request to the  \n        author. "    },    {        "title": "1806.07989v1.Data_driven_studies_of_magnetic_two_dimensional_materials.pdf",        "content": "Data-driven studies of magnetic two-dimensional materials\nTrevor David Rhone,1,\u0003Wei Chen,1Shaan Desai,1Amir Yacoby,1and Efthimios Kaxiras1, 2\n1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA\n2School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA\n(Dated: Friday 22ndJune, 2018)\nWe use a data-driven approach to study the magnetic and thermodynamic properties of van\nder Waals (vdW) layered materials. We investigate monolayers of the form A 2B2X6, based on\nthe known material Cr 2Ge2Te6, using density functional theory (DFT) calculations and machine\nlearning methods to determine their magnetic properties, such as magnetic order and magnetic\nmoment. We also examine formation energies and use them as a proxy for chemical stability. We\nshow that machine learning tools, combined with DFT calculations, can provide a computationally\ne\u000ecient means to predict properties of such two-dimensional (2D) magnetic materials. Our data\nanalytics approach provides insights into the microscopic origins of magnetic ordering in these\nsystems. For instance, we \fnd that the X site strongly a\u000bects the magnetic coupling between\nneighboring A sites, which drives the magnetic ordering. Our approach opens new ways for rapid\ndiscovery of chemically stable vdW materials that exhibit magnetic behavior.\nI. INTRODUCTION\nThe discovery of graphene ushered in a new era of stud-\nies of materials properties in the two-dimensional (2D)\nlimit [1]. For many years after this discovery only a\nhandful of van der Waals (vdW) materials were exten-\nsively studied. Recently, over a thousand new 2D crystals\nhave been proposed [2, 3]. The explosion in the number of\nknown 2D materials increases demands for probing them\nfor exciting new physics and potential applications [4, 5].\nSeveral 2D materials have already been shown to exhibit\na range of exotic properties including superconductivity,\ntopological insulating behavior and half-metallicity [6{9].\nConsequently, there is a need to develop tools to quickly\nscreen a large number of 2D materials for targeted prop-\nerties. Traditional approaches, based on sequential quan-\ntum mechanical calculations or experiments are usually\nslow and costly. Furthermore, a generic approach to de-\nsign a crystal structure with the desired properties, al-\nthough of practical signi\fcance, does not exist yet. Re-\nsearch towards building structure-property relationships\nof crystals is in its infancy [10{12].\nLong-range ferromagnetism in 2D crystals has recently\nbeen discovered [13, 14], sparking a push to understand\nthe properties of these 2D magnetic materials and to\ndiscover new ones with improved behavior [15{18]. 2D\ncrystals provide a unique platform for exploring the mi-\ncroscopic origins of magnetic ordering in reduced dimen-\nsions. Long-range magnetic order is strongly suppressed\nin 2D according to the Mermin-Wagner theorem [19], but\nmagnetocrystalline anisotropy can stabilize magnetic or-\ndering [20]. This magnetic anisotropy is driven by spin-\norbit coupling which depends on the relative positions\nof atoms and their identities. As a result, the magnetic\norder should be strongly a\u000bected by changes in the struc-\ntural arrangements of atoms and chemical composition of\n\u0003trr715@g.harvard.eduthe crystal.\nChemical instability presents a crucial limitation to the\nfabrication and use of 2D magnetic materials. For in-\nstance, black phosphorous degrades upon exposure to air\nand thus needs to be handled and stored in vacuum or un-\nder inert atmosphere. Structural stability is a necessary\ningredient for industrial scale application of magnetic\nvdW materials, such as CrI 3and Cr 2Ge2Te6[13, 14]. In\naddition to designing 2D materials for desirable magnetic\nproperties, it is important to screen for those materials\nthat are chemically stable. In our approach, we employ\nthe calculated formation energy as a proxy for the chem-\nical stability [21]. In particular, we obtain the total ener-\ngies of systems at zero temperature, and calculate the for-\nmation energy as the di\u000berence in total energy between\nthe crystal and its constituent elements in their respec-\ntive crystal phases. This quantity determines whether\nthe structure is thermodynamically stable or would de-\ncompose. This formulation ignores the e\u000bects of zero-\npoint vibrational energy and entropy on the stability.\nRecently, machine learning (ML) has been combined\nwith traditional methods (experiments and ab-initio cal-\nculations) to advance rapid materials discovery [2, 3, 21{\n27]. ML models trained on a number of structures can\npredict the properties of a much larger set of materi-\nals. In particular, there is presently a growing inter-\nest in exploiting ML for discovery of magnetic materi-\nals [17, 28]. Data-driven studies of ferromagnetism in\ntransition metal alloys have highlighted the importance\nof novel data analytics techniques to tackle problems in\ncondensed matter physics [28]. It is conceivable that tun-\ning the atomic composition could provide an additional\ndegree of freedom in the search for stable 2D materi-\nals with interesting magnetic properties [29]. Even more\ncompelling is the ability of ML tools to assist in uncov-\nering the physics underlying the stability and magnetism\nof 2D materials [30, 31]. Speci\fcally, ML methods can\nidentify patterns in a high-dimensional space revealing\nrelationships that could be otherwise missed.arXiv:1806.07989v1  [cond-mat.mtrl-sci]  20 Jun 20182\nII. METHODOLOGY\nIn order to develop a path towards discovering 2D\nmagnetic materials, we generate a database of structures\nbased on a monolayer Cr 2Ge2Te6(Fig. 1(a)) using den-\nsity functional theory (DFT) calculations [32]. The pos-\nsible structures amount to a combinatorially large num-\nber of type A 2B2X6(\u0018104) with di\u000berent elements oc-\ncupying the A, B and X sites. We select a subset of 198\nstructures due to computational constraints. We obtain\nthe total energy, magnetic order, and magnetic moment\nof each structure. The ground-state properties were de-\ntermined by examining the energies of the fully optimized\nstructure with several spin con\fgurations, including non-\nspin-polarized, parallel, and anti-parallel spin orienta-\ntions at the A sites (Fig. 1(b)).\nWe then employ a set of materials descriptors which\ncomprise easily attainable atomic properties, and are\nsuitable for describing magnetic phenomena. We employ\nadditional descriptors which are related to the formation\nenergy [33]. The performance of descriptors in predict-\ning the magnetic properties or thermodynamic stability\nsheds some light into the origin of these properties.\nTo create the database we use DFT calculations [34]\nwith the VASP code [35]. We create the di\u000berent struc-\nFIG. 1. (a) Crystal structure of the A 2B2X6lattice. (b)\nMagnetic orders considered in the A plane, labelled parallel\nand anti-parallel. (c) Elements used for substitution of A\n(blue), B (red) and X (magenta) sites.\ntures by substituting one of two Cr atoms (A site) in the\nunit cell with a transition metal atom, from the list: Ti,\nV, Cr, Mn, Fe, Co, Ni, Cu, Y, Nb, Ru. In the two B sites\nwe place combinations of Ge, Si, and P atoms, namely\nGe2, GeSi, GeP, Si 2, SiP, P 2. The atoms at X sites were\neither S, Se, or Te, that is, S 6, Se6, Te 6. Fig 1(c) shows\nthe choice of substitution atoms in the Periodic Table.\nAn example of a structure created through this process\nis (CrTi)(SiGe)Te 6.\nThe careful choice of descriptors is essential for the suc-\ncess of any ML approach [36, 37]. We use atomic proper-\nties data from the python mendeleev package 0.4.1 [38] to\nbuild descriptors for our ML models. We performed su-\npervised learning with atomic properties data as inputs,\nwith target properties the magnetic moment and the for-\nmation energy. The choice of the set of descriptors for the\nmagnetic properties was motivated by the Pauli exclusion\nprinciple, which gives rise to the exchange and super-exchange interactions. We also consider the magneto-\ncrystalline anisotropy [39] by building inter-atomic dis-\ntances and electronic orbital information into our descrip-\ntors. With respect to the formation energy, the choice of\ndescriptors was motivated, in part, by the extended Born-\nHaber model [33], and include the dipole polarizability,\nthe ionization energy and the atomic radius (see Supple-\nmental Materials for a full list of atomic properties and\ndescriptors used [40]).\nThe data were randomly divided into a training set,\na cross-validation set and a test set. Training data and\ncross-validation were typically 60% of the total data while\ntest data comprised 40% of all the data. We employed the\nfollowing ML models: kernel ridge regression, extra trees\nregression, and neural networks. Kernel ridge regression\nwith a gaussian kernel has been shown to be successful\nin several materials informatics studies. Extra trees re-\ngression allows us to determine the relative importances\nof features used in a successful model [41]. An analysis\nof hidden layers of the deep neural networks could allow\nus to identify patterns in 2D materials properties data,\nthereby guiding theoretical studies [31].\nIII. RESULTS AND DISCUSSION\nA. Magnetic properties\nWe \fnd that the non-spin-polarized con\fguration has\nthe highest energy for all the structures considered. That\nis, all structures prefer either parallel or anti-parallel or-\ndering in the A plane. Fig. 2(a) shows the energy dif-\nference of parallel and anti-parallel spin con\fgurations.\nNegative (positive) energy di\u000berence means the parallel\n(anti-parallel) is more stable. We note that, because of\nthe supercell size limit, we do not consider more com-\nplex spin con\fgurations in this study. For example, the\nlowest-energy spin con\fguration of Cr 2Si2Te6was re-\nported to be zigzag anti-ferromagnetic type [42]. Total\nmagnetic moments for the lowest energy spin con\fgura-\ntion of each structure are presented in Fig. 2(b). We \fnd\nthat only atoms in the A sites show \fnite magnetic mo-\nments, while the moments in the B and X sites are small.\nDistinct patterns for regions of high and low magnetic\nmoments are observed for X = Te, Se and S in Fig. 2(b).\nStructures created by substituting non-magnetic atoms\nat the A site, such as Cu, have small variations in their\nrelatively small magnetic moments, as seen in the rows\nof Fig. 2(b). However, substitutions of magnetic atoms,\nsuch as Mn, result in a set of structures with a large vari-\nation in the magnetic moment, with a much larger upper\nlimit to the range of values observed.\nBoth the magnetic order and magnetic moment are\nsensitive to the occupancy of B and X sites, even though\nthe atoms in these sites have negligible contribution to\nthe overall magnetic moment. Atoms in the X sites\nstrongly mediate the magnetic coupling between neigh-\nboring A sites [42]. Atoms at the B sites can a\u000bect the3\nFIG. 2. (a) Energy di\u000berence between parallel and anti-\nparallel spin con\fgurations ( Eparallel \u0000Eanti-parallel in eV=unit\ncell) of A 2B2X6structures. (b) Magnetic moment per unit\ncell (in\u0016B) for each A 2B2X6structure at the lowest energy\nspin con\fguration. The occupation of the two B sites is shown\non the horizontal axis while that of one of the A site is shown\non the vertical axis.\nrelative positions of A and X sites. Direct exchange be-\ntween \frst nearest neighbor A sites competes with super-\nexchange interactions mediated by the p-orbitals at the\nX sites. The ground state magnetic order is determined\nby the interplay between \frst, second and third nearest\nneighbor interactions. Changing the identity of one of\nthe A, B or X sites a\u000bects the interplay between the di-\nrect exchange and super-exchange interactions. Recent\nwork has shown that applying strain to the Cr 2Si2Te6\nlattice tunes the \frst nearest neighbor interaction, re-\nsulting in a change in the magnetic ground state from\nzig-zag antiferromagnetic to ferromagnetic [42]. Our\nwork demonstrates that tuning the composition of the\nA2B2X6lattice can have an equivalent e\u000bect. For in-\nstance, whereas X=Te structures show more parallel (\u0016\u0016P)\nthan anti-parallel (anti-\u0016\u0016P) spin-con\fgurations with lower\nenergy, there is a clear change when X = Se or S. As X\nmoves up the periodic table, there are increasingly more\nregions of anti-parallel spin con\fguration, as well as re-\ngions in which\u0016\u0016Pand anti-\u0016\u0016Pare degenerate. In particu-lar, we \fnd that the distance between nearest neighbor\nA and X sites, as well as two adjacent X sites is linked\nto the magnitude of the magnetic moment (see Supple-\nmental Materials for details).\nWe use extra trees regression [41] to approximate the\nrelationship between the total magnetic moment and a\nset of descriptors designed for magnetic property pre-\ndiction (see Supplemental Materials). Training and test\ndata are considered for the X = Te, Se, and S structures\nindividually. The model performance for X = Te is shown\nin Fig. 3(a). We \fnd reasonable prediction performance\nfor X = Te that deteriorates for X = Se and is even worse\nfor X = S. This suggests that our model, along with the\nset of descriptors used to predict X = Te structures, does\nnot generalize well. This could arise due to the fact that\nthere are more structures that have degenerate\u0016\u0016Pand\nanti-\u0016\u0016Pspin con\fgurations if X=Se and S than for X =\nTe. Nevertheless, subgroup discovery can be exploited\nto learn more about these systems [43], implying that\nthe identity of the X site strongly a\u000bects the magnetic\nproperties of the structures.\nFIG. 3. ML predictions of magnetic moments of A 2B2X6\nstructures. (a) Extra trees model performance for the mag-\nnetic moment (in \u0016B) prediction. A subset of structures for\nX = Te are displayed. The red squares indicate the test data,\nthe green circles show the training data. (b) Top six descrip-\ntors for the extra trees prediction of the magnetic moment.\nThe size of the bar indicates relative descriptor importance\n(see text for details).\nDetermining which descriptors are most important for\nmaking good predictions of a property can be exploited\nfor knowledge discovery, especially when a large num-\nber of descriptors are available but their relationships\nwith the target property are not known [44]. Fig. 3(b)\nshows the descriptor importances [45] as derived from\nextra trees regression. It shows that the ` the number of\nvalence electrons ' [\\nvalence max dif\" in Fig. 3(b)], ` the\naverage covalent radius ' [\\covalentrad avg\" in Fig. 3(b)]\nand the ` average number of spin up electrons ' [\\Nup avg\"\nin Fig. 3(b)], linked to the atomic dipole magnetic mo-\nment, are among the top six descriptors in the set exam-\nined. The magnetic moment per unit cell is a function\nof the magnetic moments of the individual atoms in the\nunit cell. We examine the local magnetic moments at the4\nA sites to determine how the magnetic moment per unit\ncell is constructed. The local magnetic moment at the\nA sites (A Crand A TM) can be di\u000berent from the atomic\ndipole magnetic moment of the corresponding element.\nFor instance, while the atomic magnetic moment of Cr3+\nis 3\u0016B, the local magnetic moment at A Cr\ructuates\nfrom 2.7 to 3.2 \u0016B. Fig. 4 (a) shows the local magnetic\nmoment at A TM.\nFIG. 4. (a) Local magnetic moment of the transition metal\nA site, A TM(in\u0016B). (b) Formation energy (in eV/cell) for\nA2B2X6structures at the lowest energy spin con\fguration.\nConventions are the same as in Fig. 2.\nB. Formation energy\nIn addition to identifying structures with speci\fc mag-\nnetic properties, the ability to screen for chemical stabil-\nity is also important. DFT-calculated formation energies\n(for the lowest energy spin con\fguration) are shown in\nFig. 4 (b). Structures comprising certain elements, such\nas Y, decrease the formation energy considerably in com-\nparison to those without it. Certain transition metals,\nsuch as Cu, tend to destabilize the (CrA)B 2X6struc-\ntures. The formation energy becomes more negative as\nthe substituted atom at the A site goes from the left to\nthe right of the \frst and second row of transition metal el-ements in the Periodic Table. This is linked to the \flling\nof the d-orbital, where elements with a \flled d-orbital do\nnot form chemical bonds with other elements. Varying\nthe composition at the B site does not appear to have a\nstrong impact on the formation energy (see Supplemen-\ntal Materials, Fig. S1). Changing the X site from Te to\nSe and then S results in the overall trend of decreasing\nformation energy.\nTo exploit the trends in the formation energy data, we\nuse statistical models to predict the formation energy and\nto infer structure-property relationships. We \fnd that\nsome descriptors, such as the atomic dipole polarizability,\nare strongly correlated with the formation energy, and\nare therefore important in generating good ML predic-\ntions. Since useful descriptors are not always revealed in\nan analysis of the Pearson correlation coe\u000ecient [44], we\nconsider other methods to learn descriptor importances\nsuch as the extra trees model [45]. Using the ML mod-\nFIG. 5. Formation energy prediction performance of (a) ker-\nnel ridge regression, (b) deep neural network regression and\n(c) extra trees regression. Red squares are test data and green\ncircles training data. (d) Performance of the extra trees re-\ngression model on the test data as the training set size in-\ncreases, in terms of the R2and mean absolute error (MAE)\nscores.\nels to predict the formation energy of A 2B2X6structures\npermits the quick calculation of the formation energy for\na large set of compounds. Whereas DFT calculations\nof 104structures could take up to 1 million CPU hours,\nthe ML prediction takes a few seconds. Fig. 5(a) shows\nthe prediction performance for kernel ridge regression us-\ning a gaussian kernel. Fig. 5(b) shows the performance\nof a neural network [46] while Fig. 5(c) shows the per-\nformance of the extra forests regression. Both training\nset and test set results are displayed, as well as the test5\nscores for kernel ridge regression, extra trees regression,\nand neural network regression.\nFurther analysis (see Supplemental Materials) shows\nthat the ` variance in the ionization energy of atoms '\nand the ` average number of valence electrons ' are the\ntwo most important descriptors in the set examined.\nThis demonstrates a link between the formation energy\nand the atomic ionization energy, emanating from the\nincreased atomic ionizability which produces stronger\nchemical bonding. In addition, the number of valence\nelectrons is linked to the number of electrons available\nfor bonding. For instance, substitutions by atoms with\na \flled outer orbital shell will create less stable bonds,\nleading to chemical instability. The ability of our models\nto generalize is demonstrated by the high scores on the\ntest data. We further examined how the test set perfor-\nmance varies with the training set size. Fig. 5(d) shows\ntest scores as a function of training set size using ex-\ntra trees regression. The test score reaches a plateau at\nabout a training set size of 40%, with test score (R2) as\nhigh as 0.91.\nC. High-throughput screening using ML models\nWe can use our trained ML models to make predictions\non a wide range of structures not included in the original\nDFT data set. Thus far, we have used our ML models\nto estimate the formation energy for an additional 4,223\nA2B2X6structures, constructed as follows: (i) For A site\nsubstitutions, we considered transition metals not used\nin the DFT dataset. (ii) We included Al, Sn and Pb in\nthe set of atomic substitutions for B sites (not shown).\n(iii) For the X sites, we added O to our previous choice of\nS, Se and Te. The resulting predictions, partly shown in\nFig. 6(a), provide a means to quickly screen a large data\nset of structures for chemical stability. For instance, our\nML predictions suggest that structures based on Er, Ta,\nHf, Mo, Zr, and Sc in the A site and Al in the B site are\nlikely to be stable and thus good candidates for further\nexploration.\nMagnetic moment predictions are shown in Fig. 6(b).\nFrom the results of the ML predictions we select struc-\ntures with formation energies below -1 eV and mag-\nnetic moments above 5 \u0016B(for X=Te only). From the\n4,223 predictions, we obtained 40 that satis\fed our con-\nstraints. 15 of these were randomly selected for veri-\n\fcation with DFT. 5 of these 15 structures were con-\n\frmed to have the expected properties within uncer-\ntainty. These 15 structures were then added to the\ntraining data to build an improved model for predict-\ning magnetic moment. A second iteration of predic-\ntion and veri\fcation by DFT generated three structures,\nall of which satis\fed the constraints within uncertainty:\n(CrTc)(SiSn)Te 6, (CrTc)Sn 2Te6, Cr 2(SiP)Te 6.IV. CONCLUSION\nWe presented evidence that the magnetic properties of\nA2B2X6monolayer structures can be tuned by making\natomic substitutions at A, B, and X sites. This provides\na novel framework for investigating the microscopic ori-\ngin of magnetic order of 2D layered materials and could\nlead to insights into magnetism in systems of reduced\ndimension [13, 14]. Our work represents a path toward\ntailoring magnetic properties of materials for applications\nin spintronics and data storage [47]. We showed that ML\nmethods are promising tools for predicting the magnetic\nproperties of 2D magnetic materials. In particular, our\ndata-driven approach highlights the importance of the X\nsite in determining the magnetic order of the structure.\nChanging the composition of the A 2B2X6structure alters\nthe inter-atomic distances and the identity of electronic\norbitals. This impacts the interplay between \frst, second\nand third nearest neighbor exchange interactions, which\ndetermines the magnetic order.\nOne goal of this work was to \fnd magnetic 2D ma-\nterials that are also thermodynamically stable. ML\nmodels were trained to predict chemical stability that\nallow the rapid screening of a large number of pos-\nsible structures. We showed that the chemical sta-\nbility of A 2B2X6structures based on Cr 2Ge2Te6can\nbe tuned by making atomic substitutions. Examples\nof structures that satisfy both magnetic moment and\nformation energy requirements include the following:\n(CrTc)(SiSn)Te 6and (CrTc)Sn 2Te6, not included in our\noriginal DFT database. In addition, we found structures\nin our set of DFT calculations that also satis\fed our re-\nquirements: Cr 2(SiP)Te 6, (TiCr)(SiP)Te 6, (YCr)Ge 2S6\nand (NbCr)Si 2Te6.\nThis work provides the impetus for further exploration\nof structures with other architectures not considered\nhere, that is, with more complex atomic substitutions\nbeyond 1 in 2 replacement of Cr atoms at the A site. We\nestimate a total number of at least 3 \u0002104structures of\nthe A 2B2X6type described in Fig. 1. A computation-\nally e\u000ecient estimation of the magnetic properties and\nformation energy is required to quickly explore this vast\nchemical space. We also expect the ML methods explored\nhere, with proper modi\fcation, to allow an e\u000ecient ex-\nploration of other families of 2D magnets, such as CrI 3,\nCrOCl and Fe 3GeTe 2[13, 18, 48].\nACKNOWLEDGMENTS\nWe thank Marios Mattheakis, Daniel Larson, Robert\nHoyt, Matthew Montemore, Sadas Shankar, Ekin Dogus\nCubuk, Pavlos Protopapas and Vinothan Manoharan for\nhelpful discussions. For the calculations we used the Ex-\ntreme Science and Engineering Discovery Environment\n(XSEDE), which is supported by National Science Foun-\ndation (grant number ACI-1548562) and the Odyssey\ncluster supported by the FAS Division of Science, Re-6\nFIG. 6. (a) ML predicted formation energies (in eV/cell) for a wide range of substitutions that were not included in the DFT\ndata set covering 4,223 new structures (570 are shown here). 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Jpn\n82, 124711 (2013)."    },    {        "title": "1807.08128v1.Quantum_oscillations_of_the_Stoner_susceptibility__theory.pdf",        "content": "Quantum oscillations of the Stoner susceptibility : theory\nAjeet Kumar and Raman Sharma\u0003, Navinder Singhy\nJuly 24, 2018\nAbstract\nOscillatory e\u000bects in magnetic susceptibility of free electrons in a strong magnetic \feld\nis well known phenomenon and is well captured by Lifshitz-Kosevich formula. In this paper\nwe point out similar oscillatory e\u000bects in Stoner susceptibility which makes the system to\noscillate between paramagnetic phase and ferromagnetic phase alternatively as a function of\nexternal magnetic \feld strength. This e\u000bect can happen in a material which is tuned near\nto its magnetic instability. We suggest an experimental set-up to observe this e\u000bect. We\nalso suggest that our result can be exploited to control a quantum critical system around its\nquantum critical point to study its thermodynamical or transport properties.\n1 Introduction\nThe magnetization of free electron gas consist of two parts : the \frst part is due to the intrinsic\nmagnetic moment(spin) of the electrons called Pauli paramagnetism, and the second part is due\nto orbital motion of electrons and is called Landau diamagnetism. These two are weak forms\nof magnetism. The possibility of strong form of magnetism i.e. ferromagnetism in a free elec-\ntron gas was discussed by Bloch on the basis of exchange interaction between electrons which\nis ferromagnetic in nature (parallel spin electrons reduce mutual Coulomb repulsion by staying\naway from each other and thereby inducing spin polarization [1] ). Later on Wigner pointed out\nthat correlation e\u000bects ( which also act between electrons with antiparallel spins ) destroy the\npossibility of ferromagnetism in a free electron gas. In contrast, Stoner adopted a phenomenolog-\nical approach and impressed an exchange \feld on free electrons, and discussed the possibility of\nferromagnetismi. The occurance of ferromagnetism is given by the Stoner condition\nIg(EF)>1:\nHere I is the Stoner exchange parameter and g(EF) is EDOS (Electronic Density of States) at\nthe Fermi energy. The origin of exchange \feld in Stoner model can be attributed to intra-atomic\nHund's mechanism [1].\nIn the present paper we consider Stoner model for a ferromagnetic metal which is tuned near\nto its magnetic instability ( Ig(EF)\u00181), and which is placed in a strong external magnetic \feld.\n\u0003Himachal Pradesh University Shimla-171005, India. Email: kumarajeetsuryavansi@gmail.com; bra-\nmans70@yahoo.co.in\nyPhysical Research Laboratory, Ahmedabad, India, PIN: 380009. Email: navinder.phy@gmail.com; navin-\nder@prl.res.in\niStoner approach is analogous to Weiss approach in which Weiss incorporated mean molecular \feld on atomic\nmoments thus generalizing Langevin's theory of paramagnetism to ferromagnetism.\n1arXiv:1807.08128v1  [cond-mat.str-el]  21 Jul 2018EDOS gets modi\fed due to Landau level formation in the presence of external magnetic \feld. We\nstudy the e\u000bect of modi\fed EDOS on the Stoner condition. It turns out that under the action\nof strong magnetic \feld Stoner condition becomes a function of external magnetic \feld through\nmodi\fed EDOS ( g(EF;H)). As the strength of external magnetic \feld is changed the system\noscillates between paramagnetic phase (when Ig(EF;H)<1) and ferromagnetic phase (when\nIg(EF;H)>1). We also suggest an experiment to observe this e\u000bect.\nThe plan of the paper is as follows. In section (2) we present the derivation of Pauli suscep-\ntibility to set the stage for further development. In section (3) we study the e\u000bects of magnetic\n\feld on the EDOS and present a correction term to Pauli susceptibility. In section (4) we present\nthe oscillatory e\u000bects in Pauli susceptibility using Poisson summation formula. Then generalizing\nthis treatment we presented oscillatory e\u000bects in Stoner susceptibility in section (5). We conclude\nthat under the action of external magnetic \feld there are oscillations in EDOS as a function of\nexternal magnetic \feld, which allows the system to oscillate between paramagnetic phase and\nferromagnetic phase alternatively through Stoner condition.\n2 Pauli paramagnetic susceptibility\nPauli paramagnetic susceptibility is due to the intrinsic angular momentum of the electrons.\nLetg(E) be the EDOS of free electrons with given polarization in zero magnetic \feld. When an\nexternal magnetic \feld is applied electronic energy changes by Ek\u0006\u0016BH:The total spin imbalance\nis given by\n4N=Z1\n0dEg(E) (f(E\u0000\u0016BH)\u0000f(E+\u0016BH)): (1)\nHeref(E\u0000\u0016BH) is the Fermi-Dirac distribution function for electrons that are aligned along the\n\feld direction and f(E+\u0016BH) for those electrons that are aligned in the opposite direction. The\ninduced magnetization is given by\nM=\u0016BZ1\n0dEg(E) ((f(E\u0000\u0016BH)\u0000f(E+\u0016BH)); (2)\nor\nM= 2\u00162\nBHZ1\n0dEg(E)\u0012f(E\u0000\u0016BH)\u0000f(E+\u0016BH)\n2\u0016BH\u0013\n; (3)\nM'2\u00162\nBHZ1\n0dEg(E)\u0012\n\u0000@f(E)\n@E\u0013\n; (4)\nas\u0016BH <<k BT <<E F, we have\n\u0012\n\u0000@f(E)\n@E\u0013\n'\u000e(E\u0000EF); (5)\ntherefore equation (4) can be written as\nM= 2\u00162\nBHg(EF); (6)\nor\n\u001f\u000e\nPauli = 2\u00162\nBg(EF); (7)\nasM=\u001f\u000e\nPauliH:This shows that Pauli susceptibility depends on density of states at Fermi level,\nand it is the standard derivation[3].\n23 E\u000bect of magnetic \feld on the density of states and\nhigher order term in the Pauli susceptibility\nThe expression for the density of states per unit energy per unit volume in the case of free electrons\nis given by[2]\ng(E) =p\n2Em3\n2\n\u00192~3: (8)\nUnder the action of an external magnetic \feld energy levels of a free electron gas split into\nLandau levels[3][6]:\nE= (2n+ 1)\u0016BH+p2\nz\n2m: (9)\nSincepz=~kztherefore\nkz=r\n2m\n~2(E\u0000(2n+ 1)\u0016BH)1\n2 (10)\nThe number of electronic energy levels for z-direction is kz\u0000Lz\n2\u0019\u0001\n. Including transverse plane\ndegeneracy [3], number of electronic energy levels up to energy E is given by\nN(E) =2p\n2mVeH\n(2\u0019~)2cX\nn(E\u0000(2n+ 1)\u0016BH)1\n2: (11)\nFrom the above equation EDOS is given by\ng(E;H ) =dN(E)\ndE=2p\n2mVeH\n(2\u0019~)2cd\ndE X\nn(E\u0000(2n+ 1)\u0016BH)1\n2!\n: (12)\nBefore di\u000berentiating, let us \frst simplify the summation.\nS=X\nn(E\u00002n\u0016BH\u0000\u0016BH)1\n2= (2\u0016BH)1\n2X\nn\u0012E\n2\u0016BH\u0000n\u00001\n2\u00131\n2\n: (13)\nDe\fne\u000f=E\n2\u0016BHand write S as (2 \u0016BH)1\n2f(\u000f), where\nf(\u000f) =X\nn\u0012\n\u000f\u0000n\u00001\n2\u00131\n2\n:\nFrom the Poisson summation formula (Appendix A), and under the conditioniiwhen\u0016BH <<\nkBT <<E F, we have\nX\nn\u0012\n\u000f\u0000n\u00001\n2\u00131\n2\n'2\n3\u000f3\n2\u00001\n48\u000f\u00001\n2: (14)\nUnder this approximation, EDOS takes the form:\ng(E;H )'2p\n2mVeH\n(2\u0019~)2c p\nE\n\u0016BH+1\n24\u0012\u0016BH\nE3\n2\u0013!\n; (15)\ng(E;H )'p\n2m3\n2\n\u00192~3\u0012p\nE+1\n24\u0012(\u0016BH)2\nE3\n2\u0013\u0013\n: (16)\niiFor more information see appendix B\n3As\u001fPauli = 2\u00162\nBg(EF;H) therefore we get\n\u001fPauli(H)'\u001f\u000e\nPauli \n1 +1\n24\u0012\u0016BH\nEF\u00132!\n; (17)\nwhere\u001f\u000e\nPauli = 2\u00162\nBg(EF):Here the correction term\u0010\n\u0016BH\nEF\u00112\nin the Pauli susceptibility originates\nfrom the deformed EDOS due to external magnetic \feld. It is small correction as \u0016BH <<E F.\n4 Oscillatory e\u000bects in Pauli susceptibility\nThe expression for the EDOS at low temperature i.e. ( kBT.\u0016BH <<E F) under the action of\nan external magnetic \feld is given by (after di\u000berentiating eqn.(11) w.r.t. E)\ng(E;H ) =2p\n2mVeH\n(2\u0019~)2c \n1\n2X\nn(E\u0000(2n+ 1)\u0016BH)\u00001\n2!\n: (18)\nDe\fne\u0015=2p\n2mVeH\n(2\u0019~)2c;with this the above equation takes the form:\ng(E;H ) =\u0015 \n1\n2X\nn(E\u0000(2n+ 1)\u0016BH)\u00001\n2!\n: (19)\nAs\u001fPauli(H) = 2\u00162\nBg(EF;H);therefore the expression for Pauli susceptibility is given by\n\u001fPauli(H) =\u00162\nB\u0015X\nn(EF\u0000(2n+ 1)\u0016BH)\u00001\n2 (20)\nor\n\u001fPauli(H) =\u00162\nB\u0015X\nn(2\u0016BH)\u00001\n2\u0012EF\n(2\u0016BH)\u0000n\u00001\n2\u0013\u00001\n2\n: (21)\nDe\fne ~\u000fF=EF\n(2\u0016BH). As\u0016BH <<E Ftherefore ~\u000fF>>1. The above equation can be written as:\n\u001fPauli(H) =\u00162\nB\u0015\n(2\u0016BH)1\n2X\nn\u0012\n~\u000fF\u0000n\u00001\n2\u0013\u00001\n2\n: (22)\nAs ~\u000fF>>1;the upper limit can be restricted to ~ \u000fFinstead of (~ \u000fF\u00001\n2). The summation can\nbe written as\nS=~\u000fFX\nn=0(~\u000fF\u0000n)\u00001\n2= lim\n\u000e!0Z~\u000fF+\u000e\n\u0000\u000edx~\u000fFX\nn=0(~\u000fF\u0000x)\u00001\n2\u000e(x\u0000n); (23)\non writing the delta function as Fourier sumiii, we obtain\nS= lim\n\u000e!0Z~\u000fF+\u000e\n\u0000\u000edx(~\u000fF\u0000x)\u00001\n2+1X\nk=\u00001e2\u0019ikx; (24)\ntherefore equation (22) can be written as\n\u001fPauli(H) =\u00162\nB\u0015\n(2\u0016BH)1\n2+1X\nk=\u00001lim\n\u000e!0Z~\u000fF+\u000e\n\u0000\u000edx(~\u000fF\u0000x)\u00001\n2e2\u0019ikx: (25)\niiiHere we will do the exact treatment i.e., including the oscillatory terms as we have kBT.\u0016BH <<E F. Refer\nAppendix B for details.\n4Before integrating let us \frst simplifyivP+1\nk=\u00001e2\u0019ikx:We have\n+1X\nk=\u00001e2\u0019ikx=+1X\nk=\u00001(cos(2\u0019kx) +isin(2\u0019kx)) = \n1 + 21X\nk=1cos2\u0019kx!\n: (26)\nBy using this result the equation (25) takes the form:\n\u001fPauli(H) =\u00162\nB\u0015\n(2\u0016BH)1\n2lim\n\u000e!0Z~\u000fF+\u000e\n\u0000\u000edx(~\u000fF\u0000x)\u00001\n2 \n1 + 21X\nk=1cos2\u0019kx!\n; (27)\n\u001fPauli(H) =2\u00162\nB\u0015\n(2\u0016BH)1\n2p\n~\u000fF+2\u00162\nB\u0015\n(2\u0016BH)1\n21X\nk=1Z~\u000fF\n0dx(~\u000fF\u0000x)\u00001\n2cos 2\u0019kx: (28)\nLet us simplify the integral I=R~\u000fF\n0dx(~\u000fF\u0000x)\u00001\n2cos 2\u0019kx;\nI=Z~\u000fF\n0dx(~\u000fF\u0000x)\u00001\n2cos 2\u0019kx=1p\nk\u0010\ncos(2\u0019~\u000fFk)FC(p\n4k~\u000fF) +sin(2\u0019~\u000fFk)FS(p\n4k~\u000fF)\u0011\n:(29)\nHere theFC(p4k~\u000fF) andFS(p4k~\u000fF) are the Fresnel integrals, therefore the expression for \u001fPauli(H)\nis given by\n\u001fPauli(H) =\u001f\u000e\nPauli0\n@1 +r\n2\u0016BH\nEF1X\nk=1(cos(k\u0019EF\n\u0016BH)FC(q\n2kEF\n\u0016BH) +sin(k\u0019EF\n\u0016BH)FS(q\n2kEF\n\u0016BH))\np\nk1\nA:(30)\nThis expression is Lifshitz-Kosevich formula but including only the spin part. Let us de\fne\nf\u0012EF\n\u0016BH\u0013\n=r\n2\u0016BH\nEF1X\nk=1\u0010\ncos(k\u0019EF\n\u0016BH)FC(q\n2kEF\n\u0016BH) +sin(k\u0019EF\n\u0016BH)FS(q\n2kEF\n\u0016BH)\u0011\np\nk:(31)\nWhen thekBT.\u0016BH <<E Fthen the expression for Pauli susceptibility is given by\n\u001fPauli(H) =\u001f\u000e\nPauli\u0012\n1 +f\u0012EF\n\u0016BH\u0013\u0013\n; (32)\nwhich is an oscillatory function.\n5 Oscillatory e\u000bects in Stoner susceptibility\nFrom the expression for Stoner susceptibility [1] we have\n\u001fStoner =\u001fPauli\n(1\u0000I\u001fPauli); (33)\nwhere I is an e\u000bective exchange energy.\nSubstituting the expression for \u001fPauli from equation (32), the expression for Stoner suscepti-\nbility is given by\n\u001fStoner =\u001f\u000e\nPauli\n1\u0000f(EF\n\u0016BH)\u0000I\u001f\u000e\nPauli; (34)\nivConsidering only real part as static susceptibility is a real quantity.\n5becausef\u0010\nEF\n\u0016BH\u0011\n<<1 we divided numerator and denominator by\u0010\n1 +f\u0010\nEF\n\u0016BH\u0011\u0011\n. Set\u0016B= 1\nfor simplicity.\nNow the Stoner condition Ig\u000e(EF)>1 changes to f\u0000EF\nH\u0001\n+Ig\u000e(EF)>1 (g\u000e(EF) is the total\nEDOS including both spin directions). Let us take a special case where the system is near to\nStoner instability Ig\u000e(EF)'1;we get\nf\u0012EF\nH\u0013\n>0;\nfor the condition of ferromagnetism when the metal is near its magnetic instability. But it depends\nupon external magnetic \feld strength H and it is an oscillatorty function (\fgure 1) of \u0011=EF\nHand\nit changes sign also. Thus when magnetic \feld H is varied, system oscillates between paramagnetic\nand ferromagnetic phases.\nhf(h)\n®\n10021004100610081010\n-4-3-2-1123\nFigure 1: Oscillatory behaviour of function f(\u0011=EF\nH) as a function of external magnetic \feld\nstrength. Here reactangles shows paramagnetic regions below the \u0011axis and dots above the \u0011\naxis shows ferromagnetic regions. When magnetic \feld is varied i.e. when \u0011is varied the system\noscillates between ferromagnetic regions and paramagnetic regions.\n6 Discussion and conclusion\nThe above \fgure 1 shows that there are oscillations in Stoner susceptibility as a function of external\nmagnetic \feld strength (H). An interesting behaviour is noticed when f(\u0011)>0, the system lies in\nferromagnetic phase and when f(\u0011)<0 the behaviour of system switches to paramagnetic phase.\nThis kind of oscillatory behaviour shows that the system under the in\ruence of strong external\nmagnetic \feld oscillates between paramagnetic phase and ferromagnetic phase as a function of H.\nThe oscillatory behaviour arises from the deformed EDOS of the system in the presence of strong\nexternal magnetic \feld.\nOne can observe such oscillatory behaviour experimentally with a special experimental set-up\nas shown in \fgure 2. The sample ( for example the material HfZr 2which can be tuned near to\nits Stoner instability [11]) is placed inside a coil and then strong external magnetic \feld is applied\non the sample which is along the axis of the coil. When magnitude of \feld is varied the sample's\nmagnetization oscillates as depicted in \fgure 1. Due to the changing magnetization of sample an\ninducede:m:f is produced in the coil which can be ampli\fed if needed and the output can be\ngiven to an oscilloscope.\nWe also suggest an important application of this e\u000bect. In the current topic of anomalous\ntransport and thermal properties of magnetic material tuned near to their quantum criticality it\nis important to tune and control the system near and around a QCP (Quantum Critical Point).\n6Figure 2: Experimental set-up for the study of oscillatory behaviour.\nVarious methods like doping, pressure, and magnetic \feld is used for this purpose [7][8][9][10] our\nmethod could be a new addition to such methods. The present method will lead to H-T phase\ndiagrams of the form depicted in \fgure 3. The alternative phases result due to oscillations in f(\u0011)\nas explained in \fgure 1. At high temperature oscillations will vanish and regions in \fgure 3 will\nhave dome like structures.\nFigure 3: H-T phase diagrams. Here PM stands for Paramagnetism and FM stands for Ferro-\nmagnetism.\n77 Appendix A\nWe will prove equation (14) by two methods. In the \frst method we use Poisson summation\nformula and in the second method we use Euler-Maclaurin sum formula.\nProof of equation (14) by using Poisson summation formula\nStarting from\nf(\u000f) =X\nn\u0012\n\u000f\u0000n\u00001\n2\u00131\n2\n;\nas\u000f>> 1, the upper limit of sum over ncan be restricted to n\u0014\u000finstead ofn\u0014(\u000f\u00001\n2) therefore\nthef(\u000f) is given by\nf(\u000f)'X\nn(\u000f\u0000n)1\n2: (35)\nFrom Poisson summation formula [3]\n1X\nn=0F(n) =+1X\nl=\u00001(\u00001)lZ1\n0F(x)e2\u0019ilxdx: (36)\nSummation in equation (35) can be evaluated:\nF(n) =p\n\u000f\u0000n;\n\u000fX\nn=0(\u000f\u0000n)1\n2'+1X\nl=\u00001(\u00001)lZ\u000f\n0(\u000f\u0000x)1\n2e2\u0019ilxdx: (37)\nOn integrating by parts we get\n\u000fX\nn=0(\u000f\u0000n)1\n2'Z\u000f\n0dx(\u000f\u0000x)1\n2++1X\nl=\u00001;l6=0(\u00001)l+1\nl\u000f1\n2\n2\u0019i++1X\nl=\u00001;l6=0(\u00001)l\nl2\u000f\u00001\n2\n8\u00192+(oscillatoryterms ):(38)\nThe second term in R.H.S. of above equation converges to zero[5] ( as each positive term cancel\nwith its symmetric negative term) andP+1\n\u00001(\u00001)l\nl2=\u0000\u00192\n6, therefore we have\nX\nn(\u000f\u0000n)1\n2'2\n3\u000f3\n2\u00001\n48\u000f\u00001\n2: (39)\nWe explian the issue of oscillatory terms in Appendix B.\nProof of eqn. (14) by using Euler-Maclaurin sum formula\nStarting from\nf(\u000f) =X\nn\u0012\n\u000f\u0000n\u00001\n2\u00131\n2\n: (40)\nFrom the Euler-Maclaurin formula[4] we have\n1X\nn=0g(n+1\n2)'Z1\n0g(x)dx+1\n24g0(x)jx=0: (41)\n8Asg\u0000\nn+1\n2\u0001\n=\u0000\n\u000f\u0000n\u00001\n2\u00011\n2,\n\u000fX\nn=0\u0012\n\u000f\u0000n\u00001\n2\u00131\n2\n'Z\u000f\n0(\u000f\u0000x)1\n2dx\u00001\n48(\u000f\u0000x)\u00001\n2jx=0; (42)\nas\u000f!1 we can use (41)\n1X\nn=0\u0012\n\u000f\u0000n\u00001\n2\u00131\n2\n'2\n3\u000f3\n2\u00001\n48\u000f\u00001\n2: (43)\nAppendix B\nWhen are oscillations in susceptibility important and when\nnot ?\nFigure 4: When \u0016BH <<k BT\nCase 1. Oscillations are not important: when \u0016BH <<k BT <<E Fthere are large number of\noscillations over an energy scale of kBT(\fgure 4). These oscillating terms give no contribution in\nthe Pauli susceptibility due to thermal averaging e\u000bect. This can be appreciated in the following\nway. The oscillatory term appears in a more careful treatment of the summation (14). The sum\n(from equation 38) can be exactly written as\n\u000fX\nn=0(\u000f\u0000n)1\n2=Z\u000f\n0(\u000f\u0000x)1\n2++1X\nl=\u000010\nBBB@(\u00001)l+1\nl\u000f1\n2\n2\u0019i+(\u00001)l\nl2\u000f1\n2\n8\u00192+1\n16\u00192(\u00001)l\nl2Z\u000f\n0(\u000f\u0000x)\u00003\n2e2\u0019ilxdx\n| {z }\nOscillatory term1\nCCCA:(44)\nThe last term on the RHS of the above equation is the oscillatory term. To understand when it\nis important and when it is not important consider equation (4) and substitute the expression for\ng(E;H ) from equation (19) into equation (4):\n\u001fPauli'\u00162\nB\u0015Z1\n0dE\u0012\n\u0000@f(E)\n@E\u0013X\nn(E\u0000(2n+ 1)\u0016BH)\u00001\n2: (45)\nNow the summation includes the oscillatory term in the above equation. The derivative\u0010\n\u0000@f(E)\n@E\u0011\nin the above equation is not exactly a delta function, but it is a Gaussian function of width kBT\n9centered around EF. If there are many oscillations over kBTthen they average out in the integral\nover E in the above equation. This leads to omitting of oscillations.\nCase 2. Oscillations are important: when kBT.\u0016BH << E F, in this case the temperature\nis low and\u0016BHis the same order as kBTthen the contribution of oscillations is to be taken into\naccount (\fgure 5). The treatment given in section (4) takes this into account exactly.\nFigure 5: When kBT.\u0016BH\nReferences\n[1] Kubo, R. and Knox, R.S. and Nagamiya, T. Solid State Physics , 1969, McGraw-Hill.\n[2] C. Kittel, Introduction to Solid State Physics 7TH ED, John Wiley and sons, Inc.\n[3] R.E. Peierls, Quantum Theory of Solids, Oxford University Press , 1955.\n[4] Landau, L.D. and Lifshitz, E.M., Statistical Physics , Part-1, Vol-5 , Third Edition, Elsevier\nScience.\n[5] W. L. Ferrar, A Textbook of Convergence , Oxford University Press, 1937.\n[6] D. Shoenberg, Magnetic Oscillations in Metals , Cambridge University Press, 1984\n[7] Y. Machida, K. Tomokuni, C. Ogura, K. Izawa, K. Kuga, S. Nakatsuji, G. Lapertot, G.\nKnebel, J.-P. Brison, Phys. Rev. Lett 109, 156405 (2012).\n[8] S. Nakatsuji, K. Kuga, Y. Machida, T. Sakakibara, Y. Karki, H. Ishimoto, S. Yonezawa, Y.\nMaeno, E. Pearson, G.G. Lonzarich, L. Balicas, H. Lee and Z. Fisk, Nature Phy. Vol-4, (2008).\n[9] T. Tomita, K. Kuga, Y. Uwatoko, P. Coleman, S. Nakatsuji, Science Vol-349, 6247, (2015).\n[10] Y. Matsumoto, S. Nakatsuji, K. Kuga, Y. Karki, N. Horie, Y. Shimura, T. Sakakibara, A.H.\nNevidomskyy, P. Coleman, Science , Vol-331, (2011).\n[11] V. Yu. Irkhin and Yu. P. Irkhin, arXiv: cond-mat/9812072v2 , 2000\n10"    },    {        "title": "1807.09257v1.Pushing_the_limits_of_magnetic_anisotropy_in_the_Sm_Co_system.pdf",        "content": "Pushing the limits of magnetic anisotropy in the Sm–Co syste m\nS.-Y. Jekal,1, 2,∗M. Charilaou,1,†and J. F. L¨ offler1\n1Laboratory of Metal Physics and Technology, Department of Ma terials, ETH Zurich, 8093 Zurich, Switzerland\n2Condensed Matter Theory Group, Paul Scherrer Institute, CH- 5232 Villigen PSI, Switzerland\n(Dated: July 24, 2018)\nMaterials based on the Sm–Co system exhibit remarkable magnet ic performance due to their high\nCurie temperature, large saturation magnetization, and st rong magnetic anisotropy, which are the\nresult of the electronic structure in Co and Sm and their arra ngement in the hexagonal lattice. In\nthis paper we show, using first-principles calculations, me an-field theory, and atomistic Monte Carlo\nsimulations, that slight modifications of the SmCo 5crystal structure, induced by strain or partial\nsubstitution of Sm by Ce, change the exchange interaction and increase the magnetocrystalline\nanisotropy energy drastically. This finding shows how small changes in local-structure environments\ncan generate substantial changes in macroscopic propertie s and thus enable the optimization of\nhigh-performance materials via tailoring at the atomic leve l.\nI. INTRODUCTION\nThe magnetism in intermetallic phases of rare-earth\n(RE) metals and transition metals (TM), such as the\nhigh-performance magnet SmCo 5, is a result of the syn-\nergy between the 4 fRE electrons, which provide a high\nanisotropy due to spin-orbit coupling, and the 3 dTM\nelectrons, which have large magnetic moments and pro-\nvide strong ferromagnetic exchange interactions, thus en-\nabling long-range order1,2. The synergy between 3 dand\n4felectrons depends crucially on the local atomic envi-\nronments, and thus the net performance of a magnet is\nextremely sensitive to changes of the lattice.\nIn this paper we show that slight modifications of\nthe local atomic arrangements in SmCo 5generate dras-\ntic changes of the magnetic performance, particularly\nof the magnetocrystalline anisotropy (MCA). Specifi-\ncally, we will discuss, based on first-principle electronic -\nstructure calculations and atomistic Monte Carlo simu-\nlations, that tensile strain in SmCo 5leads to enhanced\nmagnetic anisotropy. Hence, we propose a special case\nwhere partial substitution of Sm with Ce generates a\nmaterial with magnetic performance which is superior\nto that of the well-known SmCo 5.\nThe permanent magnet SmCo 5has been intensively\nstudied over the past few decades because it exhibits\nstrong uniaxial MCA, large saturation magnetization\n(Ms), and high Curie temperature ( TC) of about 1000\nK2–10. The high magnetic performance of this sys-\ntem, particularly its spectacularly strong MCA, makes\nit a key component in various precision-sensitive applica-\ntions. In fact, out of all Co- and Fe-based RE-containing\ncompounds SmCo 5has the largest magnetocrystalline-\nanisotropy energy EMCA2,11. It is thus of fundamental\ninterest to investigate the limits of magnetism in this\nhigh-performance system and to find ways of how to en-\nhance the magnetic anisotropies beyond the limit of this\nwell-known compound.\nExperimental and theoretical studies have reported\npossibilities of improving the magnetic performance of\nSmCo 5both in bulk materials12,13and in thin-film\nFIG. 1. Illustration of the crystal structure of SmCo 5and\n(Sm,Ce)Co 5. In the left panel, the ideal hexagonal CaCu 5\nprototype structure with and without strain is shown, and\ntensile and compressive strains are indicated by red and blu e\narrows, respectively. The right panel shows the crystal stru c-\nture of (Sm 1−xCex)Co5forx= 0.25, 0.50 and 0.75. Gray,\nred, light blue and dark blue spheres correspond to Sm, Ce,\nCo(2c) and Co(3g) atoms, respectively. Note that these are\nthe most stable structures obtained by total-energy minimi za-\ntion in the DFT calculations.\nstructures14–17. Promising approaches to enhance the\nmagnetic performance are e.g. the partial substitution of\nCo with other transition metals (TM), such as Cu, Ni,\nFe, or Zr in Sm(Co, TM) 517–19, but in most cases the\nTC, the saturation magnetization ( Ms), and the EMCA\ndecrease with increasing TM content due to the reduced\nexchange coupling parameters when Co is substituted by\narXiv:1807.09257v1  [cond-mat.mtrl-sci]  24 Jul 20182\nother elements17,20.\nConsidering the above, the question remains of\nwhether or not it is possible to increase the EMCA, and\nthus enhance the performance of Sm–Co without substi-\ntuting Co, but by carefully modifying the local atomic\narrangements, i.e., by introducing strain. Strain is a\nparticularly important issue in permanent magnets, be-\ncause their synthesis typically involves sintering, and\nthermal processes inevitably introduce strain on fine-\ngrained materials21,22. Strain can be applied externally\nby mechanical means, but it can also be introduced to\nthe system by partial element substitution, such as e.g.\nof Sm by another RE metal with a different atomic ra-\ndius.\nCe is a particularly promising candidate for substi-\ntuting Sm because it is the most abundant and second-\nlightest among all RE metals, and CeCo 5is isomorhpous\nto SmCo 523,24, suggesting that it is possible to substi-\ntute Sm with Ce through the entire composition, i.e.,\n(Sm 1−xCex)Co5for 0≤x≤1. SmCo 5and CeCo 5crys-\ntallize in the hexagonal CaCu 5structure, where each Sm\n(Ce) occupies a 1a site and Co occupies 2c and 3g sites in\nthe lattice25(see Fig. 1); the unit cell volume decreases\nfrom 84.71 ˚A3in SmCo 5to 83.09 ˚A3in CeCo 5.\nIt is well-known that the intrinsic magnetic material\nparameters of CeCo 5lead to a magnetic performance\nthat is substantially weaker than that of SmCo 511, i.e.,\nreduced TCof 660 K instead of 1000 K, reduced EMCA\nof 5.3 MJ/m3instead of 17.2 MJ/m3, and reduced Msof\n0.95 T instead of 1.15 T. Nevertheless, a partial substitu-\ntion of Sm by Ce can have drastic effects on the magnetic\nperformance, as we discuss below, because it introduces\nstrain in the structure and breaks the lattice symmetry\nin a way that enhances the contribution of the Co atoms\nto the MCA.\nIn the following we briefly discuss the theory and the\ncomputational methodology and then analyze our re-\nsults, which reveal a surprisingly enhanced magnetic per-\nformance of SmCo 5with strain and with partial Sm sub-\nstitution.\nII. COMPUTATIONAL METHOD\nDensity Functional Theory (DFT) calculations were\nmainly performed using plane wave basis sets and pseu-\ndopotentials, implemented in Quantum Espresso (QE)26.\nThis approximation enables time-efficient calculations,\nbut its precision in certain cases might be limited by the\nuse of pseudopotentials. Hence, results from QE were\ncomplemented by more computationally demanding all-\nelectron full-potential augmented plane-wave and local-\norbitals basis sets, implemented in Elk27and WIEN2k28,\nrespectively, which enable more precise calculations.\nThis is particularly important for the precise estimation\nof the MCA energy, which is associated with variations\non the order of meV.\nAll DFT calculations were performed on a 2 ×2×2supercell of the SmCo 5(CaCu 5) structure. For the\nexchange-correlation potential we adapted the local spin-\ndensity approximation plus Hubbard U(LSDA+ U),\nwhich can adequately describe the strongly correlated\nelectronic states of 4 felectrons9,29,30. The LSDA+ U\nmethod requires the Coulomb energy ( U) and the ex-\nchange energy ( J) as input parameters, and we de-\nfined UandJthrough the derivatives of the energy\nlevels ǫfof the f-orbital with respect to their occu-\npancies nf, described as U=∂ǫf↑/∂n f↓andJ=\n∂(ǫf↑−ǫf↓)/∂(nf↑−nf↓) for the majority (minority)\nspin↑(↓), respectively. From these expressions we ob-\ntainU= 6.0 eV and J= 1.0 eV for Sm in our DFT\ncalculations.\nWave functions were expanded by a plane-wave basis\nset with an optimized cutoff energy of 340 Ry, and the\nBrillouin zone was sampled via a 12 ×12×15k-point\nmesh. Different mesh values were tested to ensure the\nnumerical stability of our calculations, with the conver-\ngence criterion being 0 .1µeV. All structures were fully\nrelaxed prior to calculating the magnetic parameters, i.e. ,\nthe atomic arrangements in our calculations correspond\nto the most stable structures, found by minimizing the\ntotal energy.\nThe magnetocrystalline anisotropy energy EMCA was\ncalculated using the force theorem and is defined as\nthe total energy difference between the magnetization\nperpendicular to the c-plane and in the c-plane of the\nSmCo 5structure, i.e., K=E[100]−E[001], where E[100]\nandE[001]are the total energies with the magnetization\naligned along the hard- ([100]) and easy-axis ([001]) of\nthe magnetic anisotropy, respectively, with (001) corre-\nsponding to the cplane. Specifically, EMCA is calcu-\nlated in three steps: (i) collinear self-consistent calcul a-\ntion without spin-orbit coupling (SOC); (ii) global rota-\ntion of the density matrix to consider the magnetization\nalong [100] and [001] to calculate E[100] andE[001], re-\nspectively; and (iii) non-collinear and non-self-consist ent\ncalculation with SOC.\nFor the calculation of the energy-resolved distribution\nof the orbitals within second-order perturbation theory31,\nEMCA can be expressed as\nE↑↓\nMCA∝ξ2×\n/summationdisplay\no↑(↓),u↓(↑)|<o↑(↓)|Lx|u↓(↑)>|2−|<o↑(↓)|Lz|u↓(↑)>|2\nǫo↑(↓)−ǫu↓(↑),\n(1)\nwhere ξis the spin-orbit coupling constant, ↑(↓) repre-\nsent majority (minority) spins, and o(u) and ǫo(u)repre-\nsent the eigenstate and eigenvalue of occupied (unoccu-\npied) states, respectively. For contributions from the d-\norbitals, the matrix elements of the LxandLzoperators\ncan be expressed via the magnetic quantum number m.\nWhen the occupied and unoccupied states have opposite\nspin directions, mo−mu=±1 and mo−mu= 0 con-\ntribute to out-of-plane and in-plane MCA, respectively,\nwhile mo−mu=±2 is treated as zero matrix element.3\nFrom the DFT calculations we obtain the magne-\ntocrystalline anisotropy energy, as described above, and\nfrom the total energy minimization we obtain the atomic\nmagnetic moments and the inter-atomic ferromagnetic\nexchange interactions as the difference between ferro-\nmagnetic and antiferromagnetic spin configuration, i.e.,\nJ= (E↑↑−E↑↓)/2. We then use these intrinsic material\nparameters as inputs for atomistic Monte Carlo simula-\ntions to calculate finite-temperature material properties\nby mapping the material parameters onto a Heisenberg-\ntype model. Here, the Hamiltonian has the form\nH=−1\n2/summationtext\ni/negationslash=jJij/vectorSi·/vectorSj−/summationtext\njKi(Sz)2, (2)\nwhere the spin /vectorSiis the 3-dimensional vector at each lat-\ntice site, Jijis the Heisenberg exchange interaction en-\nergy between neighboring sites, and Kiis the site-specific\nuniaxial anisotropy energy constant. The numerical val-\nues of the interaction and anisotropy energies were taken\ndirectly from the DFT results in absolute values.\nThe simulations were performed by means of the\nMetropolis algorithm, i.e., by single-site updates. For th e\nthermalization the algorithm was run 5000 MCS (Monte\nCarlo steps per site) and another 5000 MCS were run to\nsample the thermal-average magnetic moment /vectorMat the\n1a, 2c, and 3g lattice sites at each temperature.\nWe considered systems with 120000 and 276000 spins,\ncorresponding to real dimensions of 5 nm ×8.5 nm ×39\nnm and 10 nm ×10 nm ×40 nm, respectively, to verify\nthe numerical stability of our results. Additionally, we\nused periodic boundary conditions to eliminate surface\neffects.\nFor the calculation of the Curie temperature we started\nthe simulations with all spins parallel and pointing along\n[001] at T= 0 and gradually increased the tempera-\nture to 1200 K, whereas for the calculation of the finite-\ntemperature domain-wall thickness, we started the sim-\nulations with half the spins pointing along [001] and the\nother half pointing along [00 ¯1] and thermalized the sys-\ntem at each given temperature, in a similar way as de-\nscribed in Ref.32.\nIII. RESULTS AND DISCUSSION\nA. Strain effects in SmCo 5\nWe calculated the total energy of the SmCo 5struc-\nture to optimize the lattice and determined the in-plane\nlattice constant to be a= 4.980˚A and the axis ratio\nto be c/a= 0.792. These values are in excellent agree-\nment with previous first-principles calculations9,29,33,34\nand experiments35, thus confirming our computation of\nthe equilibrium structure.\nFurther, for SmCo 5in equilibrium we obtain an EMCA\nof 9.09 meV/f.u. (f.u. = formula unit), which is/s45/s53 /s45/s52 /s45/s51 /s45/s50 /s45/s49 /s48 /s49 /s50 /s51 /s52 /s53/s48/s51/s54/s57/s49/s50/s49/s53/s49/s56/s49/s55/s54/s49/s55/s56/s49/s56/s48/s49/s56/s50/s49/s56/s52/s49/s56/s54/s49/s46/s51/s49/s46/s52/s49/s46/s53/s49/s46/s54/s49/s46/s55/s49/s46/s56/s49/s46/s57/s50/s46/s48\n/s32/s112/s108/s97/s110/s101/s32/s119 /s97/s118/s101/s32/s43/s32/s112/s115/s101/s117/s100/s111/s112/s111/s116/s101/s110/s116/s105/s97/s108/s115\n/s32/s112/s108/s97/s110/s101/s32/s119 /s97/s118/s101/s32/s43/s32/s102/s117/s108/s108/s32/s112/s111/s116/s101/s110/s116/s105/s97/s108/s32\n/s32/s111/s114/s98/s105/s116/s97/s108/s32/s98/s97/s115/s105/s115/s32/s115/s101/s116/s115/s32/s43/s32/s102/s117/s108/s108/s32/s112/s111/s116/s101/s110/s116/s105/s97/s108\n/s32/s112/s108/s97/s110/s101/s32/s119 /s97/s118/s101/s32/s43/s32/s111/s114/s98/s105/s116/s97/s108/s32/s98/s97/s115/s105/s115/s32/s115/s101/s116/s115/s32/s43/s32/s102/s117/s108/s108/s32/s112/s111/s116/s101/s110/s116/s105/s97/s108\n/s32/s112/s111/s108/s121/s110/s111/s109 /s105/s97/s108/s32/s102/s105/s116/s69\n/s77 /s67/s65/s32/s40/s109/s101/s86/s32/s47/s32/s102/s46/s117/s46/s41\n/s83/s116/s114/s97/s105/s110/s32/s40/s37/s41/s40/s97/s41\n/s32/s74\n/s67/s111 /s45 /s67/s111 /s74\n/s67/s111/s45/s67/s111/s32/s40/s109/s101/s86/s41\n/s40/s98/s41/s67/s111 /s32/s109/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s32 /s40\n/s66 /s41\n/s32/s67/s111/s32/s40/s50/s99/s41\n/s32/s67/s111/s32/s40/s51/s103/s41\n/s32/s83/s109\n/s40/s99/s41/s51/s50/s51/s52/s51/s54/s51/s56\n/s32/s74\n/s67/s111 /s45 /s83/s109\n/s74\n/s67/s111/s45/s83/s109/s32/s40/s109/s101/s86/s41/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53/s48/s46/s51/s48\n/s83/s109/s32 /s109/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s32 /s40\n/s66 /s41\nFIG. 2. Intrinsic magnetic properties as a function of strain in\nSmCo 5: (a) atomic magnetic moments of Co atoms at sites\n2c (diamonds) and 3g (squares), and Sm atoms (triangles);\n(b) effective total ferromagnetic exchange energy JCo−Cobe-\ntween Co atoms (hexagons) and JCo−Smbetween Co and Sm\natoms (triangles); and (c) magnetocrystalline anisotropy e n-\nergy ( EMCA), calculated by several approximations.\nagain in excellent agreement with experimental data11,36\n(corresponding to 17.18 MJ/m3) and with other DFT\ncalculations19,33. Specifically, we find that in the SmCo 5\nstructure the Co sublattice contributes 2.20 meV and\nthe Sm sublattice contributes 6.88 meV to the to-\ntal anisotropy energy per unit cell (corresponding to\n4.16 MJ/m3and 13.02 MJ/m3, respectively, in good\nagreement with experimental values for the individual\ncontributions36).\nIn order to probe SmCo 5away from equilibrium we ap-\nplied both tensile (+) and compressive ( −) strain along\nthe [001] axis (see Fig. 1), while keeping the unit cell vol-\nume constant, to investigate strain-dependent electronic\nstructure and magnetic properties.\nStrain effects on the magnetism, i.e., individual atomic\nmagnetic moments in SmCo 5, effective ferromagnetic ex-\nchange energy, and MCA energy are presented in Fig. 2.\nWe find that the magnetic moments at each crystal-\nlographic site decrease nearly linearly with increasing\nstrain (see Fig. 2a). The reason for the decrease in the\nmagnetic moments is the decreasing overlap of the d-\norbitals as the caxis grows with strain.\nFurther, we calculated the overall effective exchange\ncoupling parameters in the framework of the multi-\nsublattice mean-field approximation, and we obtained4\nJCo−CoandJCo−Sm, which are the sum of the exchange\ncoupling constants within a sphere of radius R= 5a.\nFrom these calculations we observe that the total fer-\nromagnetic exchange interaction increases monotonically\nwith increasing strain (see Fig. 2b), in contrast to the\nmagnetic moments. Note that the Curie temperature\ndoes not change significantly because the increase in the\neffective ferromagnetic exchange is offset by the decrease\nin the magnetic moments.\nWhile the overall exchange interactions increase with\nincreasing strain, the intra-plane interaction strengths\nJCo(2c) −Co(2c) andJCo(3g) −Co(3g) increase while the inter-\nplane interaction strength JCo(2c) −Co(3g) decreases with\nincreasing tensile strain (data not shown). This illus-\ntrates the dependence of the ferromagnetic exchange\non the inter-atomic distance: as the c–c and g–g dis-\ntances decrease the corresponding exchange interaction\nincreases, whereas as the c–g distance increases with in-\ncreasing tensile strain the exchange interaction decrease s.\nSimilar strain effects were calculated for YCo 5in\nRef.29,33, but with significantly weaker impact. It was,\nhowever, pointed out that the inter- vs intra-plane hy-\nbridization plays a key role in the development of the fer-\nromagnetic exchange, the magnetic moments, and EMCA.\nConsidering the effects of strain on EMCA in SmCo 5,\nwe find that the magnetocrystalline anisotropy energy\nchanges monotonically with strain. While compressive\nstrain decreases EMCA, increasing tensile strain has the\nopposite effect and EMCA along the c-axis exhibits a re-\nmarkable strengthening (see Fig. 2c). In fact, even with\nonly +1% strain EMCA increases by ∼30%, whereas with\n+5% strain EMCA increases, strikingly, by 80% compared\nto that in equilibrium at zero strain. Importantly, we\nfind that the results based on the use of pseudopotentials\nare in excellent agreement with full-electron calculation s,\nand they only start to deviate by 3–5% with increasing\nstrain, as the values of EMCA become larger. Specifically,\ncalculations based on orbital basis-sets and plane-waves\nconsistently tend to slightly underestimate EMCA.\nWe examined the cause of this remarkable enhance-\nment of EMCA and found that the major contribution\ncomes from the Co(2c) sites. Note that the 2c sites also\nplay a vital role in the MCA of the SmCo 5in equilibrium,\nas indicated by nuclear magnetic resonance studies37, and\nthat calculations of strained YCo 5also indicated the im-\nportance of 2c contributions to the anisotropy33.\nTo obtain a deeper understanding of why the lattice\ndistortion affects the MCA, we plot tensile and com-\npressive strain-induced changes in the electronic density\nof states (DOS), i.e., ∆DOS = DOS(+5%)–DOS(0%)\n(Fig. 3a) and ∆DOS = DOS( −5%)–DOS (0%) (Fig. 3b).\nWe observe that, especially in the range near the Fermi\nlevel, the changes in the DOS with m(with up and down\nspins) are mirrored depending on strain directions, i.e.,\nthem= 0,↓andm=±2,↓states lose electrons while\nthem=±1,↑state gains electrons with tensile strain,\nand vice versa with compressive strain.\nThe concurrence of the increased occupied m=±1,↑/s45/s48/s46/s51 /s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s45/s48/s46/s50/s53/s45/s48/s46/s50/s48/s45/s48/s46/s49/s53/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53\n/s32/s109 /s61/s48/s44 /s32/s32/s32 /s32/s109 /s61/s48/s44 /s32\n/s32/s109 /s61 /s49/s44 /s32 /s32/s109 /s61 /s49/s44\n/s32/s109 /s61 /s50/s44 /s32 /s32/s109 /s61 /s50/s44 /s32/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s32/s101/s86/s45/s49\n/s97/s116/s111/s109/s45/s49\n/s115/s112/s105/s110/s45/s49\n/s41\n/s69 /s32/s40/s101/s86/s41/s40/s97/s41\n/s45/s48/s46/s51 /s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51\n/s69 /s32/s40/s101/s86/s41/s40/s98/s41\nFIG. 3. Changes in DOS of the Co(2c) atom induced by (a)\n+5% (tensile) and (b) −5% (compressive) strain. Positive\nchange indicates gain of electrons, whereas negative chang e\nindicates loss of electrons for each state. Black, blue, and r ed\nlines denote m= 0,m=±1, and m=±2, respectively, while\nsolid and dashed lines represent majority and minority stat es,\nrespectively. The Fermi level is set to zero.\nstate and decreased unoccupied m= 0,↓andm=±2,↓\nstates around the Fermi level lead to larger EMCA be-\ncause of the stronger matrix elements of < m =±1,↑|\nLx|m= 0,↓>and< m =±1,↑|Lx|m=±2,↓>.\nMeanwhile, for the compressive strain shown in Fig. 3(b),\nthis process is reversed and EMCA decreases.\nThese results show that tensile stain along the c-\ndirection enhances the effective ferromagnetic exchange\nenergy notably but most importantly it causes a striking\nincrease of the MCA energy. This also suggests that off-\nequilibrium properties that exceed those of the equilib-\nrium state so drastically may be exploited in applications\nwhere materials are mechanically strained, either contin-\nuously or in pulses. Importantly these off-equilibrium\nmaterial parameters provide a deeper understanding of\nthe fundamental mechanisms of magnetic anisotropy and\nferromagnetic exchange in RE–TM systems and thus may\nenable the design of materials with enhanced properties.\nInspired by these findings, we now turn to a case study\nwhere we partially substitute Sm by Ce to induce local\nstrain in the cell and to predict changes in magnetic per-\nformance.5\n/s49/s51/s46/s56/s49/s51/s46/s57/s49/s52/s46/s48/s49/s52/s46/s49\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s86/s111/s108/s117/s109/s101/s32/s40 /s197/s51\n/s41/s40/s97/s41/s83/s116/s114/s97/s105/s110/s32/s40/s37/s41\n/s120 /s32/s40/s97/s116/s111/s109/s115/s47/s102/s46/s117/s46/s41/s32/s40/s98/s41\nFIG. 4. Change of (a) unit-cell volume and (b) effective strain\nin (Sm 1−xCex)Co5as a function of Ce concentration x.\nB. Partial substitution of Sm by Ce\nWe performed calculations of the (Sm 1−xCex)Co5(x\n= Ce/Sm) system, where we considered seven different\ncompositions: x= 0, 0.125, 0.25, 0.375, 0.5, 0.75, and\n1. For each xthe total energy was minimized to obtain\nthe most stable atomic arrangement among all possible\nconfigurations and the corresponding unit-cell volume.\nExamples of the systems with x= 0.25, 0.5, and 0.75 are\nshown in Fig. 1, where gray, red, light blue and dark blue\nspheres correspond to Sm, Ce, Co(2c) and Co(3g) atoms,\nrespectively.\nIn order to obtain a direct comparison between the\neffects of Ce-substitution on the magnetic properties with\nthose of strain, we show in Fig. 4(a) and (b) the unit-cell\nvolume and the effective strain, respectively, as function\nof the Ce concentration x. The volume decreases nearly\nlinearly with increasing x, which is due to the smaller\natomic radius of Ce compared to that of Sm, whereas\nthe strain increases correspondingly monotonically with\nincreasing x. This substitution-induced tensile strain can\nhave drastic effects on the magnetic state, which do not\nnecessarily correspond to the electronic properties of Ce\nthat would tend to decrease the anisotropy energy.\nThe calculated magnetic moments of the Co, Ce,\nand Sm atoms decrease monotonically with increasing\nCe-substitution x, which correlates with the decreasing\natomic volume, following Vergard’s law38(see Fig. 5a).\nAlso, the calculated JCo−CoandJCo−RE, which is the\naveraged interaction between Co and Sm or Ce atoms,\ndecrease strongly with increasing x, as seen in Fig. 5(b).\nFor the magnetocrystalline anisotropy, however, we\nfind a surprising non-monotonic behavior. In fact, for\nx= 0.125 the anisotropy energy increases to 10.49\nmeV/f.u. (19.8 MJ/m3) and for x= 0.25 it becomes/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s50/s48/s49/s52/s48/s49/s54/s48/s49/s56/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56/s49/s46/s50/s49/s46/s54\n/s32/s112/s108/s97/s110/s101/s32/s119 /s97/s118/s101/s32/s43/s32/s112/s115/s101/s117/s100/s111/s112/s111/s116/s101/s110/s116/s105/s97/s108/s115\n/s32/s112/s108/s97/s110/s101/s32/s119 /s97/s118/s101/s32/s43/s32/s102/s117/s108/s108/s32/s112/s111/s116/s101/s110/s116/s105/s97/s108/s32\n/s32/s111/s114/s98/s105/s116/s97/s108/s32/s98/s97/s115/s105/s115/s32/s115/s101/s116/s115/s32/s43/s32/s102/s117/s108/s108/s32/s112/s111/s116/s101/s110/s116/s105/s97/s108\n/s32/s112/s108/s97/s110/s101/s32/s119 /s97/s118/s101/s32/s43/s32/s111/s114/s98/s105/s116/s97/s108/s32/s98/s97/s115/s105/s115/s32/s115/s101/s116/s115/s32/s43/s32/s102/s117/s108/s108/s32/s112/s111/s116/s101/s110/s116/s105/s97/s108/s69\n/s77 /s67/s65/s40/s109/s101/s86/s32/s47/s32/s102/s46/s117/s46/s41\n/s120 /s32/s40/s97/s116/s111/s109/s115/s47/s102/s46/s117/s46/s41/s40/s97/s41\n/s32/s74\n/s67 /s111/s45/s67 /s111/s74\n/s67/s111/s45/s67/s111/s32/s40/s109/s101/s86/s41/s40/s98/s41/s77/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s32 /s40\n/s66 /s41\n/s32/s67/s111/s40/s50/s99/s41/s32/s32 /s32/s67/s111/s40/s51/s103/s41\n/s32/s83/s109/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s67/s101\n/s40/s99/s41/s50/s48/s50/s52/s50/s56/s51/s50/s51/s54\n/s32/s74\n/s67 /s111/s45/s82 /s69\n/s74\n/s67/s111/s45/s82/s69/s32/s40/s109/s101/s86/s41\nFIG. 5. Magnetic properties as a function of Sm-substitution\nby Ce: (a) atomic magnetic moments for Co on sites 2c (di-\namonds) and 3g (squares), Sm (triangles), and Ce (inverse\ntriangles); (b) effective total ferromagnetic exchange inte rac-\ntion energy JCo−Cobetween Co atoms (hexagons) and JCo−RE\nbetween Co and RE atoms (triangles), averaged over Co–Sm\nand Co–Ce pairs; and (c) magnetocrystalline anisotropy en-\nergy ( EMCA). The size of the symbols is larger than the error\nbars. In panel (c) the crossed circle corresponds to the EMCA\nof metastable systems with x= 0.5.\n11.63 meV/f.u. (22.0 MJ/m3), which is 30% larger than\nthat of the SmCo 5system in equilibrium (see Fig. 5c).\nAsxincreases further, however, EMCA decreases and for\nx= 1, i.e., CeCo 5, it becomes 2.96 meV/f.u. (corre-\nsponding to 5.6 MJ/m3). This EMCA for CeCo 5is also\nin excellent agreement with experimental data11.\nNote that for x= 0.5 the equilibrium structure has\nEMCA = 6.18 meV/f.u., which corresponds to only 68%\nof that of the SmCo 5compound, but we found metastable\nconfigurations that yield a magnetic anisotropy energy of\n7.56 meV/f.u., which is only slightly smaller than that of\nSmCo 5despite the fact that half of the Sm is substituted\nby Ce.\nThe 30% enhancement in EMCA forx= 0.25 is un-\nexpected, given that the electronic structure of Ce and\nits contribution would tend to substantially lower the\nanisotropy. We find, however, that this increase corre-\nlates with the strain in the unit-cell of SmCo 5induced\nby the substitution of every 4th Sm atom by a Ce atom,\nwhich corresponds to 1.13% strain (see Fig. 4b). For\nthe non-substituted SmCo 5system, 1% of strain corre-\nsponds to 22% increase in EMCA, as shown in Fig. 2c.6\nHence, this means that with 1/4 Ce substitution the ef-\nfects of the strained Sm–Co structure dominate over the\nelectronic contributions of Ce.\nAgain we find excellent agreement between the dif-\nferent approximations, with an exception at x= 0.25,\nwhich exhibits the highest EMCA and is slightly underes-\ntimated by the approximation based on orbital basis-sets\nand plane waves.\nThe main contribution to the enhancement of EMCA\ncomes from the Co(2c) sites, which is in direct analogy\nto the effects of strain (see Fig. 3). In order to in-\nvestigate in detail the origin of the enhanced EMCA of\n(Sm 0.75Ce0.25)Co5and the decreased value in high Ce\nconcentration cases, we present in Fig. 6 the DOS de-\ncomposed into dxy,x2−y2,dyz,xz anddz2orbitals, which\ncorrespond to magnetic quantum numbers of m=±2,\nm=±1, and m= 0, respectively. To provide a clear\nexplanation, only the DOS of Co(2c) atoms that drive\nthe change of EMCA is presented.\n/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52/s45/s48/s46/s54/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54/s83/s109/s67/s111\n/s53\n/s83/s109\n/s48/s46/s55/s53/s67/s101\n/s48/s46/s50/s53/s67/s111\n/s53/s32/s109 /s32/s61/s32\n/s32/s109 /s32/s61/s32\n/s32/s109 /s32/s61/s32\n/s67/s101/s67/s111\n/s53/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s32/s101/s86/s45/s49/s32\n/s97/s116/s111/s109/s45/s49/s32\n/s115/s112/s105/s110/s45/s49\n/s41\n/s69 /s40/s101/s86/s41\nFIG. 6. Comparison of the d-projected DOS of Co(2c) atoms\nin SmCo 5, (Sm 0.75Ce0.25)Co5and CeCo 5. Shaded, solid, and\ndotted lines denote m= 0, m=±1, and m=±2, respec-\ntively. The Fermi level is set to zero. Arrows emphasize\norbital shifting.\nAs shown in Fig. 6(a), in the DOS of SmCo 5we find\n< m =±1,↑|Lx|m= 0,↓>and< m =±1,↑|\nLx|m=±2,↓>couplings, as the peaks of m=±1\nandm= 0 lie directly below and above EF, respec-\ntively, and the energy difference between occupied and\nunoccupied states is reduced (see Eq. 1), leading to\nthe strong MCA along the caxis. In the substitutedsystem with (Sm 0.75Ce0.25)Co5the unoccupied minor-\nitym= 0 and m=±2 orbitals, indicated by arrows,\nshift toward the Fermi level. As a result, EMCA is en-\nhanced because the < m =±1,↑|Lx|m= 0,↓>and\nthe< m =±1,↑|Lx|m=±2,↓>coupling become\nstronger. For the CeCo 5case, however, the peaks of mi-\nnority m= 0 and m=±2 orbitals move to the occupied\nstates, and therefore couplings which contribute to the\nEMCA are strongly weakened.\nGiven the above, first-principles calculations provide\ncompelling evidence that strain in SmCo 5, either in\nthe form of mechanical strain or due to partial Sm-\nsubstitution by Ce in (Sm 1−xCex)Co5, can push the lim-\nits of magnetic anisotropy in this system. The agreement\nbetween our results and existing literature for the equilib -\nrium properties of SmCo 5, both structural and magnetic,\nconfirms our calculations and makes our predictions for\nthe strained structures very promising.\n/s45/s50/s48 /s45/s49/s53 /s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48 /s49/s53 /s50/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s32/s77\n/s122/s32/s32/s67/s111/s32/s40/s51/s103/s41\n/s32/s77\n/s122/s32/s32/s67/s111/s40/s50/s99/s41/s32/s38/s32/s82/s69/s40/s49/s97/s41/s77\n/s122/s32/s40\n/s66/s41\n/s68/s105/s115/s116/s97/s110/s99/s101/s32/s102/s114/s111/s109/s32/s100/s111/s109/s97/s105/s110/s32/s119/s97/s108/s108/s32/s40/s110/s109/s41\nFIG. 7. Monte Carlo simulation at finite temperature showing\nthe domain-wall profile of the sublattice containing Co(3g)\natoms and the sublattice containing RE(1a) and Co(2c)\natoms. A snapshot of the system containing a domain wall is\npresented at the corner.\nAll calculated material parameters so far correspond\nto the intrinsic magnetic properties at zero temperature.\nHence, we extend now our computational analysis to fi-\nnite temperature to predict both the intrinsic material\nparameters and the resulting relevant length scales, i.e.,\nthe domain-wall thickness δdw, of Ce-substituted Sm–Co\nmaterials. The computation of finite-temperature length\nscales takes into account the effects of the microstructure\nand is thus particularly relevant to high-temperature ap-\nplications.\nTaking the DFT results shown in Fig. 5 as inputs, we\nperformed Monte Carlo simulations and computed the\nlength scale of a domain wall by simulating domain-wall\nbehavior in the Sm–Ce–Co system. An example is illus-\ntrated in Fig. 7, which shows the spatially-resolved sub-\nlattice magnetization Mzas a function of the distance\nfrom the domain-wall center. The spins rotate around the7\n/s48/s46/s48/s48 /s48/s46/s50/s53 /s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s49/s50/s51/s52/s48/s46/s55/s53/s48/s46/s57/s48/s49/s46/s48/s53/s49/s46/s50/s48/s54/s48/s48/s55/s53/s48/s57/s48/s48/s49/s48/s53/s48/s49/s50/s48/s48\n/s32\n/s32/s100/s119/s32/s40/s110/s109/s41\n/s120/s32 /s40/s97/s116/s111/s109/s115/s47/s102/s46/s117/s46/s41/s32/s51/s48/s48/s32/s75\n/s32/s53/s48/s48/s32/s75/s32\n/s32/s77 /s40/s84/s41/s32 /s40\n/s66/s41\n/s40/s99/s41/s40/s98/s41/s32\n/s32/s84\n/s67/s32/s40/s75/s41/s40/s97/s41\nFIG. 8. Effect of the Ce-substitution on the finite-\ntemperature magnetic properties: (a) Curie temperature; ( b)\naverage magnetic moments at T= 300 K and 500 K; and (c)\ndomain-wall thickness at T= 300 K and 500 K, as a function\nof the Ce concentration x. The size of the symbols is larger\nthan the error bar.\ncenter of the domain wall and the profile is proportional\nto tanh ( r/λ), with rthe distance from the domain-wall\ncenter and λ=δdw/2. Note that we distinguish between\nthe magnetization of the 3g sites and the 1a and 2c sites.\nIn order to obtain a comparison between all the Ce-\nsubstituted systems, we computed the Curie Tempera-\nture TC, the average atomic magnetic moment M, and\nδdwatT= 300 K and 500 K (= 0 .5TSmCo 5\nC ). Figure 8\nshows these results as a function of the Ce concentration.\nBoth TCandMgradually decrease with increasing\nCe concentration. For SmCo 5(x= 0) we obtain\nTC≈1100 K, whereas for CeCo 5(x= 1) we obtain\nTC≈700 K, both of which are in good agreement with\nexperimental data ( ∼1050 K for SmCo 5and∼660 K\nfor CeCo 5).\nFor the domain-wall thicknesses, we find that for\nSmCo 5at room temperature δdw= 1.61(1) nm, which\nis in very good agreement with previous theoretical\ncalculations39(δdw= 1.6 nm) and experiments40(δdw=\n1.3±0.3 nm), supporting the validity of our first-\nprinciples calculations and MC simulations, consideringthat SmCo 5without Ce is our reference system.\nTheδdwreflects the tendency of EMCA shown in Fig. 5,\nwith the smallest domain wall being at x= 0.25 and\nthe largest at x= 1. Note that the decreasing domain-\nwall thickness in the Ce-substituted system suggests that\na domain wall could fit in smaller structures and be\nmore easily trapped, thus potentially increasing the co-\nercivity of a nanostructured magnet, and hence ensuring\nthat the same microstructure that makes SmCo 5a high-\nperformance magnet would also make (Sm 0.75Ce0.25)Co5\na permanent magnet with even higher performance, es-\npecially with regards to its MCA, which is the crucial\ncomponent for precision applications.\nIV. CONCLUSIONS\nWe found, based on electronic structure calculations,\nthat changes in the lattice spacing of SmCo 5have a\nstrong effect on the density of states close to the Fermi\nlevel, causing dramatic changes in the magnetic prop-\nerties. Specifically, we observed that tensile strain in\nSmCo 5monotonically enhances the magnetocrystalline\nanisotropy energy ( EMCA) and the effective ferromag-\nnetic exchange interaction ( Jeff), while reducing the\natomic magnetic moments of Sm and Co. In contrast,\ncompressive strain reduces the EMCA and the Jeff, and\nincreases the atomic magnetic moments. Further, we in-\nvestigated the effects of strain induced by partial sub-\nstitution of Sm by Ce and discovered that it has also\ndrastic effects on the EMCA. In fact, when Ce substi-\ntutes every 4th Sm atom in the lattice the EMCA in-\ncreases by more than 30%, compared to that of SmCo 5,\nbecause the density of states increases close to the Fermi\nlevel. In order to predict the potential of this system\nfor high-performance applications, we investigated the\nfinite-temperature properties of Ce-substituted SmCo 5\nand found that the magnetic length scales, the atomic\nmagnetic moments, and the Curie temperature are com-\nparable to those of the parent compound SmCo 5, suggest-\ning that the Ce-substituted material is a very promising\ncandidate for high-performance high-temperature appli-\ncations with enhanced magnetocrystalline anisotropy and\nwith 25% reduced heavy-rare earth content. Further, we\nhave shown that multi-scale modeling is a powerful strat-\negy for the detailed analysis and prediction of functional\nmaterials properties.\nV. ACKNOWLEDGMENTS\nThe authors acknowledge funding from the Swiss Na-\ntional Science Foundation via Grant No. 200021–165527\nand from the ETH Grant ETH-47 17-1, and thank P. 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Phys. 64, 1338 (1988)."    },    {        "title": "1809.09785v2.Resonant_magnetic_induction_tomography_of_a_magnetized_sphere.pdf",        "content": "Resonant magnetic induction tomography of a magnetized sphere\nA. Gloppe,1,∗R. Hisatomi,1Y. Nakata,1Y. Nakamura,1, 2and K. Usami1\n1Research Center for Advanced Science and Technology (RCAST),\nThe University of Tokyo, Meguro-ku, Tokyo 153-8904, Japan\n2Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama 351-0198, Japan\n(Dated: April 11, 2019)\nWe demonstrate the structural imaging of magnetostatic spin-wave modes hosted in a millimeter-\nsized ferromagnetic sphere. Unlike for low-dimensional magnetic materials, there is no prior tech-\nnique to image these modes in bulk magnetized solid of revolution. Based on resonant magnetic\ninduction tomography in the microwave range, our approach ensures the robust identification of\nthese non-trivial spin-wave modes by establishing their azimuthal and polar dependences, starting\npoint of magnonic fundamental studies and hybrid systems with complex spin textures well beyond\nthe uniform precession mode.\nMacroscopic magnetically ordered structures, such\nas Yttrium Iron Garnet (YIG) millimetric spheres, are\nsolid supports of extended collective spin excitations,\nmagnons [1–3], that can be cooled down to their quan-\ntum ground state and coherently coupled to a super-\nconducting quantum bit through a microwave cavity [4].\nA coherent optical control of magnons in the quantum\nregime could enable the efficient transduction of optical\nand microwave photons [5], opening the way to quan-\ntum telecommunications between superconducting quan-\ntum computers [6, 7] as well as quantum-noise limited\nmicrowave amplifiers [8]. The study of the interactions\nof magnons and photons in an optical cavity, or cavity\noptomagnonics , in a solid-state matrix has been initi-\nated by the first observations of magnon-induced Bril-\nlouin light scattering involving the uniform precession\nspin-wave mode and optical whispering gallery modes of\na YIG sphere [9–11]. Higher-order magnetostatic spin-\nwave modes, with a variety of orbital angular momenta\nand spin textures, extend the richness of this hybrid\nsystem. In particular, the exchange of orbital angular\nmomentum between magnons and optical photons has\nbeen experimentally demonstrated recently [12, 13]. The\nrelated selection rules allow a controlled non-reciprocal\nscattering, dependent on the interacting spin wave, po-\ntentially leading to the development of a new class of chi-\nral devices [14]. The optomagnonic coupling, faint with\nthe uniform precession mode, will be optimized for high-\norder modes whose spatial distribution localizes more and\nmore towards the resonator boundaries where the optical\nwhispering gallery modes spread [15].\nThese spin-wave modes can be described within the\nmagnetostatic approximation for a saturated magnetic\nellipsoid [17] and their eigenfrequencies determined nu-\nmerically. Diverse effects could alter this description: re-\nlated to the environment as the temperature dependence\nof the saturation magnetization, the non-uniformity of\nthe saturating static magnetic field or the presence of\nclose-by parasitic elements [13, 18], or related to the sam-\nple as propagation corrections [19], magneto-crystalline\nanisotropy [20] or irregularities and composition defects,potentially resulting in inter-mode coupling [21]. Increas-\ning the optomagnonic coupling by using hybrid custom\nshapes with smaller mode volumes [22] will disturb the\nfrequency distribution [23]. Associated with the density\nof modes excited by a non-uniform microwave field, the\nsimple identification of the modes beyond the uniform\nprecession mode using their expected ferromagnetic res-\nonance frequencies [19] is possibly ambiguous. An in situ\nstructural mapping of the spin-wave modes is needed to\nproperly identify them and estimate their coupling to\nothers modes, independently of the sample nature and of\nthe experimental conditions, for fundamental magnonic\nstudies [24, 25], quantum magnonics [4, 26] and hybrid\nsystem operations [12, 15, 27–30].\nPushed forward by the demands for fast recording\nand high-capacity storage devices, magnetization dynam-\nics of micro and nanostructures on a substrate have been\nintensively studied in the last years with advanced mag-\nnetic microscopy methods involving x-ray magnetic cir-\ncular dichroism [31], magneto-optical interactions [32],\nthermal effects [33], microwave near-field [34] and mag-\nnetic scanning probe [35]. At the other extreme, space\nexploration and geophysics have been implementing suc-\ncessfully magnetic field measurements around gigantic\nsolids of revolution from satellite-based loop coil mag-\nnetometers employed to understand terrestrial polar au-\nrorae from the ionosphere [36] to vector fluxgate magne-\ntometers used to determine the magnetosphere and inte-\nrior structures of Jupiter [37].\nHere, we access the spatial structure of the spin-\nwave modes of a bulk magnetized solid by measuring the\nmagnetic flux spectrum intercepted by a mobile loop coil\nfacing the sample at different azimuth-altitude positions,\nwhile the magnons are coherently excited by a fixed mi-\ncrowave antenna — realizing a magnetic resonance imag-\ning (MRI) [38] scanner for collective electron spins exci-\ntations. The spatially and spectrally resolved magnetic\nresponses of the system are carefully processed to extri-\ncate the nature of each spin-wave mode.\nA strong magnetic field HDCapplied along the z-\naxis saturates the magnetization of the sample. SolvingarXiv:1809.09785v2  [cond-mat.mtrl-sci]  10 Apr 20192\nFIG. 1. Imaging the stray field induced by the spin-\nwave modes of a magnetized sphere. a, Snapshot of\nthe calculated spatial magnetization distribution of magne-\ntostatic modes in a YIG sphere for n<4, transverse to the\nstatic magnetic field HDC. Their magnetization norm is color-\nencoded for successive sections from θ= 10◦to 170◦[16].\nb,Schematic of the experiment. A fixed loop coil (pump\ncoil) excites the magnons at microwave frequencies. These\nmodes generate a dynamic magnetic field, represented here in\nthe equatorial plane for the ( n,m,r) = (3,3,0) mode. The\nstray field spectrum is captured by a second loop coil (probe\ncoil) at different azimuth-altitude ( ϕc,zc) positions. c,Nu-\nmerical calculations of the spatially-resolved magnetic flux in-\ntercepted by a rectangular coil (∆ y= 0.5 mm, ∆ z= 0.3 mm,\n˜rc= 2 mm) on a YIG sphere for ( n,m) = (5,5), (7,3) and\n(12,6) families (i-iii), encoded in the surface colors and cor-\nrugations.Maxwell and Landau-Lifshitz equations reveals the exis-\ntence of dynamic magnetization modes in the orthogo-\nnal plane [23]. For a millimeter-sized YIG sphere, these\nspin-wave modes have typical eigenfrequency Ω k/2π∼5–\n10 GHz for HDCin the 100 mT/ µ0range (with µ0the\nmagnetic constant). In the magnetostatic approxima-\ntion [39], they are described by three indices ( n,m,r):\nmexpresses the azimuthal dependency and is linked to\nthe winding number of the spin texture [12], n−|m|\nthe polar dependency and rthe number of nodes along\nthe radial direction. Their spatial distribution is deduced\nfrom a magnetic potential solution of Laplace equation\nin spheroidal coordinates dependent on the applied static\nmagnetic field and on the considered mode. The non-\ntrivial spatial distributions of the lowest-order modes are\npictured in Fig. 1 a, their phase dependence at fixed alti-\ntude being given approximately by −(m−1)ϕ. Outside\nthe sphere, a magnetostatic mode induces a magnetic\nfieldHm\nn=∇ψm\nnsuch that close to the magnon reso-\nnance\nψm\nn(r,θ,ϕ ) =ζm\nn\nrn+1Pm\nn(cosθ)eimϕ\nwithζm\nnencapsulating the pump field projection on the\nspin-wave mode and the resonance condition, while Pm\nn\nis the Ferrers function [16]. The spherical coordinate\nsystem (r,θ,ϕ ) is depicted in the inset of Fig. 1 a.\nThe imaging scheme is illustrated on Figure 1 b. The\npump coil, fixed during the experiment, applies a mi-\ncrowave field exciting the magnons. The excitation fre-\nquency is swept to measure the spectral response of the\nwhole magnonic system. The probe coil turns around the\nsample axis of revolution along a cylindrical orbit ( ϕc,zc)\nand intercepts at each position the phase-resolved spec-\ntrum of the induced magnetic flux. Working at a cylin-\ndrical detection distance ˜ rclarge compared to the typi-\ncal width 2∆ yand height 2∆ zof the probe coil, the in-\nduced flux due to a ( n,m) mode can be approximated to\nφm\nn(ϕc,zc)∼eimϕcZm\nn(zc) with Zm\nn(zc) having a mode-\ndependent envelope whose number of nodes along the al-\ntitude axis is related to n−|m|[16]. Figure 1 cpresents\nnumerical computations of the magnetic flux induced by\nsome representative mode families. Rotating the probe\ncoil around the sample at fixed altitude grants access to\nthe azimuthal parameter m, enclosed in the mode rela-\ntive phase, while a walk along the altitude z-axis leads to\nthe polar parameter n−|m|. The radial dependency, not\naffecting the spatial distribution of the stray field, has to\nbe deduced from the suite of eigenfrequencies of a given\nfamily ( n,m) [23]. Each excited mode, with its distinct\nspectral signature, will contribute to the total stray field.\nThe extraction of these features in the measured spectra\nat all positions along these two axes leads to their spatial\nmapping and subsequently to their robust identification.3\nFIG. 2. Broadband ferromagnetic resonances (FMR)\nat a fixed azimuth-altitude coordinate. a, Typical spec-\ntrum of the microwave power reflected by a 2-mm YIG sphere\ninto the pump coil (blue line). The absorption dips corre-\nspond to the resonance of more than 50 different individual\nspin-wave modes (resolution bandwidth: 10 kHz). The ex-\ncitation microwave frequency range is chosen such that the\nspectrum encapsulates all the excited modes in one run. At\na fixed probe-coil azimuth-altitude coordinate ( ϕc,zc), the\ntransmitted microwave spectra in power (b)and phase (c)\n(black line) intercepted by the probe coil through the sphere\ngive access to the local information on each mode once fitted\nas a collection (red line) of individual harmonic oscillators\n(grey lines), in particular the local phase with respect to the\norigin fixed by the excitation field. The relative phase differ-\nences of the modes are responsible for non-trivial interference\npatterns. All the measurements are repeated after an autom-\natized vertical retraction of the sample to define robust phase\nreferences [16]. Colored stars mark the modes analyzed along\nthe azimuth in Fig. 3 (green) and along the altitude in Fig. 4\n(orange).\nThe studied sample is a 2 mm-diameter YIG sphere,\nplaced at the center of an iron magnetic circuit ended by\ntwo permanent ring magnets distanced by 15 mm. The\nstatic magnetic field ( ∼230 mT/µ0) created along their\nrevolution axis zsaturates the sphere along its [110] crys-tal axis. The two small loop coils with a sub-millimetric\ninner radius (∆ y∼0.5 mm, ∆z∼0.3 mm), made out\nof semi-rigid coaxial copper cables by terminating their\nends, are facing the sample and are respectively con-\nnected to the output (pump) and input (probe) ports of\na vector network analyzer. The fixed pump coil stands a\nfew millimeters away from the sample. Fixed to a three-\naxis linear actuator on a motorized rotation stage, the\nprobe coil takes arbitrary positions (˜ rc,ϕc,zc) along a\ncylindrical orbit around the sample. The detection dis-\ntance is set at ˜ rc= 2 mm. The reflected power from\nthe pump coil (Fig. 2 a) reveals a collection of absorption\ndips, signatures of individual resonant spin-wave modes\nwhich can be modeled as damped harmonic oscillators [1].\nWe measure in transmission (Fig. 2 b-c) the magnetic\nflux intercepted by the probe coil at a particular po-\nsitionSφ[Ω] =/summationtext\nk={n,m,r}Akeiϕk/(Ω2−Ω2\nk−iΓkΩ)\nwithkrunning on all the excited modes, Ω k/2πthe spin-\nwave mode eigenfrequency, Γ kits damping rate, and Ak\nits relative response amplitude depending on the mode-\ndependent pump efficiency and on the mode spatial struc-\nture. The relative mode phase with respect to the excita-\ntion field,ϕk=ϕ0\nk+ϕk, can be decomposed such that ϕ0\nk\nis the mode phase origin defined by the excitation field\nandϕkits spatial component.\nFor imaging the spatial structure of the spin-wave\nmodes, we assemble these spectral measurements at nu-\nmerous probe coordinates ( ϕc,zc) and adjust the local\nspin-wave mode responses. First at fixed altitude, the\ncoil travels around the sample. Figure 3 adepicts the\ntransmitted ferromagnetic resonance signal power and\nphase spectra along the azimuth in the equatorial plane\n(zc= 0). While the modes amplitude is globally con-\nstant, their relative phase ϕkindividually changes with\nthe probe azimuth. We report the evolution of the rela-\ntive phase as a function of the coil azimuthal position.\nFigure 3 b(i-vi) illustrates the extracted azimuthal de-\npendence of the phase of six different modes, with their\ntheoretical counterparts appended, insuring a clear iden-\ntification of modes with m= 0, 1, 2, 3, 4 and 7.\nNext, at fixed azimuth we record the spectra along\nthe altitude axis (Figure 4 a). As we are traveling over\nthe mode envelope, the variation of the mode transmitted\npower and phase flips are observable directly on the spec-\ntra for well-isolated dominant modes. We report the ex-\ntracted signed mode amplitude Akcosϕkas a function of\nthe altitude of four representative modes in Fig. 4 b(i-iv),\nillustrating polar mode families n−|m|= 0,n−|m|= 1,\nn−|m|= 2 and n−|m|= 3. The altitude axis of the coil\nis slightly tilted by ξz=−6◦with respect to the z-axis\ndefined by the permanent magnets [16]. This induces a\nslight imbalance in the measured flux, favoring positive\naltitudes. The measurements are in very good agreement\nwith the theoretical calculation of the flux taking into\naccount this correction. The exhibited modes could be\nidentified respectively as (2 ,2,0), (3,2,0), (3,1,0) and4\nFIG. 3. Azimuthal dependence of the induced magnetic flux. a, Microwave transmission spectra in power and phase\nas a function of the azimuthal position of the probe coil ϕcin the sample equatorial plane. These spectra are all processed\n(see Figure 2 and Supplemental Material) to extract the relative phase of each spin-wave modes ϕk, exemplified by six of\nthem in b(i-vi) (colored filled circles, top), with mean eigenfrequencies of 5 .74 GHz, 5.42 GHz, 6.82 GHz, 5.61 GHz, 7.05 GHz\nand 7.17 GHz (indicated on aand on Fig. 2 bwith green stars), exhibiting an azimuthal parameter m= 0, 1, 2, 3, 4 and 7\nrespectively, in comparison with the theoretical evolution of mϕc(solid color lines, bottom).\n(4,1,0). Their relative eigenfrequencies spacing to the\nuniform precession mode (Ω k−Ω110)/2πare computed\nin the magnetostatic approximation for comparison [16].\nThe discrepancies with the experimental observations,\nrespectively 15 %, −36 %,−8% and−8 %, most likely\ndue to magneto-crystalline anisotropy and propagation\neffects, underline the necessity to access the mode spa-\ntial properties in situ to avoid a misidentification between\nclose-by modes.\nIn the current conditions, our method reveals the po-\nlar mode families up to n−|m|= 3 and azimuthal fam-\nilies up to m= 7. This range could be further expanded\nby designing the pump antenna to maximize the exciting\nefficiency of specific spin-wave modes of interest. Detec-\ntion artifacts due to the probe coil height should appear\nonly for modes with n−|m|>11 and can be outclassed\nfurthermore by tuning ∆ z/˜rc[16], largely overcoming the\nspin texture complexity traditionally under study.\nWe have developed a new broadband resonant to-\nmography scanning method to map the spatial structure\nof spin-wave modes hosted in a magnetized solid of rev-\nolution, providing a robust mode identification. Demon-\nstrated here on a 2-mm YIG sphere saturated along its\n[110] axis, this approach straightforwardly extends to el-\nlipsoids [17], disks and rods [40, 41], provided that the\nspin-wave modes exhibit linewidths smaller than their\ntypical frequency splitting [16]. This versatile imaging\nmethod, which does not require to have a perfect knowl-edge on the sample and its environment, will be partic-\nularly relevant for studying the magnetization dynam-\nics of emergent magnetic materials and structures whose\nshape, crystallinity and composition could be challenging\nto control. Hybrid magnonic operations well beyond the\nuniform precession mode can be envisioned, broadening\nthe scope of quantum magnonics, magnomechanics and\ncavity optomagnonics towards the emergence of macro-\nscopic quantum devices.\nThe authors thank A. Osada, R. Yamazaki and Y.\nTabuchi for fruitful interactions. This work was sup-\nported by JSPS KAKENHI (Grant Nos. 16F16364 and\n26220601) and by JST ERATO project (Grant No. JP-\nMJER1601). A.G. is an Overseas researcher under Post-\ndoctoral Fellowship of Japan Society for the Promotion\nof Science.\n∗arnaud.gloppe@qc.rcast.u-tokyo.ac.jp\n[1] A. Gurevich and G. A. Melkov, Magnetization Oscilla-\ntions and Waves (CRC Press, 1996).\n[2] D. Stancil and A. Prabhakar, Spin Waves: Theory and\nApplications (Springer, 2009).\n[3] V. Kruglyak, S. Demokritov, and D. Grundler, Journal\nof Physics D: Applied Physics 43, 264001 (2010).\n[4] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya-\nmazaki, K. Usami, and Y. Nakamura, Science 349, 4055\nFIG. 4. Altitude dependence of the induced magnetic\nflux. a, Microwave transmission spectra in power and phase\nas a function of the altitude position zcof the probe coil.\nb,Extraction of the local response of the modes allowing the\nreconstruction of their structure along the altitude axis (light-\ncolor filled circles), exemplified here for the lower-order fami-\nlies (i) n−|m|= 0, (ii) n−|m|= 1, (iii) n−|m|= 2 and (iv)\nn−|m|= 3. They are illustrated respectively by the modes\nof mean eigenfrequencies 6 .82 GHz, 6.23 GHz, 5.42 GHz and\n5.23 GHz (indicated on aand on Fig. 2 bwith orange stars),\nidentified as (2 ,2,0), (3,2,0), (3,1,0) and (4,1,0). Note that\n(i) and (iii) were represented in Fig. 3, respectively labeled\n(iii) (m= 2) and (ii) ( m= 1). The solid dark lines cor-\nrespond to the expected flux (˜ rc= 2 mm, ∆ y= 0.5 mm,\n∆z= 0.3 mm) with a coil axis tilt ξz=−6◦. The altitude\nextent of these plots is depicted as a black rectangle in the\ninsets showing the magnetic flux distribution.(2015).\n[5] R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa,\nA. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura,\nPhys. Rev. B 93, 174427 (2016).\n[6] H. J. Kimble, Nature 453, 1023 (2008).\n[7] S. Wehner, D. Elkouss, and R. Hanson, Science 362\n(2018).\n[8] S. Barzanjeh, S. Guha, C. Weedbrook, D. Vitali,\nJ. Shapiro, and S. Pirandola, Phys. Rev. Lett. 114,\n080503 (2015).\n[9] A. Osada, R. Hisatomi, A. Noguchi, Y. 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Lauterbur, Nature 242, 190 (1973).\n[39] P. C. Fletcher and R. O. Bell, Journal of Applied Physics\n30, 687 (1959).\n[40] J. F. Dillon, Journal of Applied Physics 31, 1605 (1960).\n[41] R. I. Joseph and E. Schlomann, Journal of Applied\nPhysics 32, 1001 (1961).Supplemental Material\nResonant magnetic induction tomography of a magnetized sphere\nA. Gloppe,1,∗R. Hisatomi,1Y. Nakata,1Y. Nakamura,1, 2and K. Usami1\n1Research Center for Advanced Science and Technology (RCAST),\nThe University of Tokyo, Meguro-ku, Tokyo 153-8904, Japan\n2Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama 351-0198, Japan\n(Dated: April 11, 2019)\nCONTENTS\nReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1\nI Methods summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2\nII Magnetic flux induced by a magnetostatic mode of a ferromagnetic sphere . . . . . . . . . . . . . . . . . . . . . . . . . 2\nIII Evolution of the eigenfrequencies with the static magnetic field HDC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\nIV Additional details on the setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\nV Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\nVI Generalization to other shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n∗arnaud.gloppe@qc.rcast.u-tokyo.ac.jp\n1P. C. Fletcher and R. O. Bell, Journal of Applied Physics 30, 687 (1959).\n2L. R. Walker, Journal of Applied Physics 29, 318 (1958).\n3F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions , 1st ed. (Cambridge\nUniversity Press, New York, NY, USA, 2010).\n4R. L. White, Journal of Applied Physics 31, S86 (1960).\n5R. I. Joseph and E. Schlomann, Journal of Applied Physics 32, 1001 (1961).\n6L. R. Walker, Phys. Rev. 105, 390 (1957).arXiv:1809.09785v2  [cond-mat.mtrl-sci]  10 Apr 20192\nI. METHODS SUMMARY\nSpin-wave modes calculations\nThe spin-wave modes in Fig. 1 aare plotted at time t= 0.2×2π/Ωk, with a saturation magnetization Ms=\n194 mT/µ0, a static magnetic field HDC= 315 mT/µ0and a gyromagnetic ratio γ/2π= 28 GHz/T. The resonance\nfrequency Ω k/2πof a given spin-wave mode is numerically determined by solving the resonance equation (S19) in these\nconditions, a requirement to define properly the spheroidal coordinate system in which the internal magnetic potential\nhas explicit solutions, computed for each colatitude θ. Once interpolated on a regular Cartesian grid, the internal\nmagnetic potential is numerically differentiated to obtain the internal magnetic field. The transverse magnetization\nMm\nnis obtained by linear combination of the internal magnetic field components with oblate factors, as functions of\nHDCandMs1. The stray field is numerically evaluated from the external magnetic potential ψm\nn. The internal and\nexternal fields for (3 ,3,0) are joined to plot the total transverse magnetic induction field B3\n3in Figure 1 b, slightly out\nof resonance for a better visualization.\nProbe coil\nThe coil parameters ∆ y, ∆zand ˜rcare chosen carefully to ensure proper imaging. The detection distance ˜ rc\nshould be short to maximize the acquired signal ( φm\nn∝1/˜rn+2\nc) while insuring that ∆ y/˜rc/lessmuch1 and ∆z/˜rc/lessmuch1. The\nlateral semi-extension ∆ ywhile small compared to ˜ rc, should be maximized ( φm\nn∝∆y). The value of ∆ zshould be\nsmall enough to guarantee the measured structure can be safely related to n−|m|. The choice of ∆ z= 0.3 mm for\n˜rc= 2 mm insures the safe detection of spin-wave modes with n−|m|up to 11. The distance ˜ rcmay be slightly\ntuned to maximize the detection of a particular azimuth family (see Sec. II for details).\nData acquisition\nIn the magnetostatic approximation2, the spin-wave modes are expected on a frequency range of γµ0Ms/4π∼\n2.6 GHz. The information on all the observable spin-wave modes is obtained with a high resolution by acquiring\nthe spectra by pieces, for a total range of 2.89 GHz and more than 65,000 points by spectrum (microwave excitation\npower: 0 dBm). At fixed altitude, we record spectra at 101 different probe coil azimuthal positions on a 158◦range.\nThe accessible angular range is limited by the presence of the magnetic circuit and pump coil. At fixed azimuth,\nwe record spectra at 297 different probe altitudes on a 2 .2 mm extent across the equatorial plane of the sphere. A\nresidual background from the direct coupling between the coils and the parasitic response of close-by elements subsists\nwith typical linewidths ( >1 GHz) much larger than those of the spin-wave modes ( <10 MHz) and could be then\nconscientiously dismissed. A set of measurement at a given coil position lasts for 4 min with a resolution bandwidth\nof 10 kHz, resulting in a total measurement time of 6 h along the azimuth (Figure 3) and 22 h along the altitude\n(Figure 4).\nII. MAGNETIC FLUX INDUCED BY A MAGNETOSTATIC MODE OF A FERROMAGNETIC\nSPHERE\nWe compute the magnetic flux induced by a magnetostatic mode ( n,m) within a ferromagnetic sphere, inter-\ncepted by a coil facing the sample along a cylindrical orbit (˜ rc,ϕc,zc), represented in Figure S1.\nThe magnetic potential outside the sample induced by the magnetostatic mode ( n,m), excited by a pump field, is\ngiven in spherical coordinates by1:\nψm\nn(r,θ,ϕ ) =Am\nnrn/bracketleftbigg\n1 +αm\nn/parenleftBiga\nr/parenrightBig2n+1/bracketrightbigg\nPm\nn(cosθ)eimϕ, (S1)\nwithPm\nnthe Ferrers function3,athe sphere radius, Am\nnrelated to the projection of the pump field on the spherical\nharmonics and αm\nnis an amplification factor related to the mode resonance.\nThe gradient of this outer magnetic potential is\n∇ψ(r,θ,ϕ ) =∂rψer+1\nr∂θψeθ+1\nrsinθ∂ϕψeϕ,3\nFIG. S1. Frame of the problem figuring the position and orientation of the coil compared to the sample, in the ideal case in\nwhich the coil faces perfectly the sphere and its trajectory shares the same center and rotation axis, projected onto the xy-plane\n(a) and onto the xz-plane rotating with ϕc(b). The coil cylindrical trajectory is represented in color dashed lines, with the\nazimuth and altitude in green and orange respectively. The horizontal distance of the probe coil ˜ rcis pictured in blue.\nwith\n∂rψ(r,θ,ϕ ) =eimϕ∆1(r)Pm\nn(cosθ),\n∂θψ(r,θ,ϕ ) =eimϕ∆2(r)1\nsinθ/bracketleftbig\n−(n+ 1) cosθPm\nn(cosθ) + (n+ 1−m)Pm\nn+1(cosθ)/bracketrightbig\n,\n∂ϕψ(r,θ,ϕ ) =eimϕ∆2(r)imPm\nn(cosθ),\nand\n∆1(r)≡Am\nnrn−1/bracketleftbigg\nn−(n+ 1)αm\nn/parenleftBiga\nr/parenrightBig2n+1/bracketrightbigg\n, (S2)\n∆2(r)/r≡˜∆2(r)≡Am\nnrn−1/bracketleftbigg\n1 +αm\nn/parenleftBiga\nr/parenrightBig2n+1/bracketrightbigg\n. (S3)\nand will be written in the Cartesian basis Bas\n∇ψ(r,θ,ϕ ) =/bracketleftbigg/parenleftbigg\n∂rψsinθ+1\nr∂θψcosθ/parenrightbigg\ncosϕ−1\nrsinθ∂ϕψsinϕ/bracketrightbigg\nex\n+/bracketleftbigg/parenleftbigg\n∂rψsinθ+1\nr∂θψcosθ/parenrightbigg\nsinϕ+1\nrsinθ∂ϕψcosϕ/bracketrightbigg\ney\n+/bracketleftbigg\n∂rψcosθ−1\nr∂θψsinθ/bracketrightbigg\nez.\nIn the rotating frame of the probe coil Bsuch that\nx\ny\nz\n\nB=R−1\nϕc·\nx\ny\nz\n\nBwith the rotation matrix\nRϕc≡\ncosϕcsinϕc0\n−sinϕccosϕc0\n0 0 1\n.\nFor a perfectly oriented coil, the surface normal of the probe can be simply expressed as\nRϕc·dSB=dSB=dS\n1\n0\n0\n\nB(S4)\nand we can write the induced flux\nφm\nn=µ0/integraldisplay/integraldisplay\nS∇ψ(r,θ,ϕ )·dSB=µ0/integraldisplay/integraldisplay\nSdS(Ir+Iθ+Iϕ) (S5)4\nwith\nIr(r,θ,ϕ )≡sinθcosϕ∂rψ,\nIθ(r,θ,ϕ )≡1\nrcosθcosϕ∂θψ,\nIϕ(r,θ,ϕ )≡−1\nr1\nsinθsinϕ∂ϕψ,\ndefiningϕ≡ϕ−ϕc. By developing these formulae we obtain,\nIr(r,θ,ϕ ) =eimϕc/bracketleftBig\nei(m+1)ϕ+ei(m−1)ϕ/bracketrightBig\nsinθPm\nn(cosθ)∆1(r)\n2, (S6)\nIθ(r,θ,ϕ ) =eimϕc/bracketleftBig\nei(m+1)ϕ+ei(m−1)ϕ/bracketrightBigcosθ\nsinθ/bracketleftbig\n−(n+ 1) cosθPm\nn(cosθ) + (n+ 1−m)Pm\nn+1(cosθ)/bracketrightbig˜∆2(r)\n2,(S7)\nIϕ(r,θ,ϕ ) =−eimϕc/bracketleftBig\nei(m+1)ϕ−ei(m−1)ϕ/bracketrightBig\nm1\nsinθPm\nn(cosθ)˜∆2(r)\n2. (S8)\nThese expressions can be computed numerically for an arbitrary coil section.\nA. A small rectangular coil\nWe consider now that the coil has a rectangular section with a lateral extension 2∆ ymuch smaller than the\ncylindrical detection distance ˜ rc, then\nϕ= arctan/parenleftbiggy\n˜rc/parenrightbigg\n∼y\n˜rc\nand the contribution of ytoris negligible. Then Eq.(S5) could be then rewritten\nφm\nn=µ0/integraldisplayzc+∆+\nz\nzc−∆−\nzdz/integraldisplay∆+\ny\n−∆−\nydy(Ir+Iθ+Iϕ).\nWe define the integrals along the y-direction\nY±≡/integraldisplay∆+\ny\n−∆−\nydy/bracketleftBig\nei(m+1)ϕ±ei(m−1)ϕ/bracketrightBig\n(S9)\nand along the z-direction\nZr(zc)≡/integraldisplayzc+∆+\nz\nzc−∆−\nzdzsin(θ)Pm\nn(cosθ)∆1(r)\n2, (S10)\nZθ(zc)≡/integraldisplayzc+∆+\nz\nzc−∆−\nzdzcosθ\nsinθ/bracketleftbig\n−(n+ 1) cosθPm\nn(cosθ) + (n+ 1−m)Pm\nn+1(cosθ)/bracketrightbig˜∆2(r)\n2, (S11)\nZϕ(zc)≡/integraldisplayzc+∆+\nz\nzc−��−\nzdzm1\nsinθPm\nn(cosθ)˜∆2(r)\n2, (S12)\nso that\nφm\nn=µ0eimϕc/bracketleftbig\nY+[Zr(zc) +Zθ(zc)]−Y−Zϕ(zc)/bracketrightbig\n. (S13)\nB. Influence of the lateral extension\nIf the coil is centered regarding the y-direction, ∆ y≡∆+\ny= ∆−\nyand\nY±\nc= 2∆y/parenleftbigg\nsinc/bracketleftbigg(m+ 1)∆y\n˜rc/bracketrightbigg\n±sinc/bracketleftbigg(m−1)∆y\n˜rc/bracketrightbigg/parenrightbigg\n(S14)5\nFIG. S2. a, Normalized amplitude variations of the induced flux with the lateral extension, when the coil is centered along the\ny-direction following Eq.(S14) in the limit of small lateral extension, for the ten first azimuthal parameter m.b,Displayed as\na color plot on an extended range of m, quasiperiodic patterns at fixed ∆ y/˜rcappear. The experimental setting (∆ y= 0.5 mm\nfor ˜rc= 2 mm) is indicated by the vertical dashed lines.\nwhich is represented in Figure S2. While the signal amplitude globally decreases with m, it exhibits oscillations which\ndepends on the lateral extension 2∆ y, the cylindrical detection distance ˜ rcandm, suggesting that ˜ rccan be adjusted\nto optimize the detection of a particular mode family.\nThe azimuthal dependence of a considered mode can be read in the phase of the measured magnetic flux, as given\nby Eq.(S13) as mϕc. These two lateral components Y±control the contribution of the different components of the\nmagnetic field to the flux envelope along z.\nC. z-direction\nEquations (S10), (S11) and (S12) cannot be processed analytically. They can be simplified when working at a\nresonance such that αm\nn/greatermuch(r/a)2n+1, then\n∆1(r)∼−(n+ 1)˜∆2(r) =−(n+ 1)a2n+1\nrn+2αm\nnAm\nn. (S15)\nThe dependence of the flux φm\nnwith 1/˜rn+2\ncalong the equatorial plane partially conditions the choice of the detection\ndistance. In Fig. S3 we represent, for the first spin-wave mode families with n<6 andm<6, the envelopes Zr,Zθ\nandZϕfollowed by the total magnetic flux φm\nnalong the altitude axis for different values of ∆ z, the other parameters\nbeing fixed to their experimental values (∆ y= 0.3 mm, ˜rc= 2 mm). Once determined the azimuthal parameter m,\nwe access the polar parameter n−|m|through the number of zeroes along the altitude z. The envelope along the\nz-axis is different for each mode, providing an additional identification method if the extent along zexperimentally\naccessible by the probe coil does not allow to go through all the envelope zeroes. The spin-wave mode polar structure\nwith higher n−|m|gets more and more complex, requiring the height dimension 2∆ zto be small enough for the flux\nnot to be a mixture of its different features (marked informally by purple lines on Fig. S3). This limit value accounts\nfor ∆z∼0.6 mm for (7 ,0) and (7,1) families. We design the coil such that ∆ z∼0.3 mm to comfortably work in the\nreliable region for at least n−m≤7. The complete magnetic fluxes computed in Fig. 1 cand the insets of Fig. 46\nFIG. S3. Decomposed magnetic flux envelopes along the z-axis as a function of the semi-extension of the coil height (in each\npanel, respectively Zr,ZθandZϕfrom top to bottom) followed by the total magnetic flux described by Eq.(S13) for modes\nfamily with n<6 andm<6 (indicated on top right), calculated with the parameters ∆ y= 0.5 mm and ˜rc= 2 mm. The\nexperimental value ∆ z= 0.3 mm is indicated by green vertical dashed lines. The purple lines mark the apparition of artifacts\ndue to an excessive height extension compared to the stray field polar features observed at this distance.7\nFIG. S4. Frame of the problem when the axis considered as the altitude degree of freedom of the probe coil is not parallel to\nthe saturation axis defining the z-direction but tilted by an angle ξzaround the y-axis.\nin the main text result from the numerical evaluation of Eq.(S13) with the coil centered along yand the following\nparameters: ∆ y= 0.5 mm, ∆z= 0.3 mm and ˜rc= 2 mm.\nD. Non-ideality\n1. Coil miscentered along the y-direction\nEvaluating the integral with ∆−\ny= ∆+\ny+/epsilon1such that the total lateral extension is 2∆ y≡2∆+\ny+/epsilon1,\nY±\nmc=Y±\nc(∆+\ny)±/epsilon1/parenleftbigg\ne−i(m−1)∆y\n˜rcsinc/bracketleftbigg(m−1)/epsilon1\n2˜rc/bracketrightbigg\n±e−i(m+1)∆y\n˜rcsinc/bracketleftbigg(m+ 1)/epsilon1\n2˜rc/bracketrightbigg/parenrightbigg\n. (S16)\nAt fixed altitude, this simply results in a constant extra phase, innocuous for the determination of m. The phase\nvariations between Y+\nmcandY−\nmcweighted by the different envelopes Zr+ZθandZϕcould induce a varying extra\nphase along the altitude axis. The effect could be neglected when Zr+Zθ/greatermuchZϕ.\n2. Coil altitude-axis tilted compared to HDC\nA tilt ofξzof the coil altitude-axis compared to the saturation axis (Figure S4) around yimplies the coordinates\n(˜rc,zc) injected in the calculations should be replaced by\n˜rc= ˜r0\nc+zapp\ncsinξz,\nzc=zapp\nccosξz\nwithzapp\ncthe apparent altitude along the coil zc-axis and ˜r0\ncthecylindrical distance atzc= 0. The solid lines in Fig. 4\ncomes from the numerical evaluation of Eq.(S13) with the coil centered along ywith ∆y= 0.5 mm, ∆z= 0.3 mm\nand ˜rc= 2 mm and an axis tilt of ξz=−6◦. The axis tilt is determined by counterbalancing the predicted lobes\namplitudes with respect to zc= 0 to match the detected modes.\nIII. EVOLUTION OF THE EIGENFREQUENCIES WITH THE STATIC MAGNETIC FIELD H DC\nThe evolution of the magnon eigenfrequencies with the static magnetic field HDCcan be predicted coarsely by\nsolving the resonance condition in the magnetostatic approximation1,2:\nn+ 1 +ξ0Pm/prime\nn(ξ0)\nPmn(ξ0)±mν= 0 (S17)8\nFIG. S5. a,Microwave reflection spectra from the pump coil as a function of the uniform mode precession frequency Ω 110/2π\n(vertical axis) — tuned by varying the current flowing in the solenoid from −0.8 A to 0.8 A. The uniform mode precession\nfrequency also serves as a reference for the frequency horizontal axis. For readers’ comfort, the colorscale has been adapted\nby frequency region (separated by black dashed lines and numbered by kfrom 0 to 10) such that as many spin-wave modes\nas possible clearly appear (min k=−0.48,−7.1,−2.5,−1.4,−2.1,−1.2,−2.3,−4.8,−0.91,−0.41 and −0.42 dB). b,Eigen-\nfrequencies computed in the magnetostatic approximation from Eq.(S17) for a magnetization saturation Ms= 187 mT/ µ0,\nγ/2π= 28 GHz/T up to n= 7 and m= 7. The orange stars mark the modes discussed in Fig. 4 in the main text. The white\ndashed line indicates the working point used in the main text with Ω 110/2π= 6.414 GHz.9\nwith\nξ02= 1 + 1/κ,\nν=Ω\nΩ2\nH−Ω2,\nκ=ΩH\nΩ2\nH−Ω2,\nthe normalized internal field Ω H=HDC−Ms/3\nMs, the normalized magnon frequency Ω =Ωk\nγµ0Msand the gyromagnetic ra-\ntioγ. We determine the effective saturation magnetization by comparing the frequency difference between (2,1,0) and\n(2,2,0), which are affected similarly by propagation effects4. In the magnetostatic approximation1, their frequencies\nf210andf220are such that\nµ0Ms=10π\nγ(f220−f210) (S18)\nwhich leads to an effective Ms= 187 mT/µ0in our experimental condition with Ω 110/2π= 6.414 GHz.\nThe solenoid wound around the magnetic circuit allows us to change the static magnetic field such that Ω 110/2π\ncould be tuned from 6.01 to 6 .81 GHz. The microwave reflection spectra are reported on Fig. S5 aabove the computed\neigenfrequencies expected in the magnetostatic approximation on Fig. S5 b.\nWe compare the linespacing 2 π×∆fnmr = Ωnmr−Ω110between the uniform precession mode (1,1,0) and\nothers. In particular at the working point Ω 110/2π= 6.414 GHz, the modes discussed in Fig. 4 of the main text\nreading experimentally:\n∆f220= +410 MHz\n∆f320=−184 MHz\n∆f310=−994 MHz\n∆f410=−1185 MHz\nare compared to the theoretical values predicted in the magnetostatic approximation, for Ms= 187 mT/µ0:\n∆f220= + 349 MHz\n∆f320=−249 MHz\n∆f310=−1071 MHz\n∆f410=−1279 MHz\nresulting in the following relative discrepancies 14 .9%,−35.5%,−7.7% and−7.9%.\nIV. ADDITIONAL DETAILS ON THE SETUP\nThe core of the experimental setup is depicted on Fig. S6. The motorized rotation stage and the magnetic circuit\nare screwed on the aluminum master support, clamped to the experiment table. The magnetic circuit is ended by\nconical concave elements in which can fit 12-mm diameter NdFeB permanent ring magnets. A solenoid is wound\naround the magnetic circuit to tune the static field ( ∼±14 mT). The motorized rotation stage is an Optosigma\nHST-120YAW (0.1◦position accuracy). The linear motors are composed by Optosigma TAMM40-10C (10 mm travel\nrange, 6 µm position accuracy) and HPS60-20X-M5 (20 mm travel range, 15 µm position accuracy). The probe coil\nplane tilt is corrected with a small manual rotation stage Optosigma KSP-256 .\nThe sample is attached to the end of a Al 2O3rod going through the upper permanent ring magnet, on a three-axis\nlinear actuator. The centering of the sample with respect to the rotation stage axis is performed with the help of a\nCMOS camera, placed on the rotation stage to move jointly with the probe coil. The CMOS camera is on a motorized\ntranslation stage — to change dynamically the focus and evaluate distances — fixed on the rotation stage. During\nthe centering process, a white PVC screen with a carved target is fixed on the opposite side of the rotation stage. The\nsphere is centered along the rotation stage axis when its position is fixed on the target for any rotation stage angle.\nDesigned to indicate the central position between the two magnets, it allows positioning the sample vertically in the\nstatic magnetic field. The screen is removed during the measurement to insure a maximum angular range.10\nFIG. S6. Setup of the core of the experiment.\nV. DATA PROCESSING\nDefining solid phase references is an absolute necessity for this imaging method. At each coil position, we record\nthe spectra ( S11andS21) in two configurations: with the sample in position centered in the static magnetic field and\nwith the sample retracted vertically (Fig. S7 a). The latter gives access to the complex transfer functions of the pump\ncoil and of the local pump-probe system which are used to normalize the magnonic spectra measured with the sample\nin position (Fig. S7 b).\nWe gather hundreds of spectra from which we extract the spin-wave modes features, constituted by 67,242\npoints each across a span of 2.89 GHz. These high-resolution spectral data require few hours of acquisition during\nwhich small temperature variations and the probe coil positions induce slight changes in the magnon frequencies and\ndamping rates. We track and classify all the observable peaks in the dataset, before fitting them properly. Spectrum\nby spectrum, the reflection measurement (as in Fig. 2 a) is used to get a precise guess of their eigenfrequencies by\nperforming a Lorentzian fitting on a short sliding window of size comparable to the narrowest observed peak over the\nwhole spectrum. The corresponding feature is considered as a plausible spin-wave mode if its fitted amplitude and\nlinewidth fall within reasonable limits. This process leads to the pre-detection of tens of peaks. Comparing these\npeaks from spectrum to spectrum, the slightly deviating frequencies can be attributed to a particular mode by a\nnearest-neighbor approach. The process ends with a collection of eigenfrequencies over the whole dataset attributed\nto modes to be analyzed.\nInformation on the mode structures is contained in the transmission measurements. The phase of each mode is\nswamped in the background constituted by all the other modes. Great care is taken to adjust properly the response\nof the modes on each spectrum, progressing sequentially from the most dominant and correcting iteratively the fits\nto properly take into account the contribution of all the modes at a given frequency.\nOnce the amplitude of the dominant modes has been established, a position-dependent phenomenological back-\ngroundS0[Ω] = (aΩ2+bΩ +c)ei(d+eΩ), likely due to remains of direct coupling and parasitic responses, could\nbe piecewise-defined typically on 100 MHz, such that the solid red lines in Fig. 2 band Fig. 2 care respectively11\nFIG. S7. a,Typical raw spectra in reflection S11and in transmission S21with the sphere centered in the static magnetic field\nHDC(grey lines) and vertically retracted (purple lines) as measured at a particular coil coordinate ( ϕc,zc).b,Subsequent\ncalibrated spectra obtained by normalizing the magnonic response with coils’ local transfer function.\n|Sφ[Ω] +S0[Ω]|2and arg(Sφ[Ω] +S0[Ω]). This residual background with features much larger ( >1 GHz) than the\nspin-wave modes ( <10 MHz) allows to study modes appearing with small signal-to-noise ratio.\nVI. GENERALIZATION TO OTHER SHAPES\nWe generalize the technique by formally developing the expression of the magnetic flux induced by spin-wave modes\nin other ideal geometries.\nA. Magnetic flux induced by a magnetostatic mode of a ferromagnetic cylinder\nThe outside magnetic potential induced by a magnetostatic mode ( ζ,m) of a long axially-magnetized cylinder of\nradiusRis given by Joseph and Schl¨ omann5:\nψm\nζ(˜r,ϕ,z ) =Am\nζKm(|β|˜r)e−iβzeimϕ(S19)\nwithβthe propagation constant, ζ=|β|RandKmthe modified Bessel function of the second kind3.12\nThe magnetic flux intercepted by a well-centered coil is\nφm\nξ=µ0/integraldisplay/integraldisplay\nS∇ψ(˜r,ϕ,z )·dSB=µ0/integraldisplay/integraldisplay\nSdS(I˜r+Iϕ) (S20)\nwith\nI˜r(˜r,ϕ,z) =eimϕc/bracketleftBig\nei(m+1)ϕ+ei(m−1)ϕ/bracketrightBig\ne−iβz∆˜r(˜r),\nIϕ(r,θ,ϕ ) =eimϕc/bracketleftBig\nei(m+1)ϕ−ei(m−1)ϕ/bracketrightBig\ne−iβz∆ϕ(˜r)\nand\n∆˜r(˜r) =−|β|\n4[Km−1(|β|˜r) +Km+1(|β|˜r)]\n∆ϕ(˜r) =−m\n2˜rKm(|β|˜r)\nConsidering Y±previously presented, it can all be compacted as\nφm\nξ=µ0eimϕce−iβzc/parenleftbig\n2∆zsinc(β∆z)/bracketleftbig\nY+∆˜r(˜rc) +Y−∆ϕ(˜rc)/bracketrightbig/parenrightbig\n. (S21)\nThe presented imaging technique can be directly applied to cylinders by reading the azimuthal and the altitude\ndependences in the phase evolution along the cylindrical orbital of radius ˜ rc.\nB. Magnetic flux induced by a magnetostatic mode of a spheroid\nFollowing the description given by Walker6in oblate spheroidal coordinates ( ξ,η,ϕ ), the magnetic potential\noutside an oblate spheroid of transverse semi-axis aand longitudinal semi-axis breads:\nψm\nn(ξ,η,ϕ ) =Qm\nn(iξ)Pm\nn(η)eimϕ(S22)\nwithQm\nnthe Ferrers function of the second kind3and defining ι2≡a2−b2,\nx=ι(1 +ξ2)1/2(1−η2)1/2cosϕ, (S23)\ny=ι(1 +ξ2)1/2(1−η2)1/2sinϕ, (S24)\nz=ιξη. (S25)\nThese oblate coordinates can be expressed in cylindrical coordinates:\nξ2(˜r,z) =1\n2ι2[Λ(˜r,z) + Υ(˜r,z)], (S26)\nη(˜r,z) =√\n2z[Λ(˜r,z) + Υ(˜r,z)]−1\n2. (S27)\nwith\nΛ(˜r,z) = ˜r2+z2−ι2, (S28)\nΥ(˜r,z) =/bracketleftbig\n4ι2z2+ Λ2/bracketrightbig1\n2. (S29)\nWith a well-centered coil,\nφm\nn=µ0/integraldisplay/integraldisplay\nS∇ψ(˜r,ϕ,z )·dSB=µ0/integraldisplay/integraldisplay\nSdS(I˜r+Iϕ) (S30)\nwith\nI˜r=eimϕ c/bracketleftBig\nei(m+1)ϕ+ei(m−1)ϕ/bracketrightBig1\n2∂˜r[Qm\nn(iξ)Pm\nn(η)], (S31)\nIϕ=eimϕ c/bracketleftBig\nei(m+1)ϕ−ei(m−1)ϕ/bracketrightBig−m\n2˜rQm\nn(iξ)Pm\nn(η) (S32)13\nFIG. S8. Norm of the induced magnetic flux by a magnetized spheroid described by Eq. (S33) computed for modes family\nwithn<6 andm<6, in the experimental conditions (∆ y= 0.5 mm, ∆z= 0.3 mm, ˜rc= 2 mm,a= 1 mm) for values of the\nlongitudinal semi-axis bvarying from 0.1 mm (dark green) to 0.9 mm (yellow). For each mode ( n,m), the data are normalized\nby the maximum in b= 0.1 mm. Approaching the sphere limit when b∼a, the pattern is the same as observed in Sec. II.\nso that\nφm\nn=µ0eimϕ c/bracketleftbig\nY+Z˜r(zc)−Y−Zϕ(zc)/bracketrightbig\n. (S33)\nThe dependence along the azimuth is the same as seen previously. Along the altitude,\nZ˜r(zc)≡1\n2/integraldisplayzc+∆+\nz\nzc−∆−\nzdz∂ ˜r[Qm\nn(iξ[˜rc,z])Pm\nn(η[˜rc,z])], (S34)\nZϕ(zc)≡m\n2˜rc/integraldisplayzc+∆+\nz\nzc−∆−\nzdz Qm\nn(iξ[˜rc,z])Pm\nn(η[˜rc,z]) (S35)\nwith the derivatives along ˜ rreading\n∂˜rξ(˜r,z) =1√\n2ι˜r[Λ(˜r,z) + Υ(˜r,z)]1\n2\nΥ(˜r,z), (S36)\n∂˜rη(˜r,z) =−√\n2z˜r[Λ(˜r,z) + Υ(˜r,z)]−1\n2\nΥ(˜r,z)(S37)\nand\n∂˜r[Qm\nn(iξ)Pm\nn(η)] =∂˜rη\nη2−1Qm\nn(iξ)/bracketleftbig\n−(n+ 1)Pm\nn(η)η+ (n−m+ 1)Pm\nn+1(η)/bracketrightbig\n(S38)\n−i∂˜rξ\nξ2+ 1Pm\nn(η)/bracketleftbig\n−(n+ 1)Qm\nn(iξ)iξ+ (n−m+ 1)Qm\nn+1(iξ)/bracketrightbig\n. (S39)\nThese expressions can be numerically computed to provide analog patterns to Fig. S3 and be similarly used for\nidentification. For illustration, in Fig. S8 we plot the norm of the flux that would be intercepted for the first modes\nin our experimental conditions for various values of ellipticity. These developments suggest the imaging method can\nbe directly applied to any spheroid."    },    {        "title": "1810.12151v1.A_Spin_Glass_State_in_Ba3TiRu2O9.pdf",        "content": "1 \n A Spin Glass State in Ba 3TiRu 2O9 \n \nLoi T. Nguyen  and R.J. Cava  \nDepartment of Chemistry, Princeton University, Princeton, New Jersey 08544,  USA  \n \nAbstract  \nThe magnetic properties of Ba3TiRu 2O9 , whose crystal structure  is based on stacked  triangular \nplanar lattice s of MO 6 dimers and single MO 6 octahedra , are reported. The system is magnetically \ndisturbed by a substantial amount of Ti/Ru chemical disorder . The Weiss temperature and  effective \nmagnetic moment were found to be -29.5 K and 1.82 μB/f.u. respectively, and a bifurcation in the \nzero field cooled and field cooled magnetic susceptibility is observed below 4.7 K, suggesting that \nthis is a compositionally -disordered spin -glass system. The material is a semiconductor with an \nactivation energy for charge transport of approximately 0.14 eV.  \n \nKeywords : magnetic frustration, spin glass , ruthenium dimers , triangular lattice.  \n  2 \n 1. Introduction  \nSpin glass  systems , with a disordered low temperature  magnetic state often caused by atomic \ndisorder(1),(2), have been of interest for many years(3). Local small spin clusters are gradually formed \nwhen a spin -glass material is cool ed, and, at the freezing temperature, the system is stuck in one \nmetastable configuration of  many degenerate ground states(4).  Ruthenium oxides  can often display  \nunusual  magnetic and electrical properties  due to the multiple degrees of freedom of the Ru 4d \nelectrons.  Sr2RuO 4, for example, becomes a superconductor below 1 K while SrRuO 3 is \nferromagnetic at 160  K(5),(6), with strong spin orbit coupling is observed in  Ba3CoRu 2O9(7) and \nmagnetoelastic coupling in Ba 3BiRu 2O9(8). Systems that include Ru -Ru dimers can also  exhibit \ninteresting  properties  - Ba3CoRu 2O9 for example is antiferromagnetic with Weiss temperature of \n93 K , and has robust coupling among orbital, spin and charge degrees of freedom(7) while \nBa3MRu 2O9 (M=La3+, Nd3+ and Y3+)(9) materials display  ferromagnetic interactions within the Ru -\nRu dimers and antiferromagnetic interactions  between dimers. Here we report  that the related \nmaterial  Ba3TiRu 2O9, isostructural with Ba3BiRu 2O9(8), based on Ru 2O9 dimers in triangular \nplanes , shows  magnetic behavior  that is  dominated by  strong chemical disorder . The bifurcation  \nbetween  its ZFC and FC susceptibility  at 4.7 K indicates the presence of a low temperature  spin-\nglass state with a freezing temperature T f  = 4.7 K.  \n2. Experimental  \nA polycrystalline sample of Ba3TiRu 2O9 was synthesized by solid -state reaction  using BaCO 3, \nRuO 2, and TiO 2 (Alfa Aesar, 99.9 %). The mixture of reagents in stoichiometric  ratios was heated \nin air at 1100  ℃ for 12 hours, reheated at 1100  ℃ and then at 1200  ℃ for 12 hours . The phase \npurit y and crystal structure of our sample of Ba3TiRu 2O9 was determined using  powder X -ray \ndiffraction (PXRD) using a Bruker D8 Advance Eco with Cu Kα radiation and a LynxEye -XE 3 \n detector.  The structural refinement s were  performed  with GSAS .(10) Crystal structure drawings \nwere created by  using the program VESTA .(11)  \nThe magnetic susceptibilit y of Ba3TiRu 2O9 was measured by a  Quantum Design Physical Property \nMeasurement System (PPMS) Dyna Cool equipped with a VSM option.  For those measurements, \nthe sample w as ground into powder and placed  in plastic capsules . The magnetic susceptibility , \ndefined as M/H,  where M is the sample magnetization and H is the applied field, was measured at \nthe field of H = 1 kOe  from 1.7 K to 300 K. Zero-field cooled (ZFC) and field-cooled (FC) \nmagnetization data were measured from 1.7 K to 20 K at an applied  field of 30 Oe  for Ba 3TiRu 2O9. \nThe resistivit y of Ba3TiRu 2O9 was measured by the DC four -contact method in the temperature \nrange 250 K to 330 K with the PPMS. The sample w as pressed, sintered and cut into pieces with \nthe approximate  sizes  1.0 × 2.5 × 1. 5 mm3. Four Pt contact wires were connected  to the samples \nusing silver paint . \n3. Results  \nThe powder X -ray diffraction pattern  and structural refinement  results for  Ba3TiRu 2O9 are shown \nin Figure 1 a. The crystal structure of Ba 3TiRu 2O9 is shown in Figure 1b. The lattice parameters  \nand atomic positions reported in Table 1  are similar  to those reported previously(12). Ba3TiRu 2O9 \nadopts a hexagonal structure with the space group P63/mmc  (No.194). Ti occupies 5 1.2% of the \nisolated octahedra, while in the dimers  there is  an occupancy  of 17.7% Ti. This indicates  that \nalthough Ti  prefers  the isolated  octahedra, while Ru prefers  the face -shared octahedra , there is a \nsubstantial amount of chemical disorder  present . We attribute this disorder  to the similarities in \nthe charge  and radii of the Ti and Ru ions .(13)  \nThe temperature -dependent magnetic susceptibility of Ba 3TiRu2O9 and its reciprocal  are plotted \nin Figure  2. The Curie -Weiss fitting for data from 75 -300 K  yields  a Weiss temperature of -29.5 4 \n K, effective magnetic moment of 1.82 μB/f.u., and a temperature -independent paramagnetic \nsusceptibility  of 9.04×10-4 emu Oe-1 f.u.-1. The observed effective moment is 1.82 μB/f.u, which is \nsmaller than what is typically expected for S = 1 Ru4+ ions (15),(16),(17). This may be due to the effect \nof spin orbit coupling, which can partially split the three degenerate t 2g states into two states with \nthe same energy and a third one at higher energy, as is commonly seen in ruthenates and \niridates(18),(19),(20). Another possibility is that  only Ru ions  either  in the isolated octahedra or in the \ndimers are magnetic  due to different coordination environment s.  \nThe inset shows the bifurcation of FC and ZFC  magnetic susceptibility at the temperature 4.7  K in \nan applied DC field of  30 Oe.  Thee data  imply that the material  is a spin glass with the freezing \ntemperature 4.7 K. Similar spin glass  behavior  has been  seen in Ba3FeRu 2O9(14), with the disorder \nbetween Fe and Ru lead ing to a spin glass state with a freezin g temperature of 25 K. ZFC/FC DC \nmagnetic susceptibilities  were measured at different fields to investigate the spin glass behavior in \nBa3TiRu 2O9. As shown in Figure  3a, freezing temperature is determined to be  4.7 K at the applied \nfield of 60 Oe. When the field is increased to 2.5 kOe, T f is shifted down to 3.8 K. At the field of \n1 T, it is moved to 2.5 K. Above 2 T, the freezing temperature is further suppressed below 2 K and \nis not detected  in our measurements . This is similar to the cluster spin glass behavior  seen in \nCr0.5Fe0.5Ga(21). The resistivity  of Ba3TiRu 2O9 as a function of reciprocal temperature is plotted in \nFigure  3b. Ba3TiRu2O9 is nonmetallic,  similar to what is observed for  Ba3BiRu 2O9.(8) The \nactivation energy is calculated to be Ea = 0.14 eV in the temperature range from 2 50-300 K.  \n4. Conclusions  \nBa3TiRu 2O9 crystallizes in a 6-layer hexagonal unit cell in the P63/mmc  space group. The material \nhas high resistivi ty, indicating that it is semiconducting ; the measured activation energy for charge \ntransport is approximately 0.14  eV. The bifurcation  in the zero fi eld cooled (ZFC) and field cooled 5 \n (FC) magnetic susceptibilit ies indicates  the presence of  a spin glass transition at about 4.7 K.  This \nmay come from the fact that  material displays substantial structural disorder , although there may \nalso be some contribution from the magnetically frustrating triangular arrangement of the Ru -Ru \ndimers  and the single RuO 6 octahedra . The lower than expected spin on the Ru4+ present suggests \nthat the oretical modelling  should  best be performed  to understand the magnetism of this material.    \n5. Acknowledgements  \nThis work was supported by the Basic Energy Sciences of the Department of Energy grant number \nDE-FG02 -08ER46544 through the Institute for Quantum Matter at Johns Hopkins University .  6 \n 6. References  \n \n(1). J. A. Mydosh, (2014).  \n(2) K. Binder and A. P. Young, Reviews of Modern Physics 58, 801 (1986).  \n(3) J. M. D. Coey, (2009).  \n(4) S. Blundell and D. Thouless, American Journal of Physics 71, 94 (2003).  \n(5) Y. Maeno, H. Hashimo to, K. Yoshida, S. Nishizaki, T. Fujita, J. G. Bednorz, and F. \nLichtenberg, Nature 372, 532 (1994).  \n(6) A. Callaghan, C. W. Moeller, and R. Ward, Inorganic Chemistry 5, 1572 (1966).  \n(7) H. D. Zhou, A. Kiswandhi, Y. Barlas, J. S. Brooks, T. Siegrist, G. Li,  L. Balicas, J. G. Cheng, \nand F. Rivadulla, Physical Review B 85, (2012).  \n(8) W. Miiller, M. Avdeev, Q. Zhou, A. J. Studer, B. J. Kennedy, G. J. Kearley, and C. D. Ling, \nPhysical Review B 84, (2011).  \n(9) M. S. Senn, S. A. J. Kimber, A. M. A. Lopez, A. H. H ill, and J. P. Attfield, Physical Review \nB 87, (2013).  \n(10) B. H. Toby, Journal of Applied Crystallography 34, 210 (2001).  \n(11) K. Momma and F. Izumi, Journal of Applied Crystallography 44, 1272 (2011).  \n(12) D. Verdoes, H. W. Zandbergen, and D. J. W. Ijdo,  Acta Crystallographica Section C Crystal \nStructure Communications 41, 170 (1985).  \n(13) R. D. Shannon, Acta Crystallographica Section A 32, 751 (1976).  \n(14) S. Middey, S. Ray, K. Mukherjee, P. L. Paulose, E. V. Sampathkumaran, C. Meneghini, S. \nD. Kaushik, V. Siruguri, K. Kovnir, and D. D. Sarma, Physical Review B 83, (2011).  \n(15) S. V. Streltsov and D. I. Khomskii, Physical Review B 86, (2012).  \n(16) Y. Klein, G. Rousse, F. Damay, F. Porcher, G. André, and I. Terasaki, Physical Review B \n84, (2011).  \n(17) C. Dussarrat, F. Grasset, R. Bontchev, and J. Darriet, ChemInform 27, (2010).  \n(18) P. Kayser, B. J. Kennedy, B. Ranjbar, J. A. Kimpton, and M. Avdeev, Inorganic Chemistry \n56, 2204 (2017).  \n(19) M. W. Lufaso and H. -C. Z. Loye, Inorganic Chemistry 44, 9143 (2005 ). 7 \n (20) M. W. Lufaso and H. -C. Z. Loye, ChemInform 37, (2006).  \n(21) Bag, Pallab, P. R. Baral, and R. Nath. \"Cluster spin -glass behavior and memory effect in \nCr0.5Fe0.5Ga.\" arXiv preprint arXiv:1810.03890 (2018).  8 \n Table 1. Atomic coordinates and equivalent isotropic displacement parameters of \nBa3TiRu 2O9, space group P63/mmc  ( No. 194) . \nAtom  Wyckoff.  Occ.  x y z Uiso \nBa1 2b 1 0 0 ¼ 0.0015(5) \nBa2 4f 1 ⅓ ⅔ 0.091 53(8) 0.0023(3) \nTi1 2a 0.512(8)  0 0 0 0.0076( 5) \nRu2 2a 0.488(8)  0 0 0 0.0076( 5) \nTi2 4f 0.177(6)  ⅓ ⅔ 0.8419(1)  0.0014( 5) \nRu1 4f 0.823(6)  ⅓ ⅔ 0.8419(1)  0.0014( 5) \nO1 6h 1 0.5105(8) 0.021(2) ¼ 0.0059(3) \nO2 12k 1 0.8369(7) 0.674(1)  0.080 1(4) 0.0060(2) \na = 5.71392(4) Å, c = 14 .0351(1) Å  \nRwp = 12. 03%, R p = 8.76%, RF2 = 5.05% \n  9 \n  \nFigure  1: (a) Rietveld powder x -ray diffraction refinement of  Ba3TiRu 2O9 in space group  \nP63/mmc . The observed X -ray pattern  is shown in black, calculated in red, difference (Iobs-\nIcalc) in blue, the background in green , and tick marks denote allowed peak positions in pink . \nRp = 0.08 76, R wp = 0.12 03, χ2 = 1.480. (b) C rystal structure of Ba 3TiRu 2O9 in space group  \nP63/mmc . The isolated  (Ru/Ti)O 6 octahedra (blue) are corner -shared to the (Ru/Ti) 2O9 \ndimers (blue with purple  rectangles) to form what is frequently referred to as a “6 -layer” \nhexagonal perovskite structure. Barium is labeled in green and oxygen is in red.   \n10 \n  \nFigure  2: The temperature dependence of the magnetic susceptibility (black squares) and the \ninverse of the difference between the magnetic susceptibility and the temperature \nindependent magnetic susceptibility ( red circles) for Ba 3TiRu 2O9. The applied field is 1 kOe. \nThe red solid line is the susceptibility fit calculated from Curie -Weiss law from 75 -300 K.  \nThe inset shows  the bifurcation of the field cooled (FC) and zero field cooled (ZFC) DC \nsusceptibility in an applied field of 30 Oe. The spin glass transition  temperature is \ndetermined to be 4.7 K.  \n  \n11 \n  \nFigure  3: (a) The temperature dependence of the ZFC and FC  magnetic susceptibilit ies \nmeasured under different applied fields for Ba3TiRu 2O9. The freezing temperature s are \ndecreasing as the field s increase.  (b) Temperature -dependent r esistivity of a sintered pellet \nof Ba3TiRu 2O9 as a function of inverse temperature , in log -form . Data in the elevated \ntemperature regime (2 50-300 K) was fit to the model 𝝆=𝝆𝒐𝒆𝑬𝒂\n𝒌𝒃𝑻 (red line) with E a = 0.14  eV. \nInset – The raw resistivity data from 200 -350 K.  \n"    },    {        "title": "1811.03960v1.Giant_Exchange_Bias_in_the_Single_layered_Ruddlesden_Popper_Perovskite_SrLaCo0_5Mn0_5O4.pdf",        "content": "  \n \n1 \n Article type: Full Paper  \nTitle : Giant Exchange Bias in the Single -layered  Ruddlesden -Popper Perovskite \nSrLaCo 0.5Mn 0.5O4 \nRanjana R. Das1, Priyadarshini Parida2, A. K. Bera3, Tapan Chatterji4, B. R. K. Nanda2 and \nP. N. Santhosh1* \n1Low Temperature Physics Laboratory, Department of Physics, Indian Institute of Technology \nMadras, Chennai 600036 , India.  \n2Condensed Matter Theory and Computational Lab, Department of Physics, Indian Institute of \nTechnology Madras, Chennai 600036, India.  \n3Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India.  \n4Institut Laue -Langevin, 71 Avenue des Martyrs, 38000 Grenoble, France.  \nE-mail: *santhosh@iitm.ac.in   \nKeywords:  exchange bias, cluster spin glass, magnetic memory effect, density functional \ntheory  \n \nExchange bias (EB) as large as ~5.5 kOe is observed  in Sr LaCo0.5Mn 0.5O4 which is the highest \never found in any layered transition m etal oxides including Ruddlesden -Popper series. Neutron \ndiffraction measurement rules out long-range  magnetic ordering and together with dc magnetic \nmeasurements suggest formation  of short -range  magnetic domains. AC magnetic susceptibility , \nmagnetic memory effect and magnetic training effect confirm the system to be a cluster spin \nglass. By carrying out density functional calculations on several model configurations,  we \npropose that EB is originated  at the boundary between Mn-rich antiferro magnetic and Co-rich \nferromagnetic domains at the sub -nanoscale. Reversal of magnetization axis on the Co -side \nalters the magnetic coupling between the interfacial Mn and Co spins which leads to EB. Our \nanalysis infers that the presence  of competing  magnet ic interactions is sufficient to induce \nexchange bias and thereby a wide range of materials exhibiting giant EB can be engineered  for \ndesigning novel magnetic memory devices.  \n \n   \n \n2 \n  \n1. Introduction  \nComplex 3D perovskite oxides offer a rich materials platform for investigating emergent \nphenomena like ferromagnetic  insulator [1], multi -ferroic [2-4], colossal magnetoresistance [5,6]  \nthat arise due to a complex web of interactions involving spin, charge, orbital and lattice degrees \nof freedom in the three -dimensional  space  [7]. In order to  bring exotic quantum phenomena, a \nnatural  extension of this 3D cubic perovskite is to reduce the interaction dimensionality by \neither making artificial hetero superlattices or by tuning the stoichiometry  so that the active \nlayers involved in the coupling process are well separated  [8-10]. While the former i s sensitive \nto the growth condition,  the latter is thermodynamically stable and can be synthesized  under \nambient conditions.  \n \nFig. 1. (a) Ideal perovskite structure in which the alternate stacking  of A/A'O and B/B'O 2 \nlayers of perovskite block is presented  (square indicates one unit cell of perovskite). (b) \nRuddlesden -Popper (RP) single layered in which adjacent perovskite block is replaced  \nwith a new order of the form A/A'O | A/A'O | B/B'O 2. The rectangle represents one unit \ncell of RP tetragonal lattice.  \n  \n \n3 \n The layered Ruddlesden -Popper (RP) series with general formula (A) n+1 (B) nO3n+1[11] are \namong the most preferred materials to study the physics of reduced dimensionality as  they \nexhibit interesting  properties such as superconductivity  [12] [13], charge ordering (CO)  or \norbital ordering (OO)  [14], ferroelectricity  [15], and colossal magnetoresistance  [16]. The \ncrystal structure of n = 1 RP compounds, A'A(B/B')O 4, can be described by the periodic \nstacking of the layers with the order A/A'O | A/A'O  | B/B'O 2 as shown in Fig . 1. The structure \nmay also be described as magnetically active (A'/A)(B/B')O 3  perovskite blocks well separated \nby (A'/A)O  layers along [001]. Under suitable conditions, the synthesis of these single -layer \nRP compounds will produ ce B and B ' rich domains at the sub -nanoscale. The boundary \nseparating these domains can further constrain certain  long-range magnetic order, and in turn \nnovel  quantum states can be formed which may not be observed  in pristine perovskites ABO 3.  \nWhile ther e is a large number of literature available on single layered RP compounds with one \ntransition metal element, very few have reported the results with two transition metal elements \nin a single -layered RP system  [17-19]. Therefore, there is lack of evidence on emerging \nquantum states at the boundary of two different magnetic domains in these layered RP \ncompounds.  \n In the present work,  we report a novel mono -layered RP phase based layered perovskite oxide \nSrLaCo 0.5Mn 0.5O4 (SLCMO) synthesized by sol-gel method. The dc and ac magnetic \nmeasurements and along with the  powder neutron diffraction measurements point  to cluster \nspin glass (CG) like  magnetic behavior  as well as a giant  exchange bias (EB) effect. In fact the \nEB in this layered system shows fivefold increase compared to the EB reported in other layered \noxide compounds such as single layer RP Sr 0.5Pr1.5CoO 4 (1 kOe)  [20], double layer RP \nSr3FeMoO 7 (0.2 kOe)  [21], layered oxychalcogenides  La2O3Mn 2Se2 (0.5 kOe)  [22] and double \nperovskites SrLaCoMnO 6 (0.3 kOe)  [23]. The density fun ctional calculations further confirm \nthe existence of giant EB in this compound,  and it attributed to the anisotropic magnetic \ncoupling among the Mn and Co spins.     \n \n4 \n 2. Experimental Section  \nSynthesis details :  \nDespite remarkable  electronic and magnetic properties,  the synthesis of phase pure \nstochiometric single layer RP system is quite challenging. The probability of getting a \nperovskite phase was often detected  in the polycrystalline sample , which was synthesized by \nsolid -state synthesis methods  [24]. Thus a citrate gel technique is used to prepare a \npolycrystalline sample of SrLaCo 0.5Mn 0.5O4 (SLCMO) . La2O3, Mn(CH 3COO) 2.4H 2O, \nSr(NO 3)2 and Co(NO 3)2.6H2O in stoichiometric amounts were first dissolved  in dilute nitric \nacid (HNO 3), and then an excess of citric acid and ethylene glycol (CH 2OH) 2 was added.  La2O3 \nwas preheated  at 1273 K before adding in dilute nitric acid. The dissolved solution was heat ed \non a hot plate resulting in the form ation of a gel.  The gel was dried  at 523 K and then heated to \n973 K for 12 h to remove the organic components and to decompose the nitrates. SLCMO \nceramic was subsequently sintered  at 1573 K for 36 h in the air  with i ntermittent grindings for \nhomogeneity.  \nCharacterization details :  \n Laboratory X -ray powder diffraction (XRPD) measurements were conducted at room \ntemperature on a P ANalytical Xʹpert using a Cu Kα radiation source (λ = 1.54056 Å).  The \nneutron powder diffraction  (NPD)  patterns were recorded  in zero field by using a high -\nresolution powder diffractometer SPODI at the research reactor FRM -II (Garching, Germany) \nwith monochromatic neutrons of 1.5481(1 ) Å over the 2θ range of 6 – 150˚ with a step size of \n0.05˚. X-ray photoelectron spectra (XPS)  of SLCMO were  recorded by an instrument with Mg -\nKα as the X -ray source , and a PHOIBOS 100MCD analyzer (SPECS) operated under ultra -high \nvacuum (10−9mbar).  The XPS spectra  were fitted  by the Casa XPS software (Casa Software Ltd) \nusing Gaussian -Loren tzian peak functions and Shirley background subtraction.   Dc and ac \nmagnetic susceptibility measurements were carried out in a commercial superconducting \nquantum interference device (VSM -SQUID) magnetometer  70 kOe (MPMS). Dc   \n \n5 \n magnetization curves in zero -field cooled (ZFC) and field cooled (FC) cycles were performed \nat several magnetic fields. Both the ZFC and FC magnetization versus temperature (M -T) \ncurves were measured  during the warming process. Isothermal magnetization, M (H) , hyster esis \nloops  were measured  under ZFC and FC conditions in the range −70 kOe ≤ µ 0H ≤ 70 kOe  at \ndifferent temperatures . For the ZFC case, the samples were first cooled from 300 K down to the \ntemperature of measurement under zero magn etic field, and five -quadrant M(H) measurements \nwere performed  starting at H = 0. For FC analysis , the samples were cooled each time under an \napplied magnetic field  from 300 K down to measurement temperature.  \nComputational Details:  \nPseudopotential based density functional calculations are carried out using the plane waves as \nbasis sets as implemented in Quantum ESPRESSO [25]. We have used the generalized gradient \napproximation (GGA) exchange correlat ion functional, given by Perdew, Burke and Ernzerhof, \nincorporated with the Hubbard U correction to consider the strong correlation effect. We have \ntaken U = 5 eV in our calculations. In all our calculations ultrasoft type pseudopotentials are \nconsidered. The kinetic energy cut -off for the plane waves and charge densities are taken  as 30 \nRy and 250 Ry, respectively. The k -mesh of 3×9×4 with 30 irreducible k -points are used  in the \ncalculations.  \n3. Experimental results  \n3.1. Crystal structure  \n3.1.1. X -ray and Neutron diffraction  \nThe crystal structure of SLCMO is examined  through X -ray powder diffraction (XRPD) and \nneutron powder diffraction (NPD) measurements. NPD  measurements are performed  at various \ntemperatures ranging from 300 K to 4 K. The Rietveld refinement of NPD patterns collected at \n300 K, and 4 K are shown  in Fig . 2 and Rietveld  refinement  at 50 and 100 K can be seen  in the \nsupplementary file . The data indicate that the compound crystallizes in a single -layered  RP \nphase with the bo dy-centered tetragonal lattice having space group I4/mmm  No. (139) and the   \n \n6 \n schematic of this structure is presented  in Fig. 1(b) (where A/Aʹ= Sr/La as silver balls, B/Bʹ = \nCo/Mn as blue balls and O as red balls). The room temperature lattice parameters ʹ aʹ and ʹcʹ are \nfound to be 3.8420  (1) Å and 12.6211  (1) Å respectively from the NPD. The lattice parameters \na and c change by 0.23% and 0.17% respectively over the temperature range 4 - 300 K. Details \nof crystal structure parameters obtain  from XR PD and NPD are provided  in the supplementary  \nmaterial. From the nuclear structure refinement, we conclude that SLCMO is stoichiometric.  \nWe observed a large  difference between the equatorial  and apical bonds of Mn/Co -O6 octahedra \nwith bond  length 1.92 Å and 2.09 Å respectively, and the difference in bond length is close to \n0.17 Å . We may note that Jahn -Teller active LaSrMnO 4 has a similar difference in the bond \nlength  [26]. The obtained bond lengths (Table S3 supplementary) from the NPD refinement \nsignatures the compression of equatorial and elongatio n of apical bonds indicating the presence \nof  Jahn Teller active Mn3+ ion (t 2g3 eg1).  \nIn order to  get an insight into the oxidation states of the ions,  we have performed bond -valence -\nsum (BVS) calculations from the refined atomic  positions. The derived B VS values reveal a \nmixed valence state  of both Co2+/3+ and Mn3+/4+ ions.  \n \nFig. 2 Neutron powder diffraction pattern of SrLaCo 0.5Mn 0.5O4 measured at (a) 300 K  and \n(b) 4 K. We excluded the 2θ region 38 – 40˚ of the neutron diffraction  patterns during \nRietveld refinements, as it consists of two weak peaks from the sample  holder.  \n  \n \n7 \n Temperature evolution of NPD shows that there is no structural change in the measured \ntemperature range. Furthermore, no additional magnetic peak or increase in  intensity is \nobserved  which confirms the lack of long-range  magnetic ordering in the extended  temperature \nrange [4 – 300 K].  \n3.1.2 X -ray photoelectron studies (XPS)  \nXPS measurements provide information on the surface composition and therefore we \nperformed Mn -2p and Co -2p core level XPS  to assign the atomic oxidation states of Mn and \nCo in SLCMO compound. The spin -orbit splitting of Mn 2p peaks corresponds to Mn 2p 3/2 and \nMn 2p 1/2 which are located at 641.97eV and 653.38eV, respect ively (Inset: Fig. 3 (a)) where as \nthat of Co 2p peaks (Inset: Fig. 3 (b)) are found at 779.93eV (Co 2p 3/2) and  795.33eV (Co \n2p1/2). The oxidation state of Mn/Co ions was determined  by the curve fitting the corresponding \n2p spectral peaks. The experiment al peak shape for Mn 2p 3/2 and Co 2p 3/2 was modeled  by \nemploying double peaks (Gaussian -Lorentzian)  patterns and shown in Fig. 3(a) and (b) \nrespectively. Both Mn 2p 3/2 and Co 2p 3/2 spectra show  perfect fitting for mixed valence states. \nThe two binding ener gy values obtained for Mn 2p 3/2 at 641.17 eV and 643.87 eV matched well \nwith the reported values for Mn3+ and Mn4+, respectively  [27,28] . Similarly, Co 2p 3/2 spectra \nmatches with the reported binding energy values of 3+ and 2+ state at 779.8 eV and 780.2 eV \nrespectively [27,28] . Peak fitting corresponding to Mn 2p 1/2 and Co 2p 1/2 spectra are given  in \nthe supplementary  files which again confirms mixed  valance state. From the XPS fitting, we \nhave estimated the percentage of Mn3+/ Mn4+ to be 67/33 % and that of Co3+ /Co2+ to be \n65/35 %. Hence, the predominant  oxidation states in SLCMO is confirmed to be Mn3+ and Co3+ \nwhich also corroborate s the charge states obtained from the neutron diffraction data. These \nobservations lend strong support to the counter part  of DFT calculations.  \n3.2. Magnetic properties:  \n3.2.1. dc Susceptibility     \n \n8 \n Fig. 3(c) shows the temperature dependence of the dc magnetic  susceptibilities χ(T) of SLCMO \nin the zero field cooled (ZFC)  and field cooled  (FC) conditions under applied fields of 100, 200 \nand 1000 Oe. ZFC curve shows two broad peaks (humps) at ∼ 150 K and ∼ 50 K, whereas, the \nFC curve shows a sharp rise at ∼ 140 K fo llowed by an anomaly below ∼ 50 K (slope change)  \nat low temperatures. The temperature dependent derivative curves (dM ZFC/dT) (inset of  Fig . \n3c) reveal these two transitions at T C1∼ 150 K and T C2∼ 50 K. The susceptibility data under an \napplied field of H =100 Oe (ZFC) in the temperature range 180 -300 K were fitted with the \nCurie -Weiss (CW) law, i.e., χ-1 (T)= C/(T -θp), (Fig . 3(d)) where C and θ P are Curie constant and \nCW temperature respectively.  The fit provided positive values of CW constant [ θP = 99 K] \nsuggesting the presence  of dominant ferromagnetic interactions. The effective paramagnetic \nmoment calculated from the Curie constant of C = 6.42 emu.K.mol−1Oe−1 is µ eff= 4.86 µ B / f.u. \n( theoretically estimated value  4.54 µ B (µthe = √(4S(S + 1))µ B) on considering high spin (HS) \nstate of Mn3+/Mn4+  [67/33 %]  and Co3+/Co2+  [65/35 %]  from the XPS are taken into account).  \nIn general, a strong ferromagnetic compound with long range  magnetic interaction exhibits CW \ntemperature  θP to be equal or greater tha n Curie temperature T C. However, in disordered \nsystems with randomly distributed magnetic ions provide competing  magnetic interactions and \nspin frustrations resulting in θP < T C [29]. The bifurcation between ZFC and FC arms and the \nlower temp erature  transition at 50 K indicate that there might exist spin  glass or glassy like \nmagnetic interaction in addition to the FM interactions, which will be clarified  in later sections.      \n \n9 \n  \nFig. 3 X-ray Photoemission spectroscopy of (a) Mn 2p 3/2 (inset shows the Mn 2p core level  \nspectr um) and (b) Co 2p 3/2 along with fitted curves  (inset shows the Co 2p core level  \nspectr um). (c) Magnetic susceptibility  measured under 100, 200 and 1000 Oe with ZFC \nand FC protocols. The dotted vertical lines  indicate two transitions at T C1 = 150 K and \nTC2 = 50 K respectively. ( Inset  shows the derivative of magnetiz ation vs. temperature \ncurve  (ZFC) measured  under 100 Oe).  (d) The inverse magnetic susceptibility ( -1) vs. \ntemperature curve, for ZFC at 100 Oe.  \n3.2.2. Memory effect  \nThe dc t hermomagnetic analysis of SLCMO suggests glassy  magnetic transition at lower \ntemperatures. Magnetic memory effect is an experimental fingerprint of glassy magnetic \nmaterial.  We have performed the same to figure out the glassy magnetic features in SLCMO. \nWe have carried  out a detailed magnetic memory analysis using both ZFC and FC experimental \n  \n \n10 \n protocols [30]. The ZFC memory experiments were carried out by conventional procedure with \ncooling the sample without field at a co nstant rate of 2 K/min  with an intermediate halt at 40 K \nfor a duration of 1h. The magnetization data with respect to  ZFC memory were recorded  during \nthe warming cycle without any halt under an applied dc field of 100 Oe. Fig. 4(a) depicts the \ntemperature dependence of ZFC reference magnetization 𝑀𝑍𝐹𝐶𝑊𝑟𝑒𝑓 and ZFC memory \nmagnetization 𝑀𝑍𝐹𝐶𝑊𝑚𝑒𝑚 along with the differential curve. A sharp memory dip in the differential \ncurve at 40 K shows the clear  time evolution of magnetization at the stopping  temperature and \nconfirms the glassy dynamics in SLCMO below T C2. For FC memory, first the sample  was \ncooled from  300 to 5 K at a constant rate of 2 K/min  under a cooling field of 100 Oe with \nrecurring stops at 100, 40 and 10 K for a duration of 1h at each stops.  During each stop the field \nwas set to zero t o let the magnetization relax downward. After each stop and wait period  (where \nwe see a sharp step dur ing each halt ed temperature  in figure 4 (b) blue solid line with open \ndiamond sym bol with dot ), the 100 Oe field is reapplied and cooling is further resumed. This \ncooling procedure produces a sharp jumps in the M(T)  curve at the halt temperatures. T he data  \nthus obtained is considered as 𝑀𝐹𝐶𝐶𝑠𝑡𝑜𝑝. After reaching the base temperature 5  K, the sam ple \ntempera ture is raise d continuou sly at the  2K/min  rate in a constant 100  Oe field and the \nmagnetization is recorded again. This curve is called MFCWmemcurve. Fig. 4(b) represents the FC \nmemory plots where 𝑀𝐹𝐶𝐶𝑠𝑡𝑜𝑝 curve exhibits sharp jumps at halt temperatures (100, 40 and 10 K) \nwhile 𝑀𝐹𝐶𝑊𝑚𝑒𝑚curve exhibits two clear upturns near the stopping temperatures at 10 and 40 K \nwhich is below T C2.  However, 𝑀𝐹𝐶𝑊𝑚𝑒𝑚 curve portraits a continuous curve without any noticeable \nchange at the stopping temperature of 100 K  (see Fig. 4(b) solid red line)  which is above T C2. \nThus , clear magnetic memory effects is observed  at temperatures 10 and 40 K con firming that \nthe present compound remembers its previous history of zero field relaxation only below T C2 \ndue to the slow dynamics of frozen spins in this region.  Hence, it is confirmed from both ZFC \nand FC memory experiments  that the low temperature  transition exhibited by SLCMO at T C2 is   \n \n11 \n a glassy  magnetic transition and the high temperature  transition at T C1 is a ferromagnetic \ntransition .  \n \nFig. 4. (a) ZFC memory effect measured after halting for 1 h at 40 K while cooling from \n300 K to 5 K an d then measurement was done  in the warming cycle under an applied  field \nof 100 Oe. The difference curve ∆𝑴=𝑴𝒁𝑭𝑪𝑾𝒓𝒆𝒇−𝑴𝒁𝑭𝑪𝑾𝒓𝒆𝒇\n  as a function of temperature. For the \nreference curve , the s ample was cooled to 5 K from 300 K without a halt and measurement \nwas done in the warming cycle. (b) FC memory effect experiment: Intermittent -stop \ncooling magnetization 𝑴𝑭𝑪𝑪𝒔𝒕𝒐𝒑 at 100, 40 and 10 K  while cooling from 300 K to 5 K (marked \nas solid blue arrow)  and the red solid line is the continuous warming memory curve  𝑴𝑭𝑪𝑾𝒎𝒆𝒎 \n(marked as dashed red arrow) . (c) Temperature dependence of the real part of ac \nsusceptibility (χ ׳ )measured under different frequencies with ac magnetic field of 2 Oe. \n(inset  the plot of \nln  vs. 𝒍𝒏 (𝑻𝑷\n𝑻𝒇−𝟏) (red solid circles) and the best fit to Eq.2 (solid blue \n  \n \n12 \n line) shown and inset shows the zoomed portion of T C2. (d) ln  vs.  𝟏\n(𝑻𝒑−𝑻𝑽𝑭) plot f or cluster \nglass transition and the sol id line is the linear fit for Vogel -Fulcher law (Eq. 1)  \n3.2.3. ac Susceptibility : \nTo confirm the true nature of magnetic ordering in this compound, we have measured the \ntemperature -dependent  ac susceptibility (χ ac), under an ac field of 2 Oe within the frequency (f) \nrange of 3 Hz to 923 Hz. In agreement with the dc susceptibility data, the temperature \ndependence  of the real part of the ac -susceptibility curves (χ ׳ )show  two pronounced peaks at \naround ∼ 150 and 50 K as shown in Fig . 4(c). The frequency variation study reveals that the \nposition of the high-temperature  peak ( ∼150 K) does not shift , and all the curves at various \nfrequencies merge well above ∼ 150 K confirming paramagnetic to ferromagnetic phase \ntransitio n at T C1. Whereas, the peak at ∼ 50 K has a pronounced frequency dependence  shift, \ni.e., with increasing frequencies, the peak shifts towards high temperature from 49.82 K at 3 Hz \nto 55.25 K at 923 Hz, with a decrease in peak amplitude  as shown in the inset of Fig. 4( c). Such \nbehavior is a characteristic feature of the spin glass (SG) and/or  disordered magnetic systems \n[31]. In order to  differentiate between the SG and CG behavior , we have carried out the analysis  \nof the susceptibility data at T C2 using the  Mydosh parameter, Vogel -Fulcher (VF) relation, and \nslow dynamics models  [32]. First, the Mydosh parameter ( K = ∆𝑇𝑃\n𝑇𝑃∆(𝑙𝑜𝑔 10𝑓)) which is empirically \nknown as the relative frequency shift in the peak temperature TP of (χ׳ )per decade of frequency  \n[31], is found to be 0.04 (±1) and this value is compara ble to the values reported for other CG \nsystems  (K ≤ 0.08) [33,34] . \nSecondly, we used Vogel –Fulcher ( VF) relation to understand  the  characteristic relaxation \ntime, which diverges at the freezing temperature  TVF \n𝜏=𝜏0𝑒𝑥𝑝 ⌊−𝐸𝑎\n𝑘𝐵(𝑇𝑃−𝑇𝑉𝐹)⌋ ,-------------- (1)   \n \n13 \n  where  τ0 represents the characteristic relaxation time of the clusters,  Ea is the activation  energy, \nand TVF is the VF freezing temperature which provides inter -clusters interaction strength. The \nexpression  is valid for peak temperature ( TP) greater than T VF [32]. The linear fitted 𝑙𝑛𝜏 versus \n1/(T P –TVF) curve, obtained by the Souletie and Tholence method [35] with TVF = 47.3 K, is \nshown in inset of Fig. 4(d). The curve yields Ea/kB = 19.65 (4) K and 0 = 2.42 x 10-5 s, which \nfalls within the range of characteristic relaxation times for  CG [34]. \nFinally, we have fitted  experimental data (Fig 4 (d)) by the conventional critical slowing down \ndynamics model,[31]  \n 𝜏=𝜏∗(𝑇𝑃−𝑇𝑓\n𝑇𝑓)−𝑧𝜈\n,--------------------- (2) \nwhere  τ is the relaxation time corresponding to the measured frequency, τ∗ is the microscopic  \nrelaxation time, z is the dynamic critical exponents, and Tf is the stat ic finite freezing \ntemperature for f → 0 Hz . We observed that our data well fitted to Eq. (2) as shown in  Fig  4(d); \nthe best fit yields τ∗ ≈ 7.45 ×10−8 s and z ≈ 4.14 ( ± 0.13),  with Tf ≈ 48 K. The critical exponent \nfalls inside the typical range for glassy magnetic  systems ( z ∼ 4 -12), and observed fitted \nvalues are clos e to those reported for CG  (τ∗ ≈ 10−8 s and z ≈ 6) [36]. The observed lengthier \nrelaxation time  obtained from both VF, and critical slowing down relaxation exemplars  in \nconjunction  with the calculated Mydosh parameter  value confirm the CG behavior of the sample \nbelow Tf  at 48 K.  \n \n3.2.4. Isothermal magnetization and Exchange bias  \n    Fig. 5(a)   shows the  M - H plot for SLCMO sample measured at 5, 50, 100, 150 and 250 K. \nThe linear isothermal magnetization curve at 250 K indicated the paramagnetic phase of the \nsample. For T = 150 K and below, the  M-H curve diverged from the linearity, and the hysteresis \nloop becomes more prominent below T C1. An e nlarged  view of hyster esis loops at 50 , and 100 \nK are given  in the inset of Fig. 5 (a). The absence of long range  ordering between the magnetic   \n \n14 \n cations (which is confirmed  from neutron diffracti ons) along with the presence of frozen spins \nat lower temperatures confirms the coexistence of ferromagnetic –antiferromagnetic phases in \nSLCMO. Therefore , the transition below 150 K is a weak ferromagnetic transition resulting \nwith low coercivity below 150  K down  to T C2 (100 K: HC ~ 200 Oe and 50 K : H C ~ 600 Oe). \nHowever, in SLCMO an enhance coercivity is observed  below CG (T C2) transition and it \nreaches to a maximum value of about ~ 7500 Oe at 5 K. As the system enters into CG region \nthe coercivity is enha nced  due to spin freezing phenomena. When the magnetic moments freeze \nbelow T < T C2, the spins are trapped in the increased free energy barriers between multiple \nenergy states.  So, in an applied field the magnetization direction is flipped, and thus the \ncoercivity is enhanced in order to  overcome the  increased free energy barrier  [37]. The \nestimated magnetization measured at 5 K and high field of 70 kOe is 0.8 µ B / f.u. which is \nsmaller than the expected spin only saturation magnetization value of 3.66 µ B / f.u. (this value \nis calculated  by considering the spin states ratio obtained from  XPS with Mn and Co ions as \nhigh spin states). This massive reduction in observed magne tic moments at 5 K  strongly implies \nthe presence of competing  AF interactions inside the multiple ferromagnetic islands.  \nExchange bias is a phenomena  formed due to the m agnetic anisotropy created at the interface \nof AFM and FM phase s [38,39]. There are several reports regarding the presence of Exchange \nBias (EB) in magnetic oxides with compe ting magnetic interactions [40-42]. Since the present \nsystem possesses  diffe rent magnetic interactions lead to CG and FM transitions, we have \ncomprehensi vely investigated the EB effect in SLCMO. The magnetic hysteresis measurements \nat 5 K was performed  in ZFC as well as FC cycles at different values of applied cooling field \n(CF) and temperatures to understand EB for this system in detail. In FC mode the sam ple was \ncooled under a magnetic field of 50 kOe (other than specified) from 300 K to the measurement \ntemperature.  \nIn earlier reports  [43,44] , it has been noticed  that the incorrect optimization of maximum \nmeasuring fields can lead to the existen ce of a minor  loop effect. To rule out the minor  loop   \n \n15 \n effects  and to ensure the observation of a genuine  EB-shift, we have considered the following \npoint:  Anisotropy field (H A) of the system should be less than the optimal maximum applied \nfield (H max).  \nNow to obtain the anisotropy field, we have used the law of approach for saturation \nmagnetization  equation on the initial magnetization virgin curve at 5 K  [45]  i.e. \n 𝑀=𝑀𝑆∗(1−𝑎1\n𝐻−𝑎2\n𝐻2)+𝜒𝐻--------------------------- (3)  \nWhere a 1 and a 2 are the free parameters, M s is the saturation magnetization, and χ is the high -\nfield susce ptibility . According to Andreev et al. [45] the first term in Eq.  (3) is related to a local \nanisotropy which  originates from the structural defects and nonmagnetic  inclusions of local \nmagnetization. While the second term corresponds to the rotation of magnetization against the \nmagnetocrystalline energy. I n case of a high anisotropic ferromagnet compounds a 1<< a 2  (𝑎2=\n4𝐾12\n15𝑀𝑠2), where constant a 2 provides the estimation of magnetocrystalline  anisotropy.  \n  A rough estimation of the anisotropy field 𝐻𝐴=2𝐾1\n𝑀𝑠 was obtained by using E q. (3) (where K 1 \nis an anisotropy constant).  From 5 K M-H data, the anisotropy field of ∼20 kOe obtained by \nusing Eq. (3), which is comparable to the value reported for Fe -doped LaMnO 3 [46]. The value s \nof H max were considered much higher than HA ~ 20 kOe for the major hysterisis loop tracing. \nAlong with to rule out the minor loop presence, we have recorded the hysteresis loop under \npositive and negative CF of 50 kOe. The hysteresis loops are found to be shifted  towards the \nopposite direction to th e applied CF, which is a signature  of conventional EB presence in this \ncompound [Fig . 5(b) shows zoomed version s of the hyste resis loop s between ±5 0 kOe for the \nsake of clarity].  \n   \n \n16 \n  \nFig. 5 (a) M - H loops at different temperatures in the range ±70 kOe (Inset shows the \nzoom  of the interior  region in the curve at 50 and 100 K ). (b) The M -H hysteresis loops \nmeasured at 5 K after cooling the sample from 300 K under 50 kOe field. The \nmeasurement range is between \n  70 kOe . For clarity, only the data between \n  50 kOe \nare shown . (c) H EB and H C as a function of cooling fields at 5 K. (d) Temperature -\ndependent  HEB and H C in cooling field 50 kOe. (e) Magnetization curves measured at 5 K \nafter field cooling (H FC = 50 kOe) with 12 continuous loops and zoomed the left side of the \nhysteresis curves around M=0, apparently, notice the significant difference from n = 1 to \nn = 2 than any other consecutive loops.  f) Training effect of FC exchange bias field ( HEB) \nvs. no of hysteresis at 5 K. The lines are fitted curves by two different conditions.  \nTraditionally, EB and coercivity values are obtained  as 𝐻𝐸𝐵=(𝐻𝑐1+𝐻𝑐2\n2) and 𝐻𝐶=\n(|𝐻𝑐1|+|𝐻𝑐2|\n2), where \n1cH  and \n2cH  are left and right coercive fields respectively . The hysteresis \ncurves in a cooling field of 50 kOe measured at different temperatures from 5 to 75 K in the \nfield range ±70 kOe and the value of H EB and H C are plotted  as a function of temperatures [Fig . \n5(c)]. The H EB vs. temperature curve shows an exponential decay behavior. One can see that \n  \n \n17 \n the EB appears below Tf = 48 K  of the CG state. At 5 K, the displacement of FC loop becomes \nmuch more prominent with an HEB = 5.5 kOe at a CF of 50 kOe, which is ten tim es larger than \nthat of the double perovskite compound LaSrCoMnO 6 involving same magnetic ions (measured \nin the same condition, i.e., at 5 K with CF 50 kOe)  [23]. Also, we have performed the cooling \nfield from 0.01 kOe to 60 kOe dependence of the EB (Fig . 5(d)) at 5 K with a maximum field \nrange of 70 kOe. We observe a sharp increase in both HEB and Hc with increasing CF up to 40 \nkOe with a giant EB of ∼5.5 kOe followed by a more gradual saturation of this effect at higher \nCFs up to 60 kOe. When cooling the specimen to T < Tf, in the presence of a magnetic field, \nthe CG spins next to the FM/AFM spins arrange along specific direction due to the exchange \ninteraction at the frustrating interface.  As a result,  there will be strong  pinning between frozen  \nFM and AFM island of spin clusters and the FM spins producing EB effect along the CF \ndirection.  \n      Training effects are complementary characteristics of EB phenomena and occurring by the \nnon-equilibrium nature of the spin structures in the pinning layer  [47]. While cycling the system \nthrough several consecutive hysteresis loops, it  is manifested as the gradual decrease in HEB \nand showed a c lear indication of rearrangements in the pinning layer spin structure towards an \nequilibrium configuration supporting  Binek ʹs proposition  on EB effect in antiferromagnetic -\nferromagnetic heterostructures . Later Mishra et al.  [48] proposed that local spins of AFM side \nof the interface be affected  from  both the components of frozen and rotating spins by the FM \nmagnetization reversal. In this view,  a series of twelve continuous hysteresis loops were \nmeasured at 5 K over \n  70 kOe under a CF of 50 kOe as shown in Fig . 5(e), which illustrates \na close  view of the left side of hysteresis curves around M = 0 axis. The H EB values obtained \nfrom each M (H) are plotted  with the number of hysteresis  cycles (n) in Fig . 5(f). A monotonic \ndecrease of the EB effect is observed  with continuo us loops measurement. The following \npower -law function which can describe the reduction of H EB as a function of n in terms of  \nenergy dissipation of the AFM regions at the pinning interfaces :    \n \n18 \n \n1\n2\nEB EBH H n, --------------------- (4)  \nwhere n is the loop index number and \nEBH  the value at n =∞ which is 3.188 kOe for the present \nsample.  The Eq. (4) holds only for loops from n ≥ 2, and cannot explain the sheer  relaxation \nbetween the first and second loops as shown in Fig . 5(f). According to Mishra et al. [48], \ninterfacial spin frustration can occur at the magnetically disordered FM/AFM interface due to \nAFM magnetic anisotropy. This magnetic anisotropy is contributed  from  two different types of \nAFM spins after field cooling: specifically , frozen and rotatable AF M spins [48]. As this \ncompound exhibit E B below the CG transitions  with disordered FM/AFM phases, it is \nappropriate to use the model (Eq. 5) proposed by them for fitting the training effect. The \nequation which satisfies the condition is expressed  as \nn\nEB EB f r\nfrnnH H A APP      \n, ------------------ ----(5) \nwhere  f and r denote the frozen and rotatable AFM spin components at the pinning interface.  \nThe parameters‚ ' A' have dimensions of magnetic field (Oe), whereas, parameters‚ ' P' are \ndimensionless quantities identified with relaxation. As can be seen from Fig . 5(f), the FC EB \ntraining effect data fit well with Eq. 5 in comparison to Eq. 4. The parameters obtained from \nthe fit to the H EB  data are 𝐻𝐸𝐵∞ = 3.4 kOe, Af = 3.8 kOe, Pf = 0.51, Ar= 0.505 kOe, and  Pr = \n8.71 which suggest that the rotatable components are relaxed 17 times faster than the frozen \nspin component at the interface in the presence of cooling field of 50 kOe.  Similar phen omena \nhas been obseved in a spin glass system of La 1.5Ca0.5CoIrO 6 [40].  \n \n4. Estimation of exchange bias from DFT calculations  \n The experimental results conclusively show the CG  magnetic structure and a giant  exchange \nbias for SLCMO RP structure.  The  CG state of the system suggests coexistence of  AFM and \nFM rich domains and the EB occurs at the boundary between these domains. Experimentally it   \n \n19 \n has been reported that the Sr 2-xLaxCo/MnO 4 has a rich magnetic phase diagram depending  on \nthe La and Sr conc entration [49-53]. Our electronic structure calculations along with the \nreported literature  confirm that pristine LaSrCoO 4 and LaSrMnO 4 have FM [54] and AFM \n[49,55 -58] states, respectively.  However, LaSrCoO 4 is also reported  as a spin glass in a recent \narticle [59]. The earlier discussions on the magnetic measurements have revealed the \ncoexistence  of both AFM and FM phases at low temperature (< 50 K). Further, short -range  \nmagnetic ordering are inferred  from the combined study of neutron diffraction, dc - and ac - \nsusce ptibility measurements. However, the XR PD/NPD do not indicate any Co and Mn-rich \nsegregated phases. Therefore, the coexistence of the FM and AFM ordering can be explained  \nprovi ded there is a sub -nano scale ( two to three unit cell  length) Co and Mn-rich domains. The \nprimary  arrangements of such FM and AFM domains are shown  in Fig . (6). The secondary \narrangements can be obtained  through vector  combination of these primary  arrangements.  \nUsing DFT calculations, the strength of exchange  bias is estimated  by spin-flipping  mechanism \n[60]. \n \n   \n \n20 \n  \n  \n \n21 \n Fig. 6 Schematics for three different possible interfaces between FM LaSrCoO 4 (LSCO) \nand AFM LaSrMnO 4 (LSMO) to study the exchange  bias effect in SrLaCo 0.5Mn 0.5O4 \ncompound. The dotted lines represent the interface. (a) LSCO and LSMO are repeated  \nalong the z-axis, (b & c) shows the interface with compensated and uncompensated AFM \nat the interface layers  respectively , in which LSCO and LSMO are repeated  along x - or \ny-axes to make the supercell. The exchange bias energy (EB) for each of the case is \ncalculated  by the spin-flip mechanism of the Co atoms in the FM LSCO layers. The spin -\nflip for Co atoms for each case are represented using the blue arrows. The results are \nobtained  from DFT+U calculations with U = 5 eV.  (d) For interface (a), (a -I) and ( a-II) \nshow the total and partial densities of states (DOS) before and after spin -flip respectively. \nSimilarly, (b -I, b-II) and (c -I, c-II) show the total and partial DOS for interfaces (b) and \n(c) respectively.  \nUnder the collinear arrangement and the spin -flip mechanism , the EB is calculated  as follows. \nIn each of the supercell, the spin of the Co atoms in the FM LSCO layers are flipped , whereas, \nthat of Mn atoms in AFM LSMO layers remain same. The exchange bias energy (E EB) is the \ndifference (|𝐸𝐼−𝐸𝐼𝐼|) between the two cases, i.e., between the spin -up (E I) and spin -down (E II) \narrangement for Co atoms in the FM LSCO layer. A similar method has also been adopted \nearlier to study the exchange bias effect in SrRuO 3/SrMnO 3 [60]. In the first case (Fig . 6(a)), \nthe EB is zero, suggesting weak magnetic coupling among the Co and Mn layers of atom s along \nthe z -axis. This is due to the  large  layer separation of 6.31 Å.  \nIn the second case, two possible types of interfaces are considered depending on the spin \nalignment of the Mn atoms (Fig . 6(b) and 6 (c)). In these two types of interfaces, the spin \nalignment of the Mn atom at the interface layer is different. In the first case (Fig . 6(b)), the Mn \nspins at the interface layer are opposite to each other to form a compensated AFM structure. In \nthe second structure (Fig . 6(c)), the spin -alignment of the Mn  atoms in the interface layer are in \nthe same direction to produce uncompensated moments at the interface. In case of the supercells   \n \n22 \n having compensated and uncompensated LSMO AFM layers at the interface, the EBs are found \nto be 178.99 meV/f.u . and 82.33 meV/f.u . respectively . The Mn -Co distance in such supercells \nis 3.84Å. Compensated AFM layer at the interface, in principle, should not show any exchange \nbias effect [38]. However,  in our calculation, for such an interface , we found there is a \nsignificant value of the exchange bias energy.  \nOrigin of exchange bias: To explai n the origin of the exchange bias energy, we shall consider \nthe difference between the magnetic interaction energy ( Jij), given by the Hamiltonian,  \n𝐻=−∑ 𝐽𝑖𝑗𝑺𝒊.𝑺𝒋 𝑖,𝑗  ----- (6) \nFirst,  we consider the case of the uncompensated LSMO AFM at the interfac e layer (Fig . 6 (c)) \nwith the configuration I. The total magnetic interaction energy ( Jtot) for this configuration is \n8J1-8J2+8J 3, where J1, J2, and J3 are the interaction energies between Co -Co, Mn -Mn, and Co-\nMn respectively. The interaction is considered  positive for the same  spin, whereas, they are \nconsidered to be negative  for AFM ordering between two atoms. Similarly,  for the \nconfiguration II, the total magnetic exchange interaction energy is 8J1- 8J2-8J3. Therefore, the \ndifference in magnetic energy b etween the two systems is 16 J3, which further leads to exchange \nbias energy.  \nTo further analyze, in the lower panels of Fig . 6, we have plotted the total and Mn/ Co-d DOS \nbefore and after spin -flip for the aforementioned interfaces .  From the total DOS for each of \nthese interfaces, we observe finite DOS at E F in either of the spin -channels, suggesting the FM \nmetallic behavior. From the partial DOS plots, we find that Mn -d states create  (pseudo) gap at \nEF, which is similar to the pure AFM and insulating LSMO bulk compounds, [55] whereas, the \nCo-d states crosses the EF which is the case for intermediate spin state as observed earlier in \nFM and metallic LSCO compound [54]. This further confirms that the bulk magnetic phases \nare nearly maintained .  Minor deviations are due to coupli ng between the Mn and Co spins \nacross the interface.    \n \n23 \n For the interface along [001], we find that the DOS does not change with spin -flipping. This is \nbecause  the strength of the magnetic coupling across the interface remained unaltered leading \nto the absen ce of an exchange  bias effect for this interface. However, for the interface along x \nand y, the DOS at EF changes significantly with spin -flipping suggesting a variation in the \nmagnetic coupling. Therefore, the total energy of these configurations with spi n-flipping \nchanges to create an exchange bias effect.  \n5. Conclusion s \nTo summarize and conclude we have carried out a combined experimental and theoretical \ninvestigation to show that SLCMO produces an exchange bias as large as 5.5 kOe. To our \nunderstanding , this is the highest ever reported among all the transition metal layered \noxides. The glassy magnetic nature of the sample has been confirmed using ac and dc \nmagnetic measurements. As the first principles electronic calculations suggest the origin \nof this  glassy phase and subsequently, the exchange bias effect is subscribed  to the \npresence of competing  magnetic interaction at the interface between magnetic domains at \nthe sub -nanoscale. This work concludes that new layered oxides with more than one \ntransition metals can be designed to create natural/artificial magnetic interfaces so that \ntunable giant exchange bias can be observed  at the desired temperature.  \n \nAcknowledgments : \nP.N.S. acknowledges the Council of Scientifi c and Industrial Research , India for financial \nsupport (Project No. 03 (1214)/12/EMR -II). We acknowledge Nano Functional Materials \nTechnolog y Centre of Indian Institute of Technology Madras for X -ray Photoelectron \nspectroscopy measurement. We wish to thank Dr. A. Senyshyn for his help in the neutron \ndiffraction experiment. The author T.C. gratefully acknowledges the financial support \nprovided by F RM II to perform neutron scattering experiments at the Heinz Maier -\nLieibnitz  Zentrum (MLZ), Garching, Germany.  The theoretical part of this work is   \n \n24 \n supported by DST, India through grant no. EMR/2016/003791. P.P. and B.R.K.N. would \nlike to acknowledge  the High Performance  Computing Environment  facility of Indian \nInstitute of Technology Madras. P.P. acknowledges Indian Institute of Technology \nMadras for the Institute Postdoc toral fellowship. R.R.D. is grateful  to Dr. P. Neenu \nLekshmi for many fruitful discussions.  R.R.D. thank s Indian Institute of Technology \nMadras Alumni for the travel support.  \nReferences:  \n[1] S. Baidya and T. Saha -Dasgupta, Physical Review B 84, 035131 (2011).  \n[2] J. Su, Z. Z. Yang, X. M. Lu, J. T. Zhang, L. Gu, C. J. Lu, Q. C. Li, J. M. Liu, and J. 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Spaldin1\n1Materials Theory, ETH Zurich, CH-8093 Z urich, Switzerland\n(Dated: December 14, 2018)\nIn ionic materials, circularly polarized phonons carry orbital magnetic moments that arise from\ncircular motions of the ions, and which interact with other magnetic moments or \felds. Here, we\ncalculate the orbital magnetic moments of phonons in 35 di\u000berent materials using density functional\ntheory, and we identify the factors that lead to, and materials that show, large responses. We\ncompute the resulting macroscopic orbital magnetic moments that can be induced by the excitation\nof coherent phonons using mid-infrared laser pulses, and we evaluate the magnitudes of the phonon\nZeeman e\u000bect in a strong magnetic \feld. Finally, we apply our formalism to chiral phonons, in\nwhich the motions of the ions are intrinsically circular. The zoology presented here may serve as a\nguide to \fnding materials for phonon and spin-phonon driven phenomena.\nI. INTRODUCTION\nWhen the atoms of a solid body move along the path\nof a circularly polarized vibration mode, they form closed\nloops and therefore carry angular momentum. In ionic\nmaterials, the circular motions of the ions induce or-\nbital magnetic moments that are reminiscent of atom-\nistic electromagnetic coils, see Fig. 1, and that are linked\nto the angular momentum via the gyromagnetic ratio of\nthe ions. As the ions have di\u000berent masses, they move\non di\u000berent orbital radii and consequently produce mag-\nnetic moments that are unequal in size, leading to a net\nmagnetic moment produced by the phonon mode. While\nthe concept of rotational and vibrational angular momen-\ntum and magnetic moments has been discussed for over\n80 years in molecules [1{8], a rigorous microscopic [9{12]\nand quantum mechanical [13{15] treatment for solids has\nonly been developed since the turn of the century.\nIn recent years, various physical phenomena have been\nattributed to the e\u000bect of phonon angular momentum,\nsuch as the phonon Hall and phonon spin Hall e\u000bects [16{\n18], a contribution to the Einstein-de Haas e\u000bect [14, 19{\n22] and to spin relaxation [21, 23], the phonon a.c. Stark\ne\u000bect [24], and the phonon Zeeman e\u000bect [12]. Further-\nmore, terahertz sources are nowadays able to coherently\nexcite phonons to yield large vibrational amplitudes, so\nthat the e\u000bects of phonon angular momentum become\nvisible also on a macroscopic level, for example by inter-\naction with the magnetic or valley degrees of freedom of\na material [25, 26]. In order to predict the strength of\nthese e\u000bects accurately, it is necessary to calculate the\ngoverning physical parameters from \frst principles.\nIn this study, we calculate the size of the orbital mag-\nnetic moments of phonons in 35 di\u000berent materials us-\ning a recently established microscopic formalism of the\ndynamical multiferroic e\u000bect [12]. The dynamical multi-\nferroic e\u000bect describes in general the generation of mag-\nnetization from temporally varying electric polarization,\nM/P\u0002@tP[12, 27], and Pin our case corresponds\nto the dipole moment of an infrared (IR)-active phonon\nmode. We simulate the resonant excitation of coherent\nphonons with intense terahertz and mid-infrared radia-\nAnionCation|µA|>|µC|FIG. 1. Schematic of the magnetic moments produced by\nthe circular motion of ions along the eigenvectors of a circu-\nlarly polarized optical phonon mode. Here, the lighter anion\n(drawn smaller) generates a larger magnetic moment \u0016Athan\nthe heavier cation (drawn bigger) \u0016C, which in a binary ma-\nterial leads to the orbital magnetic moment of the phonon\n\u0016ph=\u0016A+\u0016Cin the unit cell.\ntion and calculate the size of the macroscopically induced\norbital magnetic moments. We further compute the mag-\nnitude of the phonon Zeeman e\u000bect in an external mag-\nnetic \feld. As the orbital magnetic moment of a phonon\nis caused by the motion of the ions, we expect it to be of\nthe order of the nuclear magneton, \u0016N, which is roughly\nthree orders of magnitude smaller than the electronic\nmagnetism, \u0016N\u00190:5\u000210\u00003\u0016B. We therefore choose\nonly nonmagnetic compounds, so that the magnetic sig-\nnature of the material is caused entirely by the phonons.\nWe choose sets of materials with each of the rocksalt,\nwurtzite, zincblende, and perovskite structures, to allow\nus to study trends within and across material classes. As\na special case, we investigate a set of monolayer transi-\ntion metal dichalcogenides, because they have recently\nbeen predicted and observed to host chiral phonons that\nare intrinsically circularly polarized [15, 28].\nII. THEORY OF ORBITAL MAGNETIC\nMOMENTS OF PHONONS\nA. Phonon angular momentum and phonon\nmagnetic moment\nWe begin by reviewing the mathematical expression for\nthe angular momentum Land the magnetic moment MarXiv:1812.05379v1  [cond-mat.mtrl-sci]  13 Dec 20182\nof a general elliptically polarized phonon mode, which\nis a superposition of two orthogonal linearly polarized\nphonon modes. LandM, de\fned per unit cell , are given\nby\nL=Q\u0002@tQ; (1)\nM=\rL; (2)\nwhere Qis the normal mode coordinate (or amplitude)\nvector in units of \u0017Apamu with amu being the atomic\nmass unit, and \ris the gyromagnetic ratio of the phonon\n[9, 12, 14]. The amplitude vector can be written without\nloss of generality as\nQ(t) =0\n@Q1(t)\nQ2(t)\n01\nA=0\n@Q1sin(\n 1t+')\nQ2sin(\n 2t)\n01\nA;(3)\nwhere \n 1;2are the eigenfrequencies of the two orthogonal\nphonon modes, and 'is their relative phase shift. The\nangular momentum and magnetic moment accordingly\nreduce to L=L^zandM=M^z, and we will in the\nfollowing always refer to the z-components LandM.\nThe gyromagnetic ratio is given by\n\r=X\ni\riqi;1\u0002qi;2; (4)\nwhere\ri=eZ\u0003\ni=(2Mi) are the gyromagnetic ratios of\nthe ionsi,Z\u0003\niare the Born e\u000bective charge tensors, Mi\nare the masses, and qi;1=2are the eigenvectors of the two\nsuperposed phonon modes [12]. edenotes the elementary\ncharge, and the index iruns over all atoms in the unit cell.\nFor circularly polarized phonon modes, \n 1= \n 2\u0011\n0,\nQ1=Q2\u0011Q,'=\u0019=2, and the angular momentum in\nEq. (1) simpli\fes to L=Q\u0002@tQ= \n 0Q2^z\u0011L^z.\nB. Oscillator model and ab-initio calculations\nIn order to evaluate Eqs. (1) and (2), we need to com-\npute the gyromagnetic ratio \rand the phonon ampli-\ntudesQ1=2. The Born e\u000bective charge tensors Z\u0003\ni, and\nthe phonon eigenvectors qiand eigenfrequencies \n 0are\ncomputed in this work using density functional theory.\nIn order to obtain the phonon amplitude Q, we solve\nnumerically the dynamical equation of motion\nQ+\u0014_Q+@QV(Q) =F(t): (5)\nFor the e\u000bects that we investigate in this study, a minimal\nmodel consists of a harmonic potential V(Q) = \n2Q2=2,\nzero damping \u0014= 0, and a resonant mid-infrared driv-\ning forceF(t) =ZE(t). Here,Z=P\niZ\u0003\niqi=pMiis\nthe mode e\u000bective charge [29], and we model the elec-\ntric \feld of an ultrashort mid-infrared laser pulse as\nE(t) =E0expf\u0000(t\u0000t0)2=[2(\u001c=p\n8ln2)2]gcos(!0t+\u001eCEP),\nwhereE0is the peak electric \feld, \u001cis the full width at\nhalf maximum pulse duration, !0is 2\u0019times the center\nfrequency, and \u001eCEP is the carrier envelope phase [30].The amplitude and frequency of the excited phonons can\nbe a\u000bected by nonlinear phonon couplings in the per-\novskites and by general anharmonicities in all investi-\ngated compounds. In Ref. [30] it was shown that the am-\nplitudes of the IR-active phonon modes remain mostly\nunchanged in the presence of anharmonicities, and we\ntherefore stay within the harmonic approximation.\nWe calculated the phonon eigenfrequencies, eigenvec-\ntors, and the Born e\u000bective charge tensors from \frst-\nprinciples using the density functional theory formalism\nas implemented in the Vienna ab-initio simulation pack-\nage (VASP) [31, 32], and the frozen-phonon method as\nimplemented in the phonopy package [33]. We used the\ndefault VASP projector augmented wave (PAW) pseu-\ndopotentials for every considered atom and converged\nthe Hellmann-Feynman forces to 50 \u0016eV/\u0017A. For the\nrocksalt, wurtzite, zincblende, and perovskite structures\nwe used plane-wave energy cut-o\u000bs of 700, 750, 750,\nand 700 eV, and 15 \u000215\u000215, 12\u000212\u00028, 12\u000212\u000212, and\n8\u00028\u00028k-point gamma-centered Monkhorst-Pack meshes\nto sample the Brillouin zone, respectively [34]. For the\nexchange-correlation functional, we chose the Perdew-\nBurke-Ernzerhof revised for solids (PBEsol) form of the\ngeneralized gradient approximation (GGA) [35].\nIII. RESULTS\nA. Orbital magnetic moments of coherent phonons\nand single phonon quanta\nWe turn to the numerical results obtained using\nEq. (2). In order to investigate comparable excitation\nstrengths in the di\u000berent materials, we scale the \ru-\nence of the laser pulse linearly with the energy ~\n0of\nthe resonantly driven phonons. We keep the number\nof cycles of the simulated laser pulse constant by \fxing\nτ(ω0)E0(ω0)0.10.20.30.0.2.4.0.10.20.\nCenter frequency,ω0/(2π)[THz]Pulse duration,τ[ps]\nElectric\u0001eld,E0[MV/cm]\nFIG. 2. Dependence of the peak electric \feld E0and of the\nfull width at half maximum pulse duration \u001con the center\nfrequency!0=(2\u0019). We used a quadratic variation of E0and\na variation of \u001caccording to the ratio \u001c!0= 2:5\u00012\u0019in our\ncalculations.3\nLiNb-0.10.00.1Displacement,d(111)x[Å]Pulse envelopeMMLiMNbMO-0.8-0.40.00.40.80.00.51.01.5\nTime,t[ps]Magnetic moment,M[μN]Pulse envelopeMMBaMO-0.8-0.40.00.40.80.00.51.0\nTime,t[ps]Magnetic moment,M[μN](b)(c)\n(d)(e)(f)\n(a)BaOLiNbOabc[111]\nOBa×5O-0.30.00.3-0.30.00.3\nDisplacement,da[Å]Displacement,db[Å]\n-0.10.00.1-0.10.00.1\nDisplacement,d(111)x[Å]Displacement,d(111)y[Å]\nFIG. 3. Visualization of the coherent phonon dynamics of the 3 THz mode in BaO and the 17 THz mode in LiNbO 3. All\ntrajectories are shown for the time interval f-1,1gps around the center of the laser pulse at t0= 0 and relative to their\nrespective equilibrium positions (set to the origin of the plot). (a) Crystal structures of BaO and LiNbO 3. (b) Displacements\nof the barium and oxygen ions in the abplane of the BaO crystal. (c) Orbital magnetic moment Mper unit cell of the 3 THz\nmode and magnetic moments produced by each of the ions MBaandMO. The envelope of the pulse is shown schematically.\n(d) Displacements of the oxygen ions in the (111) plane of the LiNbO 3crystal. (e) Displacements of the lithium and niobium\nions in the (111) plane of the LiNbO 3crystal. (c) Orbital magnetic moment Mper unit cell of the 17 THz mode and magnetic\nmoments produced by each of the ions MLi,MNb, andMO. The envelope of the pulse is shown schematically.\nthe ratio\u001c=! 0= 2:5\u00012\u0019, and we scale the peak elec-\ntric \feld quadratically with the laser frequency, taking\nE0= 15 MV/cm at !0=(2\u0019) = 20 THz as our reference\npoint, see Fig. 2. We found this to be an appropriate\nway to scale the electric \feld, as \fxing the \ruence instead\nleads to an overproportional excitation of low-frequency\nphonon modes.\nIn Fig. 3 we show the time-dependent responses of the\natoms to the coherent excitation by the laser pulse; for\nthe 3 THz mode in barium oxide (BaO) (upper row), and\nfor the 17 THz mode in lithium niobate (LiNbO 3) (lower\nrow). Their crystal structures are shown in Fig. 3a. We\nplot the trajectories of the ions for the time interval f-\n1,1gps around the center of the laser pulse at t0=0. The\ndisplacement of the ions is shown for the abplane of the\nBaO crystal in Fig. 3b and for the (111) plane of the\nLiNbO 3crystal in Figs. 3d and e. The displacements\nare shown with respect to the equilibrium positions of\neach ion, which are set to the origin of the plots. We\nfurther show the time evolution of the orbital magnetic\nmomentMof the circularly polarized phonon mode, and\nof the constituent magnetic moments Migenerated bythe circular motion of each ion iin Figs. 3c and f. The\nenvelope of the pulse is shown schematically.\nIn BaO, the radius of the circular motion of barium is\nsmall compared to that of oxygen, and the correspond-\ning magnetic moment MBa(scaling quadratically with\nthe radius) is negligible. Mis therefore almost entirely\ngenerated by the motion of the oxygen ions. Here (and\nin the binary compounds in general), the magnetic mo-\nments of the anions and cations have opposite signs. In\nLiNbO 3, the mass di\u000berences between the ions are not\nas big as in BaO, and the corresponding magnetic mo-\nments of the cations MLiandMNbcontribute notably to\nthe orbital magnetic moment of the phonon mode. Here,\nthe eigenvector of the circularly polarized phonon mode\nis such that the magnetic moment of the cations and an-\nions are cooperative.\nWe now formulate the equations of Sec. II in terms of\nsingle phonon quanta. The harmonic vibrational energy\nper unit cell of the two superposed phonon modes is given\nbyVvib= \n2\n1Q2\n1=2 + \n2\n2Q2\n2=2\u0011\n2\n0Q2and the energy of\na single phonon quantum by ~\n0. We obtain the phonon4\nTABLE I. Binary compounds: IR-active phonon frequencies \u00170= \n 0=(2\u0019) in THz, mode e\u000bective charges Zin units of the\nelementary charge, root mean square displacements drelative to the interatomic distances d0in percent, phonon population\nnumbersNper unit cell, phonon magnetons \u0016phand macroscopically induced orbital magnetic moments per unit cell M=N\u0016ph\nin units of the nuclear magneton \u0016N, and phonon Zeeman splittings \u0001\n =\n0for an external magnetic \feld of B= 50 T.\nCompound \u00170Z d=d 0N \u0016 phM\u0001\n=\n0Compound \u00170Z d=d 0N \u0016 phM \u0001\n=\n0\nRocksalt structure1Wurtzite structure3\nBaO 3.0 0.7 9 6 0.15 1.0 0.0002 BN 31.8 1.1 4 13 0.05 0.7 6 \u000210\u00006\nCsF 3.6 0.3 3 1 0.06 0.1 0.00008 AlN 20.0 1.2 5 16 0.11 1.7 0.00002\nCsH 11.9 1.1 25 15 1.12 16.8 0.0005 GaN 17.1 1.1 6 15 0.17 2.5 0.00005\nLiI 4.8 0.5 7 3 0.18 0.5 0.0002 InN 14.7 1.2 6 16 0.2 3.2 0.00006\nMgO 11.7 0.6 4 5 0.04 0.2 0.00002 BeO 21.9 1.1 7 13 0.1 1.3 0.00002\nPbO27.7 1.2 7 16 0.13 2.1 0.00008 CuH 31.0 0.7 12 6 0.49 2.7 0.00008\nPbS 2.2 0.8 8 8 0.12 1.0 0.0003 SiC 23.7 1.3 6 20 0.15 2.9 0.00003\nPbSe 1.6 0.6 4 5 0.04 0.2 0.0001 Zincblende structure4\nPbTe 1.4 0.7 3 5 0.02 0.1 0.00006 BeS 17.3 0.6 5 4 0.13 0.5 0.00004\nSnTe 1.0 1.0 5 13 0.004 0.1 0.00002 BeSe 15.3 0.5 5 3 0.15 0.5 0.00005\nBeTe 14.1 0.4 4 2 0.14 0.3 0.00005\nGaAs 7.9 0.4 1 2 0.002 0.004 1 \u000210\u00006\n1cubic (Fm \u00163m)2tetragonal (P4/nmm)3hexagonal (P6 3cm)4cubic (F \u001643m)\npopulation number Nper unit cell as\nN=Vvib\n~\n0=\n0\n~Q2; (6)\nand subsequently rewrite LandMin terms of Nas\nL=N~\u0011Nlph; (7)\nM=\rN~\u0011N\u0016ph: (8)\nHere, the quantized angular momentum of a circularly\npolarized phonon is equal to the reduced Planck con-\nstantlph=~[14, 15], and its quantized magnetic mo-\nment is given by \u0016ph=\rlph=\r~, which we will refer to\nin the following as the \\phonon magneton\". Note that\nprevious publications sometimes refer to the angular mo-\nmentum of phonons as \\phonon spin\" [15, 23, 36]. Since\nthe phonon is a quasiparticle and the angular momen-\ntum arises from circular (orbital) motions of the atoms,\nwe \fnd it more appropriate to use the term \\intrinsic\norbital angular momentum\" [37].\nIn Tables I and II we show the results of our calcula-\ntions for the binary and perovskite compounds, respec-\ntively. We show the calculated phonon eigenfrequen-\ncies\u00170= \n 0=(2\u0019) and mode e\u000bective charges Z, the\nphonon magnetons \u0016ph=\r~obtained using Eq. (4),\nthe phonon population numbers Nper unit cell obtained\nfrom Eqs. (5) and (6), and the macroscopically induced\norbital magnetic moment per unit cell Mobtained from\nEq. (8). To check that the vibrational response to the\npulsed excitation is in a meaningful range, we evaluate\nthe Lindemann criterion for the induced atomic displace-\nments. According to the Lindemann criterion, melting\noccurs when the root mean square displacement dof the\natoms reaches around 10-20% of the interatomic distanced0[38, 39]. We therefore evaluate dfor the ioniwith the\nlargest root mean square displacement di=qiQ=p2Mi\nasd= max ijdij, and we include the ratio d=d0in the\nTables I and II.\nThe phonon magneton is enhanced by high Born e\u000bec-\ntive charges and big mass di\u000berences between the ions,\nas is apparent from Eq. (4). As a result, we found the\nlargest phonon magnetons of \u0016ph= 1:2 and 0:49\u0016Nin\nthe hydride compounds CsH and CuH, because of the\nlarge mass di\u000berence due to the light H atom. Typical\nvalues for the other binary and perovskite compounds\nrange between \u0016ph\u00180:1 and 0:2\u0016N.\nWhen looking at the experimental feasibility of induc-\ning large values of M, two factors in addition to the\nphonon magneton have to be taken into account: the\nexcitability of the phonon mode and the limitation due\nto the Lindemann criterion. As we scale the \ruence of\nthe laser pulse with the energy of the phonons, the mode\ne\u000bective charge Zis the main factor in determining the\nexcitability of the phonon mode. It determines the ampli-\ntude of the driven phonon mode according to Eq. (5), and\nconsequently the population number Nper unit cell ac-\ncording to Eq. (6). For example, the exceptionally large\nmode e\u000bective charges of Z > 2eof the 6 THz modes\nin BaTiO 3and KNbO 3and the soft 1.7 THz mode in\nSrTiO 3lead to high population numbers of >50 phonons\nper unit cell under the pulsed excitation. This excitabil-\nity is however limited by the Lindemann criterion, which\npredicts instability of the lattice due to melting for val-\nues ofd=d0>10%. In most cases we obtain phonon\namplitudes within the stability limit, however the previ-\nous example shows that while the phonon population in\nKNbO 3leads tod=d0= 10%, the corresponding excita-5\nTABLE II. Perovskites in their low-temperature structures: IR-active phonon frequencies \u00170= \n 0=(2\u0019) in THz, mode e\u000bective\nchargesZin units of the elementary charge, root mean square displacements drelative to the interatomic distances d0in\npercent, phonon population numbers Nper unit cell, phonon magnetons \u0016phand macroscopically induced orbital magnetic\nmoments per unit cell M=N\u0016phin units of the nuclear magneton \u0016N, and phonon Zeeman splittings \u0001\n =\n0for an external\nmagnetic \feld of B= 50 T. We display a selection of phonons with the largest values of \u00170,N,\u0016ph, andMfor each compound.\nCompound \u00170Z d=d 0N \u0016 phM\u0001\n=\n0Compound \u00170Z d=d 0N \u0016 phM\u0001\n=\n0\nBaHfO 3115.7 0.8 5 8 0.04 0.3 0.00001 LiTaO 3217.4 1.5 5 25 0.04 1.0 0.00001\n5.9 1.1 7 14 0.12 1.7 0.0001 10.9 0.9 8 10 0.13 1.3 0.00006\nBaZrO 3115.0 0.9 6 10 0.04 0.4 0.00001 4.2 1.0 4 11 0.07 0.8 0.00008\n5.8 1.0 5 12 0.04 0.4 0.00003 BaTiO 3314.1 0.8 3 8 0.03 0.3 0.00001\n3.1 0.7 5 6 0.07 0.4 0.0001 8.9 0.01 { { 0.14 { 0.00007\nKTaO 3115.8 1.0 6 12 0.003 0.04 9 \u000210\u000076.5 2.2 9 58 0.02 1.0 0.00001\n2.3 1.4 14 24 0.12 3.0 0.0003 KNbO 3315.2 1.7 7 35 0.01 0.5 5 \u000210\u00006\nBiAlO 3218.5 0.6 2 4 0.08 0.3 0.00002 8.1 0.02 { { 0.18 { 0.0001\n12.8 0.1 { { 0.14 { 0.00005 6.0 2.2 10 55 0.13 7.0 0.0001\n11.5 1.4 6 22 0.05 1.1 0.00002 PbTiO 3414.9 0.8 6 8 0.09 0.7 0.00003\n3.9 1.1 6 14 0.05 0.7 0.00006 8.1 0.1 1 0.2 0.19 0.05 0.0001\nCsPbF 329.1 0.04 { { 0.001 3 \u000210\u000072.6 0.8 7 7 0.08 0.6 0.0001\n4.9 0.6 3 4 0.04 0.2 0.00004 SrTiO 3515.7 1.1 4 15 0.03 0.4 9 \u000210\u00006\n1.5 0.02 { { 0.04 { 0.0001 7.3 0.04 { { 0.18 { 0.0001\nLiNbO 3217.0 1.6 6 32 0.04 1.4 0.00001 1.7 2.5 21 74 0.1 7.2 0.0003\n10.6 0.8 7 7 0.13 0.9 0.00006\n4.3 1.0 6 12 0.1 1.2 0.0001\n1cubic (Pm \u00163m)2rhombohedral (R3c)3rhombohedral (R3m)\n4tetragonal (P4mm)5tetragonal (I4/mcm)\ntion of the soft mode in SrTiO 3withd=d0= 21% would\nlikely destroy the sample, and the peak electric \feld E0\nhas to be scaled down accordingly. The most convenient\nmaterials for achieving large Mare therefore PbO, AlN,\nGaN, InN, and SiC for the binary compounds (CsH can\nbe omitted for practical reasons, due to its instability un-\nder air), in which values of M\u00181:7 to 3:2\u0016Nshould be\nachievable. For the perovskites, we calculated values of\nM= 1:7 and 3:0\u0016Nin BaHfO 3and KTaO 3, and up to\nM\u00187\u0016Nfor KNbO 3and SrTiO 3, due to the presence\nof phonon modes with both high excitability and large\nphonon magnetons.\nB. Phonon Zeeman e\u000bect\nThe orbital magnetic moment of the phonon interacts\nwith an external magnetic \feld B=B^zvia Zeeman cou-\npling of the form M\u0001B. This leads to a Zeeman splitting\nof the phonon frequencies of the right (+) and left ({)\nhanded circular polarization of the phonon that is linear\nin the external magnetic \feld [9, 12]:\n\n\u0006= \n 0\u0006\rB: (9)The relative splitting of the phonon frequency is hence\ngiven by\n\u0001\n\n0=2\rB\n\n0=2\u0016phB\n~\n0: (10)\nThe e\u000bect is independent of the phonon population num-\nber. We therefore have to compute only \u0016phin order to\ncalculate the magnitude of the e\u000bect.\nWe show the calculated relative splittings of the binary\ncompounds in Table I, and of the perovskites in Table II,\nin an external magnetic \feld of B= 50 T. The split-\nting is highest for low-frequency phonons with high \u0016ph.\nThe largest splittings of \u0001\n =\n0>10\u00004occur for BaO,\nCsH, LiI, PbS, PbSe for the binary compounds, and are\ncomparably large for most of the perovskites, in which\nlow-frequency phonons are generally present.\nC. Orbital magnetic moments of chiral phonons\nWe now discuss the orbital magnetic moments pro-\nduced by chiral phonons, which have recently been pro-\nposed and observed in the valleys of the monolayer\ntransition metal dichalcogenide (TMD) WSe 2and other\ngraphene-derived materials [15, 28, 40, 41]. The motion\nof the ions along the eigenvectors of a chiral phonon mode\nis intrinsically circular and cannot be constructed by a6\nTABLE III. Monolayer transition metal dichalcogenides\n(TMDs) in their hexagonal (P \u00166m2) structure: Chiral phonon\nfrequencies \u00170in THz and phonon magnetons \u0016phin units of\nthe nuclear magneton \u0016Nat the K/K' points of the Brillouin\nzone.\nTMD\u00170\u0016phTMD\u00170\u0016phTMD\u00170\u0016ph\nWS26.7 0.0007 WSe 26.4 0.0003 WTe 26.0 0.0002\nsuperposition of linearly polarized phonon modes. Chiral\nphonons are generated through the decay of an exciton\nstate, in which a hole scatters between the K/K' points\n(valleys) of the Brillouin zone under emission of a chiral\nphonon. Because they occur at nonzero wave vectors,\nthey cannot be coherently excited via IR-absorption,\nwhich is only possible for modes at the center of the Bril-\nlouin zone.\nThe calculation of \u0016ph=\r~using Eq. (4) has to be\nadjusted: the eigenvectors qi;1andqi;2no longer cor-\nrespond to the eigenvectors of two superposed phonon\nmodes, but to the real and imaginary parts Re[ qi;chiral]\nand Im[ qi;chiral] of the eigenvector that de\fnes the cir-\ncular motions of the ions in the chiral phonon modes.\nHere, the transition metal stands still, while the entire\nvibrational motion is made by the respective chalcogen\natoms. We calculated the phonon magnetons \u0016phfor the\nseries of WS 2, WSe 2, and WTe 2, and we list the values\nin Table III. The phonon magnetons with a maximum\nvalue ofM= 0:0007\u0016Nfor WS 2are small compared to\nthe other material classes, because the chalcogen atoms\nthat make the circular motion possess only small Born\ne\u000bective charges of Z\u0003\ni\u00180:3 to 1:3e.\nD. A simple estimate of the phonon magneton\nIn this last results section, we investigate whether it is\npossible to derive a simple estimate of the magnitude of\nthe phonon magneton without detailed knowledge about\nthe phonon eigenvectors. If we write the phonon magne-\nton as given by Eq. (4) in units of the nuclear magneton\n\u0016N=e~=(2MP), with the proton mass MP\u00191 amu,\nwe remain with \u0016ph=P\niZ\u0003\ni=Mi(qi;1\u0002qi;2). Therefore,\npoor-man's estimates, which we denote Sand in which\nwe ignore the contribution of the phonon eigenvectors,\ncan be written as\nSBEC/M =X\niTr[Z\u0003\ni]\n3Mi; (11)\nSFC/M =X\niZ(f)\ni\nMi: (12)\nBoth estimates consist of a sum of the charge-to-mass\nratio of the ions. In SBEC/M , we take the average trace\nTr[Z\u0003\ni]/3 of the Born e\u000bective charge (BEC) tensor, for\nwhich literature values are often available and density\nfunctional theory calculations are inexpensive. In SFC/M ,we simply use the formal charges (FC) Z(f)\ni, where the\nentire information can be extracted from the periodic\ntable of the elements.\nIn Fig. 4 we show a comparison of the estimates\nSBEC/M andSFC/M to the calculated values of \u0016ph\nfor each of the materials classes rocksalt, wurtzite,\nzincblende, perovskite, and monolayer transition metal\ndichalcogenides. For the materials with diatomic unit\ncells (rocksalt and zincblende structures), the estimate\nfrom the sum over the Born e\u000bective charge to mass ra-\ntioSBEC/M is in excellent agreement with the calculated\nvalues, see Figs. 4a and c. For the materials with 4-\natom unit cells (wurtzite structure and PbO), SBEC/M\ndeviates from the calculated value by a signi\fcant mar-\ngin - it does however capture the relative trend within\nthe materials class and can be used to estimate an up-\nper boundary for \u0016ph, see Fig. 4b. For the perovskites\nwith unit cells consisting of 5 or 10 atoms, SBEC/M is no\nlonger a good predictor, see Fig. 4d, because the vector\nproduct of the phonon eigenvectors qi;1\u0002qi;2is di\u000ber-\nent for each of the sets of degenerate IR-active phonon\nmodes. Furthermore, the shape of the Born e\u000bective\ncharge tensors with their inhomogeneous diagonal and\nnonzero o\u000b-diagonal terms can no longer be accounted\nfor. A similar analysis can be applied to the monolayer\ntransition metal dichalcogenides with 3-atom unit cells,\nin which the trend across the series is captured, but \u0016ph\nis strongly overestimated due to the large inhomogeneity\nof the diagonal Born e\u000bective charges parallel and normal\nto the two-dimensional surface. Finally, the sum over the\nformal charge to mass ratio SFC/M predicts neither the\nmagnitude of \u0016ph, nor the trends within materials classes,\nwhich emphasizes the importance of using the Born ef-\nfective charge formalism to include the electronic rehy-\nbridization contribution to the electric dipole moment of\nthe IR-active phonon modes.\nIV. CONCLUSION\nIn summary, we \fnd the quantized orbital moments\nof circularly polarized phonons, which we call phonon\nmagnetons, to be in the order of 10\u00004\u0016Bper unit cell,\nand the macroscopically induced orbital magnetic mo-\nments of coherent phonons generated from pulsed mid-\nIR excitation to reach the order of 10\u00003\u0016Bper unit cell.\nThe phonon Zeeman splittings in an external magnetic\n\feld of 50 T reach the order of 10\u00004of the phonon fre-\nquency. We expect that the orbital magnetic moments of\nphonons should be observable with modern experimental\ntechniques, for example indirectly via Faraday rotation\nmeasurements that are able to detect small changes in\nelectronic magnetic order [25], or directly via nitrogen-\nvacancy center magnetometry [42{45]. We hope that our\nanalysis of the information generated from our database\nwill stimulate and guide future experimental studies in\nthe \feld of optical phononics.7\nXXXXXXXXXXOOOOOOOOOO++++++++++XCalculated valueOBEC/M estimate+FC/M estimateBaOCsFCsHLiIMgOPbOPbSPbSePbTeSnTe0.00.51.0\nRocksalt compoundsPhonon magneton,μph[μN]\nXXXXXXXOOOOOOO+++++++BNAlNGaNInNBeOCuHSiC0.00.51.01.52.0\nWurtzite compoundsXXXXOOOO++++BeSBeSeBeTeGaAs0.00.10.20.3\nZincblende compoundsXO+(a)(b)(c)\nXXXXXXXXXXXOOOOOOOOOOO+++++++++++\nBaHfO3BaZrO3KTaO3BiAlO3CsPbF3LiNbO3LiTaO3BaTiO3KNbO3PbTiO3SrTiO30.00.40.81.2\nPerovskitesPhonon magneton,μph[μN]×10-2(d)(e)\nXXXOOO+++WS2WSe2WTe20.4.8.12.\nMonolayer TMDs\nFIG. 4. Calculated values of the phonon magneton \u0016phand estimated values using the Born e\u000bective charge (BEC) and\nformal charge (FC) to mass ratio. (a) Rocksalt, (b) wurtzite, (c) zincblende compounds, (d) perovskites, and (e) monolayer\ntransition metal dichalcogenides (TMDs). 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Charilaou1,5\n1Laboratory of Metal Physics and Technology, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland\n2Ernst-Ruska Centre for Microscopy and Spectroscopy with Electrons and Peter Gr unberg Institute,\nForschungszentrum J ulich, 52425 J ulich, Germany\n3Arnold Magnetic Technologies, 5242 Birr-Lup\fg, Switzerland\n4Scienti\fc Center for Optical and Electron Microscopy, ETH Zurich, 8093 Zurich, Switzerland and\n5Present address: Department of Physics, University of Louisiana at Lafayette, Lafayette, 70504 LA, USA\nThe most powerful magnets for high temperature applications are Sm{Co-based alloys with a\nmicrostructure that combines magnetically soft and hard regions. The microstructure consists of a\ndense domain-wall-pinning network that endows the material with remarkable magnetic hardness.\nA precise understanding of the coupling between magnetism and microstructure is essential for\nenhancing the performance of Sm{Co magnets, but experiments and theory have not yet converged\nto a uni\fed model. Here, we combine transmission electron microscopy, atom probe tomography,\nand nanometer-resolution o\u000b-axis electron holography with micromagnetic simulations to show that\nthe magnetization processes in Sm{Co magnets result from an interplay between curling instabilities\nand pinning e\u000bects at the intersections between magnetically soft and hard regions. We also \fnd\nthat topologically non-trivial magnetic domains separated by a complex network of domain walls\nplay a key role in the magnetic state. Our \fndings reveal a previously hidden aspect of magnetism\nand provide insight into the full potential of high-performance magnetic materials.\nI. INTRODUCTION\nSm{Co-based materials are the strongest magnets\navailable today, particularly for vital high-temperature\nprecision applications thanks to their high Curie temper-\natures [1{3]. Extensive research and industrial develop-\nment in the past few decades has led to a signi\fcant im-\nprovement in their magnetic performance [4]. One such\nexample is a highly engineered Sm{Co-based system that\nconsists of a cellular microstructure with a Sm 2Co17ma-\ntrix that is enclosed by SmCo 5cells and intersected by\nthe so-called Z phase (Zr-rich platelets) perpendicular to\nthec-axis of the Sm 2Co17matrix, which is at the same\ntime the magnetic easy axis [1, 5, 6]. This characteristic\ngeometry results from tailored aging-heat treatments [1],\nand corresponds to a network of intertwined magnetically\nsoft (Sm 2Co17) and hard (SmCo 5) regions.\nMagnetic properties due to combinations of hard and\nsoft structures are highly tunable because they make use\nof the high saturation magnetization of the soft phase\nand the strong magnetocrystalline anisotropy of the hard\nphase [7, 8]. Particular attention has been devoted to\nmodeling the Sm{Co cellular microstructure [9{12] in or-\nder to predict its coercivity. The enhanced coercivity in\nthese cellular Sm{Co magnets emerges from the di\u000ber-\nence between the magnetocrystalline anisotropy of the\ntwo phases and consequently the di\u000berence in domain-\nwall energy [9]. While conventional wisdom states that\nthis microstructure constitutes a pinning system for do-\nmain walls, the exact magnetization processes remain\nelusive despite the intense activities that have been per-\nformed to understand the interaction of domain walls\nwith the SmCo 5cells [13{19].\nMagnetic imaging experiments, by means of Lorentz\ntransmission electron microscopy (LTEM), magneticforce microscopy, and Kerr microscopy, have revealed\nthat domain walls follow the SmCo 5cell morphology\n[17, 19{24], thus con\frming strong pinning at the cell\nboundaries. Theory and experiment, however, have yet\nto converge on the role of the Z phase on the magnetic\nproperties of this material [9{12]. Forward modeling of\nthe Z phase is impeded by the fact that the material\nparameters of this phase cannot be easily estimated be-\ncause the thickness of the Z phase can be as thin as 1-2\natomic layers (the thickness varies from material to ma-\nterial), and thus cannot be compared with measurements\non bulk samples [25]. Hence, high-resolution imaging of\nthe magnetization textures is crucial to elucidate the in-\nterplay between the soft, hard, and Z phases, and unveil\nthe magnetization processes that are at play in the cel-\nlular Sm{Co magnets.\nHere we present a detailed study of the magnetic state\nin a cellular Sm{Co magnet containing Fe, Cu and Zr,\nwhere we correlate atomic-resolution TEM and atom\nprobe tomography (APT) with high-resolution LTEM\nand o\u000b-axis electron-holography (EH) imaging of the\ndomain-wall structure, and systematically compare it\nwith detailed micromagnetic simulations. By match-\ning experiments and theory one-to-one we show that\nthe nanoscale magnetization processes in cellular Sm{\nCo magnets stem from an interplay between pinning at\nthe SmCo 5cell boundaries and curling instabilities at the\nintersections between all three phases.\nII. RESULTS AND DISCUSSION\nThe Sm{Co sample in our study has an overall chem-\nical composition of Sm(Co,Fe,Cu,Zr) 7:57with minor\namounts of oxygen (see the Methods section). Figure 1aarXiv:1901.01922v3  [cond-mat.mtrl-sci]  14 Oct 20192\nshows for this sample an overview of the microstructure,\nrevealing the typical soft-magnetic Sm 2Co17matrix en-\nclosed by the hard-magnetic SmCo 5cells and intersected\nby the Z-phase platelets. A close-up view in Fig. 1b\nand corresponding energy-dispersive X-ray spectroscopy\n(EDX) chemical maps in Fig. 1c{g show interfaces be-\ntween the three phases and con\frm that the Z-phase is\nrich in Zr. The thickness of the interface between the\nSm2Co17matrix and the SmCo 5cells ranges from atom-\nically sharp to 2 nm (visible as blurry contrast), whereas\nthe interfaces with the Z phase are always atomically\nsharp. The c-axis of the crystal structure (see Fig. 1 in\nthe Supplementary Material) lies inside the TEM lamella,\nand the Z platelets are always perpendicular to the c-axis\n[6].\nThe APT reconstruction in Figure 1h shows the iso-\nconcentration surfaces of Zr and Fe with concentration\nvalues of 9.8 and 13.5 at. %, respectively, and reveal four\nperfectly \rat Z platelets, where two rightmost of which\nare actually so close that they are visible only as one\nwide platelet. In the top middle part of Fig. 1h between\ntwo Z platelets a twisted SmCo 5cell can be seen. Thetwisted shape explains why in Fig. 1a di\u000berent interfaces\nbetween the 2:17 and 1:5 phases have di\u000berent sharpness.\nThe concentration pro\fles of individual elements across\na SmCo 5cell and a Z platelet are shown in Figures 1i\nand j, respectively (across the blue areas in the inset \fg-\nures). Cu is found inside the cell with a Gaussian-like\ndistribution, which might critically a\u000bect the magnetic\nproperties [24]. The increase in Sm across the SmCo 5\ncell appears to be non-symmetrical, which may be due\nto applying a one-dimensional concentration pro\fle to\nthe twisting of the cell. As expected, Zr increases across\nthe Z platelet, but, surprisingly, Cu segregates at the in-\nterfaces between the platelet and the matrix, which has a\nsigni\fcant impact on the magnetic performance. There-\nfore, further nanoscale segregation and clustering studies\nare feasible and encouraged. The SmCo 5cells are typi-\ncally around 200 nm wide across their widest region and\napproximately 15 nm thick (Fig. 1a), and the Z platelets\nare at most 5 nm thick (Fig. 1b) and thus consist of only\na few atomic layers (some platelets consist of only 1{2\natomic lattice planes). As we will discuss below, it is the\nthickness of the Z platelets that plays a crucial role in\nthe magnetic properties.\nFig. 1. Sm{Co microstructure. a Bright-\feld TEM image of a Sm{Co magnet showing the Sm 2Co17matrix (light grey)\nenclosed by SmCo 5cells (dark grey), with the entire structure intersected by the Z phase. The corresponding di\u000braction pattern,\nshown in the bottom right corner, contains re\rections from the [100] (green) and [110] (yellow) directions (see Suppl. Fig. 1).\nbHigh-angle annular dark-\feld (HAADF) scanning TEM image showing the details of the microstructure, accompanied by\nEDX chemical maps of cSm,dCo,eFe,fZr and gCu.hAtomic-resolution APT reconstruction with the isoconcentration\nsurfaces of Zr and Fe, exhibiting \rat Z-phase platelets (vertical) and a twisted SmCo 5cell (in the top middle between the two\n\rat Z phase platelets). Concentration plots, as indicated by insets, of individual elements show that iCu accumulates in the\nmiddle of the cell, while Sm increases non-symmetrically across the 1:5 cell, and jZr peaks in the middle of the the Z phase,\nwhile Cu accumulates at the interface between the Z phase and the Sm 2Co17matrix.3\nFig. 2. Magnetic structures in a Sm{Co magnet. DW structures imaged by LTEM in 0.5 mm aover- and bunderfocus are\ncharacterized by domain-wall annihilation and o\u000bsetting. cHigher magni\fcation reveals a topologically non-trivial structure\nwith branching and alternating \u0019and 2 \u0019magnetization pro\fles. A comparison of da Fresnel defocus image and ea magnetic\ninduction map extracted from o\u000b-axis EH of the same region (arrows indicate the same location) shows a \u0019domain wall\ninvolving vortex-like curling. fMagnetic induction map of a complex magnetic state consisting of DWs with a wide range of\nangles (the arrows follow the magnetic induction direction). The phase di\u000berence between adjacent contours in the induction\nmaps is 2 \u0019radians, and the di\u000berent contour spacing in panels e and f results from di\u000berent specimen thickness.\nFigure 2 shows magnetic imaging studies of the cellu-\nlar Sm{Co magnet in a thermally demagnetized (mag-\nnetically pristine) state. Figures 2a and b are over- and\nunderfocus Fresnel-mode LTEM images of the same area,\nrespectively. Domain walls (DWs) appear as alternating\nsharp and bright contrast (convergent), or blurry and\ndark contrast (divergent), depending on the sign of the\ndefocus and the sense of the magnetization pro\fle along\nthe domain wall. Changing the sign of the defocus inverts\nthe contrast, i.e. the same domain walls have opposite\ncontrast in over and underfocus images. These LTEM\nimages reveal a complex DW pattern that follows exactly\nthe microstructure of the material, similar to those shown\nin the literature [17, 19{21]. Speci\fcally, we observe that\nthe domain walls are always adjacent to the SmCo 5cells\nand are o\u000bset by approximately 50 nm via the Z phase ex-\nactly where they intersect the cells. Even though LTEM\ndoes not enable a quanti\fcation of the magnetization\ndirection along the domain-wall pro\fle, considering the\nhigh uniaxial anisotropy in the system we may assume\nthat these are \u0019Bloch-type domain walls [17]. The av-erage domain-wall thickness width was estimated from a\nseries of images taken at di\u000berent defocus values (see Fig.\n2 in the Supplementary Material) to be 4 \u00062 nm. This is\nin agreement with theory, considering the properties of\nSm2Co17from which \u000edw=p\nA=K u= 2:7 nm can be de-\nrived, where Ais the exchange sti\u000bness and Kuis the uni-\naxial magnetocrystalline anisotropy (see Methods). This\nresult suggests that the domain walls are mostly located\nin the soft Sm 2Co17phase, reminiscent of an exchange\nspring magnet [26].\nWhile a zig-zag domain-wall structure is well known\n[27], here we have observed for the \frst time unexpected\ndomain wall patterns at some of the intersections between\nthe three phases, where domain walls with opposite sense\nmeet and annihilate each other, leaving a trivial ferro-\nmagnetic state (marked with arrows in Figs. 2a and b).\nSurprisingly, topologically complex structures bounded\nwith two \u0019walls of the same sense, i.e. total winding\nof 2\u0019can also be observed, as shown in Fig. 2c. The\nunwinding of such regions is non-trivial and requires a\nviolation of topological constraints.4\nWe have complemented LTEM with o\u000b-axis EH to gain\nin-depth information on the direction of the local mag-\nnetic \feld inside the sample. Figure 2d shows an LTEM\nimage of a domain wall pinned to a SmCo 5cell with con-\ntrast that varies in intensity. Domain walls may be tilted\ninside the sample and therefore overlap with adjacent\nmagnetic domains, which might result in such contrast.\nHowever, a magnetic induction map extracted from o\u000b-\naxis EH of the same area shown in Fig. 2e gives more\ninformation on this magnetic structure. The domain wall\nhas a winding of \u0019and the magnetic \feld curls around the\nSmCo 5cell, forming closed loops reminiscent of magnetic\nvortices (indicated by the middle arrow). This curling\nmay explain the variation of contrast intensity in LTEM.\nInterestingly, Figure 2e reveals another domain wall in\nthe top left corner, which is not visible in LTEM in Fig.\n2d. Figure 2f shows that the magnetic texture in some\nareas can be so complex that some domain walls do not\nhave a well-de\fned angle; instead the angle varies be-\ntween \u0019=2 and \u0019. These exotic magnetization textures\nare closely correlated with the microstructure and indi-\ncate that topological aspects need to be considered in\norder to correctly interpret the magnetic state.\nGiven the elaborate microstructure, these observations\nraise various questions regarding the magnetic state in\ncellular Sm{Co magnets. In order to obtain further in-\nsight, we performed detailed high-resolution micromag-\nnetic simulations to elucidate the formation of the ob-\nserved complex domain patterns. To this end, it is imper-\native that we fully consider the real microstructure, and\nthus we constructed a simulation system directly from\nthe TEM images shown above. Figures 3a,b show how\nwe have truncated the microstructure in order to model\na system with the three phases at the same geometry and\nscale.\nIn our simulations, we have considered the ferromag-\nnetic exchange and uniaxial anisotropy energies, the\ndipole-dipole interactions, and the exchange energy be-\ntween the three di\u000berent phases. The material parame-\nters (A,Ku, and saturation magnetization MS) are well\nknown and were taken from the literature [6, 24, 25, 28]\n(see Methods). The exchange interaction energy between\nthe three phases is unknown. We performed parametric\nmicromagnetic studies where we varied the exchange in-\nteraction and compared the theoretical hysteresis curve\nwith the experimental data, thus deducing the correct\nvalues by matching simulations to experiments (Fig. 3c-\ne).\nAs mentioned above, the precise material properties\nof the Z phase are unknown because the exact chemi-\ncal composition is unclear and the platelets can be as\nthin as a single atomic layer (see Fig. 1b). All material\nparameters, e.g., exchange sti\u000bness, saturation magne-\ntization, and magnetocrystalline anisotropy are a\u000bected\nby the reduced dimensions of the platelets and their in-terfaces [29]. It is known from thin-\flm studies that the\nanisotropy is the most sensitive property and changes\ndrastically depending on the thickness and local atomic\narrangements [30]. We have therefore performed sim-\nulations where we considered the Z phase to be either\nanisotropic or isotropic. In the former case, we assigned\nthe bulk value of Ku[6, 25], whereas in the latter case we\nsetKu= 0. From the comparison of experimental and\ntheoretical M(H) curves, we \fnd that the simulations\nmatch the experimental observations only if Ku= 0.\nHence, for the rest of the discussion we assume that\nKu= 0 for the Z phase.\nIn order to make our simulation quantitatively compa-\nrable to the experimental results, we matched the theo-\nretically predicted coercivity with the experimental one\nby varying the exchange energy between the three phases.\nWe \fnd that the coercivity increases with decreasing ex-\nchange energy (see Fig. 3c). This supports experiments\nthat show that increasing Cu content leads to higher co-\nercivity, depending on the compositional gradient at the\nboundary [22, 24]. This is due to the formation of Cu-rich\ninterfaces between the magnetic phases that decrease the\nexchange coupling.\nWe have also found that changing the thickness of the\nSmCo 5cells does not modify the magnetic performance\nstrongly, in agreement with experiments [31] and theory\n[32], showing that the pinning \feld is saturated for a\nSmCo 5thickness of more than 4 nm. This contradicts\nthe predictions by Fidler et al. [10, 12], stating that the\nSmCo 5thickness should be at least three times the ex-\nchange length (3 \u000eexc\u001920 nm) for e\u000bective domain-wall\npinning. The pinning, however, is a complex process and\ndepends strongly on the Cu content in the SmCo 5cells\n[33]. In our experiments we have found a Cu-composition\ngradient, but because the variation of the magnetic ma-\nterial parameters as a function of Cu is unknown, we\nmodeled SmCo 5with a homogeneous Cu enrichment.\nFurthermore, we have found that the coercivity\nstrongly depends on the Z-phase thickness (see Fig. 3d).\nIn the absence of the Z phase, it has a maximum value\nof 5.7 T, and decreases signi\fcantly with increasing Z-\nphase thickness up to 5 nm, where it reaches a minimum\nof 3 T and then remains constant. These results ex-\nplain recent experimental observations, where the mag-\nnetic performance deteriorated with increasing Z-phase\nthickness [19]. However, since Zr is essential in forming\nthe Sm{Co microstructure [20], it cannot be completely\neliminated from the material, but the Z-phase platelets\nmay be designed to be as thin as possible to maximize\nthe performance. As we will discuss below, smaller thick-\nness impedes magnetization curling and hence a stronger\nexternal \feld is required to initiate the magnetization re-\nversal process, which begins at the intersections of the Z\nand Sm 2Co17phases.5\nFig. 3. Modelling the microstructure of Sm{Co magnets. a Modelling of the microstructural features in Sm{Co, as\nseen in Fig. 1a, to create ba model with the Sm 2Co17matrix (grey), the SmCo 5cells (red), and the Z phase (yellow). c\nSimulated dependence of the coercivity as a function of exchange coupling between the individual phases, illustrating that\nsmaller exchange between the phases leads to higher coercivity. Using these results, we can compare the theoretical coercivity\nwith that of real samples. dSimulated dependence of the coercivity on the thickness of the Z phase, showing that it increases\nsigni\fcantly with decreasing thickness. eComparison between an experimentally measured M(H) loop at T= 300 K and a\nsimulated loop along the easy axis. By matching the simulation to the 300 K experimental data, we have obtained the value\nof the exchange sti\u000bness in the system. The external \feld is applied parallel to the caxis, i.e., perpendicular to the Z-phase\nplatelets.\nIn the simulations that we discuss in the following sec-\ntion, the thicknesses of the cell boundary and the Z phase\nwere derived directly from the TEM images, where we\nhave 10 nm thick SmCo 5cells and 1 to 5 nm thick Z-\nphase platelets. Figure 3e shows an experimentally mea-\nsured (see Methods) and a simulated M(H) curve along\nthe easy axis, con\frming the agreement between experi-\nment and theory, speci\fcally the value of the remanence\nMR= 0:95MSand a gradual decrease of the magneti-\nzation upon reversing the external \feld prior to the full\nmagnetization switching at 3.4 T. Our simulations in-\ndicate that the gradual decrease is due to the magnet-\nically isotropic defects that reverse their magnetization\nearlier than the rest of the material, which is yet another\ncon\frmation that the Z phase does not exhibit signi\f-\ncant magnetocrystalline anisotropy. Note that the M(H)\ncurves are not identical, because the experiment was per-\nformed on a bulk sample, where we have gradual switch-\ning of parts of the material, while the theory considers\na single thin lamella. Furthermore, the lamella with a\nthickness/width aspect ratio of about 1/1000 has an ad-\nditional shape anisotropy, though much smaller than the\nmagnetocrystalline anisotropy Ku>>\u0016 0M2\ns=2.\nIn order to study the magnetization texture in cellu-\nlar Sm{Co magnets, we compare simulated domain-wall\nstructures with those observed in the experiment. Fig.\n4a shows an experimental Fresnel defocus image at 0.24mm overfocus of the region shown in Fig. 3 in the ther-\nmally demagnetized state, which contains magnetic do-\nmains separated by three domain walls with the charac-\nteristic zig-zag shape following the microstructure. In our\nsimulations, we initialized the system with three straight\nDWs and ran it for 1 ns to allow the domain walls to\nrelax into the state of minimal energy. The resulting re-\nlaxed magnetization texture is overlaid in Fig. 4b onto its\ncorresponding TEM image. The positions of the domain\nwalls in the experimental LTEM image (white intersected\nlines) and the simulated magnetization image match very\nclosely. In fact, in both cases the domain walls follow pre-\ncisely the microstructure. Additionally, small magnetic\ndomains with opposite magnetization approximately 5\nnm wide are present (white circles in Fig. 4), which form\ndue to strong pinning to SmCo 5. In order to directly\ncorrelate the micromagnetics with experimental obser-\nvation, a magnetic phase shift image has been calculated\nbased on the micromagnetic results (see Methods). From\nit, a Fresnel image and a magnetic induction map were\nreconstructed and shown in Figs. 4c and d, respectively.\nA close match between the theory and experiment is ap-\nparent, as the simulated images contain features, such\nas domain wall o\u000bsetting and curling, identical to those\nobserved in Fig. 2. Additionally, the micromagnetic sim-\nulation reveals that the curling is out-of plane (see Figure\n5).6\nFig. 4. Comparison between experimental observations and theoretical predictions of the magnetization in\nSm{Co. a LTEM image at 0.24 mm overfocus of the region shown in Fig. 1a reveals four magnetic domains separated by\nDWs. bMicromagnetic simulation of magnetization in the same microstructure superimposed on Fig. 4a, and ca Fresnel\ndefocus image and dmagnetic-induction map simulated based on Fig. 4b show a distinct resemblance of the DW network in\ntheory and experiment, namely the pinning at SmCo 5and o\u000bsetting by the Z phase. The white circles indicate one of the small\ndomains with opposite magnetization. eFresnel defocus image at 0.8 mm overfocus of the remanent state (not the same region\nas panel a). fSimulation of the remanent state of panel a. Consecutive domains of opposite magnetization are present at the\nSmCo 5cells. The corresponding gFresnel defocus image and hmagnetic induction map of the remanent state, once again\nillustrating very good agreement between experiment and theory, speci\fcally the branching DW network pinned at SmCo 5.\nThe phase di\u000berence between adjacent contours in the induction maps is 2 \u0019.\nFurthermore, we ramped up the magnetic \feld to sat-\nurate the sample in both the experiment and the simula-\ntion and then removed the \feld to observe the remanent\nstate. As we know from Fig. 3e, this corresponds to\nM= 0:95MS, meaning that one would expect the mag-\nnetization to be nearly uniform in the remanent state, but\nsurprisingly this is not the case. Figure 4e, which shows\nan experimental Fresnel image of the remanent state (af-\nter applying an external \feld of 6 T) at 0.8 mm overfocus,\nreveals a state with a complex network of domains sep-\narated by branching domain walls (note that the region\nshown in Fig. 4e is not the same as in Fig. 4a.). This\nstate represents initial nucleation stages of the magneti-\nzation process. Some domain walls have strong contrast\nwith multiple lines of three or more satellites, which are\nonly visible for \u0019domain walls that are perfectly edge-\non. Lower-degree domain walls usually form weak and\nfading satellites. In Fig. 4f, the simulated magnetization\nstate, again very similar to the experiment, contains a\nlarge number of small domains with opposite magnetiza-\ntion pinned to the SmCo 5cells. These are the smallest\npossible domains, around 5 nm wide, constrained by the\ndomain-wall width. We have again simulated a magneticphase image from the micromagnetic simulation, from\nwhich we extracted a Fresnel defocus image and a mag-\nnetic induction map, shown in Fig. 4g and h, respec-\ntively. The magnetic texture shows a good match with\nthe experiment, notably the complex DW network. The\nmagnetic induction map reveals the presence of vortex-\nlike out-of-plane curling in the remanent state.\nIn the following, we take a deeper look into our sim-\nulations beyond the experimental limitations. Figure 5\nshows the magnetization as contour plots overlaid onto\nthe microstructure. Note the prominent resemblance of\nFig. 5a with the Fresnel defocus images of Fig. 2, i.e. the\nDWs follow exactly the SmCo 5cell geometry, including\nthe o\u000bsetting by the Z phase platelets. Figure 5b shows a\nclose-up image of the magnetization texture around inter-\nsections between the three phases for the region marked\nwith a square in Fig. 5a. By \ftting a magnetization pro-\n\fle across the domain wall with tanh ( r=\u0015), whereris\nthe distance from the domain-wall center and \u0015=\u000edw=2,\nwe have deduced the DW width in the SmCo 5and the\nSm2Co17phases to be 1.5 nm and 4.7 nm, respectively.\nThese values are slightly larger than the theoretically ex-\npected values of 1.2 nm and 2.7 nm using the equation7\n\u000edw=p\nA=K u, but they agree with our experimental ob-\nservation of the domain-wall thickness of 4 \u00062 nm (see\nSuppl. Fig. 2). This con\frms our conclusion that the do-\nmain walls are mostly located in Sm 2Co17and pinned to\nthe the SmCo 5cell. The minimum domain size of 5 nm\nis also shown in Fig. 5b at the left edge of the SmCo 5\ncell that is intersected by a Z phase platelet. Notably,\nthe DWs inside the Z phase are extremely thin, and the\nmoments turn away from the c-axis due to the dominat-\ning shape anisotropy of the platelets. This indicates that\nthe DWs between the Sm{Co phases and the Z phase are\nin fact \u0019=2 DWs. To minimize the associated exchange\nenergy, the magnetic moments are twisted with oppo-\nsite handedness at the edges of the SmCo 5cell bound-\nary. This is further analyzed in Fig. 5c, which shows a\ndetailed view of a region where the three phases inter-\nsect and a domain wall propagates through all of them.\nWe observe a narrow domain wall in the SmCo 5cell, a\nbroader wall in the Sm 2Co17cell, and a curling of the\nmoments away from the c-axis inside the Z phase. The\ncurling has a signi\fcant out-of-plane component. The\ndomain wall is in fact injected into the hard phase at the\nlocation where the three phases meet. These results shed\nnew light on previous observations based on electron mi-croscopy [23], which suggested an out-of-plane tilting of\nthe magnetic \rux away from the easy axis around inter-\nsections.\nThe points in the microstructure where the three\nphases intersect play a critical role in the magnetization\nprocess because their edges with di\u000berent material prop-\nerties enable curling instabilities. We show in Fig. 5e-h\nthe process of magnetization reversal, i.e., coming from\na saturated state and applying an external \feld in the\nopposite direction. The demagnetization starts at the\nintersections between the Z phase and the soft Sm 2Co17\nmatrix in the form of nucleating domains that gradually\ngrow inside the Sm 2Co17matrix, and become pinned by\nthe hard SmCo 5cells. The domain growth then pro-\ngresses through the intersections where all three phases\nmeet and crosses the SmCo 5cells through these points.\nThis further indicates that the Z phase does indeed play\na vital role in the magnetization process, emerging as an\ninterplay between curling instabilities at intersections be-\ntween the Z phase and the Sm 2Co17matrix, and pinning\nat the hard cells. Importantly, this demagnetization pro-\ncess might also be responsible for the formation of DWs\nwith higher winding angles, such as those observed in\nFig. 2.\nFig. 5. Simulations of 3D DW structures and their nucleation in Sm{Co magnets. a Structure with four domains in a\ndemagnetized state showing that DWs are pinned by the microstructural features, speci\fcally the SmCo 5cells, and are o\u000bset by\nthe Z phase. bClose-up of the marked square area in areveals that the domain walls nucleate at the phase boundary between\nSm2Co17and SmCo 5and are mostly situated inside Sm 2Co17.cDetailed view of the magnetization texture at the intersection\nof all three phases, showing how a DW can be injected into the hard phase through a Z-phase platelet, and consequently how\nits thickness varies between the two phases. e{h Illustration of the magnetization reversal process at a slightly supercritical\n\feld, i.e., larger than the switching \feld, parallel to the c-axis: as time progresses domains with magnetization parallel to the\n\feld begin nucleating at the Z phase and spread into the Sm 2Co17matrix, but their growth is impeded by the SmCo 5cells\n(the time step between each \fgure is 0.1 ns).8\nIII. CONCLUSIONS\nWe have shown via magnetic imaging in TEM, APT\nand micromagnetic simulations that there are sharp mag-\nnetic DWs in cellular Sm{Co magnets that follow ex-\nactly the morphology of the hard-phase cells and are o\u000b-\nset by the Z-phase platelets. Based on our \fnding that\nthe domain walls are mostly located in the soft Sm 2Co17\nphase, reminiscent of an exchange spring system [34], we\npropose that the thickness of SmCo 5cells should be re-\nduced to a minimum, considering that the domain walls\nin SmCo 5are only about 1.5 nm wide. Importantly, we\nhave con\frmed that the Z phase plays a crucial role in\nthe magnetization process because curling instabilities\nat the intersections between the soft and Z phase act\nas nucleation sites for DWs upon switching the external\nmagnetic \feld. The domain walls propagate inside the\nsoft Sm 2Co17matrix and become pinned at the SmCo 5\nhard cells. In that case they can only propagate fur-\nther through the intersections between the cells and the\nZ phase. Hence, we propose that both the Zr and Sm\ncontents should be reduced in this system. This would\nresult in \fner Zr platelets, which would impede magne-\ntization switching, and the reduced Sm would lead to\nsmaller SmCo 5cells, which would in turn increase rema-\nnence, leading to a much more powerful magnet. Addi-\ntionally, we have observed topologically non-trivial do-\nmains with highly-complex DWs, and, where all three\nphases meet, out-of-plane curling of domain walls. These\nexotic magnetic structures need to be studied further in\norder to understand the physics of multi-phase magnets\nin detail and to allow fully harnessing the potential of\nthese high-performance materials.\nIV. METHODS\nA. Sample synthesis:\nFor the synthesis of the Sm{Co magnet, the alloying\nelements were melted in an induction furnace under ar-\ngon and the resulting alloy was cast in a metallic mold.\nAfter crushing the alloy with a hammer mill, the result-\ning powder was milled in a jet mill with a particle size\nof 4 { 8 \u0016m. The powder was then \flled into a rubber\nmold, aligned with magnetic pulses of \feld strength 5 T\nand pressed in an isostatic press with 3000 MPa. The\ngreen parts were sintered under vacuum at a tempera-\nture of 1200{1220\u000eC, solution annealed at 1170{1200\n\u000eC, and then quenched with an inert gas to room tem-\nperature. Subsequently, the parts were tempered at 850\n\u000eC, slowly cooled to 400\u000eC, and then quenched to room\ntemperature.\nThe material was produced by Arnold Magnetic\nTechnologies, and has an overall chemical composition\nSm(Co 0:695,Fe0:213,Cu0:07,Zr0:022)7:57with a minor addi-\ntional oxygen content in the form of Sm 2O3.B. Magnetometry:\nThe magnetization of a small sample piece along the\neasy axis was measured as a function of external \feld at\nroom temperature using a Superconducting Quantum In-\nterference Device (SQUID) in a Magnetic Property Mea-\nsurement System (MPMS3) by Quantum Design.\nC. Transmission Electron Microscopy:\nElectron-transparent specimens for TEM studies were\nprepared using Ga+ sputtering and a conventional lift-\nout method in Helios 600i dual-beam focused ion beam\n(FIB) scanning electron microscope (SEM) workstation.\nThe ion-beam induced damage on the surfaces was re-\nduced by low-energy ( <1 keV) Ar+ milling using a\nFischione Nanomill system. The thickness of the lamel-\nlae was measured on an FEI Tecnai F30 FEG transmis-\nsion electron microscope using an electron energy loss\nspectroscopy (EELS) log-ratio technique. A uniformly\nvarying range of thicknesses between 80 - 140 nm was\nachieved.\nThe Sm{Co specimens were studied at remanence in\nmagnetic-\feld-free (Lorentz mode) conditions using a\nspherical aberration-corrected FEI Titan microscope op-\nerated at 300 keV. In Fresnel-mode LTEM images, the\nintensity distribution at defocus \u0001 zis recorded to reveal\na bright (convergent) or dark (divergent) contrast at the\npositions of the magnetic domain walls. The net de\rec-\ntion of electrons from the magnetic domains is induced\nby the Lorentz force, F=\u0000ev\u0002B, whereeis the elec-\ntron charge, vis the velocity of the incident electrons\nandBis the in-plane magnetic induction in the sample.\nA conventional microscope objective lens was used to ap-\nply magnetic \felds on the specimen. TEM images were\nrecorded using a direct-electron counting Gatan K2-IS\ncamera and Gatan Microscopy Suite software. Electron\nholograms were recorded in Lorentz mode using a biprism\npositioned in one of the conjugated image planes of the\nelectron column. The biprism voltage used was typically\nin the range of 90-100 V that forms a fringe spacing of 3\nnm with a contrast of 75%.\nD. Micromagnetic simulations:\nHigh-resolution micromagnetic simulations were per-\nformed to investigate the link between the microstruc-\nture and the domain-wall network in Sm{Co magnets.\nThe total energy density of the system consists of: (i)\nferromagnetic exchange; (ii) uniaxial magnetocrystalline\nanisotropy; (iii) Zeeman coupling to an external magnetic\n\feld; and (iv) dipole-dipole interactions:9\nF=X\ni[Ai(r\u0001mi)2\u0000Ki\nu(mi\nc)2+\u00160Mi\nSmi\u0001Hext\n\u0000\u00160Mi\ns\n2mi\u0001Hi\ndip\u0000\u00160Mi\nSmi\u0001Hexc];(1)\nwhere mi=Mi=Mi\nSis the magnetization unit vector\nfor phaseiwithMi\nSthe saturation magnetization, Ai\nis the exchange sti\u000bness, Ki\nuis the \frst-order uniaxial\nanisotropy constant, Hextis the external magnetic \feld,\nHdipis the local demagnetizing \feld due to dipole-dipole\ninteractions, and Hexcis the exchange \feld at the inter-\nfaces between the di\u000berent phases. The c-component of\nthe magnetization is inside the lamella plane.\nThe material parameters were taken from the literature\n[6, 25, 28], and are Ms= 1:05 T,A= 23:6 pJ/m, and\nKu= 17:2 MJ/m3for SmCo 5;Ms= 1:25 T,A= 24:7\npJ/m, andKu= 3:3 MJ/m3for Sm 2Co17; andMs= 0:37\nT,A= 11 pJ/m, and Ku= 0 MJ/m3for the Z phase.\nNote that Cu in the SmCo 5changes the material param-\neters by lowering Ms,AandKu, but the composition\nof Cu in our material is less than 10% of the 3d metal\ncontent and the material parameters are not strongly re-\nduced [24]. We also performed tests with the material\nparameters of Sm(Co 0:9Cu0:1)5and found no qualitative\ndi\u000berence in the behavior of the system. The simula-\ntions were tested for the lamella thickness between 50\nand 100 nm, and qualitatively the same results have been\nobtained.\nThe exchange \feld between two phases iandjis pro-\nportional toAi\nMisAj\nMj\ns=\u0010\nAi\nMis+Aj\nMj\ns\u0011\n. Based on our opti-\nmization, described in the main text, we found the follow-\ning exchange values: (i) hard to soft: 16 pJ/m; (ii) soft\nto hard: 13 pJ/m; (iii) soft to Z phase: 4.4 pJ/m; (iv)\nZ-phase to soft: 1.3 pJ/m; (v) hard to Z phase: 2.7pJ/m;\nand (vi) Z phase to hard: 0.95 pJ/m.\nUsing Equation 1, we solved the Landau-Lifshitz-\nGilbert (LLG) equation of motion\n@tm=\u0000\r(m\u0002He\u000b) +\u000b(m\u0002@tm); (2)\nwhere\ris the electron gyromagnetic ratio, \u000bis\nthe dimensionless damping parameter, and He\u000b=\n\u0000@mF=\u0016 0MSis the e\u000bective magnetic \feld in the ma-\nterial consisting of external and internal magnetic \felds,\nwhich depend on the material parameters. The simu-\nlations have been done with mumax3 [35], and the vi-\nsualization of the magnetization textures was done with\nParaview [36].\nE. Atom probe tomography:\nThe needle-shaped geometry required for APT analy-\nsis was prepared by applying standard lift-out practicesusing an FEI Helios Focused Ion Beam 600i worksta-\ntion, and mounting it to a \rat-top microtip coupon sup-\nplied by Cameca. Sequential annular milling was applied\nto achieve an apex of <70 nm diameter, including low-\nkV cleaning, resulting in <0.01 at.% Ga in the top 10\nnm of the specimen. Data collection was performed us-\ning a LEAP4000X-HR instrument applying 100 pJ laser\npulse energy with 200 kHz repetition rate and a speci-\nmen temperature of 54 K (resulting in a Co charge-state\nratio (Co++/Co+) between 5 and 10). With these pa-\nrameters and a chamber vacuum level at 10-9Pa, data\nwere collected between 5 kV and 9.5 kV with a back-\nground level consistently below 20 ppm/ns. The atom-\nmap reconstruction was validated by considering that the\nZ-platelets are atomically \rat, and spatial distribution\nmaps were performed along the c-axis (normal to the\nplatelets) to measure the lattice spacings and thereby\nvalidate the accuracy of the atom-map reconstruction di-\nmensions.\nF. Magnetic phase image and LTEM simulations:\nThe electromagnetic phase shift induced in an electron\nwave by passing through a sample is described by the\nAharonov-Bohm e\u000bect and can be expressed as [37]:\n'(x;y) ='el(x;y) +'mag(x;y) (3)\n=CelZ\nV(r)dz\u0000\u0019\n\b0Z\nAz(r)dz; (4)\nwith'el(x;y) and'mag(x;y) denoting the electrostatic\nand magnetic contributions to the phase shift, the inter-\naction constant Cel=\rmele\u0015\n~2, the magnetic \rux quantum\n\b0=\u0019~=e, the Lorentz factor \r, the electron rest mass\nmeland the electron wavelength \u0015. Furthermore, Az(r)\nwithr= (x;y;z ) is thezcomponent of the magnetic vec-\ntor potential A(r), wherezcorresponds to the incident\nelectron beam direction [38, 39].\nThe magnetization M(r) in the sample is linked to the\nvector potential by the vector convolution integral [40]\nA(r) =\u00160\n4\u0019Z\nM(r0)\u0002r\u0000r0\njr\u0000r0j3dr0; (5)\nwhere\u00160is the vacuum permeability is the vacuum per-\nmeability. Using both equations, the magnetic phase\nshift can be expressed in terms of the magnetization as\n'mag(x;y) =\u0000\u00160\n2\b0Z(y\u0000y0)Mx(r0)\u0000(x\u0000x0)My(r0)\n(x\u0000x0)2+ (y\u0000y0)2dr0:\n(6)\nBy discretizing this equation and utilizing known analyt-\nical solutions for the magnetic phase of simple magne-\ntized geometries, magnetic phase images 'mag(x;y) can\nbe calculated for arbitrary magnetization distributions\nM(r) [41].10\n\"Contour maps\" are used for the visualization of the\nmagnetic phase in the form of magnetic induction. They\nare generated in Figs. 2 and 4 by taking the cosine of the\nmagnetic phase 'mag, which can be ampli\fed beforehand\nto increase the number of fringes for visualization pur-\nposes. A color scheme is superimposed on the magnetic\ninduction maps, which is determined by the gradient of\n'mag. The latter is an indicator of the direction of the\nprojected in-plane magnetic induction and is indicated in\nFigs. 2e and 4d as a color wheel. The phase di\u000berence\nbetween two neighboring contours is 2 \u0019.\nThe magnetic phase 'magcan further be utilized to\nsimulate LTEM images by convolving the corresponding\nwave function \t ( x;y) =ei'mag(x;y)with a phase plate:\n\tLTEM (x;y) =F\u00001\n2n\nF2n\nei'mag(x;y)o\n\u0001e\u0000i\u001f(qx;qy)o\n;\n(7)\nwithF2f:::gdenoting the 2D Fourier transform,\nF\u00001\n2f:::gits inverse and \u001f(qx;qy) denoting an aberra-\ntion function [42] in the di\u000braction space containing the\ndefocusC1(with positive C1referring to overfocus) given\nby:\n\u001f(qx;qy) =\u0019\u0015C 1\u0000\nq2\nx+q2\ny\u0001\n: (8)\nThe LTEM images are then calculated from the corre-\nsponding electron wave by:\nILTEM (x;y) = \t LTEM (x;y)\u0001\t\u0003\nLTEM (x;y):(9)\nV. SUPPLEMENTARY INFORMATION\nSupplementary Figure 1 shows for the sample of Fig.\n1 in the main text a di\u000braction pattern that contains the\nre\rections from [100] and [110] directions. The experi-\nmental and simulated patterns reveal an excellent agree-\nment.\nThe width of the domain-wall contrast of 4 \u00062 nm\nat zero defocus was extrapolated from a series of LTEM\nimages recorded at di\u000berent defocus values, as shown in\nSupplementary Fig. 2. It is worth mentioning that this\napproach usually gives a slight overestimate of the true\ndomain-wall width.\nAn atom-probe tomography (APT) reconstruction is\nshown in the Supplementary Video 1. Isoconcentration\nsurfaces of 9.83 at. % Zr indicate \rat Z-phase platelets,\nwhile the isoconcentration surfaces of 14.51 at. % Sm\nreveal a twisted SmCo 5cell. Note the accumulation of\nCu inside the cells (pink).\nSupplementary Fig. 1. Crystal structure of Sm{Co: a\nUnit cell of the Sm 2Co17phase showing the atomic sites for\nSm (black) 2 band 2 c, and the 4 di\u000berent atomic sites for Co:\n(blue) 12 j, (red) 12 k, (green) 6 g, and (yellow) 4 f.bshows\nthe recorded di\u000braction pattern from one Sm 2Co17cell and\ncshows the calculated pattern overlaid on the experimental\npattern, illustrating an excellent agreement. The calculated\npattern is a convolution of d[100] and e[110] re\rections,\nand the corresponding crystal structure viewed along these\ndirections is shown in fandg, respectively.\nSupplementary Fig. 2. Measurements of domain-wall\nwidth. a-b The thickness of two di\u000berent domain walls was\nobtained from a focal-series reconstruction. The apparent\nwidth of divergent domain walls (appearing as dark contrast)\nwas measured for di\u000berent values of defocus (data points).\nThe real width of the domain walls was deduced from the in-\ntersection of the linear \ft to the data points with the y-axis.\n[1] Hadjipanayis, G. C. et al. High temperature 2:17 mag-\nnets: Relationship of magnetic properties to microstruc-\nture and processing. IEEE Trans. Magn. 36, 3382 (2000).\n[2] Gut\reisch, O. et al. Magnetic materials and devices forthe 21st century: Stronger, lighter, and more energy ef-\n\fcient. Adv. Mat. 23, 821 (2011).\n[3] McCallum, R. W., Lewis, L. H., Skomski, R., Kramer,\nM. J. & Anderson, I. E. Practical aspects of modern and11\nfuture permanent magnets. Annu. Rev. Mater. Res. 44,\n451 (2014).\n[4] Horiuchi, Y. et al. E\u000bects of solution treated temperature\non the structural and magnetic properties of iron-rich\nSm(Co,Fe,Cu,Zr) zsintered magnet. IEEE Transactions\non Magnetics 49, 3221{3224 (2013).\n[5] Liu, J. F., Zhang, Y., Dimitrov, D. & Hadjipanayis, G. C.\nMicrostructure and high temperature magnetic proper-\nties in Sm(Co,Cu,Fe,Zr) z(z= 6:7\u00009:1) permanent mag-\nnets. J. Appl. Phys. 85, 2800 (1999).\n[6] Duerrschnabel, M. et al. Atomic structure and domain\nwall pinning in samarium-cobalt-based permanent mag-\nnets. Nat. Comm. 8, 54 (2017).\n[7] Kronm uller, H., Fischer, R., Seeger, M. & Zern, A. Mi-\ncromagnetism and microstructure of hard magnetic ma-\nterials. J. Phys. D: Appl. Phys. 29, 2274 (1996).\n[8] Skomski, R. et al. Predicting the future of permanent-\nmagnet materials. IEEE Transactions on Magnetics 49,\n3215{3220 (2013).\n[9] Skomski, R. Domain-wall curvature and coercivity in\npinning type Sm{Co magnets. J. Appl. Phys. 81, 5627\n(1997).\n[10] Fidler, J. & Schre\r, T. Micromagnetic modelling { the\ncurrent state of the art. J. Phys. D: Appl. Phys. 33, R135\n(2000).\n[11] Streibl, B., Fidler, J. & Schre\r, T. Domain wall pinning\nin high temperature Sm(Co,Fe,Cu,Zr) 7\u00008magnets. J.\nAppl. Phys. 87, 4765 (2000).\n[12] Fidler, J. et al. Micromagnetic modelling and magnetiza-\ntion processes. J. Magn. Magn. Mater. 272-276 , 641{646\n(2004).\n[13] Gaunt, P. Domain wall pinning as a source of magnetic\nhardening in alloys. J. Appl. Phys. 43, 637{638 (1972).\n[14] Livingston, J. D. & Martin, D. L. Microstructure of aged\n(Co,Cu,Fe) 7Sm magnets. J. Appl. Phys. 48, 1350 (1977).\n[15] Nagel, H. Coercivity and microstructure of\nSm(Co 0:87Cu0:13)7:8.J. Appl. Phys. 50, 1026{1030\n(1979).\n[16] Hadjipanayis, G. C., Yadlowsky, E. J. & Wollins,\nS. H. A study of magnetic hardening in\nSm(Co 0:69Fe0:22Cu0:07Zr0:02)7:22. J. Appl. Phys.\n53, 2386 (1982).\n[17] Fidler, J. Coercivity of precipitation hardened cobalt rare\nearth 17:2 permanent magnets. J. Magn. Magn. Mater.\n30, 58 (1982).\n[18] Wong, B. Y., Willard, M. & Laughlin, D. E. Domain\nwall pinning sites in Sm(CoFeCuZr) xmagnets. J. Magn.\nMagn. Mater. 169, 178 (1997).\n[19] Zhang, C. et al. The evolution of phase constitution\nand microstructure in iron-rich 2:17-type Sm-Co mag-\nnets with high magnetic performance. Scienti\fc Reports\n8(2018).\n[20] Hadjipanayis, G. C., Hazelton, R. C., Lawless, K. R. &\nHorton, L. S. Magnetic domains in rare-earth cobalt per-\nmanent magnets. IEEE Trans. Magn. 18, 1460 (1982).\n[21] Zhang, Y. et al. Magnetic hardening studies in sintered\nSm(Co,Cu x,Fe,Zr) z2:17 high temperature magnets. J.\nAppl. Phys. 87, 6722 (2000).\n[22] Gut\reisch, O. et al. Evolution of magnetic domain struc-\ntures and coercivity in high-performance SmCo 2:17-\ntype permanent magnets. Acta Materialia 54, 997{1008\n(2006).\n[23] Okabe, F. et al. Microstructures and magnetic domain\nstructures of sintered Sm(Co 0:720Fe0:200Cu0:055Zr0:025)7:5permanent magnet studied by transmission electron mi-\ncroscopy. Mat. Trans. 47, 218 (2006).\n[24] Sepehri-Amin, H. et al. Correlation of microchemistry\nof cell boundary phase and interface structure to the co-\nercivity of Sm(Co 0:784Fe0:100Cu0:088Zr0:028)7:19sintered\nmagnets. Acta Materialia 126, 1{10 (2017).\n[25] Katter, M., Weber, J., Assmus, W., Schrey, P. & Rode-\nwald, W. A new model for the coercivity mechanism of\nSm2(Co,Fe,Cu,Zr) 17magnets. IEEE Trans. Magn. 32,\n4815 (1996).\n[26] Fullerton, E. E., Jiang, J. S. & Bader, S. D. Hard/soft\nmagnetic heterostructures: model exchange-spring mag-\nnets. J. Magn. Magn. Mater. 200, 392-404 (1999).\n[27] Tian, Y. et al. In Situ Observation of Domain Wall\nPinning in Sm(Co,Fe,Cu,Zr) zMagnet by Lorentz Mi-\ncroscopy. IEEE Trans. Magn. 51, 11 (2015).\n[28] Nagel, H. Hard magnetic materials: rare earth-transition\nmetal and Fe-Nd-B , vol. 2A1 of Landolt-B ornstein -\nGroup VIII Advanced Materials and Technologies , chap.\nPowder Metallurgy Data (SpringerMaterials, 2003).\n[29] Hellman, F. et al. Interface-induced phenomena in mag-\nnetism. Reviews of Modern Physics 89(2017).\n[30] Charilaou, M. et al. Magnetic properties of ultrathin dis-\ncontinuous Co/Pt multilayers: Comparison with short-\nrange ordered and isotropic CoPt 3\flms. Phys. Rev. B\n93(2016).\n[31] Shen, Y., Leontsev, S., Sheets, A. O., Horwath, J. C.\n& Turgut, Z. E\u000bect of \rake thickness on coercivity of\nnanocrystalline SmCo 5bulk prepared from anisotropic\nnano\rake powder. AIP Advances 6, 056005 (2016).\n[32] Nishida, Y., Endo, M. & Sakurada, S. A modeling study\nof domain wall pinning in magnets. Journal of Magnetism\nand Magnetic Materials 324, 1948{1953 (2012).\n[33] Kronm uller, H. & Goll, D. Micromagnetic analy-\nsis of pinning-hardened nanostructured, nanocrystalline\nSm2Co17based alloys. Scripta Materialia 47, 545{550\n(2002).\n[34] Kneller, E. F. & Hawig, R. The exchange-spring magnet:\na new material principle for permanent magnets. IEEE\nTrans. Magn. 27, 3588 (1991).\n[35] Vansteenkiste, A. et al. The design and veri\fcation of\nmumax3. AIP Advances 4, 107133 (2014).\n[36] Ahrens, J., Geveci, B. & Law, C. Visualization Hand-\nbook, chap. ParaView: An End-User Tool for Large Data\nVisualization (Elsevier, 2005).\n[37] Dunin-Borkowski, R. E. et al. O\u000b-axis electron hologra-\nphy of magnetic nanowires and chains, rings, and planar\narrays of magnetic nanoparticles. Microscopy Research\nand Technique , 390{402 (2004).\n[38] Y. Aharonov and D. Bohm. Signi\fcance of electromag-\nnetic potentials in the quantum theory. Physical Review ,\n485{491 (1959).\n[39] W. Ehrenberg and R. E. Siday. The refractive index in\nelectron optics and the principles of dynamics. Proceed-\nings of the Physical Society. Section B , 8 (1949).\n[40] M. Mansuripur. Computation of electron di\u000braction pat-\nterns in lorentz electron microscopy of thin magnetic\n\flms. Journal of Applied Physics , 2455 (1991).\n[41] Jan Caron. Model-based reconstruction of magnetisa-\ntion distributions in nanostructures from electron optical\nphase images, Schriften des Forschungszentrums J ulich.\nReihe Schl usseltechnologien/Key technologies 17 . (2018).\n[42] J. Barthel. Dr. probe: A software for high-resolution\nstem image simulation. Ultramicroscopy , 1{11 (2018).12\nACKNOWLEDGMENTS\nLP, MC, RES and JFL gratefully acknowledge funding\nfrom the Swiss National Science Foundation (Grant No.\n200021{172934). We also thank L. Grafulha Morales and\nA.-G. Bitterman for the preparation of TEM samples,\nand ScopeM ETH Zurich for the use of its facilities.CORRESPONDENCE\nCorrespondence and requests for mate-\nrials should be addressed to LP (email:\nleonardo.pierobon@mat.ethz.ch), JFL (email:\njoerg.loe\u000fer@mat.ethz.ch) and MC (email:\nmichalis.charilaou@louisiana.edu)."    },    {        "title": "1901.03072v2.Ultrafast_magnetization_dynamics_in_uniaxial_ferrimagnets_with_compensation_point__GdFeCo.pdf",        "content": "Ultrafast magnetization dynamics in uniaxial ferrimagnets with compensation point.\nGdFeCo\nM. D. Davydova,1, 2,\u0003K. A. Zvezdin,1, 2A. V. Kimel,3, 4and A. K. Zvezdin1, 2,y\n1Prokhorov General Physics Institute of the Russian Academy of Sciences, 119991 Moscow, Russia\n2Moscow Institute of Physics and Technology (State University), 141700 Dolgoprudny, Russia\n3Moscow Technological University (MIREA), 119454 Moscow, Russia\n4Institute for Molecules and Materials, Radboud University, Nijmegen 6525 AJ, The Netherlands\nWe derive an e\u000bective Lagrangian in the quasi-antiferromagnetic approximation that allows to\ndescribe the magnetization dynamics for uniaxial f-d(rare-earth - transition metal) ferrimagnet\nnear the magnetization compensation point in the presence of external magnetic \feld. We perform\ncalculations for the parameters of GdFeCo, a metallic ferrimagnet with compensation point that is\none of the most promising materials in ultrafast magnetism. Using the developed approach, we \fnd\nthe torque that acts on the magnetization due to ultrafast demagnetization pulse that can be caused\neither by ultrashort laser or electrical current pulse. We show that the torque is non-zero only in\nthe non-collinear magnetic phase that can be acquired by applying external magnetic \feld to the\nmaterial. The coherent response of magnetization dynamics amplitude and its timescale exhibits\ncritical behavior near certain values of the magnetic \feld corresponding to a spin-\rop like phase\ntransition. Understanding the underlying mechanisms for these e\u000bects opens the way to e\u000ecient\ncontrol of the amplitude and the timescales of the spin dynamics, which is one of the central problems\nin the \feld of ultrafast magnetism.\nPACS numbers: 79.20.Ds, 75.50.Gg, 75.30.Kz, 75.78.-n\nINTRODUCTION\nMost of the prominent advances in the \feld of ultrafast magnetism have been achieved by using thermal mechanism\nof magnetization control [1{7]. These studies rooted from the pioneering work by Beaurepaire et al. [8] on ultrafast\nlaser-induced demagnetization of Ni. In this experiment, partial destruction of magnetic order was found at much\nfaster rates that were believed to be possible prior to that publication. Since then, the \feld of ultrafast magnetism has\nbeen rapidly growing and the possible channels of ultrafast angular momentum transfer have been studied extensively\n[9]. Ultrafast demagnetization can be achieved by applying ultrashort laser pulses [1{7], or, alternatively, by using\nshort pulses of electric currents [10, 11].\nIn the last decades, GdFeCo and other rare-earth - transition metal compounds (RE-TM) have been in the center\nof attention in this regard [12]. For example, all-optical switching has been demonstrated for the \frst time in GdFeCo\n[13]. It was found that the switching is possible due to di\u000berent rates of sublattice demagnetization, which enables\nultrafast magnetization reversal to occur because of the angular momentum conservation [1, 2].\nIn many RE-TM compounds, GdFeCo and TbFeCo being part of them, realization of the magnetization compen-\nsation point is possible. At this point, the magnetizations of the two antiferromagnetically coupled sublattices with\ndi\u000berent dependencies on temperature become equal and the total magnetization of the material turns to zero. In\nthe presence of the external magnetic \feld, a record-breaking fast subpicosecond magnetization switching was found\nin GdFeCo across the compensation point [13]. In addition, a number of anomalies in the magnetic response was\nobserved near this point [14{16], which has never been explained theoretically. All said above illustrates the impor-\ntance of understanding the role of the compensation point in the dynamics and working out an appropriate tool for\nits description.\nE\u000ecient control of the amplitude and the timescales of the response of the magnetic system to an ultrafast de-\nmagnetizing impact on a medium is one of the most important issues in the area of ultrafast magnetism nowadays\n[9, 14, 15]. Understanding of the mechanisms and of the exhaustive description of the subsequent spin dynamics is\nalso a long-standing goal that will help to promote the achievements of this area towards practical applications in\nmagnetic recording [3, 17], magnonics [18] and spintronics [19]. In this work, we expand the understanding of response\nof magnetic system of a uniaxial f-dferrimagnet near the compensation point in the external magnetic \feld to an\nultrafast demagnetizing pulse, which can be induced either by a femtosecond laser or an electric current pulse. We\npresent a theoretical model and calculations, which allow to describe the ultrafast response of the system that resides\nin an angular phase before the impact. We show that in this magnetic phase the coherent precessional response is\npossible and the subsequent magnetization dynamics may become greatly nonlinear and is governed by large inter-\nsublattice exchange \feld [20]. We derive the e\u000bective Lagrangian that governs the dynamics of the system near thearXiv:1901.03072v2  [cond-mat.mtrl-sci]  28 Jan 20192\ncompensation point and obtain the torque acting on the magnetizations of the two sublattices due to demagnetiza-\ntion. In ref. [15], the critical response of the amplitude and the time of the signal rise have been found in GdFeCo\nin external magnetic \feld along the easy axis. At given laser pump \ruences, the response was found to be negligible\nin collinear phase, but it was dramatically large in angular one. We elaborate on this example and show that the\ncritical behavior of the response is the consequence of the second-order magnetic phase transition from collinear to an\nangular in the external magnetic \feld. We \fnd that these e\u000bects are pronounced in the vicinity of the compensation\npoint, where the phase transitions cross each other[21{23]. Thus, the proposed model explains a range of important\nexperimental observations as well as allows for developments of methods and tools of magnetization control by setting\nthe temperature near the compensation point and applying magnetic \feld. Moreover, by changing the composition\nof the alloy, the [24], the position of the magnetization compensation point can be tuned arbitrary close to the room\ntemperature. Our results might open new ways for technologies for ultrafast optical magnetic memory.\nEFFECTIVE LAGRANGIAN AND RAYLEIGH DISSPATION FUNCTION\nOur approach is based on Landau-Lifshitz-Gilbert equations for a two-sublattice (RE-TM) ferrimagnet. These\nequations are equivalent to the following e\u000bective Lagrangian and Rayleigh dissipation functions:\nL=Mf\n\r(1\u0000cos\u0012f)@'f\n@t+Md\n\r(1\u0000cos\u0012d)@'d\n@t\u0000\b(Mf;Md;H); (1)\nR=Rf+Rd;Rf;d=\u000bMf;d\n2\r\u0010\n_\u00122\nf;d+ sin2\u0012f;d_'2\nf;d\u0011\n(2)\nwhere\ris the gyromagnetic ratio, MdandMfare the magnetizations, \u0012d(TM) and\u0012f(RE) are the polar, 'dand\n'fare the azimuthal angles of d- andf- sublattices correspondingly in the spherical system of coordinates with z-axis\naligned along the external magnetic \feld H. \b(Mf;Md;H) is the thermodynamic potential for the system that we\ntake in the following form:\n\b =\u0000MdH+\u0015MdMf\u0000MfH\u0000Kf(Mfn)2\nM2\nf\u0000Kd(Mdn)2\nM2\nd; (3)\nwhere\u0015is the intersublattice exchange constant, nis the direction of the easy axis and Kf;dare the anisotropy\nconstants for f- andd- sublattices, respectively.\nNext, we transfer to description in terms of the antiferromagnetic L=MR\u0000Mdand the total magnetization\nM=MR+Mdvectors. In the vicinity of the compensation point the di\u000berence between the sublattice magnetizations\njMR\u0000Mdj\u001cLis small. The two vectors are parametrized using the sets of angles \u0012;\"and';\f, which are de\fned\nas:\n\u0012f=\u0012\u0000\"; \u0012d=\u0019\u0000\u0012\u0000\";\n'f='+\f; 'd=\u0019+'\u0000\f:(4)\nIn this case the antiferromagnetic vector is naturally de\fned as L= (Lsin\u0012cos';Lsin\u0012sin';Lcos\u0012).\nFollowing the work [25] we use the quasi-antiferromagnetic approximation to describe the dynamics near the mag-\nnetization compensation point. F In this approximation the canting angles are small \"\u001c1,\f\u001c1, and we can\nexpand the Lagrangian (1) and the corresponding thermodynamic potential up to quadratic terms in small variables:\nL=\u0000m\n\r_'cos\u0012\u0000M\n\rsin\u0012\u0010\n_'\"\u0000\f_\u0012\u0011\n\u0000\b;\n\b =\u0000K(l;n)2\u0000Hmcos\u0012\u0000\"MH sin\u0012+\u000e\n2\u0000\n\"2+ sin2\u0012\f2\u0001\n:(5)\nHerem=MR\u0000Md,M=MR+Md,K=KR+Kdis the e\u000bective uniaxial anisotropy constant, l=L=Lis\nthe unit antiferromagnetic vector \u000e= 2\u0015MdMRand we assume the anisotropy to be weak K\u001c\u0015M. For GdFeCo\nwith 24% Gd and compensation point near 283 K, we assume the following values of parameters: M\u0019800 emu/cc,\nK= 1:5\u0002105erg/cc,\u0015= 18:5 T/\u0016B,\u000e\u0019109erg/cc and mchanges in the range between 150 emu/cc and \u000050\nemu/cc at \felds H\u0019H\u0003\u00194 T. The characteristic values of small angles \u000fand\fare of the order of 10\u00002.3\nNext, we exclude the variables \"and\fby solving the Euler-Lagrange equations. Substituting them into the\nLagrangian (5), we obtain the e\u000bective Lagrangian, which describes the dynamics of a uniaxial ferrimagnet in the\nvicinity of the compensation point:\nLeff=\u001f?\n2 _\u0012\n\r!2\n+mcos\u0012\u0012\nH\u0000_'\n\r\u0013\n+\u001f?\n2sin2\u0012\u0012\nH\u0000_'\n\r\u00132\n+K(l;n)2; (6)\n\beff(H) =\u0000mHcos\u0012\u0000\u001f?\n2H2sin2\u0012\u0000K(l;n)2; (7)\nReff=\u000bM\n2\r\u0010\n_\u00122+ sin2\u0012_'2\u0011\n(8)\nwhere\u001f=2M2\n\u000e. In GdFeCo \u001f\u00191:6\u000210\u00003and\u000b\u00190:05. In the derivation above we assumed the gyrotropic factor\n\rand Gilbert damping constant \u000bto be the equal for both sublattices. Taking into account the di\u000berence between\nthese values for di\u000berent sublattices will lead to the angular momentum compensation e\u000bect at certain temperature.\nThe Lagrangian, Rayleigh function and equations of motion preserve the same form in this case if we substitute the\nparameters \rand\u000bwith temperature-dependent factors e\rande\u000bde\fned as:\n1\ne\r=1\n\u0016\r\u0012\n1 +M\nm\rf\u0000\rd\n\rf+\rd\u0013\n=Md\n\rd\u0000Mf\n\rf\n(Md\u0000Mf);1\n\u0016\r=1\n2\u00121\n\rd+1\n\rf\u0013\n;e\u000b=(\u000bd\rf+\u000bf\rd)\n(\rf+\rd)1\n1 +M\nm\rf\u0000\rd\n\rf+\rd(9)\nThis allows to reproduce the angular moment compensation phenomenon, which was studied experimentally in ref.\n[14].\nEXCITATION OF THE SPIN DYNAMICS\nThe proposed approach presents a powerful tool allowing analyzing coherent magnetization dynamics in ferrimagnets\nthat occurs under a broad range of conditions. Let us consider the following example that poses an important problem\nin the \feld of ultrafast magnetism. An femtosecond laser pulse strikes the uniaxial ferrimagnet (for instance, of\nGdFeCo, TbFeCo type) in the presence of external static magnetic \feld. The impact of the laser pulse leads to the\ndemagnetization of one or both of the sublattices. What coherent magnetization dynamics will occur as a consequence\nof this impact? The proposed model can be further developed in order to answer to this question and is applicable\nfor small values of demagnetization \u000eM.\nIn our framework the spin dynamics in ferrimagnet is described by Euler-Lagrange equations of the formd\ndt@L\n@_q\u0000@L\n@q=\n\u0000@R\n@_q, whereq=\u0012; ' are the polar and azimuthal angles describing the orientation of the antiferromagnetic vector\nL, correspondingly. Let us consider a particular case when the easy magnetization axis is aligned with the external\nmagnetic \feld, which leads to the presence of azimuthal symmetry in the system. In this case n= (0;0;1). In this\nparticular case the Euler-Lagrange equations can be rewritten as:\n\u001f?\n\r2\u0012=@Leff\n@\u0012\u0000@Reff\n@_\u0012;d\ndt@Leff\n@_'=\u0000@Reff\n@_'(10)\nThe nonlinear equations that are similar to Eqs. (10) and describe the spin dynamics of two-sublattice ferrimagnets\nwere obtained in the work [26] under the conditions H= 0 andReff= 0. Over the short time of demagnetization the\nsecond equation can be approximately treated as a conservation law and the conserving quantity (angular momentum\nof magnetization precession J) stays approximately constant as @L=@'= 0 due to the Noether theorem:\nJ=@Leff\n@_'=\u00001\n\r\u0014\nmcos\u0012+\u001f?sin2\u0012\u0012\nH\u0000_'\n\r\u0013\u0015\n=const (11)\nLet the moment of time t= 0\u0000denote the moment before the laser pulse impact and system initially is in the\nground state de\fned by the ground state angles \u0012(0\u0000) =\u00120,'(0\u0000) ='0, and their derivatives _ '(0\u0000) = 0, _\u0012(0\u0000) = 0.4\nDepending on the external parameters and preparation of the sample, the system might reside in one of the two\npossible antiferromagnetic collinear phases or in angular phase, which are separated by the magnetic phase transition\nlines [27]. If the demagnetization due to the laser pulse action is small, it produces the changes in the values of Mf,Md\nandMof the order of percent or less, whereas the change of m(which is approximately equal to total magnetization\nnear the compensation point) may be of several orders of magnitude, as its value is almost compensated. In what\nfollows, we assume that the demagnetization is associated only with change of m, namelym=m0+ \u0001m(t). As we\nwill see below, the change in this quantity already leads to several drastic e\u000bect in dynamics.\nTherefore, the conservation law (11) leads to the emergence of azimuthal dynamics _ '(t) at the demagnetization\ntimescales (\u0001 t) due to demagnetization pulse \u0001 m(t):\n_'(t) =\r\u0001m(t)\n\u001f?cos\u00120\nsin2\u00120(12)\nWe see that the torque is non-zero only in the angular phase, where 0 <\u00120<\u0019. Emergence of the azimuthal spin\nprecession as a result of demagnetization of the medium is similar to the well-known Einstein-de-Haas e\u000bect, where\nthe demagnetization leads to azimuthal precession of the body. Subsequently, this azimuthal spin dynamics leads to\nthe emergence of polar dynamics \u0012(t), which is most commonly measured in pump-probe experiments of ultrafast\nmagnetism, by acting as an e\u000bective \feld Heff=H\u0000_'\n\rin the Lagrangian (5). We can then view the Lagrangian\nas depending only on variable \u0012and the e\u000bective \feld Heff. At demagnetization \u000em\u00180:01Min GdFeCo the value\nof _'can reach up to 1 THz, and the corresponding e\u000bective magnetic \feld is of the order of 10 T. Note that initial\nstate of the system corresponds to the condition@\b\n@\u00120(Heff=H) = 0. We can rewrite the Euler-Lagrange equation\nfrom eq. (10) for polar angle as follows:\n\u001f?\n\r2\u0012+@\b(Heff)\n@\u0012=\u0000\u000bM\n\r_\u0012: (13)\nOr, alternatively:\n\u001f?\n\r2\u0012+msin\u0012Heff\u0000\u001f?sin\u0012cos\u0012\u0012\nH2\neff\u00002K\n\u001f?\u0013\n=\u0000\u000bM\n\r_\u0012 (14)\nBy integrating this equation over the short demagnetization pulse duration \u0001 twe obtain the state of system after\nthe laser pulse impact at t= 0+, which is characterized by the initial conditions\n\u0012(0+) =\u00120;_'(0+); '(0+) =Z\u0001t\n0_'(t)dt;_\u0012(0+) =Z\u0001t\n0\u0012(t)dt\nThe value \u0001 tis of the order of the optical pulse length. It may also include the time of restoration of the magnetization\nlength (or the value of m). After the moment of time 0+ free magnetization precession occurs in the model. Analysis\nof the spin dynamics under laser pump excitation will lead to emergence of critical dynamics near the second-order\nphase transitions to the collinear phases where \u0012= 0;\u0019, as is already seen from (12). We will discuss this behavior\nbelow.\nCRITICAL DYNAMICS\nIn a simple case of a quick decay of demagnetization (at the exciton relaxation timescales) with \u0001 m(t) = \u0001m,\n0<t< \u0001t; \u0001m(t) = 0,t>\u0001t, we obtain the initial condition from (14):\n_\u0012(+0)\u0019\u0014\n\u0000\u0012\n2cos2\u00120\nsin\u00120+ sin\u00120\u0013\nH+m0\n\u001f?cos\u00120\nsin\u00120+\u0001m\n\u001f?cos\u00120\nsin3\u00120\u0015\r2\n\u001f?\u0001m\u0001t=B(\u00120)\u0001m+O(\u0001m2): (15)\nThis quantity de\fnes the initial angular momentum of the polar spin precession that is induced in the system due to\nthe optical spin torque created by the femtosecond laser pulse. The amplitude of oscillations is proportional to the\ninitial condition (15). Its dependence on the external magnetic \feld is illustrated in Fig. 1 for di\u000berent temperatures\nfor magnetic parameters of GdFeCo uniaxial ferrimagnet. At low values of external magnetic \felds there is only\ncollinear ground state in the ferrimagnet and above certain \feld Hsfthe transition to an angular state occurs [21].\nThe schematic of the magnetic phase diagram for GdFeCo is shown in insertions in Fig. 1. At T= 275 K and T= 2885\nFIG. 1. The amplitude of the magnetization precessional response after the demagnetization due to the femtosecond laser\npulse action in GdFeCo ferrimagnet near the compensation point at di\u000berent temperatures. Insertions: the schematic of the\nmagnetic phase diagram. There are two antiferromagnetic collinear phases with Mddirected along(opposite) to the external\nmagnetic \feld above(below) the compensation temperature TM. They are separated by the \frst-order phase transition line\n(blue). Above them, an area where the angular phase exists, which is \flled with gray color. The black solid lines are the\nsecond-order phase transition lines. The dashed lines corresponds to the \fxed temperature in the plot. The red dot is the point\nof phase transition for this temperature.\nK the phase transitions are of the second order, which corresponds to a smooth transition from angle \u00120= 0 to\u00120>0,\nand the divergence of the response occurs at Hsf. Immediately above the compensation temperature the transition is\nof the \frst order and the behavior of the response above the is more complex; however, there is no critical divergence.\nThe critical behavior of the signal amplitude was observed experimentally for GdFeCo in ref. [15].\nAnother feature in the dynamics described by the proposed model is the critical behavior of the characteristic\ntimescales that occurs in the vicinity of the second-order phase transitions. To demonstrate this e\u000bect analytically,\nwe assume small deviations of \u0012during oscillations: \u0012(t) =\u00120+\u000e\u0012(t). We obtain:\n\u000e\u0012+!2\nr(\u00120)\u000e\u0012=\u0000\u000b!ex\u000e_\u0012; (16)\nwhere!2\nr(\u00120) =\r2h\nm\n\u001f?Hcos\u00120+\u0010\n2K\n\u001f?\u0000H2\u0011\ncos 2\u00120i\n,!ex=\rM\n\u001f?. The initial conditions are \u0012(0) =\u00120and eq.\n(15). In the limit of small oscillations and !r<\u000b!ex=2 (is ful\flled near the second-order transition) the solution has\nthe form\u000e\u0012(t) =Ae\u0000\ftsinh!t, where\f=\u000b!ex=2,!2=\f2\u0000!2\nr,A=B(\u00120)=!. The rise time can be estimated from\nthe condition _\u0012(\u001crise) = 0:\n\u001crise\u0019atanh!\n\f\n!=atanhp\n\f2\u0000!2r\n\fp\n\f2\u0000!2r(17)\nThe time of the oscillations decay (relaxation time) is proportional to the imaginary part of eigenfrequency and can\nbe estimated by the following expression:\n\u001crelax\u00194\u0019\f\n!2r: (18)\nNear second-order phase transition the mode softening occurs and the eigenfrequency turns to zero: !r!0,\nand we observe growth of the both timescales. The critical behavior of the rise time has been observed in GdFeCo\nexperimentally [15] and the typical values of \u001crisewere of the order of 10 ps.6\nCONCLUSIONS\nTo sum up, the developed theoretical model based on quasi-antiferromagnetic Lagrangian formalism proved to\nbe suitable for description of the coherent ultrafast response of RE-TM ferrimagnets near the compensation point\ndue to an ultrashort pulse of demagnetization in the presence of external magnetic \feld. We have found that the\ntorque acting on magnetizations is non-zero in the noncollinear phase only. We have explained the experimentally\nobserved critical behavior of the response amplitude and characteristic timescales as the consequence of the second-\norder magnetic phase transition from collinear to an angular in the external magnetic \feld and the mode softening\nnear it. These e\u000bects are vivid in the vicinity of the compensation point in external magnetic \feld. Understanding\nthe ultrafast response to demagnetizing optical or electrical pulses and subsequent spin dynamics can facilitate future\ndevelopments in the \felds of ultrafast energy-e\u000ecient magnetic recording, magnonics and spintronics.\nACKNOWLEDGMENTS\nThis research has been supported by RSF grant No. 17-12-01333.\n\u0003davydova@phystech.edu\nyzvezdin@gmail.com\n[1] T. Ostler, J. Barker, R. Evans, R. Chantrell, U. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui, L. Le Guyader, E. Men-\ngotti, L. 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Phys. 82, 2731 (2010).\n[10] R. B. Wilson, J. Gorchon, Y. Yang, C.-H. Lambert, S. Salahuddin, and J. Bokor, Phys. Rev. B 95, 180409 (2017).\n[11] Y. Yang, R. B. Wilson, J. Gorchon, C.-H. Lambert, S. Salahuddin, and J. Bokor, Science Advances 3(2017), 10.1126/sci-\nadv.1603117, http://advances.sciencemag.org/content/3/11/e1603117.full.pdf.\n[12] R. Chimata, L. Isaeva, K. K\u0013 adas, A. Bergman, B. Sanyal, J. H. Mentink, M. I. Katsnelson, T. Rasing, A. Kirilyuk,\nA. Kimel, O. Eriksson, and M. Pereiro, Phys. Rev. B 92, 094411 (2015).\n[13] C. D. Stanciu, A. Tsukamoto, A. V. Kimel, F. Hansteen, A. Kirilyuk, A. Itoh, and T. Rasing, Phys. Rev. Lett. 99, 217204\n(2007).\n[14] C. D. Stanciu, A. V. Kimel, F. Hansteen, A. Tsukamoto, A. Itoh, A. Kirilyuk, and T. Rasing, Phys. Rev. B 73, 220402\n(2006).\n[15] J. Becker, A. Tsukamoto, A. Kirilyuk, J. C. Maan, T. Rasing, P. C. M. Christianen, and A. V. Kimel, Phys. Rev. Lett.\n118, 117203 (2017).\n[16] Z. Chen, R. Gao, Z. Wang, C. Xu, D. Chen, and T. Lai, Journal of Applied Physics 108, 023902 (2010),\nhttps://doi.org/10.1063/1.3462429.\n[17] A. Moser, K. Takano, D. T. Margulies, M. Albrecht, Y. Sonobe, Y. Ikeda, S. Sun, and E. E. Fullerton, Journal of Physics\nD: Applied Physics 35, R157 (2002).\n[18] B. Lenk, H. Ulrichs, F. Garbs, and M. Mnzenberg, Physics Reports 507, 107 (2011).\n[19] J. Walowski and M. Mnzenberg, Journal of Applied Physics 120, 140901 (2016), https://doi.org/10.1063/1.4958846.\n[20] M. Liebs, K. Hummler, and M. F ahnle, Phys. Rev. B 46, 11201 (1992).\n[21] B. Goransky and A. Zvezdin, JETP Lett. 10, 196 (1969).\n[22] A. Zvezdin and V. Matveev, JETP 35, 140 (1972).\n[23] C. K. Sabdenov, M. Davydova, K. Zvezdin, D. Gorbunov, I. Tereshina, A. Andreev, and A. Zvezdin, J. Low. Temp. Phys\n43, 551 (2017).\n[24] M. Ding and S. J. Poon, Journal of Magnetism and Magnetic Materials 339, 51 (2013).\n[25] A. Zvezdin, JETP Lett. 28, 553 (1979); arXiv preprint arXiv:1703.01502 (2017).\n[26] B. Ivanov and A. Sukstanskii, Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 84, 370 (1983).7\n[27] A. Zvezdin, Handbook of Magnetic Materials 9, 405 (1995)."    },    {        "title": "1901.11498v1.Anisotropic_ferromagnetism_and_structure_stability_in__4f___3d__intermetallics__ab_initio_structure_optimization_and_magnetic_anisotropy_for_RCo__5___R_Ce__La__and_Y_.pdf",        "content": "Anisotropic ferromagnetism and structure stability in 4f-3dintermetallics:\nab initio structure optimization and magnetic anisotropy for RCo 5(R=Ce, La, and Y)\nMunehisa Matsumoto\nInstitute for Solid State Physics (ISSP), University of Tokyo, Kashiwa 277-8581, JAPAN\n(Dated: February 1, 2019)\nElectronic mechanism in the interplay between ferromagnetism and structure stability of 4 f-3d\nintermetallics in the main phase of rare-earth permanent magnets is investigated from \frst principles.\nWe present a case study with an archetypical materials family RCo 5(R=Ce, La, Y), which was a\npart of the earliest rare-earth permanent magnets and from which other representative main-phase\ncompounds can be regarded as a derived type. Comparison with the champion magnet materials\nfamily R 2T14B and recently revisited materials family RT 12(T=Co and Fe) points to a direction\nleading to a mid-class magnet for the possible next generation materials.\nPACS numbers: 75.30.Gw, 75.50.Ww, 75.10.Lp, 71.15.Rf\nI. INTRODUCTION\nRare-earth permanent magnet (REPM) obviously\nneeds a good ferromagnet with su\u000eciently strong coer-\ncivity and robust structure. Unfortunately, cohesive en-\nergy and magnetic energy in magnetic materials do not\nquite behave in a synchronized way: too strong magneti-\nzation may stretch the lattice spacing too much via the\nmagnetovolume e\u000bect for the chemical bonds to sustain\nthe crystal structure. Thus a strong magnetization can\nruin the structure stability. This makes one of the un-\navoidable trade-o\u000b situations in the materials design for\npermanent magnets. Careful inspection is needed to \fnd\nout a best compromise with an optimal chemical compo-\nsition and crystal structure to satisfy the prerequisites for\npermanent magnets. We do this on the basis of ab initio\nelectronic structure theory, unearthing interplay in the\nmiddle of dual nature harboring both of delocalization\nand localization both in 3 d-electrons and in 4 f-electrons.\nWe focus on light rare-earth elements, R=Ce and La,\nmotivated by their abundance and also in quest for a way\nto possibly exploit the subtlety in 4 f-electron physics to\nfabricate a new type of REPM's where half-delocalized\n4f-electrons contribute positively to the bulk magnetic\nproperties. As a reference case without 4 f-electrons,\nR=Y is also addressed. The particular crystal structure\nof RCo 5makes a part of the building block for all of the\nother representative compounds, R 2M14B1, RM 12and\nR2M172where M represents Fe-group elements.\nIn the next section, we describe our methods based on\nab initio structure optimization utilizing the open-source\npackage OpenMX3. In Sec. III, we present ab initio re-\nsults on RCo 5(R=Ce, La, and Y) for their formation\nenergy, magnetization, and magnetic anisotropy, being\ncontrasted to the analogous data for the Fe-counterparts.\nStructure varieties and implications on the material-\ndesign principles for REPM's are discussed in Sec. IV.\nConclusions and outlook are given in the \fnal section.II. METHODS\nIntrinsic magnetic properties of RCo 5and RFe 5\n(R=Ce, La, Y) are calculated from \frst principles via\nab initio structure optimization utilizing the open-source\nsoftware package OpenMX3{8on the basis of pseudopo-\ntentials9,10and the local orbital basis sets.\nThe lattice constants of Fe-group ferromagnets seems\nto be best described within Generalized Gradient Ap-\nproximation (GGA) as proposed Perdew, Burke, and\nErnzerhof (PBE)11and we present the results based\non GGA-PBE. The basis set we take in OpenMX\nisCe8.0-s2p2d2f1 ,La8.0-s2p2d2f1 ,Y8.0-s3p2d2f1 ,\nFe6.0S-s2p2d1 , and Co6.0S-s2p2d2f1 for R(Co,Fe) 5\nwithin the given pseudopotential data set3. The energy\ncuto\u000b is set to be 500 Ry of which choice has been in-\nspected together with the basis sets to ensure a good\nconvergence.\nSimilar basis sets with a few more or less inclusion of\nlocal basis wavefunctions can be good as well depend-\ning on the target materials and the issue being investi-\ngated as long as the choice of the basis set is coherently\napplied in a \fxed scope of target materials and target\nobservables. For the calculations of magnetic anisotropy\nenergy presented below and elsewhere12, we have actu-\nally seen that slightly richer basis sets Ce8.0-s3p3d3f2\nand La8.0-s3p3d3f2 work on a par with the basis sets\nwritten in the previous paragraph or sometimes in a bet-\nter way especially when the target material is close to\nthe verge of a delocalization-localization transition in 4 f-\nelectrons. In the scope of the present work, presum-\nably we stay on the side where the delocalized nature\nof 4f-electrons dominates within the crystal structure\nof CeCo 5. For this purpose, either Ce8.0-s3p3d3f2 or\nCe8.0-s2p2d2f1 will do basically.\nThe starting structure is taken from the experimen-\ntally measured lattice constants for YCo 513andab initio\nstructure optimization is done for stoichiometric com-\npounds RCo 5and RFe 5to get a minimized energy\nUtot[RT 5] (T=Co, Fe) and the associated magnetization\nin the ground state. Thus extracted energy is used toarXiv:1901.11498v1  [cond-mat.mtrl-sci]  31 Jan 20192\nassess the structure stability by looking at the forma-\ntion energy referring to the elemental materials which\nare analogously addressed with ab initio structure opti-\nmization by which the reference energies are extracted.\nThe formation energy of RCo 5, which we denote by\n\u0001E[RCo 5], is de\fned as follows:\n\u0001E[RCo 5]\n\u0011Utot[RCo 5]\u0000Utot[R per atom]\u00005Utot[Co per atom]\nThe structure optimization is done allowing for magnetic\npolarization without spin-orbit interaction. Then on top\nof the optimized lattice, magnetic anisotropy energy is in-\nvestigated by fully relativistic calculations incorporating\nthe spin-orbit interaction, putting a constraint on the di-\nrection of magnetization and numerically measuring the\nenergy as a function of the angle between magnetization\nand crystallographic c-axis. Thus we look at the trends\nin the intrinsic properties focusing on the tradeo\u000b be-\ntween formation energy and magnetization, assisted by\nthe data for magnetic anisotropy. As for the other prereq-\nuisite intrinsic property, Curie temperature, some of the\nissues and \fnite-temperature magnetism are addressed\nin separate works14{17.\nIII. RESULTS\nAb initio structure optimization for RCo 5, elemen-\ntal R (fcc-Ce, dhcp-La, hcp-Y) and elemental Co, that\nis, hcp-Co gives the energy and magnetization for each\nof the target systems within the given pseudopotential\ndata sets. Calculated formation energy and magnetiza-\ntion from structure optimization runs are presented in\nSec. III A and III B, respectively. Then in Sec. III C we\nshow results from fully-relativistic calculations for mag-\nnetic anisotropy energy on top of the optimized lattice.\nA. Formation energy\nWe start with inspecting the formation energy as a clue\nfor the trend in the structure stability. Taking calculated\nenergy within the particular choice of the basis set de-\nscribed in the previous section and given standard data\nsets of pseudopotentials3, calculated formation energy for\nRT5(R=Y, La, Ce, and T=Co, Fe) is summarized in\nFig. 1. The detailed procedures for the calculation of for-\nmation energy follow those described in Refs. 18 and 19.\nOn the optimized lattice, the lattice constants and unit-\ncell volume have been read o\u000b as summarized in Table I.\nComparing with the experimental data21which is avail-\nable only for Co-based materials, it is seen that ab initio\nstructure optimization predicts the realistic lattice con-\nstants and the unit cell volume within the precision of\nthree signi\fcant digits.\nWe reproduce the known experimental fact that RFe 5\nis metastable20with the calculated formation energy for\n-1-0.5 0 0.5 1Calculated formation energy [eV/(formula unit)]\nRLa Ce YRFe5\nRCo5FIG. 1. Calculated formation energy for RT 5(R=Ce, La, Y\nand T=Fe and Co).\ncalculated results experimental data\n(a;c) [\u0017A]Vcell[\u0017A3] (a;c) [\u0017A]Vcell[\u0017A3]\nCeCo 5(4:89;4:02) 84:0 (4:93;4:02) 84:5\nLaCo 5(5:06;3:96) 87:7 (5:09;3:94) 88:3\nYCo 5(4:92;3:95) 83:2 (4:93;3:99) 84:0\nCeFe 5(5:06;4:10) 91:1 N/A\nLaFe 5(5:19;4:08) 95:0 N/A\nYFe 5(5:08;3:99) 88:8 N/A\nTABLE I. Optimized lattice constants and unit-cell volume\nVcellfor RT 5(R=Ce, La, Y and T=Co, Fe). The experimental\ndata are taken and rounded up to the 3rd digit as quoted in\nRef. 21.\nRFe 5running into the positive region. The relative trend\nbetween LaCo 5and YCo 5,\nj\u0001E[LaCo 5]j<j\u0001E[YCo 5]j\nshows that the smaller rare-earth elements in the over-\nall trend of lanthanide contraction comes with the bet-\nter structure stability. Most notable di\u000berence between\nLaCo 5and YCo 5lies in the radius of rare-earth element,\nwith La being the largest rare earth element and Y be-\ning as small as typical heavy rare earth elements like Dy\nin the trend of lanthanide contraction. It is reasonable\nfor La-based compounds having the enlarged lattice with\nLa sitting on the peak in the size of rare-earth atoms\nto become relatively fragile against the anticipated mag-\nnetovolume e\u000bect when combined with the strong 3 d-\nelectron magnetization coming from Fe-group elements,\nsince there would be relatively less margin for the chem-\nical bonds to hold on against the volume expansion as\nimposed by magnetization. In this regard, the relatively\nsmall size of Ce4+actually gives an advantage on Ce-\nbased ferromagnets at least for the structure stability.3\ncalculated results expt.\nM[\u0016B/(f.u.)]M[Tesla]M[Tesla]\nCeCo 5 6:03 0 :837 0:77\nLaCo 5 7:03 0 :934 0:91\nYCo 5 7:22 1 :01 1:09\nCeFe 5 10:6 1:36 N/A\nLaFe 5 11:7 1:43 N/A\nYFe 5 10:7 1:41 N/A\nTABLE II. Calculated magnetization for RT 5(R=Ce, La,\nY, and T=Co, Fe). Experimental data for magnetization in\nTesla is taken from Ref. 22.\nB. Magnetization\nCalculated magnetization as a result of the structure\noptimization is summarized in Table II. Total magnetic\nmoments per formula unit which occupies the unit cell\nis normalized by the volume of the unit cell to yield the\nmagnetization measured in Tesla which is of direct rele-\nvance for REPM's. The quantitative agreement between\nthe results from ab initio structure optimization and ex-\nperiments for the magnetization in Tesla is seen up to\ntwo digits. The realistic energy scales for the cohesion\nand magnetism seem to be properly taken into account\nin the present description.\nNature of ferromagnetism can be inspected from cal-\nculated total density of states (DOS) as shown in Fig. 2.\nOnly for the \fctitious YFe 5, weak ferromagnetism with\na non-negligible amount of majority-spin states on the\nFermi level is seen. All of the other compounds show\nstrong ferromagnetism with the d-electron majority-spin\nstates all below the Fermi level which means that ferro-\nmagnetic order is basically saturated there and there is\nlittle space for further enhancing magnetization e.g. by\nmixing Fe and Co to follow the celebrated Slater-Pauling\ncurve. We note that our DOS for CeCo 5look at vari-\nance with what is shown in one of the previous works21\nwhere their DOS could point to the possible weak ferro-\nmagnetism in CeCo 5. We have actually veri\fed in our\ncalculations that Slater-Pauling curve is not observed for\nCe(Co,Fe) 523so indeed weak ferromagnetism does not\nseems to be the case at least for the optimized CeCo 5\nin our calculations. Since the electronic states near the\nFermi level can sensitively depend on the details of cal-\nculation setup it could be reasonable to get some variant\nresults for CeCo 5that may reside almost on a border\nline between weak ferromagnetism and strong ferromag-\nnetism.\nC. Magnetic anisotropy energy (MAE)\nOn top of the optimized crystal structure, magnetic\nanisotropy energy is inspected from \frst principles by\nmonitoring the energy as a function of the angle between(a)\n-30-20-10 0 10 20 30\n-5 -4 -3 -2 -1  0  1  2  3  4  5Calculated total DOS [states/eV]\n(E-EFermi) [eV]CeCo5\nLaCo5\nYCo5\n(b)\n-30-20-10 0 10 20 30\n-5 -4 -3 -2 -1  0  1  2  3  4  5Calculated total DOS [states/eV]\n(E-EFermi) [eV]CeFe5\nLaFe5\nYFe5\nFIG. 2. (Color online) Calculated total density of states for\n(a) RCo 5and (b) RFe 5(R=Ce, La, and Y). The arrow in the\n\fgure points to the majority-spin states on the Fermi level.\n-4-2 0 2 4 6\n 0  10  20  30  40  50  60  70  80  90Calculated energy [meV/(formula unit)]\nθ [degrees]YCo5\nYFe5\nLaCo5\nLaFe5\nCeCo5\nCeFe5\nFIG. 3. (Color online) angle dependence of the calculated\nenergy with the zero point being set at the bottom of the\ncalculated total energy. Here the angle is made between\nthe bulk magnetization direction and the crystallographic c-\naxis. Overall RCo 5show uni-axial magnetic anisotropy while\nRFe 5's have easy-direction along the ab-plane. Dotted lines\nare \fts of Eq. (1) in the text to the calculated data points.\nThe results of such \fts are summarized in Table III.\nconstrained magnetization and the crystallographic c-\naxis under the presence of spin-orbit interaction. The re-\nsults are shown in Fig. 3. Uni-axial magnetic anisotropy\nof RCo 5is switched into easy-plane magnetic anisotropy\nwith RFe 5. Calculated energy as a function of the mag-\nnetization direction as shown in Fig. 3 is \ftted to the4\nK[meV/(formula unit)] p q\nCeCo 5 3:79(2) \u00000:0181(2) 0:004(4)\nLaCo 5 1:737(1) \u00000:162(4)q\u00110\nYCo 5 1:5171(6) \u00000:077(2)q\u00110\nCeFe 5 \u00004:39(6) \u00000:0130(3) 0:28(1)\nLaFe 5 \u00000:7471(7) 0 :073(4)q\u00110\nYFe 5 \u00001:34706(3) \u00000:00103(8)q\u00110\nTABLE III. Calculated magnetic anisotropy energy and the\ncoe\u000ecient of the higher order terms for RT 5(R=Ce, La, Y\nand T= Co, Fe).\nfollowing relation\nEaniso =\u0000K\u0002\n(1\u0000p\u0000q) cos2\u0012+pcos4\u0012+qcos6\u0012\u0003\n(1)\nto extract the magnetic anisotropy energy together with\nthe higher-order terms. The \ft results are summarized\nin Table III. The absolute values of Kshow a signi\fcant\nunderestimate as compared to experiments, which seems\nto be reasonable for ab initio calculations for 3 d-electron\nmagnetic anisotropy. Without special treatment such\nas orbital polarization, LDA/GGA+U, or self-interaction\ncorrection, it is not easy to quantitatively match the cal-\nculated MAE to the experimentally observed range24.\nFor the moment we focus on qualitative trends in the\nextracted parameters. It is to be noted for the 4 f-\nelectron part that relatively strong anisotropy is found\nfor Ce-based compounds as compared to La or Y-based\ncompounds even though the 4 f-electrons are presumably\nin a delocalized state. In Fig. 2 the majority-spin state for\nCe, seen on the opposite side to the majority-spin of Co\nor Fe, has a signi\fcant contribution above and below the\nFermi level which means 4 f-electrons are half-localized\nas are 3d-electrons in Fe and Co. Thus they contribute\npositively to the bulk anisotropy.\nTurning to the 3 d-electron part, it is seen that all Fe-\nbased compounds end up with easy-plane anisotropy:\nthis is not quite interesting in the context of REPM.\nGiven that Fe-rich compounds come with poor structure\nstability and loss of uni-axial magnetic anisotropy, rela-\ntively strong magnetization does not help by itself and\nan optimal chemical composition for REPM would rather\nlie on the Co-rich side. Quantitatively identifying the lo-\ncation of an optimal point on a continuum of chemical\ncomposition axis is addressed in a separate work23.\nRemarkably, the coe\u000ecient of the extracted higher-\norder coe\u000ecient in YCo 5is in agreement with the past\nexperimental claim stating K2=K1=\u00000:02125where\nK1\u0011K(1\u0000p) andK2\u0011Kpin our notation. Our\ndata amounts to K2=K1=\u00000:071 and the sign of the\nhigher order contribution has been correctly reproduced.\n-3-2.5-2-1.5-1-0.5 0Calculated formation energy [eV/(formula unit)]\nRLa Ce YR2Fe14B\nR2Co14BFIG. 4. Calculated formation energy for R 2T14B (R=Ce,\nLa, Y and T=Fe,Co). The data for (La,Ce) 2Fe14B and\nY2(Fe,Co) 14B are taken from Ref. 19 and Ref. 18, respec-\ntively.\nIV. DISCUSSIONS\nComparison between what we have seen for RT 5and\nanalogous data for the champion magnet compound fam-\nily, R 2T14B, is described in Sec. IV A which reveals the\ntrends with respect to the crystal structure variety where\nthe latter was reached in quest for a way to enlarge the\ninteratomic spacing between Fe26. Implications on the\nmaterials design for REPM's are discussed in Sec. IV B\ntaking the recently discussed materials family RT 1227as\nthe possible playground for the application of the pro-\nposed principles.\nA. Comparison with R 2T14B\n1. Formation energy\nCalculated formation energy for Ce 2Co14B and\nLa2Co14B are shown in Fig. 4 together with the data for\nR2Fe14B (R=Ce and La)19and Y 2T14B (T=Co, Fe)18for\nthe convenience of getting an overview. Parallel trends\nwith what is seen in Fig. 1 is obvious, with the most sig-\nni\fcant di\u000berence being the overall o\u000bset to push down\neverything down to the negative region of the formation\nenergy. Even with the intrinsic trend that Fe-based com-\npounds come with the relatively poor structure stability,\nwith the particular crystal structure of R 2T14B they can\nbe stabilized except for La 2Fe14B that looks perhaps too\nclose to being on the verge to the positive side of the\nformation energy.\nAbsence of Ce 2Co14B in the experimental literature is\na mystery as long as we look at the trends of the for-\nmation energy. The absolute value of the calculated for-\nmation energy for Ce 2Co14B is more than double of that\nfor Ce 2Fe14B, which is consistent with one of the recent\nclaims28. Presumably, other compounds may compete\nagainst Ce 2Co14B in getting bulk-stabilized, or some-\nthing beyond the present level of electronic structure cal-5\ncalculated results expt.\nM[\u0016B/(f.u.)]Vcell[\u0017A3] (a;c)\u0017A (a;c)\u0017A\nCe2Co14B 16:64 865 (8 :596;11:73) N/A\nLa2Co14B 18:67 891 (8 :635;11:96) (8:67;12:01)\nY2Co14B 18:70 855 (8 :560;11:68) (8:60;11:71)\nCe2Fe14B 29:95 936 (8 :797;12:11) (8:76;12:11)\nLa2Fe14B 31:92 961 (8 :835;12:33) (8:82;12:34)\nY2Fe14B 31:47 922 (8 :775;11:99) (8:76;12:00)\nTABLE IV. Optimized lattice constants and unit-cell vol-\nume for R 2T14B (R=Ce, La, Y and T=Co, Fe). The data\nfor (La,Ce) 2Fe14B and Y 2(Fe,Co) 14B are partly taken from\nRef. 19 and Ref. 18, respectively. Experimental lattice con-\nstants are taken from Ref. 1.\nculations might be at work in real experiments: in the\npresent calculations we have plainly put 4 f-electrons of\nCe into the valence state which is not always justi\fed.\n2. Magnetization\nCalculated magnetization for R 2T14B are summarized\nin Table IV and V together with the optimized lattice in-\nformation for the convenience of calculating magnetiza-\ntion in Tesla. Again the data for R 2Fe14B (R=Ce and La)\nand Y 2T14B (T=Co, Fe) are taken from Refs. 19 and 18,\nrespectively. It is seen that the optimized lattice con-\nstants are in quantitative agreement with the experimen-\ntal ones up to three digits while calculated magnetization\nin Tesla sometimes has an overestimate where the devia-\ntion from experimental data amounts to a few tens of %.\nPractically it has not been easy to let the calculations\nfor R 2Co14B converge. It actually took having an initial\nmagnetic state with each Co atom starting with unrealis-\ntically large magnetic polarization to reach any sensibly\nconverged results in our ab initio structure optimization\nfor R 2Co14B so far. In this way it might have been rea-\nsonable to have ended up with an overestimate of magne-\ntization and also this may indicate that our description\nof 3d-electron state may not have been as precise as has\nbeen achieved for Fe-based counterparts19. Anyways rel-\native trends e.g. between La 2Co14B and Y 2Co14B seems\nto have been satisfactorily described.\n3. Magnetic anisotropy energy\nMagnetic anisotropy energy for Ce 2Co14B and\nLa2Co14B can be inspected in an analogous way to what\nwas done for RT 5in Sec. III C. The results are shown in\nFig. 5 together with the counterpart data for Ce 2Fe14B\nand La 2Fe14B as partly taken from Ref. 19. It actu-\nally took lifting o\u000b the constraints on magnetic moments\non Ce and La for the fully relativistic runs of R 2Co14B\n(R=Ce and La) to reach convergence. The same was truecalc. expt.\nCe2Co14B 0:896 N/A\nLa2Co14B 0:977 0:741\nY2Co14B 1:02 0:848\nCe2Fe14B 1:49 1:47\nLa2Fe14B 1:55 1:48\nY2Fe14B 1:59 1:59\nTABLE V. Calculated magnetization in Tesla for R 2T14B\n(R=Ce, La, Y and T=Co, Fe). Experimental data for Mat\nT= 4:2 K is taken from Ref. 1 for R 2Fe14B and Ref. 29 for\nR2Co14B.\n-2-1.5-1-0.5 0 0.5 1 1.5\n 0  10  20  30  40  50  60  70  80  90[Utot(θ)-Utot(θ=90)] [meV/(f.u.)]\nθ [deg.]Ce2Co14B\nLa2Co14B\nCf. Ce2Fe14B\nCf. La2Fe14B\nFIG. 5. Angle-dependence of calculated energy of R 2Co14B\n(R=Ce and La). For a comparison, analogous data for\nR2Fe14B are included partly from Ref. 19.\nfor La 2Fe14B in Ref. 19 but it was not actually the case\nfor Ce 2Fe14B in Ref. 19 - here, just for a comparison on an\nequal footing, results from fully relativistic calculations\nfor Ce 2Fe14B without the constraints on magnetic mo-\nments on Ce are presented. The di\u000berence from the fully\nrelativistic calculations for Ce 2Fe14B in Ref. 19 where\nthe constraints on the direction of magnetic moments\nwere all imposed on Ce and Fe is seen only quantita-\ntively in the absolute value of Kwith a slightly smaller\nnumber coming up here. The other parameters show\nqualitatively similar behavior, with positive pand nega-\ntiveqcoming up in the same range in the absolute value\njpj'jqjfor Ce 2Fe14B, irrespectively of the constraints\non the magnetic moments on Ce being applied (Ref. 19)\nor not (here).\nIt is remarkable that we hardly observe any di\u000ber-\nK[meV/(formula unit)] p q\nCe2Co14B \u00001:21(1) \u00000:0188(4) 0:089(9)\nLa2Co14B \u00001:10974(5) \u00000:0072(2)q\u00110\nCe2Fe14B 1 :83(1) 0 :0360(6) \u00000:051(7)\nLa2Fe14B \u00000:28106(7) 0 :098(1)q\u00110\nTABLE VI. Calculated magnetic anisotropy energy and the\ncoe\u000ecient of the higher order terms for R 2T14B.6\n-1-0.5 0 0.5 1 1.5Calculated formation energy [eV/(formula unit)]\nRLa Ce YRFe12\nRCo12\nFIG. 6. Calculated formation energy for RT 12(R=Ce, La,\nY and T=Co, Fe).\ncalculated results\nM[\u0016B/(f.u.)] (a;c)\u0017AVcellM[Tesla]\nCeCo 12 17:99 (8:33;4:65) 161 1 :30\nLaCo 12 18:95 (8:40;4:67) 165 1 :34\nYCo 12 19:05 (8:28;4:66) 160 1 :39\nCeFe 12 27:15 (8:58;4:73) 174 1 :82\nLaFe 12 27:7 (8:66;4:73) 177 1 :82\nYFe 12 27:2 (8:51;4:73) 171 1 :85\nTABLE VII. Calculated magnetization and optimized lattice\nconstants and unit-cell volume for RT 12(R=Ce, La, Y and\nT=Co, Fe).\nence between Ce 2Co14B and La 2Co14B concerning their\neasy-plane bulk MAE in the absolute value, except for\nthe qualitative behavior of higher-order terms. This is\nin strong contrast to the di\u000berence between Ce 2Fe14B\nand La 2Fe14B where even the sign of bulk anisotropy is\n\ripped between them.\nB. Implications on the possible new REPM based\non RT 12\nIn the past \fve years there has been a surge in the\ninterests in RT 12compounds27that can potentially go\nbeyond the champion magnet compounds R 2T14B in the\nintrinsic properties. Here the structure stability has been\ntraded o\u000b to demonstrate the real-material implementa-\ntion of case studies to go beyond Nd 2Fe14B30{32. Recent\ninterests focus on the quest for gaining both of satisfac-\ntorily good magnetic properties and su\u000ecient structure\nstability to move on to a working REPM in industrial\napplications. Here we take a brief look at the outcome\nofab initio structure optimization for RT 12with R=Ce,\nLa, Y and T=Co, Fe and see to what extent we might beable to recycle what we have seen with RT 5and R 2T14B.\nCalculated formation energy for RT 12is shown in\nFig. 6. The overall trend in the structure stability is\nqualitatively shared among RT 5and R 2T14B in the sense\nthat a) Co-based compounds are generally more stable\nthan Fe-based compounds and b) smaller rare-earth ele-\nments give more stable structure. Combined with calcu-\nlated magnetization in Tesla as summarized in Table VII\npoints to CeCo 12and YCo 12that can provide a good\ncompromise coming with both reasonably good magne-\ntization and comfortably as stable as CeCo 5as long as\nthe calculated formation energy shows. Apparently RT 12\nresembles RT 5in several respects rather than being con-\nsidered as an alternative to R 2T14B. It might be feasible\nto pursue the possible line toward a mid-class REPM on\nthe basis of CeCo 12possibly with a small addition of Fe\nfor gaining magnetization not to the extent where the\nstructure stability is ruined, in an analogous way to the\ndevelopment of Fe-doped YCo 517,23,33.\nV. CONCLUSIONS AND OUTLOOK\nIntrinsic trade-o\u000b between structure stability and mag-\nnetization has been inspected from \frst principles for\nRT5. Comparison has been done with the champion mag-\nnet compounds R 2T14B and the recently focused mate-\nrials family RT 12. Detailed characterization of the \fc-\ntitious compound Ce 2Co14B has been done for the \frst\ntime to the best of the author's knowledge. Through-\nout the calculations covering materials family, roles of\nCe in helping structure stabilization and positive con-\ntribution to the intrinsic magnetic anisotropy has been\nelucidated on the basis of calculated electronic structure\nand we identify CeCo 12as a possible starting point for\nthe fabrication of a next-generation mid-range REPM.\nACKNOWLEDGMENTS\nMM's work in ISSP, Univ. of Tokyo is supported by\nToyota Motor Corporation. Helpful interactions with\nA. Marmodoro, A. Ernst, C. E. Patrick, J. B. Staunton,\nM. Ho\u000bmann, H. Akai, T. Miyake, Y. 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The Departm ent of Energy will provide public access to these results of federally sponsored \nresearch in accordance with the DOE Public Access Plan ( http://energy.gov/downloads/doe-pubic- access-plan\n). \nLong range antiferromagneti c order in a rocksalt high entropy oxide  \nJunjie Zhang,†,* Jiaqiang Yan,† S. Calder,‡ Qiang Zheng,† Michael A. McGuire,† D. L. Abernathy,‡ \nYang Ren,║ Saul H. Lapidus,║ Katharine Page,‡ Hong Zheng,┴ J. W. Freeland,║ John D. Budai,† R. P. \nHermann†,** \n†Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA . \n‡Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA .  \n║X-ray Science Division, A dvanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA .  \n┴Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA  \nAbstract : We report for the first time the magnetic structure of the  high entropy oxide  (Mg 0.2Co0.2Ni0.2Cu0.2Zn0.2)O \nusing neutron powder diffraction. This material exhibits a sluggish magnetic transition but possesses a long -range \nordered antiferromagnetic ground s tate, as revealed by DC and AC magnetic susceptibility, elastic and inelastic \nneutron scattering measurements. The magnetic propagation wavevector is k =(½, ½, ½) based on the cubic structure \nFm-3m, and the magnetic structure consists of ferromagnetic sheets in the (111) planes with spins antiparallel be-\ntween two neighboring planes. Inelastic neutron scattering reveals strong magnetic excitations at 100 K that survive up to room temperature. This work demonstrates that entropy -stabilized o xides represent a unique platform to study \nlong range magnetic order with extreme chemical  disorder. \n1 High entropy oxides (HEOs), a class of materials based on a concept parallel to high entropy alloys,1-2 are solid \nsolutions stabilized by  high configuration entropies. Recently, these materials have attracted much attention due to \nfundamental issues related to phase formation and materials design as well as their potential multifunctional prop-\nerties that can emerge from their high configurational entropy and spin order .3-7  \n(Mg 0.2Co0.2Ni0.2Cu0.2Zn0.2)O (hereafter MgO -HEO), the first entropy-stabilized oxide, was reported by Rost et al \nin 2015.3 Their choice of this system was based on the diversity in structures, coordination and cationic radii of \nMgO  (Fm-3m), CoO  (Fm-3m), NiO ( Fm-3m), CuO  (C2/c) and ZnO  (P63mc). This material and its substitution by \naliovalent elements were reported to exhibit colossal dielectric constants8 and superionic conductivity,9 making this \nnew family of materials very promising for applications such as large- k dielectric materials and solid state electro-\nlytes in solid -state batteries. Inspired by the superionic conductivity , Sarkar et al reported the electrochemical prop-\nerties of the rocksalt high entropy oxides, such as lithium storage capacity and the cycling stability .10 One of the \nfundamental interest s in MgO -HEO is the possib le Jahn -Teller distortion of Cu2+ environment. Electron paramag-\nnetic resonance,11 extended x- ray absorption fine structure12 and x -ray di ffraction studies11 as well as theoretical \ncalculations13-14 suggest this distortion. Besides studies of bulk samples, fundamental studies have been  carried out \non thin films .15-16 Recently, Braun et al reported the heat capacity and low thermal conductivity of  MgO -HEOs.15 \nMeisenheimer  et al reported antiferromagnetic order in MgO -HEO by studying the exchange anisotropy of perm-\nalloy/MgO -HEO heterostructures.16 However, up to date, there are no reports on the magnetic structure on bulk \nsamples. \nThe observation of new functi onality in MgO -HEO triggered the search for other type s of high entropy oxides \nbeyond the rocksalt structure. F luorite,17-18 perovskite4-6 and spinel7 high entropy oxides have been reported. Be-\nsides structural stud ies, the fluorite high entropy oxides  have been reported to have very low thermal conductivity ,17 \nmaking these materials promising candidates for thermal insulators and satisfying one of the criteria for high per-\nformance thermoelectric material s. More recently, Witte et al studied the magnetic properties of perovskite high \nentropy oxides and found predominant antiferromagnetic behavior with a small ferromagnetic contribution using \nmagnetometry and Mössbauer spectroscopy.19  \nIn this communication, we report the magnetic structure of the rocksalt high entropy oxide using neutron powder \ndiffraction. This is the first report of magnetic structure for high entropy oxides. \n2Fi\ngure 1 . Crystal structure of (Mg 0.2Co0.2Ni0.2Cu0.2Zn0.2)O. (a) Rietveld refinement of diffraction data as a func-\ntion of Q  (Q=4πsinθ/λ, λ= 0.412758 Å ) collected at room temperature  using synchrotron x- rays, (b) C rystal struc-\nture, (c) Temperature dependence of lattice parameters between 90 and 440 K. \nMgO -HEO polycrystalline powders were synthesized using stan dard solid- state reaction techniques according to \nRost et al.3 Chemical analysis was carried out using a Bruker Quantax 70 energy dispersive spectroscopy (EDS) \nsystem on a Hitachi TM- 3000 scanning electron microscope. Mg, Co, Ni, Cu and Zn were found to distribute \nhomogenous at micron level (see Figure S1). EDS measurements confirm that the ratio of Mg, Co, Ni, Cu and Zn \nis essentially 1:1:1:1:1 (see Figure S2 ). \nWe investigated the average crystal structure of MgO -HEO using high resolution synchrotron x- ray powder dif-\nfraction  at 11 -BM at the Advanced Photon Source, Argonne National Laboratory. The relative intensit ies of the \nBragg peaks  (111) and (200)  deviates from the ideal rocksalt lattice, indicating  anisotropic strain broadening. Such \na phenomenon has been observed by Berardan et al.11 The Rietveld refinement on the data collected at room tem-\nperature with a single phase of Fm -3m converged to Rwp=12.5% ( Figure 1a , see Supporting Information for details ). \nA sketch of this MgO -HEO structure is shown in Figure 1 b. In such a structure, Mg, Co, Ni, Cu, and Zn reside on \nthe same site ( 000, W yckoff  position 4a ) with octahedral oxygen coordination, and oxygen locates at (½,½,½) \n(Wyckoff position 4b ). No structural phase transition was  observed within our resolution  down to 90 K, as  no extra \npeaks appear or peak splitting occurs on cooling (see Fig ure S3 ). The temperature dependence of lattice parameter \n3 from the Rietveld refineme nts is shown in Figure 1 c, with a shrinking upon cooling . Linear thermal expansion \ncoefficient is estimated to be 11 ×10-6 K-1 at room temperature , which is comparable to that of MgO. \n \nFigure 2 . Local structure of (Mg 0.2Co0.2Ni0.2Cu0.2Zn0.2)O. (a) x- ray PDF  in the 1-4 Å range. (b) x- ray PDF in the \n1-49 Å range. (c) neutron PDF  in the 1 -4 Å range. Empty blue circles: experimental data; red line: fitted structure; \ngreen line: residual. Residuals have been shifted by -5 for (a, c) and -12 for (b). \nThe local structure of MgO -HEO obtained from total scattering of synchrotron x -rays and neutrons, which yield \nbetter contrast for Co, Ni, Cu and Zn, can be  described using the rocksalt  structure with  random cationic distribu-\ntion. The x- ray pair distribution function (PDF) fit to the Fm-3m structure in the range of 1-49 Å ( Figure 2 b) \nconverged to Rw=15.6% , demonstrating long range order of the atoms , consistent with x -ray powder diffraction \nanalysis. The x -ray PDF fit in the 1 -4 Å range converged to R w=8.9% ( Figure 2 a), indicating that also the local \nstructure can be described using Fm-3m. The neutron PDF in the 1-4 Å range was also fitted well using the cubic \nstructure ( Rw=7.6% , see Figure 2 c), confirming similar local environments of Co, Ni, Cu and Zn. The shape of the \nnearest -neighbor peak is somewhat asymmetric, deviating from a single Gaussian function with extra intensity on \nthe high -r side. This indicate s that  the local structure is not ideal rock -salt, which is consistent with a slightly \ndistorted local environment  resulting from Jahn -Teller effect of Cu2+.11-14 However, such an effect cannot be in a \nstatic, cooperative manner .  \n4 In order to characterize  the oxidation states of cations , we carried out x -ray absor ption spectroscopy on MgO -\nHEO. The oxidation states of Ni, Co, and Cu are 2+ , as evidenced by the L  edge x-ray absorption of Ni, Co and Cu \nas a function energy (see Fig ure S4). Our result is in good agreement with Rost’s E XAFS report.12 \n \nFigure 3 . Physical properties of ( Mg 0.2Co0.2Ni0.2Cu0.2Zn0.2)O. (a) DC magnetic susceptibility measured with a \nmagnetic field of 0.1 T. ZFC- W, FC -C and FC- W are short for zero -field cooling with data collection on warming, \nfield cooling with data collection on cooling, and field -cooling with data collection on warmi ng, respectively. (b) \nInverse magnetic susceptibility as a function of temperature.  The dashed line is only a guide for the eye. (c) Real \nand imaginary part of the AC magnetic susceptibility measured without external magnetic field at various frequen-\ncies. (d) Heat capacity measured under a magnetic field of 0 T.  \nWe now present the physical properties of MgO -HEO, including magnetic susceptibility, heat capacity and ther-\nmal conductivity . Figure 3a  shows the DC magnetic susceptibility as a function of temperat ure measured with a \nmagnetic field of 0.1 T on warming after zero field cooling (ZFC -W), cooling with magnetic field (FC -C) and \nwarming with field (FC -W). An anomaly is immediately identified, which suggest s an antiferromagnetic order.  The \ntransition temperature is determined to be 113 K by the peak position of  ∂(χT)/∂T. The magnetic susceptibility \ncontinues to increase below 113 K. This suggests that not all of the local moments participate in the long range \norder, which is somewhat expected for such a strongly disordered system. The inverse magnetic susceptibility, 1/χ, \nis not linear above 113 K up to 380 K (see Figure 3b), suggesting non- Curie -Weiss behavior  (see a modified Curie -\nWeiss fit in Figure S5 ). Such a deviation from the Curie -Weiss law may or iginate from short -range fluctuations, \nwhich is corroborated by the absence of an anomaly in heat capacity, the nonzero intensity of magnetic peaks above 113 K in neutron diffraction, and magnetic excitations at room temperature in inelastic neutron scatte ring, as will \n5 be discussed below. Magnetization was also measured as a function of magnetic field at 300 K and 2 K, and both \nshow linear behavior  (See Figure S6), which  is consistent with paramagnetism at room temperature and an antifer-\nromagnetic or spin g lass state at low temperature. The real and imaginary part of the AC magnetic susceptibility as \na function of temperature measured at various frequencies without magnetic field are  presented in Figure 3c . No \nfrequency dependence of the peak at 113 K is seen, and no peak in the imaginary part occurs near the transition. \nThese observations are consistent with a long -range antiferromagnetic order, and inconsistent with spin- glass freez-\ning. Figure 3d shows the heat capacity as a function of temperature without applied magnetic field. Surprising ly, \nno anomaly is observed. One plausible explanation is  that the antiferromagnetic transition is not sharp and the \nentropy release occurs in a broad temperature range, corroborate d by short range magnetic fluctuations above N éel \ntemperature and magnetic excitations at room temperature (see Figure 4d and Figure S8). The thermal conductivity \nof MgO -HEO exhibits phonon glass behavior (see Figure S7), which is expected as it contains extreme chemical \ndisorder. \nTo understand the nature of the phase transition at 113  K observed in DC magnetic susceptibility, we performed \nneutron powder diffraction at HB2A at HFIR, ORNL.20-21 The nuclear structure was Rietveld refined from data \ncollected at 160 K using the cubic structure Fm -3m with the fit converging to Rwp=12.3% . Four  extra peaks  at \nQ=1.277 Å-1, 2.455 Å-1, 3.233 Å-1 and 3.856 Å-1 that do not coincide with nuclear Bragg reflections are observed \nat 2 K (see Figure 4a ). The absence of these peaks at 160 K and the lack of a structural transition in synchrotron x-\nray diffraction data confirm that they arise from  antiferromagnetic order . Indexing of these peaks leads to a com-\nmensurate propagation vector k =(½, ½, ½). The m agnetic structure was determined based on representation analysis  \nusing SARAh22-23 and FullProf.24 For the space group Fm -3m with propagation vector (½, ½, ½) and magnetic ions \non site 4a (0,0,0), there are two irreducible representations (IRs) (Γ 3 and Γ5), following the labelling scheme of \nKovalev,25 see Supporting Information for basis vector components  and Table S 1. Both IRs were tested against the \ndata and only Γ 5 found to fit all the observed reflections  (see Figure 4 b). The magnetic structure consists of ferro-\nmagnetic sheets in the (111) planes, with spins antiparallel between two neighboring planes, as shown in Figure 4c . \nThe ordered magnetic moment is  1.4(1) μB, which is somewhat  smaller than 2  μB expected for the average pure spin \nmoment  calculat ing from 1/3 of 3 μB from  Co2+ (S=3/2), 1/3 of 2 μB from Ni2+ (S=2/2), and 1/3 of 1 μB from Cu2+ \n(S=1/2). The observation of reduced moment from experiments is understandable due to suppression of ordering \n6 throughout the lattice because of  extreme chemical disorder and a significant amount of nonmagnetic ions, cova-\nlency (some of the moment is spread out over the oxygen) , or magnetic fluctuations.  Figure 4d shows the temper-\nature dependence of the intensity at Q =1.277 Å-1 through the cooling and warming process es. Notably, intensity \nabove background is observed above T N determined from magnetic susceptibility.  A plausible explanation for this \nobservation is that neutron diffraction measurements cover a certain inelastic energy range around E =0 meV, thus \nsuggesting dynamic fluctuations. Such an observation strongly suggests that the magnetic order  occurs in a rela-\ntively wide temperature range, in good agreement with the absence of an anomaly in heat capacity.  \n \nFigure 4 . Magnetic structure and excitations of ( Mg 0.2Co0.2Ni0.2Cu0.2Zn0.2)O. (a) Neutron diffraction peaks at se-\nlected Q  (Q=4πsinθ/λ, λ= 2.41 Å ) ranges at 2 K  and 160 K. (b) Rietveld refinement for the 2 K data with fit qual-\nity R wp=11.7%. Top: magnetic; Middle: nuclear; Bottom: Al can. Inset shows the magnification for the 1.21-1.33 \nÅ-1 range. (c ) Scheme of magnetic structure. +: spin up; -: spin down. (d ) Temperature dependence of the i ntensi-\nties at Q =1.277 Å-1 on cooling and warming processes. (e) Dynamic structure factor at 100 K with Ei= 30 me V.  \n \nThe antiferromagnetic origin of the phase transition at 113  K is also evidenced from collective spin dynamics, i.e. \nmagnons, which were measured on polycrystalline powders by selecting two  incident energy of neutrons  (Ei=140 \n7 and 30 meV ) at the Wide Angular -Range Chopper Spectrometer (ARCS ) spectrometer at the Spallation Neutron \nSource ( SNS). Figure 4e  presents the dyna mic structure factor S( Q,E) for the rocksalt MgO -HEO at 100 K with E i \nof 30 meV. Strong magnons are observed  at Q=1.277 Å-1 with a steep slope, and die out quickly with the  increasing \nof Q. These magnons survive up to 300 K  (see Figure S8), indicating strong interaction between magnetic ions even \nwell above 2× TN. \nThe existence of antiferromagneti c order  in the rocksalt high entropy oxide is not unexpected . Both CoO  and NiO \nhave the NaCl -type crystal structure, and are antiferromagnet ic with propagation wavevector of k=(½, ½, ½) and \nNéel temperature of  291 K for CoO and 523 K for NiO .26 The substitution  with nonmagnetic ions such as Mg  or \nZn suppresses long range antiferromagnetic order  in these FCC antiferromagnet s, as was exemplified by the Co 1-\nxMg xO solid solution which  becomes a spin glass for x >0.53 and a paramagnet for x >0.87,27 and by Ni1-xMg xO \nwhich is a frustrated antiferromagnet (0.37< x≤0.6), spin glass (0.6< x≤0.75) and paramagnet state ( x≥0.8).28 Sup-\nposing that x=0.4 applies for the case of MgO -HEO, it would correspond to the antiferromagnetic state of Co1-\nxMg xO or the frustrated antiferromagnetic state of Ni 1-xMg xO.  \nHowever , there are several major differences between MgO -HEO and the randomly diluted FCC system. Firstly, \nthe magnitude of the spin exchange interactions in MgO -HEO are expected to be different from diluted FCC sys-\ntems due to the existence of Cu2+ (d9, S=1/2) on t he 4a  site in addition to Co2+ (d7, S=3/2)  and Ni2+ (d8, S=2/2).  In \nthe rocksalt structure, the transition metal (TM)  ions are connected via oxygen atoms with TM -O-TM bond angle \n180° . The magnetic exchange interactions are superexchange interactions. Based on Kannamori-Goodenough \nrules,29-30 the TM -O-TM coupling in MgO -HEO can be effectively AFM or FM, depen ding on the electronic con-\nfiguration of the two coupled TM ions. AFM interactions dominates for Co (or Ni, Cu) -O-Co(or Ni, Cu ). For NiO \nand CoO based diluted systems, the N éel temperature is dominated by the next -nearest -neighbor exchange interac-\ntion, J2.31-32 For example, the spin exchange interactions in CoO are reported to be J2= -25.54 meV and J 1= 8.00 \nmeV (nearest -neighbor interaction).33 The exchange interactions in MgO -HEO are the key to understand its long \nrange antiferromagnetism. To address this, theoretical calculations on spin exchange interactions are required. Sec-\nondly, no structural transition or deformation has been observed across the magnetic transition in MgO -HEO, in \nsharp contrast to the diluted FCC systems with nonmagnetic ions , which are tetragonal , rhombohedral  or mono-\nclinic  in the antiferromagnetic state .34-37 Thirdly, the absence of a λ-shape peak around the magnetic transition in \n8 heat capacity in MgO -HEO contrasts with that  in CoO.38 Such characteristics are related to the large degree of \nchemical disorder stabilized by high configuration entropy. It will be  very interesting to study the magnetic transi-\ntions  and magnetic structures of other  types of  high entropy oxides to see if these features are general . Along this \nline, the perovskite  R(Cr 0.2Mn 0.2Fe0.2Co0.2Ni0.2)O3 (R=La, Gd, Nd, Sm, Y)19 are obvious candidates. \nIn summary , the rocksalt high entropy oxide is long range antiferromagnetically  ordered below TN=113 K, as \nevidenced by a cusp in DC magnetic susceptibility, magnetic peaks in neutron powder diffraction, and strong mag-\nnetic excitations in inelastic neutron scattering. The magnetic structure consist s of ferromagnetic sheets in the (111) \nplanes with spins antiparallel between two neighboring planes , similar to NiO or CoO. The  ordered magnetic mo-\nment is 1.4(1) μB. This work opens a new chapter for understanding the magnetic properties of high entropy oxides, \na fascinating class of materials stabilized by high configuration entropy.  \n \nACKNOWLEDGMENT   \nThis material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic \nEnergy Sciences, Materials Sciences and Engineering Division.  The use of Oak Ridge National Laboratory’s Spall-\nation Neutron Source and High Flux Isotope Reactor was supported by the Scientific User Facilities Division, \nOffice of Basic Energy Sciences, U.S. Department of Energy.  Use of the Advanced Photon Source, an Office of \nScience User Facility operated for the US Department of Energy (DOE) Office of Science by Argonne National Labo ratory, was supported by the US DOE under Contract DE -AC02 -06CH11357. J.Z. would like to thank Mr. \nLiang Wang for his help with x- ray pair distribution function data at 11- ID-C. 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Moler1, 2\n1Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory,\n2575 Sand Hill Road, Menlo Park, CA 94025, USA\n2Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA\n3Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA\n4Department of Chemistry, Princeton University, Princeton, New Jersey 08540, USA\n(Dated: March 28, 2019)\nHolmium titanate (Ho 2Ti2O7) is a rare earth pyrochlore and a canonical example of a classical\nspin ice material. Despite the success of magnetic monopole models, a full understanding of the en-\nergetics and relaxation rates in this material has remained elusive, while recent studies have shown\nthat defects play a central role in the magnetic dynamics. We used a scanning superconducting\nquantum interference device (SQUID) microscope to study the spatial and temporal magnetic \ruc-\ntuations in three regions with di\u000berent defect densities from a Ho 2Ti2O7single crystal as a function\nof temperature. We found that the magnetic \rux noise power spectra are not determined by sim-\nple thermally-activated behavior and observed evidence of magnetic screening that is qualitatively\nconsistent with Debye-like screening due to a dilute gas of low-mobility magnetic monopoles. This\nwork establishes magnetic \rux spectroscopy as a powerful tool for studying materials with complex\nmagnetic dynamics, including frustrated correlated spin systems.\nKeywords: Condensed matter physics\nI. INTRODUCTION\nClassical spin ices such as holmium titanate\n(Ho2Ti2O7) have generated intense interest, both the-\noretical and experimental, in the last two decades.1{11\nThe pyrochlore lattice of corner-sharing tetrahedra hosts\na magnetic holmium atom at each shared vertex, and\nthe local crystal \feld environment causes each holmium\nmoment to point directly into one or the other of the\nadjacent tetrahedra. The term \\spin ice\" comes from\nthe analogy between the ground state manifold, which\nhas two spins pointing in and two spins pointing out of\neach tetrahedron (the ice rule ), and the freezing of water\nice, in which the two electron lone pairs on the oxygen\natom line up with the hydrogen atoms of adjacent water\nmolecules.1\nA single spin \rip from the ground state manifold re-\nsults in an excitation that is dipole-like. A subsequent\n\rip of a nearest-neighbor spin can restore the ice rule in\none of the tetrahedra while violating it in the adjacent\ntetrahedron, extending the dipole to next nearest tetra-\nhedra. Continuing this process, one end of the dipole\ncan be taken away entirely, leaving a single monopole-\nlike excitation behind.8Much of the recent theoretical\nwork has focused on these emergent magnetic monopole\nmodels.3,5,6,8,9,12{14Nevertheless, it has subsequently be-\ncome clear that defects, such as oxygen vacancies and\nstu\u000bed spins (additional spins from Ho atoms occupying\nTi sites), must be accounted for in understanding the full\nmagnetic dynamics of spin ice.15{17Progress in under-\nstanding these dynamics has been limited by a relative\nlack of suitable tools for microscopic magnetic studies.\nIn this paper, we demonstrate the utility of magnetic\n\rux noise spectroscopy as such a tool for studying frus-\ntrated correlated spin systems. We used a scanning su-perconducting quantum interference device (SQUID) mi-\ncroscope with a gradiometric SQUID magnetometer that\nhas been described previously.18With a spatial resolu-\ntion of 4:6\u0016m and a magnetic \rux noise \roor of or-\nder 1\u0016\b0=p\nHz, we measured the temperature depen-\ndence of spatial magnetic correlations and of the mag-\nnetic \rux noise spectrum in three regions with di\u000berent\ndefect densities from a Ho 2Ti2O7single crystal. We ob-\nserved qualitative deviations from simple thermally acti-\nvated behavior, which would predict a Lorentzian noise\nspectrum with a characteristic time that follows an Ar-\nrhenius law in temperature. Furthermore, we found ev-\nidence of screening at low frequencies and high tem-\nperatures, which we compare to a model for Debye-like\nscreening from the theoretically predicted gas of mag-\nnetic monopoles.19,20\nII. EXPERIMENTAL SETUP\nWe measured two samples from a single crystal grown\nvia the \roating-zone method which has previously been\ncharacterized elsewhere;17sample A is from near the cen-\nter of the growth boule while Sample B is from near\nthe edge, where the density of defects was somewhat\nhigher. Sample A was transparent pink, while Sample\nB was cloudy but translucent pink, with a dark, opaque\nregion at one corner. We measured both regions of Sam-\nple B, which we will subsequently refer to as Samples B1\nand B2, respectively. We fractured sample A to obtain\na smooth but not \rat surface with roughly [111] orienta-\ntion. For Sample B, we prepared a polished [111] surface\nwith<1\u0016m grit polishing \flm and isopropanol.\nFor two-dimensional image data, we de\fned the scan\nsurface by determining the height at which the SQUID\nwas in contact with the sample at a series of locationsarXiv:1903.11465v1  [cond-mat.str-el]  27 Mar 20192\n0.43 K 18 hours later\n145 µm\n1 hour later\n-1\n/uni0394Φ (mΦ0)1\n0\nFIG. 1. Magnetometry scans of Sample A at 430 mK taken over 18 hours. Overlay in \frst panel is scale drawing of SQUID\npickup loop (orange) and \feld coil (blue). The pickup loop size sets the spatial resolution of the images; resolution-limited\nfeatures in the scans are qualitatively similar from one scan to the next, but are di\u000berent in the details. This demonstrates\nthat magnetic dynamics persist, even over long timescales and at the lowest measured temperatures, and that the magnetic\ntexture fails to order or otherwise converge.\nand \ftting a two-dimensional, second order polynomial to\nthe surface topography. We rastered the SQUID parallel\nto this surface at a nominal height of 1 \u0016m. For one\ndimensional scans, we acquired a single row of the image\nrepeatedly to discern the time evolution.\nWe obtained magnetic \rux noise power spectra by\nplacing the SQUID in contact with the sample and col-\nlecting each spectrum with an SR760 FFT Spectrum An-\nalyzer in three overlapping segments (3.8 mHz{1.52 Hz,\n488 mHz{195 Hz, and 125 Hz{49.9 kHz), using a Han-\nning window function and 128 exponentially weighted av-\nerages.\nIII. RESULTS\nWe present magnetometry scans of Sample A taken at\nour base temperature of 430 mK in Fig. 1. At this tem-\nperature, features which are limited by the spatial resolu-\ntion of the SQUID magnetometer (4.6 \u0016m) dominate each\nscan, suggesting that the dynamics are slow compared to\nthe scan speed. Repeated scans appear as di\u000berent pan-\nels, at times indicated at the top of each panel, showing\nlong timescale \ructuations despite qualitative similarity\nfrom scan to scan. As expected for a truly frustrated spin\nsystem, the sample fails to show convergence or ordering\nof the magnetic texture even over the hold time of our\ncryostat, in excess of two days, at 430 mK.\nIn magnetometry scans conducted as a function of tem-\nperature, from base temperature to 810 mK, we observe\nthat magnetic dynamics quicken as the temperature is in-\ncreased. In Fig. 2, we show both two-dimensional scans\n[Fig. 2(a)], each acquired over several minutes, and one-\ndimensional scans as a function of time [Fig. 2(b)], each\nacquired over 200 minutes with each row taking approxi-\nmately 12 seconds, taken at various temperatures. As in\nFig. 1, scans at the lowest temperatures (\frst panels in\neach part of the \fgure) show resolution-limited features,implying that temporal dynamics occur on timescales\nlong compared to the sampling rate. The \frst panel\nin Fig. 2(a) shows this explicitly, as the features persist\nover many rows, suggesting a correlation time of order\nhours. As the temperature is increased, the features in\nthe scans in Fig. 2(a) vary on shorter length scales, indi-\ncating that there are faster temporal variations coming\ninto play. This is made manifest by comparing adjacent\nrows in the various panels of Fig. 2(b), where the corre-\nlation time falls to order seconds by 610 mK. Due to the\nlimited scan speed of the SQUID, these measurements\ncannot resolve magnetic dynamics at temperatures above\n1 K, as it is di\u000ecult to unambiguously distinguish spatial\nand temporal variations. To measure at higher tempera-\ntures, we instead \fx the position of the SQUID in contact\nwith the sample surface and measure the magnetic \rux\nas a function of time only. By taking the Fourier trans-\nform of time series data, we obtain a magnetic \rux noise\npower spectrum.\nThe key results of this paper are contained in plots of\nthe natural logarithm of the magnetic \rux noise power\nspectra, in units of \b2\n0=Hz, as a function of the standard\nlogarithm of frequency, log( f), and the inverse tempera-\nture, 1=T. Were the sample an ensemble of identical but\nnon-interacting, thermally-excited Ising spins, we would\nobserve a Lorentzian noise spectrum, S\b=c\u001c=(1+!2\u001c2),\nwithcconstant,!the measurement frequency, and \u001ca\ntemperature-dependent characteristic time. The thermal\nexcitation over the Ising barrier would yield an Arrhenius\nlaw for that temperature dependence, \u001c=\u001c0eEa=kBT,\nwhere\u001c0is a microscopic attempt time, Eais an activa-\ntion energy, kBis the Boltzmann constant, and Tis the\ntemperature. In this illustrative example, the contours\nof constant noise power would be vertical at high tem-\nperatures and linear at low temperatures. For Lorentzian\nnoise spectra, the noise power monotonically increases for\ndecreasing frequency, down to a characteristic frequency\nat which it plateaus. The line formed by the maxima of3\n0.43 K\n0.81 K 0.79 K 0.77 K 0.75 K 0.73 K0.71 K 0.69 K 0.67 K 0.65 K 0.63 K0.61 K 0.59 K 0.57 K 0.55 K 0.53 K0.51 K 0.49 K 0.47 K 0.45 Ka.\n-1\n/uni0394Φ\n(mΦ0)1\n0\nb.\n0.47 K 0.50 K 0.52 K 0.55 K 0.58 K 0.61 K40 µm\n40 µm\n200 min\nFIG. 2. Magnetometry scans of Sample A as a function of\ntemperature. (a) Two-dimensional scans, as in Fig. 1, from\nbase temperature to 810 mK. As the temperature increases,\nthe observed \ructuations become sub-resolution, suggesting\nthat there are temporal \ructuations that are fast compared\nto the scan speed. (b) One-dimensional scans vs. time from\n470 mK to 610 mK. Each series is 200 minutes long, and the\nvertical correlations of pixels from row to row characterize the\ncorrelation time, which is of order hours at 470 mK but falls\nto seconds by 610 mK.\n0.5 1 1.5 2\n1/T [1/K]-2-101234log10(f [Hz])\n-18-14-10-6\nlog10(Flux PSD [Φ02/Hz])\nTmax[21]\nTmax[22]\nTmax[23]\nfmax[23]\nTmaxFIG. 3. Magnetic \rux noise power spectra as a function of\ntemperature for Sample A. Overlaid red line is a linear \ft to\nmaxima for each row, while markers indicate ac susceptibility\ndata from previous works for comparison.\nhorizontal line cuts gives the Arrhenius law: the slope\ngivesEawhile the intercept is related to \u001c0.\nIn Fig. 3, we show the magnetic \rux noise spectra for\nSample A from 430 mK to 4 K. The data are manifestly\nnon-Arrhenius: the contours of constant noise power at\nlow temperatures \rare out, with the noise power from\n1{100 Hz at 430 mK higher than would be expected for\na Lorentzian following an Arrhenius law. For tempera-\ntures above 650 mK, the \rux noise power as a function of\ndecreasing frequency reaches a maximum and then falls\no\u000b sharply at lower frequencies, below 1 Hz at 1 K. This\nfeature suggests that, regardless of whether the observed\nmagnetic \rux noise spectra result from the dynamics of\nmagnetic monopoles or some other microscopic origin,\nthere is a source of magnetic screening within the sam-\nple. The overlaid red line in Fig. 3 is a best \ft line (Ar-\nrhenius law) to the maxima of each row. Comparing the\nextracted Arrhenius law to previously reported bulk ac\nsusceptibility data,21{23we \fnd that it is in close agree-\nment, suggesting that we are measuring the same mag-\nnetic dynamics as have previously been reported.14,21{31\nTo understand the impact on the magnetic dynamics of\ndefects, such as those introduced by additional magnetic\nholmium atoms on titanium sites (stu\u000bed spins),16,17,32,33\nwe measured two regions from an additional sample\n(Sample B) taken from nearer to the edge of the growth\nboule. The additional \rux noise spectra are shown in\nFig. 4, together with those from Sample A. The predom-\ninant Arrhenius-like feature smears out considerably and\nbecomes somewhat less steep as the defect density is in-\ncreased, implying a broadening distribution of activation\nenergies that are lower on average. This is consistent with\nexpectations for increased defect densities, as magnetic\ndisorder broadens and reduces the barrier for individual\nspins in the sample to \rip.\nThe qualitative deviations from Arrhenius behavior\nseen in Sample A can be seen more clearly in Sample B.4\n0.5 1 1.5 2\n1/T [1/K]-2-101234log10(f [Hz])\n-18-14-10-6\nlog10(Flux PSD [Φ02/Hz])\n0.5 1 1.5 2\n1/T [1/K]-2-101234log10(f [Hz])\n-18-14-10-6\nlog10(Flux PSD [Φ02/Hz])a.\nb.\nFIG. 4. Comparison of magnetic \rux noise power spectra\nfrom two locations on second sample (a. B1; b. B2), demon-\nstrating di\u000berent defect densities. The central, Arrhenius-like\nfeature seen in Sample A is also seen in both data sets; the\nfeature becomes broader, more di\u000buse, and more shallow as\nthe defect density is increased, consistent with the barrier to\nspin \rips shortening and becoming broadened with increased\nmagnetic disorder. Qualitative deviations from Arrhenius be-\nhavior, seen as excess noise at center left and bottom right\nof each panel, and screening at bottom left of each panel, ap-\npear in all data sets and are quantitatively similar in their\nfrequency and temperature dependence. These comparisons\nsuggest that the central, Arrhenius-like feature is a\u000bected by\nmagnetic defects, while the deviations from Arrhenius behav-\nior are universal.\nThe \raring out of the contours of constant noise power\nat low temperatures have corresponding features at high\ntemperatures, seen above 1.3 K from 0.1{100 Hz. The\ncorrespondence of the excess noise features at the highest\nand lowest temperatures suggests that there is an addi-\ntional source of magnetic dynamics in these samples at\nlower frequencies than has been accessible in previous ac\nsusceptibility measurements. Furthermore, we see that\nthe screening behavior in Sample A is also seen in Sam-\nple B, and that it is quantitatively comparable across all\nthree samples, independent of the defect density.IV. DISCUSSION\nHaving shown the \rux noise spectra from our samples,\nwe now turn to the question of what we expect from a\ndilute gas of monopoles. The basic form of our model is\na Lorentzian noise term (this form for the noise due to\nmagnetic monopoles was previously robustly justi\fed by\nRyzhkin20), modi\fed by a Debye-like screening term:\nS\b=C\u0012\u001cMon\n1 +!2\u001c2\nMon\u0013!2=!2\nc\n1 +!2=!2c(1)\nwhereCis an overall scaling constant, !is the angular\nfrequency,!cis a characteristic cuto\u000b frequency for De-\nbye screening as described below, and \u001cMon is the char-\nacteristic time associated with spin relaxation, \u001cMon =\n\u001cMon; 0=x(T), where\u001cMon; 0is the microscopic monopole\nhopping time and x(T) is the temperature-dependent\nmonopole concentration. This relaxation time is respon-\nsible for monopole hopping by way of the \ructuation-\ndissipation theorem.19\nThe Debye-H uckel model, as applied to the case of\nmagnetic monopoles in spin ice by Castelnovo, Moess-\nner, and Sondhi,8implies that the magnetic \felds due\nto a source magnetic charge will be screened by a cloud\nof magnetic monopoles equal and opposite in charge to\nthe source. This screening occurs over a length scale, the\nDebye length lD, which is given by:\nlD=s\nkBTV0\n\u00160Q2x(2)\nwithTthe temperature, V0the volume of the dia-\nmond lattice site, Qthe monopole charge, and xthe\nmonopole concentration. The Debye-H uckel concentra-\ntion for monopoles, x(T), can be calculated iteratively\nas described in Ref.8.\nThe Debye length is of order 50 nm for the lowest tem-\nperatures at which we performed \rux spectroscopy, and\nmonotonically decreases as the temperature rises, such\nthat it is always far smaller than the spatial resolution\nfor the SQUID. This suggests that in the presence of\nmonopoles there would be no observable magnetic \ructu-\nations whatsoever; however, because the monopoles have\na \fnite mobility, only slowly varying magnetic \felds are\nscreened.\nThe mobility, \u0016, appears in the characteristic Debye\nfrequency by way of the di\u000busivity, D, and the Einstein\nrelation:\n!c=D\nl2\nD=\u0016kBT\nl2\nD: (3)\nThe mobility itself has been calculated from Monte Carlo\nsimulations:8\n\u0016=4\n27a2\nd\nkBT\u001c(4)5\n0.5 1 1.5 2\n1/T [1/K]-2-101234log10(f [Hz])\n-18-14-10-6\nlog10(Flux PSD [Φ02/Hz])\n10/one.superior s10⁰ s10/hyphen.superior/one.superior s10/hyphen.superior/two.superior s10/hyphen.superior/three.superior s\nFIG. 5. Temperature dependence of the Debye screening criti-\ncal frequency, !c, for di\u000berent values of the monopole hopping\ntime. The contours indicate !c(T) for each hopping time, as\nlabeled.\nwhereadis the lattice constant and \u001cis the Monte Carlo\nstep time, which we identify as the same magnetic relax-\nation time as is used in the noise calculation above.\nSince the Debye-H uckel concentration can be com-\nputed directly, the noise and screening due to magnetic\nmonopoles can be modeled with only two free param-\neters, the microscopic monopole hopping time and an\noverall scale factor that takes into account geometric fac-\ntors that couple the \ructuating population of monopoles\nand the SQUID magnetometer.\nIn Fig. 5, we overlay contours that give the characteris-\ntic frequency as a function of temperature, !c(T), for the\nDebye screening for various values of the monopole hop-\nping time, as indicated. Noting that the characteristic\nfrequency is where the screening is of order unity, while\nthe blue region in the bottom left corner of Fig. 5 is where\nthe noise is already reduced by orders of magnitude, we\nidentify the monopole hopping time as 1{10 ms, in agree-\nment with some previous measurements.6,23,26,34,35We\nalso note the qualitative agreement between the data and\nplotted contours for the temperature dependence of the\nscreening.\nTaking the monopole hopping time \u001c0= 3 ms, we plot\nthe full monopole model including the noise and screen-\ning terms in Fig. 6, in arbitrary units. We see that the\nmonopole dynamics account not only for the screening at\nhigh temperatures and low frequencies, but can also qual-\nitatively account for the non-Arrhenius source of noise\nas well. A full modeling of the measured noise spec-\ntrum would also require a model for the Arrhenius-like\nnoise behavior. The defect series that we have measured\nhere suggests that this noise feature is due to defects,\nmost likely stu\u000bed spins. Previous studies have shown\nthat even nominally stochiometric Ho 2Ti2O7grown by\nthe \roating zone method contains approximately 3% Ho\nstu\u000eng on the Ti site, or roughly 0.06 stu\u000bed Ho spins\nper tetrahedron.17Given that this exceeds the calcu-\n0.5 1 1.5 2\n1/T [1/K]-2-101234log10(f [Hz])FIG. 6. A model of the expected magnetic dynamics for\nmonopoles in Ho 2Ti2O7. A band of noise, plotted in arbi-\ntrary units, is present above the critical frequency, as the\nmonopole gas is too dilute and immobile to completely screen\nitself. The monopole hopping time used here is 3 ms and is\nthe only free parameter; the resulting model is qualitatively\nconsistent with all deviations from Arrhenius behavior in our\ndata.\nlated monopole density at all but the highest temper-\natures measured (at which the calculated monopole den-\nsity reaches nearly 0.15 monopoles/tetrahedron), it is un-\nsurprising that defect dynamics would produce magnetic\n\rux noise of a similar magnitude to monopoles.\nOne possible route towards modeling these dynamics\nwould be to calculate the distribution of activation ener-\ngies for stu\u000eng defects in a transverse \feld Ising model.\nA holmium atom on a titanium site has 6 nearest neigh-\nbor holmium spins that form a closed hexagon in the py-\nrochlore lattice. In their normal Ising orientations, these\nspins each provide an in-plane \feld for the defect spin.\nIf the ice state manifold is taken into account in deter-\nmining the frequency with which di\u000berent orientations\nof these nearest neighbor spins will occur, a distribu-\ntion of activation energies could be calculated. However,\nthis would not yield the distribution of attempt times\nwhich is also necessary for a full accounting of the mag-\nnetic dynamics of the defects. Other defects, such as\nnon-magnetic substitutions on the holmium site or oxy-\ngen vacancies,16could also give rise to magnetic dynam-\nics that would not be accounted for in this model, and\n\u001c0could also be temperature dependent for other rea-\nsons not considered here, such as non-trivial spin-phonon\ncoupling.8\nV. CONCLUSION\nWe have demonstrated the utility of scanning SQUID\nmagnetic \rux spectroscopy by measuring the magnetic\n\rux noise power spectra as a function of temperature in\nthree locations on two samples of the classical spin ice6\nHo2Ti2O7. In these measurements, we observe a dom-\ninant Arrhenius-like feature that matches the behavior\nobserved in previous bulk ac susceptibility measurements\non similar samples. We identify this feature as the result\nof the magnetic dynamics of defects in the sample, which\nwe speculate are stu\u000bed spins.\nWe further identify three qualitative deviations from\nArrhenius behavior in all three datasets, namely excess\nnoise below 10 Hz at the lowest temperatures and be-\nlow 100 Hz at the highest temperatures and screening of\nthe noise at high temperatures and low frequencies. We\n\fnd that all three of these behaviors are consistent with\nthe expected dynamics of a dilute, low-mobility gas of\nmagnetic monopoles.\nOur measurements represent a new technique that is\ncomplementary to existing magnetic probes used in the\nstudy of frustrated magnetic systems. We demonstrate\nthe importance of quantitative modeling for the mag-\nnetic dynamics of defects in these systems and the utility\nof scanning SQUID magnetic \rux spectroscopy in disen-tangling the overlapping magnetic signals of such defects\nand the essential physics of the system under study, with\npotential further applications in the study of other, re-\nlated magnetic systems such as spin liquids.\nACKNOWLEDGMENTS\nWe thank C. Castelnovo, B. Gaulin, M. Gingras, G.\nLuke, R. Moessner, H. Noad, J. Rau, K. Ross, and S.L.\nSondhi for helpful discussions. This work was primar-\nily supported by the Department of Energy, O\u000ece of\nScience, Basic Energy Sciences, Materials Sciences and\nEngineering Division, under Contract No. DE-AC02-\n76SF00515; Ilya Sochnikov was partially supported by\nthe Gordon and Betty Moore Foundation through grant\nGBMF3429. The crystal growth was supported by the\nUS Department of Energy, Division of Basic Energy Sci-\nences, grant de-sc0019331.\n\u0003These two authors contributed equally\n1B. C. den Hertog and M. J. P. Gingras, Phys. Rev. Lett.\n84, 3430 (2000).\n2S. T. Bramwell, M. J. Harris, B. C. den Hertog, M. J. P.\nGingras, J. S. Gardner, D. F. McMorrow, A. R. Wildes,\nA. L. Cornelius, J. D. M. Champion, R. G. Melko, and\nT. Fennell, Phys. Rev. Lett. 87, 047205 (2001).\n3T. Fennell, P. P. Deen, A. R. Wildes, K. Schmalzl, D. Prab-\nhakaran, A. T. Boothroyd, R. J. Aldus, D. F. McMorrow,\nand S. T. Bramwell, Science 326, 415 (2009).\n4D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke,\nC. 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Thygesen,1, 2and Thomas Olsen1\n1Computational Atomic-scale Materials Design (CAMD), Department of Physics,\nTechnical University of Denmark, DK-2800 Kgs. Lyngby, Denmark\n2Center for Nanostructured Graphene (CNG), Department of Physics,\nTechnical University of Denmark, DK-2800 Kongens Lyngby, Denmark\n(Dated: February 18, 2020)\nThe recent observation of ferromagnetic order in two-dimensional (2D) materials has initiated a\nbooming interest in the subject of 2D magnetism. In contrast to bulk materials, 2D materials can\nonly exhibit magnetic order in the presence of magnetic anisotropy. In the present work we have used\nthe Computational 2D Materials Database (C2DB) to search for new ferromagnetic 2D materials\nusing the spinwave gap as a simple descriptor that accounts for the role of magnetic anisotropy.\nIn addition to known compounds we \fnd 12 novel insulating materials that exhibit magnetic order\nat \fnite temperatures. For these we evaluate the critical temperatures from classical Monte Carlo\nsimulations of a Heisenberg model with exchange and anisotropy parameters obtained from \frst\nprinciples. Starting from 150 stable ferromagnetic 2D materials we \fnd \fve candidates that are\npredicted to have critical temperatures exceeding that of CrI 3. We also study the e\u000bect of Hubbard\ncorrections in the framework of DFT+U and \fnd that the value of U can have a crucial in\ruence\non the prediction of magnetic properties. Our work provides new insight into 2D magnetism and\nidenti\fes a new set of promising monolayers for experimental investigation.\nI. INTRODUCTION\nThe nature of magnetic order in two-dimensional (2D)\nmaterials is fundamentally di\u000berent from the three-\ndimensional case. In 3D, magnetic order arises from\nspontaneously broken symmetry of the magnetization\ndirection and the magnetic anisotropy only plays a\nmarginal role. In 2D, however, the Mermin-Wagner\ntheorem1prohibits a broken symmetry phase at \fnite\ntemperatures and the spin rotational symmetry has to\nbe broken explicitly by magnetic anisotropy.\nIn 2017, two examples of 2D ferromagnetic insulators\nwere discovered experimentally. 1) A monolayer of CrI 3\nthat exhibits magnetic order below 45 K.22) A bilayer\nof Cr 2Ge2Te6with a Curie temperature of 20 K.3In\nthe case of CrI 3the the magnetic order is driven by a\nstrong out-of-plane magnetic anisotropy - a case that is\noften referred to as Ising-type ferromagnet. In contrast,\nCr2Ge2Te6has a rather weak magnetic anisotropy and\nthe magnetic order is maintained in bilayer structures\nby interlayer exchange couplings. Subsequently, ferro-\nmagnetic order at room temperature has been reported\nin monolayers of MnSe 24and VSe 2.5Both of these are\nitinerant (metallic) ferromagnets and the origin of mag-\nnetism in these materials is still not completely clari\fed.\nIn particular, VSe 2has an easy plane, which implies lack\nof magnetic order by virtue of the Mermin-Wagner theo-\nrem. However, such a two-dimensional spin system may\ncomprise an example of a Kosterlitz-Thouless phase,6\nwhich is known to display magnetic order due to \fnite\nsize e\u000bects.7More recently, Fe 3GeTe 28was reported to\nhost itinerant ferromagnetic order below 130 K, which\noriginates from strong out-of-plane magnetic anisotropy.\nSeveral other 2D materials have been predicted to exhibit\neither ferromagnetic or anti-ferromagnetic order based on\frst principles calculations, but in most cases the predic-\ntions have not yet been con\frmed by experiments9and\nestimates of the critical temperatures are often unjusti-\n\fed or very crude.\nTwo-dimensional CrI 3has proven to comprise a highly\nversatile material. For example, an applied electric\n\feld can induce Dzyaloshinskii-Moriya interactions,10\nand switch the magnetic state in bilayer samples.11,12In\naddition, it has been demonstrated that one can obtain\ncontrol of in-plane conductivity and valley polarization\nby constructing heterostructures of CrI 3/graphene13and\nCrI3/WSe 214respectively. van der Waals heterostruc-\ntures of 2D materials involving magnetic layers thus con-\nstitute a highly \rexible platform for designing spin tunnel\njunctions and could provide new ways to build nanos-\ntructured spintronics devices.9,15,16However, in order to\nmake 2D magnetism technologically relevant there is a\npressing need to \fnd new 2D materials that exhibit mag-\nnetic order at higher temperatures.\nA theoretical search for materials with particular prop-\nerties may be based on either experimental databases\nsuch as the Inorganic Crystal Structure Database (ICSD)\nor computational databases where \frst principles sim-\nulations are employed to predict new stable materials.\nThe former approach has been applied to predict new\n2D materials rooted in exfoliation energies of 3D materi-\nals in the ICSD17{19and several materials was found to\nhave a magnetically ordered ground state (at T= 0 K).\nAn example of the latter approach is the Computational\n2D Materials Database (C2DB);20,21presently contain-\ning 3712 2D materials of which 20 % are predicted to be\nstable. One advantage of using a theoretical database is\nthat the search is not restricted to materials that are\nexperimentally known in a 3D parent van der Waals\nstructure. However, materials predicted from theoreti-arXiv:1903.11466v2  [cond-mat.mtrl-sci]  17 Feb 20202\ncal databases may pose severe challenges with respect to\nsynthetization and experimental characterization; even if\nthey are predicted to be stable by \frst principles calcu-\nlations.\nRegarding the magnetic properties of materials, a ma-\njor di\u000eculty stems from the fact that standard \frst prin-\nciples methods can only predict whether or not mag-\nnetic order is present at T= 0 K. For 2D materials\nthe Mermin-Wagner theorem implies that magnetic or-\nder atT= 0 vanishes at any \fnite temperature in the\nabsence of magnetic anisotropy. A \frst principles pre-\ndiction of magnetic order at T= 0 is therefore irrele-\nvant unless other properties of the material are taken\ninto account. The question then arises: how to calculate\nthe critical temperature for magnetic order given a set\nof exchange and anisotropy parameters for a particular\nmaterial. It is clear that the Mermin-Wagner theorem\ndisquali\fes any standard mean-\feld approach because\nsuch methods neglect the \ructuations that are respon-\nsible for deteriorating magnetic order at \fnite tempera-\ntures in the absence of magnetic anisotropy. On the other\nhand, the importance of having magnetic anisotropy and\nan easy axis for the magnetization (as opposed to an\neasy plane) has led many authors to derive the mag-\nnetic properties from an Ising model for which the critical\ntemperatures are known for all Archimedian lattices.22,23\nHowever, the Ising model only provides a good magnetic\nmodel in the limit of in\fnite single-ion anisotropy and\nsimply provides an upper bound for the critical tem-\nperature in general.24For example, in the case of CrI 3,\nwhich is regarded as an Ising-type ferromagnet, the Ising\nmodel overestimates the critical temperature by a factor\nof three. The e\u000bect of \fnite anisotropy was analyzed in\nRef. 24, where Monte Carlo simulations and renormal-\nized spin-wave theory were applied to obtain a simple\nexpression for the critical temperature of Ising-type fer-\nromagnets. The expression only depends on the number\nof nearest neighbors, the nearest neighbor exchange inter-\nactions, and two anisotropy parameters. In the present\nwork we have applied this expression to search the C2DB\nfor ferromagnetic materials with \fnite critical tempera-\ntures. For some materials in the C2DB, the magnetic\nstructure is not well approximated by an Ising-type fer-\nromagnet and we have performed full Monte Carlo sim-\nulations to obtain the critical temperatures of these ma-\nterials.\nThe paper is organized as follows. In Sec. II we sum-\nmarize the computational details and discuss the Heisen-\nberg model, which forms the basis for calculations of crit-\nical temperatures in the present work. In Sec. III we\npresent the magnetic materials found by searching the\nC2DB and discuss and compare the calculated critical\ntemperatures with previous works. Sec. IV contains a\nconclusion and outlook.II. METHOD AND COMPUTATIONAL\nDETAILS\nThe materials in C2DB have been found by perform-\ning \frst principles calculations of hypothetical 2D mate-\nrials in the framework of density functional theory (DFT)\nwith the Perdew-Burke-Ernzerhof (PBE) exchange-\ncorrelation functional using the electronic structure pack-\nage GPAW.25,26The geometry of all materials are fully\noptimized and the dynamical stability is obtained based\non phonon frequencies at the the center and corners of\nthe Brillouin zone. The heat of formation is calculated\nwith respect to standard references20and a material is\nregarded as thermodynamically stable if it situated less\nthan 0.2 eV above the convex hull de\fned by the 2807\nmost stable binary bulk compounds from the OQMD.27\nWe \fnd that more than 700 materials in the C2DB are\npredicted to have a ferromagnetic ground state and \u0018150\nof these are thermodynamically and dynamically stable.\nThe DFT calculations show whether or not the mate-\nrials have a ferromagnetic ground state at T= 0 and\nfor insulators the critical temperature can then be ob-\ntained from the descriptor derived in Ref. 24 or Monte\nCarlo simulations. The procedure requires knowledge of\nexchange and anisotropy parameters, which can be ob-\ntained from an energy mapping analysis including spin-\norbit coupling non-selfconsistently.28We will brie\ry out-\nline the approach below.\nThe magnetic properties of a system of localized spins\nare commonly analyzed in terms of the Heisenberg model.\nThe most basic ingredient in the model is the isotropic ex-\nchange interactions arising between neighboring spins as\na consequence of Coulomb repulsion and Pauli exclusion.\nIn addition, spin-orbit coupling may lead to magnetic\nanisotropy, which manifests itself through anisotropic ex-\nchange interactions29as well as single-ion anisotropy. 2D\nmaterials often exhibit (nearly) isotropic magnetic inter-\nactions in the plane of the materials and in the following\nwe will restrict ourselves to models of the form\nH=\u00001\n2X\ni6=jJijSi\u0001Sj\u0000X\niAi(Sz\ni)2\u00001\n2X\ni6=jBijSz\niSz\nj;(1)\nwhere the sums over iandjrun over all magnetic sites.\nJijdenotes the isotropic exchange between spins at site i\nandj,Bijis the anisotropic exchange for spins pointing\nout of the plane (here assumed to be the z-direction),\nandAiis the single-ion anisotropy. We will also assume\nthat the model is composed of a single kind of magnetic\natom, which is characterized by a half-integer S, yielding\nthe maximum possible eigenvalue of Szfor any site. The\nmost general form of the exchange interaction between\nsitesiandjcan be written asP\n\u000b\fS\u000b\niJ\u000b\f\nijS\f\nj, where\nJ\u000b\f\nijis a 3\u00023 tensor for a given pair of iandj. This in-\ncludes the Dzyaloshinkii-Moriya interactions as the anti-\nsymmetric part as well as symmetric o\u000b-diagonal compo-\nnents that give rise to Kitaev interactions.29Such terms\nare neglected in the present work since, we are mainly in-3\nb)\nc) d)a)\nFIG. 1. Examples of spin con\fgurations for the calculation of\nHeisenberg parameters A,BandJ: (a) E?\nAFM, (b) E?\nFM, (c)\nEk\nAFM and (d) Ek\nFM.\nterested in critical temperatures which is dominated by\nthe terms included in Eq. (1). However, we emphasize\nthat the neglect of such terms as well as the assumption\nof in-plane magnetic isotropy is an approximation we will\nmake to reduce the set of parameters needed for the iden-\nti\fcation of promising candidates. Below we discuss a\nfew important exceptions exempli\fed by materials with\nlarge critical temperatures that are not well described by\nthe model (1).\nIn order to obtain the magnetic properties of a given\nmaterial based on the model (1), one needs to extract the\nparameters Jij,AiandBij. In the case of a single mag-\nnetic element we have Ai=Aand restricting ourselves\nto nearest neighbor interactions we take Jij=Jand\nBij=Bifi;jare nearest neighbors and Jij=Bij= 0\notherwise. The parameters can then be obtained by map-\nping the model to \frst principles calculations based on\ndensity functional theory.30In particular, the three pa-\nrameters can be obtained from the total energies of the\nfour spin con\fgurations E?(k)\nFM andE?(k)\nAFM, whereEFM\nis the energy of a fully ferromagnetic con\fguration and\nEAFM is an anti-ferromagnetic state that involves anti-\nparallel spin alignment. The superscripts ?andkin-\ndicates whether the spinors are lying in the plane of the\nmaterials or perpendicular to the plane. A ferromagnetic\nmaterial with E?\nFM<Ek\nFM, will thus have an out-of-plane\neasy axis. The four con\fgurations are illustrated in Fig.\n1. All energies are evaluated with the geometry obtained\nfrom the relaxed ferromagnetic ground state. For materi-\nals with a single magnetic atom in the unit cell, we have\ndoubled the unit cell in order to accommodate the anti-\nferromagnetic con\fguration. Comparing with Eq. (1)\nand approximating the spin operators by classical vec-\ntors we can obtain the parameters as\nA=\u0001EFM(1\u0000NFM\nNAFM) + \u0001EAFM(1 +NFM\nNAFM)\n2S2;(2)\nB=\u0001EFM\u0000\u0001EAFM\nNAFMS2; (3)\nJ=Ek\nAFM\u0000Ek\nFM\nNAFMS2; (4)\nwhere \u0001EFM(AFM) =Ek\nFM(AFM)\u0000E?\nFM(AFM)are theenergy di\u000berences between in-plane and out-of-plane\nspin con\fgurations for ferromagnetic(anti-ferromagnetic)\nstructures and NFM(AFM) is the number of nearest\nneighbors with aligned(anti-aligned) spins in the anti-\nferromagnetic con\fguration. For bipartite lattices\nNFM= 0 andNAFM is simply the number of nearest\nneighbors, but the triangular magnetic lattices (for ex-\nample the MoS 2crystal structure) have no natural anti-\nferromagnetic con\fgurations and one has to consider a\nfrustrated con\fguration where each atom has aligned as\nwell as anti-aligned nearest neighbors.\nOnce the parameters have been determined the Curie\ntemperatures can be calculated from an expression ob-\ntained in Ref. 24, which was derived by \ftting to classi-\ncal Monte Carlo (MC) simulations of the model (1) com-\nbined with renormalized spinwave theory. This simpli-\n\fes the procedure signi\fcantly compared to running MC\ncalculations for all materials and still assures a good re-\nliability compared to available experimental data.24The\nexpression for the Curie temperature is.\nTC=S2JTIsing\nC\nkBf\u0012\u0001\nJ(2S\u00001)\u0013\n(5)\nwhere\nf(x) = tanh1=4\u00146\nNnnlog(1 +\rx)\u0015\n(6)\nis a \ftted function with \r= 0:033 andNnnthe num-\nber of nearest neighbors. TIsing\nC is the critical tem-\nperature of the corresponding Ising model (in units of\nJS2=kB), which has been tabulated for all the Archime-\ndian lattices,22and are given by 1.52, 2.27, and 3.64 for\nhoneycomb, square and quadratic lattices respectively.\n\u0001 =A(2S\u00001) +BSNnn (7)\nis the spinwave gap derived from spinwave theory. It fol-\nlows from the Mermin-Wagner theorem that a positive\nspinwave gap is a minimal requirement for magnetic or-\nder in 2D, and \u0001 thus provides a crucial parameter that\ncan be used for a rough screening for materials that ex-\nhibit magnetic order at \fnite temperature. It should be\nnoted that for S= 1=2, the single-ion anisotropy alone\ncannot open a gap in the spinwave spectrum and mag-\nnetic order thus requires anisotropic exchange. We note\nthat the present approach can lead to situations where a\nmaterial with an out-of-plane easy axis (\u0001 EFM>0) has\na negative spinwave gap indicating that the ground state\nis unstable. For example, for a honeycomb lattice with\nNnn=NAFM = 3 andS= 1 Eqs. (2)-(4) one obtains\n\u0001<0 if \u0001EAFM>3\u0001EFM. This is due to the factor\nof 2S\u00001, which replaces a factor of 2 Swhen quantum\ncorrections to the anisotropy terms are taken into ac-\ncount in renormalized spinwave theory.3,24In principle\nthis is inconsistent with the energy mapping approach,\nwhich is based on a classical treatment of the Heisenberg\nmodel. However, a full quantum mechanical energy map-\nping analysis is beyond the scope of the present work. In4\n0 20 40 60\nT [K]0123Mz [µB]\n50100150200250\ndE/dTkB\nFIG. 2. MC calculations of the magnetic moment per atom\nand heat capacity ( dE=dT ) calculated as a function of tem-\nperature for CrI 3. The dashed vertical line at T= 31 K, in-\ndicates the predicted critical temperature obtained from Eq.\n(5).\nb)\na) c) d)\nFIG. 3. Top and side view of a (a) square, (b) honeycomb,\n(c) triangular, (d) TMHC crystal structures. Magnetic atoms\nare in blue.\nFig. 2 we compare the magnetization and heat capac-\nity obtained from MC calculations of CrI 3as well as the\nmodel result from Eq. (5). The critical temperature can\nbe obtained from the position of the peak in the heat\ncapacity.\nThe parameters A,B, andJ(Eqs. (2)-(4)) and criti-\ncal temperatures (Eqs. (5)-(7)) have been calculated for\nthe nearly 550 materials listed in the C2DB database,\nwhich display honeycomb, square or triangular magnetic\nlattices, including stable as well as unstable materials.\nThe calculations were performed with the same plane\nwave cuto\u000b and k-point sampling as used for the mag-\nnetic anisotropy calculations in the database.20Examples\nof such structures are shown in Fig. 3 and includes the\ntransition metal dichalcogenides (TMD) in the 1T phase\nand in the 2H phase (triangular magnetic lattice), com-\npounds adopting the FeSe crystal structure (square mag-\nnetic lattice), and transition metal trihalides such as CrI 3\n(honeycomb magnetic lattice). In Fig. 4 we show all the\ncalculated parameters Jand \u0001 for insulators and metals\nwith triangular, square or honeycomb lattice. The spin-\nwave gap was calculated by taking the ground state to\n0.05\n 0.00 0.05\n∆ [eV]0.1\n0.00.10.2J [eV]Metals\nInsulatorsFIG. 4. Distribution of the calculated parameters Jand \u0001 for\n87 metallic and 270 insulating materials obtained with PBE.\nhave an out-of-plane ferromagnetic magnetization and a\nnegative spinwave gap thus implies that the ground state\nmust have in-plane magnetization.\nThe transition metal halogen chalcogen (TMHC) com-\nprises another crystal structure that deserves an addi-\ntional comment here. These materials display an atomic\nstructure that resembles a distorted hexagonal magnetic\nlattice arranged over two layers. Although at least two\ncomparable - but distinct - exchange paths are identi-\n\fable, MC calculations show that we can obtain rough\nestimates of the critical temperatures from the model (5)\nby treating it as an hexagonal lattice with a single ef-\nfective nearest neighbour coupling obtained from the en-\nergy mapping analysis. For example, for CrIS we obtain\nTC= 118 K from the model (5), which is in decent agree-\nment with the MC results of 140 K including both nearest\nand next-nearest neighbour exchange interactions.\nIII. RESULTS AND DISCUSSION\nIn Fig. 5 we show the exchange coupling Jand spin-\nwave gap \u0001 for the stable ferromagnetic materials with\n\u0001>0. We have performed the calculations for insulat-\ning as well as metallic materials. For metals, the value\nofSis ill-de\fned and here we have simply used the mag-\nnetic moment localized on the magnetic atoms, which is\nobtained by integrating the magnetization density over\nthe PAW spheres. Moreover, the Heisenberg model is\nnot a reliable starting point for itinerant magnets and to\nour knowledge there is no simple method to obtain crit-\nical temperatures for metallic ferromagnetic materials.\nFor this reason we will not discuss metallic materials any\nfurther in the present work, but simply note that large\nanisotropies and exchange couplings indicate that metal-\nlic compounds such as CoBr 3, VBr 3, NiI 3, and NiBr 3\ncould potentially exhibit very high critical temperatures.5\nFIG. 5. PBE calculations of exchange coupling J(triangles) and spinwave gap \u0001 (squares) of stable ferromagnetic materials\nwith \u0001 >0. Green background indicates insulating materials.\nFormula Structure J[meV] \u0001 [meV] S[~]TC[K]\nFeCl 2 MoS 2 15.2 0.056 2.0 208\nCuCl 3 BiI3 15.3 0.058 1.0 37\nCrI3 BiI3 2.3 0.96 1.5 35\nCoCl 2 CdI2 2.0 0.058 1.5 31\nCrBr 3 BiI3 2.0 0.23 1.5 23\nMnO 2 CdI2 0.54 0.31 1.5 19\nNiCl 2 CdI2 7.2 0.001 1.0 14\nCrCl 3 BiI3 1.4 0.033 1.5 13\nRuCl 2 MoS 2 18.7 2.3 2.0 606\nRuBr 2 MoS 2 16.1 1.77 2.0 509\nTABLE I. List of 2D magnetic insulating materials with posi-\ntive exchange coupling Jand positive spinwave gap \u0001. Struc-\nture denotes the prototypical crystal structure and Sis the\nspin carried by each magnetic atom. The critical tempera-\ntureTCis obtained from Eq. (5). The top part of the table\ncontains dynamically and thermodynamically stable materi-\nals. The lower part of the table contains materials that are\nnot expected to be stable in their pristine form but exhibit\nhigh critical temperatures.\nIn general we observe that most compounds contain tran-\nsition metal atoms with 3 dvalence electrons. In partic-\nular Cr, Mn, Fe, Ni and Co, which are all well-known\nelements in magnetic materials. In addition, most of the\ncompounds contain halides, albeit with a few important\nexceptions (for example MnO 2).\nIn Tab. I we show a list of the all the insulating fer-\nromagnetic materials and the calculated critical temper-\natures. The top part of the table contains the stable\nmaterials and in the lower part we have included a few\nexamples of materials that exhibit very high critical tem-\nperatures but which are predicted to be unstable in their\npristine form.\nFeCl 2\nThe largest Curie temperature is found for FeCl 2in the\nMoS 2crystal structure where we obtain TC= 202 K. Themain reason for the high value of TCis the large magnetic\nmoment of 4 \u0016Bper Fe atom and an exchange coupling\nofJ= 15 meV, which is one of the largest values found\nin the present study. Previous ab initio calculations have\nreported that FeCl 2in the CdI 2crystal structure (which\nis metallic) is more stable compared to the MoS 2crys-\ntal structure31,32and the Curie temperature was esti-\nmated to 17 K based on mean-\feld theory.31Our cal-\nculations con\frm the stability hierarchy and predict an\neven more stable prototype GeS 2(formation energy re-\nduced by\u001830 meV/atom compared to the MoS 2phase).\nHowever, we do not expect out-of-plane long-range fer-\nromagnetic order in either the CdI 2or the GeS 2crystal\nstructures, since the spinwave gaps are negative in both\ncases. Interestingly, FeCl 2in the CdI 2crystal structure\nhas positive single-ion anisotropy ( A), which could in-\ndicate magnetic order. However, a negative anisotropic\nexchange coupling ( B) yields an overall negative spin-\nwave gap and the material thus serves as a good example\nof a case where the single-ion anisotropy is not a good\nindicator of magnetic order. To our knowledge there\nis no experimental reports of isolated 2D FeCl Xcom-\npounds. However, FeCl 3in the BiI 3crystal structure has\nbeen intercalated in bulk graphite exhibiting a ferromag-\nnetic transition at temperature T= 8:5 K.33. Recently\nit has also been employed as functional intercalation in\nfew-layer graphene compounds to weaken restacking of\ngraphene sheets34and bilayer graphene compounds35to\npromote magnetic order in graphene.36Nevertheless, ac-\ncording to our calculations, the FeCl 3crystal structure\nis less stable than the FeCl 2ones and is not expected to\nexhibit ferromagnetic order as free standing layers due\nto a negative value of the spinwave gap. In bulk form\nFeCl 2is known in the CdI 2crystal structure with in-\nplane ferromagnetic order,37but the long range order is\nstabilized by interlayer anti-ferromagnetic exchange cou-\npling, which supports our assertion that exfoliated layers\nof this type will not exhibit magnetic order. Bulk FeCl 3\nhas also been reported to form di\u000berent stacking poly-\nmormphs of the BiI 3crystal structure, but the magnetic\nproperties of these materials are not known.376\nMnO 2\nMonolayers of MnO 2in the CdI 2crystal structure have\nbeen exfoliated in 2003,38but the magnetic properties\nhave not yet been thoroughly investigated experimen-\ntally. Our calculations con\frm a ferromagnetic ground\nin agreement with previous calculations39, where a criti-\ncal temperature of 140 K was predicted. However, that\nresult were obtained from energy mapping analysis using\nPBE+U DFT calculations (U = 3.9 eV) and MC cal-\nculations based on the Ising model. From simulations\nof the Heisenberg model - explicitly including the \fnite\nanisotropy - we obtain a TC) of 63 K using Heisenberg\nparameters from a pure PBE calculations. The e\u000bect of\nHubbard correction will be discussed in the next section.\nYCl 2\nA critical temperature of TC= 55 K is found for YCl 2\nin the MoS 2prototype. There is neither experimental\nor theoretical reports on this material and it could pose\nan interesting new 2D magnetic compound. Our calcula-\ntions indicate that it is highly stable in the ferromagnetic\ncon\fguration with magnetic moment of 1 \u0016Bper Y atom.\nHowever, since the material comprises a spin-1/2 system\nthe classical MC calculations of the critical temperature\nmay not be very accurate.\nCrX 3\nAs reported in a previous study employing the same\nmethod24we predict CrI 3in the BiI 3structure to have\nTC= 31 K, while the similar compounds and CrCl 3and\nCrBr 3haveTCof 9 K and 19 K respectively. We note\nthat our calculated critical temperature for CrI 3is some-\nwhat lower than the experimental value of 45 K. This is\nmainly due to the fact that PBE tends to underestimate\nthe exchange coupling and can be improved by using a\nPBE+U scheme as discussed below. CrCl 3and CrBr 3\nhave not previously been described in their 2D form, but\nare known as ferromagnetic bulk compounds consisting\nof layers in the BiI 3crystal structure with out-of-plane\nmagnetization.37The experimental Curie temperatures\nof bulk CrCl 3, CrBr 3, and CrI 3are 27 K, 47 K, and 70 K\nrespectively. Our calculated values show the same hierar-\nchy, but are reduced compared to the bulk values due to\nthe lack of stabilization by interlayer exchange coupling.\nCuCl 3\nFor CuCl 3in the BiI 3crystal structure we \fnd a criti-\ncal temperature of 33 K, which is similar to the calculated\nvalue of CrI 3. The material does, however, lie above the\nconvex hull by 0.15 eV per atom, which could complicate\nexperimental synthesis and characterization.XCl2\nBulk CoCl 2and NiCl 2are both known to display anti-\nferromagnetic interlayer coupling and in-plane ferromag-\nnetic order37. As seen in Tab. I, our calculations predict\nthe materials to exhibit out-of-plane order. For NiCl 2,\nhowever, one should be a bit cautious due to the ex-\ntremely small value of the spinwave gap \u0001 and more\naccurate calculations could lead to a ground state with\nin-plane magnetic order. Experimental measurements on\nbulk samples indicate anomalies in the heat capacity re-\nlated to magnetic phase transitions at 24 K and 52 K.\nWhile the \frst result is in good agreement with our pre-\ndicted properties, the second one is signi\fcantly higher\nand could be related to an additional phase transition in\nthe 3D structure.\nMetastable high- TCcompounds\nThe lower part of Tab. I shows two materials that\nwe do not predict to be stable, but may be of interest\ndue to the large predicted critical temperatures. Here\nwe comment brie\ry on the case of RuCl 2in the MoS 2\ncrystal structure, which we predict to be a dynamically\nstable insulator with a critical temperature of 598K. It\nis, however, situated 0.5 eV above the convex hull, which\nwill mostly likely pose an obstacle to experimental syn-\nthesis. Nevertheless, the calculations show that very\nhigh values of critical temperatures are indeed possible\nin 2D materials with realistic atomic-scale parameters.\nIt should be mentioned that monolayers of RuCl 3in the\nBiI3crystal structure have been exfoliated and character-\nized experimentally.40Moreover, in a recent study41the\ncritical temperature of monolayer RuCl 3was calculated\nusing DFT and MC simulations based on the Heisenberg\nmodel and found TC= 14:21 K.41However, a Hubbard\nterm is required to open a gap and RuCl 3in the BiI 3\ncrystal structure is metallic within PBE,42,43which is\nwhy we do not include it in Tab. I.\nIn-plane anisotropy\nAs mentioned above, materials with the TMDH crys-\ntal structure have been considered as e\u000bective triangular\nmagnetic lattices with a single nearest-neighbour cou-\npling. However, this model can only be used for a rough\nscreening of materials. For example, CrIS exhibits a\nstrong in-plane anisotropy and the axis of magnetiza-\ntion are ordered (from the hardest axis to the easiest)\nas:x,zandy. In Eq. (1), in-plane anisotropy is not\nconsidered and we thus extend the model with the full\nset of anisotropy parameters Ax,Ay,Bx, andBythat\nmeasures the single-ion anisotropy and anisotropic ex-\nchange with respect to both xandydirections (relative\nto thez-direction). These parameters can be found by\ngeneralizing the energy mapping analysis Eqs. (2)-(3) to7\nJ1J2Ax AyBxByEasy axis TC\nCrIS 5.71 4.85 0.084 -0.223 0.025 0.033 x,z, y140\nMnClN 2.66 5.76 0.023 0.044 0.022 0.012 x,y, z 75\nCrClO 1.08 0.74 -0.010 0.034 0.004 0.001 y,x, z 15\nTABLE II. Ferromagnetic materials in the TMDH crystal\nstructure. The \frst two columns show nearest neighbour and\nnext-nearest neighbour exchange coupling constants in meV.\nColumns three to six display anisotropy parameters calculated\nwith respect to the two in-plane directions xandyin meV.\nIn the second last column we state the crystallographic direc-\ntions of magnetization listed from the hardest to the easiest\naxis. The last column shows the critical temperature in K\nobtained from MC calculations with these parameters.\ninclude di\u000berent in-plane directions. We then run MC\ncalculations using the full set of parameters to \fnd the\ncritical temperatures, including nearest and next-nearest\nneighbours couplings J1andJ2. We \fnd three insulat-\ning materials in this crystal structure that shows ferro-\nmagnetic order. The results are shown in Tab. II. In\nparticular, CrIS is predicted to have a critical tempera-\nture of 140 K. We also note that we obtain a Curie tem-\nperature of 15 K for CrClO, which has previously been\npredicted to have a Curie temperature of 160 K based\non an Ising model approach.44Again, this comparison\nemphasizes that the magnetic anisotropy cannot simply\nbe regarded as a mechanism that \fxes the magnetization\nto the out-of-plane direction: approximating magnetic\nproperties by the Ising model may yield a critical tem-\nperature that is wrong by an order of magnitude.\nA. Hubbard U\nAlmost half of the materials present in C2DB contain\nat least one element with a partially \flled d-shell. Local\nand semi-local xc-functionals such as PBE are known to\noverestimate delocalization of correlated electrons, due to\nthe uncompensated Coulomb self-interaction of the elec-\ntron. In the Hubbard model a term is introduced that\nacts as an e\u000bective electronic on-site repulsion and pro-\nvides a penalty to delocalization. In order to determine\nthe in\ruence of the Hubbard correction we have recalcu-\nlated exchange and anisotropy parameters for CrI 3for a\nrange of U values in the PBE+U scheme. The structure\nwas fully relaxed for each value of U and the results are\nshown in Fig. 6. We observe that an increasing value of\nU leads to an overall increase of both \u0001 and J, which\nresult in higher critical temperatures. The dependence\nofTCon U is roughly linear with TCincreasing by 5 K\nper eV that U is increased.\nIn order to gain more insight into the general in\ru-\nence of U for the calculations of magnetic properties, we\nhave recalculated the magnetic parameters and critical\ntemperature for all magnetic materials in the C2DB con-\ntaining 3dvalence electrons. We used the optimal values\ndetermined in Ref. 45 and listed in Tab. III. For each\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0J\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0A\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0B\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0∆\n0 1 2 3 4\nU [eV]TC34\n0.00.1\n0.20.3\n1.001.25\n304050\nFIG. 6. Calculated magnetic parameters of CrI 3as a func-\ntion of U. J,AandB, \u0001 are in units of meV and critical\ntemperatures TCare in units of K.\nElement FeMnCrCoNiVCu\nU [eV] 4.03.83.53.36.43.14.0\nTABLE III. Hubbard parameters employed in the PBE+U\ncalculations.\nmaterial the structure was relaxed with the given value\nof U, but the stability analysis was based on bare PBE.\nThe inclusion of U can have a rather dramatic e\u000bect\non the results. For example, the magnetic con\fguration\nof the ground state or the magnetic moment localized on\nthe transition metal ion may change. The results for the\nstable materials are shown in Fig. 7, while the insulating\nsystems are listed in Tab. IV. Including a Hubbard term\nin the entire work\row (from the relaxation step onward)\na\u000bects quantitatively and in some cases also qualitatively\nthe ground-state. This means that the magnetic moment,\nthe energy gap or the sign of \u0001 may change, making the\ncomparison with Tab. I meaningful only for a subset of\nmaterials. Compounds that are also present in Tab. I\nare shown in bold face for comparison.\nE\u000bect of U on TC\nFor MnO 2, the main e\u000bect of adding a Hubbard correc-\ntion is to increase the exchange parameter Jby a factor\nof two. Interestingly the anisotropy parameters AandB\ndecrease and the spinwave gap \u0001 becomes less than half\nthe value obtained with PBE. Nevertheless, the overall\ne\u000bect is an increase of the critical temperature from 63\nK to 82 K. This number can be compared to the result\nin Ref. 39 where a critical temperature of 140 K was\nestimated from the Ising model.\nFor CrI 3we predict a critical temperature of 50 K with\nPBE+U. This is signi\fcantly closer to the experimental8\nFIG. 7. PBE+U calculations of exchange coupling J(triangles) and spinwave gap \u0001 (squares) of stable ferromagnetic materials\nwith \u0001 >0. Green background indicates insulating materials.\nFormula Structure J[meV] \u0001 [meV] S[~]TC[K]\nMnO 2 CdI2 6.43 0.125 1.5 82\nCoCl 2 CdI2 3.21 0.249 1.5 57\nCrI 3 BiI3 3.95 1.280 1.5 50\nCrBr 3 BiI3 2.82 0.185 1.5 24\nMnI 2 CdI2 0.40 0.081 2.5 21\nMnBr 2 CdI2 0.41 0.024 2.5 16\nCrCl 3 BiI3 2.19 0.016 1.5 10\nNiCl 2 CdI2 5.69 \u001810\u000041.0 7\nFeBr 3 BiI3 0.04 0.124 2.5 2\nCoO FeSe 106.54 0.199 1.5 520\nFeS FeSe 28.99 0.591 2.0 413\nTABLE IV. List of 2D non-metal materials with positive\nexchange coupling Jand spin wave gap \u0001, obtained from\nPBE+U calculations. Structure denotes the prototypical\ncrystal structure and Sis the spin carried by each magnetic\natom. The structures in bold are present also in I for com-\nparison. The top part contains stable materials, whereas the\nlower part contains materials with large critical temperatures\nthat may be unstable in their pristine form.\nvalue than the 31 K obtained with bare PBE. Similarly\nTCincreases from 19 K to 24 K for CrBr 3while theTC\nof CrCl 3is increased from 9 K to 10 K. The critical tem-\nperature of CoCl 2is almost una\u000bected. These results\nindicate that it is non-trivial to predict how the inclu-\nsion of a Hubbard U in\ruences the calculated critical\ntemperatures in general.\nFor the compounds MnI 2, MnBr 2, and FeBr 3, which\nall have large magnetic moment of 5 \u0016Bper magnetic\natom we obtain rather low critical temperatures of 21,\n16 and 2 K respectively. This is mainly due to small\nvalues of exchange coupling Jfor these materials. The\ninclusion of U in MnI 2and MnBr 2increases the elec-\ntronic gap as well as the spinwave gap. But most impor-\ntantly, it yields a ferromagnetic ground state, while the\nground state is anti-ferromagnetic without the inclusion\nof U.46For MnI 2The result appears to be in qualita-\ntive agreement with neutron scattering experiments on\nthe bulk compounds, which reports a helical magnetic\nstructure below a critical temperature of 3.4 K, with themoments being aligned in the individual planes.37,47This\ncould indicate that PBE+U provides a more accurate\ndescription than PBE, which does not predict magnetic\norder for MnI 2. For MnBr 2in the CdI 2crystal struc-\nture, neutron scattering experiments on the bulk par-\nent structure revealed an anti-ferromagnetic order below\nT= 2:16 K with magnetic moments lying in-plane.48\nHowever, this is not necessarily in contradiction with our\ncalculations since the observed anti-ferromagnetic con\fg-\nuration is \"double-striped\", a con\fguration that has not\nbeen considered in the present study. For FeBr 3the Hub-\nbard term makes the spin jump from ~=2 to 5 ~=2per Fe\natom and opens a spinwave gap. A previous investigation\nof this material showed that it is predicted to be a quan-\ntum spin Hall insulator with PBE while PBE+U predicts\na trivial insulator above a critical value of U= 0 :18 eV.21\nMetastable high- TCcompounds\nThe lower part of Tab. IV lists materials, which are not\npredicted to be completely stable in their pristine form\naccording to PBE calculations (we have not performed\na full stability analysis with PBE+U). Bulk CoO has an\nanti-ferromagnetic rock-salt structure with a critical tem-\nperature of 293 K49. According to PBE calculations the\nmost stable 2D phase is a metallic CdI 2crystal structure\n(parameters Jand \u0001 are shown in Fig. 5). In the FeSe\ncrystal structure, CoO has a low dynamic stability but\nwe report it here due to the very high critical tempera-\nture of 520 K originating from the extraordinarily large\nexchange coupling predicted by PBE+U.\nFeS in the FeSe crystal structure has a non-magnetic\nground state with PBE, but is predicted to be highly\nstable and is situated on the convex hull. With PBE+U\nthe ground state becomes ferromagnetic and we predict a\nhigh critical temperature of 413 K. According to previous\ncalculations,50however, the true ground state is a striped\nanti-ferromagnetic con\fguration, which is not taken into\naccount in this work.9\nJ1J2AxAyBx By Easy axis TC\nCrBrO 1.12 0.78 0.043 -0.010 0.001 0.001 x,z, y 35\nCrIO 0.49 -1.46 0.586 -0.123 -0.008 -0.003 x,z, y 25\nCrClO 1.38 1.27 0.007 0.016 0.001 \u0018 \u000010\u00004y,x,z 20\nTABLE V. Parameters and results for TMHC structures ob-\ntained from PBE+U calculations. Symbols and units are the\nsame as in Tab. II. Materials in bold are the ones listed in\nboth tables.\nIn-plane anisotropy\nIn Tab. V we list Heisenberg parameters and crit-\nical temperatures for TMHC structures obtained from\nPBE+U calculations and MC calculations following the\nsame procedure as in the previous section where no Hub-\nbard correction was included. Comparing the results\nwith Tab. II, it is noted that MnClN and CrIS are not\npredicted to be ferromagnetic insulators with PBE+U. In\nparticular, MnClN is predicted to be a metal and CrIS\nexhibits a negative spinwave gap. On the other hand, two\nnew materials - CrIO and CrBrO - are predicted to ex-\nhibit ferromagnetic order at 25 K and 35 K respectively.\nIV. CONCLUSIONS\nWe have presented a high throughput computational\nscreening for magnetic insulators based on the Compu-\ntational 2D Materials Database. In contrast to several\nprevious studies of magnetism in 2D, we have empha-\nsized the crucial role of magnetic anisotropy and used the\nspinwave gap as a basic descriptor that must necessarily\nbe positive in order for magnetism to persist at \fnite\ntemperatures. This criterion severely reduces the num-\nber of relevant candidates and we end up with 12 stable\ncandidate materials for which the critical temperatures\nwere calculated from classical MC simulations. Seven of\nthe materials were predicted to have Curie temperatures\nexceeding that of CrI 3.\nThe classical MC simulations appear to comprise an\naccurate method for obtaining the critical temperatures\nfor insulating materials with S > 1=2. However, the\nHeisenberg parameters that enter the simulations may be\nsensitive to the approximations used to calculate them.\nIn the C2DB all calculations are performed with the PBE\nfunctional, which may have shortcomings for stronglycorrelated systems. We have thus tested how the re-\nsults are modi\fed if the parameters are evaluated with\nPBE+U instead and we \fnd that the predictions do in-\ndeed change in a non-systematic way. For the hexagonal\nand honeycomb systems three materials that were pre-\ndicted to be ferromagnetic (at \fnite temperature) are\nno longer predicted to show magnetic order when the\nPBE+U scheme is employed and three materials that\nwere not magnetic with PBE become magnetic with\nPBE+U. For the \fve materials that are magnetic with\nboth PBE and PBE+U the critical temperatures are\nslightly di\u000berent in the two approximations. The biggest\ndi\u000berence is seen for CrI 3where inclusion of U increases\nthe critical temperature from 31 K to 50 K, which is\ncloser to the experimental value of 45 K.\nIn the present work we have mainly focused on insu-\nlators. This restriction is rooted in the simple fact that\nwe do not have a reliable way to estimate Curie temper-\natures of metallic 2D magnetic materials. Metallic ferro-\nmagnetism in 2D is, however, a highly interesting subject\nand we note that room-temperature magnetism has re-\ncently been reported in the 2D metals VSe 25and MnSe 2.4\nMoreover, Figs. 4 and 5 indicate that in the C2DB the\nlargest values of both spinwave gaps and exchange cou-\nplings are found in metallic materials. Clearly, there is\npressing need for theoretical developments of 2D itinerant\nmagnetism that can be applied in conjunction with \frst\nprinciples simulations to provide accurate predictions of\nthe magnetic properties of 2D metallic materials.\nFinally, we have restricted ourselves to ferromag-\nnetic order. Nevertheless, the C2DB contains 241 anti-\nferromagnetic entries - 50 of which are predicted to be\nstable. 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Samsel-Czeka la, Com-\nputational Materials Science 142, 372 (2018)."    },    {        "title": "1903.11922v1.Micromagnetics_of_rare_earth_efficient_permanent_magnets.pdf",        "content": "Micromagnetics of rare-earth e\u000ecient permanent\nmagnets\nJohann Fischbacher1, Alexander Kovacs1, Markus\nGusenbauer1, Harald Oezelt1, Lukas Exl2;3, Simon Bance4,\nThomas Schre\r1\n1Department for Integrated Sensor Systems, Danube University Krems, Viktor\nKaplan Stra\u0019e 2, 2700 Wiener Neustadt, Austria\n2Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1,\n1090 Wien, Austria\n3Institute for Analysis and Scienti\fc Computing, Vienna University of\nTechnology, Wiedner Hauptstra\u0019e 8-10, 1040, Wien, Austria\n4Seagate Technology, 1 Disc Drive, Springtown, Derry, BT48 0BF Northern\nIreland, UK\nE-mail: thomas.schrefl@donau-uni.ac.at\nJanuary 2018\nAbstract. The development of permanent magnets containing less or no rare-\nearth elements is linked to profound knowledge of the coercivity mechanism.\nPrerequisites for a promising permanent magnet material are a high spontaneous\nmagnetization and a su\u000eciently high magnetic anisotropy. In addition to the\nintrinsic magnetic properties the microstructure of the magnet plays a signi\fcant\nrole in establishing coercivity. The in\ruence of the microstructure on coercivity,\nremanence, and energy density product can be understood by using micromagnetic\nsimulations. With advances in computer hardware and numerical methods,\nhysteresis curves of magnets can be computed quickly so that the simulations\ncan readily provide guidance for the development of permanent magnets. The\npotential of rare-earth reduced and free permanent magnets is investigated using\nmicromagnetic simulations. The results show excellent hard magnetic properties\ncan be achieved in grain boundary engineered NdFeB, rare-earth magnets with\na ThMn 12structure, Co-based nano-wires, and L1 0-FeNi provided that the\nmagnet's microstructure is optimized.\nKeywords : micromagnetics, permanent magnets, rare eartharXiv:1903.11922v1  [cond-mat.mtrl-sci]  28 Mar 2019Micromagnetics of rare-earth e\u000ecient permanent magnets 2\n1. Introduction\n1.1. Rare-earth reduced permanent magnets\nHigh performance permanent magnets are a key tech-\nnology for modern society. High performance mag-\nnets are distinguished by (i) the high magnetic \feld\nthey can create and (ii) their high resistance to op-\nposing magnetic \felds. A prerequisite for these two\ncharacteristics are proper intrinsic properties of the\nmagnet material: A high spontaneous magnetization\nand high magneto-crystalline anisotropy. The inter-\nmetallic phase Nd 2Fe14B [1, 2] ful\flls these properties.\nToday NdFeB-based magnets dominate the high per-\nformance magnet market. In the following we will use\n\\Nd 2Fe14B\" when we refer to the intermetallic phase\nand \\NdFeB\" when we refer to a magnet which is based\non Nd 2Fe14but contains additional elements. There\nare six major sectors which heavily rely on rare-earth\npermanent magnets [3]. The usage of NdFeB is sum-\nmarized in \fgure 1 based on data given by Constan-\ntinides [3] for 2015. Modern acoustic transducers use\nNdFeB magnets. Speakers are used in cell phones, con-\nsumer electronic devices, and cars. The total number\nof cell phones that are shipped per year is reaching 2\nbillion. Air conditioning is a growing market. Around\n100 million units are shipped every year. Each unit\nuses about three motors with NdFeB magnets. Nd-\nFeB magnets are essential to sustainable energy pro-\nduction and eco-friendly transport. The generator of a\ndirect drive wind mill requires high performance mag-\nnets of 400 kg/MW power; and on average a hybrid\nand electric vehicle needs 1.25 kg of high end perma-\nnent magnets [4]. Another rapidly growing market is\nelectric bikes with 33 million global sales in 2016. For a\nlong time NdFeB magnets have been used in hard disk\ndrives. Hard disk drives use bonded NdFeB magnets\nin the motor that spins the disk and sintered NdFeB\nmagnets for the voice coil motor that moves the arm.\nThere are around 400 million hard disk drives shipped\nevery year.\nIn many applications the NdFeB magnet are used\nat elevated temperature. For example, the operating\ntemperature of the magnet in the motor/generator\nblock of hybrid vehicles is at about 450 K. Though\nNd2Fe14B (Tc= 558 K) shows excellent properties\nat room temperature its Curie temperature Tcis\nmuch lower than those of SmCo 5(Tc= 1020 K) or\nSm2Co17(Tc= 1190 K) magnets [5]. Therefore the\nFigure 1. Usage of NdFeB magnets in the six major markets\nin the year 2015. Data taken from [3].\nanisotropy \feld and the coercive \feld of Nd 2Fe14B\nrapidly decays with increasing temperature. In order\nto compensate this loss, some of the magnet's Nd is\nreplaced with heavy rare earths such as Dy. Figure\n2 compares the coercive \feld of conventional Dy-free\nand Dy-containing NdFeB magnets as a function of\ntemperature. (NdDy)FeB magnets, containing around\n10 weight percent Dy, can reach coercive \felds \u00160Hc>\n1 T at 450 K. However, since the rare-earth crisis [6] the\nrare-earth prices have become more volatile. During\n2010 and 2011 the Dy price peaked and increased by\na factor of 20 [7]. Only four percent of the primary\nrare-earth production comes from outside China [6].\nBecause of supply risk and increasing demand, Nd and\nDy are considered to be critical elements [8]. In order\nto cope with the supply risk, magnet producers and\nusers aim for rare-earth free permanent magnets. With\nrespect to the magnet's performance, rare-earth free\npermanent magnets may \fll a gap between ferrites and\nNdFeB magnets [9]. An alternative goal is magnets\nwith less rare earth than (NdDy)FeB magnets but\ncomparable magnetic properties [10].\nPossible routes to achieve these goals are:\n\u000fShape anisotropy based permanent magnets;\n\u000fGrain boundary di\u000busion;\n\u000fImproved grain boundary phases;\n\u000fNanocomposite magnets;\n\u000fAlternative hard magnetic compounds.\nIn this work we will use micromagnetic simulations,\nin order to address various design issues for rare-earth\ne\u000ecient permanent magnets. Micromagnetic simula-\ntions are an important tool to understand coercivityMicromagnetics of rare-earth e\u000ecient permanent magnets 3\nFigure 2. Coercive \feld of Nd 15Fe77B8and\n(Nd0:77Dy0:33)15Fe77B8magnets as function of tempera-\nture. Data taken from [17].\nmechanisms in permanent magnets. With the advance\nof hardware for parallel computing [11{13] and the im-\nprovement of numerical methods [14{16], micromag-\nnetic simulations can take into account the microstruc-\nture of the magnet and thus help to understand how\nthe interplay between intrinsic magnetic properties and\nmicrostructure impacts coercivity.\n1.2. Key properties of permanent magnets\nThe primary goal of a permanent magnet is to create\na magnetic \feld in the air gap of a magnetic circuit.\nThe energy stored in the \feld outside of a permanent\nmagnet can be related to its magnetization and to its\nshape. According to Maxwell's equations the magnetic\ninduction Bis divergence-free (solenoidal): r\u0001B=\n0 and in the absence of any current the magnetic\n\feldHis curl-free (irrotational): r\u0002H= 0. The\nvolume integral of the product of a solenoidal and\nirrotational vector \feld over all space is zero, when the\ncorresponding vector and scalar potentials are regular\nat in\fnity [18]. This is the case when\nB=\u00160(M+H) (1)\nis the magnetic induction due to the magnetization\nMof a magnet. Here \u00160= 4\u0019\u000210\u00007Tm/A is the\npermeability of vacuum. The magnetostatic energy in\na volumeVaof free space, where M= 0 and B=\u00160H,\nisEmag;a= (\u00160=2)R\nVaH2dV. Splitting the space into\nthe volume inside the magnet, Vi, andVa, we haveR\nB\u0001HdV=R\nVa\u00160H2dV+R\nViB\u0001HdV= 0 or\nEmag;a=\u00001\n2Z\nViB\u0001HdV: (2)\nSince the left-hand side of equation (2) is positive, B\nandHmust point in opposite directions inside the\nmagnet. Approximating the magnetic induction Band\nFigure 3. The maximum energy density product ( BH)maxis\ngiven by the area of the largest rectangle that \fts below the 2nd\nquadrant of the B(H) curve. Left: M(H) loop, right: B(H)\nloop of an ideal magnet.\nthe magnetic \feld Hby a uniform vector \feld inside\nthe magnet, we can write Emag;a= (1=2)R\nVi(BH)dV,\nwhereB=jBjandH=jHj. We see that we can\nincrease the energy stored in its external \feld either\nby increasing the magnet's volume Vior by increasing\nthe product ( BH), which is referred to as energy\ndensity product [19]. It is de\fned as the product\nof the magnetic induction Band the corresponding\nopposing magnetic \feld H[20] and is given in units\nof J/m3. When there are no \feld generating currents,\nthe magnetic \feld inside the magnet\nH=\u0000NM (3)\ndepends on the magnet's shape which can be expressed\nby the demagnetizing factor N. We further assume\nthat the magnet is saturated and there are no\nsecondary phases so that jMj=Ms, whereMsis\nthe spontaneous magnetization of the material. Using\nequations (1) and (3) we express the energy density\nproduct as ( BH) =j\u00160(M\u0000NM)jj\u0000NMj=\u00160(1\u0000\nN)NM2\ns[9, 21]. When maximized with respect to N\nthis gives the maximum energy density product of a\ngiven material\n(BH)max=1\n4\u00160M2\ns (4)\nforN= 1=2. It is worth to check the shape of a\nmagnet with a demagnetizing factor of 1 =2. Let us\nassume a magnet in form of a prism with dimensions\nl\u0002l\u0002plwhich is magnetized along the edge with\nlengthpl. Then a simple approximate equation for the\ndemagnetizing factor is N= 1=(2p+1) [22]. Therefore,\nthe optimum shape of a magnet that results in the\nmaximum energy density product is a \rat prism with\ndimensions l\u0002l\u00020:5l, which is twice as wide as high.\nMany modern magnets have this shape.\nWhen there is no drop of the magnetization with\nincreasing opposing \feld until H > M s=2, the energy\ndensity product reaches its maximum value given by\nequation (4). In this case, the magnetic induction BMicromagnetics of rare-earth e\u000ecient permanent magnets 4\nas function of \feld is a straight line. For an ideal loop\nas shown in \fgure 3 the remanent magnetization, Mr,\nequals the spontaneous magnetization, Ms. In some\nmaterials, magnetization reversal may occur at \felds\nlower than half the remanence. When Hc<M r=2, the\nenergy density product is limited by the coercive \feld,\nHc. Similarly, the maximum value for ( BH)maxis not\nreached, when M(H) is not square but decreases with\nincreasing \feld H.\nA higher energy density product reduces volume\nand weight of the permanent-magnet-containing device\nmaking it an important \fgure of merit. Other decisive\nproperties are the remanence, the coercive \feld, and\nthe loop squareness.\n1.3. Permanent magnets and intrinsic magnetic\nproperties\nA magnetic material suitable for a permanent magnet\nmust have certain intrinsic magnetic properties. From\ninspection of \fgure 3 we see that a good permanent\nmagnet material requires a high spontaneous magneti-\nzationMsand a uniaxial anisotropy constant Kthat\ncreates a coercive \feld\nHc>Ms\n2: (5)\nThe theoretical maximum for the coercive \feld is the\nnucleation \feld [23]\nHN=2K\n\u00160Ms(6)\nfor magnetization reversal by uniform rotation of a\nsmall sphere. Equations (5) and (6) give the condition\nK > \u0016 0M2\ns=4. In other words, the anisotropy\nenergy density, K, should be larger than the maximum\nenergy density product, ( BH)max. For most magnetic\nmaterials this condition is not su\u000ecient [9]. There\nare two stronger conditions for the magnetocrystalline\nanisotropy constant.\nTo be able to make a permanent magnet or its\nconstituents in any shape, the nucleation \feld must\nbe higher than the maximum possible demagnetizing\n\feld. The demagnetizing factor of a thin magnet\napproaches 1 and the magnitude of the demagnetizing\n\feld approaches Mswhich gives the condition K >\n\u00160M2\ns=2. This is often expressed in terms of the\nquality factor Q= 2K=(\u00160M2\ns), which was introduced\nin the context of bubble domains in thin \flms [24,\n25]. For Q > 1 stable domains are formed and\nthe magnetization points either up or down along\nthe anisotropy axis perpendicular to the \flm plane.\nOtherwise, the demagnetizing \feld would cause the\nmagnetization to lie in plane.\nThe maximum possible coercive \feld is never\nreached experimentally. This phenomenon is usually\nreferred to as Brown's paradox [26, 27]. Imperfections\nFigure 4. Both the intrinsic magnetic properties and the\nphysical/chemical structure of the magnet determine coercive\n\feld, remanence and energy density product.\nare one reason for the reduction of the coercive \feld\nwith respect to its ideal value. In the presence of\ndefects with zero magnetocrystalline anisotropy, the\ncoercive \feld may reduce to Hc=HN=4. Plugging\nthis limit for the coercive \feld into equation (5) gives\nthe condition K >\u0016 0M2\nsfor the anisotropy constant.\nThis corresponds to the empirical law \u0014>1 for many\nhard magnetic phases of common permanent magnets\n[9], where\u0014=p\nK=(\u00160M2s) is the hardness parameter\n[28].\nThe key \fgures of merit of permanent magnets\nsuch as the coercive \feld, the remanence, and\nthe energy density product are extrinsic properties.\nThey follow from the interplay of intrinsic magnetic\nproperties and the granular structure of the magnet\nwhich is schematically shown in \fgure 4. Thus,\nin addition to the spontaneous magnetization Ms,\nmagnetocrystalline anisotropy constant K, and the\nexchange constant A, a well-de\fned physical and\nchemical structure of the magnet is essential for\nexcellent permanent-magnet properties.\nEmpirically, the e\u000bects that reduce the coercive\n\feld with respect to the ideal nucleation \feld are often\nwritten as [29, 30]\nHc=\u000bK\u000b HN\u0000Ne\u000bMs\u0000Hf: (7)\nThe coe\u000ecients \u000bKand\u000b express the reduction\nin coercivity due to defects and misorientation,\nrespectively [31]. The microstructural parameter Ne\u000b\nis related to the e\u000bect of the local demagnetization\n\feld near sharp edges and corners of the microstructure\n[32]. The \ructuation \feld Hfgives the reduction of the\ncoercive \feld by thermal \ructuations [33].\nLet us look at an example. Figure 5a shows\nthe microstructure of a nanocrystalline Nd 2Fe14B\nmagnet used for micromagnetic simulations to identify\nthe di\u000berent e\u000bects that reduce coercivity. For the\nset of material parameters used ( K= 4:3 MJ/m3,\n\u00160Ms= 1:61 T,A= 7:7 pJ/m), the ideal\nnucleation \feld is \u00160HN= 6:7 T. The 64 grainsMicromagnetics of rare-earth e\u000ecient permanent magnets 5\nFigure 5. (a) Microstructure used for the simulation of mi-\ncrostructural e\u000bects that decrease the coercive \feld. (b) 1 mis-\nalignment (solid line) : non-magnetic grain boundary phase, de-\nmagnetizing \feld switched o\u000b; 2 Defects(dashed line): Weakly\nferromagnetic grain boundary phase and demagnetization \feld\nswitched o\u000b; 3 Demagnetizing e\u000bects: Weakly ferromagnetic\ngrain boundary phase and demagnetization \feld switched on.\nThe vector \felds in (c) and (d) show magnetization and the de-\nmagnetizing \feld before switching for case 3.\nwere generated from a centroid Voronoi tessellation\n[34]. The average grain size was 60 nm. The\nanisotropy directions were randomly distributed within\na cone with an opening angle of 15 degrees. The\ngrain boundary phase of NdFeB magnets contains\nFe and is weakly ferromagnetic [35{37]. In addition\nto magnetostatic interactions between the grains, the\ngrains are also weakly exchange coupled. In our\nsimulations the grain boundary phase was 3 nm thick.\nThe magnetocrystalline anisotropy constant Kof the\ngrain boundary phase was zero. Its magnetization\nand exchange constant were \u00160Ms= 0:5 T andA=\n7:7 pJ/m for cases 2 to 4.\nIn numerical micromagnetics we can arti\fcially\nswitch physical e\u000bects on or o\u000b and thus gain\na deeper understanding of how the various e\u000bects\nimpact magnetization reversal. We start with the\ngranular system whereby the grains are separated\nby a nonmagnetic grain boundary phase and the\nmagnetostatic energy term is switched o\u000b. Thus,\nthere are no demagnetizing \felds and no magnetostatic\ninteractions. The grains are isolated and there are\nno defects. The solid line (case 1) of \fgure 5b\nshows the in\ruence of misalignment on magnetization\nreversal. Owing to the di\u000berent easy directions the\ngrains switch at slightly di\u000berent values of the external\n\feld. The coercive \feld is \u00160Hc= 4:77 T. With\n\u000bK= 1,Ne\u000b= 0, and Hf= 0 which hold for\ncase 1 per de\fnition, we obtain \u000b = 0:71 from\nequation (7). For the computation of the dashed\nline (case 2) we assume a weakly ferromagnetic grainboundary phase. The magnetostatic terms are not\ntaken into account. The grain boundary phase acts\nas a soft magnetic defect and reduces coercivity. The\ncorresponding microstructural parameter is \u000bK= 0:67.\nOwing to exchange coupling between the grains all\ngrains reverse at the same external \feld. For the\ndotted line (case 3) we switch on the demagnetizing\n\feld. The magnetization and the demagnetizing \feld\nare shown in a slice through the grains in \fgures 5c and\n5d, respectively. The reduction of the coercive \feld\nowing to demagnetizing e\u000bects equates to Ne\u000b= 0:2.\nFinally, we take into account thermal activation by\ncomputing the energy barrier for the nucleation of\nreversed domains as function of \feld [38]. The decrease\nof coercivity by thermal activation is \u00160Hf= 0:23 T.\n2. Micromagnetics of permanent magnets\n2.1. Micromagnetic energy contributions\nMicromagnetism is a continuum theory that handles\nmagnetization processes on a length scale that is small\nenough to resolve the transition of the magnetization\nwithin domain walls but large enough to replace the\natomic magnetic moments by a continuous function of\nposition [39]. The state of the magnet is described\nby the magnetization M, whose magnitude jMj=\nMsis constant and whose direction m=M=Msis\ncontinuous. A stable or metastable magnetic state can\nbe found by \fnding a function m=m(r) withjmj= 1\nthat minimizes the Gibbs free energy of the magnet\nE(m) =Z\nVh\nA\u0010\n(rmx)2+ (rmy)2+ (rmz)2\u0011\n(8)\n\u0000K(m\u0001k)2(9)\n\u0000\u00160Ms\n2(m\u0001Hd) (10)\n\u0000\u00160Ms(m\u0001Hext)i\ndV: (11)\nThe di\u000berent lines describe the exchange energy, the\nmagnetocrystalline anisotropy energy, the magneto-\nstatic energy, and the Zeeman energy, respectively.\nThe coe\u000ecients A,K, andMsin equation (8) to\n(11) vary with position and thus represent the mi-\ncrostructure of the magnet. The unit vector along the\nanisotropy direction, k, varies from grain to grain re-\n\recting the orientation of the grains. The anisotropy\nconstantKwill be zero in local defects or within the\ngrain boundary phase. The grain boundary phase may\nbe weakly ferromagnetic with magnetization Msand\nthe exchange constant Aconsiderably reduced with re-\nspect to the bulk values. In \u000b-Fe inclusions Kis neg-\nligible and MsandAare high. Composite magnets\ncombine grains with di\u000berent intrinsic properties. The\ndemagnetizing \feld Hdarises from the divergence of\nthe magnetization. The factor 1/2 in equation (10)Micromagnetics of rare-earth e\u000ecient permanent magnets 6\nindicates that it is a self-energy which depends on the\ncurrent state of M. It can be calculated from the static\nMaxwell's equations. One common method is the nu-\nmerical solution of the magnetostatic boundary value\nproblem for the magnetic scalar potential Uwhere the\ndemagnetizing \feld is derived as Hd=\u0000rU. The\nmagnetic scalar potential ful\flls the Poisson equation\nr2U=r\u0001M (12)\ninside the magnet, the Laplace equation\nr2U= 0 (13)\noutside the magnet, and the interface conditions\nU(in)=U(out)(14)\u0010\nU(in)\u0000U(out)\u0011\n\u0001n=M\u0001n (15)\nat the magnet's boundary with unit surface normal\nn. Equation (14) follows from the continuity of the\ncomponent of the magnetic \feld Hparallel to the\nsurface (which follows from r\u0002H= 0). Equation\n(15) follows from the continuity of the component of\nthe magnetic induction Bnormal to the surface (which\nfollows fromr\u0001B= 0) [40].\n2.2. Numerical methods\n2.2.1. Hysteresis There is no unique constrained\nminimum for equation (8) to (11) for a given\nexternal \feld. The magnetic state that a magnet\ncan access depends on its history. Hysteresis in a\nnon-linear system results from the path formed by\nsubsequent local minima [41]. In permanent magnet\nstudies we are interested in the demagnetization\ncurve. Thus, we use the saturated state as initial\nstate and compute subsequent energy minima for a\ndecreasing applied \feld, Hext. The projection of\nthe magnetization onto the direction of the applied\n\feld integrated over the volume of the magnet,\nthat isR\nVMs(m\u0001Hext=jHextj)dV, as function of\ndi\u000berent values of Hextgives the M(Hext)-curve.\nFor computing the maximum energy density product\nwe need the M(H)-curve, where His the internal\n\feldH=Hext\u0000NM(Hext). Similar to open\ncircuit measurements [42] we correct M(Hext) with the\nmacroscopic demagnetizing factor Nof the sample, in\norder to obtain M(H).\n2.2.2. Finite element and \fnite di\u000berence discretiza-\ntion The computation of the energy for a permanent\nmagnet requires the discretization of equations (8) to\n(11) taking into account the local variation of Ms,K,\nandA, according to the microstructure. Common dis-\ncretization schemes used in micromagnetics for perma-\nnent magnets [43] are the \fnite di\u000berence method [13]\nor the \fnite element method [14, 15].Each node of a \fnite element mesh or cell of\na \fnite di\u000berence scheme with index iholds a unit\nmagnetization vector mi. We gather these vectors into\nthe vector xwhich has the dimension 3 n, wherenis the\nnumber of nodes or cells. Then the Gibbs free energy\nmay be written as [14]\nE(x) =1\n2xTCx\u0000\u00160\n2hT\nd\u0016Mx\u0000\u00160hT\next\u0016Mx: (16)\nThe three terms on the right-hand side of (16) from left\nto right are the sum of the exchange and anisotropy\nenergy, the magnetostatic self-energy, and the Zeeman\nenergy, respectively. The sparse matrix Ccontains\ngrid information associated with the discretization of\nthe exchange and anisotropy energy. The matrix\n\u0016Maccounts for the local variation of the saturation\nmagnetization Mswithin the magnet. It is a diagonal\nmatrix whose entries are the modulus of the magnetic\nmoment associated with the node or cell i[14]. The\nvectors x,hd(x), and hexthold the unit vectors of\nthe magnetization, the demagnetizing \feld, and the\nexternal \feld at the nodes of the \fnite element mesh\nor the cells of a \fnite di\u000berence grid, respectively.\nFor computing the demagnetizating \feld, equa-\ntions (12) to (15) can be solved using an algebraic\nmultigrid method on the \fnite element mesh [14, 44,\n45].\nIn \fnite di\u000berence methods the magnetization\nis assumed to be uniform within each cell. Then\nthe magnetic \feld generated at point rby the\nmagnetization in cell jis given by an integration over\nthe magnetic surface charges \u001bj=Msjmj\u0001n[46],\nHd;j(r) =\u00001\n4\u0019r Z\n@Vj\u001b0\nj\njr\u0000r0jdS0\nj!\n; (17)\nwhere nis the unit surface normal. The magnetostatic\nenergy is a double sum over all computational cells\nEm=\u0000\u00160\n2X\niMs;imi\u0001Z\nViX\njHd;j(r)dV: (18)\nApplying integration by parts we can rewrite the\nmagnetosatic energy as [47]\nEm=\u00160\n8\u0019X\ni;jZ\n@ViZ\n@Vj\u001bi\u001b0\nj\njr\u0000r0jdS0\njdSi: (19)\nIntroducing the demagnetization tensor Nijreduces\nequation (19) to\nEm=\u00160\n2VcellX\ni;jMsimiNijMsjmj; (20)\nwhereVcellis the volume of a computational cell.\nThe term\u00160VcellMsimiNijMsjmjis the magnetostatic\ninteraction energy between cells iandj. From equation\n(20) we can compute the cell averaged demagnetizing\n\feld [48, 49]\nhi=\u0000X\njNijMsjmj: (21)Micromagnetics of rare-earth e\u000ecient permanent magnets 7\nThe demagnetization tensor Nijdepends only on the\nrelative distance between the cells iandj. The\nconvolution (21) can be e\u000eciently computed using\nFast Fourier Transforms. Special implementations of\nthe Fast Fourier Transform with low communication\noverhead makes large-scale simulations of permanent\nmagnets possible on supercomputers with thousands\nof cores [13].\n2.2.3. Energy minimization The intrinsic time scale\nof magnetization processes is related to the Lamor\nfrequencyf=\r\u00160H=(2\u0019). The gyromagnetic ratio is\n\r= 1:76086\u00021011/(Ts). For example, let us estimate\nthe intrinsic time scale for precession in Nd 2Fe14B. The\nmagnitude of typical internal \felds, \u00160Hare about\na few Tesla. The Lamor frequency is 28 GHz per\nTesla. This gives a characteristic time scale smaller\nthan 10\u000010s. Such time scales may be relevant\nfor magnetic recording or spin electronic devices. In\npermanent magnet applications the rate of change of\nthe external \feld is much slower. For example, low\nspeed direct drive wind mills run at about 20 rpm\n[50] and motors of hybrid vehicles run at 1500 rpm to\n6000 rpm [51], which translate into frequencies ranging\nfrom 1/3 Hz to 100 Hz. The magnetization always\nreaches metastable equilibrium before a signi\fcant\nchange of the external \feld. Based on this argument\nmany researchers use energy minimization methods\nfor simulation of magnetization reversal in permanent\nmagnets, taking advantage of a signi\fcant speedup as\ncompared to time integration solvers [11, 15].\nMinimizing equation (16) subject to the unit\nnorm constraint for decreasing external \feld gives the\nmagnetic states along the demagnetization curve of the\nmagnet. The sparse matrix Cand the diagonal matrix\n\u0016Mdepend only on the geometry and the intrinsic\nmagnetic properties. The vector hddepends linearly\non the magnetization. Evaluation of the energy,\nits gradient, or the Hessian requires the solution of\nthe magnetostatic subproblem. The magnetic \feld\ndepends linearly on the magnetization. Thus, (16)\nis quadratic in x. However, the condition M=Ms\nimplies that each subvector miofxis constrained\nto be a unit vector. The optimization problem is\nsupplemented by nnonlinear constraints jmij= 1; i=\n1;:::;n .\nIn its most simple form an algorithm for energy\nminimization (see algorithm 1) is an iterative process\nwith the following four tasks per iteration: The\ncomputation of the search direction, the computation\nof the step length, the motion towards the minimum,\nand the check for convergence. These computations\nmake use of the objective function E(x) and its\ngradients, possibly second derivatives and maybe\ninformation gathered from previous iterations. Thesuperscript+marks quantities which are computed\nfor the next iteration step. The superscript\u0000marks\nquantities which have been computed during the\nprevious iteration step.\nAlgorithm 1 minimizeE(x)\nrepeat\ncompute search direction d\ncompute step length \u000b\nproceed x+=x+\u000bd\nuntilconvergence\nVariants of the steepest-descent method, [11, 52{\n54], the non-linear conjugate gradient method [14,\n15], and the quasi-Newton method [55{57] are most\nwidely used in micromagnetics for permanent magnets.\nIn steepest-descent methods the search direction is\nthe negative gradient g=rE(x) of the energy:\nd=\u0000g. The nonlinear conjugate-gradient method\nuses a sequence of conjugate directions d=\u0000g+\n\fd\u0000. The Newton method uses the negative gradient\nmultiplied by the inverse of the Hessian matrix: d=\n\u0000(r2E)\u00001rgas search direction. In quasi-Newton\nmethods the inverse of the Hessian is approximated\nby information gathered from previous iterations.\nThe step length \u000bis obtained by approximate line\nsearch minimization. To that end the new point is\ndetermined along the line de\fned by the current search\ndirection and should yield a su\u000eciently smaller energy\nwith a su\u000eciently small gradient (approximate local\nminimum along the line). By expanding and shrinking\na search interval for \u000b, line search algorithms [58] \fnd\nan appropriate step length. Owing to the solution\nof the magnetostatic subproblem, evaluations of the\nenergy are expensive. In order to reduce the number\nof energy evaluations Koehler and Fredkin [59] apply\nan inexact line search based on cubic interpolation.\nTanaka and co-workers [15] propose to interpolate the\nmagnetostatic \feld within the search interval if it is\nsu\u000eciently small. Fischbacher et al. [14] showed that\nlong steps should be avoided, in order to compute\nall metastable states along the demagnetization curve.\nThey suggest applying a single Newton step in one\ndimension to get an initial estimate for the step\nlength which then may be further reduced to ful\fll\nthe su\u000ecient decrease condition. For steepest descent\nmethods Barzilai and Borwein [60] proposed a step\nlength\u000bsuch that \u000bmultiplied with the identity\nmatrix, 1, approximates the inverse of the Hessian\nmatrix:\u000b1\u0019(r2E)\u00001. Thus, the Barzilai-Borwein\nmethod makes use of the key idea of limited memory\nquasi Newton methods applied to step length selection.\nThe step length is computed from information gathered\nduring the last 2 iteration steps. This method was\nsuccessfully applied in numerical micromagnetics [11,Micromagnetics of rare-earth e\u000ecient permanent magnets 8\n61, 62].\nWhen applied to micromagnetics the update rule\nx+=x+\u000bdwill not preserve the norm of the\nmagnetization vector. A simple cure is renormalization\nm+\ni= (mi+\u000bdi)=j(mi+\u000bdi)j. When certain\nconditions for d[63] and the computational grid [64]\nare met the normalization leads to a decrease in the\nenergy. One condition [63] for an energy decrease\nupon normalization is that the search direction is\nperpendicular to the magnetic state of the current\npoint: mi\u0001difor alli. Following Cohen et al. [65] we\ncan replace the energy gradient by its projection onto\nits component perpendicular to the local magnetization\n^gi=gi\u0000(gi\u0001mi)mi=\u0000mi\u0002(mi\u0002gi): (22)\nIn nonlinear conjugate gradient methods the search di-\nrections are linear combinations of vectors perpendic-\nular to the magnetization (the current projected gra-\ndient and the previous search directions initially being\nthe projected gradient). Instead of updating and nor-\nmalization, the vectors mimight also be rotated by\nan angle\u000bjdij[66] or a norm conserving semi-implicit\nupdate rule [67] may be applied.\nThe right-hand side of equation (22) follows from\nthe vector identity a\u0002(b\u0002c) = (a\u0001c)b\u0000(a\u0001b)cand\njmij= 1. On computational grids with a uniform mesh\nthe energy gradient is proportional to the e\u000bective \feld\nhe\u000bi=\u00001\n\u00160MsiVcellgi: (23)\nBy inspecting the right-hand side of equation (22),\nwe see that the search direction of a steepest-descent\nmethod is proportional to the damping term of\nthe Landau-Lifshitz equation [68]. Time integration\nof the Landau-Lifshitz-Gilbert equation without the\nprecession term [52] is equivalent to minimization by\nthe steepest-descent method. Furaya et al. [52] use\na semi-implicit time integration scheme. They split\nthe e\u000bective \feld into its local part and the long-\nranging magnetostatic \feld which not only depends on\nthe nearest neighbor cells but on the magnetization\nin the entire magnet. By treating the local part\nof the e\u000bective \feld implicitly and the magnetostatic\n\feld explicitly, much larger time steps and thus faster\nconvergence toward the energy minimum is possible.\nThere are several possible termination criteria\nfor a minimization algorithm. Koehler and Fredkin\n[59] used the relative change in the energy between\nsubsequent iterations. Others [13, 52] use the di\u000berence\nbetween the subsequent magnetic states. Gill et al. [69]\nrecommend a threefold criterion taking into account\nthe change in energy, the change in the magnetic\nstate, and the norm of the gradient for unconstrained\noptimization. This ensures convergence of the sequence\nof the magnetic states, avoids early stops in \rat regions\nand ensures progress towards the minimum.2.2.4. Time integration The torque on the magnetic\nmoment MVcellof a computational cell in a magnetic\n\feld HisT=\u00160MVcell\u0002H. The angular\nmomentum associated with the magnetic moment\nisL=\u0000MVcell=j\rj. The change of the angular\nmomentum with time equals the torque, @L=@t=T.\nApplying the torque equation for the magnetic volume\nwhich is divided into computational cells gives\n@mi\n@t=\u0000j\rj\u00160mi\u0002he\u000bi; (24)\nwhich describes the precession of the magnetic moment\naround the e\u000bective \feld. In order to describe the\nmotion of the magnetization towards equilibrium,\nequation (24) has to be augmented with a damping\nterm. Following Landau and Lifshitz [68] we can add\na term\u0000\u0015mi\u0002(mi\u0002he\u000bi) to the right-hand side\nof equation (24) which will move the magnetization\ntowards the \feld. Alternatively we can - as suggested\nby Gilbert [70] - add a dissipative force \u0000\u000b@mi=@t\nto the e\u000bective \feld. The precise path the system\nfollows towards equilibrium depends on the type of\nequation used and the value of the damping parameters\n\u0015or\u000b. In the Landau-Lifshitz equation precession is\nnot changed with increasing damping, whereas in the\nGilbert case an increase of the damping constant slows\ndown precessional motion. Only for small damping the\nLandau Lifshitz equation and the Gilbert equation are\nequivalent. This can be seen if the Gilbert equation is\nwritten in Landau-Lifshitz form [71]\n@mi\n@t=\u0000j\rj\u00160\n1 +\u000b2mi\u0002he\u000bi\n\u0000\u000bj\rj\u00160\n1 +\u000b2mi\u0002(mi\u0002he\u000bi): (25)\nIn the limit of high damping only the second term of\nequation (25) remains and the time integration of the\nLandau-Lifshitz-Gilbert equation becomes equivalent\nto the steepest-descent method. For coherent rotation\nof the magnetization the minimum reversal time occurs\nfor a damping parameter \u000b= 1. In turn, fast\nreversal reduces the total computation time. This\nis the motivation for using a damping parameter\n\u000b= 1 for simulation of magnetization reversal in\npermanent magnets [66, 72] by numerical integration\nof equation (25). Several public domain micromagnetic\nsoftware tools use solvers for the numerical solution of\nequation(25) based on the adaptive Euler methods [73],\nRunge-Kutta schemes [62], backward-di\u000berentiation\nmethods [74], and preconditioned implicit solvers [75].\n2.2.5. Energy barriers Permanent magnets are used\nat elevated temperature. However, classical micromag-\nnetic simulations take into account temperature only\nby the temperature-dependent intrinsic materials prop-\nerties. Thermal \ructuations that may drive the sys-\ntem over a \fnite energy barrier are neglected. BeforeMicromagnetics of rare-earth e\u000ecient permanent magnets 9\nFigure 6. Energy landscape as function of the magnetization\nangle. (a) Small hard magnetic sphere in an external \feld. The\n\feld is applied at an angle of \u0019\u0000 = 45 degrees. (b) to (d)\nWith increasing \feld the energy barrier decreases.\nmagnetization reversal, a magnet is in a metastable\nstate. With increasing opposing \feld the energy bar-\nrier decreases. The system follows the local minima\nreversibly until the energy barrier vanishes and the\nmagnetization changes irreversibly [76]. If the height\nof the energy barrier is around 25 kBT, thermal \ruc-\ntuations can drive the system over the barrier within\na time of approximately one second [77]. To illus-\ntrate this behavior, let us look at the energy land-\nscape of a small hard magnetic sphere with volume\nV(see \fgure 6). The energy per unit volume is\nE(';H ext)=V=Ksin2(')\u0000\u00160MsHextcos('\u0000 ). The\nexternal \feld is applied at an angle  with respect to\nthe positive anisotropy axes. For small external \felds\nthe energy shows two minima as function of the magne-\ntization angle '. The state before switching is given by\n'1= (\u0019\u0000 )MsHext=(2K\u0000MsHext) and the state after\nswitching is given by '2=\u0019\u0000(\u0019\u0000 )MsHext=(2K+\nMsHext). The maximum energy occurs at the saddle\npoint at'0=3p\u0000tan [78].\nIn permanent magnets magnetization reversal\noccurs by the nucleation and expansion of reversed\ndomains [79]. Similar to the situation depicted in\n\fgure 6 the nucleation of a reversed domain or the\ndepinning of a domain wall is associated with an\nenergy barrier that is decreased by an increasing\nexternal \feld. Using the elastic band method [80]\nor the string method [81] the energy barrier can be\ncomputed as function of the external \feld. The\ncritical \feld at which the energy barrier EB(Hext)\ncrosses the 25 kBT-line is the coercive \feld of the\nmagnet taking into account thermal \ructuations. The\nelastic band method and the string method are well-\nestablished path \fnding methods in chemical physics\n[82, 83]. In micromagnetics they can be used to\nFigure 7. Thermally induced magnetization reversal in a\nNd2Fe14B cube. Left: Computed demagnetization curve\nby classical micromagnetics. Right: Energy barrier for the\nformation of a reversed nucleus as a function of the applied\n\feld. The inset shows the saddle point con\fguration of the\nmagnetization. Data taken from [30].\ncompute the minimum energy path that connects the\nlocal minimum at \feld Hextwith the reversed magnetic\nstate. A path is called a minimum energy path if for\nany point along the path the gradient of the energy is\nparallel to the path. In other words, the component\nof the energy gradient normal to the path is zero.\nThe string method can be easily applied by subsequent\napplication of a standard micromagnetic solver. It is\nan iterative algorithm: The magnetic states along the\npath are described by images. Each image is a replica\nof the total system. A single iteration step consists\nof two moves. (1) Each image is relaxed by applying\na few steps of an energy minimization method or by\nintegrating the Landau-Lifshitz-Gilbert equation for\na very short time. (2) The images are moved along\nthe path such that the distance between the images is\nconstant. Within the framework of the elastic band\nmethod images may only move perpendicular to the\ncurrent path and the distance between the images\nis kept constant with a virtual spring force. For\nan accurate computation of the energy barrier for a\nnucleation process [83] variants of the string method\nexists which keep more images next to the saddle point.\nThis can be achieved by an energy weighted distance\nfunction between the images [84] and truncation of the\npath [85].\nFigure 7 compares the coercive \feld of a Nd 2Fe14B\ncube with an edge length of 40 nm obtained by\nclassical micromagnetic simulations and computing\nenergy barriers as discussed above. In both methods\nthe intrinsic magnetic material parameters for T=\n300 K were used. The non-zero temperature coercive\n\feld, which takes into account thermal \ructuations,\nis de\fned as the critical value of the external \feld\nat which the energy barrier reaches 25 kBT. By\ninspecting the magnetic states along the minimum\nenergy path we can see how thermally inducedMicromagnetics of rare-earth e\u000ecient permanent magnets 10\nmagnetization reversal happens. At the saddle point\na small reversed nucleus is formed. If there is no\nbarrier for the expansion of the reversed domain, the\nreversed domain grows and the particle will switch.\nThe simulations are self-consistent: The coercive \feld\ncalculated by classical micromagnetics equals the \feld\nat which the energy barrier vanishes. For nearly ideal\nparticles such as the cube without soft magnetic defects\ndiscussed above, the reduction of the coercive \feld by\nthermal \ructuations may be as large as 25 percent\n[86]. However, the presence of defects reduces the\ndecay of coercivity owing to thermal \ructuations [30].\nFor example for the magnetic structure in \fgure 2,\nwhich contains a weakly ferromagnetic grain boundary,\nthermal \ructuations reduce the coercive \feld by only\n8 percent.\nEnergy barriers for reversal may also be computed\nby atomstic spin dynamics. Miyashita et al. [87, 88]\nsolved equation (25) numerically for atomic magnetic\nmoments augmented by a stochastic thermal \feld.\nFrom the computed relaxation time, \u001c, at a \fxed\nexternal \feld the energy barrier can be computed\nby \ftting the results to an Arrhenius-Neel law \u001c=\n(1=f0) exp(EB=(kBT)) or to Sharrock's law [89], which\ngives the coercive \feld as function of Hextand\u001c.\nAlternatively, Toga et al. used the constrained Monte-\nCarlo method [90] to compute \feld dependent energy\nbarriers for an atomistic spin model.\nIn equation (7) we attributed the reduction of co-\nercivity to the \ructuation \feld Hf. The energy barrier\nfor magnetization reversal is related to this \ructuation\n\feld byHf=\u000025kBT=(@E=@H ext). Experimentally,\nthe energy barrier or the \ructuation \feld can be ob-\ntained by measuring the magnetic viscosity which is\nrelated to the change of magnetization with time at a\n\fxed external \feld. It was measured by Givord et al.\n[91], Villas-Boas [92], and Okamota et al. [93] for sin-\ntered, melt-spun, and hot-deformed magnets, respec-\ntively.\n2.3. Microstructure representation\n2.3.1. Grain size and particle shape The discretiza-\ntion of the Gibb's free energy by \fnite di\u000berences or\n\fnite elements poses a question concerning the required\ngrid size. The required grid size is related to the char-\nacteristic length scale of inhomogeneities in the magne-\ntization, which is related to the relative weight of the\nexchange energy to other contributions of the Gibb's\nfree energy.\nUpon minimization the exchange energy favors\na uniform magnetization with the local magnetic\nmoments on the computational grid parallel to each\nother. The accurate computation of the critical\n\feld for the formation of a reversed nucleus requires\nthe energy of the domain wall, which separates the\nFigure 8. (a) Coercive \feld of a sphere, a cube, and a\npolycrystalline magnet as function of grain size. The grains are\nmade of Nd 2F14B and surrounded by a 3 nm thick, weakly-\nferromagnetic grain boundary phase. The grain size is de\fned\nas3p\nVgrain, whereVgrain is the volume of the grain. (b) Coercive\n\feld of the 100 nm pure Nd 2F14B cube as function of the\nexpansion factor ain a geometrical mesh. The solid, dashed,\ndot-dashed, and dotted line refer to a mesh size of 1.3 nm, 2 nm,\n2.7 nm, and 4 nm, respectively, de\fned at the surface of the\ncube.\nnucleus from the rest, to be known with high accuracy.\nTherefore, we should be able to resolve the transition\nof the magnetization within the domain wall on the\ncomputational grid. The width of a Bloch wall is\n\u000eB=\u0019\u000e0. The Bloch wall parameter \u000e0=p\nA=K\ndenotes the relative importance of the exchange energy\nversus crystalline anisotropy energy.\nWhereas in ellipsoidal particles the demagnetizing\n\feld is uniform, it is inhomogeneous in polyhedral\nparticles. The non-uniformity of the demagnetizing\n\feld strongly in\ruence magnetization reversal [94].\nNear edges or corners [32] the transverse component of\nthe demagnetizing \feld diverges. Owing to the locally\nincreased demagnetizing \feld, the reversed nucleus\nwill form near edges or corners [95] (see also \fgure\n7). We have to correctly resolve the rotations of\nthe magnetization that eventually form the reversed\nnucleus. For the computation of the nucleation \feld\nthe required minimum mesh size has to be smaller\nthan the exchange length lex=p\nA=(\u00160M2s=2) at the\nplace where the initial nucleus is formed. It gives\nthe relative importance of the exchange energy versus\nmagnetostatic energy. Please note that sometimes the\nexchange length is also de\fned as llex=p\nA=(\u00160M2s)\n[5, 96]. In order to keep the computation time low and\nresolve important magnetization processes, Schmidts\nand Kronm uller introduced a graded mesh that is\nre\fned towards the edges [97].\nThe relative importance of the di\u000berent energy\nterms also explains the grain size dependence of\ncoercivity. The coercive \feld of permanent magnets\ndecreases with increasing grain size [97{101]. The\nsmaller the magnet the more dominant is the exchange\nterm. Thus, it costs more energy to form a domain\nwall. To achieve magnetization reversal, the Zeeman\nenergy of the reversed magnetization in the nucleus\nneeds to be higher. This can be accomplished by a\nlarger external \feld.Micromagnetics of rare-earth e\u000ecient permanent magnets 11\nIn the following numerical experiment we com-\nputed the coercive \feld of a sphere, a cube, and a\nmagnet consisting of 27 polyhedral grains. The poly-\ncrystalline magnet is shown in \fgure 9. We computed\nthe coercive \feld as function of the size of the mag-\nnet for di\u000berent \fnite element meshes. We used the\nconjugate gradient method [14] to compute the mag-\nnetic states along the demagnetization curve. The\nNd2Fe14B particles ( K= 4:9 MJ/m3,\u00160Ms= 1:61 T,\nA= 8 pJ/m [5]) were covered by a soft magnetic phase\nwith a thickness of 3 nm. In the polycrystalline sample\nthe grains are also covered by a 3 nm soft phase which\nadds up to a 6 nm thick grain boundary. The mate-\nrial parameters of the grain boundary phase K= 0,\n\u00160Ms= 0:477 T, and A= 6:12 pJ/m correspond to\na composition of Nd 40Fe60[102]. The characteristic\nlengths for the main phase are \u000e0= 1:3 nm,\u000eB= 4 nm,\nandlex= 2:8 nm. The exchange length for the bound-\nary phase is lex= 8:2 nm. The results are summarized\nin \fgure 8a which gives the coercive \feld as function of\ngrain size. The three sets of curves are for the sphere,\nthe cube, and the polycrystalline magnet. The di\u000ber-\nent curves within a set correspond to a mesh size of\n1.3 nm, 2 nm, 2.7 nm, and 4 nm, de\fned at the sur-\nface of each grain and a mesh expansion factor of 1.05\nfor all models. As compared to the sphere the coercive\n\feld of the cube is reduced by about 1 T/ \u00160. For the\ncube the coercive \feld decreases with the particle size.\nIn the sphere the demagnetizing \feld is uniform. The\nratio of the hard (core) versus the soft phase (shell)\ndetermines coercivity. With increasing grain size the\nvolume fraction of the soft phase decreases and coerciv-\nity increases. In all samples the coercive \feld decreases\nwith decreasing mesh size. For all simulated cases, the\nrelative change in the coercive \feld is less than two\npercent for a change of the mesh size from 1.3 nm to\n2.7 nm.\nIn \fgure 8b we present the results for the coercive\n\feld obtained by graded meshes. In a geometrical mesh\n[103] the mesh size is gradually changed according to\na geometric series. Towards the center of the grain\nthe mesh size hincreases according to h\u0002an; wherea\nis the mesh expansion factor and nis the distance to\nthe surface measured by the number of elements. The\ncoercive \feld increases with increasing n. However, for\na<1:1 there is almost no change in the coercivity. On\nthe other hand, the number of \fnite element cells is\nreduced from 3.2 million for a= 1:01 to 1.6 million for\na= 1:09 and a mesh size of 1.3 nm at the boundary. In\nthis case, the runtime of the simulation was reduced by\na factor 4, with both simulations computed on a single\nNVidia Tesla K80 GPU. The situation is di\u000berent if the\ncube contains a soft magnetic inclusion in the center\nwhich will act as nucleation site. Then a \fne mesh is\nalso required at the interface between the hard and the\nFigure 9. (a) Polycrystalline model of a Nd 2Fe14B magnet.\nThe edge length of the cube is 300 nm. (b) The computed\ndemagnetization curve is the same for (c) an almost uniform\nmesh (a= 1:01) and (d) a geometric mesh with expansion factor\na= 1:09.\nsoft phase.\n2.3.2. Representation of multi-grain structures\nComputer programs for the semi-automatic generation\nof synthetic structures are essential to study the\nin\ruence of the microstructure on the hysteresis\nproperties of permanent magnets. Microstructure\nfeatures that need to be taken into account are the\nproperties of the grain boundary phase [36, 104, 105],\nanisotropy enhancement by grain boundary di\u000busion\n[53, 57, 86, 106{108], and the shape of the grains [12,\n16, 107, 109, 110]. The grain boundary properties may\nbe anisotropic based on the orientation of the grain\nboundary with respect to the anisotropy direction [37,\n111, 112].\nSoftware tools for the generation of synthetic\nmicrostructures include Neper [34] and Dream3d [113].\nThey generate synthetic granular microstructures with\ngiven characteristics such as grain size, grain sphericity,\nand grain aspect ratio based on Voronoi tessellation\n[114]. The grain structure has to be modi\fed\nfurther, in order to include grain boundary phases.\nAdditional shells around the grains with modi\fed\nintrinsic magnetic properties may be required in order\nto represent soft magnetic defect layers or grain\nboundary di\u000busion. These modi\fcations of the grain\nstructure can be achieved using computer aided design\ntools such as Salome [115]. In particular, boundary\nphases of a speci\fed thickness can be introducedMicromagnetics of rare-earth e\u000ecient permanent magnets 12\nby moving the grain surfaces by a \fxed distance\nalong their surface normal. When the \fnite element\nmethod is used for computing the magnetostatic\npotential, the magnet has to be embedded within\nan air box. The external mesh is required to treat\nthe boundary conditions at in\fnity. As a rule of\nthumb the problem domain surrounding the magnet\nshould have at least 10 times the extension of the\nmagnet [116]. The polyhedral geometry, the grain\nboundary phase, and the air box that surrounds the\nmagnet are then meshed using a tetrahedral mesh\ngenerator. Public domain software packages for mesh\ngeneration include NETGEN [117], Gmsh [118], and\nTetGen [119]. Ott et al. [120] and Fischbacher et al.\n[14] used NETGEN for meshing nanowires with various\ntip shape and for meshing polyhedral NdFe 12based\nmagnets. Zighem et al. [121] and Liu et al [122] used\nGmsh to mesh complex shaped Co-nanorods and to\nmesh polyhedral models for Cerium substituted NdFeB\nmagnets, respectively. Fischbacher et al. [107] used\nTetGen to mesh polyhedral core-shell grains separated\nby a grain boundary phase.\nFigure 9a shows a synthetic grain structure\ncreated with Neper [34]. The grain size follows a\nlog-normal distribution. The edge length of the cube\ncontaining the 27 grains is 300 nm. The thickness\nof the grain boundary phase is 6 nm. The material\nparameters for the main phase and the grain boundary\nphase were the same as used previously. Figure 9c\nand 9d show slices through the tetrahedral mesh of the\nmagnet.\nIn both cases the mesh size at the boundary is 2.7\nnm. In (c) an almost uniform mesh ( a= 1:01) was\ncreated with an average edge length of 2.9 nm and a\nmaximum of 6 nm in the center of the grains. The\nnumber of elements in the magnet is 11.8 millions. In\n(d) a graded mesh with an expansion factor of 1.09\nwas created. The average mesh size is then 3.3 nm\nwith a maximum of 11.6 nm. The number of elements\nis reduced by almost 42 percent to 6.9 million elements.\nFor both meshes we obtain identical demagnetization\ncurves shown in \fgure 9b. For the preconditioned\nconjugate gradient [123] used in this study, the time\nto solution scales linearly with the problem size. Thus\nthe use of geometric meshes reduces the computation\ntime by a factor of 1/2.\n3. Rare-earth e\u000ecient permanent magnets\n3.1. Shape enhanced coercivity\nShape-anisotropy based permanent magnets have a\nlong history. AlNiCo permanent magnets contain\nelongated particles that form by phase separation\nduring fabrication [124{126]. AlNiCo magnets were\nusurped as the magnets with the highest energy density\nFigure 10. (a) Coercive \feld as function of the aspect ratio\nof a Co nanorod for di\u000berent diameters. (b) Coercive \feld\nof a Co wire with an aspect ratio of 10:1 and a diameter of\n10 nm as a function of the magnetocrystalline anisotropy of a\nhard magnetic shell. (c) Reduction of the nucleation \feld of\nFePt-coated Co nanorods as function of packing density. (d)\nFormation of reversed domains in three interacting FePt-coated\nCo nanorods depending on their relative position.\nproduct before the development of permanent magnets\nbased on high uniaxial magnetocrystalline anisotropy;\n\frst ferrite magnets and then those based on rare\nearths [124]. In fact, some have suggested that the\ndevelopment of iron-rare-earth magnets was initially\nmotivated by the perceived need to replace Co, at that\ntime considered strategic and critical [127]. If true,\nthis situation ironically mirrors our current plight.\nToday shape-anisotropy magnets are again sought as\ncandidates for rare-earth free magnets.\nLivingston [125] and Ke et al. [54] discussed\nthe coercivity mechanisms of shape-anisotropy based\npermanent magnets. In ellipsoidal particles the\ndemagnetizing \feld is uniform. If the particles are\nsmall enough to reverse by uniform rotation [128] the\nchange of the demagnetizing \feld with the orientation\nof the magnet leads to an e\u000bective uniaxial anisotropy\nKd= (\u00160=2)M2\ns(N?\u0000Nk) [78], whereby N?and\nNkare the demagnetizing factors perpendicular and\nnormal to the long axis of the ellipsoid. However, if\nthe particle diameter becomes too large magnetization\nreversal will be non-uniform and coercivity drops\n[125]. Coercivity also decreases with increased packing\ndensity of the particles [129].\n3.1.1. Magnetic nanowires High aspect ratio Co,\nFe, or CoFe nanowires can be grown via a chemical\nnanosynthesis polyol process or electrodeposition [130{\n133]. Key microstructural features of nanowires and\nnanowire arrays such as particle shape [120], packing\ndensity and alignment [54, 134, 135], and particle\ncoating [135] have been studied using micromagneticMicromagnetics of rare-earth e\u000ecient permanent magnets 13\nsimulations.\nThe shape of the ends of magnetic nanowires\na\u000bects the coercivity. An improvement in the coercive\n\feld of between 5 and 10 percent is found when\nthe ends are rounded, as opposed to being \rattened\nlike an ideal cylinder [130, 134]. This enhancement\nof the coercive \feld is due to the reduction of high\ndemagnetizing \felds which occur at the front plane\nof the cylinder [136]. One of the important results\nfrom the shape anisotropy work is that the width\nof elongated nanoparticles is more crucial than the\nlength. Assuming that a particular aspect ratio of\n5:1 has been reached, increasing the length will give\nno further increase in coercive \feld. Figure 10a\ngives the computed coercive \feld of Co cylinders\n(K= 0:45 MJ/m3,\u00160Ms= 1:76 T,A= 1:3 J/m)\nwith rounded ends as function of the aspect ratio\nfor di\u000berent cylinder diameters. The smaller the\ndiameter the higher is the coercive \feld. Ener et\nal. [133] measured the coercive \feld of Co-nanorods\nfor diameters of 28 nm, 20 nm, and 11 nm to be\n0.36 T/\u00160, 0.47 T/\u00160, 0.61 T/\u00160, respectively. When\ncomparing with micromagnetic simulations we have\nto consider misorientation, magnetostatic interactions,\nand thermal activation which occur in the sample but\nare not taken into account in the results presented in\n\fgure 10a. Viau et al. [137] measured a coercive \feld\nof 0.9 T/\u00160atT= 140 K for Co wires with a diameter\nof 12.5 nm.\n3.1.2. Nanowires with core-shell structure The coer-\ncivity of Fe nanorods can be improved by adding anti-\nferromagnetic capping layers at the end. Toson et al.\n[135] showed that exchange bias between the antifer-\nromagnet caps and the Fe rods mitigates the e\u000bect of\nthe strong demagnetizing \felds and thus increases the\ncoercive \feld by up to 25 percent. Alternatively, a Co\ncylinder may be coated with a hard magnetic material.\nFigure 10b shows the coercive \felds of a Co-nanorod\nwith a diameter of 10 nm and an aspect ratio of 10:1\nwhich are coated by a 1 nm thick hard magnetic phase.\nThe coercive \feld increases linearly with the magne-\ntocrystalline anisotropy constant of the shell.\nThe demagnetizing \feld of one rod reduces\nthe switching \feld of another rod close-by. The\ncloser the rods, the stronger is this e\u000bect. We\nsimulated two bulk magnets consisting of either a\nhexagonal close-packed (h.c.p.) or a regular 3 x\n3 arrangement of Co/FePt core-shell rods. As the\n\flling factor increases, the separation of the nanorods\nbecomes smaller, meaning that the demagnetizing\ne\u000bects on neighboring rods increase, so the nucleation\n\feld leading to reversal is reduced (see \fgure 10c).\nDepending on the arrangement of the nanorods, the\nmagnetostatic interaction \feld nucleates reversal in\nFigure 11. (a) Energy barrier for magnetization reversal as\nfunction of the applied \feld for (i) a perfect Nd 2Fe14B grain,\n(ii) a Nd 2Fe14B grain with a surface defect with zero anisotropy,\nand (iii) a system with a defect and a (Dy 47Nd53)2Fe14B shell.\nThe critical \feld value at which the energy barrier becomes\n25kBis the temperature dependent coercive \feld. T= 450 K.\nData taken from [38]. (b) Coercivity of a Nd 2Fe14B particle as\nfunction of the percentage of coverage with a Tb-containing shell\nfor the continuous coverage model and the percolation model.\nneighboring nanowires (see \fgure 10d). The reversed\nregions start to grow in the core of the wire owing to\nthe hard magnetic shell.\n3.2. Grain boundary engineering\nThere is evidence from both micromagnetic simulations\n[86, 106, 138] and experiments [106] that magnetization\nreversal in conventional magnets starts from the\nsurface of the magnet or the grain boundary. An\nobvious cure to improve the coercivity of NdFeB\nmagnets is local enhancement of the anisotropy \feld\nnear the grain surface [86]. This may be achieved\nby adding heavy rare-earth elements such as Dy in\na way that (Dy,Nd) 2Fe14B forms only near the grain\nboundaries, creating a hard shell-like layer. Possible\nroutes for the latter process are the addition of Dy 2O3\nas a sintering element [139] or by grain boundary\ndi\u000busion [140, 141]. These production techniques\nreduce the share of heavy rare-earth elements while\nmaintaining the high coercive \feld of (Dy,Nd) 2Fe14B\nmagnets. In addition, grain boundary di\u000bused magnets\nshow a higher remanence, because the volume fraction\nof the (Dy,Nd) 2Fe14B phase, which has a lower\nmagnetization than Nd 2Fe14B, is small. Similarly,\nthe coercive \feld has been enhanced by Nd-Cu grain\nboundary di\u000busion which reduced the Fe content in the\ngrain boundaries [104].\n3.2.1. Core-shell grains We used the string method\n[84, 86] to compute the temperature-dependent hys-\nteresis properties of Nd 2Fe14B permanent magnets in\norder to assess the in\ruence of a soft outer defect and\na hard shell created by Dy di\u000busion. Dodecahedral\ngrain models, approximating the polyhedral geometriesMicromagnetics of rare-earth e\u000ecient permanent magnets 14\nof grains observed in actual rare earth permanent mag-\nnets, are prepared in three varieties: (i) a pure NdFeB\n(K= 2:1 MJ/m3,\u00160Ms= 1:3 T,A= 4:9 pJ/m) grain\nwith no defect and no shell, (ii) a NdFeB core with a\nsoft outer defect ( K= 0) of 2 nm thickness and (iii)\na Nd 2Fe14B core with a (Dy 47Nd53)2Fe14B hard shell\n(K= 2:7 MJ/m3,\u00160Ms= 1 T,A= 6:4 pJ/m) of 4 nm\nplus an outer defect (2nm). The outer grain diameter\nis constant at 50 nm. Figure 11a shows how the energy\nbarrier decreases as a function of applied \feld. In all\nmodel variations, at T= 450 K the thermal activa-\ntion reduces the coercivity by around 25 percent. The\nreduction in coercivity from the soft defect in (ii) is\ncanceled out by the hard shell in (iii).\nIn a real magnet the di\u000busion shell will not\nnecessarily be of uniform thickness or fully cover the\ngrain. We investigate this e\u000bect for a Nd 2Fe14B grain\nwith a diameter of 250 nm and a Tb-containing shell.\nIn order to investigate the e\u000bects of imperfect shells we\nsimulate systems where parts of the material in a 20 nm\nthick shell are replaced with the core material, in order\nto calculate the change in coercive \feld. A number of\napproaches are possible. First, a continuous island of\nvarying size is formed where the (Tb 0:5Nd0:5)2Fe14B is\nreplaced by Nd 2Fe14B. A 2 nm outer defect layer is still\npresent, with material properties of elements matching\nthose of the material they cover, except that K= 0. A\nsecond approach for an imperfect hard magnetic shell\nis percolation. Random shell elements are switched to\nthe core material. In the beginning the islands with\nthe core materials are very small until the number\nof switched elements increase and the islands join up.\nDepending on the type of the Tb-containing shell the\nbehavior is di\u000berent. The coercive \feld is plotted\nagainst the percentage of Tb-containing material in\nthe shell in \fgure 11b for continuous coverage and\nfor percolated coverage. For the continuous coverage\nmodel there is an exponential relationship, with a more\ncomplete covering leading to the highest coercivity\nvalues. As soon as the covering is reduced, the\ncoercivity drops rapidly. The trend is not smooth\nsince at various points the growing island's boundary\nreaches the edges and corners of the dodecahedral\ngrain, locations of importance where reversal begins.\nFor the percolation model the coercive \feld increases\nlinearly with the amount of (Tb,Nd) 2Fe14B in the shell.\n3.2.2. Grain boundary properties By NdCu di\u000busion\nhigh performance Nd 2Fe14B magnets without any\nheavy rare earths can be achieved [105, 142]. Energy-\ndispersive X-ray spectroscopy and atom probe analysis\n[104] showed the formation of a Nd-rich intergranular\nphase upon in\fltration. The Nd rich grain boundary\nphase predominantly forms at the grain surfaces\nperpendicular to the anisotropy axes [104]. The\nFigure 12. Coercive \feld (left) and energy density product\n(right) as function of the grain boundary properties for\na nanocrystalline magnet with a grain size of 50 nm.\nThe computed demagnetization curves are corrected with a\ndemagnetizing factor of N= 1=3. The coercive \feld is given\nwith respect to the internal \feld H.\ncoercive \feld of hot deformed NdFeB magnets increases\nfrom 1.5 T/ \u00160to 2.3 T/\u00160by Nd-Cu in\fltration.\nIn\fltration increases the Nd concentration from 38\nat.% to 80 at.% in grain boundary perpendicular to\nthe anisotropy axes, whereas the Nd content in grain\nboundaries parallel to the anisotropy axes remains low\nreaching only 25 at.% after in\fltration.\nIn order to understand the in\ruence of the grain\nboundary properties on coercivity and energy density\nproduct we compute demagnetization curves of a\nnanocrystalline magnet. We vary the thickness of\nthe grain boundary from 1.5 nm to 6 nm, keeping\nthe size of the magnet constant. We also change\nthe Nd content of the grain boundary phase and\nadjust the magnetization and the exchange constant\nof the grain boundary phase according to the data\npublished by Sakuma et al. [102]. Please note that\nthe magnetization of Nd xFe100\u0000xshows a maximum\natx= 20. We correct the demagnetization curve\nwith the macroscopic demagnetization factor N=\n1=3 and extract the coercive \feld and the energy\ndensity product (see \fgure 12). Clearly the maximum\ncoercive \feld is reached for a thin Nd-rich grain\nboundary phase. For nanocrystalline grains the\nmagnetization of the grain boundary phase contributes\nto the total magnetization. Therefore, the maximum\nenergy density product occurs for a Nd content of 20\npercent and a grain boundary thickness of 1.5 nm. A\nsimilar result was reported by Lee et al. [143] who\nsimulated the hysteresis properties as a function of the\nmagnetization and the exchange constant in the grain\nboundary phase.\n3.3. Alternative hard magnetic compounds\nIn the following we describe how coercive \feld,\nremanence, and energy density product change with\ntypical microstructural features for several possibleMicromagnetics of rare-earth e\u000ecient permanent magnets 15\nFigure 13. Magnetic properties as function of the aspect\nratio of the grains in a L1 0-FeNi granular system with K=\n0:35 MJ/m3. Top: Nanostructures with di\u000berent aspect ratios\nof the grains. Left: Coercivity. Right: Energy density product.\nThe computed demagnetization curves are corrected with a\ndemagnetizing factor of N= 1=3. The coercive \feld is given\nwith respect to the internal \feld H. Data taken from [110].\nalternative hard magnetic phases.\nFor all simulations we assume aligned grains.\nThe alignment factor f= cos(\u001e) is always close to\nunity. Here \u001eis the average misalignment angle. To\naccount for higher misalignment the Mr-values and\nthe (BH)max-values need to be multiplied with f\nandf2, respectively. Let us consider the following\nexample: The simulated values are \u00160Mr= 1:22 T\nand (BH)max= 251 kJ/m3for a L1 0FeNi magnet.\nAssuming\u001e= 20 degrees the expected values for\nthe remanence and the energy density products are\n\u00160Mr= 1:15 T andBHmax= 222 kJ/m3.\n3.3.1. L1 0-FeNi based permanent magnets The rare-\nearth free FeNi with a tetragonal L1 0structure has\na large saturation magnetization of \u00160Ms= 1:5 T,\nwhich translates to a theoretically possible energy\nproduct of ( BH)max = 448 kJ/m3. However, such\na high energy density product requires a su\u000eciently\nlarge magnetocrystalline anisotropy. The empirical law\n\u0014>1 suggests an anisotropy constant K > 1:8 MJ/m3.\nLewis and co-workers [144, 145] studied the crystal\nlattice, microstructure and magnetic properties of the\nmeteorite NWA 6259. Its L1 0-FeNi phase is highly\nordered and therefore regarded as a possible candidate\nfor use in permanent magnets. They estimated\nthe magneto-crystalline anisotropy of the meteorite\nto beK= 0:84 MJ/m3. Edstr om et al. [146]\npredicted an anisotropy in the range of 0.48 MJ/m3\nto 0.77 MJ/m3, using density functional theory. The\nmagnetocrystalline anisotropy linearly depends on the\nchemical order parameter [147].\nFigure 14. Magnetic properties of Nd 0:2Zr0:8Fe10Si2and\nSm0:7Zr0:3Fe10Si2without and with \u000b-Fe inclusions.Top:\nGranular structure used for the simulation. Left: Coercivity.\nRight: Energy density product. The computed demagnetization\ncurves are corrected with a demagnetizing factor of N= 1=3.\nThe coercive \feld is given with respect to the internal \feld H.\nWhereas chemical ordering is much smaller in\nmost other attempts to fabricate L1 0FeNi [148],\nGoto et al. [149] synthesized L1 0FeNi powder with\na degree of order of 0.7 through nitrogen insertion and\ntopotactic extraction. They measured a coercive \feld\nof 0.18 T/\u00160.\nWe investigated how nanostructuring may help to\ncreate reasonable hard magnetic properties with a low-\nanisotropy L1 0-FeNi phase. In L1 0-FeNi thin \flms\nmade by combinatorial sputtering an anisotropy con-\nstantK= 0:35 MJ/m3was measured by ferromag-\nnetic resonance [110]. We computed the demagnetiza-\ntion curves for three di\u000berent nanostructures consist-\ning of platelets, equiaxed grains, and columnar grains.\nThe grains have approximately the same volume of\n72\u000272\u000234 nm3, 56\u000256\u000256 nm3, and 34\u000234\u0002146 nm3,\nfor the platelets, polyhedra, and columns, respectively.\nThe macroscopic shape of the magnet is cubical with\nan edge length of 300 nm. The volume fraction of the\nnon-magnetic grain boundary phase is 18 percent.\nThe coercive \feld increases with increasing aspect\nratio. The data summarized in \fgure 13 shows that\nthe coercivity can be tuned by 120 mT/ \u00160through a\nchange in the shape of the grains. The grain shape\nhas no in\ruence on the remanent magnetization which\nwas computed to be \u00160Mr= 1:21 T. For platelet-\nshaped grains the energy density product is coercivity\nlimited with ( BH)max = 201 kJ/m3. For higher\ncoercivity such as in equiaxed and columnar grains the\nexpected energy density product is close to ( BH)max=\n255 kJ/m3.\n3.3.2. ThMn 12based permanent magnets Possible\ncandidate phases are NdFe or SmFe compounds inMicromagnetics of rare-earth e\u000ecient permanent magnets 16\nthe ThMn 12structure, which were discussed already\nin the late 1980s [150, 151]. However, NdFe 12and\nSmFe 12are not stable without any stabilizing elements\nsuch as Ti, Mo, Si, or V [152, 153]. At 450 K\nNd and Sm based magnetic phases in the ThMn 12\nstructure show a higher magnetization and a higher\nanisotropy \feld than Nd 2Fe14B [154, 155]. In addition,\nthe rare earth to transition metal ratio of the 1:12\nbased magnets is lower. Therefore, magnets based on\nthis phase are considered as a possible alternative to\nNd2Fe14B magnets [87, 156]. The rare-earth content\nis further reduced if some Nd or Sm is partially\nreplaced with Zr [156]. The fabrication of a magnets\nin the 1:12 structure is di\u000ecult. In contrast to\nNd2Fe14B, phases in the vicinity of R(Fe,M) 12(R rare\nearth, M stabilizing element) in the equilibrium phase\ndiagram are ferromagnetic. As a consequence there\nis no isolation of the grains with a non-magnetic or\nonly weakly ferromagnetic grain boundary phase [156].\nGabay and Hadjipanayis [157] measured a coercive\n\feld of 1.08 T/ \u00160in Sm 0:3Ce0:3Zr0:4Fe10Si2oriented\nparticles prepared by a mechano-chemical route.\nHere we look at the potential of the very rare-\nearth lean compounds Nd 0:2Zr0:8Fe10Si2[158] and\nSm0:7Zr0:3Fe10Si2[159]. Experiments show that the\nmagnets contain \u000b-Fe as a secondary phase with a\nvolume fraction of about 6 percent [158]. Therefore,\nwe investigate the in\ruence of the \u000b-Fe content on\nthe hysteresis properties. The synthetic microstructure\nused for the simulations is shown in \fgure 14. The\nvolume fraction of the grain boundary phase is 8\npercent. The grain boundary phase was assumed to\nbe moderately ferromagnetic with \u00160Ms= 0:56 T\nandA= 2:5 pJ/m. Nd 0:2Zr0:8Fe10Si2shows uniaxial\nanisotropy [158] with an anisotropy constant of K=\n1:16 MJ/m3and magnetization of \u00160Ms= 1:12 T\n[158]. For Sm 0:7Zr0:3Fe10Si2we useK= 3:5 MJ/m3\nand\u00160Ms= 1:08 T [159]. We compare two scenarios:\n(i) A sample without any \u000b-Fe as secondary phase,\nand (ii) a sample in which each grain contains an \u000b-Fe\ninclusion so that the total volume fraction of \u000b-Fe is 6\npercent.\nThe presence of \u000b-Fe reduces the coercive \feld.\nIn Nd 0:2Zr0:8Fe10Si2it decreases by about a factor of\n1/2 from\u00160Hc= 1:04 T to\u00160Hc= 0:5 T. Similarly,\nin Sm 0:7Zr0:3Fe10Si2the coercive \feld changes from\n\u00160Hc= 2:28 T to\u00160Hc= 1:2 T when\u000b-Fe inclusions\nare taken into account. The remanent magnetization\nfor the Nd and Sm compound is \u00160Mr= 1:07 T and\n\u00160Mr= 1:04 T, respectively. The presence of \u000b-Fe\nreduces the remanent magnetization by 4 percent and\n3 percent in the Nd and the Sm magnet, respectively.\nWith\u000b-Fe the energy density product reduces from\n228 kJ/m3to 198 kJ/m3and from 214 kJ/m3to\n194 kJ/m3in Nd 0:2Zr0:8Fe10Si2and Sm 0:7Zr0:3Fe10Si2,respectively.\n4. Summary\nHard magnetic phases for rare-earth free or rare-\nearth reduced permanent magnets may show a\nlower magnetocrystalline anisotropy than Nd 2Fe14B.\nTherefore a detailed understanding of the in\ruence\nof the microstructure on the magnetic properties is\nof utmost importance for the development of new\npermanent magnets. Computational micromagnetics\nreveals the main microstructural e\u000bects on the coercive\n\feld, the remanence, and the energy density product.\n4.1. Grain boundary phase\nThe grain boundary phase signi\fcantly in\ruences\nthe coercive \feld. If the grain boundary phase\nis ferromagnetic, the coercive \feld decreases with\nincreasing thickness of the grain boundary. Dy\nor Tb di\u000busion recovers the coercivity of magnets\nwith ferromagnetic grain boundary phases. A\nheavy rare-earth containing shell with a thickness\nof 10 nm doubles the coercive \feld which keeps\nincreasing moderately with further increasing thickness\nof the hard magnetic shell. High energy products\nand reasonable coercive \felds can be achieved for\nferromagnetic grain boundaries with thicknesses below\n3 nm. In nanocrystalline magnets the remanence\nand energy density product increase with increasing\nmagnetization of the grain boundary phase.\n4.2. Grain shape\nIn magnets based on CoFe nanorods coercivity is\nmostly governed by the thickness of the rods. The\nhighest coercive \felds can be obtained if the rod\ndiameter is comparable with the exchange length of\nthe material. Nanostructuring is essential and helps\nto improve the hysteresis loop squareness in materials\nwith low magnetocrystalline anisotropy. For the\nmagnetic properties of commonly synthesized L1 0FeNi\nthe energy product is coercivity limited. A change from\nplatelet-shaped grains to columnar grains may increase\nthe energy density product by 25 percent.\n4.3. Soft magnetic secondary phases\nSoft magnetic inclusions may reduce the coercive \feld\nby up to a factor of 1/2. If the magnetocrystalline\nanisotropy is su\u000eciently high, such as in SmFe or NdFe\ncompounds in the ThMn 12structure, still excellent\nhard magnetic properties can be achieved despite\nthe presence of \u000b-Fe. The reduction of the energy\ndensity product by soft magnetic inclusions is about\n10 percent.Micromagnetics of rare-earth e\u000ecient permanent magnets 17\n5. Conclusion\nThe coercivity and the energy density product were\ncomputed for several rare-earth reduced and rare-\nearth free permanent magnets using micromagnetic\nsimulations. For some materials the theoretically\npredicted values are higher than those currently\nachieved in experiments. This discrepancy emphasizes\nthe importance of the microstructure. A small grain\nsize, thin non-magnetic grain boundary phases that\nseparate the grains, and elongated grains for phases\nwith low magnetocrystalline anisotropy are essential\nto achieve a high coercive \feld.\nAcknowledgments\nThis work was support by the EU FP7 project\nROMEO (Grant no 309729), the EU H2020 project No-\nvamag (Grant no 686056), the Austrian Science Fund\nFWF (Grant no F4112 SFB ViCoM), the Japan Sci-\nence and Technology Agency (JST, CREST), Siemens\nAG, Toyota Motor Corporation, and the future pio-\nneering program Development of Magnetic Material\nTechnology for High-e\u000eciency Motors commissioned\nby the New Energy and Industrial Technology Devel-\nopment Organization (NEDO).\nReferences\n[1] Sagawa M, Fujimura S, Togawa N, Yamamoto H and\nMatsuura Y 1984 J. Appl. Phys. 552083{2087\n[2] Croat J J, Herbst J F, Lee R W and Pinkerton F E 1984\nJ. Appl. 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Schrefla,* \n \na Department for Integrated Sensor Systems, Danube University Krems, Austria  \nb Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden  \nc ICCRAM, University of Burgos, Burgos, Spain  \ndIT4Innovations, VSB -Technical University of Ostr ava, Ostrava, Czech Republic  \n* Corresponding author. E-mail: thomas.schrefl@donau -uni.ac.at , Tel: +43 2622 23420 20 , Fax:  43 2622 23420 99 , Address: Donau -Universität \nKrems , Department für Integrierte Sensorsysteme , Viktor Kaplan Str. 2 E , 2700 Wiener Neustadt , Austria  \nColumn  of article ( Research)  Type of article ( Article)  Subject:  Rare earth permanent magnets   \n \nABSTRACT  Multiscale simulation  is a key research tool for the quest for new permanent magnets. Starting with first \nprinciples  methods,  a sequence of simulation methods can be applied to calculate the maximum possible coercive field \nand expected energy density product of a magnet made f rom a novel magnetic material composition.  Fe-rich magnetic \nphases suitable for permanent magnets can be found by adaptive genetic algorithms. The intrinsic properties computed \nby ab initio simulations are used as input for micromagnetic simulations of the hysteresis properties of permanent \nmagnets with realistic structure. Using machine learning techniques,  the magnet’s structure can be optimized so that the \nupper limits for coercivity and energy density product for a given phase can be estimated. Structur e property relations \nof synthetic permanent magnets were computed  for several candidate hard magnetic phases . The following pairs \n(coercive field (T), energy density product (kJ/m³)) were obtained  for Fe 3Sn0.75Sb0.25: (0.49, 290), L1 0 FeNi: (1, 400), \nCoFe 6Ta: (0.87, 425), and MnAl: (0.53, 80).  \n \nKEYWORDS  rare-earth, permanent magnets, micromagnetics  \n \n1 Introduction  \nPermanent magnets are widely used in modern society. Important markets for permanent magnets [1] are wind power, \nhybrid and electric  vehicles, electric bikes, air conditi oning, acoustic transducer, and hard disk drives. With the growing \ndemand for permanent magnets in environmental friendly transport and power generation [2] there is an ongoing quest \nto reduce the rare -earth content or use alternative rare -earth efficient or rare -earth free hard magnetic phases. Some of \nthe considered hard magnetic phases may bridge the gap between ferrites and high perform ance NdFeB magnets [3]. \nIn this work we present an overview on how the magnetic properties of a virtual magnet can be predicted starting from \nfirst principles. The materials modelling workflow in this paper is an example for traditional multiscale si mulations \nwith parameter passing. Several physical models are linked together in order to compute the hysteresis properties of \npermanent magnets: Genetic algorithms in combination with density functional theory guide the search for stable \nuniaxial ferromagnetic phases. This process may be assisted by mining materials databases. Then density functional \ntheory is applied, in order to compute intrinsic  magnetic properties such as spontaneous magnetization, magneto -\ncrystalline anisotropy energy, and exchange integrals. The results feed into atomistic spin dynamics models for the \ncomputation of the magnetization, the anisotropy constant, and the exchange constant as function of temperature. Th ese \ntemperature dependent properties are then used as input for micromagnetic simulations. Numerical optimization tools \nhelp to tune the microstructure such that the coercive field or the energy density product is max imized for a given set \nof intrinsic magnetic properties.   \nIn addition, we computed the reduction of coercivity owing to thermal fluctuations  [4]. Analyzing the results of the \nmicromagnetic simulations, we can identify how much different eff ects such as misorientation, demagnetizing fields, \nand thermal fluctuations reduce the coercive field with respect to the anisotropy field of the material.  \nThe main focus of this paper is to predict the potential for various rare -earth free or rare -earth reduced permanent \nmagnetic phases with respect to the expected extrinsic magnetic properties such as coercivity and energy density \nproduct. A sufficiently hi gh coercive field and a sufficiently high energy density product are the key for the application \nof a new phase. These properties result from the interplay between the intrinsic magnetic properties of the magnet, the \nmagnet’s microstructure, and thermal fl uctuations. Therefore , the main part of the paper will cover micromagnetic \nresults  for the hysteresis properties , which will be presented for well -know n magnetic phases (L10 FeNi ,   \n2                            Nd0.2,Zr0.8Fe10Si2, Sm 0.7Zr0.3Fe10Si2) as well as for phases predicted by ge netic algorithms and density functional theory  \n(Fe 5Ge, CoFe 6Ta). For some of the phases the intrinsic magnetic properties were computed by first principle simulations  \n(Fe 5Ge, CoFe 6Ta, Fe3Sn0.75Sb0.25) and atomistic spin dynamics (MnAl) . \nThe results presented in this paper are centered on the micromagnetic computation of the expected performance of \nvarious hard magnetic phases.  For details on the adaptive genetic algorithms for  the search for new magnetic phases we \nrefer the reader to recent articles applying the method to Fe 3Sn [5], CoFe 2P [6], and magnetic phases in the L10 structure \n[7]. First principle simulations of magnetic properties are review in [8]. An overview of essential micromagnetic \ntechniques to compute the influence of microstructure on the coercivity and on the energy density product is given by \nFischbacher et al. [9]. \n \n2 Methods  \nAn adaptive genetic algorithm  [7] in combination with the ab initio package VASP [10] was used to scan the phase \nspace for Fe rich compounds that are non -cubic and stable.  The magnetic properties were calculated  with help of the \nfull-potential lin ear muffin -tin orbital (FP -LMTO)  method implemented in the RSPt code  [8]. Synthetic micro -structures \nwere constructed with the open -source 3D polycrystal generator tool Neper  [11].  \nA Python script controlling the open -source CAD software Salome [12] introduces the grain boundary phase with a \nspecific thickness and produces the finite element mesh.  For these synthetic microstructures the  demagnetization curve \nis computed through minimization of the micromagnetic energy with a preconditioned nonlinear conjugate gradient \nmethod  [13]. The search for higher coercive fields , µ0Hc, and energy density products , (BH)max, is managed via the \nopen -source optimization framework Dakota  [14]. Thus, the optimal structure for a given ha rd-magnetic phase can be \nfound. For characteri zing the magnet, we use the M(Hext)-loop, which gives the magnetization as function of the external \nfield. The demagnetization cur ve is then corrected by the demagnetizing field of the sample. A similar procedure is \ndone in experiments when the hysteresis curves are not measured in a closed circuit. Then we transform the \nmagnetization to the magnetic induction, B, in order to obtain the B(Hint)-loop and the energy density product . Here Hint \nis the internal field.   \nFinally, we consider the reduction of coercivity by thermal activation. We compute the critical value of the external \nfield that reduces the energy barrier for nucleation to  25 kBT. The system is assumed to overcome this energy barrier \nwithin a waiting time of one second owing to thermal fluctuations  [15]. We use a modified string method [16] to \ncompute the energy barriers for different values of the external field. The computation of reduction of the coercive field \nthrough thermal activation gives the limits of the coercive field [4] of a certain hard magnetic phase.  \n \n3 Results  \n3.1 Rare -earth free phases  \nUsing an adaptive genetic algorithm [7] the crystal phase space of Fe -Co-Ta was searched for non -cubic systems with \nhigh stability. For CoFe 6Ta we performed the two simulations starting from the scratch with 8 and 16 atoms/cell . \nVarious non -cubic stable phases could be ident ified. Some of the most stable non -cubic phases were tetragonal (space \ngroup 115), rhombohedral (space group 160), orthorhombic (space group 38), and orthorhombic (space group 63, where \na and b lattice parameters are very similar), and with an enthalpy of formation of -0.07033 eV/atom, -0.06353 eV/atom, \n-0.06025 eV/atom and -0.05929 eV/atom, respectively. Data and calculations details of these theoretical phases can be \nfound in the Novamag database  [17] , see the following links  [18]. The lowest ground state energy was found for a \nmonoclinic system (space group 8) with enthalpy of formation of -0.07488 eV/atom  [19]. These results correspond  to a \nhigh-throughput DFT calculations (at zero -temperature) using AGA, where similar default settings  were used for all of \nthem with the Generalized Gradient Approximation (GGA). To analyze  in more detail the stability of these phases is \nrecommended  to compute the free energy at finite temperature including electronic, phononic and magnetic terms  [20]. \nIn space groups 63 and 160, CoFe 6Ta shows a uniaxial magnetocrystalline aniso tropy. The complete theoretical study \nof these phases is in progress and it is planned to be reported in the near future, so here we just selected and mentioned \nsome preliminary results.  Using the RSPt code  [8] we calculated  the anisotropy constant and the spontaneous \nmagnetization for CoFe 6Ta in space group 63 to be  K = 1 MJ/m³ and µ 0Ms = 1.82 T . \nFigure 1 shows the micromag netically computed B(Hint) loop for different nanostructures made of CoFe 6Ta. The grains \nhave approximately the same volume of 34 × 34 × 146 nm3, 56 × 56 × 56 nm3, and 72 × 72 × 34 nm3 for the columns , \nequiaxed polyhedra, and platelets , respectively. The macroscopic shape of the magnet is cubical with an edge length of \n300 nm. The volume fraction of a non-magnetic grain boundary phase is 18 percent.  The energy density product is \n425 kJ/m³.  \n    \n  3 \n  \nFigure 1 Magnetic induction as func tion of the internal field  for nanostructured CoFe 6Ta. Nanostructuring is essential to obtain a high \ncoercive field.  The coercive field increases with increasing aspect ratio of the grains.  The aspect ratios of the columnar, equiaxed, and \nplatelet shaped grains 4.3, 1, and 0.47, respectively  \nFe-rich materials with non -cubic uniaxial crystal structures  are promising can didates for rare -earth free permanent \nmagnets. Because of the hexagonal crystal structure and its high spontaneous magnetization, Fe3Sn compounds were \nconsidered. However,  Fe3Sn shows an easy -plane anisotropy  [21] both in simulations and experiment.  Substituting Sn \nby Sb changes the easy -plane anisotropy  to uniaxial  anisotropy . The results show a uniaxial anisotropy constant \nK = 0.33 MJ/m³  and spontaneous magnetization µ 0Ms = 1.52  T for Fe3Sn0.75Sb0.25 [22]. These properties were assigned \nto the grains of a synthetically generated structure whereby the average grain size was 50  nm. An exchange stiffness \nconstant A = 10 pJ/m was used.  The grains were separated by a weakly ferro magnetic grain boundary (gb) phase with \na magnetization of µ 0Ms,gb = 0.81 T and an exchange stiffness constant Agb = 3.7 pJ /m. The micromagnetic simulation \nof the reversal process (see Figure 2) shows that multidomain states remain stable after irreversible switching owing to \ndomain wall pinning at the grain boundaries. The computed energy density product is coercivity limited . Its maximum \nvalue of 290 kJ/m³ may only be achieved for nanostructured systems with a grain size smaller than 50 nm.  Unfortunately, \nFe3Sn0.75Sb0.25 is not stable. Attempts to stabilize the phase by small additions of Mn were successful. However, owing \nto the change of the electronic stru cture and the number of valence electrons  the anisotropy flipped back to in -plane \nagain [22]. \n \n \n4                             \nFigure 2 Domain wall – microstructure interaction in a Fe3Sn0.75Sb0.25 magnet. Left hand side: Grain structure. Right hand side: At an \ninternal  field of µ0Hint = 0.4 9 T the flower like magnetic state (1) breaks into a two-domain  state (2) with a domain wall pinned at the grain \nboundaries. Images reproduced from [22]. \n3.3 Microstructure optimization  \nFor computing the influence of the microstructure on the hysteresis properties we varied the grain size, the grain shape, \nthe thickness of the grain boundary phase, and the magnetization in the grain boundary phase. The design space was \nsampled with the help of the software tool Dakota  [14].  \nIn order to obtain a general trend on how microstructura l features influence the coercive field we use dimensionless \nunits. The coercive field is given in units of the anisotropy field , 2K/(µ0Ms). The grain boundary magnetization is \nmeasured in units of the magnetization of the main hard magnetic phase , Ms,bulk. Grain size and grain boundary thickness \nare measured in units of the Bloch parameter 0 = (A/K)1/2, which is the characteristic length in hard magnetic materials.  \nThe results presented in the Figure 3 and Figure 4 were obtained by varying the microstructure for magnets made of the \nL10 FeNi (bulk), MnAl, and Nd0.2Zr0.8Fe10Si2 (see Table 1 ). The granular structure used for the simulations is shown in \nthe top row of Figure 3. Because we used dimensional units, the influence of grain boundary phase , grain size, and grain \naspect ratio on coercivity for other hard magnetic phases can be derived from the presented data.    \n \n \nFigure 3 Coercive field (left hand side) and the energy density product (right hand side) as function of grain  boundary properties.  \nThe design space for analysis of the influence of grain boundary properties on coercivity and energy density product \nwas spanned by th e grain boundary thickness and the magnetization of the grain boundary. We varied the thickness of \nthe grain boundary from 1.1  δ0 to 4.4 δ0, while keeping the size of the magnet constant. The magnetization of the grain \nboundary phase was varied from 0.05  Ms,bulk to 0.55  Ms,bulk. The exchange stiffness constant of the grain boundary phase \nis assumed to be proportional to its magnetization squared [23] according to Agb = Abulk(Ms,gb/Ms,bulk)². Thus, the grain \nboundary phase changes from almost non -magnetic to ferromagnetic. The polycrystalline structure used for the \nsimulations is shown in Figure 3. The average g rain size is 37 δ 0. \n   \n  5 \n Clearly the maximum coercive field is reached for a  thin, almost non -magnetic  grain boundary phase.  Both, increasing \nthe grain boundary thickness or increasing the grain boundary magnetization reduces the coercive field.  The \nmagnetization of the grain boundary phase contributes to the total magnetization. Therefore, the maximum energy \ndensity product oc curs for  thin grain boundaries and a moderately high magnetization in the grain boundary . We can \nconclude that excellent hysteresis properties can be achieved even for ferromagnetic grain boundaries, given that its \nthickness is sufficiently small. For exam ple, a coercive field of 0.4  × 2K/(µ0Ms) is reached for a grain boundary thickness \nof 2 δ0, when the magnetization in the grain boundary phase i s about ½ of its bulk value.   \nThe weakly soft magnetic grain boundary phase acts as soft magnetic defect. Detail ed micromagnetic studies show that \nat such grain boundaries magnetization reversal is initiated [24]. We see that the coercive field decreases with increasing \nspontaneous magnetization of the grain boundary phase. Furthermore, the coercive f ield decreases with increasing \nthickness of the grain boundary phase. Though the structure is more complicated for polycrystalline magnets with a \nweakly soft magnetic grain boundary phase, the effect is similar to that reported by Richter et al. [25] who showed  a \nsimilar dependence of the nucleation field on the size of a soft defect in a one -dimensional micromagnetic model.  The \nenergy to form the domain wall of the r eversed nucleus increases with decreasing thickness of the soft magnetic defect. \nIn magnets with a thin grain boundary phase the domain wall of the nucleus extend s into the main hard magnetic phase  \nand the domain wall energy increases.  Therefore, magnets w ith a thin ner grain boundary phase show a high er coercive \nfield.  \n   \n \n \nFigure 4 Influence of grain size and grain shape. The contours give the coercive field as function of grain size and aspect ratio.  The different \npanels refer to different saturation magnetization of the grain boundary phase with a thickness of 0. \nWe now modified the design space. We kept the grain boundary thickness at δ0 and varied the magnetization in the \ngrain boundary phase, the size of the grains, and the aspect ratio of the grains. An aspect ratio greater 1 refers to \nelongated, needle like gr ains; an aspect ratio smaller 1 refers to platelet like grains.  \nThe panels of Figure 4 show the coercive field as function of the grain size and the aspect ratio for different \nmagnetization in the grain boundary phase. For an almost non -magnetic grain boun dary phase the coercive field \nincreases with increasing aspect ratio. This means that  magnets with needle like grains show a higher coercive field \nthan magnets with platelet like grains. This effect diminishes when the magnetization of the grain boundary p hase is \nincreased. For Ms,gb = 0.4  Ms,bulk, there is hardly any change of the coercive field with aspect ratio. For large \nmagnetization in the grain boundary phase the trend is reversed and platelet -shaped grains show a slightly higher \ncoercive field than needle like grains. The grain size effect on coercivity is more pronounced in platelet shaped grains.  \n \n \n6                            The results of Figure 3 also show that the highest coercive field can be achieved for an almost non -magnetic grain \nboundary ( 0.05 Ms,bulk). The coer cive field is a factor of 4.5 higher than for a grain boundary phase with a spontaneous \nmagnetization of 0.55 Ms,bulk. Figure 4 shows that the coercivity increases with decreasing grain size. We can conclude \nthat magnets with small, exchange -decoupled grai ns show the highest coercive field. Indeed, the highest coercive field \nis found for the top left point on the top left subplot for Figure 4 with Ms,gb = 0.02  Ms,bulk: Here we have a nanostructured \nsystems with exchange isolated grains with a grain size whi ch is smaller than 20  0.  \n \n3.4 Coercivity limits  \nUsing numerical micromagnetics  we can separate the effects that lead to a reduction of coercivity with respect to the \nanisotropy field of the magnet. We compute the demagnetization curve but switch off the magnetostatic field.  When the \ncomputed coercive field is less than the anisotrop y field the reduction has to be attributed to misalignment of the grains \nor secondary soft magnetic phases. In a second step, we switch on magnetostatic interactions  and simulate the \ndemagnetization curve again.  The resulting decrease of the coercive field  has to be attributed to demagnetizing effects. \nFinally, we can simulate how the system escapes from a metastable state over the lowest energy barrier. This gives the \ntemperature dependent coercive field [4]. \n \n \nFigure 5 Limits of coercivit y. Effects that reduce the coercive field in permanent magnets for different candidate phases . The symbols gi ve \nthe coercive field. stars: anisotropy field, x: micromagnetic s without magnetostatics, +: full micromagnetics, o: micromagnetics  with \nthermal activation.  Please note the different scale for  the µ 0H axis for  (Sm,Zr)Fe 10Si2. \nIn the following analysis we did not assume any soft magnetic secondary phases. The external field was oriented one \ndegree off the easy axes of a small cube with an edge length of 40 nm. The computed effects that reduce the anisotropy \nfields are (1) misorientation, (2) demagnetizing effects, and (3) thermal fluctuations.  Here t he coercive field was \ncomputed for an ideal structure : The grain size is very small (40 nm) and there are no  defects. T hus, the computed \ncoercive field is an upper limit for coercivity for a given hard magnetic phase.  \nWe applied this procedure to several candidate phases for rare -earth free or rare earth reduced magnets. For each phase \nwe show the anisotropy field, the reduction owing to misorientation, the reduction by demagnetizing effects, and the \nreduction by thermal fluctuations (see Figure 5). The intrinsic magnetic properties used for the simulations are listed in \nTable 1.  The aniso tropy constant, the spontaneous magnetization, and the exchange constant for MnAl were obtained \nfrom atomistic spin dynamics at T = 300 K. Fe5Ge is an Fe -rich binary phase predicted by an adaptive genetic algorithm. \nThe anisotropy constant and the spontane ous magnetization for Fe 5Ge, Fe 3Sn0.75Sb0.25, and CoFe 6Ta were obtained from \nfirst principle simulations at T = 0. The exchange constant for Fe5Ge and CoFe 6Ta was taken to be proportional to the \nspontaneous magnetization squared  (A = cMs²) whereby c was taken from Ms and A of -Fe. The intrinsic material \nparameters for L1 0 FeNi, Nd0.2Zr0.8Fe10Si2, and Sm 0.7Zr0.3Fe10Si2 are experimental data for T = 300 K  were  taken from \nliterature . If no other source for the value of the exchange constant was availab le we used A = 10 pJ/m  [26].  \nThe results clearly show that we cannot expect a coercive f ield greater than 1 T in most rare earth free magnets. For \nFeNi (bulk) a high degree of uniform chemical order was assumed . Experimentally synthesized L1 0 FeNi particles may \ncontain patches where the chemical order is reduced locally. The corresponding loc al reduction of magnetocrystalline \nanistropy will deteriorate coercivity . Similarly, crystal defects such as twins or antiphase boundaries reduce the coercive \nfield in MnAl magnets [27]. Rare -earth magnets in the ThMn 12 structure  with Zr substitution have a low rare -earth \ncontent . Moreover,  the magnetocrystalline anisotropy – especially that of the (Sm,Zr)Fe 10Si2 magnet  – is sufficiently \n   \n  7 \n high to support a reasonable coercive field.  For Nd0.2Zr0.8Fe10Si2, and Sm 0.7Zr0.3Fe10Si2 the coercive field computed \nwith thermal activation (dots in Figure 5) is 70 percent of the anisotropy field.  \n \nTable 1  Anisotropy constant K, spontaneous magnetization Ms, and exchange constant A used for the simulations presented in Figure 5.  \nPhase  K (MJ/m³)  µ0Ms (T) A (pJ/m)   \nFe5Ge 0.23 1.8 14.7  \nL10 FeNi (Si substrate)  0.38 1.5 10 [28] \nFe3Sn0.75Sb0.25 0.33 1.52 10 [22] \nCoFe 6Ta 1 1.82 14.9  \nL10 FeNi (bulk)  1.1 1.38 10 [29] \nMnAl  0.7 0.8 7.6 [30] \nNd0.2Zr0.8Fe10Si2 1.16 1.12 10 [31] \nSm 0.7Zr0.3Fe10Si 2 3.5 1.08 10 [32] \n \n4 Conclusions  \nWe showed how  to exploit materials simulations for the computational design of the next generation rar e-earth reduced \npermanent magnets.  Based on the results presented above we can draw the following conclusions.  \n• Nanostructuring is essential to achieve a high coercive field in rare -earth free compounds with moderate  magneto -\ncrystalline anisotropy.   \n• Coercivity decreases with increasing magnetization in the grain boundary phase and with increasing thickness of the \ngrain boundary phase.  \n• However, excellent permanent magnetic properties can be achieved even for moderately ferromagnetic grain \nboundary phases provided that the grain boundary is thin enough.  \n• The shape of the grains is only important for nearly non -magnetic grain boundaries. For systems in which \nferromagnetic Fe containing grain boundaries are expected, the grain shape plays a minor role.  \n• Thermal fluctuations may considerably reduce the coercive field. Thus , even in perfect structures the coercive field \nis well below the anisotropy field.  \n \nAcknowledg ements \nThis work was support ed by the EU H2020 project NOVAMAG  (Grant no 686056) , the Austrian Science Fund FWF \n(I3288 -N36), and by the European Regional Development Fund in the  IT4Innovations national supercomputing center \n- path to exascale  project,  project number CZ.02.1.01/0.0/0.0/16_013/0001791 within the Operational  Programme \nResearch, Development and Education.  \n \nCompliance with ethics guidelines  \nAll authors declare that they have no conflict of interest or financial conflicts to disclose.  \n \nReferences  \n[1] Constantinides S. Magn Mag 2016;Spring 2016:6.  \n[2] Nakamura H. The current and future status of rare earth permanent magnets. Scr Mater 2017.  \n[3] Coey JMD. Permanent  magnets: Plugging the gap. 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Checkelsky1c\n1Department of Physics, Massachusetts Institute of Technology,\nCambridge, Massachusetts 02139, USA\n2Department of Physics, Hong Kong University of Science and Technology,\nClear Water Bay, Hong Kong, China\n(Dated: April 5, 2019)\naThese authors contributed equally to the work.\nbliuj@ust.hk\nccheckelsky@mit.edu\n1arXiv:1904.02582v1  [cond-mat.str-el]  4 Apr 2019Abstract\nRealizing quantum materials in few atomic layer morphologies is a key to both\nobserving and controlling a wide variety of exotic quantum phenomena. This includes\ntopological electronic materials, where the tunability and dimensionality of few layer\nmaterials have enabled the detection of Z2, Chern, and Majorana phases. Here, we\nreport the development of a platform for thin \flm correlated, topological states in the\nmagnetic rare-earth monopnictide ( RX) system GdBi synthesized by molecular beam\nepitaxy. This material is known from bulk single crystal studies to be semimetallic\nantiferromagnets with Neel temperature TN= 28 K and is the magnetic analog of the\nnon-f-electron containing system LaBi proposed to have topological surface states.\nOur transport and magnetization studies of thin \flms grown epitaxially on BaF 2reveal\nthat semimetallicity is lifted below approximately 8 crystallographic unit cells while\nmagnetic order is maintained down to our minimum thickness of 5 crystallographic\nunit cells. First-principles calculations show that the non-trivial topology is preserved\ndown to the monolayer limit, where quantum con\fnement and the lattice symmetry\ngive rise to a C= 2Chern insulator phase. We further demonstrate the stabilization\nof these \flms against atmospheric degradation using a combination of air-free bu\u000ber\nand capping procedures. These results together identify thin \flm RXmaterials as\npotential platforms for engineering topological electronic bands in correlated magnetic\nmaterials.\n2Introduction\nThe rare-earth monopnictides ( RX, whereRandXdenote the rare-earth and pnictogen\nelements) are a class of materials which host a rich variety of magnetic and electronic phases\nranging from ferromagnetic semiconductors to antiferromagnetic semimetals [1, 2]. Recent\nstudies have shown that for X= Bi the large spin-orbit coupling results in band inver-\nsion and topologically protected surface states [3{12]. Given the strong coupling between\nthe conduction electrons and localized magnetic moments of RX systems, this o\u000bers the\nopportunity to study the non-trivial electronic topology of correlated electrons [13{17]. A\nchallenge for these systems is the signi\fcant semimetallic band overlap of the conduction\nand valence bands (described below), which for transport studies precludes the isolation of\nthe surface response. However, while realization of few monolayer morphologies has proven\nto be a powerful method to remove parasitic bulk conductance from unintentional doping in\nconventional three-dimensional topological insulators (TIs) [18{21], theoretical calculations\nofRXsystems in this limit suggest further that quantum con\fnement has the potential to\nenergetically isolate the topologically non-trivial bands [22].\nHere we report the synthesis and study of epitaxial thin \flms of the R= Gd compound\nGdBi and show that it can be engineered toward an insulator by two-dimensional (2D) quan-\ntum con\fnement while retaining its magnetic and topological properties. As shown in Fig :1\n(a), GdBi has a rock-salt structure (space group Fm 3m) and hosts type-II antiferromagnetic\n(AFM) order below TN= 28 K [1, 23] with Gd moments ferromagnetically aligned within\nthef111gplanes and antiferrromagnetically stacked along the h111idirections [24]. The\nelectronic band structure in the vicinity of the Fermi level EFconsists of a Gd 5 d-derived\nconduction band around the Xpoints and two Bi 6 p-derived valence bands around the \u0000\npoint (see Fig :1 (h)). These bands overlap by approximately 1 eV, thus constituting a\nsemimetal. Midway in momentum between \u0000andXa band crossing occurs, which when\nhybridized by spin-orbit coupling is proposed to give rise to a topologically non-trivial gap\n[3]. This spin-orbit induced gap also o\u000bers a possibility for realizing Weyl nodes in the\nmagnetic phase of GdBi in an applied \feld, taking advantage of strong exchange splitting\nin a canted con\fguration (Fig :1 (h) inset) [25{28].\nWhile photoemission studies of bulk single crystals have proven instrumental for deter-\nmining the bulk and surface electronic structure of the RX systems [2, 4{6, 8, 9, 29], as\nnoted above it is of signi\fcant interest to realize few monolayer morphologies of these ma-\n3terials to study their transport properties as well as enable in-situ control via electrostatic\ngates. Given their cubic rock-salt structure, this is most appropriately achieved by thin\n\flm growth, which we employ here via molecular beam epitaxy (MBE). Subsequent struc-\ntural, electrical transport, and magnetic characterization demonstrate the high quality of\nthese materials. First principle band structure calculations con\frm that nontrivial topology\nremains upon con\fnement and leads to a C= 2 Chern insulator phase in the monolayer\nlimit. These suggest that thin \flm GdBi may be an ideal platform to investigate correlation\nbetween magnetism and topologically non-trivial surface states [13].\nFabrication of GdBi epitaxial thin \flms by MBE\nHigh quality epitaxial thin \flms of GdBi (111) were synthesized by MBE. Single crys-\ntalline BaF 2(111) is used for the substrate, which has a cubic lattice constant aBaF2= 0:620\nnm well-matched with that of GdBi aGdBi = 0:632 nm. Prior to GdBi deposition, 200\nnm of an epitaxial BaF 2(111) bu\u000ber layer was deposited to improve surface \ratness. As\nshown in Fig :1 (e) and (f), atomic force microscope (AFM) images of the BaF 2bu\u000ber layer\nshow a smooth surface with atomically \rat terraces and steps with height corresponding to\nthe spacing between successive (111) planes of BaF 2. The GdBi (111) layer was grown at\ntemperature T= 400\u000eC with the thickness varied from tGdBi = 5 nm to 40 nm. Finally,\nthe structure was capped with 40 to 100 nm of epitaxial BaF 2(see Methods for details).\nThe overall structure is summarized in Fig :1 (b) with an optical photograph of a typical\n\flm shown in Fig :1 (c). For a magnetic characterization by SQUID magnetometer and\nfor structural characterizations, samples were capped with an additional AlN layer using\natomic layer deposition.\nStructural characterization\nGrowth of single crystalline GdBi was con\frmed by X-ray di\u000braction as shown in Fig :1\n(g), where the peak at 2 \u0012= 49:90\u000eis identi\fed as GdBi (222). A slight shift of this peak from\nthe bulk reference position 2 \u0012ref= 50:02\u000eimplies an epitaxial lattice strain of approximately\n0.2%. The AFM image (Fig :1 (d)) and the X-ray re\rectivity (XRR) oscillations (Fig :1 (g)\ninset) con\frm a smooth surface on the top BaF 2and AlN cap layers. The thickness of the\nGdBi layer was calibrated by \ftting the XRR oscillations to a model structure simulation\n(see supplementary materials). We found that the GdBi thin \flms are extremely sensitive\n4to oxygen and moisture; air exposure of the samples degrades their magnetic and electrical\nproperties within a time scale of seconds even with a BaF 2capping layer (see supplementary\nmaterials). Therefore, we added an additional coating of non-aqueous liquid after direct\ntransfer of the \flm from ultrahigh vacuum to an inert Ar atmosphere for magnetic torque\nand electrical transport measurements. This allows for the \flms to preserve their intrinsic\nphysical properties for an adequate amount of time for transfer to characterization appara-\ntus (>10 minutes air exposure).\nMagnetic characterization\nFigure 2 (a) shows a comparison of the temperature Tdependence of magnetic suscep-\ntibility\u001fBulkof a GdBi bulk single crystal and the magnetic moment mof a 40 nm thick\nGdBi thin \flm measured by a commercial SQUID magnetometer. As previously reported\n[1, 23], the bulk crystal exhibits an AFM transition with Neel temperature TN= 28 K as\nidenti\fed by the kink in \u001fBulk(T) (measured here with the applied \feld Hparallel to [100]).\nWhile for the thin \flm samples there is a relatively large background response arising from\nthe substrate, bu\u000ber, and capping layers, there is a discernible peak for T\u0018TNforH\nperpendicular to [111]. For Hparallel to [111] this feature is largely suppressed, suggestive\nof the anisotropic magnetic susceptibility in the AFM phase.\nIn order to study the magnetic response of the \flms with higher resolution, we performed\ntorque magnetometry experiments. As shown in the inset of Fig :2 (b), we mounted the\nsample directly to the end of a metal cantilever with a small angle \u0012\u001915\u000ebetweenH\nand the sample normal. The magnetic torque \u001c(H) is shown in Fig :2 (b); atT= 100\nK we observe a quadratic response typical of a paramagnetic susceptibility. This response\nis enhanced at T= 30 K while a prominent dip at intermediate Hdevelops at the lowest\nT= 4 K. This is reminiscent of the W-shaped negative torque response originating from\ndiamagnetism in superconducting states of high- Tccuprates [30].\nIn Fig:2 (c) we plot the corresponding torque magnetization M\u001c\u0011\u001c=\u0016 0H=Mplane\u0000\nMnorm, where\u00160is the vacuum permeability, Mplane, andMnorm are the magnetization\nparallel and normal to the sample plane, respectively. Here the trend of an approximately\nlinear susceptibility giving way to a strong non-linear response at low Tcan also be observed.\nThe non-linear M\u001c(H) at lowTcan be naturally explained by the development of mag-\nnetic anisotropy upon entering the AFM phase as observed in the SQUID measurements.\n5In the absence of magnetic \feld, the GdBi thin \flm forms antiferromagnetic domains, and\nthe Gd spins point to symmetrically equivalent directions. Therefore the susceptibility is\nnearly isotropic for H\u00180. However, application of magnetic \feld H < 5 T \rops the spins\ndue to anisotropic susceptibility of an antiferromagnet, and con\fne them within the sample\nplane. In this con\fguration, the spin susceptibility are anisotropic depending on magnetic\n\feld directions parallel or perpendicular to the sample plane. This generates the torque\nresponse as in Fig :2 (b) acted by the tilted magnetic \feld.\nAs the anisotropy of the magnetic susceptibility develops in the AFM state, it can be\nused to probe the Neel temperature of the \flms T\flm\nN. We plot the observed anisotropy of\ne\u000bective magnetic susceptibility \u0001 \u001fe\u000b\u0011\u001fplane\u0000\u001fnorm=M\u001c=\u00160Hin Fig:2 (d) for di\u000berent\nHand identify T\flm\nN= 30 K. The results for a thinner \flm with tGdBi= 9 nm are shown in\nFig:2 (e) where a similar T\flm\nNis observed.\nElectrical characterization\nThe coupling between the conduction electron and localized magnetic moments of GdBi\nallows further characterization of the magnetic transition using electrical transport. Figure\n3 (a) shows the Tdependence of longitudinal resistivity \u001axx(T) for di\u000berent tGdBi. For the\nthickest \flm with tGdBi = 40 nm we observe a metallic response for all Twith a kink in\n\u001axx(T) nearT= 30 K, the latter more clearly observed in the second derivative d2\u001axx=dT2\nshown in the inset of Fig :3 (a). At all thicknesses down to tGdBi = 5 nm we observe this\nkink nearT= 30 K. This approximately matches the observed T\flm\nNshown in Figs. 2 (d)\nand 2 (e), suggesting that it is associated with the AFM transition. Such a \u001axx(T) feature\nhas been previously reported in GdBi bulk single crystals and ascribed to suppressed spin\ndisorder scattering in the AFM phase [31]. The observation of this feature in d2\u001axx=dT2for\nall \flms down to tGdBi= 5 nm therefore suggests that T\flm\nNremains unchanged down to at\nleast 5 crystallographic unit cells of GdBi.\nAs can be seen in Fig :3 (a), the overall electrical response of the \flms changes from\nmetallic to non-metallic with decreasing tGdBi; the lowTsloped\u001axx=dTchanges from positive\nfortGdBi= 40 nm to negative for tGdBi= 5 nm. This suggests a possible shift of the bulk\nband edges in the system with decreasing tGdBi, which we address further below.\nThe magnetotransport response of this series of \flms at T= 2 K is shown in Fig :3\n(b), where a non-saturating ( \u001axx(H)\u0000\u001axx(H= 0))=\u001axx(H= 0) of approximately 10% at\n6\u00160H= 9 T for the thickest \flms gives way to smaller, saturating behavior for thinner \flms.\nBulk single crystals of GdBi and other RX systems have been reported to show extreme\nmagnetoresistance (XMR) [23, 32{35] of similar form to that seen here for the thickest \flms,\nbut with signi\fcantly larger amplitude in the bulk case. As we discuss below, since XMR is\nin\ruenced by band parameters including carrier density, mobility, and position of chemical\npotential [32], the qualitative change of this response in the thin limit probes the evolution\nof these parameters as well as the band structure itself.\nTo further investigate the tGdBi dependence of transport, we measured the transverse\n(Hall) resistivity \u001ayx(H) across a broad range of T. As shown in Fig :4 (a), fortGdBi= 40\nnm atT= 2 K we observe a non-linear \u001ayx(H) consistent with that expected from a\nsemimetallic band structure. With decreasing tGdBi (Figs. 4 (b)-(d)), a linear response\nemerges suggestive of hole-like single-band transport. At elevated T,\u001ayx(H) for thicker\n\flms (tGdBi\u00159 nm) evolves towards a linear response whereas for thinner \flms ( tGdBi\u00146\nnm) it remains unchanged.\nFirst-principles Calculations\nThe electrical transport and the torque data in GdBi \flms suggests suppressed metal-\nlicity in the thin limit, while long-range magnetic correlation is una\u000bected. If GdBi has a\ntopologically non-trivial band structure, the consequence of broken time reversal symmetry\nmay manifest itself as a topologically distinct phase such as a Chern insulating state char-\nacterized by its Chern number C[36], or antiferromagnetic topological insulator state with\nchiral edge modes on the step edges [13]. Here we will show by theoretical calculations that\nGdBi has a topologically non-trivial band structure in both the bulk and thin \flm limit.\nFigure 5 (a) shows the calculated band structure of bulk GdBi (111) with type-II AFM\norder in the hexagonal unit cell with spin-orbit coupling (SOC). It exhibits an indirect neg-\native band gap of about \u00001 eV between the Bi-derived valence band at \u0000-point and the\nGd-derived conduction band at X-points, forming a semimetal. From the systematic depen-\ndence of the calculated band structure as a function of the lattice constant, we con\frmed\nthat the bands are inverted at the X-points. We also calculated the surface states and \fnd\nthey connect the bulk valence bands and conduction bands. All calculations con\frm that\nbulk GdBi has a non-trivial band topology corresponding to an AFM topological insulator\ngap degenerate with trivial bulk electronic states (see supplemental information).\n7In the extreme thin \flm limit, monolayer GdBi (111) consists of a pair of single atomic\nlayers of Gd and Bi, separated by lattice spacing d1from each other (Fig :5 (b)). To\ndetermine the topology of GdBi (111) in the 2D limit, the band structure of monolayer\nGdBi (111) slab was calculated using d1= 0:182 nm (the bulk value) as shown in Fig :5\n(c). We observe that the size of the negative indirect band gap is nearly lifted owing to the\nquantum con\fnement. The orbital-projection analysis reveals band inversion between the\nGddz2orbital and Bi pxorbital at the \u0000point, implying a nontrivial band topology.\nSymmetry Analysis and Edge Modes\nIn order to determine the topological properties relevant to monolayer GdBi (111), we\nperformed symmetry analysis of the low energy band structure. Monolayer GdBi (111) has\n3-fold rotation symmetry C3. Due to the magnetic order, both time-reversal symmetry T\nand mirror symmetry with the mirror plane perpendicular to y-directionMyare broken, but\nthe combined TMyis preserved. To characterize the low-energy properties near \u0000, we use\nthe little group containing C3andTMysymmetries to build a k\u0001pHamiltonian\nH(kx;ky) =m1\u001bz+m2(k2\nx+k2\ny)\u001bz+v1(kx\u001by\u0000ky\u001bx)\n+v2\u0000\n2kxky\u001by\u0000\u0000\nk2\nx\u0000k2\ny\u0001\n\u001bx\u0001\n+v3\u0000\nk2\nx+k2\ny\u0001\nI; (1)\nwhere\u001bis the pseudo-spin representing the conduction and valence bands, m1andm2are\nmass parameters, kxandkyare the crystal momenta, and v1;v2, andv3are the velocity\nparameters. The v3term gives the same energy shift for both conduction and valence\nbandsand thus will not a\u000bect the topological properties. We therefore set v3= 0 in the\nfollowing.\nFor the simplest case with m2=v2= 0, thek\u0001pHamiltonian reduces to the typical\nmassive Dirac Hamiltonian. For m16= 0, a gap will open at \u0000(red circles in Fig :5 (d)).\nAcross this band inversion, a topological phase transition occurs with Chern number changed\nby \u0001C=\u00001. In the case of small v26= 0, Dirac cones appear at four di\u000berent points: one at\n\u0000and the other three at equivalent points along the \u0000\u0000Mlines (blue circles in Fig :5 (d)).\nAsm1changes from negative to positive, the total Chern number now changes by \u0001 C= 2.\nForv2>> v 1, asv2increases the three Dirac points along the \u0000\u0000Mline converge at \u0000\nand transform into a quadratic band touching. As in the case of small v2, \u0001C= 2 when\nm1changes sign. For m26= 0,m2(k2\nx+k2\ny) = 0 at \u0000andm2will not a\u000bect the gap closing\n8or reopening there. However, m2does a\u000bect the gap along the \u0000\u0000Mlines. By \fne-tuning\nm2, we can also realize an intermediate phase with C=\u00001 or 3 between C= 0 andC= 2\nphases; the parameter regime of the intermediate phase is \u000e=jm2j(v1\nv2)2.\nTo con\frm the Chern insulating state with C= 2, we show in Fig :5 (e) the edge\nstates along Zigzag direction, which are calculated in ab-initio tight-binding models with\nall the parameters \ftted from the \frst-principles calculations through Wannier90 [37]. It\nshows two chiral edge modes connecting the valence and the conduction bands around the \u0000\npoint. These modes always appear as a pair, re\recting the topological band character C= 2.\nCalculations of Topological Phase Transition\nWe performed systematic calculations for the topological phase diagram for monolayer\nGdBi by varying the interlayer distance d1and the SOC strength \u0015. As shown in the Fig.\n5 (f), the phase diagram contains only phases with C= 0 andC= 2. As shown in the top\npanel of Fig. 5 (g), the conduction band and valence band have quadratic touching at the\nband inversion (similar behavior happens when we vary SOC strength \u0015as shown in Supple-\nmentary Information), which indicates a high-order topological phase transition. Fitting all\nthe parameters in the k\u0001pmodel to the band structures around the critical distance d= 1:34\n(inset of Fig. 5 (g)), we \fnd v1=\u00000:3,v2= 37:2,v3=\u00007:1, andjm2j<0:1. This is\nconsistent with the phase diagram obtained by \frst-principles calculations (the intermediate\nphase\u000e=jm2j(v1\nv2)2\u001810\u00005is extremely small). In Fig :5 (g), we show the evolution of\nthe energy gap \u0001 at the \u0000point as a function of d1. \u0001 monotonically decreases when d1\nincreases for d1<0:134 nm. At d1= 0:134, \u0001 = 0 and the gap reopens for d1>0:134 nm\nindicating a topological phase transition.\nDiscussion\nThe GdBi thin \flms studied here appear to retain the magnetic properties of bulk crystals\ndown totGdBi = 5 nm, suggesting that the symmetry breaking AFM order of these mate-\nrials is not signi\fcantly altered. The transport response, on the other hand, does evolve\nsigni\fcantly on decreasing thickness particularly across tGdBi = 9 nm. By quantitatively\nanalyzing the magnetotransport results, we can connect this behavior with that reported\nfor bulk single crystals as well as that predicted by our theoretical calculations in the thin\nlimit.\n9Starting with the thickest \flm with tGdBi= 40 nm, we expect a minimal role of quantum\ncon\fnement and therefore transport behavior similar to bulk single crystal materials. While\nwe do observe a non-saturating magnetoresistivity (Fig :3 (b)), the overall magnitude of\nthis response is signi\fcantly smaller than the XMR behavior reported in bulk single crystals\n[34]. This can be understood in terms of the compensation model for XMR which requires\na balance of density and mobility of the conduction and valence bands [38, 39]. While\ncompensation has been observed in bulk RX systems [23, 29, 33, 35, 40{42], two band\nanalysis (see supplementary materials) of the magnetotransport results for our tGdBi= 40 nm\n\flm (Fig:4 (a)) yields ne= 2:5\u00021020cm\u00002,nh= 3:0\u00021020cm\u00002,\u0016e= 389 cm2/Vs,\u0016h= 349\ncm2/Vs fortGdBi= 40 nm at T= 2 K, where ne,nh,\u0016e, and\u0016hare the electron density, the\ntotal hole density, the electron mobility and the hole mobility, respectively. While the carrier\ndensities are similar to those observed in bulk GdBi, along with reduced overall mobility they\nare detuned enough from perfect compensation to explain the signi\fcantly reduced XMR\nresponse [43{45]. The origin of this di\u000berence of parameters could arise from a number of\nsources including charge transfer from the substrate, defect chemistry di\u000berences in bulk\nand thin \flm synthesis, or epitaxial strain.\nUpon decreasing tGdBi, the compensation is further removed such that by tGdBi = 6\nnm theT= 2 K Hall e\u000bect is captured by a purely hole-like \u001ayx(H) (see Fig:4 (c)).\nA plausible explanation for this is a con\fnement induced [22, 46] upward shifting of the\nelectron band. This scenario is depicted schematically in the upper insets of Fig :4 (a)-\n(d) and corresponds to a decrease of the semimetallic band overlap at EF. This is further\nconsistent with the change from metallic to mildly insulating behavior in \u001axx(T) shown in\nFig:3 (a): here the hole-like bands remain metallic but the upwards shifted electron band\nhas a parallel thermally activated conductivity with a relatively minor contribution to the\no\u000b-diagonal response. Stabilizing \flms with further reduced tGdBi would then be expected\nto further enhance quantum con\fnement toward the realization of C= 2 Chern insulator\nstate as predicted in our ab-initio calculations. We note that in the ultrathin limit Anderson\nlocalization is also relevant [47]; this will act to eventually gap out the non-trivial electronic\nstates and, in the case of a time-reversal symmetry breaking by canting of the AFM order,\npotentially isolate chiral modes across a broad energy range [48].\nAnalyzing our theoretical calculations o\u000bers insight in to the underlying mechanism by\nwhich monolayer GdBi may realize a C= 2 phase. This appears to be a result of the\n10collapse of the intermediate phase corresponding to more conventional \u0001 C= 1 transitions\nfound whenjm2jis small and v2is very large (and further that this is the physically relevant\nregime for monolayer GdBi). This approach to realizing C= 2 Chern insulating state is\ngeneral and may be relevant to the recently proposed C= 2 state in the Dice lattice with\nC3vsymmetry [49] and also to the topological crystalline insulator SnTe with the mirror\nChern number 2 [50]. Our further demonstration that this phase is sensitive to the layer\nspacing parameter d1suggest the opportunity to engineer band topology using lattice strain,\nwhich can be controlled either by external pressure or by epitaxial strain [51]. Therefore\nmonolayer GdBi (111) is an attractive system to realize a tunable C= 2 Chern insulator\nstate.\nConclusion\nWe have reported the structural, magnetic, and transport properties of the correlated\ntopological insulator candidate GdBi thin \flms. We \fnd bringing materials to the thin \flm\nlimit preserves the antiferromagentic properties of bulk single crystals but with modi\fed\nelectrical properties consistent with a lifting of the semimetallic band overlap. Together\nwith our \frst principles calculations that show the band topology is preserved in the mono-\nlayer limit, this demonstrates that GdBi and other magnetic RXsystems in thin \flm form\nare candidates for hosting intrinsic correlated topological phases, including antiferromag-\nnetic TIs [13], and C= 2 Chern insulating phase tunable by strain in the monolayer limit\n[15, 36, 52, 53]. We also note that the \flms here have a (111) orientation that has thus\nfar not been stabilized for spectroscopic studies in bulk single crystals [2, 4{6, 8, 9, 29].\nTheoretical calculations for RBi predict that three surface Dirac cones are each separately\nprojected on to distinct points in the Brillouin zones on the (111) surface, but that two\nare degenerate and obscured by bulk bands for the (currently available) (001) surface [3].\nTherefore, the high quality (111)-oriented \flms reported here even in the thick limit may\nserve an important role for improved spectroscopic characterization. Finally, following the\nmethodology presented herein, we expect that other members of the RXfamily with more\ncomplex magnetic phase diagrams including CeBi [54] and HoBi [55, 56] should also be\nreadily synthesized, allowing for exploration of a wide variety of novel symmetry broken\nphases in this new class of topological electronic materials.\n11Methods\nSample fabrication\nThin \flm samples were deposited on BaF 2(111) substrates using an MBE system with\nbase pressure 4\u000210\u00008Pa. Prior to the BaF 2bu\u000ber layer deposition, the BaF 2substrate\nwas annealed at T= 450\u000eC for 90 minutes. The BaF 2bu\u000ber was then grown at T= 765\n\u000eC for 150 minutes at a growth rate of 1.33 nm/min. Next, the growth stage was brought to\nT= 400\u000eC and GdBi was grown at a rate of 0.22 nm/min. The ratio between the Gd and\nBi \rux was 1 : 4 :34. Finally an epitaxial BaF 2cap layer was grown at T= 765\u000eC for 60\nminutes at a growth rate of 1.33 nm/min. Samples were transferred into a glovebox directly\nafter the growth without air exposure to prevent sample degradation before subsequent\nprocessing.\nMagnetometry using superconducting quantum interference device\nWe characterized the magnetization of GdBi thin \flms using a commercial magnetometer\nwith a superconducting quantum interference device (SQUID). To prevent degradation of\nthe \flms, samples were ex-situ coated with an additional 30 nm of AlN layer at 120\u000eC by\natomic layer deposition at a rate of 0.10 nm per cycle. Samples were immersed in acetone\nprior to deposition in order to prevent degradation. We con\frmed that the magnetic prop-\nerties of the samples stayed unchanged during the measurements.\nTorque magnetometry\nAfter deposition of the GdBi thin \flms, samples were transferred into an inert environ-\nment from the MBE chamber and then attached to a 10 µm-thick Au torque cantilever. A\ncryogenic insert \ftted with a small vacuum chamber was loaded into the inert environment,\nand the magnetometer was placed in the vacuum chamber. Before extracting the cryogenic\ninsert from the inert environment and loading it into a cryostat, the vacuum chamber was\nevacuated using an external pump. The vacuum chamber was kept under vacuum during\nthe measurements to prevent degradation of the \flm. The de\rection of the cantilever in\nmagnetic \feld was measured via capacitance between the \rexible cantilever and a \fxed\nelectrode using a capacitance bridge.\n12Electrical transport measurement\nElectrical transport properties of the GdBi thin \flms were characterized using the Van der\nPauw geometry in a commercial cryostat. We applied a non-aqueous liquid electrolyte ( N,N-\nDiethyl-N-methyl-N-(2-methoxyethyl)ammonium bis(tri\ruoromethanesulfonyl)imide) and\nmade electrical contacts to the thin \flms in an inert environment to prevent degradation\nof the thin \flms after direct transfer from the MBE chamber. There was no measurable\nchanges in the electrical properties after deposition of the liquid.\nFirst-principles calculations\nWe performed \frst-principles calculations of the electronic structures of bulk and ultra-\nthin GdBi (111) in the type-II AFM phase. The calculations are performed in the framework\nof density functional theory (DFT) as implemented in the Vienna ab initio simulation pack-\nage (VASP) [57] by the Perdew-Burke-Ernzerhof type of generalized gradient approximation\n(GGA) [58] and the projector augmented wave (PAW) method [59] with cuto\u000b energy as\n400 eV. The band structures were calculated using the Monkhorst-pack (6 \u00026\u00026) and\n(6\u00026\u00021)k-mesh to sample the Brillouin zones of the bulk and thin \flms. The Hubbard\ncorrelation term Uwas introduced into the density functional theory framework (DFT+U)\nin order to account for the electron-electron correlation. The experimental lattice constant\nof bulk GdBi ( a= 0:632 nm) was used for bulk calculations. For the value of U, we found\nthat the calculated lattice constant reproduces the experimental value when Uaround 12\neV. Therefore we set U= 12 eV for the calculations. The spin orientation is set parallel\nto [111] in a type-II AFM order. All the parameters are tested to con\frm the calculations\nconverged, and the calculated bulk band structure is consistent with previous calculations\n[60, 61].\n[1] Duan, C.-G. et al. Electronic, magnetic and transport properties of rare-earth monopnictides.\nJ. Phys. Condens. Matter 19, 315220 (2007).\n[2] Wu, Y. et al. 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Investigation of electronic and mag-\nnetic properties of antiferromagnetic GdBi system by \frst principle and series expansions\ncalculations. Comput. Mater. Sci. 84, 45 (2014).\nAcknowledgments\nWe are grateful to L. Fu and C. Fang for fruitful discussions. This research was funded, in\npart, by the Gordon and Betty Moore Foundation EPiQS Initiative, Grant No. GBMF3848\n17to J.G.C. and ARO Grant No. W911NF-16-1-0034. J.G.C. acknowledges support from the\nBose Fellows Program at MIT. M. Hu and J. Liu acknowledge \fnancial support from the\nHong Kong Research Grants Council (Project No. ECS26302118).\n18FIGURES\n19[111](a) (b)\n(g) (h)GdBi (111) ( tGdBi nm)\nBaF 2 (111)\nsubstrateBaF 2 (111) buffer\n(200 nm)BaF 2 (111) cap \n(40 - 100 nm)\ntGdBi = 40 nmGd Bi\na\nbc\n3\n2\n1\n0\nHeight ( nm )\n15\n10\n5\n0\n30\n20\n10\n0\n1 µm(e)\n(f)(d)\n102 104 106 \nIntensity ( counts )\n2.0 1.0 0.0\n2θ ( degree ) Exp.\n Fit(c)\nW Γ X W\n1 mm\n100 102 104 106 \nIntensity ( count )\n52 51 50 49 48 47\n2θ ( degree )\nBaF2 (222)GdBi (222) bulkGdBi (222) filmΓ X\nW\nEFEnergyFIG. 1. Basic characterization of GdBi thin \flms. (a) Crystal structure of antiferromagnetic\nGdBi. The shaded areas denote the (111) planes, where spins are ferromagnetically aligned. The\nspin orientations are opposite between the red and blue shaded areas. (b) Schematic and (c) optical\nmicroscope image of GdBi thin \flm structure. (d)(e)(f) Atomic force microscope image of (d) the\nBaF 2and AlN cap layers, (e) the BaF 2bu\u000ber layer, and (f) the surface of an annealed BaF 2\n(111) substrate. (g) X-ray di\u000braction data measured on \flm with tGdBi = 40 nm. Inset: X-ray\nre\rectometry data measured on the same sample. The red curve is a \ft result to a model structure\n(see text). (h) Schematic band structure of bulk GdBi. Red and blue curves denote the valence\nand conduction bands, respectively. The upper inset shows the Brillouin zone and high symmetry\nlines corresponding to the band dispersion shown in the main panel. The lower insets shows an\nexpanded view of the band structure around conduction and valence band anticrossing. Exchange\nsplitting (shown in the dashed curve) may push the conduction and valence bands with opposite\nspins toward each other to generate Weyl nodes (marked with the circles).\n20(a)(b)\n(c) (e) (d)5\n4\n3\n2\n1\n0\n-1 ( nN m )\n-15-10 -5051015\nµ0H ( T )tGdBi = 40 nm\nT = 30 K\n100\n4H\nθ[111]\n-4.0-3.5-3.0-2.5-2.0m( 10-5emu )\n200 100 0\nT ( K )0.10\n0.08\n0.06\n0.04\n0.02\n0.00Bulk ( cm3 / mol )tGdBi = 40 nm TN ( Bulk )\nH [111]\n µ0H = 0.3  TH[100]\n µ0H = 0.1  T\n µ0H = 0.3  TH[111]\n2.0\n1.5\n1.0\n0.5\n0.0eff( 10-11N m T-2 )\n100 50 0\nT ( K )µ0H ( T ) :\n3\n5\n7\n10\n14tGdBi = 9 nm\n-8-6-4-2024eff( 10-11N m T-2 )\n100 50 0\nT( K )µ0H ( T ) :\n3\n5\n7\n10\n14tGdBi = 40 nm\n3\n2\n1\n0\n-1\n-2M( 10-10N m T-1 )\n15 105 0\nµ0H ( T )tGdBi = 40 nm\n4100T = 30 KFIG. 2. AFM order in GdBi thin \flms. (a) Temperature dependence of the magnetic moment\nmfortGdBi = 40 nm (red and blue curves, left axis) and magnetic susceptibility \u001fBulkfor a GdBi\nsingle crystal (green curve, right axis) measured by a SQUID magnetometer with \feld orientations\nas indicated. (b) Magnetic \feld dependence of magnetic torque \u001cof the GdBi thin \flm sample\n(tGdBi = 40 nm) grown on a 0.5-mm-thick BaF 2substrate measured at di\u000berent temperatures. The\nmeasurement geometry is shown in inset. (c) Magnetic \feld dependence of torque magnetization\nM\u001cat di\u000berent temperatures. (d)(e) Magnetic \feld dependence of \u0001 \u001fe\u000bcalculated from M\u001cfor\n(d)tGdBi = 40 nm and (e) tGdBi = 9 nm as a function of temperature (see text).\n21(a) (b)\ntGdBi =  5 nm\n6 nm\n9 nm\n40 nm\n5 nm\n6 nm9 nmtGdBi = \n 40 nm\n (× 0.1)\nT = 2 K ( µΩ cm K-2 )d2ρxx\nd T2\n0.0010.010.11\nρxx ( mΩ cm )\n200 150 100 50 0\nT ( K )T ( K )1.0\n0.8\n0.6\n0.4\n0.2\n0.0∆ρxx / ρxx ( H = 0 ) ( % )\n-10 -5 0 5 10\nµ0H ( T )0.4\n0.2\n0.0\n60 40 205 nmtGdBi =\n6 nm\n9 nm\n40 nm×2×2FIG. 3. Change of metallicity with thickness for GdBi thin \flms. (a) Temperature dependence of\nlongitudinal resistivity \u001axx(T) for various tGdBi. Inset: Second derivative of \u001axx(T). (b) Magnetic\n\feld dependence of longitudinal resistivity \u001axx(H) measured at T= 2 K for various tGdBi. The\ndata fortGdBi = 40 nm are scaled by a factor of 0.1.\n22(d) (c) (b) (a)\ntGdBi = 5 nm tGdBi = 6 nm tGdBi = 9 nm tGdBi = 40 nm\n5\n4\n3\n2\n1\n0ρyx ( µΩ cm )\n-5 0 5\nµ0H ( T )2 K5 K10 K20 K30 K40 K50 K75 K100 KT =5\n4\n3\n2\n1\n0ρyx ( µΩ cm )\n-5 0 5\nµ0H ( T )2 K5 K10 K20 K30 K40 K50 K75 K100 KT =5\n4\n3\n2\n1\n0ρyx ( µΩ cm )\n-5 0 5\nµ0H ( T )2 K5 K10 K20 K30 K40 K50 K75 K100 KT =5\n4\n3\n2\n1\n0ρyx ( µΩ cm )\n-5 0 5\nµ0H ( T )2 K5 K10 K20 K30 K40 K50 K75 K100 KT =EFElectron\nHoleEFElectron\nHoleEFElectron\nHoleEFElectron\nHoleFIG. 4. Evolution of Hall e\u000bect as a function of GdBi \flm thickness. (a)(b)(c)(d) Transverse\nresistivity\u001ayxas a function of magnetic \feld at various temperatures for (a) tGdBi = 40 nm, (b)\ntGdBi = 9 nm, (c) tGdBi = 6 nm, and (d) tGdBi = 5 nm. The traces are o\u000bset vertically for clarity.\nThe upper inset in each panel is a schematic depiction of the overlapping bulk bands (see text).\n23(a)\nM K Γ Γ1.02.0\n0.0\n-2.0-1.0E ( eV )Gdd Bip\n0.50\n-0.500.25\n0.00\n-0.25E ( eV )\nK’ Γ K(e)\n0.8 -0.2 0.6 0.4 0.2 0.0SOC(f)\nC = 0\n0.10 0.12 0.14 0.16 0.18 0.201.0\n0.20.40.60.8\n40\n0.0\n-40\n-80E ( meV )\nΓ K M\nd1 ( nm )C = 0C = 2Gap ( eV )\nK MΓ0.00.3\n-0.3E ( eV )\nK MΓ K MΓ K MΓ K MΓ\nFundamental gap\nGap at Γ point0.6\n0.4\n0.2\n0.0\n-0.2\n0.10 0.12 0.14 0.16 0.18 0.20\nd1 ( nm )Gap ( eV )(g)\na1\na2d1\nTop viewSide view Gd\nBi\nυ2 = 0\n∆C= -1υ(b)\n(d)\n-1Γ-1+1\n+1\n+1ΓΚ\nK’M\n∆ 0\n= 22 ≠0.51.0\n0.0\n-1.0-0.5E ( eV )(c)\nM K Γ ΓCBipx Gddz2FIG. 5. First-principle calculations of GdBi band structures. (a) Band structure of bulk GdBi\n(111) with the spin-orbit coupling (SOC). (b) Illustrated crystal structure of the monolayer GdBi\n(111).d1is the distance between Bi and Gd layers and a1;a2are the in-plane lattice constants. (c)\nTypical band structure of the monolayer GdBi (111). (d) Schematic Brillouin zone of the monolayer\nGdBi (111). Color of the circle denotes the change in the Chern number \u0001 Cacross the topological\nphase transition, contributed by each Dirac point: \u00001 for red and +1 for blue. (e) Edge states of\nmonolayer GdBi (111) along Zigzag direction, exhibiting a pair of edge modes around the \u0000point.\n(f) Topological phase diagram of monolayer GdBi (111) varies with inter-layer distance d1and SOC\nstrength\u0015. The color represents the direct gap size and negative value indicates inverted band\nordering. The inset shows the \ftting results of k\u0001p(solid line) to the \frst-principles calculations\n(empty circle) around the critical point d1= 0:134 nm. (g) Strain induced topological phase\ntransition of the monolayer GdBi (111). The top and bottom panels respectively show the band\nevolution and energy gap as a function of d1. In the bottom panel, the red line represents the direct\ngap at \u0000point, and negative value means Chern insulator with C= 2; the blue line represents the\nfundamental gap in the whole Brillouin Zone and the negative value denotes semi-metallic.\n24"    },    {        "title": "1904.08148v1.Two_orbital_effective_model_for_magnetic_Weyl_semimetal_in_Kagome_lattice_shandite.pdf",        "content": "Typeset with jpsj3.cls <ver.1.0> Letter\nTwo-orbital e \u000bective model for magnetic Weyl semimetal in Kagome-lattice shandite\nAkihiro Ozawa1, Kentaro Nomura1;2\u0003\n1Institute for Materials Research, Tohoku University, Katahira, Aoba-ku, Sendai 980-8577\n2Center for Spintronics Research Network, Tohoku University, Katahira, Aoba-ku, Sendai 980-8577\nWe construct a two-orbital e \u000bective model for a ferromagnetic Kagome-lattice shandite, Co 3Sn2S2, a\ncandidate material of magnetic Weyl semimetals, by considering one dorbital from Co, and one porbital\nfrom interlayer Sn. The energy spectrum near the Fermi level, and the configurations of the Weyl points,\ncomputed by using our model, are similar to those obtained by first principle calculations. We show also\nthat nodal rings appear even with spin-orbit coupling when the magnetization points in-plane direction.\nAdditionally, magnetic properties of Co 3Sn2S2and other shandite materials are discussed.\nKEYWORDS: Weyl semimetal, anomalous Hall effect, Kagome lattice, shandite, ferromagnetism\nIntroduction — Weyl semimetals are gapless semiconduc-\ntors with non-degenerate point-nodes called Weyl points.1–3)\nThese nodes generate a fictitious magnetic field, the Berry\ncurvature,4)in momentum space. Weyl semimetals with mag-\nnetic order, magnetic Weyl semimetals, attract attention be-\ncause of the nontrivial charge-spin coupling, represented by\nthe large anomalous Hall e \u000bect.5)Additionally, some electro-\nmagnetic responses such as the charge induced spin torque6, 7)\nand electric-field-driven domain wall motion8, 9)are theoreti-\ncally predicted. These phenomena suggest a potential to im-\nplement highly e \u000ecient magnetic devices by using the topo-\nlogical characters of electronic states. As candidates, some\nmaterials such as Mn 3Sn10–17)and Heusler alloys18, 19)have\nbeen studied. However, these materials have large Fermi sur-\nfaces in addition to the Weyl points. This metallicity may sup-\npress functionalities of Weyl semimetals mentioned above by\nthe screening e \u000bect. In order to realize these functionalities,\nit is important to find magnetic Weyl semimetals with small\nFermi surfaces.\nVery recently, it was suggested that a ferromagnetic\nKagome-lattice shandite Co 3Sn2S2is a strong candidate of\nthe magnetic Weyl semimetal from first principle calcula-\ntions and experiments.20–22)This material possesses a rela-\ntively large anomalous Hall angle \u0012AHE=\u001bAHE=\u001bxx\u001920%\nwhich is much larger than those of other candidates such as\nMn 3Sn (\u0012AHA\u00193%)12, 20). This indicates the semimetallic\ncharacter with a small longitudinal conductivity \u001bxxand small\nFermi surfaces, suggesting an ideal magnetic Weyl semimetal.\nAccording to first principle calculations,20, 21)in the absence\nof spin-orbit coupling, nodal rings appear near the Fermi\nlevel, while spin-orbit coupling opens energy gaps on the\nnodal rings except at some points, the Weyl points.\nUtilizing Co 3Sn2S2, we expect electromagnetic functional-\nities of Weyl semimetals. However, it is di \u000ecult to study the\nmagnetic response using first principle calculations, because\nwe have to introduce magnetic field via the Peierls phase of\nelectrons that causes a huge matrix of the Hamiltonian with\nmany orbitals. Therefore, it is desirable to construct a mini-\nmal model describing the low energy excitations with only a\nfew orbitals.\nIn this work, we construct an e \u000bective two-orbital tight-\nbinding model by using one of the dorbitals from Co and\n\u0003nomura@imr.tohoku.ac.jpone of the porbitals from interlayer Sn. We show that con-\nfigurations of nodal rings and the Weyl points in the Bril-\nlouin zone in our model are similar to those in first princi-\nple calculations.20, 21)Additionally, we show that nodal rings\nappear even with spin-orbit coupling when the magnetization\npoints in-plane direction. We discuss the origin of magnetism\nin shandite materials and magnetic anisotropy.\nModel Hamiltonian — In the following, we construct an ef-\nfective tight-binding model Hamiltonian by considering the\ncrystal field splitting and relevant orbitals close to the Fermi\nlevel of the system. Figures 1(a) and 1(b) compare the orig-\ninal unit cell of Co 3Sn2S2and the unit cell of our model.\nCo3Sn2S2consists of primitive rhombohedral unit cells in-\ncluding three Co atoms, two Sn atoms, and two S atoms as\nshown in Fig. 1(a). One layer has the Kagome lattice struc-\nture of Co atoms with Sn at the center of hexagons as shown in\nFig. 1(c). We refer to it as Sn2 to distinguish another Sn atom,\nSn1, which form a triangular lattice. There are also two layers\nof triangular lattices of S atoms. We assume that the crystal\nfield splitting in shandite is as shown in Fig. 1(d). Splitting en-\nergies in the Kagome layer are larger than those in the triangu-\nlar lattices of Sn and S atoms because the interatomic distance\nin the Kagome lattice is shorter than those in the triangular\nlattices. In this crystal structure, the five-fold degeneracy of\nCo’s dorbitals is lifted, although the energy-level relationship\nis not clear. A neutral Co atom, Sn atom, and S atom have 9,\n4, and 6 valence electrons, respectively. When the crystal field\nsplitting energies are large enough compared to the Hund cou-\npling energies, low spin states are favored; low energy orbitals\nare occupied by the valence electrons. The electron’s config-\nuration is assumed as shown in Fig. 1(d), where the fourth of\nfivedorbitals of Co atoms are partially occupied; there is one\nelectron in 3\u00022=6 states on three Co atoms per unit cell.\nWhen spins are fully polarized this configuration is consis-\ntent with the magnetic moment MCo\u00190:3\u0016B=Co suggested\nby first principle calculations20)and experiments.22, 23, 25)We\nassume that d3z2\u0000r2is the partially occupied orbital and that\nthe occupied pzorbital of Sn1 atom is close to the Fermi\nlevel. On the other hand, all other orbitals are far from the\nFermi level and thus neglected in the following. In this model,\nthere are three electrons in eight bands, 6 bands from three\nCo atoms and 2 bands from one Sn1 atom. The Fermi level\nis determined by this 3 /8 filling condition. The unit cell of\n1arXiv:1904.08148v1  [cond-mat.mes-hall]  17 Apr 20192 J. Phys. Soc. Jpn. L etter A. Ozawa, K. Nomura\nES0.3𝜇\"/Co𝑝𝑧𝑑&'()*(SSnSnCoCoCoCoSn2SSn1(c)(b)\nCoSSn1Sn2\nCoSn1\n(a)\n𝒂𝟐𝒂𝟏𝒂𝟑\n(d)\nFig. 1. (a) Original unit cell of Co-shandite and (b) unit cell of our model.\nCo is responsible for ferromagnetic order. (c) Each layer of Co-shandite.\nKagome layer contains Co atoms and Sn2 atoms. Those Kagome layers\nsandwich two types of triangle layers of Sn1 atoms and S atoms. (d) The\nenergy relation and occupied electrons assumed in our model.\nour model has a rhombohedral lattice structure as the origi-\nnal one. The primitive translation vectors are a1=(a\n2;0;c),\na2=(\u0000a\n4;p\n3a\n4;c),a3=(\u0000a\n4;\u0000p\n3a\n4;c) as shown in Fig. 1(b).\nIn the following, we set c=p\n3a\n2for simplicity. In our model,\nthe unit cell includes three Co atoms on the Kagome lattice\nand one Sn1 atom on the triangular lattice.\nOur e \u000bective model Hamiltonian consists of three terms,\nH=Hd-p+Hexc+Hso; (1)\nwhere Hd-pis the hopping term, Hexcis the exchange coupling\nterm, and Hsois the spin-orbit coupling term. Detailed expla-\nnations of each term are given in the following.\nWe start with the hopping term Hd-p. We consider the\nfirst and second-nearest-neighbor hopping, t1andt2, in the\nKagome layer, inter-Kagome-layer hopping, tz, and dphy-\nbridization tdp. We neglect hopping between Sn atoms be-\ncause the interatomic distance is longer than others. Hd-pis\nwritten as follows,\nHd-p=\u0000X\ni j\u001bti jdy\ni\u001bdj\u001b\u0000X\ni j\u001b(tdp\ni jdy\ni\u001bpj\u001b+tdp\ni jpy\ni\u001bdj\u001b)\n+\u000fpX\ni\u001bpy\ni\u001bpi\u001b: (2)\nHere di\u001bandpj\u001bare the annihilation operators of delectrons\non the Kagome lattice and pelectrons on the triangular lattice.\nti jdescribes the hopping between Co sites and is either of t1,\nt2,tzor zero, depending on the relative position. tdp\ni j=tdp\n(a)(b)yx𝒅\"𝒅#𝒅$𝒃\"𝒃$𝒃#𝒄\"𝒄$𝒄#yxzzFig. 2. (a) Intralayer lattice vectors of a Kagome layer. b1,b2,b3are the\nfirst-nearest-neighbor vectors. d1,d2,d3are the second-nearest-neighbor\nvectors. (b) Interlayer lattice vectors of Kagome layers. c1,c2,c3are the\nfirst-nearest-neighbor vectors.\nbetween nearest Co and Sn1 sites, otherwise tdp\ni j=0.\u000fpis the\nenergy di \u000berence between porbital and dorbital.\nHexcdescribes the ferromagnetic ordering derived from the\nonsite Hubbard coupling within the mean field approxima-\ntion.26)The Hamiltonian is given in the following form,\nHexc=\u0000JX\ni\u001b\u001b0m\u0001dy\ni;\u001b\u001b\u001b\u001b0di;\u001b0: (3)\nHere, we neglect the onsite Coulomb energy of porbital\nof Sn1 because that of pelectrons is smaller than that of\ndelectrons. In the mean field theory mis determined self-\nconsistently. On the other hand, in the following, we set the\nstrength of Jjmjby comparing to the first principle calcula-\ntions,21)where m=m(0;0;1).\nWe introduce the Kane-Mele type spin-orbit coupling in the\nKagome layer27, 28)as given as\nHso=\u0000itsoX\n<ij>\u001b\u001b0\u0017ijdy\ni\u001b\u001bz\n\u001b\u001b0dj\u001b0: (4)\nHere tsois the hopping amplitude and the sign \u0017i j=\u00061 de-\npends on the orientation of the two nearest neighbor bonds.\nWhen an electron traverses in going from site jtoi,\u0017i j=\u00061\n, if the electron makes a left (right) turn to get to the sec-\nond bond. The substantial strength of spin-orbit coupling\noriginates from the presence of Sn2 atoms at the center of\nhexagons in the original lattice structures of Co 3Sn2S2. The d\nelectrons in the hexagons are susceptible to the strong poten-\ntials from Sn2’s nucleuses.\nNodal rings and Weyl points — The Hamiltonian Eq. (1)\nis diagonalized by using the Fourier transformation di\u001b=\n1p\nNP\nkeik\u0001Ridk\u000b\u001bandpi\u001b=1p\nNP\nkeik\u0001Ripk\u001b. Here kis the\ncrystal momentum, and \u000b=A;B;orCis the sublattice index\nof the Kagome lattice. For each kthe Bloch wave function is\nan eight component eigenvector junkiof the Bloch Hamilto-\nnian matrixH(k) which is given by\nH=X\nk;\u001bCy\nk\u001bH(k)Ck\u001b; (5)\nwhere Cy\nk\u001b=(dy\nkA\u001b;dy\nkB\u001b;dy\nkC\u001b;py\nk\u001b).H(k) consists of the\nfollowing terms:H(k)=H1+H2+Hdd+Hdp+Hp+Hexc+\nHso. Here, each term is given as below.J. Phys. Soc. Jpn. L etter A. Ozawa, K. Nomura 3\nH1=\u00002t10BBBBBBBBBBBB@0 cos( kb\n1)\u001b0cos(kb\n3)\u001b00\ncos(kb\n1)\u001b0 0 cos( kb\n2)\u001b00\ncos(kb\n3)\u001b0cos(kb\n2)\u001b0 0 0\n0 0 0 01CCCCCCCCCCCCA;\nH2=\u00002t20BBBBBBBBBBBB@0 cos( kd\n1)\u001b0cos(kd\n3)\u001b00\ncos(kd\n1)\u001b0 0 cos( kd\n2)\u001b00\ncos(kd\n3)\u001b0cos(kd\n2)\u001b0 0 0\n0 0 0 01CCCCCCCCCCCCA;\nHz=\u00002tz0BBBBBBBBBBBB@0 cos( kc\n1)\u001b0cos(kc\n3)\u001b00\ncos(kc\n1)\u001b0 0 cos( kc\n2)\u001b00\ncos(kc\n3)\u001b0cos(kc\n2)\u001b0 0 0\n0 0 0 01CCCCCCCCCCCCA;\nHdp=2itdp0BBBBBBBBBBBB@0 0 0 \u0000sin(ka\n1)\u001b0\n0 0 0 \u0000sin(ka\n2)\u001b0\n0 0 0 \u0000sin(ka\n3)\u001b0\nsin(ka\n1)\u001b0sin(ka\n2)\u001b0sin(ka\n3)\u001b0 01CCCCCCCCCCCCA;\nHp=\u000fp0BBBBBBBBBBBB@0 0 0 0\n0 0 0 0\n0 0 0 0\n0 0 0 \u001b01CCCCCCCCCCCCA;\nHexc=\u0000Jjmj0BBBBBBBBBBBB@\u001bz0 0 0\n0\u001bz0 0\n0 0\u001bz0\n0 0 0 01CCCCCCCCCCCCA;\nHso=\u00002itso0BBBBBBBBBBBB@0\u0000cos(kd\n1)\u001bz cos(kd\n3)\u001bz 0\ncos(kd\n1)\u001bz 0\u0000cos(kd\n2)\u001bz0\n\u0000cos(kd\n3)\u001bz cos(kd\n2)\u001bz 0 0\n0 0 0 01CCCCCCCCCCCCA:\n(6)\nHere, kb\ni=k\u0001bi,kc\ni=k\u0001ci,kd\ni=k\u0001di, and ka\ni=k\u0001ai=2,i=\n1;2;3. These lattice vectors are shown in Fig. 1(b), Fig. 2(a)\nand Fig. 2(b).\nBy solving the eigenvalue equation H(k)junki=Enkjunki,\nwe obtain eigenstates junkiand eigenvalues Enk, where n\n(from 1 to 8) being the band index labeled from the bot-\ntom. The energy eigenvalues of the system calculated with-\nout and with spin-orbit coupling along high-symmetry lines\nare shown in Fig. 3(a). Here we set t1as a unit of energy,\nt2=0:6t1,tdp=1:0t1,tz=\u00001:0t1,\u000fp=\u00003:5t1,J=2:0t1. We\nfocus on n=3 band and n=4 band crossing the Fermi level.\nWhen spin-orbit coupling is absent, nodal rings between the\nn=3 band and n=4 band appear around the L point. The\npositions of the nodal rings in momentum space are shown as\ngreen lines in Figs. 4(a) and 4(b). The above hopping param-\neters were chosen so that the configurations of the nodal rings\nare similar to those obtained by first principle calculations.21)\nThe nodal rings appearing in the absence of spin-orbit\ncoupling are gapped out in the presence of spin-orbit cou-\npling except two points on each ring. The energy spectrum\nshown in Fig. 4(c) is linear around the band touching points,\nwhich is consistent with the result of first principle calcula-\ntions.20)To characterize these nodal points we calculate the\n-10-50510Energy𝜎\"#[𝑒&/ℎ𝑎]𝐸𝑡-⁄TLWUΓWithoutWithSOCSOC-5-1005\n-0.51.500.51(a)(b)10\n-10-50510\n-0.500.511.5Fig. 3. (a) Band structure on high symmetry lines. There is a band inver-\nsion between the n=3 band and n=4 band in the absence of spin-orbit\ncoupling (green lines). Spin-orbit coupling opens energy gaps on high sym-\nmetry lines (red lines) (b) Energy dependence of the anomalous Hall con-\nductivity. A peak structure near the energy of the Weyl points is observed.\nBerry curvature4)bnk=r\u0002ankof the n=3 band. Here\nank=\u0000ihunkjrkjunkiis the Berry connection.4)Figure 4(d)\nshows the Berry curvature distribution in the ky=0 plane.\nThere are sources and sinks of the Berry curvature corre-\nsponding to the Weyl points with positive chirality and nega-\ntive chirality, respectively.\nNext, we examine the intrinsic anomalous Hall conductiv-\nity by using the Kubo formula,5)\n\u001bxy=e2\nhX\nnZ\nBZd3k\n(2\u0019)2bz\nnkf(Enk\u0000\u0016): (7)\nHere, nis the occupied band index, bz\nnkis the zcomponent\nof the Berry curvature, fis the Fermi-Dirac distribution func-\ntion and\u0016is the Fermi level. Figure 3(b) shows the Fermi\nlevel dependence of the anomalous Hall conductivity. Near\nthe energy of the Weyl points, the anomalous Hall conductiv-\nity has a large peak. The value near the Fermi energy \u001bxy\u0019\n1059\n\u00001cm\u00001is very close to the result of first principle cal-\nculation and experiment.20, 22)In an ideal Weyl semimetal\nwhere Fermi surfaces reside only at the Weyl points, we can\ncompute the anomalous Hall conductivity as the summation\nof the distance of the Weyl points separated by \u0001K(\r)\nzas fol-\nlowing,1–3)\n\u001bWeyl\nxy=e2\n2\u0019hX\n\r\u0001K(\r)\nz: (8)\nHere,\rindicates the pair of the Weyl points. The value of the\nanomalous Hall conductivity computed by Eq. (8) is \u001bxy\u0019\n0:86[e2=ha]. This value is in reasonable agreement with that\nat the energy of the Weyl points computed by Eq. (7) with our\nmodel,\u001bxy\u00190:61[e2=ha].\nIn the above calculation, we showed that the Weyl points\nappear in the presence of spin-orbit coupling when the mag-\nnetization is parallel to the z-axis. Here, we study the energy\nspectrum when the magnetization is perpendicular to the z-\naxis. In this situation, nodal rings appear even in the pres-\nence of spin-orbit coupling. In-plane magnetizations are set\nasmA=mB=mC=m(1;0;0). Figure 4(e) shows nodal\nrings between the n=3 band and n=4 band which corre-4 J. Phys. Soc. Jpn. L etter A. Ozawa, K. Nomura\n!\"=+1!!\"=−1!\"!#!$!\"!#!$!\"!#!$-10123\n33.544.555.5\n(a)\n(c)\n𝑘\"𝑎𝑘$𝑎(d)(b)\n(f)\n(e)\n𝛴𝛴𝛴𝑘\"𝑘&\n𝑘\"𝑘&𝑘\"𝑘$𝑘&\nWith SOC\nFig. 4. (a)(b) Band touching points between n=3 and n=4 band without\nSOC (green lines) and with SOC (red and blue points). (c) Energy spectrum\ninky=0 plane. Gapless linear dispersions appear near the Fermi level.\n(d) Berry curvature distribution in the ky=0 plane. The sink and source\nof the Berry curvature correspond to the Weyl points. (e) Nodal rings with\nspin-orbit coupling when the magnetization is perpendicular to the z-axis.\n(f) The relation between the Weyl points configuration and Chern number\ndefined in the plane \u0006. Chern number changes when the magnetization\nflips.\nspond to nodal rings in Fig. 4(b). This appearance of nodal\nrings with spin-orbit coupling can be understood in terms of\nthe Chern number. We consider a certain plane in momentum\nspace as shown in Fig. 4(f). For simplicity, we consider a sys-\ntem with two Weyl points.3)The Hamiltonian in this plane,\nsay\u0006, can be regarded as that of a two-dimensional quan-\ntum anomalous Hall state.3)The Chern number of nth band,\n\u0017n(\u0006)=R\n\u0006bz\nn(k)dk, can be defined in this plane \u0006, if the plane\ndoes not contain any Weyl points.3)The sign of the Chern\nnumber changes, when it is finite, with the flip of the magne-\ntization. The change of the Chern number occurs only when\nthe band gap closes. Therefore, when the magnetization flips\nfrom one direction to the opposite direction, nodal rings need\nto appear as shown in Fig. 4(f).\nMagnetic order — Next, we discuss the origin of mag-\nnetism in shandite materials based on the Stoner theory.26)Al-\nthough most shandite materials are non-magnetic, Co 3Sn2S2\nshows ferromagnetic order.29, 31)Additionally, this ferromag-\nnetism is suppressed by substituting, for example, Co for Fe\nor Ni, Sn for In.23, 24, 30, 34–36)According to the Stoner theory,\nferromagnetism of itinerant electrons is characterized by the\nfollowing criterion,26)\nD(EF)U>1: (9)\n-10-50510\n-0.3-0.2-0.100.10.20.3-10-50510\n-0.3-0.2-0.100.10.20.3(a) Non-magnetic(b) Ferromagnetic𝐸/𝑡$Density of states𝐸%𝐸%Fig. 5. The density of states of (a) non-magnetic state Jjmj=0. A peak\nnear the Fermi level stabilizes ferromagnetic order and of (b) ferromag-\nnetic state Jjmj,0. A minimum of the density of states near the Fermi\nlevel is shown.\nHere, D(EF) is the density of states in non-magnetic state at\nthe Fermi level and Uis the onsite Coulomb interaction. In or-\nder to examine this criterion, we calculate the density of states\nin non-magnetic state as shown in Fig. 5(a). The Fermi level\ncan be calculated by the 3 /8 filling condition. The density of\nstates has a peak structure near the Fermi level. This peak is\nsignificant to satisfy the Stoner criterion Eq. (9). When the\nnumber of electrons changes due to the chemical substituent,\nthe Fermi level shifts from the peak and the density of states\ndecreases, suppressing ferromagnetic order. Contrary, in fer-\nromagnetic state, the density of states has a minimum near the\nFermi level as shown in Fig. 5(b).\nMagnetic anisotropy — We examine the easy-\naxis anisotropy of our model. We calculate the\nmagnetization-angle dependence of the total energy\nwith two-types of tilted configurations. In the first\ncase, magnetizations on each sublattice are given as\nmA=mB=mC=(sin\u00121;0;cos\u00121)as shown in Fig. 6(a). In the\nsecond case, magnetizations on each sublattice are given as\nmA=m(sin\u00122;0;cos\u00122),mB=m(\u00001\n2sin\u00122;p\n3\n2sin\u00122;cos\u00122),\nmC=m(\u00001\n2sin\u00122;\u0000p\n3\n2sin\u00122;cos\u00122) as shown in Fig. 6(b).\nHere,\u00121and\u00122are tilting angles. The first tilting case Fig. 6(a)\ncorresponds to the situation when the applied magnetic filed\npoints in the direction perpendicular to the z-axis. The second\ntilting case Fig. 6(b) corresponds to umbrella structure of\nthe magnetization suggested by experiment.37)We compute\nthe total energy of electrons, E=1\nNP\nn;kEnkf(Enk\u0000\u0016) as\na function of \u00121and\u00122with two cases of magnetizations\nFigs. 6(a) and 6(b). Here Nis the number of the unit cells and\n\u0016is the Fermi level calculated under the 3 /8 filling condition\nat each\u00121,\u00122. Figures 6(c) and 6(d) show the energy shifts\nfrom the total energy with out-of-plane magnetization as\nfunctions of \u00121and\u00122, respectively. In both case, the total\nenergy has a minimum at \u00121=\u00122=0. This behavior shows\nthe easy-axis ferromagnetic anisotropy, which is consistent\nwith experiment.32, 33)\nConclusion — In this work, we constructed an e \u000bec-\ntive tight binding model for the ferromagnetic Co-shandite\nCo3Sn2S2. The configurations of nodal rings and the Weyl\npoints are similar to those obtained by first principle calcu-\nlations.20, 21)When the magnetization is perpendicular to the\nz-axis, nodal rings appear even with spin-orbit coupling. WeJ. Phys. Soc. Jpn. L etter A. Ozawa, K. Nomura 5\n00.0050.010.0150.02\n-1.5708-0.78539800.7853981.570800.050.10.150.20.250.3\n-1.5708-0.78539800.7853981.5708!\"!\"!\"#$#%#&!\"!\"!\"#$#%#&\n𝜃\"\t[rad](c)(d)(a)(b)\nΔ𝐸/𝑡-\n𝜃-\t[rad]Δ𝐸/𝑡-\n−𝜋2−𝜋4𝜋40𝜋2−𝜋2−𝜋4𝜋40𝜋2\nFig. 6. (a) Uniformly tilted magnetizations. 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B 96,\n014429 (2017)."    },    {        "title": "1904.09719v1.Strain_Induced_Reversible_Manipulation_of_Orbital_Magnetic_Moments_in_Ni_Cu_Multilayers_on_Ferroelectric_BaTiO3.pdf",        "content": "1 \n Strain -Induced Reversible Manipulation of Orbital \nMagnetic Moments in Ni /Cu Multilayers  on \nFerroelectric BaTiO 3 \n \nJun Okabayashi1*, Yoshio Miura2, and Tomoyasu Taniyama3 \n1Research Center for Spectrochemistry, The University of Tokyo, Bunkyo -ku, Tokyo  113-0033 , Japan  \n2Research Center for Magnetic and Spintronic Materials, National Institute for Materials Science \n(NIMS), Tsukuba 305- 0047, Japan  \n3Department of Physics, Nagoya University, Furo -cho, Chikusa- ku, Nagoya 464- 8602, Japan  \n(2019.4.12)  \n \n \n*jun@chem.s.u- tokyo.ac.j p \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 2 \n  \nABSTRACT  \nControlling magnetic anisotropy by orbital magnetic moments related to interfacial strains  \nhas considerable  potential for the development of future devices using spin s and orbitals . For \nthe fundamental physics, the relationship between strain and orbital magnetic  moment is still \nunknown,  because there are few  tools to probe changes of orbital magnetic moment. In this \nstudy , we developed an electric- field - (E)-induced X-ray magnetic circular dichroism \n(EXMCD) technique to  apply  E to a ferroelectric BaTiO 3 substrate . We reversibly tune d the \ninterfacial lattice  constants  of Ni/Cu multilayers  on BaTiO 3 using this technique . As the \ndomain structures in BaTiO 3 are modulated  by E, EXMCD measurements reveal that the \nchange s in the magnetic anisotropy of Ni/Cu films are induce d through  the modulation of \norbita l magnetic moments in  Ni with  magneto- elastic contributions . The strain ed Ni layer that \ninduc es the perpendicular magnetic anisotropy  without E is released at  E = 8 kV/cm , and in-\nplane magnetization  also occurs . We observed that EXMCD measurements clarif ied the origin \nof the reversible changes in  perpendicular magnetic aniso tropy and established the \nrelationship between  macroscopic inverse magnetostriction effects and microscopic orbital \nmoment anisotropy.  \n \n \nKey words  \nMagnetic anisotropy, Orbital magnetic moments, Multi -ferroics, X -ray magnetic circular \ndichroism, First -principles calculation  \n \n \n \n \n \n 3 \n Introduction  \nThe coupling between  ferromagnetic and ferroelectric properties has recently attracted \nconsiderable attention toward the creation of novel devices  using multiferroic  controlling of their \nproperties1–7. In particular , the hetero -interfaces in thin films comprising both ferromagnets and \nelectrical ly polarized materials produce  a rich variety of possibilities  for creating multifunctional  \nproperties8–18. Modulation of interfacial lattice constants by an electric field ( E) induces interfacial \nchanges in magnetism. The i nterfacial lattice distortion produces  variations  in magnetic properties , \nwhich  are recognized as inverse magnetostriction effects19–23. Moreover, m agnetic an isotropy is \ntuned by lattice distortion s. Recently, the  magnetic anisotropy controlled by  E has become an \nimportant subject in spintronics , which is the  study  aiming at the  realization of devices operating \nwith low energy consumption24–28. Recent attempts have been  focused on the modulation of the \nnumber of charge carriers  at the interface between an ultrathin ferromagnetic layer and an oxide -\nbarrier insulator  in magnetic tunnel junctions . Other approaches , which are our  focus in this study , \nare based on  the interfacial mechanical -strain coupling between ferromagnetic and ferroelectric \nlayers using multiferroic hybrid structures. As one of the candidate  approaches , applying  E to \nBaTiO 3 provides the possibility  to tune the lattice constant s by modulating the domain structures \nalong the a - and c -axes directions  by 3.992 Å and 4.036 Å, respectively, at room temperature29,30.  \nThere exist some reports related to deposit ing magnetic thin-film layers onto BaTiO 3 for E-\ninduced magnetism  with a brupt interfaces between BaTiO 3 and Fe or Co12,14 –16,21,22,24. In other case s, \nthin Ni layers sandwiched by Cu layers exhibit perpendicular magneti c anisotropy (PMA) because \nof the interfacial tensile strain  in the Ni layers31. Recently,  E-control  of the magnetic properties of \nNi/Cu multilayers on BaTiO 3 was achieved ; the magnetization  was switched from the perpendicular \naxis to the in -plane easy ax is by tuning the lattice distortion through the application of  E32. These \neffects may be  explained phenomenologically by inverse magnetostriction effects. Although  \nanisotropic energies depend on orbital magnetic moments, the microscopic origin of the control  of \nthe anisotropic energy dependent on strain is still not known explicitly. Theoretical approaches that \nconsider spin–orbit interaction s, as well as  crystalline potentials , as perturbative treatments have \nbeen  developed33. Moreover, strain -induced orbital magnetic moments have been  discussed 4 \n alongside  calculations on the strained Ni layers34, but the relationship between strain and orbital \nmoments is still unknown.  \nElement -specific magnetic properties and their origins should be investigated  by applying E \nexplicitly  to clarify the relationship between lattice distortion and magnetic properties . In particular , \nmagnetic anisotropy is related to the anisotropy of orbital magnetic moments35. A unique tool to \ndeduce spin moments as well as  orbital moments is X -ray magnetic circular dichroism (XMCD) \nwith magneto -optical sum rules36,37. Recent developments of XMCD by applying E have  focused \non charge accumulati on at the interface of FePt/MgO38, interfacial oxidation reaction of Co/Gd 2O339, \nand other cases40–42. In the case of FePt/MgO , the difference in the Fe XMCD by applying E  is \nnegligibly small because of the quite  small amounts of charge accumulation at the interfaces . In the \ncase of Co/Gd 2O3, the chemical reaction at the oxide interfaces  becomes  dominant. Investigating \nthe orbital moment anisotropy (OMA) by applying E is a challenging approach to initiate  novel \nresearch into the  physics of the relationship between lattice distortion and orbital magnetic moments , \nwhich is  a fundamental and unsolved problem  in the scientific  research field . Considering t he \nrelationship s between  spin magnetic moments ( ms), orbital  magnetic moments ( morb), and strain  (ε), \nthe spin–orbit interaction links m s and morb, and the magnetostriction links ms and ε . However, the \nrelationship between  morb and distortion is still unexplored. To  understand this relationship and the \nelastic phenomena from the view point of m orb, we developed a  technique by applying an electric \nfield in XMCD measurements  (EXMCD) to clarify the mechanism of the electric- field-induced \nchanges in the magnetic anisotropy  of Ni /Cu multilayer s on BaTiO 3 hetero -structure s through lattice \ndistortion s. In this study , we aim to clarify the relationship between strain and orbital magnetic \nmoments using the EXMCD method; we also present first -principles calculation  of the changes in  \nmagnetic anisotropy .  \n \nResults  \nStrain introduced into the samples  \nFirst, we mention the values of the strain  introduced into the  sample s, as illustrated in Fig. 1a . \nWithout applying E , the Ni layer possesses a  tensile strain of 2% through the sandwiched -Cu layers , 5 \n and the Ni layer exhibits PMA  as shown in Fig. 1b . When  E is zero, the a- and c -domain s tructures  \nare mixed in BaTiO 3. By applying E, the c-domain  structures  become dominant, from  which it may \nbe inferred  that the application of the electric field , E, compresses  the lattice constant of BaTiO 3 \nand releases  the strain in the Ni layer , thereby resulting in the magnetization in the in -plane easy \naxis in the Ni layers. Therefore, in the a-  and c-domain  structures  of BaTiO 3, the Ni layer exhibit \nPMA  and in-plane anisotropy , respectively32. Figure 1c display s the differential interference \nmicroscopy images of  the a- and c -domains before and during  the application of  E. The b right area \nindicates  the a-domain structure. Both  a- and c-domain  structures  without E are clearly observed , \nand they align to the c-domain structure by applying ± 5 kV/cm , which is consistent with the \nmagnetization . The area of a- domain structure is estimated from the images in Fig. 1c  as \napproximately 50 % without E . We emphasize that the strain propagation proceeds into all \nmultilaye rs by the modulation of lattice constants of BaTiO 3 substrates  due to the high strain- transfer \nparameter value.32,43 In the case of 20 -nm-thick Fe film on BaTiO 3, strain propagation into the \nsurfaces is also detected.44 Further, w e confirmed that t his process is reversible  by remov ing E .     \nXAS and XMCD \nResults from X -ray absorption spectroscopy  (XAS ) and XMCD in the total- electron -yield  \n(TEY) mode , which probe the depth beneath 3 nm from the surface  by collecting drain currents of \nsecondary photoelectrons , without applying E , are shown in Fig. 2. The intensity ratio between  the \nL-edges  of Ni and Cu guarantees the amounts of Ni and Cu within the detection limits in TEY . We \nobserved c lear XMCD at  the L-edge of Ni but not at  the L-edge of Cu . This phenomenon suggest s \nthat the magnetic moments at the interface are not induced into the Cu layers. Considering the \nmagneto -optical sum rules36,37, the spin and orbital magnetic moments in the Ni sites are estimated \nto be 0.50 and 0.04 µB, respectively , by assuming that the hole number i s 1.75. These values are \ncomparable to previous reports of the Ni/Cu interface44. Angular -dependent XMCD measurements \ndeduce the spin dipole term m T of smaller than 0.005 µB. \nFigure 3 shows the Ni L-edge XAS and XMCD spectra obtained in the partial  fluorescence  \nyield  (PFY) mode by applying an electric field, E, of positive bias of 8  kV/cm . The PFY mode \nprobes a depth beneath  approximately  100 nm from  the surface because of the photon- in and photon-6 \n out processes. Sample surfaces are connected to ground and E  is applied to the back side of the \nBaTiO 3 substrates. The s pectral line shapes of the XAS and XMCD are modulated by  E in normal \nincident (NI) case  in spite of fixed sample measurement position . By comparing the results with and \nwithout E, we observed a  slight variation in the  peak asymmetries  between the L3 and L2 edges . \nMoreover, the integrals of the L2,3-edge XMCD peaks are plotted on the same panel  to emphasize \nthe difference in the spectral line shapes with the application of  E. As the convergent values of the \nintegrals of the XMCD are proportional to the orbital magnetic moments  within the framework of \nthe XMCD sum rule36,37, these results reveal that the orbital moments are modulated by applying E , \nthereby resulting in changes of magnetic anisotropy . As the vertical  beam size is approximately  1 \nmm, the contribution from only the a-domain cannot be detected  in the  0 kV/cm condition, which \nunderestimates the perpendicular component of the orbital moments. The values of spin and orbital \nmoments are estimated to be 0.56 and 0.055 µB, respectively, for an electric field of 0 kV/cm, and \n0.56 and 0.045  µB, respectively, for an electric field of 8 kV/cm  with error bars of ± 20% for each \nvalue  considering estimated ambiguities applying sum rules  as listed in Table I. The modulation of \nthe orbital magnetic moments by  0.01 µB upon applying E is related to the  induced lattice distortion \nof 2% from the BaTiO 3 substrates. Moreover, after releasing E  to zero, spectral line shapes also \nrevert to the pristine state. For grazing incident (GI) case, XAS and XMCD spectra with and without \nE are displayed in Figs. 3c and 3d. The values of m s and morb are also listed in Table I. Effects of \nelectric field in GI are  smaller than  those in NI because oblique  configuration of 60°  from sample \nsurface normal detects the half of in -plane components  (cos60°=1/2) . Angular dependence depicts \nthe changes of mseff which includes m s+7mT and the  magnetic dipole  term of  mT cancels in the magic \nangle 53.7° of near GI  set up. Details are explained in supplemental material. Thus, the value of m T \nis estimated less than 0.001 µ B. Therefore, these results originate from the  modulation of not spin \nmoments but orbital moments, suggesting that the inverse magneto- striction effects are derived from \nthe changes of orbital moments .  \nThe element -specific magnetization curves ( M–H curves) at the L3-edge of Ni  during the \napplication of E in the normal incidence setup are shown in Fig. 4. As the normal of the  sample \nsurface  is parallel to both incident beam  and magnetic field , the contribution from the easy axis in 7 \n PMA is observed. By applying an electric field,  E, of ±8 kV/cm, the M –H curves change to those \nof the in -plane easy-axis behavior , which is related to the changes in the orbital magnetic moments \nin Ni . After switching  off E, the M–H curves exhibit  the PMA characteristics again , as shown in Fig. \n4b. Moreover, t he reversible changes observed by applying E in XMCD are confirmed  by the \nchanges in the XMCD line shapes . The amounts of  the changes in the M–H curves at  the Ni L 3-edge \nXMCD are a little  similar to those measured by the magneto -optical Kerr effect32. This phenomenon \nsuggest s that the domain structures in the observed area in  the EXMCD measurements can be \nchanged from  the a-domain to the  c-domain by applying E .     \nFirst -principles density -functional -theory calculation  \nWe performed first -principles calculations of magneto -crystalline anisotropy (MCA) \nenergies for fcc Ni as a function of the in- plane lattice constant ( a||). Assuming the motion of free \nelectrons as a ground state, spin- orbit interaction is adopted as a per turbation term for the \nestimation of MCA energy. The MCA energy is defined as  the difference between  the sums of \nthe energy eigenvalues for magnetizations oriented along the in- plane [100] and out -of-plane \n[001] directions ( ΔEMCA=E[100]− E[001] ). We employed the spin–orbit coupling constant ξ of \nthe Ni  atom, 87.2 meV , in the second- order perturbation calculation of the spin–orbit interaction \nto obtain MCA energies for each atomic site.46 \nFigure 5a shows the Δ EMCA and the anisotropy of orbital moments ( Δmorb=morb[001] \n−morb[100]) as a function of a ||, where a perpendicular lattice parameter for each a || is optimized \nfrom the equilibrium value of a 0 = 3.524 Å. As shown in Fig. 5a , both Δ EMCA and Δmorb increase \nwith the tensile in -plane distortions, which are consistent with the XMCD results. In the \nequilibrium condition in the fcc structure, orbital and spin moments of 0.0 483 µB and 0.625 µ B, \nrespectively,  are estimated. This estimation is in good agreement with previous band- structure \ncalculation s for Ni34,47. The slope in Fig. 5a  results in a  modulation of the orbital moment of \n0.002 µ B per 1% strain . We display in Fig. 5b the spin- resolved MCA energies for the four cases \nΔE↑↑, ΔE↓↓, ΔE↑↓, and ΔE↓↑ as a function of a ||, where ↑ and ↓ indicate majority(up) - and \nminority(down) -spin states, respectively ; the left arrow indicates an initially  occupied spin- state \nand the right arrow indicates an intermediate spin -state in the second order perturbation of the 8 \n spin–orbit interaction46. First, the spin -conservation term  ΔE↓↓ increases with increasing a || of fcc  \nNi and qualitatively reproduces the a || dependence of  the MCA energies and Δ morb. Second, the \nspin- flip term Δ E↑↓ decreases with increasing a ||, indicating that the origin of the change of the \nMCA energies of fcc Ni by the tetragonal distortion can be attributed to the Δ E↓↓. We confirmed \nthe strain dependence of spin magnetic moments is ten times smaller than Δmorb. Therefore, this \nmeans that the MCA of fcc Ni can be described mainly by Bruno’ s relation through the orbital \nmoment anisotropy35. Since ΔE↓↓ and ΔE↑↓ can be described as the energy differences between  \nthe z and x  directions,   \nΔ𝐸𝐸↓↓=𝐸𝐸↓↓(𝑥𝑥)−𝐸𝐸↓↓(𝑧𝑧)=−ξ2∑|〈𝑜𝑜↓|𝐿𝐿𝑥𝑥|𝑢𝑢↓〉|2−|〈𝑜𝑜↓|𝐿𝐿𝑧𝑧|𝑢𝑢↓〉|2\n𝐸𝐸𝑢𝑢↓−𝐸𝐸𝑜𝑜↓𝑜𝑜↓,𝑢𝑢↓   \nΔ𝐸𝐸↑↓=𝐸𝐸↑↓(𝑥𝑥)−𝐸𝐸↑↓(𝑧𝑧)=ξ2∑|〈𝑜𝑜↑|𝐿𝐿𝑥𝑥|𝑢𝑢↓〉|2−|〈𝑜𝑜↑|𝐿𝐿𝑧𝑧|𝑢𝑢↓〉|2\n𝐸𝐸𝑢𝑢↓−𝐸𝐸𝑜𝑜↑𝑜𝑜↑,𝑢𝑢↓  .            (1)  \nThe matrix elements of Lx and Lz, depending on in- plane strain , provide the orbital -resolved \ncontributions ; o(u) represents occupied  (unoccupied) states.48 We adopted spin- orbit coupling \nconstant ξ of 78 meV for Ni. The matrix elements of L z and Lx favor the out -of-plane and in-\nplane contributions, respectively. The matrix elements of L z between d(xy) and d(x2-y2) orbitals \nare large positive contributions to the MCA energies and increase with a ||. Each element is \nestimated as a function of the strain, as shown in Fig. S 2 (See supplemental material). Thus, the \nchanges of orbital hybridization in d(xy) and d(x2-y2) orbitals directly contribute to the change \nof the MCA energ y in Δ E↓↓ through the tetragonal distortions. These pictures appear in the band \ndispersions of strained fcc  Ni. Figures 5c and 5d show the minority band dispersion of fcc  Ni \nwithout distortion and with 2% extensive distortion. The color map of the band dispersion \nindicates the magnitude of  the projection of  the d(x2-y2) and d (xy) orbital s. As shown in Figs. 5c  \nand 5d, the d(x2-y2) bands around the M point approach the Fermi energy, while the d(xy) bands \nmove very little in response to  the tensile  distortion. Since the d(x2-y2) orbital spreads in the \ndirection of the nearest neighbor atoms, it is strongly affected by in- plane distortion; whereas , \nthe influence of the distortion is relatively small for the d(xy) orbital because of the distribution \nbetween the nearest neighbor atoms. Therefore, first -principles calculations also capture the \ntrends of the modulation in  the d(xy) - and d(x2-y2)-orbital states. Furthermore, the electron 9 \n occupation number in Ni is  estimated to be 8.25 which remains unchanged by the introduction \nof strain because of the compensation of the occupancy dependence of each 3d orbital.    \n \nDiscussion  \nConsidering the above results, we discuss  the relationship between  the OMA  and the magneto -\nelastic energy . In particular, we analyze the magnetic anisotropy energies  in the strained Ni layers  \ndepending on the magnetization and the lattice distortion . Microscopically , the OMA  can be \ndescribed by  Bruno’ s relation35 through the second- order  perturbation of the spin–orbit interaction . \nMoreover, the OMA  produces  the crystalline anisotropy ∆Κ = αξ∆morb, where ∆morb is the \ndifference between the orbital moment of the component perpendicular  to the film and that of in-\nplane  component , with the coefficient of the spin–orbit coupling constant  ξ and the band -structure \nparameter α=1/4 for a more -than-half-filled 3 d transition metal Ni . The XMCD shown  in Fig. 3 \nclearly  exhibits  OMA  that depends  on the applied electric fields . The value of ∆morb is estimated to \nbe 0.01 µB for a strain modulation of 2%, which results in the anisotrop y energy of 6.8 ×105 J/m3 by \nassuming fcc -Ni lattice constants of 3.524 Å. As the hysteresis curves in Fig. 1 and the EXMCD \nresults  in Fig. 4 are almost identical, similar anisotrop y energies can be obtained.  The first -principles \ncalculation also reproduces the ∆ K of the order of 105 J/m3. The anisotropy energy  ∆K is formulated \nin the scheme of OMA by including  the magneto -elastic energy  as a function of strain ( ε);  \nε αξε αξ 005.0 )(= ∆=∆orbm K .               (2 ) \nAs the interfacial strain  modulates the orbital magnetic moment, the strength  of the distortion  \nll/∆=ε , quantified as a ratio of the length difference,  is scaled to  ∆morb, which is deduced from \nthe band- structure calculation. In general, eq. (2) consists of two terms of ∆morb and mT. However, \nthe contribution of m T is much smaller than ∆ morb in 3d TMs. Then, we focus on only the first term.     \nFigure 5a confirms the linear relationship between ε and ∆morb. The slope in Fig. 5 a quantitatively \ndescribes the dependence  of ∆morb on ε in Eq. (2). By applying E  to BaTiO 3, the released lattices \nshorten the in-plane lattice distances, thereby resulting in the decrease of perpendicular orbital \nmoments.  Quantitatively, the 2% modulation of the lattice generates the OMA of 0.01 µB, which is  \nof the same order as that deduced from the first -principles calculation s.  10 \n     Next, we discuss the discrepancy between XMCD and first -principles calculation. The \ndiscrepancy might be understood as the underestimating of m orb in the calculation because of the \nlack of considering Hund ’s second rule in the electron correlation49. By cons idering orbital \npolarization, ∆morb estimated from the first- principles calculation becomes similar to that deduced \nfrom XMCD. Other reason s for the discrepancy might originate in the spin -flipped contribution \nthrough ΔE↓↑. On the other hand, EXMCD measurements also deduce d the magnetic dipole term in \nNi, the order of which is smaller than  ∆morb. However, this term proposed by van der Laan50 which \nis deduced from Δ𝐸𝐸↑↓ in eq. (1) is smaller than OMA. Therefore, the modulation of the magnetic \nanisotropy introduced by the macroscopic strain in the Ni layers is connected mainly with the OMA \nas a microscopic origin.  Further, EXMCD and first -principles calculation explain qualitatively  that \nthe ms values are less sensitive  to the strain and orbital hybridization, resulting in the changes of \nmorb.   \nIn conclusion, by using the novel EXMCD technique , we clarified that the reversible PMA \nchanges  in the Ni /Cu film on BaTiO 3 are induced by the modulation of orbital magnetic moments \nin Ni. The strained Ni layer that  induces the PMA  without E is released upon the application of an \nE-field and is modulated to produce in-plane magnetization.  Moreover, the m agnetization curves in \nthe Ni L3-edge EXMCD measurements are modulated between out -of-plane and in- plane \nmagnetization. We revealed that the changes of magnetic anisotropy by E, which were explained by \nthe phenomenological magneto -elastic description, may be  understood microscopically by OMA . \nThese results introduce  the concept of orbital -striction or orbital -elastic effects at the hetero -\ninterfaces  beyond established magnetostriction effects .  \n \nMethods  \nSample preparation . The s amples were grown by using ultra -high vacuum molecular beam epitaxy \non [100] -oriented 0.5- mm- thick BaTiO 3 single crystal substrates.  Therefore, the bias voltage of 400 \nV applied between  top and bottom electrodes means the electric field of 8 kV/cm.  The stacked \nstructures are shown in Fig. 1 a. Before the deposition of face- centered -cubic  (fcc) Ni/Cu stacked \nmultilayers, a 1 -nm-thick Fe buffer layer was deposited onto  the substrate at 300 °C . The multilayers 11 \n of [Cu (9 nm)/Ni (2 nm)] 5 were grown at room temperature and covered by 1 -nm-thick Au to prevent \noxidization. The details of the surface and interface conditions and the fabrication procedures are \nreported in Ref. 32. The magnetic properties were characterized by magneto -optical K err effect \n(MOKE) measurements and magnetometry. The ferroelectric domain structures were observed by \ndifferential interference microscop y before the EXMCD measurements.  \nXAS and XMCD measurements.  The XAS and XMCD measurements for the Ni and Cu L -edges \nwere performed at the KEK -PF BL -7A beamline, Japan, at room temperature. M agnetic fields of \n±1.2 T were applied along the incident polarized soft X -rays to saturate the magnetization \nsufficiently along the normal direction of the surface  of the samp le. The TEY mode was adopted by \ndetecting the drain currents from the samples for the case of the measurements recorded without \napplying a magnetic field. The electrodes were mounted at the surface of the  sample and at the rear \nof the substrate  to perform the EXMCD measurements . The EXMCD measurements were  \nperformed using the PFY mode to probe the signals beneath more than 10 nm below the surfaces of \nthe samples using a bipolar electric power source (Keithley 2410) to  apply E to BaTiO 3. The \nfluorescence signals were detected by a silicon drift detector (Princeton Gamma -Tech. Instrument \nInc. SD10129) , mounted at 90° to the incident beam. The XAS and XMCD measurements were \nperformed in the normal -incidence setup, in which the normal of the sample ’s surface is parallel to \nthe incident beam and the magnetic field, detecting  the signals of perpendicular components to the \nfilms. We changed the magnetic field directions to obtain the right - and left -hand- side polarized X -\nrays while fixing the pol arization direction of the incident X -ray. To avoid saturation effects in the \nPFY mode, the intensities in  the XAS measurements were carefully examined by comparing them \nwith those obtained in the TEY mode. Analysis method of m orb and m s using sum rules is described \nin Supplemental material and ref. 51.   \nFirst -principles study.  The DFT calculation code of the Vienna ab initio simulation package \n(V ASP) , including the spin–orbit interaction with the spin -polarized generalized gradient \napproximation, was employed52-54. The plane -wave cutoff energy was set to 500 eV and a 25 × 25 \n× 17 k-point mesh was used for sampling the Brillouin zone. The coordinate system of the Ni crystal \nlattice used in the calculation is as follows. The z -axis is the c -axis direction (perpendicular 12 \n magnetization direction), and the x -axis is the direction of the nearest neighbor atom in the in -plane \ndirection. With the changing of  the in -plane lattice constant,  the c-axis length wa s optimized in the \nfirst-principles calculations.   \n \nAcknowledgments  \nThis work was partly supported by JSPS KAKENHI (Grant Nos. 15H03562, 16H06332, \n15H01998, 16K14381, and 17H03377),  Spintronics Research Network of Japan, JST CREST Grant \nNumber JPMJCR18J1,  Japan,  and the Asahi Glass Foundation. The s ynchrotron radiation \nexperiments were performed under the approval of the Photon Factory Program Advisory \nCommittee, KEK (No. 201 7G060). The authors acknowledge Dr. Seiji Mitani for fruitful discussion.  \n \nAuthor  Contributions  \nJ.O. and T.T. planned the study. T.T . prepared the samples and performed the magnetization \nmeasurements. J.O. set up the E XMCD measurement apparatus at Photon Factory and collected and \nanalyz ed the data. Y. M .  performed the first -principles calculation. All authors discussed the results \nand wrote the manuscript.  \n \nAdditional Information \nSupplemental information accompanies the paper on the npj Quantum Materials website \n(http://*** ). \n \nCompeting interests  \nThe authors declare  no Competing Financial or Non- Financial Interests . \n \nData availability  \nThe data that support the findings of this study are available from the corresponding author upon \nreasonable request.  \n \nReferences  \n1. Heron,  J. T., Schlom,  D. G., &  Ramesh,  R. Electric field control of magnetism using BiFeO 3-based heterostructures, Appl. \nPhys. Rev. 1, 021303 (2014).  \n2. Cherifi,  R.O. et al.  Electric -field control of magnetic orde r above room temperature, Nature  Mater.  13, 345-351  (2014) . \n3. Matsukura , F., Tokura , Y.  &  Ohno,  H. Control of magn etism by electric fields, Nature  Nanotech.  10, 209 -220 (2015) . \n4. Kimura, T.  et al.  Magnetic control of ferroelectric polarization, Nature  426, 55-58 (2003).  \n5. 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Microscopic origin of magnetocrystalline anisotropy in transition metal thin films, J. Phys.: Condens. Matter \n10, 3239 -3253  (1998).   \n51. Stöhr, J. Exploring the microscopic origin of magnetic anisotropies with X -ray magnetic circular dichroism (XMCD) \nspectroscopy, J. Mag. Mag. Mater. 200, 470 -497 (1999).  \n52. Kresse, G. & Hafner, J.  Ab initio  molecular dynamics for liquid metals, Phys. Rev. B  47, 558(R)- 561(R)  (1993).  \n53. Kresse, G. & Furthmuller, J. Efficiency of ab -initio total energy calculations for metals and semiconductors using a plane -\nwave basis set , Comput. Mater. Sci.  6, 15-50 (1996).  \n54. Kresse, G. & Furthmuller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set , \nPhys. Rev. B  54, 11169 -11185  (1996).  \n \n \n \n \n \n \n \n \n \n \n \nFigure Legend s \n \nFigure 1  | Sample growth and characterization by applying electric fields. (a) Schematic \ndiagram of the structure of the sample indicating with film thicknesses  indicated . The \nelectrodes for applying the electric field are mounted at  the top (Ni/Cu layer) and the bottom \n(BaTiO 3 substrate) of the films. ( b) M–H loops measured by vibrational sample magnetometer \n(VSM) in the easy - and hard- axes directions , without an applied  electric field. ( c) Top-view \ndifferential interference microscope images recorded before and during the application of the \nelectric field. The white area indicates  the a- domain that exhibits PMA in the Ni layers.  15 \n  \n \nFigure 2  | XAS and XMCD spectra obtained at Ni and Cu L -edges in total -electron -yield \nmode without an applied electric field.  The spectra for NiCu/BaTiO 3 were measured in the \nnormal -incident geometry. ( a) XAS results with different ly polarized X-rays σ+ (red) and σ- \n(blue). ( b) XMCD result obtained from  the difference of the  σ+ and σ- XAS spectra.  \n \n \nFigure 3  | XAS and XMCD spectra obtained at Ni L -edges in partial -fluorescence- yield \nmode with and without an applied electric field. The spectra for NiCu/BaTiO 3 were  \nmeasured in the normal -incident geometry  (a) and (b), in the grazing- incident  geometry ( c) \nand ( d). Spectra recorded  without  an applied field E ((a), (c)) and with E  of 8 kV/cm  ((b), (d)); \nXAS with different ly polarized X-rays (σ+and σ- are indicated by the red and blue  lines \nrespectively ); XMCD obtained as  the difference between the σ+ and σ- XAS spectra. The \nintegrals of the XAS and XMCD spectra are also shown in the same panel with the ir axis \nindicated  on the right. Note that the scales of vertical axis are fixed in all panels. Illustrations \nin the XMCD panel show the schematic view of apply ing electric field and the angle between \nsurface normal and incident beam direction. \n \n \nFigure 4  | Element -specific magnetization curves for  the Ni L3-edge of XMCD spectra \nwith  applied electric field. The photon energy was fixed at the Ni L 3-edge in the PFY mode. \nNormal incidence set up was adopted. ( a) Increasing the electric field from 0 kV/cm up to 8  \nkV/cm, and ( b) decr easing the electric field from 8  kV/cm to 0 kV/cm.  \n \n \nFigure 5  | First -principles calculations.  (a) MCA  energ ies and anisotropy of orbital moment s \nof fcc  Ni as a function of the in- plane lattice constant a || (tetragonal distortion  is defined as  \n(a||−a0)/(a0×100), where a0 = 3.524  Å. (b) The second- order perturbative contribution of the \nspin–orbit interaction to the MCA  energy of fcc  Ni as a function of  a||. The minority band \ndispersions  of fcc  Ni along the high-symmetry line without (left) and with (right) 2% tetragonal \ndistortion  for the (c) d(x2−y2) and ( d) d(xy) orbital s. The color maps  of the band dispersion \nindicate  the magnitude s of projection of the d (x2−y2) and d(xy) orbital s.  16 \n  \n \n \nTable I, The spin and orbital magnetic moments with and without E . The values are in the units of \nµB. Experimental error bars are estimated about 20 % for the applications of XMCD sum rules.  \n \n 0 kV /cm 8 kV/cm  \nms (NI) 0.56 0.56 \nmorb (NI)  0.06 0.04 \nms (GI) 0.54 0.55 \nmorb (GI)  0.04 0.05 \n \n \n 0 kV/cm 5 kV/cm\n100 µmb\nc\na\nFig. 1a\nb\nA\nFig. 2a b\nc d\nFig. 30 kV/cm 8 kV/cm\n0 kV/cm 8kV/cma\nb\nFig. 4a‖[Å]Distortion [ %]∆EMCA [J/m3]∆morb[µB/atom]∆E↓↓\n∆E↑↑\n∆E↓↑\n∆E↑↓\na‖[Å]Distortion [ %]Energy [eV/atom]∆EMCA\n∆morba b\nc\nd\nd(xy) 0% d(xy) 2%d(x2-y2) 0% d(x2-y2) 2%\nFig. 5"    },    {        "title": "1904.11468v1.Magnetic_field_induced_quantized_anomalous_Hall_effect_in_intrinsic_magnetic_topological_insulator_MnBi__2_Te__4_.pdf",        "content": "Page 1 of 11 \n Magnetic -field -induced quant ized anomalous Hall effect in \nintrinsic magnetic topological insulator MnBi 2Te4 \n \nYujun Deng1,3†, Yijun Yu1,3†, Meng Zhu Shi2,4†, Jing Wang1,3*, Xian Hui Chen2,4* and Yuanbo \nZhang1,3* \n \n1State Key Laboratory of Surface Physics and Department of Physics, Fudan University, \nShanghai 200438, China  \n2Hefei National Laboratory for Physical Science at Microscale and Department of Physics, \nUniversity of Science and Techn ology of China,  and Key Laboratory of Strongly -coupled Quantum \nMatter Physics, Chinese Academy of Sciences, Hefei, Anhui 230026, China  \n3Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, Shanghai \n200433, China  \n4Key Laboratory of Strongly Coupled Quantum Matter Physics, University of Science and \nTechnology of China, Hefei, Anhui 230026, China  \n \n† These authors contributed equally to this work.  \n* Correspondence should be addressed to Y .Z. ( zhyb@fudan.edu.cn ), X.H.C. \n(chenxh@ustc.edu.cn ) and J.W. ( wjingphys@fudan.edu.cn ). \n Page 2 of 11 \n In a magnetic topological insulator, nontrivial band topology conspires with magnet ic order \nto produce exotic states of matter that are best exemplified by quantum anomalous Hall \n(QAH) insulators  (1–5) and axion insulators  (2, 6–8). Up till now, such magnetic topological \ninsulators are obtained by doping topological insulators with magnetic atoms  (9). The \nrandom magnetic dopants, however, inevitably introduce disorders that hinder further \nexploration of topological quantum effects in the material. Her e, we resolve this dilemma by \nprobing quantum transport in MnBi 2Te4 thin flake —a topological insulator with intrinsic \nmagnetic order. In this layered van der Waals crystal, the ferromagnetic layers couple anti -\nparallel to each other, so MnBi 2Te4 is an antiferromagnet  (10–13). A magnetic field, however, \naligns all the layers and induces an interlayer ferromagnetic order; we show that a quantized \nanomalous Hall response emerges in atomically thin MnBi 2Te4 under a moderate magnetic \nfield. MnBi 2Te4 therefore becomes the first intrinsic magnetic topological insulator exhibiting \nquantized anomalous Hall effect. The resul t establishes MnBi 2Te4 as an ideal arena for \nfurther exploring various topological phenomena.  \n \n The discovery of topological quantum materials demonstrates the importance of band \ntopology that is increasingly recognized in condensed matter physics  (9, 14, 15). A distinct feature \ncommon to all topological materials is the presence of topologically protected quantum states that \nare robust against local perturbations. For example, in a topological insulator (TI) such as Bi 2Te3, \nthe bulk band topology guarantees the existence of gapless two -dimensional (2D) surface states \nwith gapless Dirac dispersion  (16, 17). Introducing magnetism into the initially time -reversal \ninvariant TIs brings about profound changes in their electronic structure. Specifically, the long -\nrange magnetic order bre aks the time -reversal symmetry, and causes an exchange gap in the \ngapless Dirac dispersion of the surface states  (2, 18). The gap opening is accompanied by the \nemergence of a chiral edge mode that was predicted to give rise to a QAH effect when the Fermi \nlevel is situated in the exchange gap  (2). The dissipationless QAH edge chan nel, combined with \nthe spin-momentum lockin g inherent in topological materials , may lead to new device concepts \nfor topological electronic  applications  (9).  Page 3 of 11 \n The experimental observation of the QAH effect in chromium -doped (Bi,Sb) 2Te3 (5, 19–21) \nrepresents  a triumph  in topological quantum material research —the ratio of the multiple  elements \nin the non -stoichiometric material has to be precisely controlled to ac complish  such a feat. The \nfine-tuning required to reconcile conflicting demands , i.e. large magnetization and low initial \ncarrier doping,  poses a great challenge  for material  growth . To make matters worse, the randomly \ndistributed magnetic dopants used to achieve ferromagnetism also act as impurities  that limit the \nquality of the magnetic TIs.  As a result, the exact quantization of the anomalous Hall effect appears  \nonly at low temperatures ; the current record is limited to  2 K (in penta -layer sandwich structure \nof topological insulators ) (22), far below the ferromagnetic Curie temperatu re (a few tens of K elvin) \nand exchange gap ( ~ 30 meV ) (23) in the material.  Further explo ration of rich topological \nphenomena and their potential applications calls for intrinsic magnetic TIs—stoich iometric  TIs \nwith an innate magnetic order such that topological effects  can be studied in pristine crystals.  \nHere we  probe the quantum transport in atomically thin flakes of intrinsic magnetic TI  \nMnBi 2Te4. MnBi 2Te4 is layered ternary tetradymite compound that consists of Te-Bi-Te-Mn-Te-\nBi-Te septuple layers (SL s), so the material  can be  viewed as  layered  TI Bi2Te3 with each of its \nTe-Bi-Te-Bi-Te quintuple layer intercalated by an additional Mn -Te bilayer  (Fig. 1A) . The \nresultant MnBi 2Te4 crystal remains  a TI, but now becomes  intrinsic ally magneti c (10–13). The \nmagnetism originates from t he Mn2+ ions in the crystal , which  have a high spin of 𝑆=5/2 and \na large magnetic moment of ~ 5𝜇𝐵 (𝜇𝐵 is the Bohr magneton ) (10, 12). Below a Neel \ntemperature of 𝑇𝑁=25 K, the spins couple ferromagnetically in each SL  with an out -of-plane \neasy axis , but adjacent  SLs couple anti-parallel to each other ; bulk MnBi 2Te4 is therefore an \nantiferromagne t (AFM ) (10, 12). In this  work, we focus on thin flakes of MnBi 2Te4 to minimize \nthe parallel bulk conduction. Because the SLs are separated by van der Waals gaps  in bulk \nMnBi 2Te4, the extensive arsenal of fabrication techniques  developed for 2D materials enable us to \nobtain few -layer samples  that preserve the high quality of the crystals. As we show below, a \nmoderate perpendicular magnetic field  is able to sequentially flip the ferromagnetic SL s, and \neventually drive  the material into a ferromagnetic ally ordered  state; a well quantized anomalous \nHall effect  is realized at a temperature as high as 4.5 K. \nWe start with high quality MnBi 2Te4 single crystals that are grown by self-flux method  (12, \n24), and obtain atomically thin  MnBi 2Te4 using an  Al2O3-assisted exfoli ation technique  described \nin (25). Specifically, w e first thermally evaporate Al2O3 thin film  onto the fresh ly-prepared  surface  Page 4 of 11 \n of bulk crystal . We then lift the Al 2O3 thin film, along with MnBi 2Te4 thin flakes cleaved  from the \nbulk, using  a thermal release tape. The Al 2O3/MnBi 2Te4 stack is subsequently released onto a piece \nof transparent polydimethylsiloxane  (PDMS) , and inspected under optical microscope  in \ntransmission mode . Fig. 1B displays the optical image of few-layer MnBi 2Te4 flakes on the Al 2O3 \nfilm attached to PDMS. The t ransmittance  of various number  of SLs follows the Beer-Lambert  \nlaw (Fig. 1C) , which enables us to precisely determine the layer number . The thin flakes are finally \nstamped  onto Si wafer covered with 285 nm -thick SiO 2, followed by deposition of metal contacts \nfor transport measurements . The degenerately doped Si serves as a back gate , so a voltage bias 𝑉g \napplied between Si and the sample can electrostatically dope electron or hole carriers into \nMnBi 2Te4 flakes, depending on the polarity of 𝑉g. The entire device fabrication process is \nperformed  in an argon -filled glove box where  O2 and H 2O content is kept below 0.5 parts  per \nmillion to mitigate  sample degradation.  We focus on flakes with odd number  of SLs, wh ich are \nintrinsically ferromagnetic due to unbalanced layer polarization . \nFigure 1D displays temperature -dependent resistance , 𝑅𝑥𝑥, of few -layer MnBi 2Te4. A \nprominent feature in the data sets is the resistance  peak (or kink for the seven -layer sample) at low \ntemperature. Similar  peak was also observed in t emperature -dependent resistance  of the bulk \ncrystal , where it is attributed to increased spin-scattering at the AFM transition  (26). The location \nof the peak, therefore, gives a measur e of the Neel temperature  𝑇𝑁 in the few -layer specimens . \nWe find that the seven -layer sample has a 𝑇𝑁 of 24 K, comparable to  the value in the bulk. The \n𝑇𝑁 is however slightly suppressed in thinner samples ( 23 K for the five-layer  sample and 18 K \nfor the three -layer  sample ), which can be ascribed to increased thermal fluctuations as the samples \napproach the 2D limit.  The resistances  rise again at lowest temperatures, probably because of \nlocalization of the carriers in the presence of defects in the  samples.  \nHall measurement s provide further information on the quality of the thin flakes  (Fig. 1E) . First \nof all, the slope of the Hall resistance 𝑅𝑦𝑥 near zero magnetic field, referred to as Hall coefficient \n𝑅H, yields the initial carrier concentration in the sample , 𝑛=1/𝑒𝑅H (𝑒 is the  charge of an \nelectron ). We find that all samples are initially electron doped ; the five-layer  and three -layer  \nsamples ha ve initial dopings of 7×1011 cm−2 and 2×1012 cm−2, respectively, whereas the \ninitial doping level of the seven -layer sample is more than one order of magnitude higher.  Such  \nelectron doping is probably induced by antisite defects and/or Mn vacancies in the crystal  (27), \nand the doping level gives a  qualitative measure  of the defect density.  In the following experiments, Page 5 of 11 \n we mainly focus on the five-layer  and three -layer  samples that have low carrier concentration s and  \nhigh quality . Second , the Hall mobility  𝜇H of the samples can be determined from  𝜇H=𝜎/𝑛𝑒 , \nwhere  𝜎=𝑅𝑥𝑥𝑊/𝐿; 𝑊 and 𝐿 are the width and length of the sample, respectively . All our \nsamples exhibit mobil ity values ranging from 100 - 1000 cm2 V−1 s−1, on the same order as the \nmagnetic TI thin films grown by molecular beam epitaxy  (5, 19, 28). The high quality of the \nsamples is also reflected in  the sharp  ferromagnetic transition, which implies that the s amples (with \na typical size of 10×10 μm2) flip as a single domain  at the coercive field . Finally, w e note that \nthe zero-magnetic -field anomalous Hall response  in the five-layer  and three -layer  samples became  \na significant fraction of the quantum resistance ℎ/𝑒2 (ℎ is the plank constant) , but did not \nquantize ; a larger  exchange gap is needed to overcome disorder and temperature fluctuations.  \nAn external magnetic field applied perpendicular to the sample align s the magnetization in  all \nSLs to induce a large exchange gap  (Figs. 2C and 2D) . We observe the exact quantization of \nanomalous Hall  effect in a pristine five-layer  MnBi 2Te4 flake under a moderate magnetic field . \nFigs. 2A and 2B display  𝑅𝑥𝑥 and 𝑅𝑦𝑥 of the sample as a function of magnetic field  recorded at \nvarious temperature s. At the lowest te mperature of 𝑇=1.6 K, 𝑅𝑦𝑥(𝜇0𝐻) reaches a quantized \nplateau at ℎ/𝑒2 above 𝜇0𝐻 ~ 6 T, while 𝑅𝑥𝑥 approaches zero at the same time; both features \nare hallmarks of quant ized Hall effect  that signif ies the emergence of a chiral 1D dissipationless \nstate at the edge s of the sample  (Fig. 2D).  The magnetic -field-induced quantized Hall effect in \nfive-layer  MnBi 2Te4 is robust at elevated temperatures.  We observe that |𝑅𝑦𝑥| stays above  \n0.97ℎ /𝑒2 (and 𝑅𝑥𝑥 remains below 0.017ℎ/𝑒2) under a magnetic field of 𝜇0𝐻=12 T at \ntemperatures up to 𝑇=4.5 K (Fig. 2E) . This quantization temperature is significantly higher than \nthe highest value found in modulation -doped magnetic TI thin films (22, 29). At higher \ntemperature s, 𝑅𝑦𝑥 deviates from exact quantization , and meanwhile  𝑅𝑥𝑥 exhib its a thermally \nactivated behavior , 𝑅𝑥𝑥∝exp (−𝛥/2𝑘B𝑇) (𝑘B is the Boltzmann  constant and 𝛥 is the \nactivation  gap; Fig. 2E). Line fit to the Arrhenius plot of ln𝑅𝑥𝑥 as a function of 1/𝑇 yields 𝛥=\n21 K (Fig. 2E, black line) . Even though t his energy scale is again much larger than that in magnetic \nTI thin films (21), we note that 𝛥, which measures the energy gap between the mobility edges,  is \nstill much low er than  the exchange gap (up to ~ 88 meV ) in MnB i2Te4 (11, 12, 24, 26). There is  \ntherefore much room for further increasing 𝛥 in pure crystals of MnBi 2Te4. Page 6 of 11 \n It may be asked that why the quantized Hall effect observed here is not caused by Landau \nlevel quantization as in the case of the ordinary quantum Hall effect —after all, a fully -filled first \nLandau level also produces exact quantization of |𝑅𝑦𝑥|=ℎ/𝑒2. Close examination on the origin \nof the quantization , however, reveals fundamental difference between the two variants of quantized \nHall effects that can be probed experimentally. In both cases, the Hall conductivity 𝜎𝑥𝑦 is \ndetermined by a topological invariant integer 𝐶 known as Chern number, 𝜎𝑥𝑦=𝐶𝑒2/ℎ. For the \nordinary quantum Hall effect, 𝐶 corresponds to the occupancy of the Landau levels, so the sign \nof 𝐶 is uniquely determined by the sign of the charge carriers (electrons or holes). In contrast, the \nsign of 𝐶 depends only on the sign of the  exchange coupling and magnetization direction, but not \non the  charge  carriers, in a QAH insulator (2, 9). Probing the quantized Hall response while \nswitching the sign of charge carriers —but leaving the exchange coupling and magnetization \nintact —is a good way to unambiguously distinguish the two cases. To this end, we sweep the \nmagnetic field, and record 𝑅𝑦𝑥(𝜇0𝐻) and 𝑅𝑥����(𝜇0𝐻) of a five-layer  sample under various back \ngate biases 𝑉g (Fig. 3A and 3B). As 𝑉g is tuned from zero to −90 V, the initially electron -doped \nsample becomes hole doped, as judged from the sign of the Hall coefficient 𝑅H (Fig. 3C).  (The \ncharge neutral point (CNP) is identified at 𝑉gCNP≈−50 V, where 𝑅H switches sign, and zero-\nfield 𝑅𝑥𝑥 exhibits a peak as shown in Fig. 3D.)  We observe t hat the quantized 𝑅𝑦𝑥 does not \nchange sign on either side of the CNP, and therefore rule out Landau level quantization as a \nprobable cause of the quantized Hall effect in MnBi 2Te4. We note that the 𝑅𝑦𝑥 quantization \nbecomes  better as 𝑉g is tuned close to 𝑉gCNP (Fig. 3 A). This observation further  indicates  that \n𝑅𝑦𝑥 quantization  is linked to the  chiral edge state located inside of the exchange gap . \nThe gate -dependent 𝑅𝑥𝑥(𝜇0𝐻) and 𝑅𝑦𝑥(𝜇0𝐻) contains important information on the \nevolution of the magnetic states in the five-layer  MnBi 2Te4 flake. There are two main points to \nnotice. First, t he peak in the magnetoresistance 𝑅𝑥𝑥(𝜇0𝐻) at the coercive field transforms into a \ndip as the charge c arriers in the sample switch from electrons to holes.  The peak on the electron \nside can be understood as a result of enhanced electron -magnon scattering when the external field \nis antiparallel to magnetization (30). This theory, however, does not explain the dip observed on \nthe hole side. Here we note that the dip feature resembles the butterfly -shaped magneto -resistance \nobserved in another magnetic TI Mn -Bi2Te3 at the dimensional cross -over regime (31), where \nmagnetic Skyrmions may form. The same mechanism may be at work in MnBi 2Te4 thin flakes. Page 7 of 11 \n Second, both 𝑅𝑥𝑥(𝜇0𝐻) and 𝑅𝑦𝑥(𝜇0𝐻) undergo a sequence of transitions as the magnetic field \nincreases, before the sample settles into the quantized anomalous Hall state. The transitions are \nmanifestations of the complex intermediate magnetic states that are precursors of the QAH \ninsulator at high magnetic fields.  \nImportant insight into thes e magnetic states is gained when  we juxtapose  the magneto -\ntransport in  the five-layer  sample  with that in a three -layer  sample  (Fig. 4). The 𝑅𝑦𝑥(𝜇0𝐻) in the \nthree -layer  sample exhibits two plateaus that indicate two magnetic states, in contrast with t hree \nmagnetic states in the five-layer  sample. It now becomes clear that the magnetic states are in fact \ninitially antiferromagnetically coupled SLs with individual SLs flipped, one SL at a time, by an \nincreasing external magnetic field. These magnetic sta tes are schematically illustrated in Fig. 4. \nThe transitions between the states can be described by the Stoner -Wohlfarth model with bipartite \nAFM  (24). The model further points out that eac h layer -flip takes place in two steps via a spin flop \ntransition , wher e the layer  magnetization is free to rotate in direction s approximately perpendicular \nto the easy axis  (24). Close examination of the transitions in 𝑅𝑦𝑥(𝜇0𝐻) indeed reveal signs of \nsuch two-step proces ses. \n In summary, we observe a quantized anomalous Hall effect in atomically thin flakes of  \nintrinsic magnetic TI —MnBi 2Te4. The quantized anomalous Hall effect  is induced by an external \nmagnetic field that drives the initially antiferromagnetically coupled layers to align \nferromagnetically; dissipationless 1D chiral edge conduction emerges in the QAH insulator. The \nquantization is robust at 𝑇=4.5 K, and the tran sport gap is  𝛥=21 K. Both values are expected \nto improve in pure MnBi 2Te4 crystals. Because MnBi 2Te4 is also a 2D material, the techniques \ndeveloped for other 2D materials can be readily applied to MnBi 2Te4. We anticipate that van der \nWaals heterostructur es integrating MnBi 2Te4 with other magnetic/superconducting 2D materials \nwill provide fertile ground for exploring exotic topological quantum phenomena.  \n \nReferences  \n \n1.  F. D. M. Haldane, Model for a Quantum Hall Effect without Landau Levels: Condensed -Matter \nRealization of the “Parity Anomaly.” Phys. Rev. Lett.  61, 2015 –2018 (1988).  \n2.  X.-L. Qi, T. L. Hughes, S. -C. Zhang, Topological field theory of time -reversal invaria nt \ninsulators. Phys. Rev. B . 78, 195424 (2008).  \n3.  C.-X. Liu, X. -L. Qi, X. Dai, Z. Fang, S. -C. Zhang, Quantum Anomalous Hall Effect in Hg 1-\nyMn yTe Quantum Wells. Phys. Rev. Lett.  101, 146802 (2008).  Page 8 of 11 \n 4.  R. Yu et al. , Quantized Anomalous Hall Effect in Magnetic Topological Insulators. Science . \n329, 61–64 (2010).  \n5.  C.-Z. Chang et al. , Experimental Observation of the Quantum Anomalous Hall Effect in a \nMagnetic Topological Insulator. 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Chen et al. , Experimental Realization of a Three -Dimensional Topological Insulator, \nBi2Te3. Science . 325, 178 –181 (2009).  \n18.  Y . L. Chen et al. , Massive Dirac Fermion on the Surface of a Magnetically Doped \nTopological Insulator. Science . 329, 659 –662 (2010).  \n19.  J. G. Checkelsky et al. , Trajectory of the anomalous Hall effect towards the quantized \nstate in a ferromagnetic topological insulator. Nat. Phys.  10, 731 –736 (2014).  \n20.  X. Kou et al. , Scale -Invariant Quantum Anomalous Hall Effect in Magnetic Topological \nInsulators beyond the Two -Dimensional Limit. Phys. Rev. Lett.  113, 137201 (2014).  \n21.  A. J. Bestwick et al. , Precise Quantization of t he Anomalous Hall Effect near Zero \nMagnetic Field. Phys. Rev. Lett.  114, 187201 (2015).  \n22.  M. Mogi et al. , Magnetic modulation doping in topological insulators toward higher -\ntemperature quantum anomalous Hall effect. Appl. Phys. Lett.  107, 182401 (2015).  \n23.  I. Lee et al. , Imaging Dirac -mass disorder from magnetic dopant atoms in the \nferromagnetic topological insulator Cr x(Bi 0.1Sb0.9)2-xTe3. Proc. Natl. Acad. Sci.  112, 1316 –\n1321 (2015).  \n24.  See Supplementary Materials.  \n25.  Y . Deng et al. , Gate -tunable room -temperature ferromagnetism in two -dimensional \nFe3GeTe 2. Nature . 563, 94–99 (2018).  \n26.  S. H. Lee et al. , Spin scattering and noncollinear spin structure -induced intrinsic \nanomalous Hall effect in antiferromagnetic topological insulator MnBi 2Te4. arXiv , 27 (2018).  Page 9 of 11 \n 27.  J. Cui et al. , Transport properties of thin flakes of the antiferromagnetic topological \ninsulator MnBi 2Te4. Phys. Rev. B . 99, 155125 (2019).  \n28.  C.-Z. Chang et al. , High -precision realization of robust quantum anomalous Hall state i n \na hard ferromagnetic topological insulator. Nat. Mater.  14, 473 –477 (2015).  \n29.  Y . Ou et al. , Enhancing the Quantum Anomalous Hall Effect by Magnetic Codoping in a \nTopological Insulator. Adv. Mater.  30, 1703062 (2018).  \n30.  J. G. Checkelsky, M. Lee, E. Morosan, R. J. Cava, N. P. Ong, Anomalous Hall effect and \nmagnetoresistance in the layered ferromagnet Fe 1/4TaS 2: The inelastic regime. Phys. Rev. B . \n77, 014433 (2008).  \n31.  C. Liu et al. , Dimensional Crossover -Induced Topological Hall Effect in a Magnetic  \nTopological Insulator. Phys. Rev. Lett.  119, 176809 (2017).  \n \n \nAcknowledgements  \nWe thank Xiaofeng Jin and Yizheng  Wu for helpful discussions. Part of the sample fabrication \nwas conducted at Nano -fabrication Laboratory at Fudan University. Y .D., Y .Y ., J.W. and Y .Z. \nacknowledge support from National Key Research Program of China (grant nos. \n2016YFA0300703, 2018YFA03056 00), NSF of China (grant nos. U1732274, 11527805, \n11425415 and 11421404), Shanghai Municipal Science and Technology Commission (grant no. \n18JC1410300), and Strategic Priority Research Program of Chinese Academy of Sciences (grant \nno. XDB30000000).  Y .Y . als o acknowledges support from  China Postdoctoral Science Foundation  \n(grant no. BX20180076  and 2018M641907 ). J.W. also acknowledges support from the NSF of \nChina (grant no. 11774065)  and NSF of Shanghai  (grant no. 17ZR1442500) . M.Z.S. and X.H.C. \nacknowledge support from the National Natural Science Foundation of China (grant nos. \n11888101, 11534010), the National Key R&D Program of China (grant nos. 2017YFA0303001 \nand 2016YFA0300201), Strategic Priority Research Program of the Chinese Academy of Sciences \n(grant no. XDB25000000), and the Key Research Program of Frontier Sciences, CAS (grant no. \nQYZDY -SSW -SLH021).   \n \nAuthor Contributions  \nY .Z., X. H. C. and J. W. supervised  the project. M.Z.S. and X.H.C. grew the MnBi 2Te4 bulk  \ncrystal s. Y .D. and Y .Y . fabricated devices and performed measurements. Y .D., Y .Y ., Y .Z. , J.W. and \nX. H. C. analyzed the data. J.W. carried out theoretical calculations. Y .D., Y .Y ., J.W. and Y .Z. wrote \nthe manuscript  with input from all authors.  Page 10 of 11 \n Figure captions  \nFig. 1. Fabrication and characterization of few-layer  MnBi 2Te4 devices. (A) Crystal structure \nof MnBi 2Te4. The septuple atomic layers are stacked together by van der Waals interactions. Red \nand blue arrows denote magnetic moment of each Mn2+ ion. Under  zero exter nal magnetic field, \nthe neighboring  SLs are antiferromagnetically coupled  with an out-of-plane magnetocrystalline \nanisotropy . (B) Optical image of few-layer  flakes of MnBi 2Te4 cleaved by  thermally -evaporated \nAl2O3 thin film. The MnBi 2Te4/Al2O3 stack is supported on a piece of PDMS. Number of SLs are \nlabelled on selected flakes . Scale bar: 20 μm. (C) Transmittance  as a function of number of SLs. \nThe data follow the Beer-Lambert  law (solid line), which enables us to precisely determine the \nnumber of layers. (D) Temperature -dependen t sample resistance of  few-layer MnBi 2Te4. The \nantiferromagnetic transition manifests as resistance peaks in the three -layer and five -layer samples \n(and a kink in the seven -layer sample).  (E) Hall resistance  of few-layer MnBi 2Te4 under a low \nexternal magnetic fiel d. The flakes with odd number of layers are ferromagnetic at low \ntemperatures. Data were obtaine d at 𝑇=1.6 K. \n \nFig. 2. Magnetic -field -induced quantiz ed anomalous Hall effect in  a five-layer  MnBi 2Te4 \nsample. (A and B) Magnetic -field-dependent 𝑅𝑥𝑥 and 𝑅𝑦𝑥 acquired at various temperatures. \nFrom bottom: 𝑇=1.6,3.0,5.0,7.0,10,13,18,24,30,40 K. Quantized Hall plateaus are \nobserved above 𝜇0𝐻 ~ 6 T at low temperatures, and the quantization is accompanied by \nvanishing 𝑅𝑥𝑥. (C) Schematic illustration of the gapped Dirac -like dispersion of the surface state  \nin ferromagnetic MnBi 2Te4. A chiral edge state (gray line) emerges inside of the exchange gap. \nArrows mark the direction of the spins that are locked with crystal momentum. (D) Real-space \npicture  of the chiral edge  state when an external magnetic field 𝜇0𝐻 forces all SLs to align  \nferromagnetically. (E) 𝑅𝑦𝑥 (right axis) and Arrhenius plot  of ln𝑅𝑥𝑥 (left axis) as a function of \n1/𝑇. External magnetic field is fixed at 𝜇0𝐻=12 T. Black solid line is a line fit to the Arrhenius  \nplot, from which we obtain a transport gap of 𝛥=21 K. \n \nFig. 3. Gate -tuned Magneto -transport a five-layer  MnBi 2Te4 sample. (A and B) Magnetic -\nfield-dependent 𝑅𝑦𝑥 and 𝑅𝑥𝑥 measured under various gate biases  𝑉g. Here 𝑅𝑥𝑥 (𝑅𝑦𝑥) curves Page 11 of 11 \n are symmetrized (anti -symmetrized) to separate the two component s. (C) Hall coefficient 𝑅𝐻 \nplotted as a function of 𝑉g. 𝑅𝐻 is extracted from fitting 𝑅𝑦𝑥 near zero magnetic field. As 𝑉g is \ntuned from zero to −90 V, the initially electron doped sample becomes hole doped.  𝑉g=−50 V \nmarks the CNP where  𝑅𝐻 (and the charge carriers) changes sign . (D) Zero -field 𝑅𝑥𝑥 plotted as \na function of 𝑉g. 𝑅𝑥𝑥 exhibit a peak around the CNP. All d ata were obtained at 𝑇=1.6 K. \n \nFig. 4. Magnetic transitions in few-layer MnBi 2Te4 induced by  an external magnetic field. (A \nand B) Symmetrized 𝑅𝑦𝑥 and anti-symmetrized  𝑅𝑥𝑥 of a three -layer  sample ( A) and a five -layer \nsample (B) shown as a function of external magnetic field applied perpendicularly.  All data were \nobtained at 𝑇=1.6 K. The magnetic state at specific locations (marked by black dots) are \nschematically illustrated. Red marks the spin up SLs and blue marks the spin down SLs. For \nsimplicity, only one of the possible configurations are shown when there is degeneracy.  \n 0 50 100 150 200100101\n-2 -1 0 1 20 2 4 6 80.40.60.81\n5 SL3 SLRxx(kΩ)\nT(K)7 SL5 SL3 SLRyx(h/e2)\n0H(T)7 SL\n×1000.2h/e2Transmittance\nIndependent Variable\n1276\n3\n5\nTe Bi MnB A C\nD E\nDeng et al., Figure 1-1.0-0.50.00.51.0\n-10 -5 0 5 10024Ryx(h/e2)\n40 K\nBA\nRxx(kΩ)\n0H(T)1.6 K\n1.6 K40 K\n48E\nlnRxx(0H=12 T) (ln Ω)\n∆\u0001= 21 K\n0.0 0.2 0.4 0.601Ryx(0H=-12 T) ( h/e2)\nT-1(K-1)50 10 5 3 1.6T(K)DC\n0H\nDeng et al., Figure 2Deng et al., Figure 3-10 -5 0 5 10\n-100 -80 -60 -40 -20 002040\n-100 -80 -60 -40 -20 0-100100 V\n-10 V\n-20 V\n-30 V\n-40 V\n-50 V\n-60 V-70 V\n-80 VRyx\n0H(T)-90 V1h/e2\n-10 -5 0 5 1050 kΩBRxx\n0H(T)A\nVg=Rxx(0H=0 T) (kΩ )\nVg(V)RH/e(×10-11cm2)\nVg(V)D C050100-101\n-10 0 1002040-101A\nRxx(kΩ)\n5 SLRyx(h/e2)3 SLRxx(kΩ)\n0H(T)B\nRyx(h/e2)\nDeng et al., Figure 4\n"    },    {        "title": "1905.02587v1.Vapor_Pressure_in_a_Paramagnetic_Solid_.pdf",        "content": "¿Vapor  Pressure in a Paramagnetic Solid ? \n \nManuel Malaver  \nBijective Physics Institute, Bijective Physics Group, Idrija, Slovenia                                       \nMaritime University of the Caribbean, Catia la Mar, Venezuela .  \nEmail Address :  mmf.umc@gmail.com  \nAbstract:  In this paper, we obtain an analytical expression for the vapor pressure of a \nparamagnetic solid  for high temperatures . We have considered the behavior of magnetic \nmaterials in the presen ce of an external magnetic field using the thermodynamical analysis \nand the elements of statistical mechanics in microscopic systems. We found that the vapor \npressure  depend s on the magnetic susceptibility of material and the external field applied.  \nKeywords: Vapor pressure , Paramagnetic solid, Magnetic field, Statistical mechanics, \nMagnetic susceptibility .   \n1. Introduc tion \n    The magnetism is a type of associated physical phenomen a with the orbital and spin  \nmotions of electrons and the interaction of electrons with each other  [1]. The electric \ncurrents and the atomic magnetic moment of  elementary particles generate magnetic fields \nthat can act on other currents and magnetic materials . The most common effect occurs in \nferromagnetic solids  which is possible when atoms are rearrange in such a way that \nmagnetic atomic moments can interact to align parallel to each other  [2,3]. In quantum \nmechanics, the ferromagnetism can be described as a parallel alignment of magnetic  \nmoments what depends of the interaction between neighbouring moments  [4].  \n     Although ferromagnetism is the cause of many frequently observed magnetism effects , \nnot all materials respond equally to magnetic fields  because the interaction between \nmagnetic atomic moments is v ery weak  while in other materials this interaction is stronger  \n[1]. In the paramagnetic materials there are unpaired electrons  which freely align their \nmagnetic moment in  any direction  and when an external field is applied these magnetic \nmoments align in the same direction  of applied field making it more  intense  [2]. \n     Diamagnetism is an intrinsic property of all materials  and is the tendency of materials to \nbe repelled by an applied magnetic field  because of the presence  no unpaired electrons  so \nthat the atomic magnetic moments not produce any effect  [5]. The other forms of \nmagnetism  as  paramagnetism or ferromagnetism are much stronger  in a material  and  the \ndiamagnetic contribution is  very negligible  [1]. \n    In the statistical and microscopic description of any system, the partition function plays \ndeterminant role and is defined as the total sum of states of the system [ 6]:                                     \nkTE\nnn\neEg Z )(                                              (1) \n \nwhere n labels the total energy En , k is the Boltzman´s constant and g(E n) it is the number \nof degenerate states with the same energy En. The use of the equation (1) allows to obtain a \nstatistical and microscopic description of the energy  and the entropy  [6,7].  \n    The concepts of the statistical mechanics must be considered if we want to do a microscopic \ndescription of a   physical system. Recently Mäkelä [ 8,9] constructed a microscopic model of \n\"Stretched Horizon \" of a Schwarzschild and Reissner -Nordström black holes  and obtained an \nanalytical expression for the  partition function from the point of view of an observer on its \nstretched horizon.  Malaver [ 10,11,12] studied the behavior of the thermal capacity CV for \nSchwarzschild and  Reissner-Nordström  black holes when T>>T C  and T<<T C  where TC is the \ncharacteristic temperature and found that the value for CV  if T>>T C  is the same that  would be \nobtained in an ideal diatomic gas if are considered the rotational and translational  degrees of \nfreedom, respectively. Viaggiu [13] present a statistical analysis in gravitons and derived equations \nfor the partition function and  the mean energy.  Malaver [14] obtained an analytical expression for \nthe thermal capacity for gravitons and study the behavior of CV  in the limit of high and low \ntemperature , Mandl  [7] used the concepts of statistical thermodynamics to understand the behavior \nof paramagnetic solids  and  García -Colín Scherer [15] presents a thermodynamic analysis in \nmagnetic systems . Also Zemansky and Dittman [16] apply statistical mechanics in order to study \ndifferent types of magnetic materials .   \n     In this paper we have deduced an analytical expression for the vapor pressure in a paramagnetic \nsolid. We found that the vapor pressure will depend on the magnetic susceptibility of material \nand the external magnetic field applied.  This work is outlin ed in the following manner: the \nsection II the behavior of paramagnetic solids are studied; in section III, we present the \nstatistical thermodynamics of magnetic materials; in section IV  is obtained an expression \nfor the vapor pressure for a paramagnetic s olid; in Section V, presents the conclusions of \nthis study .  \n \n2. Behavior of Paramagnetic  Solids  \n \n      In a paramagnetic material, the atoms contain permanent magnetic moments  and the \ninduced magnetic field s are aligned in the direction of the applied magnetic field  [3]. \nParamagnetism is due to the presence of unpaired electrons in the materials and the atoms \nwith semifilled atomic orbitals are paramagnetic  [1]. Examples of paramagnetic materials \nare aluminium, oxygen and titanium.  The tendency of the magnetic moments to align with \nthe direction of the field is counteracted by the thermal movement that disorders the \nmagnetic dipoles . In the magnetic materials, the magnetization M is proportional to \nintensity of magnetic field H and the mathematical expression is [2] \n                                                  M=χH                                            (2)   \nwhere is χ  the magnetic susceptibility of material  and depends on the temperature T as \nfollow  \n                                                        \nTC                                                             (3)   \nThe eq. (3)   is known as Curie´s law and C is a positive constant characteristic of material.  \n \n    The paramagnetic susceptibility can be calculated for a type of system that contains \nseparate atoms where there are unpaired electrons in an external magnetic field  B. \nFollowing  Eisberg -Resnick  [2]  if n is the number of unpaired electrons per unit volume,  n- \nis the volumetric density of moments that are parallel to the field and n+ represents the \ndensity for the moments that are antiparallel ,  then n = n-  +  n+  . For a parallel alignment \nof the magnetic moment   μ  the magnetic potential energy is  -μB and for an antiparallel \nalignment   the energy is   μB.  From the Boltzman´s distribution for each state of energy     \n \n                           n- = neμB/kT               and         n+= ne-μB/kT                     (4)      \n \n where   k  is the Boltzman´s  constant .  The resultant magnetization is given by  \n \n                                  M = μ(n-    n+ ) = μn(eμB/kT  -  e-μB/kT  )                              (5) \n \nand the average magnetic moment can be written as  \n \n                                      \nkTB\nkTBkTB\nkTB\ne ee e\nnM\n  \n\n                                               (6) \n \nAs the limiting case , let us consider the situation at very high temperature  and weak field , \nthen when    μB<<kT and with the development of exponential terms we obtain  \n \n                   \n\nkTB\nkTB kTBkTB kTB\nnM2\n/ 1 / 1/ 1 / 1 \n                                 (7)     \n \nand the paramagnetic  susceptibility  is given for  \n                                         \n                                  \nkTHBn\nHn\nHM2                                                            (8)                        \nBut the magnetization also can be written as [2]   \n \n                                \n\n10BM                                                                         (9) \n \nwhere \n0 is the vacuum permeability . Then when \n  << 1 and with eq . (2) we have  \n       \n                                      \nH B0                                                                            (10) \n \nSubstituting (10) in (8) we can obtain  \n \n                                     \nkTn2\n0                                                                          (11) \n \nThe eq.(11) is an approximation of  Curie’s law with \nk n C /2\n0  and the paramagnetic \nsusceptibility varies inversely with the temperature  . According to (7) the alignment of the atomic \nmagnetic moments depends on the presence of field and if B=0  the thermal motion orients the \nmoments randomly and the net magnetization is null  [2,7].        \n \n3. Statistical Thermodynamics of Paramagnetic Materials  \n \n    Following Mandl [ 7] we have that in separate atoms with unpaired electrons y that can \nexist in two states with energy  \nB  , the partition function for solid of N dipoles    \n                                      \n                                               \nN\nkTB\nkTB\ne e Z\n\n\n                                                     (12) \n \nDefining the average statistical energy with the following expression [4] \n                                                            \n                                                       \nZEln                                                              (13) \n \nWith the eq. (12), for the magnetic energy of N dipoles we obtain  \n                                          \n\n\n\n\n\n\n\n\n\nkTB\nkTBkTB\nkTB\ne ee e\nBNE \n                                                       (14) \n \n   For the determination of the entropy in the paramagnetic solid can be use the expression \nfor the Helmholtz free energy  A= E -TS  . In statistical mechanics, the work function o \nHelmholtz free e nergy A is defined as  \n \n                                                    \nZ kT A ln                                                             (15)                                \n \n      Substituting eq. (12) in (15) we have for the work function  \n \n \n                                           \n\n\n\n kTB\nkTB\ne e NkT A \nln                                              (16) \n \nWith (14), (16) and the expression for A, the entropy of the system can be written as   \n \n \n \n                           \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nkTB\nkTB\nkTB\nkTBkTB\nkTB\ne e\ne ee e\nNkS \n \nln                                    (17) \n \n    In the limit  of  high temperature and weak field  μB<<kT   and with  the development \nof exponential terms , we have for (17)  \n \n \n                      \n\n\n\n\n\n\n\n\n\n\n\n\nkTB\nkTB\nkTB\nkTBkTB\nkTB\nNkS\n\n1 1ln\n1 11 1                        (18)       \n    For these approximations  we obtain for the entropy of system  \n \n                                                   \n2lnNkS                                                                  (19) \n \n  The eq.(19) implies that for N dipoles all states are equally possible and the orientation of \natomic magnetic moments is disordered.    \n \n4. Determination of the Vapor Pressure in a Paramagnetic Solid  \n \n     Based on Einstein's model, c onsider now that a paramagnetic material behaves as a \ncrystalline solid  and assume that the atoms of solid behave as oscillators that vibrate at \nsame frequency ν. If the vibrations are harmonic then the energy of harmonic oscillator is \ngiven by  \n \n                                        \n h\n\n21                                                                  (20) \n \n \n      According Nash [ 6] the eq.( 20) then permits us to express in terms of   ν and Planck´s \nconstant h , the energy εν associated with any one of the permissible quantum states \ncorresponding to integral values of the quantum  number \n .  If as usual we identify our \nreference zero of energy with the ground -state energy ε0=0, we obtain ε1=hν , ε2=2hν \nand so on . We can then easily formulate the partition function for a harmonic oscillator in \nterms of the empirically determinable value for ν which characterizes that species of \noscillator  \n \n                                  \n....... 13 2\n  kThv\nkTh\nkTh\ne e e z                                           (21) \n                                                  \nThis expression has the form  \n                               \n                                 \n....... 13 2  x x xe e e z                                                 (22) \n      \n  and this series is equal to the function  [3,4]  \n                                                     \nxexf11)(                                                          (23)          \n               \n             \n       Then with the harmonic oscillator we can express the partition function in a closed \nanalytic form                                        \n                                                                                                                \n \n                                                     \nkTh\nez\n11                                                           (24) \n                \n     Given this simple expression for the partition function , with eq. (13) we can calculate the energy \nE of   N harmonic oscillators of solid  \n \n \n                       \nkTh\nVedTdNkTdTzdNkTE\n\n11lnln2 2                                  (25) \n \n \n      and is obtained that   \n                                         \n11\n\n\n\nkTh\nekhNkE                                                          (26) \n                                       \n \n   From the eq.(26) we have the following expression for the energy of 3N harmonic \noscillator s of crystalline solid      \n                                         \n113\n\n\n\nkTh\nekhNk E                                                        (27) \n \n  With (15), (24), (27) and from  A= E -TS can be determined the entropy of the vibrations \nof crystalline solid [ 6,7]  \n \n \n                          \n\n\n\n\n\n\n\n\n\n\nkTh\nkTh\ne ekThNk S \n11ln\n113\n                                            (28) \n         \n  The eq.(28) is rewritten in the following manner   \n \n \n                            \n\n\n\n\n\n\n\n\nkTh\nkThe\nekThNk S\n1ln\n113\n                                            (29) \n \n   We now define a parameter \nkh/  where  \n  has dimensions of temperature and  \n(29) can be express in a form more compact  \n \n                            \n\n\n\n\n\n\n\n\nT\nTe\neTNk S\n1ln\n113\n                                                  (30) \n \n    If N is Avogadro´s number then Nk =R where  R is the ideal gas constant and we obtain \nfrom eq.(30)  for the molar entropy of the solid       \n \n                    \n\n\n\n\n\n\n\n\nT\nTe\neTR S\n1ln\n113                          (31) \n \n  Let us consider the situation at very high temperature when T >> \n  . As the function \nxe  \nis equivalent to the sum of the infinite   series  \n.....!3 !213 2 x xx  can be developed \nthe exponential terms in (8) and we find [7] \n \n                                  \n 3 ln3 ln3   T RS\n                                                           (32) \n \n    Total entropy for the solid paramagnetic for high temperatures will be given by the \nvibrational contribution  (32) and the paramagnetic contribution  (19) that is  \n \n                               \n 2ln3 ln3 ln3   T RS\n                                                    (33) \n \n \n    For the determination of the solid’s vapor pressure, it must be assumed that the vapor \nbehaves as an ideal monoatomic gas and its partition function will be given by  [6] \n \n                           \n                                   \nN\nVhmkT\nNZ\n\n\n\n\n\n23\n22\n!1\n                                                  (34)    \n \n                                       \n   Where m is the mass of the gas particles and V is the volume of the container containing it . \nSubstituting  (34) in equation (15), we express the Helmholtz free energy of an ideal  \nmonoatomic gas as follows  \n \n                                     \n\n\n\n\n\n\n\n\nNV\nhmkTNkT A23\n22ln1                                   (35) \n \n      For the thermodynamic relation  \n \n                                                        \nSdT PdV dA                                                (36) \n \n     When the temperature is constant, (36) is reduced to  \n \n                                                                      \n                                                        \nTdVdAP\n                                                          (37) \n \n       then we have now for the pressure P \n \n \n                         \nThmkT\nNVdVdNkTP\n\n\n\n 23\n221ln ln                                     (38) \n \n      and we obtain  for an ideal monoatomic gas       \n                                      \n                           \nVNkTP1                                                                                        (39)                    \n     or for one mole of gas  PV=RT   and with the equation (13) we can now express the \nenergy  of an ideal gas  as  \nVN\nVVhmkT\nN dTdkTdTZdkTE\n\n\n\n\n\n\n\n\n23\n22 2 2\n!1lnln \n                               (40) \n \n  and the equation  (40)  can be written as follows  \n \n           \n                                     \nNkT TdTdkTEN\n23ln23\n2                                                   (41) \n \n    For a paramagnetic gas the Helmholtz free energy A is given by [16]  \n \n                                        \nMH TSEA0                                                             (42) \n \n  Substituting (35) and (41) in eq. (42) we find   \n \nTMHNV\nhmkTNkT NkT\nTMH AES023\n2\n02ln123\n\n\n\n\n\n\n\n\n\n\n\n         (43)   \n           \n    and by rearranging of (43) we can write  \n     \n  \nTMH\nhmkTNVNkS0\n22ln23ln23ln25 \n\n                                           (44) \n \n \n    Assum ing approximation of ideal gas , the eq. (44) allows obtain the dependence of the   \ntemperature  with the  pressure of saturated  vapor , then replacing (39) in (44) \n                                               (45) \n \nand for P we obtain   \n \nNkS\nNkTMHkhmT Pvapor\n\n\n\n\n\n0 2523\n22ln ln25\n25ln \n                                   (46) \n \n    The  equation  (46) will now be applied the case of  saturated vapor in equilibrium with its  \ncondensed ph ase which in this study  is solid  phase  . According Mandl  [7], if L is the molar \nenthalpy of sublimation at temperature T , Svapor and Ssolid  are the molar entropies of vapor  \nand solid respectively we have  \n  \n                                              \nTLS Ssolid vapor                                                      (47) \n \nand the molar entropy of vapor phase Svapor is rewritten as  \n \n       \nTMHkhmP T R Svapor0 2523\n22ln ln ln25\n25  \n\n\n\n\n\n\n\n\n                 (48) \nWhere   R=Nk   and N is  Avogadro's Number .  \nSubstituting (33) and (48)  in equation (47) we obtain  \n \n TLT RTMHkhmP T R \n\n\n\n\n\n\n\n\n 2ln3 ln3 ln32ln ln ln25\n250 2523\n2 \n        (49) \n \n   and  lnP  is given by  \nTMH\nhmkTPkTNk Svapor0\n22ln23ln23ln25 \n\nRTL\nRTMHkhmT P \n\n\n\n\n\n 2ln2ln ln3 ln21\n21ln0 2523\n2  \n                                                                                                                                       (50) \nThen the expression for the vapor pressure of paramagnetic solid can be written as  \n \n                        \nRTLRTMH\ne\nTehe m kP\n1\n22\n21\n3323\n250\n                                                  (51) \nRearranging  (51) and  for the eq. (2),  we have for P    \n \n                          \nRTLRTH\neTehe mkP\n2\n21\n3323\n252\n0\n                                                  (52)    \n5. Conclusions  \n  \n     In this paper has been deduced an expression for the vapor pressure of a paramagnetic \nsolid under conditions of high temperatures and weak field, which is a function of the \nmagnetic susceptibility of solid and the external magnetic field applied. It i s considered that \nvapor behaves as an ideal monoatomic gas and the entropy of the vibrations of crystalline \nsolid was calculated in the limit of high temperature using the Einstein's model.  \n     The statistical mechanics can enrich the courses of thermodynamics, which contributes \nto a better compression of the thermal phenomena. The thermodynamic equations deduced \nfor the paramagnetic materials from the postulates of the statistical mechanics are tractable \nmathematically and offer a wide explanation of many physical systems of interest.\n References   \n \n[1] S.A.E.Ahmed, N.A.M.Babker, M.T.Fadel,  A study on classes of magnetism,         \nInternational Journal of Innovative Science, Engineering & Technology, 6 (4), 25-33, \n2019 . \n \n[2]  R. Eisberg, R.Resnick, Física Cu ántica, Editorial Limusa, S éptima Edición, ISBN 968 -  \n18-0419 -8, 1991.  \n [3]  S. Chikazumi, Physics of ferromagnetism , 2nd ed ition, Oxford: Oxford University \nPress , ISBN  9780199564811 , 2009.  \n \n[4] R.P.Feynman,  R. Leighton , M. Sands , The Feynman Lectures on Physics, Vol. 2 , \nAddison -Wesley , 1963.  \n \n[5]  R. Jackson, John Tyndall and the Early History of Diamagnetism , Annals of Science , \n72 (4), 435–489, 2014.  \n \n[6] L.Nash, Elements of Classical and Statistical Thermodynamics, Addison Wesley \nPublishing Company, 1970.  \n \n[7]  F. Mandl, Física Estadística, Editorial Limusa, Primera Edición, ISBN 968 -18-0146 -6, \n1979.   \n \n[8]  J. Mäkelä , Partition function of the Schwarzschild black hole,  Entropy , 13, 1324 -1357 , \n2011.   \n \n[9]   J. Mäkelä , Partition function of the Reiss ner-Nordstr öm black hole, Int. J.Mod.Phys,   \nD23, 1450001, 2014 .  \n \n[10]  M. Malaver ,  ¿Law of Einstein´s thermal capacity for Schwarzschild´s black hole?,       \nIJRRAS , 11(1), 31-36, 2012.   \n \n[11]   M. Malaver , Comparative Analysis of Thermal Capacity in  Schwarzschild and   \nReissner -Nordström black holes , World Applied Programming , 3(2), 61-67, 2013.   \n  \n[12]   M. Malaver , Black Holes, Wormholes and Dark Energy Stars in General Relativity.   \nLambert Academic Publishing, Berlin , ISBN 978 -3-659-34784 -9, 2013.  \n \n[13]   S. Viaggiu, Statistical mechanics of gravitons in a box and the black hole entropy, \nPhysica A, 473, 412 -422, 2017.  \n \n[14]   M.Malaver, An analysis of the thermal capacity in gravitons, World Scientific News, \n72, 34 -40, 2017.  \n \n[15]  L. García -Colín Scherer, Introducci ón a la termodi námica clásica, Editorial Trillas,  \nSegunda Edición, 1972 .  \n \n   [16]  M.W. Zemansky, R.H.Dittman, Calor y Termodinámica, McGraw -Hill Interamericana, \nSexta Edición, ISBN 968 -451-631-2, 1985.  \n \n \n \n     \n \n \n \n \n  "    },    {        "title": "1905.08722v1.Commensurate_to_incommensurate_magnetic_phase_transition_in_Honeycomb_lattice_pyrovanadate_Mn2V2O7.pdf",        "content": "Commensurate to incommensurate magnetic phase transition in Honeycomb-lattice\npyrovanadate Mn 2V2O7\nJ. Sannigrahi,1,\u0003D.T. Adroja,1, 2,yR. Perry,1M. J. Gutmann,1V. Petricek,3and D. Khalyavin1\n1ISIS Facility, Rutherford Appleton Laboratory, STFC, Chilton, Didcot, OX11 0QX, United Kingdom\n2Highly Correlated Matter Research Group, Physics Department,\nUniversity of Johannesburg, P.O. Box 524, Auckland Park 2006, South Africa\n3Institute of Physics, Czechoslovak Academy of Sciences, Na Slovance 2, 18040 Praha 8, Czechoslovakia\n(Dated: May 22, 2019)\nWe have synthesized single crystalline sample of Mn 2V2O7using \roating zone technique and\ninvestigated the ground state using magnetic susceptibility, heat capacity and neutron di\u000braction.\nOur magnetic susceptibility and heat capacity reveal two successive magnetic transitions at TN1=\n19 K and TN2= 11.8 K indicating two distinct magnetically ordered phases. The single crystal\nneutron di\u000braction study shows that in the temperature ( T) range 11.8 K\u0014T\u001419 K the magnetic\nstructure is commensurate with propagation vector k1= (0;0:5;0), while upon lowering temperature\nbelow TN2= 11.8 K an incommensurate magnetic order emerges with k2= (0:38;0:48;0:5) and the\nmagnetic structure can be represented by cycloidal modulation of the Mn spin in ac\u0000plane. We\nare reporting this commensurate to incommensurate transition for the \frst time. We discuss the\nrole of the magnetic exchange interactions and spin-orbital coupling on the stability of the observed\nmagnetic phase transitions.\nPACS numbers: 75.25.+z, 75.50.-y, 75.50.Ee, 75.30.Kz, 74.62.-c, 75.47.-m, 71.70.Gm\nI. INTRODUCTION\nBulk crystals are inherently three dimensional; how-\never, they may consist of magnetic ions whose spins in-\nteract only along one or two certain crystallographic di-\nrections. Such type of compounds are known as low di-\nmensional magnets which have aroused considerable at-\ntention in solid state chemistry and physics due to their\nnon-typical behaviour which mainly deal with electrical\nand magnetic properties. Two dimensional honeycomb-\nlattice systems are one of the interesting low dimensional\nmagnetic systems which have been intensively investi-\ngated because of novel ground states induced by frustra-\ntion and quantum \ructuations.1{5Magnetic order can be\nobtained on a regular two dimensional lattice assuming\nantiferromagnetic interactions between adjacent spins;6\nor on a bilayer lattice with frustrating interlayer interac-\ntions.7In recent times, these honeycomb lattice systems\nhave been extensively investigated both theoretically and\nexperimentally as potential candidate for Kitaev quan-\ntum spin liquid state, for example Na 2IrO3,\u000b-RuCl 3.8,9\nMoreover, superconductivity has also been observed in\npnictide SrPtAs with honeycomb lattice.10\nThe title compound Mn 2V2O7is a member of the fam-\nily of transition metal-vanadium oxides M 2V2O7(M =\nCu, Ni, Co, Mn) which have attracted much interest due\nto their rich structural features and magnetic proper-\nties.11{19Mn2V2O7is composed of magnetic Mn2+(3d5,\nS= 5=2) and non-magnetic V5+(3d0,S= 0) ions and\nwas reported to possess quasi two dimensional distorted\nhoneycomb lattice.20,21It exhibits two di\u000berent struc-\ntural phases; namely \f\u0000phase (above\u0019310 K) with\nmonoclinic ( C2=m) symmetry and \u000b\u0000phase (below\u0019\n284 K) with triclinic ( P\u00161) symmetry. As temperature\ndecreases, the high temperature monoclinic symmetryof Mn 2V2O7reduces to triclinic one and this structural\ntransition is completely reversible with reasonable ther-\nmal hysteresis, in line with the martensitic-like \frst or-\nder nature of this structural transition involving lattice\ndistortion but without atomic exchange.18,21Depending\non the synthesis procedure these transition temperatures\nvary because of the change in O 2pressure and cooling\nrate. The honeycomb networks of Mn-atoms in the \f\nphase are parallel to (001) plane and those in the \u000b\nphase are parallel to the (110) plane. Further the sample\nundergoes paramagnetic to antiferromagnetic transition\nbelow around 19 K. It is noted that such properties of\nMn2V2O7were mainly based on macroscopic character-\nization methods on polycrystalline as well as \rux grown\nsingle crystalline samples. But a detailed microscopic\nmodel of the magnetism has not yet been developed.\nSo far there are few reports on the single crystal growth\nof Mn 2V2O7where only \rux growth technique has been\nfollowed.22,23The crystals obtained are small and more-\nover the grown crystals may incorporate traces of the\nmolten \rux or the crucible materials, which is highly un-\ndesirable. Keeping this view in mind, we decided to grow\nMn2V2O7crystals using the traveling solvent \roating\nzone (TSFZ) method associated with an optical image\nfurnace. Mn 2V2O7melts congruently upon heating at\n(1080\u00063)\u000eC.24In order to understand the structural\nand magnetic properties of the ground state of Mn 2V2O7\nat the macroscopic as well as microscopic level, we carried\nout x-ray di\u000braction (XRD), magnetization and heat ca-\npacity measurement along with the single crystal neutron\ndi\u000braction study.arXiv:1905.08722v1  [cond-mat.str-el]  21 May 20192\n/s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48\n/s83/s101/s101/s100\n/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s50 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s32/s69/s120/s112/s116/s46/s32/s100/s97/s116/s97\n/s32/s83/s105/s109/s117/s108/s97/s116/s101/s100/s83/s101/s101/s100\n/s90/s111/s110/s101\n/s71/s114/s111/s119/s105/s110/s103\n/s67/s114/s121/s115/s116/s97/s108/s91/s97/s93\n/s91/s98/s93\nFIG. 1. (a) Main panel shows the XRD pattern of crushed sin-\ngle crystal measured at T= 100 K. Black scattered points and\nred solid line represent respectively the experimental data and\nthe simulated curve obtained from Rietveld re\fnement. Blue\nticks indicate the Bragg positions for triclinic P\u00161 structure\nand the cyan line shows the di\u000berence between experimental\nand calculated pattern. Right inset shows the stable solvent\nzone formation during growing, while left inset depicts the\npicture of the as grown crystal. (b) depicts the perspective\nview of the crystal structure of Mn 2V2O7at 100 K where\nblue, yellow and red spheres represent Mn, V and O atoms\nrespectively.\nII. EXPERIMENTAL DETAILS\nThe single crystalline sample of Mn 2V2O7has been\ngrown in a four-mirror optical \roating zone furnace from\nCrystal System Corporation, Japan. The furnace con-\nsists of four halogen lamps. Every lamp takes the focus\nposition of an ellipsoidal re\rector whereas the second fo-\ncus of the re\rectors coincide at a point inside growth\nchamber on the vertical axis of the furnace. A molten\nzone between the polycrystalline feed rod and the seed\nrod is formed at this common focal point where the en-\nergy \rux of the lamps converge after re\recting from the\nre\rectors. The feed rod is suspended from the upper\nshaft with a nickel wire while the seed rod is clamped\nto the lower shaft using an alumina holder. Polycrys-\ntalline sample of Mn 2V2O7was prepared by standard\nsolid state reaction route in air. Highly pure MnO 2andV2O5were mixed thoroughly in a stoichiometric ratio\nin an agate mortar.The mixture was pressed into pellets\nand sintered at 600\u000eC for one week with several interme-\ndiate grindings. The phase analysis of the \fnal product\nwas performed using powder x-ray di\u000braction (Mini\rex).\nThe single phase powder of Mn 2V2O7was \flled in rubber\ntube and cold pressed under 700 bar isostatic pressure to\nobtain uniform rod. The rods formed in this process are 5\nmm in diameter and 70 - 90 mm in length. Then the rods\nwere sintered in the platinum boat at the temperature\n900\u000eC overnight to obtain very dense and homogeneous\nrod. This process ensures the homogeneous melting of\nthe feed rod during growth process. A part about 20\nmm long was cut from the rod and used as the seed rod.\nThree di\u000berent growth experiments were performed\nwith di\u000berent oxygen pressure and growth speed em-\nployed; (i) 60% O 2+ 40% Ar and growth speed 5 mm/hr\n(ii) 100 % O 2\row (P= 7 bar) and growth speed 5 mm/hr\nand (iii) 100 % O 2\row (P= 7 bar) and growth speed\n1 mm/hr. The growth process is initiated by fusing the\nbottom end of the feed rod inside the optical furnace.\nIn our case 300 watt lamps have been installed. The\nupper and lower shafts of the furnace were rotated in op-\nposite directions with rotation speed 30 rpm to improve\nthe temperature and compositional homogeneity of the\n\roat zone. In the \frst two attempts, upper shaft was\ntranslated at a speed of 1 mm/hr and in the third at-\ntempt it is translated at a speed of 0.5 mm/hr in order\nto control the steady state of \roating zone. In the \frst\nattempt molten zone was not fully stable and formation\nof bubbles were observed near the upper edge of the zone\npossibly due to the reduction of Mn ions. Then we de-\ncided to use 100 % O 2pressure to prevent the reduction\nof Mn2+and with this environment we got signi\fcantly\nstable zone throughout the growth process as shown in\nthe right inset of \fgure 1(a).\nThe grown crystals were characterized using powder x-\nray di\u000braction of crushed crystal (Bruker D8 Advance),\nLaue x-ray di\u000braction and single crystal neutron di\u000brac-\ntion in SXD di\u000bractometer, ISIS neutron and muon facil-\nity. The magnetization of the crystals was measured by\napplying an external \feld along the three di\u000berent crys-\ntallographic directions using a vibrating sample magne-\ntometer (SQUID VSM, Quantum Design) in the temper-\nature range T= 2 to 320 K and applied magnetic \feld\n(H) ranging from -70 to +70 kOe. Heat capacity has\nbeen measured in Quantum Design PPMS. The x-ray\ndi\u000braction (XRD) pattern of crushed crystal measured\nat 100 K is shown in the main panel of \fgure 1(a). Ri-\netveld re\fnement on powder XRD data was performed\nusing Fullprof software package and the data is well \ft-\nted using triclinic structure with P\u00161 space group. The\ncrystal structure (shown in \fgure 1(b)) is consistent with\nthe previously reported literature.3\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48 /s49/s49/s48/s48/s46/s48/s54/s48/s46/s48/s55/s48/s46/s48/s56/s48/s46/s48/s57/s48/s46/s49/s48/s48/s46/s49/s49/s48/s46/s49/s50/s48/s46/s49/s51/s48/s46/s49/s52\n/s50/s55/s48 /s50/s56/s48 /s50/s57/s48 /s51/s48/s48 /s51/s49/s48 /s51/s50/s48/s48/s46/s48/s50/s54/s48/s46/s48/s50/s55/s48/s46/s48/s50/s56/s48/s46/s48/s50/s57\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50/s49/s46/s53/s49/s46/s56/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48/s72 /s73/s73 /s32/s91/s48/s48/s49/s93/s32\n/s72 /s32/s91/s49/s49/s48/s93/s32/s32/s40/s101/s109/s117/s47/s109/s111/s108/s41\n/s84/s32/s40/s75/s41/s72 /s73/s73 /s32/s91/s49/s49/s48/s93/s32/s72 /s32/s61/s32/s49/s48/s48/s32/s79/s101\n/s91/s97/s93/s72 /s73/s73 /s32/s91/s49/s49/s48/s93/s32\n/s51/s48/s54/s32/s75/s40/s101/s109/s117/s47/s109/s111/s108 /s41\n/s84/s32/s40/s75/s41/s50/s56/s52/s32/s75\n/s91/s98/s93\n/s72 /s73/s73 /s32/s91/s49/s49/s48/s93/s32/s72 /s73/s73 /s32/s91/s48/s48/s49/s93/s32/s72 /s32/s91/s49/s49/s48/s93/s32/s77/s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s72/s32/s40/s107/s79/s101/s41/s84 /s32/s61/s32/s50/s32/s75/s72 /s73/s73 /s32/s91/s49/s49/s48/s93/s32/s49/s47 /s40/s101/s109/s117/s47/s109/s111/s108/s41\n/s84/s32/s40/s75/s41/s32/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116/s97/s108/s32/s100/s97/s116/s97\n/s32/s67/s117/s114/s114/s105/s101/s45/s87/s101/s105/s115/s115/s32/s102/s105/s116/s116/s105/s110/s103\n/s72 /s32/s61/s32/s49/s48/s48/s32/s79/s101\nFIG. 2. (a) Main panel shows the temperature dependent\nmagnetization data in the zero \feld-cooled heating protocol\nwhere magnetic \feld applied along three crystallographic di-\nrections: Hk(110), H?(110) and Hk(001). Upper inset\ndepicts the thermal hysteresis observed between 284 K and\n306 K on \feld cooling and \feld-cooled heating measurements\nwhereas lower inset represents the Curie-Weiss \ftting in the\nparamagnetic region. (b) Mversus Hcurves are displayed\nmeasured at T= 2 K along three di\u000berent crystallographic\ndirections.\nIII. MAGNETIZATION AND HEAT CAPACITY\nThe main panel of \fgure 2(a) shows the temperature\ndependence of magnetic susceptibility ( \u001f) from 2 K to\n110 K, which was measured in zero \feld cooled heating\nprotocol under an applied magnetic \feld of H= 100 Oe\nalong the parallel and perpendicular to (110) plane and\nparallel to (001) plane of \u000b\u0000Mn2V2O7. A clear signa-\nture of antiferromagnetic (AFM) transition is observed\natTN1\u001919 K which matches quite well with the pre-\nviously reported data.20,21,25Additionally we found an-\nother anomaly at around TN2= 11.8 K in all three crys-\ntallographic directions. Although the TN1andTN2are\npresent at the same temperatures, but di\u000berent histories\nare clearly observed below 19 K dependent on direction\n/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48 /s50/s49/s48 /s50/s52/s48/s48/s51/s48/s54/s48/s57/s48/s49/s50/s48/s49/s53/s48\n/s49/s51 /s49/s52 /s49/s53 /s49/s54 /s49/s55 /s49/s56 /s49/s57 /s50/s48 /s50/s49 /s50/s50/s49/s50/s49/s53/s49/s56/s50/s49/s50/s52/s50/s55/s51/s48/s51/s51\n/s49/s48 /s49/s49 /s49/s50 /s49/s51 /s49/s52/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52/s49/s53\n/s57 /s49/s48 /s49/s49 /s49/s50 /s49/s51 /s49/s52/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s49/s48/s50/s48/s51/s48/s72/s32 /s61/s32/s48/s32/s79/s101\n/s72/s32 /s91/s49/s49/s48/s93/s91/s97/s93\n/s84\n/s78/s49\n/s72/s32 /s91/s49/s49/s48/s93/s91/s98/s93/s67\n/s112/s32/s40/s74/s46/s109/s111/s108/s45/s49\n/s46/s75/s45/s49\n/s41/s32/s72 /s32/s61/s32/s48/s32/s79/s101\n/s32/s49/s48/s32/s107/s79/s101\n/s32/s50/s48/s32/s107/s79/s101\n/s32/s50/s53/s32/s107/s79/s101\n/s32/s51/s48/s32/s107/s79/s101\n/s32/s53/s48/s32/s107/s79/s101\n/s32/s55/s48/s32/s107/s79/s101\n/s32/s57/s48/s32/s107/s79/s101\n/s84\n/s78/s50\n/s72/s32 /s91/s49/s49/s48/s93/s91/s99/s93\n/s84/s40/s75/s41/s32/s72 /s32/s61/s32/s48/s32/s79/s101\n/s32/s49/s48/s32/s107/s79/s101\n/s32/s50/s48/s32/s107/s79/s101\n/s32/s50/s53/s32/s107/s79/s101\n/s32/s51/s48/s32/s107/s79/s101\n/s32/s53/s48/s32/s107/s79/s101\n/s32/s55/s48/s32/s107/s79/s101\n/s32/s57/s48/s32/s107/s79/s101\n/s100/s67\n/s112/s47/s100/s84/s84\n/s78/s50/s84\n/s78/s49FIG. 3. (a) Main panel shows the temperature dependent\nheat capacity measured at zero applied magnetic \feld while\ninset shows the enlarged view of the low- Tregime indicat-\ning the two transitions at TN1andTN2. The magnetic \feld\ndependence of TN1andTN2are depicted in (b) and (c) re-\nspectively. dCp=dTvsTis plotted in the inset of (c) to show\nclearly the change in TN2withH.4\nof the applied magnetic \feld. This indicates the presence\nof magnetic anisotropy in this system. Upper inset of \fg-\nure 2(a) shows the thermal hysteresis within 284 K and\n306 K between \feld-cooling and \feld-cooled-heating pro-\ntocols of measurement. The existence of hysteresis sug-\ngests that this reversible structural transition is a \frst or-\nder martensitic-like transition where thermo-elastic solid-\nsolid phase transition occurs involving lattice distortion\nbut without atomic exchange. The 1 =\u001f(T) curve follows\nthe Curie-Weiss law above 50 K in the \u000b\u0000phase of the\ncompound as shown in the lower inset of \fgure 2(a). We\nobtained Curie-Weiss T,\u0012C= -34 K indicating the pre-\ndominant AFM interaction and an e\u000bective paramagnetic\nmoment,\u0016eff= 5.86\u0016B/Mn2+, which is nearer to the\nspin-only value 5.92 \u0016Bof Mn2+. Figure 2(b) shows the\nisothermal magnetization measured at T= 2 K under\napplied magnetic \feld up to 70 kOe where Hk(110),\nH?(110) andHk(001). Linear nature of the Mver-\nsusHcurves in all three directions indicates again the\npresence of predominant AFM interaction in the studied\ncompound. From the magnetic susceptibility and magne-\ntization isotherm measurements it is clear that the c-axis\nis an easy magnetization axis with weak anisotropy.\nThe heat capacity ( Cp) as a function of temperature\nmeasured at zero applied magnetic \feld is depicted in\n\fgure 3(a). At high temperature, Cpis completely dom-\ninated by phonon excitations. At low temperature, two\n\u0015\u0000type anomalies are observed at TN\u001919 K andTN2\u0019\n11.8 K indicating two magnetic phase transitions consist-\ning with the magnetization measurement. To gain more\ninformation about magnetic ordering, Cp(T) has been\nmeasured in di\u000berent applied magnetic \feld from 0 Oe\nto 90 kOe which is shown in \fgure 3(b) and (c). With\nincreasingH,TN1shift towards lower temperature sug-\ngesting antiferromagnetic nature of this transition. In\naddition to this, the intensities of the peaks are decreas-\ning with increasing magnetic \feld because of the redis-\ntribution of magnetic entropy. On the other hand, the\npeak atTN2shifts slightly towards higher temperature\nfromH= 0 Oe to 30 kOe up to a maximum TN2\u001912.8\nK and decreases more clearly with further increase of H.\nThis peak also becomes broader with increasing Hand\n\fnally completely suppressed above 50 kOe.\nIV. SINGLE CRYSTAL NEUTRON\nDIFFRACTION\nSingle-crystal neutron di\u000braction study was carried out\non the SXD di\u000bractometer at the ISIS Neutron and Muon\nfacility which utilizes the time-of-\right Laue technique.26\nA Mn 2V2O7single crystal of approximate dimensions 2\n\u00021\u00020.5 mm3(m\u001950 mg) was mounted at the end\nof an Aluminium pin using the adhesive Al tape. For\nreliable determination of the propagation vector single\ncrystal neutron di\u000braction data were collected at di\u000ber-\nent temperatures ranging from 4 K to 30 K with special\nattention at temperatures T= 4, 12, 13 and 30 K using\n[a] T = 30 K\n[c] T = 4 K[b] T = 13 KQz(Å-1)2.0\n1.8 \n1.6 \n1.4 \n1.2 \n1.0 \n2.0\n1.8 \n1.6 \n1.4 \n1.2 \n1.0 \n2.0 \n1.8 \n1.6 \n1.4 \n1.2 \n1.0 \n-3.0   -2.8    -2.6    -2.4   -2.2   -2.0 \nQx(Å-1)-3.0   -2.8    -2.6    -2.4   -2.2   -2.0 -3.0   -2.8    -2.6    -2.4   -2.2   -2.0 FIG. 4. 2D intensity maps of the reciprocal ( h;0:5; l) planes\nat 30 K, 13 K and 4 K are depicted in (a), (b) and (c) respec-\ntively. The peak for commensurate propagation vector k1=\n(0,0.5,0) at 13 K is shown by blue circle in \fgure(b) while the\nmagnetic peak for incommensurate propagation vector k2=\n(0.38,0.48,0.5) at 4 K is indicated by red circle in \fgure(c).\na closed cycle He refrigerator. Five exposures in di\u000ber-\nent crystal orientations with respect to incident neutron\nbeam were collected for 1 hr each. The data were pro-\ncessed using locally available SXD2001 software.27Struc-\ntural re\fnements of the neutron data were performed\nwith the JANA 200628software using standard scattering\nlength densities and the Mn2+form factor. The neutron\nintensity in a part of the reciprocal ( h;1=2;l) scatter-\ning plane in \u000b-Mn 2V2O7is shown as two dimensional\nintensity maps in \fgure 4 for T= 30 K, 13 K and 4\nK. To con\frm the nuclear structure a re\fnement of the\ndata set collected at 30 K (well above the peak in the\nmagnetic susceptibility and heat capacity) has been per-\nformed which con\frms the triclinic structure with the\nP\u00161 space group. The crystal structures parameters are\nconsistent with the already reported in the literature. In\nthe intermediate phase (in between TN2andTN1) we\nobserved a set of magnetic superstructure peak corre-\nsponding to the commensurate propagation vector k1=5\n/s51 /s54 /s57 /s49/s50 /s49/s53 /s49/s56 /s50/s49 /s50/s52\n/s73/s110/s99/s111/s109/s109/s101/s110/s115/s117/s114/s97/s116/s101/s73/s110/s116/s101/s103/s114/s97/s116/s101/s100/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s84/s32/s40/s75/s41/s32/s40/s48/s45/s50/s51/s41/s43/s107\n/s49\n/s32/s40/s48/s45/s49/s51/s41/s45/s107\n/s50\n/s67/s111/s109/s109/s101/s110/s115/s117/s114/s97/s116/s101/s84 /s32/s61/s32/s49/s50/s32/s75/s84/s42\nFIG. 5. (a) Main panel shows the temperature dependence\nof integrated intensity of two magnetic superstructure peaks\nindicating the thermal evolution of commensurate and incom-\nmensurate magnetic phases. Inset shows the two dimensional\nintensity map of the reciprocal ( h;0:5; l) at 12 K where in-\ndicating the presence of both propagation vectors k1(blue\narrow) and k2(red arrow).\n(0,0.5,0) which are marked by blue circle in \fgure 4(b) at\nT= 13 K. But in the ground state, below TN2= 11.8 K\nthis commensurate phase disappears and a di\u000berent set\nof magnetic superstructure peak arises as shown in the\n\fgure 4(c) marked with red circle. Taking the positions\nof these new sets of magnetic peaks, it is evident that the\nmagnetic order at lowest temperature is incommensurate\nwith propagation vector k2= (0.38, 0.48, 0.5). No change\nin the incommensurate value of k2with temperature has\nbeen detected within the measurement accuracy.\nThe two magnetic superstructure peaks observed in\nneutron di\u000braction pattern of Mn 2V2O7, (0\u001623)+and\n(0\u001613)\u0000satellite peaks for commensurate and incommen-\nsurate phase respectively, have been recorded at several\ntemperatures. Here, ( hkl)\u0006= (hkl)\u0006k1=k2denotes a\nsatellite peak of the nuclear ( hkl) peak. The integrated\nintensities of these peaks are plotted as a function of tem-\nperature as shown in the \fgure 5. Magnetic intensity\nfully vanishes above TN1= 19 K in close agreement with\nother bulk measurements. The intensity of the commen-\nsurate peak disappears below TN2= 11.8 K. Moreover,\nas displayed in the inset of \fgure 5, the pattern recorded\natT= 12 K, shows the presence of magnetic superstruc-\nture peaks of both commensurate as well as incommen-\nsurate phases which means phase separation takes place\nindicating a \frst order transition from low temperature\nincommensurate to commensurate intermediate temper-\nature phase. This is consistent with the \frst order nature\nof this transition observed from heat capacity and mag-\nnetization measurements.\nTo reveal the magnetic structure, re\fnement of two\ndata sets collected at 4 K and 13 K, has been performed\n/s91/s98/s93\n/s91/s97/s93FIG. 6. (a) Perspective view of spin con\fguration in the com-\nmensurate phase ( T= 13 K) is shown. (b) shows the arrange-\nment of Mn2+spins on the distorted honeycomb layer for the\nsame temperature.\nwith JANA2006. Magnetic representation analysis sets\nsymmetry restrictions only on the high-temperature com-\nmensurate phase. The restrictions imply that the mag-\nnetic structure involves either ferromagnetic (\u0000 1) or an-\ntiferromagnetic (\u0000 2) coupling of the spins related by in-\nversion in the 2i Wycko\u000b position. There are no restric-\ntions on the moments direction and Mn in the symmetry\nindependent positions can adopt di\u000berent size and orien-\ntation. The re\fnement however was constrained to keep\nthe structure collinear with the same ordered moment\nfor the all four Mn sites. The obtained solution is shown\nin Figure 6, which corresponds to \u0000 2irreducible repre-\nsentation of the P\u00161 space group. Magnetic moments lie\nmostly in the ac-plane with a small component along the\nb-axis. The size of the ordered moment at T= 13 K\nhas been determined to be 2.83 \u0016Bper Mn. In the in-\ncommensurate phase, which onsets below TN2= 11.8 K,\nsymmetry sets no restrictions and we tried to \ft the low-\ntemperature di\u000braction data considering three di\u000berent\nmodels: (a) amplitude modulated structure derived from\nhelical structure, (b) cycloidal structure where moments\nare rotating with respect to a plane and (c) proper he-\nlical structure. The statistically best \ft is obtained for\nthe cycloidal model with equal amplitude of magnetic6\n/s91/s98/s93/s91/s97/s93\nFIG. 7. (a) Perspective view of the spin con\fguration of Mn2+\nin the unit cell is shown for the incommensurate phase below\nTN2= 11.8 K. (b) depicts schematic view of spin con\fgura-\ntion in the low- Tincommensurate phase obtained from the\nre\fnement of 4 K data.\nmoment on each Mn atom. It yields for the ordered mag-\nnetic moment of \u00195.19\u0016Bper Mn. Figs. 7(a) and (b)\nshow the proposed cycloidal magnetic structure in the\nlow temperature incommensurate phase from the c\u0000axis\nand honeycomb plane respectively. The rotation of the\nspins is clearly visible in these two projections while pro-\njection from the honeycomb plane shown the rotation of\nthe moment about the honeycomb plane.\nV. DISCUSSION\n\u000b\u0000Mn2V2O7reveals two successive magnetic tran-\nsitions with two distinct magnetically ordered states,\nTN1\u001919 K andTN2\u001911.8 K, from susceptibility as\nwell as heat capacity measurements. Here, Mn2+ions\nhave high-spin state and the system exhibit magnetic\nanisotropy due to admixture of orbital moment. Actu-\nally,TN1of\u001919 K is rather low compared to Curie-Weiss\nconstant\u0012C=\u000034 K which indicates the presence of\nsome frustration in the system which might be the re-\nsponsible criteria to suppress the 3D magnetic ordering.\nThe second anomaly at TN2was not reported previously\nand it is interesting to \fnd out the nature of this transi-\ntion. At \frst, the indication of another magnetic phase\nwas observed from neutron powder di\u000braction study (al-though no anomaly was found at this TN2in magneti-\nzation), but it was nontrivial to determine the magnetic\nstructure solely based on powder data because of very low\nsymmetry of \u000b\u0000Mn2V2O7with four di\u000berent Mn sites.\nHence to solve these two magnetic phases single crystal\nhas been prepared and from our careful neutron di\u000brac-\ntion study on the single crystalline sample of Mn 2V2O7,\nwe found commensurate magnetic structure in the inter-\nmediate state below TN1and incommensurate cycloidal-\ntype structure at low temperature.\nMicroscopically, \u000b\u0000Mn2V2O7can be viewed as quasi-\n2D, with distorted honeycomb layer consisting of Mn2+\nions parallel to [1 \u001610] plane which is the plane along\nbody diagonal of the unit cell. But the absence of dif-\nfuse scattering in the neutron di\u000braction data indicates\nstrong interaction between the honeycomb layers. Neu-\ntron di\u000braction experiment have shown that this com-\npound adopts a collinear commensurate magnetic struc-\nture below TN1followed by a incommensurate structure\nwith cycloidal modulation of Mn spins at low temper-\nature and both phases can be described by the same\ntype of irreducible representation. The incommensurate\nstructure has a predominant antiferromagnetic compo-\nnent, giving rise to satellite peaks in the vicinity of the\nfundamental AFM Bragg re\rection. Now, the existence\nof this type of commensurate to incommensurate phase\ntransition in insulators is (i) either due to competing\nexchange interactions, (ii) or due to relativistic e\u000bects\nsuch as spin-orbit coupling. While the \frst one can be\nfound accidentally, the second mechanism depends on lat-\ntice symmetry.29,30In case of\u000b\u0000Mn2V2O7, relativistic\ne\u000bects can be discarded as no signature was observed\nfrom our experimental data indicating the presence of\nspin-orbit coupling. Hence, competing exchange interac-\ntions appear to be the main driving mechanism of the\nincommensurate-commensurate phase transition in our\ncase which can generate due to anisotropies in the spin\nHamiltonian and/or frustration in the lattice.31\nThe reduced magnetic moment in the commensurate\nphase (\u00192.83\u0016B) can be taken as a hallmark of quan-\ntum \ructuations triggered by the frustrated couplings\nof Mn spins while the moment increases rapidly below\nTN2(\u00195.3\u0016B). Now, frustration in the crystal lat-\ntice is expected because of very low symmetry and 4\nnon-equivalent crystal sites of Mn where all positions\nare general. The nature of Mn - Mn interaction is su-\nperexchange via O atoms. Because of the distorted na-\nture of Mn-honeycomb layers, there exist di\u000berent types\nof Mn - Mn bond lengths with variable O-environment.\nHence, the Mn - Mn coupling is random on a honeycomb\nand supposed to create competing interaction. Further\nwe did not observe any structural incommensurability or\nstructural transition between 280 K and 2 K. So, we can\nhypothesize that the incommensurate phase below TN2\noriginate from sudden appearance of quantum \ructua-\ntions which is also supported from our magnetic \feld de-\npendent heat capacity study. Magnetic \feld up to 30 kOe\nincreases the \ructuations and supports to overcome the7\ncollinear AFM ordering and the anomaly at TN2shifts\nto the higher T. But the period of the spin modulation\ndoes not change with Tin the incommensurate regime\nwhich indicates some frozen magnetic phase below TN2.\nTheoretical predictions suggest that frozen spin modula-\ntion is possible in a 2D system with random distribution\nof coupling.32The studied compound is quasi-2D and the\ntheoretical predictions are also in line with our hypothe-\nsis.\nVI. CONCLUSIONS\n\u000b\u0000Mn2V2O7reveals two successive magnetic transi-\ntions atTN1= 19 K and TN2= 11.8 K indicationg two\ndistinct magnetically ordered phases. The single crys-\ntalline sample has been grown in the \roating zone fur-\nnace. No further structural chages are observed betwen\n280 to 2 K in \u000b\u0000Mn2V2O7phase. By using single crys-\ntal neutron di\u000braction study, we have shown that in the\nTrange 11.8\u0014T\u001419 K the magnetic structure is\ncommensurate with propagation vector k1= (0;0:5;0),\nwhile upon lowering temperature below TN2= 11.8 K\na second magnetic phase towards an incommensurate\n(k2= (0:38;0:48;0:5)) 3D magnetic order emerges. The\nincommensurate modulation of the Mn2+spins of the\nlow-Tphase is driven by the presence of competing ex-\nchange interactions of the next nearest neighbour Mn2+spins. The magnetic model at low temperature can\nbe represented by cycloidal modulation along ac\u0000plane\nwhile in the intermediate commensurate phase collinear\narrangement is observed. Our study will stimulate theo-\nretical interest and microscopic studies on the nature and\norigin of incommensurate to commensurate phase tran-\nsition observed in frustrated insulator systems.\nNeutron data were taken on the SXD di\u000bractometer at\nthe ISIS Neutron and Muon Source. Information on the\ndata can be accessed through STFC ISIS Facility33.\nACKNOWLEDGEMENT\nJ.S. would like to thank the European Unions Hori-\nzon 2020 research and innovation programme under\nthe Marie Skodowska-Curie grant agreement (GA) No\n665593 awarded to the Science and Technology Facilities\nCouncil. V. Petricek likes to acknowledge the Czech Sci-\nence Foundation (project no. - 18-10504S) for developing\nthe JANA software. We thank Dr G. Stenning for help on\nmagnetization and heat capacity measurements and ISIS\nFacility for providing beam time on SXD (RB1810466).\nREFERENCES\n\u0003jhuma.iacs@gmail.com\nydevashibhai.adroja@stfc.ac.uk\n1Z. Y. Meng, T. C Lang, S. Wessel, F. F. Assaad, and A.\nMuramatsu, Nature 464, 847 (2010).\n2S. Ladak, D. E. Read, G. K. Perkins, L. F. Cohen, and W.\nR. Branford, Nat. Phys. 6, 359 (2010).\n3S. Nakatsuji, K. Kuga, K. Kimura, R. Satake, N.\nKatayama, E. Nishibori, H. Sawa, R. Ishii, M. Hagiwara,\nand F. Bridges, Science 336, 559 (2012).\n4F. 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B 65, 054418 (2002).\n33DOI: https://doi.org/10.5286/ISIS.E.RB1810466."    },    {        "title": "1905.08982v2.Effect_of_doping_of_Co__Ni_and_Ga_on_magnetic_and_dielectric_properties_of_layered_perovskite_multiferroic_YBaCuFeO5.pdf",        "content": "Effect of doping of Co, Ni and Ga on m agnetic  and dielectric properties of \nlayered perovskite multiferroic YBaCuFeO 5 \nSurender Lal *, C. S. Yadav  and K. Mukherjee  \nSchool of Basic Sciences, Indian Institute of Technology Mandi, Mandi 175005, Himachal Pradesh, \nIndia \n \n \n \n \n \n \n \n \n \n \n \n \n*Cor responding Author: Surender Lal  \nSchool of Basic Sciences,  \nIndian Institute of Technology Mandi,  \nMandi 175005, Himachal Pradesh, India  \nTel: +919418304396  \nE-mail: surenderlal30@gmail.com  \n  Abstract  \nYBaCuFeO 5 is one of the  interesting  multiferroic compound s, which exhibits  magnetic \nordering and dielectric anomaly above 200 K. Partial substitution of Fe with other magnetic and \nnon-magnetic ion affects the magnetic and the structural properties of the system.  We report \ndetailed investigation of  structural, magnetic and dielectric properties of YBaCuFe 0.85M0.15O5 \n(M=Co, Ni and Ga ). We observed that the partial replacement of Ni and Co in place of Fe , results in  \nmagnetic dilution and broadening of the magnetic transition and shifting  towards  lower \ntemperature . The r eplacement of Fe with non -magnetic Ga also results in shifting of the  magnetic \ntransition  to the lower temperature side.  The observed dielectric relaxation behavior in these \ncompounds  is due to the charge carrie r hoping. This study  highlights the impacts of magnetic and \nnon-magnetic doping at the magnetic site on magnetic and dielectric properties in layered \nperovskite compound YBaCu FeO 5. \n \n \nKey words:  Magnetic properties , glassy dynamics, dielectric relaxation  \n  1. Introduction  \nThe coexistence of magnetic and ferroelectric ordering in the same crystallographic phase  \nmakes the m ultiferroic materials quite interesting  [1-3]. These  materials are classified in two \ncategories: Type-I, and Type -II. In type-I multiferroics , the magneti c and dielectric transition s are  \nfar away and independent of each other. Therefore, t he possibility of coupling  between  magnetic \nand electric order parameters is very weak . On the other hand , in type II materials , the dielectric \nordering  follows the magnetic ordering and there is a possibility of strong coupling between the se \ntwo orders  parameters  [1]. Further,  the electric polarization can be induced in these materials by \nmagnetic field or vice versa  and this phenomenon is known as magnetoelectric coupling  [2,3] . In \nrecent year the compounds belonging to L nBaM'M\" O5 family  (where  Ln is the rare-earth  ions, and \nM', M\"  are the  transition metal ions) have  been widely studied . In such compound s, two layers of \nM'O5 and M\" O5 are present and it is pos sible to combine different type s of metal element in an \nordered or disordered network. Also depending upon the size of rare-earth ions, these compounds \nhave  the various interesti ng physical properties  [4–7]. YBaCuFeO 5 is quite interesting,  as it exhibits  \nmagnetic and diele ctric transitions above ~ 200 K [8]. This compound undergoes a paramagnetic to \ncommensurate antiferromagnetic (CM -AFM) transition temperature below  TN1 ~ 440 K, followed \nby a commensurate to incommensurate antiferromagnetic (ICM -AFM) transition at TN2~ 200 K [8]. \nRecent investigations  revealed  that the chemical pressure has  influence on the magnetic and the \ndielectric prope rties due to the presence of  cation (Cu/Fe) disorder in the system [9]. This disorder \ncan be induced in the system by varying the preparation conditions  [10]. Generally, from the \nviewpoint of technological application, the multiferroics with magnetoelectric coupling near room \ntempera ture are useful. For tuning transition,  temperature of the compound, disorder -induced \nstudies has surfaced as an enormously useful technique.  The disorder in th is compound  can also be \ninduced by the replacement of Y with some other rare earth ions  and such studies have already been \ncarried out . A detailed investigation on LaBaCuFeO 5 and LuBaCuFeO 5 have revealed contrasting \nphysical properties [11]. Studies on LnBaCuFeO 5 (Ln = Nd, Yb, Gd and Ho) reveals magnetic \ntransitions in the c ompound s is not visible,  where moments due to  the rare-earth ions dominate the \neffect arising out of Cu/Fe ions  [12], however the nature of magnetic transition is clearly probed by \nthe Neutron diffraction in  the recent investigations in spite of paramagnetic signa l [13].  \nInvestigations  on HoBaCuFeO 5, GdBaCuFeO 5 and YbBaCuFeO 5 reveal ed that the observed upturn \nin the heat capacity at low temperatures in these compounds is due to Schottky anomaly  [12]. One of the effective ways  to introduce disorder in  YBaCuFeO 5 is through partial replacement of Fe by \nother transition metal ions. Doping the Fe -site will introduce random impurities, which  will directly \naffect the  interaction s arising out of FeO 5/CuO 5 bipyramids and hence might cause a n alteration in \nthe physical p roperties of YBaCuFeO 5. \nIn this article  we present a detailed study of structural, magnetic and dielectric properties of \nYBaCuFe 0.85M0.15O5 (where M=Co, Ni and Ga) . We substituted Fe by both magnetic and non-\nmagnetic ions because the latter (Ga) might leads to the structural deformation only whereas the \nformer (Co, Ni) will result  in the change in magnetic properties along with structural deformation. \nAs compared with Fe, Ga has higher ionic radii  where as Co and  Ni have  the comparably smaller  \nionic radii , in its + 3 oxidation  state. Here we would like to mention that the maximum solubility of \nM in YBaCuFeO 5 is about 15% of Fe, as revealed by our investigations.  The p artial replacement of \nFe by  Co and Ni leads to broadening in magnetic transition  temperature . Doping with  non-magnetic  \nion Ga leads to the expansion in the unit cell  and influence s the magnetic ordering.  The dielectric \nrelaxation has  also been observed in these compounds. However, the interaction s between the \nelectric dipoles are  not strong enough for collective freezing of the electric dipoles . As compared to \nthe parent compound,  the change in magnetodielectric coupling is found to be  insignificant in  the \ndoped compounds.  \n \n2. Experimental details  \n Polycrystalline sample s of YBaCuFe 0.85Co0.15O5 (Co_0.15 ), YBaCuFe 0.85Ni0.15O5 (Ni_0.15) \nand YBaCuFe 0.85Ga0.15O5 (Ga_0.15)  are synthesized by solid -state reaction metho d similar to that \nreported in literature  [9,14 ]. Power x -ray diffraction is performed using Regaku  smart lab \ndiffractometer using monochromatized  CuKα1 radiation at room temperature. DC magnetization \nmeasurements are carried out in  a magnetic property measurement system (Quantum Design USA) . \nHioki LCR meter is used for temperature dependence of dielectric  constant measurements \nintegrated with physica l property measurement system ( Quantum, Design  USA ) with a setup  from  \nCryonano Labs . For electric measurements, silver paint contacts are made to the polycrystalline \npellets with typical electrode are a A = 15 mm2 and thickness d= 0.506 mm.  \n3. Results and Discussion  \n    3.1 Structural properties  Rietveld refined x -ray diffraction pattern s of all the studied  compounds are shown in  figure \n1 and calculated  parameters are tabulated in table 1. These  compounds crystallize in the tetragonal \nstructure (space group: P4mm) . All the peaks fit well with the theoretical curve and no impurity  \npeak is observed  which indicates  that all the  compounds  are formed in crystallographic single \nphase. The structure of these compounds is similar to that reported in literature  [9,15,16 ]. The \npartial replacement of Fe  with Co, Ni and Ga results in small  changes  in the structural parameters of \nthe unit cell in comparison to YBaCuFeO 5 due to variation in the ionic radii.  \n \nFigure 1: (a-c) Rietveld refined x -ray diffraction patterns of YBaCuFe 0.85M0.15O5 (M=Co, Ni, Ga) compounds.  Inset of \n(b) shows the pattern in an expanded form for one  peak for all the compounds, to bring out that the peaks shift with  \nsubstitution. The data of  the YBaCuFeO 5 is taken from Ref [9] for comparison.  \nInset of figure 1 (b) show the shifting of x -ray diffraction peak with substitution . The x -ray \ndiffraction curve of  YBaCuFeO 5 for compar ison is taken from Ref [9]. The peak  position is shifted \ntowards higher angle side  with Co and Ni whereas  Ga substitution shifts the peak position towards \nlower angle side;  indicating that the former substitutions results in  lattice contraction whereas the \nlater one leads to lattice  expa nsion . Co and Ni substitution leads to slight  increase in a, but c lattice \nparameter  decreases as compared to the YBaCuFeO 5. The partial replacement of Fe with  Ga leads \nthe increase in both a, and c parameters . The replaced ions sits at the Fe site at (1/2, 1/2, x) position . \nThe distance between the pyramids in Co_0.15 and Ni_0.15  compounds is smaller as compared  to \nthe YBaCuFeO 5 whereas in Ga_0.15  it increases . \n \nTable 1: Lattice parameter calculated from the Rietveld refinement of x -diffraction of  YBaCuFe 0.85M0.15O5 (M=Co,Ni, \nGa)  \n \n3.2 Magnetic properties  \nThe fig ure 2(a-c) shows the temperature dependence of the DC magnetic susceptibility of \nCo_0.15, Ni_0.15 and Ga_0.15 in the zero field cooled  (ZFC)  and field cooled (FC) conditions \nmeasured at 100 Oe.  The figure  2(d-f) shows  the temperature dependence of magnetization at 5 kOe \nof magnetic field . The commensurate to incommensurate antiferromagnetic transition is seen in the \ntemperature range of 200 -230 K  for YBaCuFeO 5 compounds  [8,17,18 ]. It is noted that the  \ntransition is shifted towards lower temperature ~ 200, 190 K and 1 10 K for Co_0.15, Ni_0.15 and \nGa_0.15 respectively.  \nFor Co_0.15 compound, a  significant bifurcation between the ZFC and FC curves is noted \nbelow 100 K. Also as the temperature is decreased a weak kink is noted in the ZFC curve.  Here, it \nis to be noted that no kink (at low temperatures) or significant bifurcation between ZFC and FC \ncurves is observed in  YBaCuFeO 5 [9]. For Ni_0.15 compound the bifurcation starts from 200 K and \nit becomes more prominent at lower temperature. A sharp peak at ~12.5 K is observed  in the ZFC \ncurve. In contrast, for Ga_0.15 compounds, no significant bifurcation between ZFC and FC curves \nis obs erved ; however, a weak signature of a peak is pre sent at ~11.5 K. Insets of figure 2 (d-f) shows \nthe M (H) at 2 and 300 K  which show the linear behavior . It is to be noted that the unit cell volume Parameters  YBaCuFeO 5 [9] YBaCuFe 0.85M0.15O5 \nM  Co_0.15  Ni_0.15  Ga_0.15  \na(Å) 3.871  3.873(1)  3.873(2)  3.873(1)  \nc(Å) 7.662  7.652(1)  7.648(1)  7.676(2)  \nV(Å3\n) 114.83  114.80  114.69  115.17  \nR-factor  4.92 9.20 11.8 17.7 \nRF-factor  4.18 8.23 15.1 13.1 \nχ2 2.18 1.80 1.95 1.85 \nInter Pyramid distance  2.833(1)  2.829(2)  2.827(3)  2.838(2)  of Ni _0.15  and Co _0.15  show very small change in comparison to YBaCuFeO 5. However, in \ncontrast to Fe3+ (which have five elections in 3 d shell), Co3+ and Ni3+\n have  six and seven  electrons \nrespectively.  The Co3+\n have the three possible spin configurations, low spin, intermediate  spin and \nhigh spin configuration [19]. The replacement of Fe3+ by Co3+ and Ni3+ results in the broadening of  \nmagnetic transition and shifting of the transition towards the lower temperature.  The observed \nbehavior in the magnetization may be due to presence of different electronic  states such as low  spin \nand high spin configuration of  Co3+ and Ni3+\n ions. The partial replacement of Fe3+ with Co3+\n or Ni3+\n \nseems to weakening the magnetic state and the  magnetic interaction s within the bipyramid . These \nstates may give complex magnetic behavior and the magnetic transition is shifted to low \ntemperature side. The substitution of Ga in place of Fe3+ leads to expansion in the unit cell as \ncompared to YBaCuFeO 5, which  in turn increases the distance between the magnetic ions within  \nthe bipyramids. This change s the antiferromagnetic interaction s within the unit cell and \ncommensurate incommensurate antiferro magnetic transition shifts towards the lo wer temperature \n[11]. At low temperature, a kink/peak is observed in all the three compounds. T his peak/kink \nobserved at low temperature and low field is suppressed under higher applied field (shown in figure \n2 (d-f)). Huge bifurcation between the ZFC and FC curve is noted below this kink/peak . This effect \nis more pronounced in Co, and Ni compounds. This bifurcation is more pronounced in the \ncompounds with volume lower than the unit cell volume of YBaCuFeO 5 [11]. As stated before, Co \nand Ni substitution leads the decrease in lattice parameter c, thereby resulting in an increase of \ninteraction between the magnetic ions. As the temperature is reduced, the crystal structures \ncontracts and the magnetic interaction fu rther changes , resulting in a possible change of magnetic \nanisotropy, which  may contribute to the observation of these features. The suppression of these \nfeatures with increasing the magnetic field also gives an indication that the origin of these \ncharacte ristics is due to the weakening of the magnetic structure  of the system . This feature in the \nZFC and FC is less prominent in Ga_0.15 due to larger volume of the unit cell as compared to \nYBaCuFeO 5 due to the lattice expansion.   \nFigure 2: (a-c) Temperature dependence of dc susceptibility measured the ZFC and FC conditions for \nYBaCuFe 0.85M0.15O5 (M=Co, Ni and  Ga) at 100 Oe of magnetic field  and (d -f) at 5 kOe . Inset of (a -c) shows the dc \nsusceptibility at low temperature. Inset (d -f) shows the isothermal response of magnetic field at 2 K and 300 K . \n \nHowever, to resolve this new low temperature anomaly, temperature dependent neutron diffraction \nis warranted . \n \n   3.3 Dielectric analysis  \nThe temperature variation of real () and imaginary part ( ) of dielectric  constant  () measured at \nselected  frequencies for the  series  YBaCuFe 0.85M0.15O5 (M= Co, Ni, Ga ) in the temperature range of \n10 to  300 K  (shown in fig ure 3). The temperature variation of  for Co_0.15 , Ni_0.15  and Ga_0.15  \nshows the sharp increase  near 56, 105 and 62 K respectively  at 10  kHz followed by a plateau in the  \nhigh temperature  region . It is noted that  value of   also increases in this temperature range. \nInterestingly, for this series, a frequency dependent behavior of  and  is observed . To analyze \nthis feature,  temperature derivative of  is plotted as function of temperature.  The curve shows a \npeak  (not shown) , the temperature of which increases with the increase in frequency. In compounds \nwhere electric dipoles exhibits glassy  dynamics s uch shift in the peak temperature are observed. \nThis variation of peak temperature can be analyzed by Arrhenius, Vogel Fulcher and/or power law  \n[20]. We tried to fit the data with the above -mentioned la ws. In our case the best fit is obtained with \nthe Arrhenius law  [21] of the form  \na\n0\nBτ τ expE\nKT\n   ..... (1)  \nWhere 0 is the pre -exponential factor  and Ea is the activation energy and  kB is the Boltzmann \nconstant . Upper Insets of figure 3  (a-c) shows the temperature variation of ln . The obtained values \nof the Ea are 0.044 eV, 0.147 eV , 0.0729 eV 8.7010-9 s and 0 are 1.0910-11 s, 4.70710-10 s for \nCo_0.15, Ni_0.15 and Ga_0.15 respectively.  The values of Ea are smaller as compared to Bi - \n \nFigure 3: Temperature d ependence of dielectric susceptibility in the temperature range of 2 -300 K of \nYBaCuFe 0.85M0.15O5 (M= Co, Ni and  Ga).  The lower inset shows imaginary part of dielectric constant  and the upper \ninset shows the Arrhenius fit of the peak temperature obtained from the derivative of the real part of the dielectric \nconstant.  \nDoped  SrTiO 3 (0.74 to 0.86 eV)  [22], Bi4Ti3O12 (0.87 eV)  [23] and Bi 5TiNbWO 15 (0.76 eV)  [24] \nwhere the relaxation mechanism is ascribed  to the thermal motion of oxygen vacancies . However, \nthe values of Ea is comparable to  that observed for La2CoIrO 6 (0.056 eV) [25], Ca3Co1.4Rh0.6O6 \n(0.071 eV) [26], BiMn 2O5 (0.065 eV) [27] and PrFe 0.5Mn 0.5O2.9 (0.19 eV)  [28] where the relaxation  \nbehavior is due to the charge carrier hoping . The values of the activation energy of the studied \ncompounds are comparable to the values observed for the charge carrier hoping.  The relaxation in  \nelectric dipoles is due  to the charge carrier hoping among the transition metal ion . However, the \ninteraction between the electric dipoles is  not strong  enough  for collective freezing of the electri c \ndipoles. The partial replacement of Ba with Sr leads the dipolar glass behavior  at low temperature  \n[9]. Howe ver, this low temperature behavior is absent in Co, Ni and Ga doped compounds.  This \nmay be due to the effect of the transition metal ion, which may leads to the change  in the interaction \nand affecting the collective freezing of electric dipoles.  \n    3.4 Magneto -dielectric properties  \n YBaCuFeO 5 shows th e magneto -dielectric coupling  at different temperatures  [8,9] . To see \nthe effect of MDE coupling due to these substitutions , the magnetic field response of the dielectric \nconstant is measured at  200 K at a  fixed frequency of 50 kHz.  The MDE effect is defined as Δ (%) \n= 100  ['(H)-'(0)]/'(0), where '(H) and '(0) are the dielectric constant in the presence and the \nabsence of magnetic field respectively [9,29,30 ]. Figure 4 shows the magnetic field response of \nΔ(%).The data for Y BaCuFeO 5 is taken from  Ref [9] for comparison . It is observed that  MDE \ncoupling persists in all the doped compounds. However, the change in magnitude of Δ (%) is \ninsignificant when compared with YBaCuFeO 5. \n \nFigure  4: The magnetic field response of relative dielectric permittivity measured at different compou nds \nmeasured at 200  K. \n4. Summary  \nIn summ ary, we have investigated the structural, magnetic dielectric and magneto -dielectric \nproperties of YBaCuFe 0.85M0.15O5 (M=Co, Ni and Ga). Partial replacement of Fe with Co and Ni \nleads to broadening in the magnetic transition may be due to the c hange  in magnetic structure of \nsystem.  Doping with the non -magnetic ion shifts the magneti c transition towards the low \ntemperature side  as compared to YBaCuFeO 5. The dielectric relaxation behavior is observed . The \nobserved  may be due to the charge carrier hoping.  However, glass -like behavior of electric dipoles \nin not observed,  as the interaction between the electric dipoles are not stron g enough. 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Alloys Compd. 693 (2017) 188 –193.  \n "    },    {        "title": "1905.10541v1.Magnetic_2D_electron_liquid_at_the_surface_of_Heusler_semiconductors.pdf",        "content": "Magnetic 2D electron liquid at the surface of Heusler semiconductors\nS. Keshavarz,1I. Di Marco,1, 2, 3D. Thonig,1L. Chioncel,4, 5O. Eriksson,1, 6and Y. O. Kvashnin1\n1Uppsala University, Department of Physics and Astronomy,\nDivision of Materials Theory, Box 516, SE-751 20 Uppsala, Sweden\n2Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Korea\n3Department of Physics, POSTECH, Pohang, Gyeongbuk 790-784, South Korea\n4Theoretical Physics III, Center for Electronic Correlations and Magnetism,\nInstitute of Physics, University of Augsburg, D-86135 Augsburg, Germany\n5Augsburg Center for Innovative Technologies, University of Augsburg, D-86135 Augsburg, Germany\n6School of Science and Technology, ¨Orebro University, SE-701 82 ¨Orebro, Sweden\n(Dated: July 9, 2021)\nConducting and magnetic properties of a material often change in some confined geometries.\nHowever, a situation where a non-magnetic semiconductor becomes both metallic and magnetic at\nthe surface is quite rare, and to the best of our knowledge has never been observed in experiment.\nIn this work, we employ first-principles electronic structure theory to predict that such a peculiar\nmagnetic state emerges in a family of quaternary Heusler compounds. We investigate magnetic and\nelectronic properties of CoCrTiP, FeMnTiP and CoMnVAl. For the latter material, we also analyse\nthe magnetic exchange interactions and use them for parametrizing an effective spin Hamiltonian.\nAccording to our results, magnetism in this material should persist at temperatures at least as high\nas 155 K.\nIntroduction. In several classes of materials, surfaces\nare known to exhibit properties distinctly different from\nthose of the bulk. At quantum mechanical level, sur-\nface electronic states were first characterized almost a\ncentury ago, in the pioneering works of Tamm [1] and\nShockley [2]. A direct probe of surface states became\npossible only much later, with the improvements in ex-\nperimental techniques, as e.g. in angular resolved photo-\nelectron spectroscopy (ARPES) [3–6]. The arising of\nsurface physics led to the observation of several key\nphenomena. Materials characterized by a band gap in\ntheir bulk electronic structure can possess a metallic\nsurface hosting a two-dimensional electron gas (2DEG),\nas e.g in SrTiO 3[7], or the much celebrated topolog-\nical edge modes [8]. Differences between surface and\nbulk electronic structures are also the foundation of\ninterface-induced effects, as those observed when com-\nbining LaAlO 3and SrTiO 3. The interface of these two\ninsulators does not only host a peculiar 2DEG [9–11],\nbut may also harbor a quasi-2D magnetic order driven\nby oxygen defects [12, 13]. These studies associate the\ninterface magnetism with off-stoichiometry modelled by\nartificially ordered supercells, sometimes limited to high\ndefect concentrations [14, 15].\nConfined geometries have always been of primary im-\nportance for Heusler and half-Heusler compounds [16].\nFor instance, understanding interfaces is crucial for build-\ning Heusler-based devices for giant and tunneling mag-\nnetoresistances [16–18]. Interfaces have not only a prac-\ntical importance, but may also host non-collinear spin\nordering [19, 20] resulting into complex magnetic struc-\ntures. Recently, materials like LaPtBi, LuPtSb, ScPtBi,\nYPdBi, and ThPtPb were shown to represent a new class\nof three-dimensional topological insulators [21]. Further-\nmore, a new Weyl system was demonstrated to exist in a\nfamily of magnetic Heuslers [22].Due to the high tunability and versatility of Heusler\ncompounds, new phenomena are expected to emerge\nwhen more and more materials are suggested and syn-\nthesized. Precisely, in this letter we predict that a rel-\natively unexplored (but already existing) family of qua-\nternary Heusler compounds hosts a rare, if not unique,\nphenomenon, where a semiconducting and non-magnetic\nbulk coexists with a metallic surface with robust mag-\nnetism. By means of first-principles electronic structure\ncalculations, we show this peculiar state to emerge in\nCoMnVAl, CoCrTiP and FeMnTiP. Using calculated ex-\nchange interactions, we illustrate that the magnetic long-\nrange order at the surface is stable up to sizeable tem-\nperatures, which is a fundamental prerequisite for ex-\nperimental verification. We discuss the results both in\nterms of fundamental understanding of this new class of\nelectronic structures, as well as in terms of technological\naspects, that may emerge when spin degrees of freedom\nenter semiconductor-based electronic devices [23–27].\nTo the best of our knowledge, the coexistence of a semi-\nconducting and non-magnetic bulk with a metallic and\nmagnetic surface has never been observed in experiment.\nHere we identify a promising prototype of such materi-\nals, which are also characterized by a spin-polarized 2D\nelectron liquid emerging at the surface. It is important\nto stress that differentiating liquid and gas is not only\na naming convention. Localized 3 dorbitals, which give\nrise to the surface metallicity and magnetism here, are\nlikely to harbour strong electronic correlations, which\nmay lead to a plethora of exotic effects. Half-metallic\nHeusler compounds are known to host a wide array of\nsuch features. For example, genuine many-body states,\nnamed non-quasiparticle states (NQP), may appear in\nhalf-metallic ferromagnets, leading to a reduction of spin\npolarization [28–31]. Another example is given by the\nmass enhancement and non-Fermi liquid behavior ob-arXiv:1905.10541v1  [cond-mat.str-el]  25 May 20192\nserved in Fe 2VAl and related materials [16]. These ef-\nfects, as well as others, are expected to emerge in the re-\nported systems, due to the considerable localization and\nreduced dimensionality.\nTheory and methods. Density functional theory (DFT)\ncalculations were carried out using the full-potential\nlinear muffin-tin orbital (FP-LMTO) code RSPt [32,\n33]. We used the generalized gradient approximation\n(GGA), in the formulation of Perdew, Burke, and En-\nzerhof [34]. The crystal structure of the considered class\nof Heusler semiconductors is described by a chemical for-\nmula XX/primeYZ, where X, X/primeand Y are transition metals\nand Z is an spelement. The surface was modelled in\na supercell geometry consisting of four unit-cells stacked\nalong the [001] direction. This means that eight (or nine)\nXX/primelayers mediated by eight alternating YZ layers were\nconsidered [35]. A vacuum of 25 ˚A thickness has been\nadded to ensure no interaction between the surfaces. The\ncomputed equilibrium lattice parameters for all consid-\nered materials were taken from Ref. [36]. Selected cal-\nculations of the FeMnTiP slabs were performed with a\nfull relaxation of the atomic positions. The effects of\nthe relaxation were found to be minor, and therefore we\nrestricted our most extensive analysis, presented in the\nfollowing, to the unrelaxed structures.\nIn addition to standard GGA, we also performed calcu-\nlations where the effects of strong electron-electron repul-\nsion were included explicitly, at the level of a mean-field\nDFT+ Uapproach [37, 38]. For simplicity, we adopted\nthe same values of the Coulomb interaction parameters\nfor the d-states of all X and X/primeelements, namely U= 2\neV and J= 0.8 eV. These values are situated in between\nthe partly and fully screened estimates [39], obtained for\nsimilar compounds using the constraint random phase\napproximation (cRPA) [40].\nThe thermal stability of the predicted magnetic long\nrange order at the surface was investigated by means of\nan effective Heisenberg model. The latter is described\nby a Hamiltonian ˆH=−/summationtext\ni/negationslash=jJij/vector ei·/vector ej, where Jijis the\nexchange interaction between the spins located at the site\niandj, and /vector eis a unit vector along the spin direction\nat the corresponding site. The Jij’s were extracted from\na GGA(+ U) calculation by means of the magnetic force\ntheorem [41–43]. Based on these values of the Jij’s, we\ncalculated the ordering temperature ( Tc) by means of a\nclassical Monte Carlo (MC) simulation, as implemented\nin the UppASD code [44, 45]. The adiabatic magnon\nspectra (AMS) were calculated for a considered magnetic\nground state via Fourier transforms of the Jij’s.\nResults and Discussion. We first focus on the physical\nproperties of three Heusler semiconductors: CoMnVAl,\nCoCrTiP and FeMnTiP. According to the Slater-Pauling\nrule [46, 47], Heusler compounds with 24 valence elec-\ntrons are expected have zero net moment. They can ei-\nther represent compensated ferrimagnetic half-metals or\nsemiconductors with antiferromagnetic or non-magnetic\norders [48, 49]. The results of our calculations indicate\nthat the selected materials belong to the latter category,in agreement with previous literature [36]. The calcu-\nlated band gaps in GGA (GGA+ U) are 0 (0.2) for CoM-\nnVAl, 0.15 (0.3) for CoCrTiP and 0.5 (0.8) for FeMn-\nTiP, where all values are given in eV. The previous val-\nues show that GGA predicts CoMnVAl to be metallic,\nwhich is remedied by the inclusion of the Uterm. The\nsituation is reminiscent of Co 2FeSi, where on-site corre-\nlations are also needed to obtain the true half-metallic\nsolution [50, 51]. CoCrTiP and FeMnTiP are already\nsemiconducting in standard GGA, but the band gaps are\nfurther enhanced when Uis added.\nNext, we focus on the finite-size slabs, starting with\nthe analysis of their spectral properties. The projected\ndensity of states (DOS) for the atoms closest to the sur-\nface is reported in Fig. 1, for all three considered systems.\nThe DOS of the deeper-lying layers quickly converges to\nthat of bulk materials, as shown in the Supplementary\nMaterial (SM) for CoMnVAl [52]. The innermost layers\nof the 16-layer slab are indistinguishable from the bulk,\nwith perfect spin-degenerate bands, as was also reported\nin Ref. 20. The fundamental feature of Fig. 1 is that all\nsystems are characterized by magnetic and metallic DOS\nat the surface and sub-surface layers. Since the surfaces\nhave a XX/prime-termination, the electronic states crossing the\nFermi level primarily originate from the relatively local-\nized 3 dorbitals of the transition metals. These localized\nsurface states are the ones responsible for the arising of\na finite magnetization.\nThe projected magnetic moments in the surface and\nsub-surface layers are reported in Table I. Transition\nmetal atoms at the surface exhibit pronounced magnetic\nmoments, which are aligned ferromagnetically with each\nother. The Y atoms (V or Ti) belonging to the sub-\nsurface layer show weak magnetic moments, which are\nanti-parallel to the large moments at the surface, thus\nresulting in a ferrimagnetic order. The third layer be-\nlow the surface is nearly non-magnetic, showing moments\nthat are 10 to 300 times smaller than those at the sur-\nface. Going deeper into the slab, the bulk properties are\nrecovered.\nWe note that the inclusion of the strong electron-\nelectron repulsion via the local Uterm does not change\nthe qualitative picture obtained in plain GGA. In the\nprojected DOS of Fig. 1, the localization induced by the\nUterm leads to slightly narrower bands and larger spin\nsplittings. Accordingly, the magnetic moments reported\nin Table I are moderately enhanced in the DFT+ Uap-\nproach, when compared to the values obtained in GGA.\nIn general, the surface of a non-magnetic material can\nbecome magnetic because of a reduced number of neigh-\nbouring atoms, which leads to a narrower bandwidth and\nthe fulfillment of the Stoner criterion. In fact, calcula-\ntions for the non-magnetic phase reveal that the surface\nDOS shows high peaks at the Fermi level, due to the\naforementioned surface states. Thus, the Stoner crite-\nrion is fulfilled, and the spin-degeneracy of the bands\nis removed to lower the total energy (see SM [52] for\nmore details). In addition to the Stoner mechanism, this3\n−202DOS (eV−1)GGACoMnVAl\n−5.0−2.5 0.0 2.5\nEnergy (eV)−202DOS (eV−1)GGA+U\nCo-3d\nMn-3d\nV-3d\nAl-2pGGAFeMnTiP\n−5.0−2.5 0.0 2.5\nEnergy (eV)GGA+U\nFe-3d\nMn-3d\nTi-3d\nP-2pGGACoCrTiP\n−5.0−2.5 0.0 2.5\nEnergy (eV)GGA+U\nCo-3d\nCr-3d\nTi-3d\nP-2p\nFIG. 1. Projected density of states of the surface atoms (blue and red lines) and the subsurface atoms (green and pink lines)\nobtained with GGA (top panel) and GGA+ U(bottom panel). Fermi level is indicated by the dashed line.\nTABLE I. Calculated site-projected spin moments ( µB) for\nthe considered Heusler compounds of XX/primeYZ family. The\nGGA+U-derived results are shown in parentheses. The in-\ndices refer to the layer numbers from the surface: surface (1),\nsubsurface (2) and sub-subsurface (3).\nCoMnVAl CoCrTiP FeMnTiP\nX11.36 (1.48) 1.11 (1.50) 2.31 (2.52)\nX/prime\n13.37 (3.66) 3.02 (3.31) 3.14 (3.50)\nY2-0.51 (-1.03) -0.12 (-0.18) -0.31 (-0.41)\nX30.14 (0.16) 0.20 (0.14) 0.14 (0.08)\nX/prime\n30.11 (0.26) 0.01 (-0.08) 0.02 (0.06)\nclass of materials exhibit another peculiarity, which is the\nSlater-Pauling rule. The latter dictates the lack of mag-\nnetism in the bulk, as a consequence of having 24 valence\nelectrons in the unit cell. However, this constraint does\nnot hold any more at the surface, due to the dangling\nbonds caused by the reduced dimensionality. Therefore,\nit is almost natural for magnetism to arise. An analogous\neffect was previously reported for Heusler ferromagnets,\nwhose surfaces were shown to exhibit a strongly mod-\nified electronic structure, lacking the half-metallicity of\nthe bulk [53].\nThe previous analysis illustrates that magnetism and\nmetallicity are basically confined to the surface and sub-\nsurface layers. Both features are due to localized sur-\nface states of 3 dcharacter, which are expected to ex-hibit a substantial electron-electron interaction. There-\nfore, we refer to the observed situation as the formation\nof a (quasi) 2D electron liquid, to be distinguished from\nthe 2DEG observed at certain interfaces, as discussed in\nthe introduction.\nTo get a deeper insight into the origin of the magnetic\norder at the surface, we have calculated the inter-atomic\nexchange parameters Jij’s for all three systems. The cal-\nculated magnetic interactions are qualitatively similar in\nall cases, and therefore the following discussion will be fo-\ncused only on CoMnVAl. The structure of the considered\nslab as well as the calculated Jij’s are reported in Fig. 2.\nThe important couplings are those involving Co, Mn and\nV atoms that are closest to the surface. The interac-\ntions between other atoms were not computed, since the\nmagnetic moments are too small and of induced nature,\nwhich makes the application of the magnetic force the-\norem doubtful. The dominant interactions correspond\nto the nearest-neighbour (NN) Co-Mn, Co-V and Mn-\nV bonds. Their signs are perfectly consistent with the\nground state magnetic order, where surface and sub-\nsurface spins are antiparallel to each other. Taking the\nUterm into account does not change this picture, but re-\nsults into an enhancement of all couplings, except the one\nbetween Co and Mn. The largest variation is observed for\nthe interactions involving V atoms, and is partly induced\nby the large increase of the V moment (see Table I). The\nNN couplings between the atoms of the same element4\nzxyCoMnVAl\n\u00001.0\u00000.50.0Jij(meV)Co-Co2.50.0\u00002.5\u00005.0Jij(meV)Mn-Mn\n5101520Distance (bohr)\u00000.50.00.51.0Jij(meV)V-V051015Co-Mn\u00007.5\u00005.0\u00002.50.0Co-V\n5101520Distance (bohr)\u000020\u0000100Mn-V\nFIG. 2. Top panel: Structure of the CoMn-terminated slab of\nCoMnVAl. Bottom panel: Inter-atomic exchange parameters\nJij’s as a function of the distance between the atoms. The\nresults obtained with GGA (GGA+ U) are shown with red\nsquares (cyan circles).\n(e.g. Mn-Mn) are strongly bond-dependent. This is a\nmanifestation of the C2vsymmetry of the slab, which\nmakes the xandydirection nonequivalent. In practice,\nthis implies that the corresponding bonds have different\nenvironments and two of them have Al atoms positioned\nbelow and for the other two there are V atoms. These\nvastly different exchange paths affect and the sign and\nthe magnitude of the Jij. The antiferromagnetic NN\nCo-Co, Mn-Mn and V-V interactions are, in fact, frus-\ntrated. They are reduced in GGA+ Ucalculation, which\n0100200300ω(meV)\n(0,0)(0,1\n2) (1\n2,1\n2) (1\n2,0)(0,0) (1\n2,1\n2)FIG. 3. Adiabatic magnon spectrum of the CoMn-terminated\nCoMnVAl slab along the high-symmetry lines in the plane of\nthe surface. Solid (dashed) lines correspond to GGA(+ U)-\nderived data.\nenhances the overall stability of the ferrimagnetic order,\nas will be shown below.\nElectronic structure calculations suggest the formation\na spontaneous surface magnetization at zero tempera-\nture. To assess the stability of the magnetic state at finite\ntemperature, we performed MC simulations for the pa-\nrameterized Heisenberg model. Ordering temperatures\nwere identified from the specific heat. For CoMnVAl,\nTcis predicted to be in the range between 155 and 290\nK, as obtained in GGA and GGA+ Urespectively. Fur-\nther information can be obtained through the AMS. The\nspectrum of CoMnVAl, reported in Fig. 3, shows that the\npredicted reference state is indeed stable. Again we see\nthe manifestation of the C2vsymmetry, since the disper-\nsion at (1\n2,0) and (0,1\n2) points is clearly different. The\nsoftest magnons are found for the case of GGA calcula-\ntion along (1\n2,0) direction, which is primarily caused by\nthe frustrated NN Mn-Mn couplings. The overall sta-\nbility of the ferrimagnetic state is drastically increased\nin the GGA+ Ucalculations, as revealed by the larger\nspin-stiffness constant in the AMS. This result and the\naforementioned difference in Tcare both consistent with\nenhanced Jij’s and suppressed frustration, obtained in\nGGA+ Uwith respect to GGA. Due to localized nature\nof the surface-derived states, correlation effects beyond\nDFT are expected to be important. Thus, the actual\nmagnetic and electronic properties should be closer to\nthose obtained with GGA+ U.\nConclusion and outlook. We have demonstrated that\nXX/primeYZ Heusler materials, which are semiconducting in\nthe bulk, host a metallic and ferrimagnetic (quasi) 2D\nelectron liquid at their [001] XX/prime-terminated surface. The\nordering temperatures are expected to be within exper-\nimental range and maybe even close to room tempera-\nture, as predicted for CoMnVAl. Our predictions can\nbe verified through several surface-sensitive experimen-\ntal techniques. The magnetically ordered state appear-\ning at reduced temperatures can be investigated using x-\nray circular dichroism [54, 55] or spin-polarized scanning\ntunnelling microscopy/spectroscopy [56]. Metallicity at5\nthe surface can unambiguously be verified using ARPES\nwith varied photon energies[57]. Finally, surface magnon\nstates can potentially be measured with spin-polarized\nelectron energy loss spectroscopy [58] and then compared\nto our predicted spectrum.\nThe peculiar magnetic state predicted in this work has\na potential of being important for technological appli-\ncations. Having an intrinsic metallic magnetism at the\nsurface of a semiconducting interior implies that these\nmaterials may provide a practical realization of a mag-\nnetic tunnel junction or a spin-injector within a single\nentity. Overall, these systems seem very promising for\nthe growing field of spintronics.\nOur discovery requires further theoretical and experi-\nmental investigation. It is interesting to study the effect\nof antisite disorder, which might lead to the appearance\nof further exotic magnetic states. The spin-orbit cou-\npling has been neglected in the present analysis and its\ninclusion might lead to new interesting features of these\nmaterials, such as non-trivial topology, Dzyaloshinskii-Moriya interactions at the surface, and an enhanced mag-\nnetocrystalline anisotropy. Finally, we speculate that\nthe dynamical correlation effects, which are relevant for\nHeusler half-metals [30], might be even more interesting\nfor the surface-derived states discussed here.\nACKNOWLEDGMENTS\nL.C. gratefully acknowledges the financial support pro-\nvided by the Augsburg Center for Innovative Tech-\nnologies, and by the Deutsche Forschungsgemeinschaft\n(DFG, German Research Foundation) - Projektnummer\n107745057 - TRR 80/F6. O.E. acknowledges support\nfrom the Swedish Research Agency, the Knut and Alice\nWallenberg Foundation, the Foundation for Strategic Re-\nsearch, The Swedish Energy Agency and eSSENCE. The\ncomputer simulations are performed on computational\nresources provided by NSC allocated by the Swedish Na-\ntional Infrastructure for Computing (SNIC).\n[1] I. Tamm, Phys. Z. Sowjetunion 1, 733 (1932).\n[2] W. Shockley, Phys. Rev. 56, 317 (1939).\n[3] F. J. Himpsel and D. E. Eastman, Phys. Rev. Lett. 41,\n507 (1978).\n[4] P. O. Gartland and B. J. 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Lett.\n102, 177206 (2009).Supplemental material for Magnetic 2D electron liquid on the surface of Heusler\nsemiconductors\nS. Keshavarz1, I. Di Marco1,2, D. Thonig1, L. Chioncel3,4, O. Eriksson1,5, and Y. O. Kvashnin1\n1Uppsala University, Department of Physics and Astronomy,\nDivision of Materials Theory, Box 516, SE-751 20 Uppsala, Sweden\n2Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Korea\n3Theoretical Physics III, Center for Electronic Correlations and Magnetism,\nInstitute of Physics, University of Augsburg, D-86135 Augsburg, Germany\n4Augsburg Center for Innovative Technologies, University of Augsburg, D-86135 Augsburg, Germany and\n5School of Science and Technology, ¨Orebro University, SE-701 82 ¨Orebro, Sweden\nHere we show the projected density of states (DOS) in CoMnVAl, obtained in various computational setups. The\nresults of non-magnetic GGA calculations (left panel) reveal high density of states at the Fermi level ( EF). It primarily\noriginates from the localized Mn and Co 3 dorbitals located in the surface layer. Sub-surface states already shows a\nsubstantially reduced DOS at this energy and the bulk electronic structure properties are quickly recovered, as one\napproaches the innermost layer.\nThus, it is clear that the surface states show the tendency to an instability of the non-magnetic solution. The latter\nresults in the emergence of pronounced magnetic moments in the spin-polarized GGA calculations, as seen in the\nresults shown in the middle panel of Fig. S1. Strong exchange splitting of the Mn and Co states is apparent in the\nmagnetic calculation. Even though the splitting causes the reduction of the density of spin-up states near the EF, a\nsubstantial amount of spin-down states remains there.\nTaking into account Coulomb Uterm leads to a further enhancement of the spin polarization (see right panel of\nFig. S1) and consequently pushes the 3 dstates away from the EF. As a result, a combined effect of spin-polarization\nand strong local interactions, the DOS at the EFis reduced, but remains finite. The metallic bands are confined to\nthe first few layers closest to the surface of the material.\n\u00006\u00004\u00002024024Co-3dMn-3dV-3d\n\u00006\u00004\u00002024024\n\u00006\u00004\u000020240.02.55.0\n\u00006\u00004\u00002024024\n\u00006\u00004\u00002024Energy (eV)024\u00006\u00004\u00002024\u0000202\n\u00006\u00004\u00002024\u0000202\n\u00006\u00004\u00002024\u0000202\n\u00006\u00004\u00002024\u0000202\n\u00006\u00004\u00002024Energy (eV)\u0000202\u00006\u00004\u00002024\u00002.50.02.5\n\u00006\u00004\u00002024\u0000202\n\u00006\u00004\u00002024\u0000202\n\u00006\u00004\u00002024\u0000202\n\u00006\u00004\u00002024Energy (eV)\u0000202non-magnetic GGAspin-polarized GGAspin-polarized GGA+U\nInnermost layersSurface\nFIG. S1. Calculated layer-resolved density of states projected onto 3 dstates of transition metals in the slab of CoMnVAl.arXiv:1905.10541v1  [cond-mat.str-el]  25 May 2019"    },    {        "title": "1905.13219v2.Evolution_of_Magnetic_Order_from_the_Localized_to_the_Itinerant_Limit.pdf",        "content": "Evolution of Magnetic Order from the Localized to the Itinerant Limit\nD. G. Mazzone,1, 2,\u0003N. Gauthier,3, 4,yD. T. Maimone,3R. Yadav,3M. Bartkowiak,3J. L. Gavilano,1\nS. Raymond,5V. Pomjakushin,1N. Casati,6Z. Revay,7G. Lapertot,8R. Sibille,1and M. Kenzelmann1\n1Laboratory for Neutron Scattering and Imaging,\nPaul Scherrer Institut, 5232 Villigen PSI, Switzerland\n2National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA\n3Laboratory for Scienti\fc Developments and Novel Materials,\nPaul Scherrer Institut, 5232 Villigen PSI, Switzerland\n4Stanford Institute for Materials and Energy Sciences,\nSLAC National Accelerator Laboratory, Menlo Park, California 94025, USA.\n5Univ. Grenoble Alpes, CEA, IRIG, MEM, MDN, F-38000 Grenoble, France\n6Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland\n7Technische Universit at M unchen, Heinz Maier-Leibnitz Zentrum, 85747 Garching, Germany\n8Univ. Grenoble Alpes, CEA, IRIG, PHELIQS, IMAPEC, F-38000 Grenoble, France\n(Dated: August 28, 2019)\nQuantum materials that feature magnetic long-range order often reveal complex phase diagrams\nwhen localized electrons become mobile. In many materials magnetism is rapidly suppressed as elec-\ntronic charges dissolve into the conduction band. In materials where magnetism persists, it is unclear\nhow the magnetic properties are a\u000bected. Here we study the evolution of the magnetic structure in\nNd1\u0000xCexCoIn 5from the localized to the highly itinerant limit. We observe two magnetic ground\nstates inside a heavy-fermion phase that are detached from unconventional superconductivity. The\npresence of two di\u000berent magnetic phases provides evidence that increasing charge delocalization\na\u000bects the magnetic interactions via anisotropic band hybridization.\nCharge carriers in a periodic array of atoms are found\neither close to the nuclei or assume a delocalized char-\nacter where they move freely throughout the crystal.\nThese extreme cases are often well-described in the Mott,\nKondo or Fermi liquid theory framework. Materials in in-\ntermediate regimes with strong electronic \ructuations are\nmore di\u000ecult to describe and can stabilize novel quantum\nground states that keep fascinating the condensed mat-\nter physics community. Examples include the transition-\nmetal oxides, pnictides or the heavy-fermion metals [1{4].\nIn materials with partly delocalized electrons, orbital\nand spin degrees of freedom can trigger magnetic long-\nrange order. Magnetism is often suppressed by increasing\ncharge delocalization, and it is thought that the asso-\nciated \ructuations are essential for macroscopically co-\nherent phases such as unconventional superconductivity\n[1{3]. It is not clear how the magnetic interactions, and\nthe resulting magnetic structure, are modi\fed as the de-\ngree of itineracy changes in a material. Ab-initio calcu-\nlations are at present not accurate enough to deal with\nthe small energy di\u000berences that are relevant, particu-\n\u0003daniel.mazzone@psi.ch\nynicolas.gauthier4@gmail.comlarly in heavy-fermion metals. In addition, it is an open\nquestion how the magnetic interactions are a\u000bected in\nexperimental realizations where magnetic order persists\nover localized-to-itinerant charge transitions [5{7].\nKondo materials are ideal systems to shed light onto\nthis scienti\fc problem. The materials possess energy\nscales that are up to four orders of magnitude smaller\nthan in transition-metal oxides [8], and are, thus, highly\nsusceptible to external tuning parameters, such as mag-\nnetic \felds or chemical doping [9{11]. The low energy\nscale arises from partially-\flled electronic f-states that\nare partly screened by mobile charge carriers. This\nso-called Kondo coupling can lead to collective singlet\nstates, where localized f-electrons admix with the con-\nduction band and become mobile. The development of\ncoherentf-bands near the Fermi surface triggers renor-\nmalized e\u000bective masses below a coherence temperature\nTcohthat allows to e\u000bectively probe the degree of the\nsystem's itinerancy. In addition, heavy-fermion materi-\nals can feature localized magnetic order that is in direct\ncompetition with Kondo screening, providing an accessi-\nble and sensitive measure of the nature of magnetism.\nHere, we study the evolution of magnetic properties\nin Nd 1\u0000xCexCoIn 5. CeCoIn 5is a highly itinerant heavy-\nfermion material with a large quasiparticle mass enhance-arXiv:1905.13219v2  [cond-mat.str-el]  27 Aug 20192\nFIG. 1. (Color online) a)Neutron powder di\u000braction results\non NdCoIn 5revealing an Ising-like magnetic structure, as\nshown in b). Single crystal neutron di\u000braction intensity of\nNd0:17Ce0:83CoIn 5along (h,h, 0.5) in c)and (0.44, 0.44, l)\nind), given in reciprocal lattice units (r.l.u). e)Amplitude\nmodulated magnetic structure, such as observed for 0.75 \u0014x\n\u00140.95.\nment [12, 13]. In contrast, isotroctural compounds con-\nsisting of Nd ions are known to generate non-hybridized\nlocal magnetic moments [14{17]. Previous macroscopic\ntransport measurements have shown that chemical sub-\nstitution of Nd for Ce in Nd 1\u0000xCexCoIn 5allows to drive\nthe system from the highly itinerant to the completely lo-\ncalized limit [17]. The series displays zero-\feld magnetic\norder forx < 0.95 that competes with superconductiv-\nity forx\u00150.83, and heavy-electron bands are thought\nto exist for x > 0.5. Hitherto, magnetic order has been\nexplored in detail for x= 0.95 only, where a spin-density\nwave (SDW) is modulated with ~QIC= (q,\u0006q, 0.5) in re-\nciprocal lattice units (r.l.u) and q\u00190.45 [18, 19]. The or-\ndered magnetic moment, \u0016= 0.13(5)\u0016B, is aligned along\nthe tetragonal c-axis. As we will now show, the magnetic\nsymmetry is modi\fed within the heavy-fermion ground\nstate upon doping, providing evidence for an evolution of\nthe magnetic exchange couplings upon band hybridiza-\ntion.\nSingle crystalline Nd 1\u0000xCexCoIn 5withx= 0, 0.16,\n0.4, 0.61, 0.75, 0.83, 0.95 and 1 was grown in indium self-\n\rux [12]. The quality of experimental realization with x\n= 0, 0.16, 0.4, 0.61 and 1 was probed via X-ray powder\ndi\u000braction at the Material Science (MS-X04SA) beam-\nline of the Swiss Light Source at the Paul Scherrer In-stitut (PSI), Villigen, Switzerland using a photon wave-\nlength\u0015= 0.56491 \u0017A [20]. The actual Nd concentra-\ntion of these samples was determined via high-resolution\nneutron di\u000braction on HRPT at the Swiss Neutron Spal-\nlation Source (SINQ) at PSI with \u0015= 1.886 \u0017A. The Nd\nconcentration in single crystals with x= 0.95 was checked\nby means of in-beam neutron activation analysis at MLZ-\nGarching, Munich, Germany [21, 22]. The macroscopic\nproperties of members with x= 0, 0.16, 0.4, 0.61, 0.75,\n0.83 and 1 were investigated via four-probe electrical re-\nsistivity and AC/DC magnetization measurements in a\nQuantum Design PPMS or in cryogenic magnets. The\nmacroscopic and microscopic properties of x= 0.95 are\nreported in Ref. 19. The magnetic structure of x=\n0, 0.16, 0.4, 0.61 was determined via neutron powder\ndi\u000braction at HRPT and experimental realizations with\nx= 0.75 and 0.83 were investigated by means of sin-\ngle crystal neutron di\u000braction on Zebra at SINQ and on\nthe triple-axis spectrometer IN12 at the Institut Laue-\nLangevin, Grenoble France, respectively. While a neu-\ntron wave-length \u0015= 1.177 \u0017A was employed on Zebra,\n\u0015= 3.307 and 4.83 \u0017A were used on IN12. All magnetic\nstructures were re\fned with the FullProf suite [23]. Fur-\nther detailed information are given in the Supplemental\nMaterial.\nNeutron powder di\u000braction results on NdCoIn 5\nare shown in Fig. 1a and are representative for\nNd1\u0000xCexCoIn 5withx= 0, 0.16, 0.4 and 0.61 (see Sup-\nplemental Material). We observe magnetic Bragg peaks\nat low temperatures, consistent with two symmetry-\nequivalent commensurate propagation vectors ~QC=\n(1/2, 0, 1/2) and (0, 1/2, 1/2). They correspond to dif-\nferent domains that are indistinguishable in a powder\ndi\u000braction experiment. The evolution of the magnetic\nBragg peak intensity at higher scattering angles shows\nunambiguously that the magnetic moment is oriented\nalong thec-axis. We \fnd an Ising-like structure that\nis displayed in Fig. 1b for the (1/2, 0, 1/2)-domain.\nIn contrast, a ground state with incommensurate mag-\nnetic order is observed for x\u00150.75. The single crystal\nneutron di\u000braction results reveal a magnetic wave-vector\n~QIC= (q,\u0006q, 0.5) with q\u00190.44, for Nd 1\u0000xCexCoIn 5\nwith 0.75\u0014x\u00140.95 (see Fig. 1c and d for x= 0.83\nand Supplemental Material for x= 0.75). The propa-\ngation vector is similar to the one of Nd 0:05Ce0:95CoIn 5\n[18, 19], and the absence of intensity at ~QCexcludes a\nscenario where di\u000berent types of magnetic order coex-\nist. The magnetic moment remains aligned along the\ntetragonalc-axis, but features a sinusoidally modulated\nstructure (see Fig. 1e).\nThe doping dependence of the magnetic moment am-\nplitude is shown in Fig. 2a. In the commensurate\nphase, it monotonically decreases from \u0016p= 2.56(3)\u0016B\nfor NdCoIn 5to 0.90(5)\u0016Bfor a Ce concentration of 61%.\nThe modulation direction experiences a rotation in the\ntetragonal plane and propagates along ~QICforx\u00150.75.\nHere\u0016pis modi\fed weakly before it is strongly sup-\npressed for x > 0.83. The incommensuration, q(x), in-3\nFIG. 2. (Color online) a)Evolution of the magnetic moment\namplitude as a function of Ce content in Nd 1\u0000xCexCoIn 5(the\ndata point at x= 0.95 was taken from Ref. 19). Inset: Incom-\nmensuration ( q,q, 0.5) as a function of x.b)Temperature\ndependence of the magnetic Bragg peak intensity at (0.44,\n0.44, 0.5) in Nd 0:17Ce0:83CoIn 5.\ncreases with increasing Ce content (see Fig. 2a inset),\nwhich may be attributed to small changes in the Fermi\nsurface. The xT-phase diagram of Nd 1\u0000xCexCoIn 5is\nshown in Fig. 3. The series features persistent mag-\nnetism up to x= 0.95 that competes with superconduc-\ntivity atx\u00150.83, and a localized charge state for x <\n0.5. The key result of our study is that magnetic order is\nmodi\fed between x= 0.61 and 0.75, shifted with respect\nto the onset of coherent heavy bands and superconduc-\ntivity.\nThe superconducting phase at x > 0.8 is believed to\narise from magnetic \ructuations of a nearby SDW critical\npoint [24]. The Cooper-pairs in CeCoIn 5feature d x2\u0000y2-\nsymmetry that is robust under small Nd substitution\n[25, 26]. The pairing symmetry reveals nodes along the\nreciprocal (1, 1, 0)-direction, where low-energy quasipar-\nticles can mediate magnetic order without directly com-\nFIG. 3. (Color online) Antiferromagnetic (AFM) phase for\nx\u00140.95, superconductivity (SC) for x\u00150.83 and heavy-\nfermion (HF) properties for x\u00150.5. CM denotes the com-\nmensurate structure with ~QC= (1/2, 0, 1/2) and (0, 1/2,\n1/2). ICM is the incommensurate order along ~QIC= (q,\u0006q,\n0.5) withq\u00190.44. Left legend represent data from Ref. 17,\nthe right legend are our data (the phase boundaries of x=\n0.95 were taken from Ref. 19).\npeting with the condensate. In consequence, the d-wave\norder parameter is compatible with the incommensurate\nwave-vector ~QIC, but not with ~QC. This is supported by\nthe temperature dependence of the magnetic Bragg peak\nintensity in Nd 0:17Ce0:83CoIn 5that shows no anomaly\nas the temperature is tuned across the superconducting\nphase boundary (see Fig. 2b).\nThe interplay between superconductivity and mag-\nnetism observed here is very di\u000berent from the behavior\nin isostructural CeCo yRh1\u0000yIn5, where the magnetic mo-\nment orientation is altered at the superconducting phase\nboundary [27{29]. In Nd 1\u0000xCexCoIn 5magnetic order\nalong~QCis established between 0.61 < x < 0.75 where\nsuperconductivity is suppressed (see Fig. 3). In contrast,\nsimilarities with the xT-phase digram of the iron-based\nsuperconductor Fe 1+yTe1\u0000zSezare recognized [30]. The\nmaterial hosts two di\u000berent types of antiferromagnetic\ncorrelations, whose relative weight can be tuned via Se\nsubstitution. The dominant correlations at low Se con-\ncentrations are associated to weak charge carrier localiza-\ntion that triggers magnetic long-range order. In contrast,\nantiferromagnetic correlations with a di\u000berent modula-\ntion become important at larger Se concentrations and\nare thought to be closely related to emergence of su-\nperconductivity. Similarly, inelastic neutron scattering\nstudies on Nd 1\u0000xCexCoIn 5withx= 1 and 0.95 have\nshown that the magnetic \ructuations related to super-\nconductivity possess a symmetry distinct from the mag-\nnetic modulation in the localized limit [31, 32]. Thus,\nan intimate link between magnetic correlations along4\n~QICand d x2\u0000y2-wave superconductivity is expected in\nNd1\u0000xCexCoIn 5.\nFIG. 4. (Color online) a)Field dependent DC magnetization,\nM, of NdCoIn 5andb)real part of the AC magnetic suscep-\ntibility,\u001f0in Nd 1\u0000xCexCoIn 5withx= 0, 0.16, 0.4 and 0.61\nfor~Hjj[0 0 1] and T < T N. The two transitions correspond\nto a spin-\rip ( H1) and a ferromagnetic transition ( H2).c)\nField dependence of the two critical \felds, H1;2, normalized\nby the ordered moment, \u0016p. The solid lines are guides to the\neye.\nA change of magnetic correlations points towards a\ndoping dependent evolution of the magnetic interactions\nthat is related to the degree of electron mobility. This is\nobserved, for instance, in some layered transition-metal-\noxides, as they establish a charge- and spin-stripe or-\nder that allows to accommodate coexisting local antifer-\nromagnetic spin correlations and itinerant charge carri-\ners [33]. In La 2\u0000yBayCuO 4, for instance, the antiferro-\nmagnetic Mott-ground state is suppressed for y\u00150.05,\nbut the inclusion of mobile hole carriers a\u000bects the mag-\nnetic exchange couplings stabilizing a stripe order. This\nground state competes with superconductivity that is\nalso present for y\u00150.05, causing a profound reduction\nof the critical temperature around y\u00191/8. Similarly,\nthe magnetic interactions in Nd 1\u0000xCexCoIn 5may evolve\nwith increasing Ce concentration, and trigger a transition\nfrom localized ~QCto itinerant ~QIC.\nThe scenario is further clari\fed by a comparison with\nisostructural localized-moment magnets that also show a\ncommensurate structure along ~QC[14{16]. The materi-\nals feature magnetic order that can be described using an\nIsing-type Hamiltonian with antiferromagnetic nearest-,\nnext-nearest-neighbor and an inter-layer exchange cou-\npling [16]. Field dependent magnetization and suscepti-\nbility measurements inside the antiferromagnetic phase\nfor \felds along the tetragonal c-axis can reveal valu-\nable information about the exchange couplings. Theyshow two critical \felds, H1;2, and a plateau region with\nincreasing magnetic \feld. The ratio of H1;2and the\nordered moment, \u0016p, is directly related to the mag-\nnetic exchange couplings [16]. Similar measurements on\nNd1\u0000xCexCoIn 5withx= 0, 0.16, 0.4 and 0.6 are shown\nin Fig. 4a and b and the doping dependence of H1;2/\u0016p\nis displayed in Fig. 4c. These results reveal a signi\fcant\nchange of the magnetic interactions for x > 0.5, coin-\nciding with the onset of the heavy-fermion ground state\nwhere localized f-electrons become mobile via hybridiza-\ntion with the conduction band.\nThe underlying microscopic process of the heavy-\nfermion ground state is driven through collective screen-\ning of the conduction electrons, minimizing the total an-\ngular momentum of the local 4 f-moments. While this is\nfavorable for Ce-ions (Ce3+,J= 5/2!Ce4+,J= 0),\nNd3+remains in a stable J= 9/2 con\fguration. Such\na case can be described best using a phenomenological\ntwo-\ruid model, in which two coexisting contributions\nof eitherf-electrons that are hybridized at low tempera-\ntures or residual local moments are assumed [13, 34, 35].\nSince the Ce-4 fground state wave-function in this family\nof compounds is hybridized mainly with In- porbitals [36{\n38], it is conceivable that the antiferromagnetic nearest-,\nnext-nearest-neighbor and inter-layer exchange coupling\nare a\u000bected di\u000berently as band hybridization becomes\nstronger with increasing Ce concentration (see Fig. 1b).\nThis scenario can naturally explain a change in magnetic\nsymmetry inside the heavy-fermion ground state, as op-\nposed to a case where all interactions change on an equal\nfooting.\nIt is noted that the evolution of the hybridized 4 f-\nwave function in doped CeCoIn 5strongly depends on\nthe chemical element that is substituted. While Ce-\nsubstitution simply yields a decrease in the \ruid account-\ning for hybridized f-electrons, doping at the other chem-\nical sites have a more complex impact on the electronic\nground state [36{39]. It has been shown, for instance,\nthat substitution of the transition metal a\u000bects the shape\nof the Ce-4 fground state wave-function [38]. Upon in-\ncreasing Rh doping the 4 f-orbital is squeezed into the\ntetragonal basal plane, mainly decreasing the hybridiza-\ntion with the out-of-plane In- porbitals. This drives the\nsystem away from the superconducting ground state and\ninto di\u000berent magnetic phases [27{29]. Substitution on\nthe In-site can also a\u000bect the shape of the Ce-4 fground\nstate wave-function, but a recent study has shown that\nit remains unchanged under Cd substitution [39]. In this\ncompound magnetic order is thought to arise via a local\nnucleation process, without altering the global electronic\nstructure [40].\nA similar controversy has also arisen lately for\nmagnetic order in Nd 1\u0000xCexCoIn 5at smallx[41{\n44]. De Haas-van Alphen e\u000bect measurements on\nNd1\u0000xCexCoIn 5, predictions on the behavior of the static\nspin susceptibility upon (non-)magnetic substitution of\nCe in CeCoIn 5and the observation of magnetic order in\n5% Gd-doped CeCoIn 5suggest that magnetic order is5\ntriggered by an instability in the band structure [41, 42].\nIn contrast, some theoretical models argue that local\nmagnetic droplets play a decisive role [43, 44]. In this\nscenario magnetism is mediated inside a d-wave super-\nconducting background via strong magnetic \ructuations\ntriggered by the nearby SDW critical point. Since we\n\fnd that magnetic order along ~QICpersists down to x\n= 0.75 that is both, outside the superconducting dome\nand far from the SDW critical point, our results are in\nline with a non-local picture involving the Fermi surface\ntopology.\nIn summary, we report the evolution of magnetism in\nthe series Nd 1\u0000xCexCoIn 5from the localized ( x= 0) to\nthe highly itinerant limit ( x= 1). We observe two di\u000ber-\nent magnetic structures with moments along the tetrag-\nonalc-axis. Magnetic order is Ising-like with ~QC= (1/2,\n0, 1/2) and (0, 1/2, 1/2) for materials with x\u00140.61,\nand amplitude modulated with ~QIC= (q,\u0006q, 0.5) with\nq\u00190.44 forx\u00150.75. We \fnd that delocalization of Ce-\n4felectrons at x>0.5 a\u000bects the magnetic interactions,\nproviding evidence for anisotropic hybridization e\u000bects.\nIncreasing charge itinerancy leads to a magnetic transi-\ntion between x= 0.61 and 0.75. This occurs far away\nfrom the emergence of unconventional superconductivity\nand is thus unrelated. Our results demonstrate that the\nmagnetic interactions strongly depend on the degree of\n4f-electron hybridization with the conduction electrons.\nThis emphasizes the need to include hybridization depen-\ndent magnetic interaction in theories describing quantum\nmaterials close to the Kondo breakdown, and o\u000bers new\nperspectives for the interpretation of the physical prop-\nerties in heavy-fermion materials.\nWe thank the Paul Scherrer Institut, the Institut\nLaue-Langevin and the Forschungsneutronenquelle Heinz\nMaier-Leibnitz for the allocated beam time. We acknowl-\nedge Mark Dean, Maxim Dzero and Priscila Rosa for\nfruitful discussions, and Maik Locher for his help with\nthe DC magnetization measurements. We thank the\nSwiss National Foundation (grant No. 200021 147071,\n200021 138018 and 200021 157009 and Fellowship No.\nP2EZP2 175092 and P2EZP2 178542) for \fnancial sup-\nport. This work used resources of the National Syn-\nchrotron Light Source II, a U.S. Department of Energy\n(DOE) O\u000ece of Science User Facility operated for the\nDOE O\u000ece of Science by Brookhaven National Labora-\ntory under Contract No. de-sc0012704.\nREFERENCES\n[1] N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois,\nJ. Bonnemaison, R. Liang, D. A. Bonn, W. N. Hardy, L.\nTaillefer. Quantum oscillations and the Fermi surface in\nan underdoped high-T csuperconductor. Nat. Phys 447,\n565 (2007).\n[2] A. A. Kordyuk. Iron based superconductors: Magnetism,\nsuperconductivity, and electronic structure Low Temp.\nPhys 38, 888 (2012).[3] N. D. Mathur, F. M. Grosche, S. R. Julian, I. R.\nWalker, D. M. Freye, R. K. W. Haselwimmer, G. G.\nLonzarich. 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Raymond5, V . Pomjakushin1, N. Casati6, Z. Revay7, G. Lapertot8, R. Sibille1, M. Kenzelmann1. Correspondence to: daniel.mazzone@psi.ch or nicolas.gauthier4@gmail.com. This PDF file includes: Materials and Methods Supplementary Results Figs. S1 to S3 \n! 18\nMaterials and Methods Sample synthesis Single crystalline Nd1-xCexCoIn5 was grown in an In self-flux with 3% Nd1-xCex, 3% Co and 94% In [1]. The high purity starting elements were placed in alumina crucibles and heated to 1150 ◦C using evacuated and sealed quartz tubes. After a 30 minutes motorized rotation of the ampoule, the liquid was cooled to Tix with 100 ◦C/h. The subsequent cooling was performed with a rate of 3 ◦C/h and interrupted at Tfx, where the sample was quenched to room temperature. The excess of indium flux was separated from the single crystals by means of centrifugation and the remaining In flux was dissolved with HCl acid. The sequence was found to be optimized with  Ti,fx = Ti,f0 - (Ti,f0-Ti,f1)x where Ti0 = 800 ◦C, Tf0 = 550 C, Ti1 = 750 ◦C  and Tf1 = 450 ◦C. Material characterization and neutron powder diffraction The sample quality of Nd1-xCexCoIn5 with x = 0, 0.16, 0.4, 0.61, and 1 was investigated by X-ray powder diffraction at the Material Science (MS-X04SA) beamline of the Swiss Light Source at the Paul Scherrer Institut (PSI), Villigen Switzerland. Ground single crystals were filled into quartz capillaries with diameter d = 0.1 mm and exposed to a photon wave-length λ = 0.56491 Å [2]. Diffracted photons were detected by a Mythen-II detector. The actual Nd concentration and magnetic structure of these samples were determined via high-resolution neutron powder diffraction on HRPT at the Swiss Neutron Spallation Source (SINQ) at PSI. Ground single crystals of about m = 3 g were filled in vanadium double-walled cans with inner and outer diameter d = 8 and 9.2 mm, respectively. The samples were inserted in an orange cryostat and exposed to a neutron wave-length λ = 1.886 Å at T = 1.8 and 15 K. The actual Nd concentration for x = 0.95 was determined using in-beam neutron activation analysis in MLZ Garching, Munich, Germany [3,4].  Macroscopic measurements The macroscopic properties of members with x = 0, 0.16, 0.4, 0.61, 0.75, 0.83, and 1 were investigated via electrical resistivity and AC/DC magnetization measurements in a Quantum Design PPMS or in cryogenic magnets. Electrical resistivity was measured via a four-probe setup. DC magnetization was measured on powder samples using a field of µ0H = 100 mT. The Kondo coherence temperature Tcoh and superconducting critical temperature Tc were defined via the center of the broad coherence peak and at 50% loss of the electrical resistivity, respectively. The Néel temperature TN was determined via DC magnetization at the maximal gradient of M/H*T.  Additional field dependent DC magnetization and AC magnetic susceptibility were performed on single crystals of x = 0, 0.16, 0.4 and 0.61 at T ≈ 2 K for H||[0 0 1]. The transition fields H1 and H2 were obtained from the peak position in AC magnetic susceptibility measurements with frequency f = 1000 Hz and amplitude A = 1.7 mT. ! 29\nSingle crystal neutron diffraction Single crystal neutron diffraction was carried out on Zebra at SINQ for x = 0.75 at T = 0.05 and 2 K and on the triple-axis spectrometer IN12 at the Institut Laue-Langevin, Grenoble France for x = 0.83 at temperatures between 0.4 and 1.8 K. A Nd0.25Ce0.75CoIn5 single crystal of m = 14 mg was oriented in the scattering plane perpendicular to [0 1 0]. The lifting-arm detector of Zebra allowed to investigate QIC and QC in a single experiment. The sample was placed in a Variox cryostat with dilution insert and a neutron wave-length λ = 1.177 Å was used. The  Nd0.17Ce0.83CoIn5  single crystal consisted of m = 20 mg and was placed in an orange cryostat with 3He insert. The sample was oriented perpendicular to [1 -1 0]. Clean wave-lengths of λ = 3.307 and 4.83 Å were derived from a velocity selector and a double-focussing pyrolytic graphite monochromator. The neutrons were collimated with 80' in front of the sample and the experiment was performed in a W-configuration.  Magnetic structure determination The diffraction results were analyzed with the FullProf suite [5]. Three irreducible representations with magnetic moment orientations along the tetragonal major axes were found for QC in P4/mmm. The best refinement for NdCoIn5 was obtained for a moment along the c-axis with Rf = 15% and Rf > 35% for other representations. Similar results were found for x = 0.16, 0.4 and 0.61. The magnetic refinement on Nd0.17Ce0.83CoIn5 was performed on seven independent magnetic Bragg peaks and provides evidence for an amplitude modulated structure with a magnetic moment oriented along the c-axis (Rf = 4%). Other irreducible representations predict magnetic moments in the tetragonal plane, for which Rf > 30% was obtained. An equal domain population was assumed in all refinements and Ce3+/Nd+3 magnetic form factors were used in concentrations respecting their content in the samples. The discrepancy in the Rf values between powder and single-crystal refinements arise mainly from technical challenges. Neutron powder diffraction on samples in vanadium double-walled cans substantially decreases the signal to noise ratio when compared to single crystal diffraction experiments on a triple-axis instrument. The neutron powder diffraction results were refined using three contributing phases, accounting for structural and magnetic Bragg peaks of the sample and structural Bragg peaks of the can. In contrast, structural and magnetic refinements were performed independently in the single crystal diffraction experiment. Finally it is noted that the main contribution to the magnetic powder refinements arise from the (0.5, 0, 0.5),  (0.5, 1, 0.5) and (0.5, 1, 1.5) Bragg peaks, as opposed to the single crystal refinement of Nd0.17Ce0.83CoIn5 where seven magnetic Bragg peaks were used. The microscopic properties of Nd0.25Ce0.75CoIn5 were investigated around the (1.44, 0.44, 0.5), (1.56, 0.56, 0.5), (0.44, 0.44, 1.5),  (1.5, 0, 0.5) and (1.5, 0, 1.5) reciprocal lattice positions. The relative magnetic Bragg peak intensities along QIC (and absence thereof for QC) provide evidence for a c-axis oriented amplitude modulated structure, without signatures of a QIC-QC phase coexistence. No magnetic moment refinement was carried out for this compound, because the magnetic Bragg peaks along QIC were investigated solely via reciprocal (h, h - 1, 0.5) and (h, h, ! 310\n1.5) scans. The absolute magnetic moment amplitude for the other compositions was determined via precise measurements of the integrated Bragg peak intensity, which require additional θ-scans (sample rotation around the axis perpendicular to the scattering plane). Supplementary Results Sample characterization The sample quality of Nd1-xCexCoIn5 with nominal concentrations xnom = 0, 0.2, 0.4, 0.6, and 1 was investigated using high-resolution X-ray powder diffraction. The structural analysis provides evidence that all compositions were synthesized in a clean form with residual impurity contributions smaller than 2%. The linear shift of the tetragonal lattice parameters as a function of Ce substitution is consistent with previous reports [6]  and shown in Fig. S1a.  This provides evidence that the Ce substitution is uniform in the series and that no supplementary inter-grown phases are stabilized during the synthesis. No differences in the diffraction patterns among batches with identical nominal Ce concentration were observed (see Fig. S1a). Figure S1b shows the experimentally determined Ce concentration, xobs, in Nd1-xCexCoIn5. The neutron powder diffraction patterns of xnom = 0.2, 0.4 and 0.6 were refined against the Ce concentration and reveal xobs = 0.16(2), 0.40(2) and 0.61(2). An uncertainty concentration of about 1% is found among different single crystals with xnom = 0.95 using in-beam neutron activation analysis. Macroscopic measurements The electrical resistivity and DC magnetization measurements on Nd1-xCexCoIn5 with x = 0, 0.16, 0.4, 0.61, 0.75, 0.83, and 1 are shown in Fig. S2. They reveal Kondo coherence temperatures Tcoh = 46, 30 and 26 K for x = 1, 0.83 and 0.75, superconducting critical temperatures Tc = 2.3 and 1.17 K for x =  1 and 0.83 and Néel temperatures TN = 8.2, 6.3, 4.4 and 2.3 K for  x = 0, 0.16, 0.4 and 0.61, respectively. No superconducting phase has been observed in Nd0.25Ce0.75CoIn5 down to T = 50 mK. Neutron Diffraction Figure S3 shows neutron powder diffraction results on Nd1-xCexCoIn5 for x = 0.16, 0.4 and 0.61 measured at  T = 15 and 1.8 K below the Néel temperature. The low temperature data establish magnetic Bragg peaks that are consistent with the two arms QC = (1/2, 0, 1/2) and (0, 1/2, 1/2). Single crystal neutron diffraction at  T = 2 and 0.05 K on Nd0.25Ce0.75CoIn5 features magnetic order at QIC = (q, q, 0.5) with q  ≈ 0.44. References [1] C. Petrovic, P. G. Pagliuso, M. F. Hundley, R. Movshovich, J. L. Sarrao, J. D. Thompson, Z. Fisk, and P. Monthoux. Heavy-fermion superconductivity in CeCoIn5 at 2.3 K. J. Phys.: Condens. Matter. 13, L337 (2001) ! 411\n[2] P. Willmott, D. Meister, S. J. Leake, M. Lange, A. Bergamaschi, M. Böge, M. Calvi, C. Cancellieri, N. Casati, A. Cervellino, Q. Chen, C. David, U. Flechsig, F. Gozzo, B. Henrich, S. Jäggi-Spielmann, B. Jakob, I. Kalichava, P. Karvinen, J. Krempasky, A. Lüdeke, R. Lüscher, S. Maag, C. Quitmann, M. L. Reinle-Schmitt, Schmidt, B. Schmitt, A. Streun, I. Vartiainen, M. Vitins, X. Wang, and R. Wullschleger. The Materials Science beamline upgrade at the Swiss Light Source. J. Synchrotron Radiat. 10, 667 (2013) [3] Zs. Revay, R. Kudejova, and K. Kleszcz. In-beam activation analysis facility at MLZ, Garching. Nucl. Instrum. Meth. A. 799, 114 (2015) [4] Zs. Revay. Determining elemental composition using prompt gamma activation analysis. Anal. Chem. 81, 6851 (2009) [5] J. Rodriguez-Caravajal. Recent advances in magnetic structure determination by neutron powder diffraction. Physica B 192, 55 (1993) [6] R. Hu, Y . Lee, J. Hudis, V . F. Mitrovic, and C Petrovic. Composition and field-tuned magnetism and superconductivity in Nd1-xCexCoIn5. Phys, Rev. B. 77, 165129 (2008)\n! 512\n ! 6\nFig. S1. a) Linear shift of the lattice parameters a and c for Nd1-xCexCoIn5 with nominal Ce concentrations xnom = 0, 0.2, 0.4, 0.6 and 1.  b) Observed Ce content, xobs, against the nominal concentration xnom = 0, 0.2, 0.4, 0.6, 0.95 and 1.\nFig. S2: Electrical resistivity in a) and DC magnetization M/H in b) on Nd1-xCexCoIn5 for x = 1, 0.83 and 0.75 and x = 0, 0.16, 0.4 and 0.61, respectively. The Kondo coherence temperature Tcoh and superconducting critical temperature Tc were defined via the center of the broad coherence peak and at 50% loss of the electrical resistivity. The Néel temperature TN was determined at the maximal gradient of M/H*T.0.111010000.40.81.2\nT[K]ρ[a. u.]CeCoIn5Nd0.17Ce0.83CoIn5Nd0.25Ce0.75CoIn5\na)1100.91.11.3\nT[K]M/H[a.u]NdCoIn5Nd0.84Ce0.16CoIn5Nd0.39Ce0.61CoIn5Nd0.6Ce0.4CoIn5\nb)13\n! 7Fig. S3: Neutron powder diffraction at T = 15 and 1.8 K on Nd1-xCexCoIn5 for x = 0.16, 0.4 and 0.61. Single crystal neutron diffraction on x = 0.75 was measured at T = 2 and 0.05 K along (h, h - 1, 0.5) in reciprocal lattice units. The dashed line represents a double Gaussian fit to the experimental data.\n"    },    {        "title": "1907.01817v2.Spin_wave_excitations_of_magnetic_metalorganic_materials.pdf",        "content": "arXiv:1907.01817v2  [cond-mat.mtrl-sci]  19 Dec 2019Spin wave excitations of magnetic metalorganic materials\nJohan Hellsvik,1,∗Roberto Díaz Pérez,1R. Matthias Geilhufe,1Martin Månsson,2and Alexander V. Balatsky1\n1Nordita, KTH Royal Institute of Technology and Stockholm Un iversity,\nRoslagstullsbacken 23, SE-106 91 Stockholm, Sweden\n2Department of Applied Physics, KTH Royal Institute of Techn ology, Electrum 229, SE-164 40 Kista, Sweden\n(Dated: December 20, 2019)\nThe Organic Materials Database (OMDB) is an open database ho sting about 22,000 electronic\nband structures, density of states and other properties for stable and previously synthesized 3-\ndimensional organic crystals. The web interface of the OMDB offers various search tools for the\nidentification of novel functional materials such as band st ructure pattern matching and density\nof states similarity search. In this work the OMDB is extende d to include magnetic excitation\nproperties. For inelastic neutron scattering we focus on th e dynamical structure factor S(q,ω)\nwhich contains information on the excitation modes of the ma terial. We introduce a new dataset\ncontaining atomic magnetic moments and Heisenberg exchang e parameters for which we calculate\nthe spin wave spectra and dynamic structure factor with line ar spin wave theory and atomistic spin\ndynamics. We thus develop the materials informatics tools t o identify novel functional organic and\nmetalorganic magnets.\nI. INTRODUCTION\nMagnetism and magnetically ordered materials have\nplayed a crucial role in the development of the technol-\nogy used in our every-day life. The identification of novel\nmaterials with desired magnetic target properties as well\nas the investigation of coupling mechanisms, the result-\ning order and its excitations are therefore of great impor-\ntance. While basic concepts are usually explored in ma-\nterials with feasible complexity, materials with complex\nunit cells and dominant interaction effects quite often ex-\nhibit the more desirable properties with respect to tech-\nnological applications. In particular metal organic frame -\nworks and organic molecular crystals exhibit promising\nstructures for electron-spin-based devices. Magnetic or-\nganics have attracted attention with respect to spintronic\ndevices [1–3] and magnon spintronics [4], multiferroic\nphases [5, 6], molecular qubits [7, 8], and spin-liquid\nphysics [9–11]. Local magnetic moments in organic mate-\nrials can arise due to transition metal and rare earth ions\nembedded in the molecules or due to local unsaturized\nbonds as they occur in stable organic radicals [12–14].\nSome organics establish stable magnetic order of vari-\nous kind up to room temperature [15–17]. For device\nengineering and technological applications, an important\nadvantage of organic materials over other functional ma-\nterials such as transition metal oxides, is that organics\ncan be synthesized with relative low cost and large-scale\nproduction methods.\nThe vast increase of experimental and theoretical data\nobtained over the past century has opened a novel ap-\nproach to materials research based on computer science\nmethods and the construction of materials databases [18–\n25]. Such databases were successfully applied in min-\ning for functional materials [21, 26–32] and as training\n∗hellsvik@kth.sedata for machine learning algorithms predicting com-\nplex materials properties [33–36], bypassing computa-\ntionally demanding ab initio calculations. In this article\nwe report about the implementation of a novel dataset\nfor organic magnets into the organic materials database\nOMDB [19, 37], available at https://omdb.mathub.io .\nThe data was calculated by means of ab initio calcu-\nlations and comprises information about local magnetic\nmoments, magnetic exchange coupling, expected mag-\nnetic ground state, as well as spin-wave excitation spectra\nand the dynamical structure factor S(q,ω)for hundreds\nof previously synthesized organic molecular crystals and\nmetal organic frameworks. This data is embedded into\nthe existing framework of the OMDB and can be ex-\nplored using interactive statistics and non-trivial searc h\ntools such as pattern matching [38, 39].\nThe implementation of such a database is closely re-\nlated to the recent enhancement of neutron flux [40–42]\nand detector technology, allowing for inelastic neutron\nscattering experiments to be performed with higher sig-\nnal to noise ratio, and higher energy resolution [43, 44].\nFor example, for inelastic neutron scattering experi-\nments, the central entity is the dynamical structure fac-\ntorS(q,ω), which contains information on the excitation\nmodes of the material. The vast majority of inelastic\nneutron scattering (INS) measurements of magnetic ma-\nterials are analyzed by means of fitting the experimental\ndynamical structure factor S(q,ω)to the dispersion and\nstructure factor of a spin wave Hamiltonian as provided\nin our dataset.\nThe remaining of the article is organized as follows.\nIn Section II we present the models and approximations\nused, as well as the scheme developed for high throughput\ncalculation of magnetic properties and excitation spectra .\nResults are presented in Sec. III, followed by a conclusion\nin Sec. IV and outlook in Sec. V.2\nCrystal structures from di\u0001raction\nCrystallographic Information Files (CIF)\nDensity Functional Theory (DFT)\nground state calculations\nExtraction of magnetic moments mi\nCalculation of exchange interactions J ij\nDetermination of magnetic ground\nstate and phase diagram\nCalculation of magnon dispersion \u0002(q)\nand dynamic structure factor S( q,\u0000)INPUT\nTARGETM\nL\nFigure 1. The workflow for high throughput calculation of\nmagnetic ground states and spin wave spectra. The blue\nboxes indicate the steps we use for the results presented in\nthis article. The black arrows on the left hand side indi-\ncate steps which can be performed with machine learning\n(ML) techniques, to replace computationally expensive DFT\nand ASD simulations, ultimately enabling for prediction of\nmagnon spectra (green TARGET box) directly from the crys-\ntal structure (red INPUT box).\nII. MODELS AND METHODS\nA state-of-the-art approach for ab initio investigations\nof spin dynamics in magnetic solids is to solve the full\nelectronic dynamics with time-dependent density func-\ntional theory (TD-DFT) and calculating the energies and\nlifetimes of the spin waves without resorting to the adia-\nbatic approximation [45, 46]. However, both the real time\nand the linear response variants of TD-DFT are compu-\ntationally demanding and currently not feasible for or-\nganic materials with large chemical unit cells containing\nin average ≈80atoms for the materials stored within the\nOMDB. The calculation of the coupling parameters for\nmagnetic Hamiltonians is nowadays supported by vari-\nous DFT softwares, such as RSPt [47], SPR-KKR [48],\nKKRnano [49], and HUTSEPOT [50, 51]. This opens the\npath for a more tractable approach in context of the high-\nthroughput computations performed by us, allowing for\nanab initio modeling of spin wave excitation spectra in\nform of a two-step approach: first, the coupling parame-\nters of an effective magnetic Hamiltonian are calculated\nfrom DFT calculations; second, analytical or numerical\nspin wave theory calculations are performed to obtain the\nspin wave spectra. Furthermore, thermodynamic prop-\nerties of the spin Hamiltonian can be investigated with\nMonte Carlo and atomistic spin dynamics (ASD) simu-\nlations [52].\nThe description of our methodology is contained in a\nnumber of sections. The magnetic model Hamiltonian\nand details about the density functional theory calcula-tions go into Sec. II A, followed by a brief description of\nlinear spin wave theory for collinear magnets in Sec. II B,\nand the atomistic spin dynamics method in Sec. II C. The\nhigh throughput workflow is described in Section II D and\nis shown schematically in Fig. 1. The pattern matching\nmethod for magnon dispersions is described in II E.\nA. Magnetic Hamiltonian\nFor a magnetic solid with NAatoms in the crystallo-\ngraphic unit cell and a total of Nccells, we use the nota-\ntionRiµ=Ri+rµto specify atomic positions, where rµ\nis the position of the basis atom µin the unit cell, and\nRiis the position of the crystallographic unit cell i. The\nmagnetic model is a Heisenberg Hamiltonian\nH=−1\n21\nNcNANcNA/summationdisplay\niµNcNA/summationdisplay\njνJiµjνˆeiµ·ˆejν, (1)\nformulated in terms of unit vectors ˆeiνfor the directions\nof the local magnetic moment at site iµ. Note the sign\nconvention with a leading minus sign for the first term,\nand that the double summation/summationtextNcNA\niµ/summationtextNcNA\njνcount\neach bond twice. The magnetic moment of an atom is\nµiµ=µBmµˆeiµ, wheremµis the size of the magnetic\nmoment of atomic type µin terms of multiples of Bohr\nmagnetons µB.\nIn order to formulate spin Hamiltonians for magnetic\norganic materials we have used the full-potential lin-\near muffin-tin method (FP-LMTO) software RSPt [47]\nto calculate Heisenberg exchange parameters. The full-\npotential basis set allows for accurate electronic calcula -\ntions irrespective of the geometry of the crystal structure .\nThe latter is important for the sparse unit cells typically\npresent in organics. The accuracy and reproducibility of\nDFT results when using different basis sets have recently\nbeen investigated in Ref. [53]. The implementation of\nFP-LMTO in RSPt is described in detail in Ref. [47].\nFurthermore, in RSPt Heisenberg exchange interactions\ncan be calculated from Green functions for a reference\nspin structure, e.g.a ferromagnetic or collinear anti-\nferromagnetic ordering, by means of the Liechtenstein-\nKatsnelson-Antropov-Gubanov (LKAG) formalism [54],\nwithout the need to set up supercells with different spin\nconfigurations. Details on the implementation of the\nLKAG formula in RSPt can be found in Ref. [55].\nThe calculations were made with the local density ap-\nproximation (LDA) for the exchange-correlation poten-\ntial and using a 6 6 6 k-point mesh. The charge density\nand the potential inside the muffin-tin spheres were rep-\nresented using an angular momentum decomposition up\ntolmax= 8. One energy set was used for the valence\nelectrons. For the description of the states in the inter-\nstitial region three kinetic energy tails were used: -0.3,\n-2.3, and -1.5 Ryd.3\nB. Adiabatic magnon theory\nThe excitation spectra of a spin Hamiltonian can be\nobtained by various analytical and semi-analytical meth-\nods. The traditional approach relies on the introduction\nof Holstein-Primakoff (HP) operators for the bosonic op-\nerators, and expanding the spin Hamiltonian to quadratic\nor higher order in these HP operators. The expansion to\nquadratic order leads to linear spin wave theory which\nhas established itself as the main formalism to model\ninelastic neutron scattering (INS) data of magnetic ma-\nterials. This technique and aspects of how it can be im-\nplemented in software is described for the general non-\ncollinear case of multisublattice magnets in Refs. [56–\n58]. In the following we briefly review the main steps\nof the so called adiabatic magnon theory, a formalism\nthat naturally connects to the frozen magnon technique\nfor obtaining spin wave energies from ab initio calcula-\ntions. More details can be found in Ref. [59]. The spatial\nFourier transform of the exchange interaction is defined\naccording to\nJµν(q) =Nc/summationdisplay\nj/negationslash=0J0µjνeiq·R0µjν, (2)\nwhereR0µjν=R0+rµ−Rj−rν, andi= 0is the central\nunit cell. After linearizing the equation of motion for the\nHeisenberg Hamiltonian in Eq. (1) for small cone angle\nexcitation relative to the magnetization quantization di-\nrection, here chosen to be along ˆz, we can introduce the\nquantity\n˜Jµν(q) =−ez\nν\nmµJµν(q)+δµν1\nmµNA/summationdisplay\nλez\nλJµλ(0)(3)\nin which ez\nµ={1,−1}specify the collinear\n(anti)ferromagnetic groundstate of the system. The spin-\nwave dispersion as a function of wave vector qcan then\nbe obtained by diagonalizing ˜Jµν(q).\nC. Atomistic spin dynamics\nIn common with spin wave theory a core element in\nthe atomistic spin dynamics method [52, 60] is that it is\npossible to parametrize the energies and dynamics of the\nmagnetic system to a magnetic Hamiltonian formulated\nin terms of local spins (or magnetic moments) as in Eqn.\n(1). The other core element is the stochastic Landau-\nLifshitz-Gilbert equation (SLLG),\ndmi\ndt=−γ\n(1+α2)mi×[Bi+Bfl\ni] (4)\n−γ\n(1+α2)α\nmimi×{mi×[Bi+Bfl\ni]},\nwhich is a semi-classical equation used to model the mo-\ntion of the atomic magnetic moments at zero or finite\nconventional magnetic\ncell (4 atoms)\nprimitive cell\n(1 atom)primitive magnetic\ncell (2 atoms)\nFigure 2. Reduction to the primitive magnetic cell for the\ncollinear antiferromagnetic material C 20H10CoN12(OMDB-\nID 11913). The primitive chemical cell has only one magnetic\nsite. In the ground state search with ASD simulation a con-\nventional magnetic cell with four magnetic sites is found, a\ncell which can be reduced to a primitive magnetic cell with\ntwo magnetic sites.\ntemperature. The first term of the equation is the pre-\ncessional motion, while the second term describes the\ndamping motion. mi=mi(t)is the magnetic moment\nand experiences an effective magnetic field Bi=Bi(t),\ncalculated from the Hamiltonian as Bi=−∂H\n∂mi.γis\nthe gyromagnetic ratio, αis a scalar (isotropic) Gilbert\ndamping constant. Temperature is included in the form\nof Langevin dynamics with the power of the stochastic\nmagnetic field Bfl\ni=Bfl\ni(t)related to the damping con-\nstantαvia a fluctuation-dissipation relation [52]. For the\nASD simulations we use the UppASD software [61].\nIn numerical simulations of the stochastic Landau-\nLifshitz-Gilbert equation there is no need for a lineariza-\ntion of the equations of motion wherefore the full dy-\nnamics of a magnetic system described by a bilinear-in-\nspin-operators Hamiltonian such as Eqn. (1) is retained\n[52, 62]. Furthermore, also the full dynamics of a mag-\nnetic Hamiltonian which contains higher order coupling\nsuch as biquadratic exchange can be studied without the\nneed for mean-field treatment of these couplings, see e.g.\nRef. [63]. Spatial and temporal fluctuations are sampled\nusing a time-dependent correlation function\nCαβ(r,t) =1\nN/summationdisplay\ni,jwhere\nri−rj=r/an}bracketle{tmα\ni(t)mβ\nj(0)/an}bracketri}ht, (5)\nwhereαandβare the Cartesian components of the mag-\nnetic moments. Fourier transforming C(r,t)over space4\nand time yields the dynamic structure factor\nSαβ(q,ω) =1√\n2πN/summationdisplay\nreiq·r/integraldisplay∞\n−∞eiωtCαβ(r,t)dt,(6)\nwhich can be related to the scattering intensity measured\nin inelastic neutron or electron scattering on magnetic\nmaterials. Only the magnetic excitations perpendicular\nto the scattering wave vector contribute to the scattering\nintensity [58, 64]\nS(q,ω) =/summationdisplay\nα,β/parenleftbigg\nδαβ−qαqβ\nq2/parenrightbigg\nSαβ(q,ω). (7)\nFor the measurement of the dynamic structure factor\nASD simulations were run at T= 1K using a time step\ndt= 5·10−16s and a small Gilbert damping α= 0.0001.\nThe correlation function in Eq. (5) was measured using a\nsampling step of tsamp= 5·10−15s, over a sampling win-\ndow oftwin= 5·10−11s. The corresponding frequency\nrange for the dynamic structure factors in Eqs. (6) and\n(7) isω/(2π) = [0.02,0.04,...,200]THz (0.0827 meV to\n827 meV). In the figures contained in Section III, the\nscattering intensity is shown relative to the correlation\nS0(q,ω= 0)for the fully ordered system.\nD. High throughput calculation\nIn this section is described the methodology that we\nhave developed for high throughput calculation of mag-\nnetic ground states and spin wave spectra. The main\nsteps are shown in Figure 1.\nInput files for the RSPt ground state calculations\nare prepared from the crystallographic information files\n(CIF) using the cif2cell program [65]. The calculations\nare performed for a ferromagnetic ground state config-\nuration, without any a priori consideration of what the\nmagnetic ground state could be. In order to classify the\natoms as magnetic or non-magnetic, a simple criteria is\nused, namely that the spin-polarized charge density inte-\ngrated over the muffin-sphere of the atomic sites is larger\nthan 0.1 µB. For these atomic sites Heisenberg exchange\ninteractions are calculated using the approach described\nin Sec. II A.\nThe linear spin wave calculation require as input not\nonly the parameters of the magnetic Hamiltonian, but\nalso ground state spin configuration. In order to search\nfor the ground state we use an atomistic spin-dynamics\nquenching scheme for simulation cells with edge length\nL, corresponding to a number Natom=NAL3of sites,\nwhereNAis the number of magnetic atoms in the prim-\nitive chemical cell. Starting from having all magnetic\nmoments initially in a ferromagnetic configuration, the\nsystem evolves in Langevin dynamics simulation at finite\ntemperature. This is followed by a sequence of simula-\ntions at zero temperature but with finite Gilbert damping\nα. For zero or static external magnetic field, the deter-\nministic Landau-Lifshitz equation has the property thatenergy is a non-increasing function of time, and conse-\nquently that the energy of the spin Hamiltonian will be\nminimized [52]. For systems with competing magnetic\nphases at low temperatures, there is a finite probabil-\nity that the system will get caught in a local minimum\nso that the true ground state of the magnetic Hamil-\ntonian is not found, an issue that could be of partic-\nular concern for systems with highly degenerate (free)\nenergy landscapes such as frustrated magnets and spin\nglasses [66]. We do not have quenched chemical disorder\nwherefore a spin glass phase is not possible, furthermore\na pure Heisenberg Hamiltonian as in Eq. 1 cannot sta-\nbilize a skyrmion spin texture. For the present data set,\nwe classify the systems as ferromagnetic, collinear anti-\nferromagnetic, noncollinear antiferromagnetic, or param -\nagnetic. An overview and statistics of the dataset will be\npresented in Section III. Another important distinction is\nwhether the magnetic ordering is commensurate with the\nprimitive chemical cell or not. Knowledge of the shape\nand size of the primitive magnetic cell, allows for the lin-\near spin wave calculation to be performed for the size\nof the spin wave Hamiltonian that will give the correct\nnumber of magnon dispersion eigenvalues and eigenvec-\ntors. In order to obtain from a size NAL3simulation cell\nthe smallest possible magnetic cell, we make use of the\ncapability of the Elk software [67] to reduce from an in-\nput crystallographic structure to the smallest primitive\ncell, considering also the magnetic ordering.\nFor the DFT ground state calculations a typical value\nof the computation time was 125 hours on a compute\nnode with 32 cores (4 000 core hours). The calculations\nof the magnetic ground states and the dynamic struc-\nture factor with atomistic spin dynamics are more than\none order of magnitude faster than the DFT calculations.\nThe sampling of the dynamic structure factor takes of\nthe order of 10 hours using 8 cores (80 core hours). The\ncalculation of the magnon dispersion for collinear spin\nconfigurations is quasi instantaneous, as it only involves\nthe diagonalization of small matrices.\nE. Magnon matcher\nDispersion relations for magnons are commonly calcu-\nlated along lines in reciprocal space between high sym-\nmetry points in the Brillouin zone. We have chosen to\nuse for each material the same paths for magnons dis-\npersions as has been used for the creation of the OMDB\nelectronic band structure data set [19, 37].\nSimilar to the previously described pattern matching\nfor electronic band structures [38], we have imple-\nmented a pattern matching algorithm for magnon\nbands. The pattern matching algorithm is based on\na moving window approach which is best described\nfor a pattern containing two bands (it can be ex-\ntended to an arbitrary number of bands). A user can\ndraw a two-band pattern in the OMDB user interface\nhttps://omdb.mathub.io/search/pattern_magnon .5\nThe pattern is vectorized and distributed in two vectors\nvupper andvlowerwhich are concatenated into a query\nvectorvquery=vupper,vlower).\nFor each calculated magnon spectrum we consider\npairs of magnon branches and choose a window of a user\nspecified width. The two bands within the window are\nvectorized in a similar way as the query vector and com-\npared to the query vector in terms of a vector distance,\nhere the cosine distance. The window is moved by a spec-\nified stride and the resulting vectors are compared again.\nFinally, from all magnon spectra and all possible win-\ndows the ones which have a suitable vector similarity to\nthe initial pattern are selected. This algorithm is imple-\nmented on the OMDB web interface, where a ball tree\nnearest neighbour search is used to accelerate the com-\nparison between initial pattern and respective windows\nwithin the magnon spectra. More details on the method\ncan be found in Ref. 38.\nIII. RESULTS\nThe main dataset of the OMDB constitute of spin-\npolarized electron band structures and density of states\n[19, 37]. The first step in the production of the current set\nof magnetic Hamiltonians and excitation spectra was to\nchoose a subset of the magnetic materials on the OMDB.\nFor these materials we have performed modeling using\nthe high throughput methodology described in Sec. II D.\nWe have obtained the magnetic excitation spectra for 118\nmaterials, containing one or more sites with a magnetic\nmoment of at least 0.1 µBaccording to the RSPt LDA\ncalculations. The majority of the materials in this mag-\nnetic excitations dataset have noncollinear ground states .\nRather remarkably, none of our currently considered ma-\nterials displays a ferromagnetic groundstate. The ASD\nquenching simulations to obtain magnetic ground states\nused simulation supercells with edge length L= 6, and\nthe ASD simulations for sampling S(q,ω)at finite tem-\nperature used supercells with L= 38.\nWe will in the following present our magnetic ex-\ncitations dataset (OMDB-SW1), closely connecting to\nhow the results are presented on the Organic Materi-\nals Database. We refer to the materials using their\nOMDB-ID numbers. Our first example is 1,1′,2,2′,4,4′-\nhexaisopropylnickelocene C 28H46Ni (OMDB-ID 855)\nwhich has very low symmetry and crystallizes in space\ngroupP-1 (space group number 2). The compound is\nformed from isopropyl groups and of cyclopentadienyl\nrings that sandwich divalent Ni atoms [68]. The prim-\nitive nonmagnetic cell has 75 atoms. The Ni atom has\nWyckoff position 1a which has inversion symmetry. For\nthis site we obtained a magnetic moment µNi= 0.84µB.\nHeisenberg interactions were calculated between the Ni\nsites for distances up to a maximum of three multiples of\nthe lattice constant a. All the leading exchange couplings\nare antiferromagnetic. The shortest Ni to Ni distance of\n8.75 Å is found along the crystallographic aaxis with theNi spins coupled antiferromagnetically, J8.75=−0.012\nmeV. The next nearest neighbour coordination shell of\nNi atoms is along the baxis with distance 9.08 Å and the\nvanishingly small value coupling J9.08=−0.002meV.\nThe strongest exchange coupling J9.16=−0.056meV is\nfound for the third nearest neighbor coordination shell\nin theab-plane. Three dimensional coordination of ex-\nchange between Ni atoms is achieved when considering\nthe coupling J9.26=−0.012meV along the caxis. On\nthe OMDB [37], the exchange couplings are listed, or-\ndered to show the strongest couplings first, in a table in\nthe lower left part of the magnetic properties panel and\nare also displayed in the JSmol viewer. A listing of the\nmagnetic moments can be found in the lower right part\nof the panel.\nThe outcome of the ASD quenching calculation was\na collinear antiferromagnetic spin configuration that is\nshown in Fig. 3(a). With regard to the original lattice\nvectors for the primitive chemical cell, a 2×1×2super-\ncell of the primitive cell can accommodate the antiferro-\nmagnetic ordering. Calculating the spin wave spectra for\nthis conventional magnetic cell, two doubly degenerate\nmagnon bands come out (not shown). The conventional\nmagnetic cell can be reduced to a two-atom magnetic\ncell as shown schematically in Fig. 2. For this primitive\nmagnetic cell one doubly degenerate magnon band comes\nout. Figure 3(b) displays the T= 0K magnon disper-\nsion (black dashed line) along a high symmetry path in\nthe Brillouin zone as well as the corresponding T= 1K\ndynamical structure factor intensity (colorplot), sample d\nusing Eqs. (5)-(7).\nAlso the azido-Ni compound C 10H10N8Ni [69]\n(OMDB-ID 19963) has a structure that belongs to space\ngroupP-1 (space group number 2). The primitive non-\nmagnetic cell has 58 atoms, corresponding to two formula\nunits. For the Ni site we obtain a magnetic moment\nµNi= 1.05µB. The magnetic ground state is collinear,\nand can be accommodated in a 2×1×2supercell of the\nprimitive cell. Given that the primitive nonmagnetic cell\nhas two Ni sites, the supercell contains a total of eight\nmagnetic sites. Similarily as for C 20H10CoN12, the su-\npercell can be reduced to a primitive magnetic cell, this\ntime with four atoms. The linear spin wave calculations\nyield two doubly degenerate magnon bands, one pair be-\ning an acoustic mode, and the other pair being an optical\nmode with finite values of the dispersion at the zone cen-\nter. Our results for C 10H10N8Ni are displayed in Fig. 4.\nThe color plot for the dynamic structure factor gives an\nindication on the relative occupancy of higher and lower\nenergy magnons at the given temperature of T= 1K\n[70].\nAs a third example, we will discuss the non-\ncollinear antiferromagnet material manganese dicarboxy-\nlate C2H3MnO3[71] (OMDB-ID 11283), with results dis-\nplayed in Fig. 5. The material crystallizes in the non-\nsymmorphic space group P121/n1 (space group number\n14). The primitive chemical cell contains 36 atoms, cor-\nresponding to four formula units. A magnetic moment is6\nX ΓY, L ΓZ, N ΓM, R Γ\nqvector0.00.10.20.30.40.50.60.70.8Energy (meV)(b)\n−9.0−7.5−6.0−4.5−3.0−1.50.01.53.0\nLog 10[S(q, ω)/S0(q, ω)]\nFigure 3. Magnetic properties of the collinear antiferroma g-\nnetic hexaiso-propylnickelocene C 14H23Ni0.5(OMDB-ID 855)\n(a) The crystallographic unit cell with its magnetic sites a nd\nground state spin configuration shown as red arrows. (b) The\nT= 0K magnon dispersion (black dashed line) and the T= 1\nK dynamical structure factor (colorplot).\ncarried by the Mn sites for which our DFT calculations\ngave a magnetic moment µMn= 2.20µB. The outcome\nof the ASD quenching simulation was a canted antifer-\nromagnetic phase. Our current implementation of lin-\near spin wave theory does not support fully automated\ncalculations of the spin wave spectra for systems with\nnoncollinear ground states. In contrast, the ASD sim-\nulation method allows for semiclassical sampling of the\ndynamic structure factor for magnetic systems with any\nspin ordering, including also the paramagnetic regime\nwhere long range ordering is absent. Figure 5 shows the\ndynamical structure factor at T= 1K for C 2H3MnO3,\nrevealing two main bands of dispersion.\nFigure 6 shows the application of the magnon matcher\nfor the example of identifying Dirac nodes within magnon\nspectra. Using as a query Dirac-type crossing of linear\nbands, the pattern matching algorithm here identified\nsuch a crossing along a line in reciprocal space at the high\nsymmetry point Y, for the space group P121/n1 (space\ngroup number 14) collinear antiferromagnet C 6H14NiO8\n(OMDB-ID 21042). The antiferromagnetic ground state\nX ΓY, L ΓZ, N ΓM, R Γ\nqvector0102030405060Energy (meV)(b)\n−10−8−6−4−202\nLog 10[S(q, ω)/S0(q, ω)]\nFigure 4. Magnetic properties of the collinear antiferroma g-\nnetic azido-Ni compound C 10H10N8Ni (OMDB-ID 19963).\n(a) The crystallographic unit cell with its magnetic sites a nd\nground state spin configuration shown as red arrows. (b) The\nT= 0K magnon dispersion (black dashed line) and the T= 1\nK dynamical structure factor (colorplot).\nbreaks the monoclinic symmetry of the chemical unit\ncell. The corresponding magnetic space group symmetry\nwas determined using FINDSYM [72, 73] and is given by\nPS−1, having the coset representatives E={1|0},I=\n{−1|0},T={1′|t},TI={−1′|t}, witht={0,0,1/2}in\nunits of the real space lattice. Here, 1′and−1′denote\nthe identity (x→x,y→y,z→z)and the inversion\n(x→ −x,y→ −y,z→ −z)connected with a flip of the\nmagnetization (mx→ −mx,my→ −my,mz→ −mz).\nThe magnetic unit cell contains four atoms, with two mo-\nments pointing in zand two moments in −zdirection.\nDue to the antiferromagnetism, each mode of frequency\nωand momentum k, belonging to magnetic sites point-\ning inz-direction, comes with a corresponding mode of\nopposite frequency and momentum belonging to the mag-\nnetic sites pointing in opposite direction. However, both\nmodes are similar in energy leading to two-fold degener-\nacy of the magnon energy momentum dispersion within\nthe entire Brillouin zone. At the Ypoint, inversion sym-\nmetry is present and eigenstates of the linear spin wave\nHamiltonian can be written in a basis where they are\neigenvectors of the inversion operation with respective\nparity eigenvalues λi. As inversion Isquares to 1, i.e.,7\nΓYHCEMAXH, M DZ, Y D\nqvector0510152025Energy (meV)\n−10.4−9.6−8.8−8.0−7.2−6.4−5.6−4.8−4.0\nLog 10[S(q, ω)/S0(q, ω)]\nFigure 5. The T= 1K dynamical structure factor (colorplot)\nfor manganese dicarboxylate C 2H3MnO3(OMDB-ID 11283).\nUnlike for the collinear magnets shown in Figs. 3 and 4,\nwe here do not include any T= 0K magnon dispersion,\nas the current workflow does not support high throughput\ncalculation of linear spin wave spectra for noncollinear sp in\ntextures.\nΓYHCEM1AXH1\nqvector0.00.20.40.60.8Energy (meV)\nFigure 6. Example of use of the magnon matcher for the\nidentification of Dirac nodes of the magnon bands. The black\ncurves show the spin wave dispersion for the space group P\n121/n1 (space group number 14) collinear antiferromagnet\nC6H14NiO8(OMDB-ID 21042), with the green rectangle in-\ndicating a linear crossing of magnon bands at the high sym-\nmetry point Y.\nI2= 1, the eigenvalues are λ+= 1andλ−=−1. How-\never, at this point the operation TIsquares to −1, i.e.,\nTI2=−1, leading to a Kramers-like degeneracy of oppo-\nsite parity modes. In Figure 6 a green rectangle indicates\nthis linear crossing of magnon bands at the high symme-\ntry point Y.IV. CONCLUSIONS\nIn summary we have developed a method for high-\nthroughput calculation of the magnetic excitation spec-\ntra of organometallic materials. To this end we have\ndeployed a multiscale modeling approach in which first\nthe atomic magnetic moments and interaction coupling\nconstants of magnetic Hamiltonians are calculated, fol-\nlowed by ground state determination, and calculation of\nthe spin wave dispersion and the dynamic structure fac-\ntor. For the present results we have used Heisenberg\nHamiltonians with interactions ranging up to three lat-\ntice constants. Frustration among the Heisenberg inter-\nactions can lead to noncollinear magnetic ground states,\nand we have among our materials encountered collinear\nantiferromagnets as well as noncollinear spin orderings.\nCurrently we have used an implementation of linear spin\nwave theory for collinear magnets, wherefore dispersions\nwere calculated only for the collinear systems, however,\nwe have sampled the dynamic structure factor at T= 1K\nfor all materials using atomistic spin dynamics.\nThe method can be naturally extended to include\nother interactions such as Dzyaloshinskii-Moriya inter-\naction, Ising interaction, and single-site magnetocrys-\ntalline anisotropy energy to the magnetic Hamiltonian,\nof relevance for the low-energy excitation spectra of\nquantum organometallic materials. Related to this we\nexpect an even higher fraction of noncollinear ground\nstates, for which the need for calculation of the spin\nwave dispersions with the more general framework of lin-\near spin wave theory for noncollinear magnets is desired\n[57, 58]. Furthermore, recent developments of machine\nlearning techniques for lattice models and spin Hamil-\ntonians, as for instance a profile method for recogni-\ntion of three-dimensional magnetic structures [74], de-\ntermination of phase transition temperatures by means\nof self-organizing maps [75], and a support vector ma-\nchines based method for multiclassification of phases [76],\nwill be most useful for identification and classification of\ncompeting magnetic phases at finite temperature, and\nthe corresponding phase transition temperatures.\nV. OUTLOOK\nA high-throughput database of magnetic properties\nof organometallic materials opens the path towards es-\ntablishing general concepts of materials statistics. The\nsearch tools that we have developed for electronic prop-\nerties [19, 39] with the new data for magnetic prop-\nerties, will enable search for user-specified patterns in\nthe magnon dispersion relations and magnon density of\nstates.[19, 39]. Such tools provide a novel approach to-\nwards identifying functional magnetic materials, such as\nmagnon Dirac materials [77–79], topological magnon in-\nsulators [80–82], and magnon Hall materials [83, 84].\nWhile there is a huge presence of theoretical concepts\nof such novel topological magnon phases, only very few8\nmaterials are known exhibiting these bosonic quantum\nphases. We see strong indications that the tools pre-\nsented by us during this article will help us to efficiently\nexplore the realm of organic materials in this context,\nand note that the very same method can also be used\nfor the calculation of magnon spectra of inorganic mag-\nnetic materials. With the database grown to a decent\nsize we furthermore see great opportunities in training\nmachine learning models. We have recently shown that\nsuch machine learning tools can be applied to predict\nbasic materials properties for extremely complex organic\nmaterials hosting thousands of atoms in the unit cell and\nwith that are outside the realm of materials which can\nbe calculated straightforwardly using density functional\ntheory [36]. In a similar manner, we see great opportu-\nnities in training models towards automatically formu-\nlating effective Heisenberg models for arbitrary organicmaterials, significantly accelerating the accumulation of\nmagnon and S(q,ω)spectra, as needed, for instance by\nlarge-scale experimental facilities [40–42].\nVI. 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Muthsam,1,a)F. Slanovc,1C. Vogler,1and D. Suess1\nUniversity of Vienna, Physics of Functional Materials, Boltzmanngasse 5, 1090 Vienna,\nAustria\n(Dated: 12 July 2019)\nWe optimize the recording medium for heat-assisted magnetic recording by using a high/low\nTcbilayer structure to reduce AC and DC noise. Compared to a former work, small Gilbert\ndamping\u000b= 0:02 is considered for the FePt like hard magnetic material. Atomistic simu-\nlations are performed for a cylindrical recording grain with diameter d= 5 nm and height\nh= 8 nm. Di\u000berent soft magnetic material compositions are tested and the amount of hard\nand soft magnetic material is optimized. The results show that for a soft magnetic material\nwith\u000bSM= 0:1 andJij;SM= 7:72\u000210\u000021J/link a composition with 50% hard and 50% soft\nmagnetic material leads to the best results. Additionally, we analyse how much the areal\ndensity can be improved by using the optimized bilayer structure compared to the pure hard\nmagnetic recording material. It turns out that the optimized bilayer design allows an areal\ndensity that is 1 Tb/in2higher than that of the pure hard magnetic material while obtaining\nthe same SNR.\nI. INTRODUCTION\nHeat-assisted magnetic recording (HAMR) [1{7] is a\npromising recording technology to further increase the\nareal storage densities (ADs) of hard disk drives. Con-\nventional state-of-the-art recording technologies are not\nable to overcome the so-called recording trilemma [8]:\nHigher ADs require smaller grains. These grains need\nto have high uniaxial anisotropy to be thermally sta-\nble. However today's write heads are not able to pro-\nduce \felds that are strong enough to switch these high\nanisotropy grains. In the HAMR process a heat pulse\nis included in the recording process to locally heat the\nrecording medium. This leads to a drop of the coercivity,\nmaking the high anisotropy recording medium writeable.\nThe medium is then quickly cooled and the information\nreliably stored.\nTo reach high linear densities it is necessary to reduce\nAC and DC noise in recording media [9]. AC noise de-\ntermines the distance between neighboring bits in bit-\npatterned [10{12] media or the transition between grains\nin granular media. DC noise restricts the maximum\nswitching probability of grains away from the transition.\nIt has been shown, that pure hard magnetic grains do\nnot switch reliably [13] if bit-patterned media are con-\nsidered whereas non-optimized exchange coupled bilayer\nstructures [14{19] of hard and soft magnetic material ex-\nperience high AC noise [20]. A work to reduce noise\nin recording media by optimizing a high/low Tcbilayer\nstructure (see Ref. [21]) showed that an optimial bilayer\nstructure consists of 80% hard magnetic and 20% soft\nmagnetic material. However, in the former work the\nGilbert damping was assumed to be \u000bHM= 0:1 which\nis hard to achieve in a FePt like hard magnetic material\nin reality. In realistic hard magnetic recording materi-\nals, the damping constant is \u000b= 0:02, according to the\nAdvanced Storage Technology Consortium (ASTC) [22].\na)Electronic mail: olivia.muthsam@univie.ac.atSince it has been shown that the damping constant has\na strong in\ruence on the maximum switching probabil-\nity and the down-track jitter, we follow the optimization\napproach and optimize a bilayer structure for the ASTC\nparameters. After the optimization, we study how the\noptimized material di\u000bers from that with \u000bHM= 0:1.\nAdditionally, we investigate how much the areal storage\ndensity (AD) can be improved when using the optimized\nrecording material instead of the pure hard magnetic one.\nThis is done with the help of the signal-to-noise ratio\n(SNR), which gives the power of the signal over the power\nof the noise and is a good indicator for the quality of writ-\nten bits.\nThe structure of this work is as follows: In Section II,\nthe HAMR model and the material parameters are pre-\nsented. In Section III, the results are shown and they are\ndiscussed in Section IV.\nII. HAMR MODEL\nThe optimization simulations are performed with the\natomistic simulation program VAMPIRE [23] which\nsolves the stochastic Landau-Lifshitz-Gilbert (LLG)\nequation. In the simulations, a cylindrical recording\ngrain with a diameter d= 5 nm and a height h= 8 nm\nis used. It can be considered as one recording bit in\nbit-patterned media. A simple cubic crystal structure is\nused and only nearest neighbor interactions are consid-\nered. The e\u000bective lattice parameter aand the exchange\ninteraxtion Jijare adjusted in order to lead to the exper-\nimentally obtained saturation magnetization and Curie\ntemperature. [24; 25]. The write head is assumed to\nmove with a velocity of v= 15 m/s. A continuous laser\npulse is assumed with the Gaussian temperature pro\fle\nT(x;y;t ) = (Twrite\u0000Tmin)e\u0000x2+y2\n2\u001b2+Tmin (1)\n=Tpeak(y)\u0001e\u0000x2\n2\u001b2+Tmin (2)arXiv:1907.05027v1  [physics.app-ph]  11 Jul 20192\nwith\n\u001b=FWHMp\n8 ln(2): (3)\nThe full width at half maximum (FWHM) is assumed\nto be 60 nm. Both, the down-track position xand\nthe o\u000b-track position yare variable in the simulations.\nThe initial and \fnal temperature is Tmin= 300 K. The\napplied \feld is modeled as a trapezoidal \feld with\na write \feld duration of 0.57 ns and a \feld rise and\ndecay time of 0.1 ns. The \feld is applied at an angle of\n22 deg with respect to the normal. The \feld strength is\nassumed to be +0.8 T and -0.8 T in z-direction. Initially,\nthe magnetization of each grain points in + z-direction.\nThe trapezoidal \feld tries to switch the magnetization\nof the grain from + z-direction to\u0000z-direction. At the\nend of every simulation, it is evaluated if the bit has\nswitched or not.\nA. Material parameters\nThe material parameters for the hard magnetic\nmaterial can be seen in Table I. For the soft magnetic\nmaterial, the atomistic spin moment is assumed to be\n\u0016s= 1:6\u0016Bwhich corresponds to a saturation polariza-\ntionJs= 1:35 T. The uniaxial anisotropy constant ku;SM\nin the soft magnetic layer is initially set to 0 but later\nvaried. The Gilbert damping \u000bSMand the exchange\ninteraction Jij;SM within the soft magnetic material are\nvaried. Experimentally, it is possible to increase the\ndamping constant by doping the soft magnetic material\nwith transition metals like Gd or Os [26{30]. Thus, also\nenhanced damping constants \u000bSMlarger than 0 :02 are\nconsidered in the simulations.\nIII. RESULTS\nA. Hard magentic grain\nFirst, a switching probability phase diagram for the\npure hard magnetic material is computed where the\nswitching probability is depending on the down-track po-\nsitionxand the o\u000b-track position y. With eq. (2) each\no\u000b-track position ycan be transformed into an unique\npeak temperature Tpeak, if the write temperature Twriteis\n\fxed, and vice versa. Thus, the switching probability in\nFigure 1 is shown as a function of the down-track position\nxand the peak temperature Tpeakthat corresponds to y.\nThe resolution of the phase diagram in down-track direc-\ntion is \u0001x= 1:5 nm and that in temperature direction\nis \u0001Tpeak= 25 K. In each phase point, 128 trajectories\nare simulated with a simulation length of 1 :5 ns. Thus,\nthe phase diagram contains more than 30.000 switching\ntrajectories. From the phase diagram it can be seen that\nFIG. 1. Switching probability phase diagram of a pure FePt\nlike hard magnetic grain. The contour lines indicate the\ntransition between areas with switching probability less than\n1% (red) and areas with switching probability higher than\n99.2% (blue). The dashed lines mark the switching probabil-\nity curves of Figure 2.\nthe pure hard magnetic grain shows only two small ar-\neas with switching probability larger than 99 :2%. This\nthreshold is used, since 128 simulations per phase point\nare performed and a switching probability of 100% corre-\nsponds to a number of successfully switched trajectories\nlarger than 1\u00001=128 = 0:992.\nTo determine the down-track jitter \u001b, a down-track\nswitching probability curve P(x) for\u000020 nm\u0014x\u00146 nm\nat a \fxed temperature Tpeak= 760 K is determined for\npure hard magnetic material (see Figure 2). The switch-\ning probability curve is \ftted with a Gaussian cumulative\nfunction\n\b\u0016;\u001b2=1\n2(1 + erf(x\u0000\u0016p\n2\u001b2))\u0001P (4)\nwith\nerf(x) =2p\u0019Zx\n0e\u0000\u001c2d\u001c; (5)\nwhere the standard deviation \u001b, the mean value \u0016and\nthe mean maximum switching probability P2[0;1] are\nthe \ftting parameters. The standard deviation \u001bdeter-\nmines the steepness of the transition function and is a\nmeasure for the transition jitter. In the further course\nit will be called \u001bdown:The \ftting parameter Pis a\nmeasure for the average switching probability for su\u000e-\nciently high temperatures. The resulting \ftting parame-\nters of the hard magnetic material can be seen in Table V.\nNote, that the calculated jitter values only consider the\ndown-track contribution of the write jitter. The so-called\na\u0000parameter is given by3\nCurie temp. TC[K] Damping\u000bUniaxial anisotropy. ku\n[J/link]Jij[J/link] \u0016s[\u0016B]\n693.5 0.02 9:124\u000210\u0000236:72\u000210\u0000211.6\nTABLE I. Material parameters of a FePt like hard magnetic granular recording medium.\n−15 −10 −5 000.20.40.60.81\ndown-track x[nm]switching probability\nHM\nFIG. 2. Down-track switching probability curve P(x) at a\npeak temperature Tpeak = 760 K for a pure hard magnetic\ngrain.\na=q\n\u001b2\ndown+\u001b2g (6)\nwhere\u001bgis a grain-size-dependent jitter contribution\n[31]. The write jitter can then be calculated by\n\u001bwrite\u0019ar\nS\nW(7)\nwhereWis the reader width and S=D+Bis the\ngrain diameter, i.e. the sum of the particle size Dand\nthe nonmagnetic boundary B[32; 33].\nB. Media Optimization\nTo \fnd the best soft magnetic material composition,\ndown-track switching probability curves P(x) similar to\nFigure 2 are computed for 50/50 bilayer structures with\ndi\u000berent damping constants \u000bSMand di\u000berent exchange\ninteractions Jij;SM. The range in which the parameters\nare varied can be seen in Table II. Note, that P(x) is\ncomputed at di\u000berent peak temperatures for the di\u000berent\nexchange interactions, since there holds\nJij=3kBTC\n\u000fz; (8)\nwherekBis the Boltzmann constant, z is the number\nof nearest neighbors and \u000fis a correction factor from the\nmean-\feld expression which is approximately 0.86 [23].\nThe temperature at which P(x) is calculated is chosen\nto beTC+ 60 K. The down-track switching probability\nFIG. 3. Down-track jitter \u001bdown as a function of the damp-\ning constant and the exchange interaction. The contour line\nindicates the transition between areas with down-track jitter\nlarger than 0.5 nm (light red, blue) and areas with down-track\njitter smaller than 0.5 nm (dark red).\ncurves are then \ftted with eq. (4). The down-track jitter\nparameters as a function of the damping constant and\nthe exchange interaction can be see in Figure 3. The\nmaximum switching probability is 1 for \u000b\u00150:1.\nFrom the simulations it can be seen that a Gilbert\ndamping\u000bSM= 0:1 together with Jij;SM= 7:72\u0002\n10\u000021J/link leads to the best results with the smallest\ndown-track jitter \u001bdown = 0:41 nm and a switching proa-\nbilityP= 1.\nThe last soft magnetic parameter that is varied, is the\nuniaxial anisotropy ku;SM. It is known that the small-\nest coercive \feld in an exchange spring medium can be\nachieved if KSM= 1=5KHM[34; 35]. Here\nKi=natku;i\na3i2fSM;HMg (9)\nare the macroscopic anisotropy constants in J/m3\nwith the unit cell size a= 0:24 nm and the number of\natomsnatper unit cell. ku;SMis varied between 0 and\n1=2ku;HM= 4:562\u000210\u000023J/link. The damping constant4\nParameter min. value max.value\n\u000bSM 0.02 0.5\nJij;SM[J/link] 5:72\u000210\u0000219:72\u000210\u000021\nku;SM[J/link] 0 1=2ku;HM= 4:562\u000210\u000023\nTABLE II. Range in which the di\u000berent soft magnetic material parameters are varied.\nku;SM\u000210\u000023[J/link]\u001bdown[nm]P\n0 0.41 1.0\n0:562 0.919 1.0\n1:8428 [= 1=5ku;HM] 1.04 1.0\n3:124 0.898 1.0\n4:562 [= 1=2ku;HM] 1.01 1.0\nTABLE III. Resulting down-track jitter parameters and mean maximum switching probability values for soft magnetic materials\nwith di\u000berent uniaxial anisotropy constants ku;SM.\nis\u000bSM= 0:1. The resulting \ftting parameters are sum-\nmarized in Table III. It can be seen that the switching\nprobability is one for all varied ku;SM. However, the\ndown-track jitter increases for higher ku;SM. Since for\nku;SM= 0 J/link the jitter is the smallest, this value is\nchosen for the optimal material composition.\nIn conclusion, the material parameters of the optimized\nsoft magnetic material composition can be seen in Ta-\nble IV.\nNext, simulations for di\u000berent ratios of hard and soft\nmagnetic material are performed. Down-track switching\nprobability curves P(x) are computed for di\u000berent ratios\natTpeak= 780 K and the down-track jitter and the mean\nmaximum switching probability are determined. The re-\nsults are listed in Table V.\nIt can be seen that a structure with 50% hard magnetic\nand 50% soft magnetic materials leads to the smallest\njitter and the highest switching probability. This result\ndi\u000bers from the optimized material composition in Ref.\n[21], where the optimal composition consists of 80% hard\nmagnetic and 20% soft magnetic materials. In Figure 4,\na switching probability phase diagram of the optimized\nbilayer structure with 50% hard and 50% soft magnetic\nmaterial can be seen.\nIt is visible that the switching probability of the\nstructure is larger than 99 :2% for a bigger area of down-\ntrack positions and peak temperatures. This shows the\nreduction of DC noise in the optimized structure.\nC. Areal Density\nTo analyse the possible increase of areal density by us-\ning the optimized bilayer structure instead of the pure\nhard magnetic recording medium, the signal-to-noise ra-\ntio is calculated. With the help of an analytical model of\na phase diagram developed by Slanovc et al [33] it is pos-\nsible to calculate a switching probability phase diagram\nfrom eight input parameters. The input parameters are\nPmax,\u001bdown;the o\u000b-track jitter \u001bo\u000b;the transition cur-\nvature, the bit length, the half maximum temperature\nand the position of the phase diagram in Tpeak direc-\ntion and the position of the phase diagram in down-track\ndirection. The \u001bdown andPmaxvalues are those result-\nFIG. 4. Switching probability phase diagram of recording\ngrain consisting of a composition of 50% hard magnetic ma-\nterial and 50% soft magnetic material with ku;SM= 0 J/link\nandJij;SM= 7:72\u000210\u000021J/link. The contour lines indicate\nthe transition between areas with switching probability less\nthan 1% (red) and areas with switching probability higher\nthan 99.2% (blue).\ning from the simulations for pure hard magnetic material\nand the optimized bilayer structure. All other model in-\nput parameters are obtained by a least square \ft from\na switching probability phase diagram computed with\na coarse-grained LLB model [36]. The phase diagram\nis mapped onto a granular recording medium where the\nswitching probability of the grain corresponds to its po-\nsition. The writing process is repeated for 50 di\u000berent\nrandomly initialized granular media. The SNR is then\ncomputed from the read-back process with the help of a\nSNR calculator provided by SEAGATE [37].\nThe SNR is analysed for areal densities of 2 to 5 Tb/in2.\nFor the bitsize ( bs) at a certain areal density, there are\ndi\u000berent track width and bit length combinations ( t;b)5\nDamping\u000bSMUniaxial anisotropy. ku\n[J/link]Jij[J/link] \u0016s[\u0016B]\n0.1 0 7:72\u000210\u0000211.6\nTABLE IV. Resulting material parameters for the optimal soft magnetic material composition.\nHM/SM\u001bdown[nm]P\nHM 0.974 0.95\n90/10 1.06 0.969\n80/20 0.813 0.998\n70/30 0.6 0.988\n60/40 0.8 0.999\n50/50 0.41 1.0\nTABLE V. Resulting down-track jitter parameters and mean maximum switching probability values for hard magnetic material\nand three di\u000berent hard/soft bilayer structures with di\u000berent damping constants in the soft magnetic material.\nthat yield\nbs=t\u0001b: (10)\nTo compute the SNR for a certain ( t;b) combination,\nthe reader was scaled in both the down-track and the o\u000b-\ntrack direction according to the bit length and the track\nwidth, respectively. The reader resolution Rin down-\ntrack direction is scaled by\nR=R0\u0001b\nb0(11)\nwherebis the bit length, R0= 13:26 nm is the initial\nreader resolution and b0= 10:2 nm denotes the mean ini-\ntial bit length according to ASTC. In o\u000b-track direction,\nthe reader width is scaled to the respective track width\nt. The initial track width is 44 :34 nm. In Figure 5(a) and\n(b) the SNR is shown as a function of the bit length and\nthe track width for pure hard magnetic material and the\noptimized bilayer structure, respectively. Additionally,\nthe phase plots include the SNR curves for ( t;b) combina-\ntions that yield areal densities from 2 to 5 Tb/in2. From\nthe phase diagram it is visible that higher SNR values can\nbe achieved for the optimized structures than for the pure\nhard magnetic material in the same bit length \u0000track\nwidth range. For example, the SNR for an areal density\nof 2 Tb/in2for the bilayer structure is larger than 15 dB\nwhereas it is between 10 dB and 15 dB for pure hard mag-\nnetic material. For each AD there is a ( t;b) combination\nfor which the SNR is maximal and which is marked by\na dot in the phase plot. In Figure 6 the maximum SNR\nover the areal density is displayed for both structures.\nThe results show that the SNR that can be achieved with\nthe optimized structure is around 2 db higher than that\nof the hard magnetic material, if the same areal density\nis assumed. To get the same SNR, the optimized design\nallows for an areal density that is 1 Tb/in2higher than\nfor the hard magnetic one. Summarizing, the bit length\n\u0000track width combinations at which the maximum SNR\nis achieved are given in Table VI.\nFIG. 5. Signal-to-noise ratio (in dB) as a function of the bit\nlength and the track width for (a) pure hard magnetic ma-\nterial and (b) the optimized hard/soft bilayer structure. The\nred lines indicate the bit length \u0000track width combinations\nthat yield 2, 3, 4 and 5 Tb/in2areal density. The dots indicate\nthe combination at which the SNR is maximal.\nIV. CONCLUSION\nTo conclude, we optimized a recording medium with\nhigh/lowTCgrains for heat-assisted magnetic record-\ning with a low Gilbert damping in the hard magnetic\npart\u000bHM= 0:02. The simulations for a cylindrical\nrecording grain with d= 5 nm and h= 8 nm were6\nAD [Tb/in2]Max. SNR [dB] (HM) x[nm] (HM) y[nm] (HM) Max. SNR [dB] (HM/SM) x[nm] (HM/SM) y[nm] (HM/SM)\n2 13.85 10.0 32.26 16.08 8.06 40.02\n3 11.07 6.23 34.52 13.37 5.37 37.53\n4 9.46 5.0 32.26 11.55 5.0 32.26\n5 7.16 4.3 30.01 9.16 4.69 27.51\nTABLE VI. Resulting bit length xand track width ycombinations for the maximum SNR at di\u000berent areal densities (AD) for\npure hard magnetic material (HM) and the optimized bilayer structure (HM/SM).\nFIG. 6. Maximum SNR for di\u000berent areal densities for pure\nhard magnetic material and the optimized bilayer structure.\nperformed with the atomistic simulation program VAM-\nPIRE. The damping constant of the soft magnetic mate-\nrial was assumed to be enhanced by doping the soft mag-\nnetic material with transition metals. The simulations\nshowed that larger damping constants lead to smaller jit-\nter and higher switching probabilities. A damping con-\nstant\u000bSM= 0:1, in combination with an exchange in-\nteractionJij;SM= 7:72\u000210\u000021J/link and an uniaxial\nanisotropy constant ku;SM= 0 J/link, led to the best re-\nsults in terms of small down-track jitter and high switch-\ning probability in a wide range of down-track and o\u000b-\ntrack positions. Interestingly, the soft magnetic com-\nposition is almost the same as for the structure with\n\u000bHM= 0:1 obtained in a previous work [21].\nIn further simulations the amount of hard and soft\nmagnetic material was varied. Surprisingly, the results\nshowed that a higher amount of soft magnetic material\nleads to smaller down-track jitter. This is not as expected\nsince for\u000bHM= 0:1 an increase of the soft magnetic ma-\nterial led to larger AC noise [21]. However, it can be\neasily explained why a higher amount of soft magnetic\nmaterial leads to better jitter results. Studying the in-\n\ruence of the damping constant on the down-track jitter\nshows that an increase of the damping constant from 0 :02\nto 0:1 reduces the down-track jitter by almost 30%. Ad-\nditionally,the maximum switching probability increases\nto 1. Since it can be seen that higher damping leads to\nsmaller jitter and higher maximum switching probability,\nit is reasonable that a higher amount of soft magnetic\nmaterial with \u000bSM= 0:1 leads to a better recording per-formance. In the former work the improved performance\ndue to higher damping was not an issue since the damp-\ning constant was 0.1 in both layers. This explains the\ndi\u000berent ratios of hard and soft magnetic material.\nFurthermore, we analyzed the increase of the areal den-\nsity can be improved if the optimized bilayer structure\nis used instead of pure hard magnetic recording mate-\nrial. This was done by analyzing the signal-to-noise ra-\ntio (SNR). The results showed that the areal density of\nthe optimized bilayer structure could be increased by\n1 Tb/in2to achieve the same SNR as for the pure hard\nmagnetic structure. In other words, that means that at\na certain areal density, the SNR was increased by 2 dB\nby using the optimized structure. Concluding, the opti-\nmized bilayer structure is a promising design to increase\nthe areal storage density by just modifying the recording\nmaterial.\nV. ACKNOWLEDGEMENTS\nThe authors would like to thank the Vienna Sci-\nence and Technology Fund (WWTF) under grant No.\nMA14-044, the Advanced Storage Technology Consor-\ntium (ASTC), and the Austrian Science Fund (FWF)\nunder grant No. I2214-N20 for \fnancial support. The\ncomputational results presented have been achieved us-\ning the Vienna Scienti\fc Cluster (VSC).\n1Hiroshi Kobayashi, Motoharu Tanaka, Hajime Machida, Takashi\nYano, and Uee Myong Hwang. Thermomagnetic recording .\nGoogle Patents, August 1984.\n2C. Mee and G. Fan. 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Multilayer exchange spring media for magnetic record-\ning.Applied Physics Letters , 89(11):113105, September 2006.\n36Christoph Vogler, Claas Abert, Florian Bruckner, and Dieter\nSuess. Landau-Lifshitz-Bloch equation for exchange-coupled\ngrains. Physical Review B , 90(21):214431, December 2014.\n37S. Hernndez, P. Lu, S. Granz, P. Krivosik, P. Huang, W. Eppler,\nT. Rausch, and E. Gage. Using Ensemble Waveform Analysis to\nCompare Heat Assisted Magnetic Recording Characteristics of\nModeled and Measured Signals. IEEE Transactions on Magnet-\nics, 53(2):1{6, February 2017."    },    {        "title": "1907.06758v1.Study_of_CoFe2O4_CoFe2_nanostructured_powder.pdf",        "content": "Study of CoFe\n2\nO\n4\n/CoFe\n2\n \nnanostructured \npowder\n \nE. S\n. \nFerreira\n1\n, E. F. Chagas\n1\n, A. P. Albuquerque\n1\n, R. J. Prado\n1\n \nand E. Baggio\n-\nSaitovitch\n2\n \n1\nInstituto de Física, Universidade Federal de Mato Grosso, 78060\n-\n900 Cuiabá\n-\nMT, Brazil.\n \n2\nCentro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud, 150 Urca Rio de Janeiro, Brazil\n \n \nAbstract\n \n \nW\ne report a\nn experimental\n \nstudy \nof the\n \nCoFe\n2\nO\n4\n/CoFe\n2\n \nnanocomposite\n, \na \nnanostructured material formed by \nhard\n \n(CoFe\n2\nO\n4\n) and \nsoft\n \n(CoFe\n2\n)\n \nmagnetic \nmateria\nls\n. The \nprecursor material, \ncobalt ferrite\n \n(CoFe\n2\nO\n4\n),\n \nwas prepared \nusing \nthe\n \nconventional s\ntoichiometric \ngel\n-\ncombustion method.\n \nT\nhe nanocomposite \nmaterial was obtained\n \nby\n \nreducing part\nially the precursor material using activated \ncharcoal as reducing agent \nin\n \nair and argon atmospheres, at 800 and 900\no\nC \nrespectively\n.\n \nThe magnetic hysteresis loops \ndemonstrate\n \nthat\n, in general, \np\nrepared nanocomposite samples \ndisplay \nsingle magnetic behavior\n,\n \nindicating \nexchange coupling between the \nsoft and hard \nmagnetic phases\n. \nHowever, for \nnanocomposite samples prepared at higher temperatures\n,\n \nthe hysteresis \nmeasurements \nshow\n \nsteps \ntypical of \ntwo\n-\nphase\n \nmagnetic behavior\n, suggesting \nthe existence of two non\n-\ncoupled magnetic phases. \nThe \nstudied \nnanocomposite\ns\n \npresent\ned\n \ncoercivity\n \n(H\nC\n)\n \nof \nabout 0.7 kOe\n,\n \nwhich is \nconsiderably \nlower than the \nexpected value for\n \ncobalt ferrite. \nA\n \nhuge increase \nin\n \nH\nC\n \n(\n>4\n4\n0%\n) \nand maximum \nenergy product (about 240%) \nwas obtained for \nthe nanocomposite \nafter \nhigh \nenergy milling \nprocessing\n.\n \n \n \nIntroduction\n \n \nThe \nmagnetic ferrites like \nM\n2+\nFe\n2\n3+\nO\n4\n \n(M = Ni; Co, Fe, Li, Mn, Zn, etc.) \nhave been\n \nused \nfor\n \nseveral \napplication\ns\n \nsuch as \nhigh\n-\ndensity\n \nmagnetic storage [1], electronic devices, biomedical applications [2\n-\n4], \npermanent magnets [5] and hydrogen production [6]. Among the hard ferrite\ns\n \nthe \nCoFe\n2\nO\n4\n \n(cobalt ferrite) \nplays an important role\n, \npresent\ning\n \npromising \ncharacteristic\ns\n \nsuch as high mag\nnet\n-\nelastic effect [7], \nchemical stability, electrical insulation, moderate saturation magnetization (M\nS\n), tunable coercivity (H\nC\n) \n[8\n-\n11] and thermal chemical reduction [6, 12, 13]. \nHowever,\n \nfor\n \npermanent magnet application\ns\n \nthe \nparameters M\nS\n \nand H\nC\n \nare \nof\n \nfundamental\n \nimportance\n, defining \nthe quantity called as\n \nthe figure of merit \nfor permanent magnets, the\n \nenergy product\n \n(BH)\nmax\n.\n \nThis quantity tends to increase with increasing of both quantities\n. The (BH)\nmax\n \nis an energy density (independent of the mass) \nthat can be simplified as a \nmeasure of the maximum amount of magnetic energy stored in a magnet.\n \nThe tunable \nbehavior of \ncoercivity \nin\n \ncobalt ferrite allows \nthe\n \nincrease\n \nin\n \nH\nC\n \n[8,\n \n11]\n, and \nn\numerous\n \nmethods\n \nas\n \nthermal annealing [\n9\n], capping [\n10\n] and \nmechanical milling treatment [\n8, 11\n]\n \nhave been used \nto this purpose. \nFor powdered cobalt ferrite materials, the highest coercivity achieved so far is \n9.5 kOe\n,\n \nreported by Limaye\n \net al.\n \n[\n10\n] \nthrough \ncapping the nanoparticles with oleic acid\n. \nH\nowever, \nin tha\nt work, \nsaturation magnetization decreased to 7.1 emu/g\n, a \nvalue about 10 times less than that expected \nfor\n \nuncapped nanoparticles. \nOn the other hand\n, Liu \net al.\n \n[\n11\n]\n \nP\nonce\n \net al.\n \n[\n8\n] \nand \nGalizia\n \net al.\n \n[\n14\n] \nobtained \nhigh \ncoercivity \nCoFe\n2\nO\n4\n \n(5.\n1,\n \n4.2 and \n3.7\n \nkOe respectively) with \nrelatively \nsmall decrease \nin\n \nM\nS\n \nusing \nhigh energy \nmechanical milling.\n \nFurthermore,\n \nthe thermal chemical reduction \nenable\ns\n \nus \nto \nincrease the M\nS \nthrough the partial \nconversion of hard ferrimagnetic\n \ncompound \nCoFe\n2\nO\n4\n \nin\nto\n \nthe soft \nferromagnetic intermetallic alloy \niron \ncobalt (CoFe\n2\n), producing the \nCoFe\n2\nO\n4\n/CoFe\n2\n \nexchange coupled \nnanocomposite \n[1\n5\n-\n1\n7\n].\n \nThe hysteresis \ncurve for \na\n \nnanocomposite with effective\n \nmagnetic \ncoupl\ning\n \nshould have\n \na \nsingle material \nmagnetic \nbehavior \n(called by \nZeng \net al.\n \n[18] \nas single\n-\nphase\n-\nlike behavior)\n \nwith M\nS\n \nand H\nC\n \nbetween the values \nexpected to CoFe\n2\nO\n4\n \n(\nM\nS\n \nand H\nC\n \nabout 70 emu/g and 1.0 kOe) and CoFe\n2\n \n(M\nS\n \nabout 230 emu/g, though \nvery small H\nC\n).\n \nTh\nis interesting\n \ncombination\n,\n \nof \na\n \nhard magnetic\n \nmaterial (\nCoFe\n2\nO\n4\n)\n \nwith \na soft one \n(\nCoFe\n2\n)\n \nin\nto\n \nan exchange coupled nanocomposite\n,\n \npresent\ns\n \nenormous potential to hard magnetic \napplications.\n \nIn a previous work\n \n[13]\n,\n \nof our \ngroup\n, hard/soft CoFe\n2\nO\n4\n/CoFe\n2\n \nnanocomposites were obtained by\n \nthermal treatment of \na \ncobalt\n \nferrite\n/\ncarbon (activated charcoal)\n \nmixture \nin argon atmosphere, as indicated \nby equation below\n:\n \n \n\u0000\u0000\n\u0000\u0000\n\u0000\n\u0000\n\u0000\n+\n2\n\u0000\n∆\n→\n\u0000\u0000\n\u0000\u0000\n\u0000\n+\n2\n\u0000\n\u0000\n\u0000\n      \n \n(1)\n \n \nThe symbol Δ indicates that thermal energy is necessary in the process.\n \nAccording \nto \nthe reduction\n \nprocess indic\nated\n \nin\n \neq. (1)\n,\n \ntwo moles of carbon \nare\n \nenough to convert \nall CoFe\n2\nO\n4\n \nin CoFe\n2\n \nbut, in practice, this \ndoes \nnot happen.\n \nIn\n \nthis previous\n \nwork\n \n[13]\n, \nt\nhe reduction \nprocess was \ndone\n \nusing \na \npowder of the mixture\n \n(cobalt ferrite plus carbon)\n \nin air and\n/or\n \ncontroll\ned\n \ninert \natmosphere (argon)\n. I\nn all case\ns\n \nthe\n \nmaster\ning of\n \nthe process\n \nwas difficult\n \nand, i\nn\n \na sample treatment in \nargon atmosphere with 2 molar of C\n,\n \nonly about 40%\n \nof CoFe\n2\nO\n4\n \nwas converted in\n \nCoFe\n2\n. The main \nreason \nmaking\n \ndifficult \nthe\n \ncontrol \no\nf \nthe reduction process was attributed to the reaction of carbon with \noxygen \nin\n \nair or \nwith \nresi\ndual oxygen in \nthe “\ninert\n”\n \natmosphere\n \n[13]\n.\n \nTrying to solve these problems, i\nn this w\no\nrk, we use\nd\n \na \nslightly different\n \nprocedure \nto obtain the \nCoFe\n2\nO\n4\n/CoFe\n2\n \nnanocomposites. Here, t\nhe mixture of cobalt ferrite with activated charcoal\n \nwas pressed\n \nto prepare a disk\n, aiming \nto avoid the contact \nbetween\n \ninternal mixture (inside the disk) with the \natmosphere of the furnace during the thermal treatment. \nT\nhis method \nc\nould \nfacilitate the control of CoFe\n2 \ncontent in the nanocomposite CoFe\n2\nO\n4\n/CoFe\n2\n. \nMoreover\n, \nthe same \nmilling \nprocedure \nused \npreviously \nto \npure nanostructured \ncobalt ferrite [8]\n \nwas performed\n \nin this work \nto increase the \nH\nC\n \nof the \nCoFe\n2\nO\n4\n/CoFe\n2\n \nnanocomposite, with\n \nexcellent\n \nresults.\n  \nExperimental\n \n \nThe \nnanostructured cobalt ferrite \nprecursor material was prepared using a conventional \ngel\n-\ncombustion method as \ndescribed \nin \nreference [\n19\n]\n. \nHigh\n-\npurity (99.9%) raw compounds were used. \nCobalt nitrate and iron nitrate (VETEC, Brazil) were dissolved in 450 ml of distilled water in a ratio \ncorresponding to the selected final composition. Glycine (VETEC, Brazil) was added in a proportion of \none an\nd half moles per mole of metal atoms, and the pH of the solution was adjusted with ammonium \nhydroxide (25%, Merck, Germany). The pH was tuned as closely as possible to 7, taking care to avoid \nprecipitation. The resulting solution was concentrated by evapor\nation using a hot \nplate at 300\n \n°\nC until a \nviscous gel was obtained. This hot gel finally burnt out as a result of a vigorous \nexothermic reaction. The \nsystem remained homogeneous throughout the entire process and no \nprecipitation was observed. Finally, \nthe \nas\n-\nreacted material was calcined in air at 700\n \n°\nC for 2 h in order \nto remove the organic residues.\n \nThe nanocomposite was obtained mixing cobalt ferrite with activated charcoal for three different \nratios \n1:1; 1:2 and 1:5 (molar mass of CoFe\n2\nO\n4\n:C).\n \nThe mixtu\nres were\n \npressed to form small disk\ns of 10 \nmm diameter and about 1 mm \nthickness\n, thermally treated at 800 °C in air and at 900\n \n°\nC in argon \natmosphere. \nAiming to obtain a total\nly\n \nreduc\ned\n \nsample (transforming all \nCoFe\n2\nO\n4\n \ninto \nCoFe\n2\n) a mixture \npowder with \nexcessive quantity of carbon (1:\n24\n)\n \nwas also prepared and thermally \ntreated in a \ntubular \nfurnace at \n900\n \n°\nC\n \nin argon atmo\ns\nphere\n.\n \nFor elucidation, the sample naming used in this work follow a \nsimple labelling rule, for example, the name CFO\n-\n5C\n-\n800 indicate \nthat 5 moles of charcoal was used \nduring thermal reduction process of the sample and that it was thermally treated at 800 °C.\n \nA\n \nSpex 8000 high\n-\nenergy mechanical ball miller\n,\n \nwith 6 mm diameter zirconia balls, was employed \nfor \nthe \nmilling processing of all \nsamples\n, aiming\n \nexclusively\n \nto increase the\nir\n \nH\nC\n. The processing time \nwas 1.5 h for all samples\n, using \nball/sample mass ratio of about 1/7. Detailed milling conditions are \ndescribed in Ponce \net al.\n \n[8].\n \nThe crystalline phases of the nano\ncomposite\n \nwere identi\nfied by X\n-\nray diffraction (XRD)\n, using a \nShimadzu XRD\n-\n6000 diffractometer installed at\n \nthe\n \nLaboratório Mult\niusuário de Técnicas Analíticas\n \n(\nL\na\nM\nu\nTA\n/ UFMT\n–\nCuiabá\n-\nMT\n–\n \nBrazil). It \nis\n \nequipped with graphite monochromator and conventional \nCu tube (0.1541\n84\n \nnm), and work\ns\n \nat 1.2 kW (40 kV, 30 mA), using the Bragg\n-\nBrentano geometry.\n \nMagnetic measurements (hysteresis loops at 300 and 50 K) were carried out using a vibrating sample \nmagnetom\neter (VSM) model VersaLab Quantum Design, insta\nlled at CBPF, Rio de Janeiro, \nBrazil. \n \n \nResults and discussion\n \n \nIt is well known that the magnetic behavior of CoFe\n2\nO\n4\n \nis quite different from that found for CoFe\n2\n. \nThe former is a hard ferrimagnetic material \nwith maximum M\nS\n \nof about 70 emu/g while the second is \nknown to be a soft ferromagnet with M\nS\n \nof about 230 emu/g [\n20\n]. Consequently, in the case of \nnanocomposite formation presenting \nmagnetic\n \ncoupling, one can assume an intermediate magnetic behavior that\n \ndepends of the relative amount of CoFe\n2\n \nformed during the reduction process and, also, of \nthe microstructure of the material.\n \nThe hysteresis curves obtained for samples CFO\n-\n5C\n-\n800; CFO\n-\n2C\n-\n800; CFO\n-\n1C\n-\n800; CFO\n-\n5C\n-\n900; CFO\n-\n2C\n-\n900; CFO\n-\n1C\n-\n900 are shown in th\ne figure 1. Measurements reveal clear differences \nbetween M\nS\n \nof the samples prepared at 800 \n°\nC, as seen in figure 1(A). However, coercivity values \nobtained are very close and the shape of the hysteresis loop is quite similar. Samples prepared at 900 °C \nals\no present similar H\nC\n \nbut different M\nS\n \nvalues. On the other hand, the shape of the hysteresis loop \nobtained for sample CFO\n-\n5C\n-\n900 is different from those obtained for the two other samples prepared at \n900 °C, as shown in figure 1(B).\n \n \nFig\nure\n \n1 \n–\n \nHysteresis\n \ncurves at room temperature for samples treated at (A) 800\n \n°C and (B) 900 °C.\n \n \nThere are only two reasonable possibilities to explain the different M\nS\n \nvalues obtained for the \nsamples presented in figure 1: (i) an intense cationic redistribution caused by t\nhe thermal process used \nto \nprepare the \nsample or (ii) the formation of CoFe\n2\nO\n4\n/CoFe\n2\n \ncomposite. \n \nTo understand how the cationic redistribution could affect the M\nS\n \nis\n \nnecessary to know the cobalt \nferrite crystalline structure \nand its \ndistribution of \nmagnetic ions\n.\n \nThe \nCoFe\n2\nO\n4\n \nis a ferrimagnetic material \nthat has an inverse spinel structure with three magnetic sites per \nformula unit. The CoFe\n2\nO\n4\n \nstructure can \nbe summarized by [\n21\n]:\n \n(\n\u0000\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\u0000\n\u0000\n\u0000\n\u0000\n)\n[\n\u0000\u0000\n\u0000\n\u0000\n\u0000\n\u0000\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n]\n\u0000\n\u0000\n\u0000\n\u0000\n \n(2)\n \nIn this representation, \nthe round and the square brackets indicate A \n(tetrahedral coordination of four \nO\n2\n-\n \nanions site) \nand B sites\n \n(two octahedral coordination of six O\n2\n-\n \nanions sites)\n, respectively,\n \nand\n \ni \n(the \ndegree of inversion) describes the fraction \nof the tetrahedral sites occupied by Fe\n3+\n \ncations. The ideal \ninverse spinel structure has \ni \n= 1\n \nand\n \nthe normal spinel has\n \ni \n= 0\n. A\n \nmixed spinel structure present \ni\n \nvalues \nbetween 0 and 1. \n \nIn an ideal inverse spinel cobalt ferrite, half of the Fe\n3+\n \ncations\n \n(magnetic moment of 5μ\nB\n) occupy \nthe A\n-\nsites and the other half the B\n-\nsites, together with Co\n2+\n \ncations (see that there are two B sites per \nformula unit). Since the magnetic moments of the ions in the A and B sites are aligned in an anti\n-\nparallel \nway, ther\ne is no magnetic contribution of the Fe\n3+\n \ncations in this case. Therefore, the net magnetic \nmoment of the ideal inverse spinel cobalt ferrite is due exclusively to the Co\n2+\n \ncations (magnetic moment \nof 3μ\nB\n). However, normally, the cobalt ferrite presents a \nmixed spinel structure and, consequently, there \nare Co\n2+\n \nand Fe\n3+\n \ncations in both tetrahedral and octahedral sites, \ni.e. \n, when \ni \ndecreases toward 0 (normal \nspinel), it means swap of Fe\n3+\n \ncations from site A with Co\n2+\n \ncations from site B, increasing the net \nmagnetic moment of the material and, consequently, promoting an increase of the \nM\nS\n. \nTherefore, the \nmagnetic behavior\n \nof the CoFe\n2\nO\n4\n, as \nsaturation magnetization\n, is strongly affected by structural changes \nand/or che\nmical disorder/substitution.\n \nAs an example, to an ideal inverse spinel (\ni = \n1) a magnetic moment of 3.0 μ\nB\n \nper formula unit \n(equivalent to M\nS\n \n=71.4 emu/g) is expected. For sample \nCFO\n-\n5C\n-\n800 we obtained \nM\nS \n= 90 emu/g, that is \nequivalent to a magnetic moment\n \nof\n \n3.8 μ\nB\n \nper formula unit and \ni = \n0.8, which means a swap of 20% of \nthe Co cations in the material. This value to \ni\n \nsounds reasonable, however highest values of M\nS\n \nfound in \nthe literature were 83.1 and 83.12 emu/g by Sato Turtelli \net al.\n \n[2\n1\n] and \nKumar \net al\n. [\n22\n], respectively. \nThese literature values are equivalent \nto \nμ = 3.5 μ\nB\n \nper\n \nformula unit and \ni = \n0.88 (12% of Co cations were \nswapped by Fe). For this reason we do not consider the cationic redistribution \nis \nresponsible for the \nincrease in M\nS\n. \nThus\n,\n \nafter thermal chemical reduction, \nfor the samples studied in this work\n, \nwe consider \nthe formation of CoFe\n2\nO\n4\n/CoFe\n2\n \nnanocomposite as responsible for \nthe increase in M\nS\n.\n \nUsing the extrapolation \nto zero \nof the M versus 1/H plot to estimate the M\nS\n \nvalues\n \nof each sample, \nand c\nonsidering the M\nS\n \nexpected to pure inverse spinel cobalt ferrite (71.4 emu/g) and pure CoFe\n2\n \n(230 \nemu/g) [2\n0\n], one can estimate the content of CoFe\n2\n \nin the composite (see table \n1\n). \nWe do not consider the \neffect of canted magnetic mome\nnt from surface cations.\n \n \nTable \n1\n \n–\n \nMagnetic parameters of the samples, obtained at room temperature.\n \nSample\n \nM\nS\n \n(emu/g)\n \nM\nR \n(emu/g)\n \nM\nR\n/\nM\nS\n \nH\nC\n \n(kOe)\n \nContent of \nCoFe\n2\n \n(%)\n \n(BH)\nmax\n \n(MGOe)\n \nCFO\n-\n5C\n-\n800\n \n90\n \n21\n \n0.23\n \n0.7\n \n13\n \n0.35\n \nCFO\n-\n2C\n-\n800\n \n84\n \n29\n \n0.35\n \n0.7\n \n9\n \n0.32\n \nCFO\n-\n1C\n-\n800\n \n75\n \n18\n \n0.24\n \n0.7\n \n3\n \n0.26\n \nCFO\n-\n5C\n-\n900\n \n79\n \n29\n \n0.37\n \n0.7\n \n6 \n \n0.27\n \nCFO\n-\n2C\n-\n900\n \n87\n \n21\n \n0.24\n \n0.6\n \n11 \n \n0.15\n \nCFO\n-\n1C\n-\n900\n \n72\n \n18\n \n0.25\n \n0.7\n \n2\n \n0.14\n \n \nThe CoFe\n2\n \ncontent in the nanocomposite was found to be small in all cases, indicating \nas false our\n \ninitial assumption that carbon of the mixture inside the disk reacts only with oxygen from CoFe\n2\nO\n4\n \n(generating CoFe\n2\n). Two reasons can cause this effect: (i) residual oxygen inside the mixture sample disk \nreacted with the carbon and (ii) certain content o\nf carbon do not reacts with cobalt ferrite during the \nthermal treatment. The second option probably occurs due to the morphology of the material, which \npermits the formation of a CoFe\n2\n \nshell around the CoFe\n2\nO\n4\n \ncore of the grain, shielding it and limiting t\nhe reduction power of the activated charcoal during thermal treatment. In any case, this \neffect makes difficult \nto control the quantity of CoFe\n2\n \nin the nanocomposite CoFe\n2\nO\n4\n/CoFe\n2\n. \n \nThe small amount of CoFe\n2\n \nformed in samples treated in both air and argon atmospheres, as \nindicated in table \n1\n, suggests that changes in the atmosphere of the furnace do not cause significant \ndifferences on the CoFe\n2\n \ncontent of the composite.\n \nThe coercivity observed for all sample\ns was very similar, and close to 0.7 kOe\n, a va\nlue smaller than \nthose obtained by Cabral \net al.\n \n[12] and Leite \net al.\n \n[13]. In other words, this value is lower than that \nexpected for a pure cobalt ferrite and higher than that expected for pure CoFe\n2\n \nand, co\nnsequently, in \nagreement with the formation of the composite CoFe\n2\nO\n4\n/CoFe\n2\n.\n \nTo perform a more detailed investigation about the differences between the shapes of the hysteresis \ncurves obtained from samples prepared at 900\n \n°\nC, the hysteresis loops were also \nmeasured at low \ntemperature (50 K). These measurements are shown in figure \n2\n. The low temperature hysteresis enhances \nthe visualization of \nthe behavior observed at room temperature (300 K). For example, the hysteresis \ncurves obtained for samples CFO\n-\n2C\n-\n900\n \nand CFO\n-\n1C\n-\n900 \npresent a typical\n \ntwo\n-\nphase magnetic \nbehavior, and the hysteresis curve obtained for sample CFO\n-\n5C\n-\n900 presents a \nsingle\n-\nphase\n \nmagnetic \nbehavior, suggesting the coupling between the different magnetic phases\n \npresent\n \ninto the \nsample\n. The \nderivative of the hysteresis curve (first quadrant decreasing field only) confirms our assumption (figure \n2\nC), because the two\n-\nphase \nmagnetic \nbehavior is characteristic of two magnetic phases coexisting without \nexchange \ncoupling\n \nbetween them\n. Similar\n \nbehavior was observed by Sun \net al.\n \n[\n23\n]. \n \nNanocomposites with \nexchange coupling or exchange spring present single magnetic phase behavior \n[15\n-\n17, \n24\n, 25\n]\n, h\nowever\n,\n \nthese effects \ndepend on\n \nthe \ncrystallite \nsize of the compounds. Zeng \net al.\n \n[18] \nshowed the single magnetic phase behavior can vanish\n,\n \nto the same nanocomposite\n,\n \ndepending of the \nrelative size \nof\n \nthe compounds\n,\n \nresulting in two\n-\nphase behavior.\n \nInterestingly, the two\n-\nphase magnetic behavior was not observed in samples prepared at 800\n \n°\nC\n, \neven in the low temperature hysteresis\n \ncurves\n, see figure \n2B\n. This result reinforces our assumption that \nnanoparticles coalescence is the driving force towards the two\n-\nphase \nmagnetic \nbehavior\n. \nMore \nvisible \neffects of coalescence \na\nre expected \nto samples t\nreated \nat\n \nhigher temperatures\n, \nwith consequent increase \nin\n \ncrystallite size.\n \nAn additional aspect observed in 50 K hysteresis loop is the huge increase \nin\n \ncoercivity. This effect \nwas \nalso \nobserved by other authors \n[\n13, \n26\n]. However, it\n \nis more significant \nin\n \nsamples with single \nmagnetic behavior presenting H\nC\n \nclose to\n \n5kOe (see figures \n2\nA and \n2\nB). In samples presenting two\n-\nphase \nbehavior the coercivities were considerably smaller, \ni.e.\n, 2.4 kOe \nfor\n \nCFO\n-\n2C\n-\n900 and H\nC\n \n= 1.8 kOe \nfor\n \nsample\n \nCFO\n-\n1C\n-\n900\n \n(figure 2A)\n. \nA work studying this effect is under development.\n \n  \n0\n5\n1 0\n1 5\n2 0\n2 5\n0\n2\n4\n6\n8\n C F O - 2 C - 9 0 0\n C F O - 5 C - 9 0 0\n C F O - 1 C - 9 0 0\nDerivative\n(\nx\n10\n-3\n)\nH ( k O e )\n( C )\n \nFig\nure\n \n2\n \n–\n \nHysteresis curves (A) at 50K for samples treated at 900 \n°\nC and (B) at 300 and 50K to sample \nCFO\n-\n5C\n-\n800\n. \n(C) Derivative of the first quadrant\n \n(only decreasing field) M versus H, at 50K, for samples \ntreated at \n900\n \n°\nC.\n   \n \n \nLiu and Ding [11] have shown that is possible to obtain a noteworthy increase \nin\n \ncoercivity of cobalt \nferrite powder via high\n-\nenergy mechanical milling. Moreover, Ponce \net al.\n \n[8] extended this method to \nnano\nstructured \npowder\ns\n \nusing specific milling parameters. Although, we could have had milled the cobalt \nferrite to obtain a high coercivity CoFe\n2\nO\n4\n, but the thermal treatment at 800 or 900\n \n°\nC used to obtain the \nnanocomposite CoFe\n2\nO\n4\n/CoFe\n2\n \nis known to decrease strain and structu\nral defects and, consequently, also \ndecreasing H\nC\n \n[19].\n \nAiming to obtain a similar effect \nfor\n \nthe nanocomposite CoFe\n2\nO\n4\n/CoFe\n2\n, the samples CFO\n-\n2C\n-\n800 \nand CFO\n-\n1C\n-\n800 were milled using the same milling conditions described in the reference [8]. The result \nof\n \nthe milling process to the sample CFO\n-\n2C\n-\n800 is shown in figure 4\n,\n \na huge increase in H\nC\n \nof samples \nCFO\n-\n1C\n-\n800 (not shown) and CFO\n-\n2C\n-\n800 was obtained\n, showing that \nthe effect obtained \nafter milling \nin pure CoFe\n2\nO\n4\n \n(a decrease of M\nS\n \nand an enormous increase of H\nC\n) can also be achieved in\n \nthe \nCoFe\n2\nO\n4\n/CoFe\n2\n \nnanocomposite. \nSpecifically, in the case of sample CFO\n-\n2C\n-\n800\n,\n \nthe coercivity \nat room \ntemperature increased from 0.7 to 3.8 kOe\n \nand, a\ns a consequence of the change in H\nC\n, (BH)\nmax\n \nin\ncreased \nfrom 0.32 to 1.1 MGOe at room temperature and to 2.6 MGOe at 50K\n. The magnetic parameters for the \nsample CFO\n-\n2C\n-\n800 before and after milling are presented in table \n2\n. This is a very interesting result, \nsince the high\n-\nenergy milling procedure used i\nn this work allows to obtain \nnanostructured powder\n \nwith \nhigh (BH)\nmax\n.\n \n \n \nTable 2 \n–\n \nMagnetic parameters \nobtained \nfor sample CFO\n-\n2C\n-\n800, \nat 300 K before milling and at 300 and \n50\n \nK \nafter milling process.\n \nSample\n \nT (K)\n \nM\nS\n \n(emu/g)\n \nM\nR \n(emu/g)\n \nM\nR\n/\nM\nS\n \nH\nC\n(kOe)\n \n(BH)\nmax\n \n(MGOe)\n \nCFO\n-\n2C\n-\n800\n \n300\n \n84\n \n32\n \n0.38\n \n0.7\n \n0.32\n \nCFO\n-\n2C\n-\n800\n(milled)\n \n300\n \n60\n \n34\n \n0.57\n \n3.8\n \n1.1\n \nCFO\n-\n2C\n-\n800\n(milled)\n \n50\n \n64\n \n48\n \n0.75\n \n11.1\n \n2.6\n \n \n \nFigure 3 presents a clear decrease of M\nS \nafter milling\n, t\nhis effect is associated to the decrease of the \nmean crystallite size due to the milling process [8, 11, 14]. \nThe reduction of the crystallite size increases \nthe number of surface magnetic ions and the cant\ning\n \neffect of these surface ions is \nresponsible fo\nr the\n \ndecrease in\n \nM\nS\n.\n \n \n \nFig\nure\n \n3\n \n–\n \nHysteresis curves for sample CFO\n-\n2C\n-\n800, obtained at 300 K before and after milling and at \n50 K after milling process.\n \n \nAn additional interesting property was observed in the hysteresis loop at 50K (see figure \n3\n). Milled\n \nCoFe\n2\nO\n4\n/CoFe\n2\n \nnanocomposites present H\nC\n \nmore than twice higher than those of non\n-\nmilled samples \n(coercivity changes from 5.0 to 11 kOe after milling). This enormous increase in H\nC\n \nwith the decrease of \ntemperature requires further investigation.\n \nXRD \nexperi\nments were also performed, in order to obtain \ndiffraction \npatterns\n \nfrom cobalt ferrite \nsamples before (pristine sample) and after thermal treatment at 800 \nand\n \n900\n \n°\nC \nin air and argon \natmosphere\n \nrespectively\n,\n \nas\n \nsho\nwn in figure \n4\n.\n \nThe \ninset of figure \n4\n \npresents\n \nthe \nx\n-\nray diffraction\n \npatter\nn\n \nof\n \nthe pristine sample and a refinement using \nMatch\n \nsoftware (solid line).\n \nThe only objective of this \nrefinement is to show that pristine cobalt ferrite is monophasic.\n \nThe \nc\nomparison between XRD patterns\n \nbefore and\n \na\nfter thermal treatment at 800\n \n°\nC\n \nshow\n \nno visible \nsignificant differences\n, except \nfor \nthe narrowing \nof \nthe peaks\n \nafter thermal treatment\n. Such n\narrowing is \nusually related\n \nto\n \nthe reduction in quantity of defects, \nimprovements\n \nin the crystal lattice distortion \naround defects \n(local strain) \nand/or\n, consequently,\n \nto the increase of the mean cry\nstallite size of the \nmaterial [\n27\n]\n.\n \n \nFig. \n4 \n-\n \nXRD patterns \nof several\n \nanalyzed\n \nsamples\n,\n \nas labeled\n.\n \n \nThe mean crystallite size of the cobalt ferrite phase (see table \n3\n) was estimated by the Scherrer \nmethod for the (311) diffraction peak, at 35.5 ° in 2\nθ\n. \nThe \nresults, as expected, indicate \nthat\n \ncrystallite size\n \nincreases\n \nwith the tempe\nrature of the thermal\n \ntreatment.\n \n \n \nTable \n3\n \n–\n \nMean \ncrystallite \nsize\n \nestimated \nfrom \nthe \nScherrer equation for\n \nthe cobalt ferrite\n \n(\n311\n)\n \nXRD \npeak\n \nof the samples\n.\n \n \n \nHowever,\n \nsamples \nthermally treated at 900 °C \npresented three \nnew\n \nsmall\n \ndiffraction\n \npeaks (indicated \nby arrow\ns\n \nin figure \n4\n). These peaks \ndo\n \nnot \nmatch\n \nwith \nthe XRD patterns expected \nfor\n \nCoFe\n2\n \n[13\n], \nbeing \nidentified as CoO\n \ncrystalline phase (ICSD Card # 9865). \nThe existence of these small peaks indicates \nphase separation due to a small lack in stoichiometry of the synthesized material. This small difference in \nSample\n \nD (nm)\n \nP\nristine\n \n12\n \nCFO\n-\n5C\n-\n800\n \n39\n \nCFO\n-\n2C\n-\n800\n \n34\n \nCFO\n-\n1C\n-\n800\n \n38\n \nCFO\n-\n5C\n-\n900\n \n57\n \nCFO\n-\n2C\n-\n900\n \n45\n \nCFO\n-\n1C\n-\n900\n \n49\n stoichiometry is common in a chemical process of synthesis, due \nto the hygroscopic nature of the reagents \nused in the process and, if necessary, can be eliminated changing in a few percent the cobalt and/or iron \nnitrate masses used in the synthesis. \nConsequently, these small peaks do not indicate partial reduction of \nt\nhe \nmaterial\n \n(formation of \nCoFe\n2\n)\n \nand, a\nlso, the CoO phase does not contribute to hysteresis curve \nbecause it is antiferromagnetic [\n28\n]. \n \nTherefore\n,\n \nwe associate the two\n-\nphase behavior to \nthe increase in \nthe mean crystallite size. This\n \nassumption \nagrees\n \nwith the \nbehavior of the m\nean crystallite \nsize \nin function of the thermal treatment \ntemperature \nobtained from Scherrer equation (\ntable \n3\n). \n \nThe absence of \nx\n-\nray diffraction\n \npeaks related with CoFe\n2\n \nphase in the patterns shown \nin figure 1 \nagrees with results obtained by Zhang\n \net al.\n \n[15]. These authors,\n \nusing a similar process, could not \nobserve the presence\n \nof CoFe\n2\n \nin the nanocomposite CoFe\n2\n/CoFe\n2\nO\n4\n,\n \nby XRD, \nuntil the content reached \nabout 30 %, relating the absence of CoFe\n2\n \nphase diffraction peaks to its poor crystallinity. Therefore, the \nabsence of C\noFe\n2\n \ndiffraction peaks in the diffractograms shown in figure \n4\n \ndoes not exclude the \nformation of this phase.\n \nFor information, the thickness of the CoFe\n2\n \nshell around the \nnanostructured \nCoFe\n2\nO\n4\n \ncore was \nestimated for sample \nCFO\n-\n5C\n-\n800\n, that with the higher \nCoFe\n2\n \ncontent, \nsupposing\n \nspherical \nCoFe\n2\nO\n4\n \nnanoparticles with diameter equal to the mean crystallite size obtained from XRD measurements (see \ntable \n3\n)\n \ncovered with a uniform CoFe\n2\n \nshell. M\nass density of both CoFe\n2\nO\n4 \nand CoFe\n2\n \ncrystalline \nphases\n \nwere used\n. \nThe result indicates that only a very thin CoFe\n2\n \nshell\n, having around 1nm, was formed\n.\n \nOne \nhas\n, also, to consider that (i) if disordered CoFe\n2\n \nshell and CoFe\n2\nO\n4\n \ncore can result in respectively larger \nand/\nor smaller values for the CoFe\n2\n \nshell thickness\n, \nreducing the importance of this consideration in a \nmore precise calculation\n, (ii) a nanostructured material (as that studied in this work) can present much \nmore relative surface/volume area than that of a perfect sphere, further reducing the estimated thic\nkness \nof the CoFe\n2\n \nshell. Therefore, the thickness of the CoFe\n2\n \nshell must be \nless than the estimated 1 nm. This \nestimative is fully consistent with the absence of diffraction peaks from a highly disordered CoFe\n2\n \nphase.\n \nAiming\n \nto maximize the reduction of \na CoFe\n2\nO\n4\n \nduring thermal treatment \n(transforming \nmore\n \nCoFe\n2\nO\n4\n \ninto CoFe\n2\n)\n,\n \na mixture powder with \nan extreme \nquantity of carbon (1:24) was also prepared and \nthermally treated in a tubular furnace at 900 °C in argon atmosphere. \nThe \nXRD pattern \nof\n \nsample \nCFO\n-\n24\n-\n900\n \nis\n \nshown in figure \n5\n \ntogether with the theoretical\n \ndiffractogram expected for a pure CoFe\n2\n \nsample\n. \nThe result \nsuggest\ns \nthat something between an almost total and a\n \ntotal conversion \nof\n \nCoFe\n2\nO\n4\n \ninto CoFe\n2\n \ntook place\n. \n \n \nFigure \n5\n \n-\n \nXRD pattern\n \nobtained\n \nfrom\n \nsample \nCFO\n-\n24\n-\n900\n \nand theoretical fit \nsupposing\n \nCoFe\n2\n \ncrystalline \nstructure\n.\n \n \nFinally, on the difficulty reported \nhere, and in previous work [13], \nin attempting to master the \nreduction \nproces\ns\n, this last result indicate that a greater amount of \ncarbon has to be used to achieve larger \nconversion efficiency, even in inert atmosphere.\n \nFuture investigations are needed to improve reduction \nprocess efficiency of CoFe\n2\nO\n4\n \nin CoFe\n2\n, in order to evaluate the magnetic behavior of the nanocomposite \nwith high\ner CoFe\n2\n \ncontent.\n \n \nConclusion\n \nWe use a new procedure to obtain the CoFe\n2\nO\n4\n/CoFe\n2\n \nnanocomposite\n. Despite the XRD patterns \nsuggesting that CoFe\n2\n \nis not formed, magnetic measurements indicate that nanocomposite \nare\n \nobtained, \nhowever with small \namount\n.\n \nThe pre\nparation method of the nanocomposite at 900\no\nC produces samples \nwith two magnetic phase behavior\n \nand a third phase was observed the antiferromagnetic cobalt oxide \nCoO\n. This behavior was not observed in samples prepared at 800\no\nC, indicating an\n \nexchange coupling \nbetween the magnetic phases.\n \n \nThe most important result of this work is to show that H\nC\n \nof the CoFe\n2\nO\n4\n/CoFe\n2\n \nnanocomposite \ncan \nbe \nincrease\nd\n \nusing the same \nmilling \nmethod \nemployed \npreviously\n \nto cobalt ferrite\n.\n \nAs a consequence,\n \nwe \nobtain\ne\nd\n \na sample that reached\n \nan increase in \n(BH)\nmax\n \nof \nabout 240%\n, but f\nuture investigations are needed \nto evaluate the magnetic behavior of the nanocomposite with higher CoFe\n2\n \ncontent.\n \n \nAcknowledgments\n \n \nThe authors would like to thank the\n \nCAPES Brazilian \nfunding agency\n \nto the master students grant.  \nIn addition, E. Baggio\n-\nSaitovitch acknowledges support from FAPERJ through several grants including \nEmeritus Professor fellow and CNPq for BPA and corresponding grants\n.\n \nR\neferences\n \n \n1.\n \nV. Pillai, D. O. 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App. Phys. \n117\n \n(2015) 7A7361.\n \n "    },    {        "title": "1907.10587v1.Understanding_Magnetic_Properties_of_Actinide_Based_Compounds_from_Machine_Learning.pdf",        "content": "(Report: LA-UR-19-25837)\nUnderstanding Magnetic Properties of Actinide-Based Compounds from\nMachine Learning\nAyana Ghosh,1, 2, a)Filip Ronning,3, 4Serge Nakhmanson,1, 5and Jian-Xin Zhu2, 6, b)\n1)Department of Materials Science &Engineering and Institute of Materials Science, University of Connecticut, Storrs, CT,\n06269 - USA\n2)Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545 - USA\n3)Institute of Materials Science, Los Alamos National Laboratory, Los Alamos, NM, 87545 - USA\n4)Condensed Matter and Magnet Science, Los Alamos National Laboratory, Los Alamos, NM,\n87545 - USA\n5)Department of Physics, University of Connecticut, Storrs, CT 06269 - USA\n6)Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, NM,\n87545 - USA\n(Dated: 25 July 2019)\nActinide and lanthanide-based materials display exotic properties that originate from the presence of itinerant or\nlocalized f-electrons and include unconventional superconductivity and magnetism, hidden order; and heavy fermion\nbehavior. Due to the strongly correlated nature of the 5 felectrons, magnetic properties of these compounds depend\nsensitively on applied magnetic field and pressure, as well as on chemical doping. However, precise connection between\nthe structure and magnetism in actinide-based materials is currently unclear. In this investigation, we established\nsuch structure-property links by assembling and mining two datasets that aggregate, respectively, the results of high-\nthroughput DFT simulations and experimental measurements for the families of uranium- and neptunium-based binary\ncompounds. Various regression algorithms were utilized to identify correlations among accessible attributes (features or\ndescriptors) of the material systems and predict their cation magnetic moments and general forms of magnetic ordering.\nDescriptors representing compound structural parameters and cation f-subshell occupation numbers were identified as\nmost important for accurate predictions. The best machine learning model developed employs the Random Forest\nRegression algorithm and can predict magnetic moment sizes and ordering forms in actinide-based systems with 10–\n20% of root mean square error.\nI. INTRODUCTION\nBig-data driven approaches employing supervised, semi-\nsupervised or unsupervised machine learning algorithms are\nbecoming tools of choice in materials physics, chemistry\nand engineering for the task of establishing yet unknown\nstructure-property-performance relationships that may exist\nwithin a given family or class of materials.1–5The success of\nthese tools in elucidating hidden connections between the ma-\nterial or molecular structure and the resulting behavior can be\nattributed to growing availability of databases collating theo-\nretical and experimental materials data across disciplines. In\nparticular, databases aggregating the results of density func-\ntional theory (DFT) computations, which provide a reason-\nable compromise between high accuracy and computational\ncosts and can also process fictitious materials structures, are\nespecially popular as components of prediction-driven strate-\ngies for materials design and discovery.6,7A non-exhaustive\nlist of examples demonstrating applications of machine learn-\ning algorithms in materials science includes multiple inves-\ntigations conducted for the families of technologically criti-\ncal (energy harvesting, storage and efficiency,8–18catalysis,19\nphotovoltaics,20etc.) and pharmaceutical (drug design,21,22\na)Electronic mail: ayana.ghosh@uconn.edu\nb)Electronic mail: jxzhu@lanl.govreaction mechanisms,23–26etc.) compounds.\nIn addition to more generic traits originating from their\ngeneral chemistry and radioactive behavior, lanthanide and\nactinide-based materials exhibit a range of interesting prop-\nerties associated with the filling of the 4 fand 5 felectron\nsubshells. In particular, the interplay of the hybridization of\n5felectrons with itinerant conduction electrons and the on-\nsite Coulomb repulsions among those electrons is responsi-\nble for the behavior exhibited by actinides. Such properties\nmay include an emergence of magnetism27and colossal mag-\nnetoresistance at partial subshell fillings, as well as uncon-\nventional superconductivity and magnetism,28metal-insulator\ntransitions,29hidden magnetic order30and the presence of\nheavy fermions.31Due to strong correlation effects involving\n5f-electrons and their interactions with itinerant conduction\nelectrons, magnetic behavior of actinide-based systems is sen-\nsitive to applied pressure, magnetic field and chemical doping.\nAs a result, actinide-based materials are not only useful in nu-\nclear applications but also constitute an interesting playground\nto push our fundamental understanding of correlated materi-\nals to the limit. So far, only the 4 f-electron magnetism has\nbeen studied with DFT-based Machine Learning (ML) tools\nin a general context of ternary oxide compounds.32A focused\nstudy of magnetic properties in 5 f-electron materials has not\nyet been reported.\nThe main purpose of this work involves a systematic in-\nvestigation of possible connections between the structure and\nmagnetic properties for a variety of different actinide-basedarXiv:1907.10587v1  [cond-mat.mtrl-sci]  24 Jul 20192\nRAW DATA\nCURATED DATASETS\nMACHINE LEARNING MODELS\nFURTHER V ALIDATION AND PREDICTIONSURANIUM-BASED COMPOUNDSDATASET IDFT Simulations\nDATA ANALYTICSCorrelation MapsConditional Inference TreesImportant Descriptors SelectionTrends Analysis\nPERFORMANCE OPTIMIZATIONSelection of AlgorithmsError AnalysisDATASET IIExperimental ReportsENDPOINT: MOMENT SIZE & ORDERINGENDPOINT : MOMENT SIZE\nMODELS DEVELOPMENT\nCOMPILATION OF PREDICTIONSDataset I & Dataset IIEXTENSION OF MODELSExternal Validation on Other Actinide SystemsINTERNAL V ALIDATION OF MODELS\nFIG. 1. Flowchart outlining the main development stages in-\nvolved in construction and validation of machine learning models\nfor predicting cation magnetic moment size and magnetic ordering\nin actinide-based binary systems. Primary stages are shown as grey\nrectangles, while any necessary secondary stages are represented by\nwhite rectangles. Some of the diagram elements introduced here are\nanalyzed in further detail in Fig. 2(b) and accompanying text.\ncompounds in an attempt to establish a general prescription\nfor constructing families of machine learning models that in-\ncorporate computational and experimental knowledge. Com-\nplementary utilization of data (See SM33for references) orig-\ninating from both of these sources is necessary for accurate\nassessment and prediction of the magnetic properties of in-\nterest: average cation moment sizes (but not their ordering,\nwhich may be quite complex) can be easily extracted from\nDFT calculations, while magnetic ordering can be straightfor-\nwardly characterized by experiments. Therefore, the task of\ncompilation and curation of existing experimental reports on\nactinide-based systems is a critical part of this study. The ac-\nquired data serves not only as a base for constructing machine\nlearning models capable of predicting magnetic ordering, but\nin some instances can also provide the necessary validation for\nthe models utilizing only computational data. We must, how-\never, point out the limitations of the experimental data, which\nnaturally translates into the capabilities of machine learning\nmodels to make predictions: in most cases, only the major\nforms of magnetic ordering, such as paramagnetism (PM), fer-\nromagnetism (FM) or antiferromagnetism (AFM) ( as denotedby 1, 2 and 3, respectively in our work ), are reported, while\ninformation about the specific types of AFM or PM, or the ori-\nentation of magnetic moments with respect to crystallographic\naxes is not given. The flowchart shown in Fig. 1, outlines the\nmain stages of this study, which include compilation and cu-\nration of appropriate datasets, performing data analysis with\nstandard data mining tools, construction of machine learning\nmodels and their following internal and external validation.\nThe rest of the paper is organized as follows: Section II\nbriefly reviews the standard methodology utilized for dataset\nacquisition and curation as well as for the machine learning\nmodel building. Section III presents the results of statisti-\ncal analysis for the contents of the both datasets followed by\na discussion of predictive capabilities of the developed ma-\nchine learning models in evaluating cation magnetic moment\nsize and magnetic ordering in uranium- and neptunium-based\ncompounds. Finally, some concluding remarks are provided\nin Section IV .\nII. METHODS\nA. Datasets\nFirst-principles calculations of average spin and orbital mo-\nments were performed using the projector augmented plane-\nwave (PAW) method implemented in the Vienna Ab initio\nSimulation Package (V ASP).34,35The generalized gradient ap-\nproximation (GGA) was adopted to represent the exchange\nand correlation interactions, with the GGA+ Ue\u000b36approach\nutilized to capture the strongly correlated nature of the 5 felec-\ntrons. All computations were carried out with a 500 eV plane-\nwave cutoff energy using tetrahedron method with Blochl cor-\nrections with appropriate Monkhorst-Pack37k-point meshes,\nwhich produced well converged results.\nFor the construction of Dataset I, which is built only on the\ndata extracted from the DFT simulations, the following twelve\nuranium-based binary compounds were utilized: UO 2, U3O8,\nUO3, UN, UC, UP, UP 2, U3P4, UAs, UBi 2, USb 2and UCl 3.\nThe magnetic structures of these compounds are well docu-\nmented in the literature2–5which is the primary motivation be-\nhind choosing them to build Dataset I. Geometrical structures\nfor all of these compounds are shown in Fig. 2(a). For each\ncompound, initial lattice parameters and ionic positions were\nobtained from Inorganic Crystal Structure Database (ICSD),42\nafter which eight individual variants were created by varying\nthe Hubbard parameter Ue\u000bbetween 0 and 6 eV in 2 eV in-\ncrements in the presence or absence of spin-orbit coupling.\nUe\u000bvalues from the same range have been used previously in\na number of DFT-based investigations of actinide compounds\n(See SM33for additional references). Electronic and magnetic\nproperties for each of the eight variants were evaluated both\nfor the ICSD provided structural parameters and after opti-\nmization, which included relaxing the unit-cell shape and vol-\nume to stresses below 0.1 kbar and all the ionic positions until\nthe Hellman-Feynman forces below 10\u00003eV/˚A. Inclusion of\ndata generated using structural parameters as obtained from\nboth ICSD and DFT computations allows us to use informa-3\nUO2UO3U3O8UX (X=N, C, P, As)\nUX2 (X=P, Bi, Sb)U3P4UCl3U\nXUU\nUUUUOOX\nClPO\nCURATED DATASETS\nMACHINE LEARNING MODEL BUILDING (LR, LASSO, KRR, RFR, SVMR)\nMODEL TESTING / INTERNAL V ALIDATION (Size: 20)DATASETS I & IIDATASET ISize: 192\nMAGNETIC MOMENT SIZE &  ORDERINGSELECTED DESCRIPTORS(Primary and Compound)TRAINING + TEST  SET(Sizes: 172 for Dataset I, 203 for Dataset II) PERFORMANCE OPTIMIZATION(Selection of Algorithm, hyperparametersusing Learning Curves Approach)DATASET IISize: 737FILTERING CRITERIA:üAvailable reports on ordering üRemove duplicate entries REDUCED DATASET IISize: 223RAW DATA(a)\n(b)\nFIG. 2. (Color Online) (a) Structural models of 12 uranium-based binary compounds that were assumed in DFT computations for creating\nDataset I. (b) Flowchart showing the stages of development of machine learning models to predict both moment size and ordering.4\ntion of varying fidelity43levels which is helpful to improve\naccuracy in ML-based models.\nFor all computations to generate Dataset I, we consider an-\ntiferromagnetic ordering (AFM I configuration) for U cations\nwith out-of-plane magnetic orientation along the caxis. We\nrestrict ourselves to studying only one type of magnetic con-\nfiguration due to our interest in estimating moment sizes\nwhich do not vary significantly, if other configurations (e.g.,\nFM) are selected instead. The choice of these initial AFM\nmagnetic configurations for all 12 compounds is further dis-\ncussed in SM33along with energy trade-offs between choos-\ning in-plane vs. out-of-plane magnetic orientations. We report\nnominal differences ( <0.3\u0016B) between these two AFM con-\nfigurations as compared to average spin (1.64 \u0016B) and orbital\n(2.82\u0016B) moment sizes for all of these compounds in the list,\nwhich supports our approach in building Dataset I. Overall,\nDataset I is built solely using DFT-simulations and comprises\n16 cases for every compound, giving rise to a total of 192 en-\ntries.\nDataset II was constructed by curating the results of 737\nexperimental reports of standard quality on uranium-based\nbinary compounds, as found in the ICSD.42Only structures\nthat are stable at low temperature were considered, while\ndata on any metastable high-temperature configurations\n(that cannot be straightforwardly characterized by DFT\ncalculations) was discarded. After the removal of duplicate\nentries, 223 data points including information on magnetic\nproperties (cation moment size and ordering) were obtained.\nThese compounds were also categorized into three numbered\ngroups according to the nature of the reported magnetic\nordering in them: paramagnetic (compounds with local\nmagnetic moment but no long range order present) group 1,\nferromagnetic (compounds with magnetic spins aligned in\nthe same directions) group 2, and antiferromagnetic (A-type\nor G-type) group 3.\nThe sizes of both datasets before and after curation, as well\nas the filtering criteria for Dataset II are shown in the top part\nof the machine learning model development flowchart pre-\nsented in Fig. 2(b). For each dataset, 20 entries are kept aside\nfor internal validations and the rest (172 for Dataset I and 203\nfor Dataset II ) are used for training (95 %) and testing (5 %)\nof ML-based models.\nB. Descriptors\nFor Dataset I, the following eight primary de-\nscriptors were considered: lattice parameters\n(magnetic unit cell parameters\n3p\nnumber of actinide elements)alatt,blatt and clatt (˚A), atomic\nvolume (magnetic unit cell lattice parameters\nnumber of actinide elements)volume (˚A3), Hubbard\nparameterUe\u000b(eV), spin-orbit coupling strength SOC (eV),\ncation 5 f-subshell occupation number (number of total\nvalence electrons of actinide element – valence of anion)\nNocc(5f)and system Fermi energy level EF(eV). For\neach primary descriptor x, additional compound descriptors\nwere generated using 10 prototypical functions, namely,\nx2,x3,exp(x),sin(x),cos(x),tan(x),sinh(x),cosh(x),tanh(x), and ln(x), to allow for possible non-linearities in the\nconnections between the descriptor and endpoint properties.\nThe descriptor space (exp. descriptors) for Dataset II con-\ntains all structural parameters (as defined before), number of\nformula units and Nocc(5f), as extracted from the respec-\ntive experimental reports. Furthermore, for every entry in\nDataset II, a matrix representation called Orbital field matrix\n(OFM)44,45as implemented in a Python library46was com-\nputed using distances between coordinating atoms, valence\nshells and V oronoi Polyhedra weights, providing information\non the chemical environment of each atom in the unit cell. The\nOFM elements are defined in literature44,45as following:\nX0p\nij=npX\nk=1op\niok\nj\u0012p\nk\n\u0012p\nmax\u0010(rpk); (1)\nwhere,i;j2D= (s1;s2;:::;f14), for central iand coordi-\nnatingjorbitals, respectively; op\niandok\njare elements of one-\nhot vectors ( i;j) of the electronic configuration pand neigh-\nboring atom as indexed by k. The weight of the atom kin the\ncoordination of the central atom at site pis given by\u0012p\nk=\u0012p\nmax,\nwhere\u0012p\nkis the solid angle determined using V oronoi poly-\nhedron. The number of nearest-neighbor atoms surrounding\natom sitepisnp. The size and distance-dependent weight\nfunction is also included by \u0010(rpk).\nTwo sets of ML-based models were constructed to predict\nmagnetic ordering using experimental descriptors and OFM\nrepresentation for which the results are discussed later in Sec-\ntion III.\nC. Data Analytics\nPrior to the development of machine learning models, both\ndatasets were analyzed using some standard data analytics\ntechniques to determine the existence of any relationships\nbetween the structural descriptors and the targeted magnetic\nproperties. The resulting inferences drawn from these anal-\nyses are complementary in choosing appropriate descriptors\nspace and algorithms in later stages to develop ML-based\nmodels. The primary descriptor sets for Dataset I were sub-\njected to a Pearson correlation filter to remove features that\nexhibit a high correlation with the other descriptors in each\nset. The same approach was also applied to the combined set\nof primary and compound descriptors as further discussed in\nSM.33Additionally, for Dataset I we have employed a condi-\ntional inference procedure with Bonferroni-corrected signif-\nicance (p-value <0.05) value, used as the splitting criteria\n(stopping rules) for each node while constructing trees as im-\nplemented in R version 3.4.2 via CTree algorithm.47The split-\nting process is continued recursively throughout the whole\nDataset I. The results of this analysis are presented in the\nSM.33\nWe have utilized all of 223 entries in Dataset II to per-\nform median analysis. The dataset was divided into two sub-\nsets (below and above the median value) based on the median\nvalue for each descriptor. For every descriptor, the magnetic\nordering was assigned to the respective entries belonging to5\n(c)(d)\n(e)(f)(g)012345678\nAFMFMParaLattice Parameter a (Å)012345678\nAFMFMParaLattice Parameter b (Å)\n0123456789\nAFMFMParaLattice Parameter c (Å)050100150200250300350400450\nAFMFMParaVolume (Å3)01234567\nAFMFMParaFUAverage Descriptor value < MedianAverage Descriptor value ≥ MedianDifference between both the aboveHorizontal Axis: Magnetic OrderingPMPM\nPMPMPM\n(a)(b)\nNocc(5f)Nocc(5f)EFEFCorrelation Maps for primary features forDataset I\nMedian-Variance Analysis for experimental features for Dataset II\nFIG. 3. (Color Online) Correlation matrices representing Pearson correlation coefficients for primary descriptors applicable to Dataset\nI for (a) spin moment size and (b) orbital moment size endpoints. The primary descriptor space for Dataset I consists of eight features:\nlattice parameters alatt,blatt andclatt, and volume , Hubbard parameter Ue\u000b, spin-orbit coupling strength SOC, cation 5 f-subshell occupation\nnumberNocc(5f)and system Fermi energy level EF. (c-g) Dataset II is divided into two sets: below median value (grey), above median\nvalue (orange), difference (blue) for all experimental descriptors. The considered features are (c-e) lattice parameters ( ˚A) in all directions, (f)\ncell volume ( ˚A3), (g) formula units (FU). The grouping is done based on the respective medians and type of magnetic ordering reported. The\nmedian values for these features are (c) 5.588 ˚A, (d) 5.5 ˚A, (e) 5.62 ˚A, (f) 191.85 ˚A3and (g) 4.\ntwo resultant subsets. The bar charts (Fig. 3 (c-g)) also report\nthe differences between the mean values of the descriptors.\nLarger difference for a particular descriptor should lead to a\nbigger variance when used in decision-tree type algorithms\n(such as Random Forest Regression).D. Algorithms\nFive algorithms, including Linear Regression ( LR), Least\nAbsolute Shrinkage and Selection Operator ( LASSO ), Kernel\nRidge Regression ( KRR ), Random Forest Regression ( RFR)\nand Support Vector Machine Regression ( SVMR ), as imple-\nmented in R47version 3.4.2 were employed to construct\nmachine-learning models from Datasets I and II. Regression-6\nbased algorithms were chosen over classification-based ones,\nas both endpoints of interest (magnetic moment size and or-\ndering) were either computed for or assigned numerically to\neach entry in respective datasets. Optimized hyperparameters\nused for all five algorithms are presented in the SM.33\nE. ML-based Model development and validation\nOne of the main concerns of conventional machine learn-\ning based models development is the selection of data-points\nused in each of these steps. For both sets of models, the num-\nber of entries in each dataset is restricted to less than 230\nentries, which may limit the accuracy of the models to be\nbuilt. The standard deviations in the predicted values for the\ntest sets may be as large as 5-10%, due to which any com-\nparisons originating from a single set of predictions may be\nmisleading. This has been further explained in the SM,33\nwhere two such cases are compared. In order to avoid any\nstatistical bias, we have built a learning curve by varying\ntraining set size to estimate performance of all models. For\neach point plotted in a learning curve diagram,21the average\nRMSE is reported as calculated using 1000 randomly gener-\nated (sampling with replacement) training and test set eval-\nuations. The average RMSE for a training set size of Nis\ndenotedETrain(N), whereas for the corresponding test set it is\ndenotedETest(N), however, in this case the size of the set is\nnow the total number of points minus N. The test set RMSE\ngives the expected error for a given model whereas the differ-\nence between ETest(N) andETrain(N) is an estimation of how\nmuch variance or overfitting the model contains.\nBoth models predicting moment size and ordering built on\nDataset I and II were tested using the internal validation set,\ni.e. 20 entries (Fig. 2(b)) in each case that are kept aside on the\nmodel development stage as mentioned earlier. Moreover, we\napplied these models to three external sets of actinide-based\nbinary and ternary compounds, for which information on ei-\nther ordered moments or ordering is scarcely available in lit-\nerature. Validation on a dataset composed of materials that\nare dissimilar to that used in model development stage helps\ntest the robustness and applicability of our conventional ML\nmodels.\nWe note that the descriptor SOC is the strength of such cou-\npling present in these systems as determined by the DFT based\ncomputations for Dataset I. Hence, this is unavailable for all\ncompounds present in the validation set. As an alternative way\nto include this interaction in our models as applicable to the\nvalidation sets, we consider SOC by the form of its presence\n(1) or absence (0) instead of including its strength value. The\nUe\u000bcan also be varied manually in the descriptor space de-\npending on the estimate of this parameter to capture the strong\ncorrelation effect among the f-orbital electrons. In addition,\nthe valence electrons of other atoms and OFM as applicable to\nthe ternary set are also included in the feature space for testing\nmodels built on Dataset I and II.\n(a)\n(b)\n(c)Endpoint: spin moment size (µB)\nEndpoint: orbital moment size (µB)\nEndpoint: orderingTraining Set SizeRMSE (µB)\nTraining Set SizeRMSERMSE (µB)FIG. 4. (Color Online) Learning curves for three machine learning\nmodels predicting (a) spin moment size, (b) orbital moment size and\n(c) magnetic ordering as constructed using the RFR algorithm.ETrain\nandETestrefer to average training and test set root mean square er-\nrors. Additionally, we report the average % mean absolute test errors\nin predicting spin (a) and orbital (b) moment size are 14%, 17%. For\nordering as represented in (c) this % mean absolute test error is 8%.\nIII. RESULTS AND DISCUSSION\nHere, we discuss the results obtained from preliminary data\nanalysis and ML-based models as applicable to both Dataset\nI, II, internal and external validation sets in sequential order,\nwith each primary step highlighted throughout the section.\nFrom the correlation analysis as presented in Fig. 3(a) on\nthe set of primary features, it is evident that volume ,Nocc(5f)\nandEFhave large Pearson coefficients with respect to the7\nTABLE I. Test set error ETest(N) and difference (overfitting) between the training ETrain(N) and test set errors for the training set size N= 50\nand the largest training test size Nmax, as obtained from the learning curves constructed with five different regression algorithms. The predicted\nendpoints are either size of spin or orbital magnetic moment in \u0016B, or magnetic ordering. Errors in prediction of magnetic ordering are with\nrespect to the indices (1 - PM, 2 - FM, 3 - AFM) as introduced earlier in Section II.\nEndpoint Algorithm ETest(50)ETest\u0000ETrain (50)ETest(Nmax)ETest\u0000ETrain (Nmax)\nSpin moment LR 0.61 0.44 0.33 0.12\nLASSO 0.44 0.24 0.30 0.05\nKRR 0.30 0.08 0.28 0.03\nSVMR 0.84 0.04 0.83 0.01\nRFR 0.32 0.13 0.17 0.04\nOrbital moment LR 1.46 0.88 0.96 0.24\nLASSO 0.90 0.14 0.80 0.10\nKRR 1.07 0.4 0.96 0.21\nSVMR 1.01 0.06 0.94 0.05\nRFR 0.41 0.22 0.19 0.03\nOrdering LR 1.77 1.14 1.02 0.50\nLASSO 1.32 0.82 0.98 0.41\nKRR 1.76 1.29 1.01 0.49\nSVMR 0.86 0.03 0.85 0.02\nRFR 0.41 0.22 0.12 0.04\nendpoint (spin moment size), suggesting that these features\nshould be included in the descriptor space when building ML-\nbased models on the later stage to predict spin moment size.\nHowever, since Nocc(5f)andEFare highly correlated to\neach other, we only keep Nocc(5f)as one of the primary\ndescriptors. We note that the EFas obtained from the DFT\nbased computations is relative and can be placed anywhere\nin the energy band gap between the occupied and unoccupied\nstates. Therefore the estimation of charges as accounted by the\nNocc(5f)feature is more appropriate to retain information on\nthe electron density of states which is important in predicting\nmagnetic moment size. The SOC has the highest correlation\ncoefficient with respect to the endpoint in the matrix, where\nthe endpoint is orbital moment size as shown in Fig. 3(b). This\nis expected for the primary feature that accounts for strong\nSOC originating from 5 fshell electrons. This approach as ex-\ntended to space of primary and secondary features shown in\nthe SM33has allowed us to shortlist 61 features by excluding\nhighly-correlated ones (correlation factor >0.85).\nNext, the conditional inference procedure as employed on\nDataset I and presented in the SM33shows that alatt,volume ,\nUeffand SOC are the top features capable of grouping the data\nwell for spin and orbital moment size as endpoints, respec-\ntively.\nWe have used a slightly different approach ( median analy-\nsis) to analyze Dataset II by determining medians and differ-\nences in average values of the features as explained in Sec-\ntion II.C. Based on the median values of the five features, the\nDataset II is divided into subgroups followed by assigning the\ncorresponding ordering and calculating the difference (vari-\nance) in average feature values for the subgroups as shown in\nFig. 3(c-g). The median values for each of these experimental\ndescriptors ( lattice parameters ,volume ,number of formulaunits ) are 5.588 ˚A, 5.5 ˚A, 5.62 ˚A, 191.85 ˚A3and 4 respec-\ntively. There is total 112 entries in the subgroup where alatt\n>=median value of alatt, out of which 24 are AFM, 44 are\nFM and 44 are PM. For the similar subgroups formed by other\nfeatures such as blatt,clatt,volume ,number of formula units ,\nthe number of entries are listed as: (26 - AFM, 50 - FM, 36 -\nPM), (30 - AFM, 46 - FM, 36 - PM), (24 - AFM, 48 - FM, 40\n- PM) and (42 - AFM, 70 - FM, 51 - PM), respectively.\nBoth conditional inference tree and median-variance meth-\nods suggest that the decision tree type of algorithm may have\nbetter performance compared to other regression algorithms,\nif used to build ML models for predicting moment size and\nordering.\nOverall, these investigations performed using standard data\nanalytics techniques are useful for cultivating some advance\nknowledge about the available data and identification of fea-\ntures important to predicting endpoints as well as any incon-\nsistencies present in the datasets.\nThe primary results produced by the ML-based models are\npresented as the learning curves in Fig. 4. For all three cases,\ntheRFR algorithm outperforms the other algorithms signifi-\ncantly, as shown in both Fig. 4 and Table I. The average spin\nmoment size of compounds used for training these models is\n1.64\u0016B. For the orbital moment size, the average is 2.82 \u0016B,\ncounting only the training data points computed by including\nthe SOC. The total moment size can be obtained using the vec-\ntor sum of both spin and orbital moments, pointed in the op-\nposite direction to each other due to Hund’s rule for f-electron\nshell that is less than half filled. The average ETestvalues in\npredicting spin and orbital moment size are 0.17 \u0016Band 0.19\n\u0016B, respectively as mentioned in Table I. In both cases, RFR\noverfits the data by approximately 4%. The LASSO, KRR\nand SVMR algorithms on the other hand show minimal over-8\n(e)(f)(g)\n00.511.522.533.544.55\n010203000.511.522.533.544.555.56\n05010000.511.522.533.544.555.5\n010203040Moment Size (µB)Compounds in External Validation set ICompounds in External Validation set IICompounds in External Validation set III\nOrbital MomentSpin Moment(h)(i)(j)Predicted OrderingCompounds in External Validation set ICompounds in External Validation set IICompounds in External Validation set III\nReported Ordering\nReported Ordering\nReported Ordering(a)\n(b)\n(c)\nUNUSb2UO2UBi2UP\nPredicted OrderingReported Ordering\n(d)\nFIG. 5. (Color online) ML-based models predictions of (a) spin and (b) orbital moment size on the internal validation set from Dataset I. A\ncomparison between experimentally reported and predicted total moment size is represented in (c) for selected (for which moment sizes are\navailable in literature) compounds in this set. Prediction on the internal validation set from Dataset II is shown in (d). The radius of each\nmarker represents the absolute error in each prediction for each entry present in the set as clustered together based on their ordering. (e-g)\nIndependent predictions of spin and orbital moment size for all compounds present in three external validation sets using two models solely\nbuilt on Dataset I. The error bars in each case represent the average root mean square errors in these predictions as obtained by averaging\nover 1000 such ML-based models predictions. (h-j) Comparative predictions of magnetic ordering for all compounds listed in three external\nvalidation sets using OFM-based model built on Dataset II. The values of each prediction is obtained by averaging over 1000 such respective\nML-models based predictions.\nfitting but plateau at higher ETest(N) as shown by comparative\nlearning curves (see SM33for more details).\nThe average RMSE in predicting magnetic ordering using\nDataset II is 0.12. The OFM representation plays a key role\nin significantly improving the model performance (RMSE re-\nduced by\u001830%) to predict magnetic ordering by including\ninformation on the valence shells and local coordination envi-\nronment of the system. The details of the errors evaluation in\nthe learning curves for both models are discussed in the SM.33\nThe most important features (structural parameters and f-\nsubshell occupation numbers) identified by RFR algorithm\nbased models comply with that found earlier by the data ana-\nlytics techniques for both moment size and ordering.\nWe have also compared the average nearest neighbor\ndistances (see SM33) for every entry in Dataset II to the\nHill limit6that provides restrictions, under which magneticordering occurs in actinide systems. Our results are in\ngood agreement with the Hill limit for uranium ions (3.4 ˚A-\n3.6˚A). This establishes a physical significance behind lat-\ntice parameters, which are identified as important features\nto predict both moment size and ordering. The average\nlattice parameters obtained from Dataset I are compara-\nble within 15% to entries present in Dataset II that re-\nportedly exhibit antiferromagnetic ordering at low tempera-\ntures. This along with results from median-variance analy-\nsis performed on Dataset II also provides quantitative mea-\nsure for structural parameters to observe a specific type\nof magnetic ordering (in high likelihood). For example,\na uranium-based binary compound with alatt>=5.58 ˚A,\nis more likely to exhibit antiferromagnetic ordering at low\ntemperatures.\nThree of the models were tested first on the internal valida-9\ntion sets kept aside in Dataset I and Dataset II. We employed\na comparable approach to the learning curve by reporting the\naverage moment size and ordering for each entry as obtained\nby averaging over 1000 ML-based models predictions. Fig-\nure 5 (a and b) shows predictions made on the internal valida-\ntion set acquired using Dataset I. The average RMSE for the\nspin, orbital moment size predictions on the internal valida-\ntion sets (for Ue\u000b= 4 eV) are 0.20 \u0016B, 0.25\u0016B, respectively.\nFor compounds such as UN, USb 2, UO 2, UBi 2and UP, total\nmoment sizes are available in the literature, and can be used to\ncompare the ML-based models predictions for Ue\u000b= 4 and 6\neV as represented in Fig. 5(c). The average RMSE for the pre-\ndiction of total moment size for these compounds are 0.32 \u0016B\nand 0.35\u0016BforUe\u000b= 4 and 6 eV , respectively. This analysis\nalso signifies the dependence of moment size on Ue\u000bas cap-\ntured by the ML models. Our ML-based regression algorithm\ncan predict numerical values marking AFM, FM and PM with\nan average RMSE of 0.15 as shown in Fig. 5(d), where the\npredictions are compared with that reported within the inter-\nnalvalidation set of Dataset II.\nFinally, to assess the performance and applicability of these\nmodels to other actinide systems, we have compiled three ex-\nternal validation sets, first two containing uranium-based bi-\nnary and ternary compounds13and a third dataset consisting\nof neptunium-based binary compounds. The results of these\npredictions using the same averaging technique as applied be-\nfore to the internal validation sets are shown in Figs. 5(e-g)\nand 5(h-j). We note that most of the predictions (Fig. 5 (e-\ng)) of the moment size cannot be validated due to the scarcity\nof experimental information. However, predictions of mag-\nnetic ordering indices that classify compounds into three sim-\nple groups of AFM, FM and PM are comparable with those\nreported in the literature13without accounting for the exact\nspin textures.\nTheexternal Set I of binary compounds include 34 differ-\nent uranium-based compounds U 2C3, U3As4, U3Bi4, U3Sb4,\nUAl2, UAl 3, UAs 2, UB 2, UB 4, UBi, UCo 2, UFe 2, UGa 3,\nUGe 2, UGe 3, UIn 3, UIr 2, UIr 3, UIr, UMn 2, UNi 2, UPb 3,\nUPd 3, UPt 2, UPt 5, UPt, URh 3, US, USb, USe, USi 3, USn 3,\nUTe and UTl 3. These are not present in either of the Datasets\n(I & II) used in model development reported above. The top 5\nmost common structure types in this list belong to cubic crys-\ntal family with space group numbers 221, 220, 225, 227 and\n191. There are 9 compounds that exhibit AFM, 10 with FM\nand the rest have PM ordering at low temperatures. The aver-\nage RMSE (based on ML-model built on Dataset I) in predict-\ning moment size for these 9 compounds are 0.21 \u0016Band 0.30\n\u0016Brespectively for spin and orbital moment size. For order-\ning, the average RMSE is 0.23 (based on ML-model built on\nDataset II) as predicted for all 34 compounds, shown in Fig. 5\n(h).\nExternal Set II containing ternary compounds has 141 dif-\nferent entries, out of which 51 exhibit AFM ordering, 31\nshow FM ordering and rest are paramagnetic at low temper-\natures. These compounds commonly belong to families of\northorhombic, tetragonal, hexagonal and cubic crystal sys-\ntems with the five most common space groups being 62, 139,\n123, 127 and 189. We predict moment sizes (based on ML-model built on Dataset I) for these 51 compounds with average\nRMSE of 0.23 \u0016Band 0.28\u0016Bfor spin and orbital moments\nrespectively, whereas this error is 0.24 as depicted in Fig. 5 (i)\nfor estimating ordering (based on ML-model built on Dataset\nII).\nExternal Set III has a total of 44 entries of 35 unique\nneptunium-based compounds NpAl 2, NpAl 3, NpAl 4, NpGa 2,\nNpGa 3, NpIn 3, NpIr 2, Np 2N3, NpNi 2, Np 2O5, Np 2Se5,\nNp3S5, Np 3Se5NpAs, NpAs 2, NpB 2, NpC, NpCo, NpFe 2,\nNpGe 3, NpIn 3, NpMn 2, NpN, NpN 2, NpNi 2, NpO 2, NpOs 2,\nNpP, NpPd 3, NpS, NpSb, NpSb 2, NpSi 2, NpSi 3and NpSn 3.\nThe most common structure type among these compounds\nis rocksalt cubic followed by Laves phase cubic and Au-\nricupride. These three structure types are also common among\ncompounds in Dataset I and II. Hence, it suggests that the\nmodels may be capable of predicting the endpoints with sim-\nilar accuracy as reported earlier in the section. In external\nSet III, 17 entries have AFM ordering as reported in litera-\nture. These 17 entries are used to validate the models (based\non ML-model built on Dataset I) predicting spin and orbital\nmoment size with average prediction RMSE of 0.19 ( \u0016B) and\n0.27 (\u0016B), respectively. All 44 entries are considered to pre-\ndict magnetic ordering using model built on Dataset II. The\naverage prediction error for ordering is reported as 0.14 for\nthis set as shown in Fig. 5 (j).\nOverall, the ML-based models built on Dataset I and II\nare capable of delivering reasonable predictions of moment\nsize and ordering for actinide-based binary and ternary com-\npounds.\nIV. CONCLUSIONS\nIn conclusion, we have compiled two datasets contain-\ning both computational and experimental reports on magnetic\nproperties of uranium-based binary compounds. Through var-\nious data analytics techniques, we have identified several de-\nscriptors that are critical for understanding magnetism occur-\nring in such systems, even before building any predictive mod-\nels. Later, these were used in developing sets of machine\nlearning models to predict moment size and ordering. We\nhave also extended this approach to other actinides and as-\nsessed the performance of the models. Currently the models\ntrained on Dataset I can only predict moment size for AFM\nordered structures. Predicting magnetic ordering using esti-\nmation of nearest, next nearest neighbor exchange parameters\nrequire additional DFT computations for other magnetic con-\nfigurations, which is beyond the scope of the current work.\nOverall, this general prescription employing both computa-\ntional and experimental results to construct machine learn-\ning models describing magnetism in actinide-based materials\nhelps understand structure-property relationships that may ex-\nist in such complicated structures.10\nACKNOWLEDGMENTS\nA.G. acknowledges the hospitality of Los Alamos National\nLaboratory, where this project was initialized. She is also\nthankful to Dr. Lydie Louis, Dennis P. Trujillo, Dr. Ghan-\nshyam Pilania and Dr. Geoffrey P.F. Wood for their helpful\ncontributions to code development and discussions on imple-\nmentation of various machine learning techniques. 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Mater. 18, 756 (2017).11\nUO2\nUO3\nU3O8\nUX (X=N, C, P, As)\nUX2(X=P, Bi, Sb)\nU3P4\nUCl3U\nUUU\nUUOO\nPXCompoundAtomsUO2U = 4, O = 8U3O8U = 6, O = 16UO3U = 2, O = 6UX(X=N,C,P,As)U = 4, X = 4UX2(X=P,Bi,Sb)U = 2, X = 4U3P4U = 12, P = 16UCl3U = 2, Cl = 6\nX\nUO\nFIG. 6. (Color online) Structural Models (with AFM I configuration)\nof 12 compounds from Dataset I.\n46Ward, L., Dunn et al. Matminer: An open source toolkit for materials data\nmining. Comput. Mater. Sci., 152, 60-69 (2018).\n47R: A language and environment for statistical computing. R Foundation for\nStatistical Computing, Vienna, Austria (2013). http://www.R-project.org/.\n48Hill, H.H. The Early Actinides: the Periodic System’s f Electron Transition\nMetal Series, in Plutonium and Other Actinides, AIME, New York, (1970).\n49Sechovsky, V . and Havela, L. Magnetism of ternary intermetallic com-\npounds of uranium. Handbook of magnetic materials, 11, 1-289 (1998).\nA. Contributions\nA.G. performed all computations and analysis and prepared\nall the figures. J.-X.Z. guided the project. F.R. provided in-\nsights on experimental reports and participated in the discus-\nsion of the results. A.G. and J.-X.Z. wrote the paper with\ninputs from F.R. and S.N.\nB. Additional Information\nThe authors declare no competing interests.\nV. SUPPLEMENTARY MATERIALS\nA. Magnetic configurations\nWe note that these magnetic configurations are in ac-\ncordance with that reported in literature.1–5For U 3P4,\nferromagnetic configuration (FM) has a lower energy as\ncompared to that in AFM I. This is also confirmed by\nadditional DFT + Hubb U eff(eV) computations where the\nenergy difference between these two magnetic configurations\nis\u00183.26 eV . The magnetic spins are assumed to be aligned\nalong x axis in our results.TABLE II. Structural parameters and symmetry groups of 12 com-\npounds from Dataset I as computed using DFT + Hubb U eff(eV).\nAll of these compounds have the same AFM I (self-consistently con-\nverged) configuration.\nCompound lattice parameters ( ˚A) Space Group Hubb U eff(eV)\n(1) UO 2 a = b = c = 5.4682 225, Fm-3m 0, 2, 4, 6\n(2) U 3O8 a = 6.7039, b = 11.9499, c = 4.1420 21, C222 0, 2, 4, 6\n(3) UO 3 a = 8.3299, b = 4.1649, c = 4.1649 221, Pm-3m 0, 2, 4, 6\n(4) UN a = b = c = 4.4889 225, Fm-3m 0, 2, 4, 6\n(5) UC a = b = c = 4.9597 225, Fm-3m 0, 2, 4, 6\n(6) UP a = b = c = 5.5869 225, Fm-3m 0, 2, 4, 6\n(7) UAs a = b = c = 5.7767 225, Fm-3m 0, 2, 4, 6\n(8) UP 2 a = b = 3.8099, c = 7.7639 129, P4/nmm 0, 2, 4, 6\n(9) UBi 2 a = b = 4.4450, c = 8.9079 129, P4/nmm 0, 2, 4, 6\n(10) USb 2 a = b = 4.2719, c = 8.7410 129, P4/nmm 0, 2, 4, 6\n(11) U 3P4 a = b = c = 8.2119 220, I-43d 0, 2, 4, 6\n(12) UCl 3 a = b = 7.4429, c = 4.3210 176, P6 3/m 0, 2, 4, 6\nTABLE III. Differences in energy and moment sizes between spin-\norientations chosen in-plane and out-of-plane.\nCompound \u0001E (eV) \u0001spinmom (\u0016B) \u0001orbitmom (\u0016B)\n(1) UO 2 0.032 0.001 -0.009\n(2) U 3O8 0.044 0.071 -0.011\n(3) UO 3 0.096 0 0\n(4) UN 0.007 0.015 -0.037\n(5) UC 0.069 0.054 -0.074\n(6) UP -0.003 -0.090 -0.068\n(7) UAs -0.002 -0.006 -0.02\n(8) UP 2 -0.206 0.008 -0.036\n(9) UBi 2 0.032 0.029 0.023\n(10) USb 2 0.024 0.028 0.038\n(11) U 3P4 0.067 -0.011 0.087\n(12) UCl 3 0.279 -0.009 -0.011\nB. Hyperparameters for ML models\nThe optimized hyperparameters as obtained using a grid-\nsearch method for the regression algorithms are listed below.\nThe amount of penalization ( \u000b) is 0.01 for LASSO. KRR uses\na linear kernel with a regularization constant of 0.1 SVMR\nbased models has used cost and epsilon parameters of 500 and\n0.04 respectively. RFR has used 60 decision trees and all the\nalgorithms are also subjected to 10 cross-validations for each\nmodel.12\nC. Statistical Bias\nRun 1:Training : RMSE = 0.165, MAE = 0.105 Test : RMSE = 0.347, MAE = 0.188Run 2:Training : RMSE = 0.149, MAE = 0.098 Test : RMSE = 0.457, MAE = 0.307(a)(b)\nFIG. 7. (Color online) The root mean square and mean absolute er-\nrors vary between runs even when same model parameters are uti-\nlized due to statistical bias arising from small dataset size.\nD. Correlation Maps for Dataset I for all features\nEndpoint:\tSpin\tmoment\tSize\t(μB)Endpoint:\tOrbit\tmoment\tSize\t(μB)\nFIG. 8. (Color online) Correlation maps for combined set of primary\nand compound features.E. Decision Trees\nEndpoint: Spin moment Size (µB)\nFIG. 9. (Color online) Conditional inference trees showing catego-\nrization of spin moment sizes based on primary features of all entries\nin Dataset I. Each node is described by the features used at the split.\nThe split is chosen according to Bornferroni-corrected significance\n(p-value) which is also given in the diagrams. Vertical axes of the box\nplots signify the moment sizes whereas thick lines are median, boxes\nare quartiles, whiskers are non-outlier minimum and maximum and\noutliers are specified by dots. The number of response variables at\neach terminal node is also given.\nEndpoint: Orbit moment Size (µB)\nFIG. 10. (Color online) Conditional inference trees showing catego-\nrization of spin moment sizes based on primary features of all entries\nin Dataset I. Each node is described by the features used at the split.\nThe split is chosen according to Bornferroni-corrected significance\n(p-value) which is also given in the diagrams. Vertical axes of the box\nplots signify the moment sizes whereas thick lines are median, boxes\nare quartiles, whiskers are non-outlier minimum and maximum and\noutliers are specified by dots. The number of response variables at\neach terminal node is also given.13\nF. ML models results\n(a)(b)(c)\nFIG. 11. (Color online) Comparative learning curves for machine\nlearning models build using linear, LASSO, KRR, RFR and SVMR\nalgorithms to predict end properties such as (a) spin moment size, (b)\norbit moment size and (c) ordering respectively.\nTABLE IV . Test set errors and difference (overfitting) between the\ntraining and test set errors for the training set size of 50 and the\nlargest training test size as obtained from learning curves for two\nmachine learning models predicting ordering using RFR algorithms.\nEndpoint E Test(50) E Test\u0000ETrain(50) E Test(max) E Test\u0000ETrain(max)\nOrdering 0.52 0.32 0.20 0.12\nOrdering (using OFM) 0.41 0.22 0.12 0.04\nG. Hill Limit Comparison\n0123\n01234567dU-U(Å)Magnetic Ordering(a)(b)\nFIG. 12. (Color online) (a) Reference6Hill plot for a selected number\nof superconducting, paramagnetic, ferromagnetic and antiferromag-\nnetic uranium compounds. (b) Average nearest neighbor distances\nfor all entries in Dataset II plotted according to the corresponding\nobserved ordering.\nH. Information of Validation Set III\nExternal Validation Set III7–21has compounds : U 2Co2In,\nU2Co2Sn, U 2Co3Si5, U 2Fe2Sn, U 2Ir2Sn, U 2Mo3Ge4,\nU2Mo3Si4, U 2Nb3Ge4, U 2Ni2In, U 2Ni2Sn, U 2Pd2In,\nU2Pd2Sn2, U2Pt2In, U 2Pt2Sn, U 2PtC2, U2Rh2Sn, U 2RhIn 8,\nU2Ru2Sn, U 2Ta3Ge4, U 2Wi3Si4, U 3Al2Si3, U 3Au3Sn4,\nU3Co3Sb4, U3Cu3Sb4, U3Cu3Sn4, U3Cu4Ge4, U3Ir3Sb4,\nU3Ni3Sb4, U 3Ni3Sn4, U 3Ni4Si4, U 3Pd3Sb4, U 3Pt3Sb4,\nU3Pt3Sn4, U3Rh3Sb4, U3Rh4Sn13, U3Ru4Al12, U4Os7Ge6,\nU4Re7Si6, U 4Ru7Ge6, U 4Tc7Ge6, U 4Tc7Si6, UAsSe,\nUAsTe, UAu2A,l UAu2In, UAu2Sn, UAuAl, UAuGa,\nUAuGe, UAuSi, UAuSn, UCo 2Ge2, UCo 2P2, UCoAs 2,\nUCoGa 5, UCoGa, UCoGa, UCoGe, UCoP 2, UCoSi, UCoSn,UCoSn, UCr 2Si2, UCrC 2, UCu 2As2, UCu 2Ge2, UCu 2P2,\nUCu 2Si2, UCu 2Sn, UCuBi 2, UCuGa, UCuGe, UCuP 2,\nUCuSb 2, UCuSi, UCuSn, UFe 2Ge2, UFe 2Si2, UFeAl,\nUFeAs 2, UFeGa 5, UFeGa, UFeGe, UFeSi, UIr 2Ge2, UIr 2Si2,\nUIrAl, UIrGe, UIrSi 3, UIrSi, UIrSn, UMn 2Ge2, UMn 2Si2,\nUNi 1:6As2, UNi 2Al3, UNi 2Ga, UNi 2Ge2, UNi 2Si2, UNi 2Sn,\nUNiAl, UNiGa, UNiGa 3, UNiGa 5, UNiGe, UNiSb 2, UNiSi,\nUNiSn, UOsGa 5, UPd 2Al3, UPd 2Ga, UPd 2Si2, UPd 2Sn,\nUPdGa 5UPdGe, UPdIn, UPdSb, UPdSi, UPdSn, UPt 2Si2,\nUPt4Au, UPtAl, UPtGa 5UPtGe, UPtIn, UPtSi, UPtSn,\nURh 2Ge2, URhAl, URhGa 5, URhGe, URhIn 5, URhSi,\nURhSn, URu 2P1:894, URu 2Si2, URu 2Si2, URu 4B4, URuAl,\nURuGa 5, URuSb, URuSn.\n1Burlet, P., Rossat-Mignod, J., V ogt, O., Spirlet, J.C. &Rebivant, J. Neutron\ndiffraction on actinides. J. Less Common Metals 121, 121-139 (1986).\n2Troc, R., Leciejewicz, J. %Ciszewski, R. Antiferromagnetic Structure of\nUranium Diphosphide. Phys. Status Solidi B 15, 515-519 (1966).\n3Zhou, F. &Ozolins, V . Crystal field and magnetic structure of UO 2. Phys.\nRev. B 83, 085106 (2011).\n4Wen, X.D., Martin, R.L., Scuseria, G.E., Rudin, S.P., Batista, E.R. &Bur-\nrell, A.K. Screened hybrid and DFT+ U studies of the structural, elec-\ntronic, and optical properties of U 3O8. J. Phys. Condens. Matter 25, 025501\n(2012).\n5Lebegue, S., Oppeneer, P.M. &Eriksson, O. Ab initio study of the elec-\ntronic properties and Fermi surface of the uranium dipnictides. Phys. Rev.\nB73, 045119 (2006).\n6Hill, H.H. The Early Actinides: the Periodic System’s f Electron Transition\nMetal Series, in Plutonium and Other Actinides, AIME, New York, (1970).\n7Leciejewicz, J., Troc, R., Murasik, A. &Zygmunt, A. Neutron diffraction\nstudy of antiferromagnetism in USb 2and UBi 2. Phys. Status Solidi B 22,\n517-526 (1967).\n8Brodsky, M.B. Magnetic properties of the actinide elements and their metal-\nlic compounds. Rep. Prog. Phys 41, 1547 (1978).\n9Gurtovoi, K.G. &Levitin, R.Z. Magnetism of actinides and their com-\npounds. Phys.-Uspekhi 30, 827 (1987).\n10Smith, J.L., Fisk, Z. %Hecker, S.S. Actinides: from heavy fermions to\nplutonium metallurgy. Physica B, 130,151-158 (1985).\n11Mentink, S.A.M., Nieuwenhuys, G.J. &Mydosh, J.A. Crystal structure\nand magnetic properties of U-Os intermetallic compounds. J. Magn. Magn.\nMater 104, 697-698 (1992).\n12Griveau, J.C. and Colineau, E. Superconductivity in transuranium elements\nand compounds. C R Phys. 15, 599-615 (2014).\n13Sechovsky, V . and Havela, L. Magnetism of ternary intermetallic com-\npounds of uranium. Handbook of magnetic materials, 11, 1-289 (1998).\n14Troc, R. et al. 2012. Single-crystal study of the kagome antiferromagnet\nU3Ru4Al12. Phys Rev B. 85, 064412 (2012).\n15Gukasov, A.G., Rogl, P., Brown, P.J., Mihalik, M. &Menovsky, A. Site sus-\nceptibility tensors and magnetic structure of U 3Al2Si3: a polarized neutron\ndiffraction study. J. Phys. Condens. Matter 14, 8841 (2002).\n16Purwanto, A. et al. Magnetic ordering in U 2Pd2In and U 2Pd2Sn. Phys Rev\nB.50, 6792 (1994).\n17Samsel-Czeka?a, M., Talik, E., Troc, R. and St?pien-Damm, J. Electronic\nstructure and bulk properties of the single-crystalline paramagnet URuGa.\nPhys Rev B. 77, 155113 (2008).\n18Matsumoto, Y . et al. Single-crystal growth and physical properties of\nURhIn 5. Phys Rev B. 88, 045120 (2013).\n19Ikeda, S. et al. Magnetic properties of U 2RhGa 8and U 2FeGa 8. J. Phys.\nCondens. Matter 15, S2015 (2003).\n20Bartha, A. et al. Single crystal study of layered U nRhIn 3n+2materials:\nCase of the novel U 2RhIn 8compound. J. Magn. Magn. Mater 381, 310-\n315 (2015).\n21Yarmolyuk, Y . Crystal Structure of U 3Ni4Si4. Sov. Phys . Crystallogr. 24,\n332-333 (1979).\nAdditional References for Sections I &II14\n22Yu, J., Devanathan, R. &Weber, W.J. First-principles study of defects and\nphase transition in UO 2. J. Phys. Condens. Matter 21, 435401 (2009).\n23Gryaznov, D., Heifets, E. &Sedmidubsky, D., 2010. Density functional\ntheory calculations on magnetic properties of actinide compounds. Phys.\nChem. Chem. Phys. 12, 12273-12278 (2010).\n24Andersson, D.A., Uberuaga, B.P., Nerikar, P.V ., Unal, C. &Stanek, C.R. U\nand Xe transport in UO 2\u0006xDensity functional theory calculations. Phys.\nRev. B 84, 054105 (2011).\n25Seth, P., Hansmann, P., van Roekeghem, A., Vaugier, L. &Biermann, S.\nTowards a first-principles determination of effective Coulomb interactions\nin correlated electron materials: Role of intershell interactions. Phys. Rev.\nLett. 119, 056401 (2017).\n26Brincat, N.A., Parker, S.C., Molinari, M., Allen, G.C. &Storr, M.T. 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Machine learning for molecular dynamics with strongly correlated elec-\ntrons. Phys. Rev. B 99, 161107 (2019).\n33Nomura, Y ., Darmawan, A.S., Yamaji, Y . &Imada, M. Restricted Boltz-\nmann machine learning for solving strongly correlated quantum systems.\nPhys. Rev. B, 96, 205152 (2017).\n34Carrasquilla, J. &Melko, R.G. Machine learning phases of matter. Nat.\nPhys. 13, 431 (2017).\n35Li, L., Baker, T.E., White, S.R. and Burke, K. Pure density functional\nfor strong correlation and the thermodynamic limit from machine learning.\nPhys. Rev. B, 94, 245129 (2016).\n36Moller, J.J., Korner, W., Krugel, G., Urban, D.F. &Elsasser, C. Composi-\ntional optimization of hard-magnetic phases with machine-learning models.\nActa Mater. 153, 53-61 (2018).\n37D. Fourches, E. Muratov &A. Tropsha, Trust, But Verify: On the Im-\nportance of Chemical Structure Curation in Cheminformatics and QSAR\nModeling Research, J. Chem. Inf. Model. 50, 1189-1204 (2010).\n38Scarel, G., Svane, A. &Fanciulli, M. Scientific and technological issues\nrelated to rare earth oxides: An introduction. In Rare Earth oxide thin films.\nSpringer, Berlin, Heidelberg (2007)."    },    {        "title": "1907.12249v1.Controlling_Magnetization_of_Gr_Ni_Composite_for_Application_in_High_Performance_Magnetic_Sensors.pdf",        "content": "1 \n Controlling Magnetization of Gr/Ni Composite  for Application  in High \nPerformance Magnetic Sensors  \nM. R. Hajiali1, *, L. Jamilpanah1, Z. She ykhifard1, M. Mokhtarzadeh1, H. F. Yazdi , B. Tork1,  \nJ. Shoa e Gharehbagh1, B. Azizi1, S. E. Roozmeh2, G. R. Jafari1 and Majid Mohseni 1, * \n1 Faculty of Physics, Shahid Beheshti University, Evin, 19839 Tehran, Iran  \n2Department of Physics, University of Kashan, 87317 Kashan, Iran  \n \n \nABSTRACT      \nGraphene  (Gr), a well -known 2D material, has been under intensive investigation in  the last decade due to \nits high potential application s in industry and advanced technological elements. The Gr, while composed \nwith magnetic materials, has opened new opportunities for further developments of magnetic based devices. \nHere, we report a  mass production of Gr/Ni composite powders using electrochemical \nexfoliation /deposition  method  with different magnetic strengths of the final composite  material . We applied \nthe magnetic composite materials in a magnetoimpedance (MI) based sensor and  observed significant \nenhance ment in the MI  effect  and its field sensitivity.  Such magnetic composites with controlled \nmagnetization strengths are coated on the MI-ribbon sensor surface and different MI responses are \nobserved. The MI response of a ribbon coated with a Gr/Ni layer  is theoretically determined based on an \nelectrodynamic model  with a qualitative  consistency between the experimental results and the theoretical \nmodel . Our comprehensive study can be applied in high performance functionalized MI based magnetic \nsensors and devices .  \n  \nKEYWORDS:  surface modification , electrochemical exfoliation , magnetoimpedance sensor, Graphene /Nickle  \ncomposite   \n \n \n \n \n \n \n \n 2 \n 1. INTRODUCTION   \nIn recent years, graphene, a two -dimensional (2D) hexagonal lattice of sp2 hybridized carbon atoms, has \nattracted comprehensive research interest because of its  fascinating electrical, mechanical, chemical and \noptical properties  and its poten tial application in next -generation electronic 1,2, energy storage devices  3,4,5 \nand composite based materials  6–8. Integration of graphene or graphene  family  with various nanoparticles \nallows the development of new nanocomposite materials with novel properties  and highly promising \napplication s in bioscience 9–11, microwave elements  12,13, sensors , 14 etc.15.  \nThe combination of graphene with nanoparticles, thereby forming graphene −nanoparticle hybrid structures, \noffers a number of additional unique physiochemical  properties and functions that are both highly desirable \nand markedly advantageous for various applications when compared to the use of either material alone  16. \nThese graphene − nanoparticle hybrid structures are especially alluring because n ot only do they display the \nindividual properties of the nanoparticles  which can already possess beneficial optical, electronic, magnetic, \nand structural properties that are unavailable in bulk materials, but they also exhibit additional advantages  \nand often synergistic properties that greatly augment their potential use for bio -applications  17. On the other \nhand, in sensing applications, the combination of magnetic nanoparticles  (MNPs) , with alternative \nfunctionalization  and catalytic properties, w ith graphene materials allows for the enhancement of sensitivity \nand selectivity over either graphene or nanoparticle magnetic  based sensors alone  16–19.  \nA wide range of magnetic sensors, such as anisotropic magnetoresistance, giant magnetoresistance, \ntunneling magnetoresistance, the Hall effect and magnetoimpedance (MI)  sensors are now available . \nAmong them,  MI effect has been considered as a n effect with higher field sensitivity and appropriate  signal \nintensity for magnetic sensors  20–22. The MI effect is defined as the change of  the electrical impedance of a \nconducting FM with high transverse magnetic permeability ( 𝜇𝑡) in the presence of a static magnetic field  \n23–29. By applying an external magnetic field, the skin depth ( 𝛿) changes due to the change in 𝜇𝑡, thus \nvarying the impedance of the FM . In the case of the ribbon, with width 𝑙 and length 𝐿, the impedance is \napproximately  \n𝑍=(1−𝑖)𝜌𝐿\n2𝑙𝛿=(1−𝑖)𝐿\n2𝑙(𝜋𝜌𝑓𝜇𝑡)1\n2  \n(1) \nwhere 𝜌 is electric resistivity, 𝑓 is frequency of the current and 𝑖 = imaginary unit. Therefore, the impedance \nof the ribbon is a function of frequency, driving current and the external dc magnetic field ( Hdc) through 𝜇𝑡 \nand 𝛿. This phenomenon has two aspects , one is related to improving the magnetic field sensor performance, \nand anothe r one is related to the environmental sensing ability. There are reports on the magnetic field \nsensitivity enhancement by coating layers with different magnetization and conductivities on the surfa ce of \nMI sensors  28,30 –34. The physical origins are well known through magnetic proximity effect  8, decrease of 3 \n surface roughness and magnetic exchange interactions between sensors and coated magnetic layers. Also, \nsurface modification of the ribbons has b een reported intensively for detection of environmental elements \nthrough magnetic and nonmagnetic interactions with the environment  21. \nRecently, we applied electrochemical exfoliation of graphene using a solution containing Ni ions which \nleads to simultan eous exfoliation  of Gr  and deposition of Ni  on Gr sheets  15. Here, we extend  this method \nfor fabrication of samples with controlled amount of magnetization of the samples  and introduce a useful \ntechnological application of such magnetic Gr/Ni  composite s. We investigate the application of the \nsynthesized materials in MI sensors with observed enhanced ratio and sensitivity when applied atop a n MI \nsensor.  \nIn order to obtain a comprehensive study and description of the experimental results, w e proposed an \nelectrodynamic  model to describe the MI  in a ribbon  covered  by solving the  linea rized Maxwell equations \nfor the electromagnetic fields coupled with the Landau -Lifshitz -Gilbert (LLG)  equation for the \nmagnetization dynamics  of the sensor . These results can b e used in interpreting the experimental results  of \nMI based sensors.  Both theoretical prediction s and experimental observations confirm the MI response \nquality change due to coating with magnetic composite s with dif ferent magnetization strengths. The \nfunda mental physical phenomenon described in this work can be used for development of magnetic field \nand environmental sensors based on the MI effect.  \n \n2. SAMPLE  PREPARATION AND EXPERIMENTAL METHODS   \n \n2.1. Preparation  of Gr/Ni Composite  by Electrochemical Exfoliation   \n \nElectrochemical exfoliation of graphite was performed in a two -electrode system using platinized silicon  \n(100 nm thickness and lateral dimension of 0.5 × 10 cm2) as the cathode electrode and a graphite foil (2 × \n10 cm2) as the anode electrode. The distance between the two electrodes was kept constant  at 2.7 cm. \nElectrolyte solutions were prepared by dissolving NiCl 2.6H 2O powder in water with three different molar \nratios of 0.05, 0.075 and 0.1 . A constant voltage (+10 V) was applied to the electrode s to provide  expansion, \nexfoliation of graphite and deposition of Ni. The voltage was maintained constant for 20 min to complete \nthe exfoliation /deposition  process. Afterward, the product was collected  using vacuum filtration and \nrepeatedly washed with  DI water. The resulted product was dispers ed in water for sonication . 4 \n  \nFigure 1. (a) Schematic illustration of the proposed mechanism for magnetic Gr /Ni production, (b) Situation of OH- \nand Cl- ions between graphite sheets which increases the interlayer  distance . Cl gas  can exert excessive force to \ngraphite layers which results in separation of the graphite layers  and then Gr sheets distributed in the solution can trap \nNi2+ ions. (c) Concentration of electrolyte solution and final mass of magnetic Gr/Ni composite .  \n \nThe mechanism of electrochemical exfoliation is depicted in Figure  1a, b. First, by applying a voltage \nbetween the electrodes, hydroxyl ions (OH−) are produced in the cathode region and then accelerate towards \nthe anode and hit the graphite surface. The collision of graphite with OH− ions initially occurs at each side \nand grain boundaries. The oxidation at the edge side and grain boundaries leads to the expansion of the  \ngraphite layers ; therefore,  Cl− ions can penetrate through them and  the reduction of Cl− ions produce Cl gas. \nThe gas can exert an excessive force to graphite layers which results in separation of the layers  35. In \ncontinue,  the distributed Gr sheets  in the solution can trap Ni2+ ions. Since such sheets have been partially \ncharged  positive ly, they can accelerate towards the negative electrode under electric field and create a black \ncomposite on  the Pt electrode. The OH− generation together with other elec trons and ions can form Ni and \nNi(OH) 2 on Gr sheets. Based on all the above process es, we finally have Ni and Ni(OH) 2 deposited as \ncrystalline layers on Gr flakes.  All above processes are confirmed by the effect of initial NiCl 2/water \nconcentration on exfoliation, as depicted in Figure  1c. In this figure, adding NiCl 2 results in higher \nproduction of Gr sheets, but below and above these values, final products do not show  magnetic properties.  \nBecause OH- and Cl- are responsible for produc tion of Gr flakes, therefore increasing molarity of NiCl 2 \nresults in more exfoliated production content.  As we are mainly interested in magnetic properties of our \nfinal products, we therefore investigate all of our samples made with 0.05, 0.075 and 0.1 M. In continue, \nwe represent a comprehensive study  of products and investigate their conductivity, magnetiza tion and M I \nmeasurements and discuss the mechanism of M I theoretically.  \n0.050 0.075 0.1000.10.20.30.4Mass (gr)\nConcentration\n  \n  2    \nH2O\nOH-\nGraphite (+)Platinum ( -)\n    \n         \n5 \n 2.2. Characterization  \nThe crystalline structure of samples was characterized using X -ray diffractometer (XRD)  w th  u Kα (λ = \n0.154 nm) radiation. Fourier transform infrared (FT -IR) sp ectra were recorded via a Bruker (Tensor 27) \nFT-IR spectrometer with resolution of 1 cm-1 in transmission mode at room temperature. X -ray \nphotoelectron spectroscopy (XPS) was done in an ESCA/AES system equipped with a concentric \nhemispherical analyzer (CHA , Specs model EA10 plus). The size and morphology of elements were \nobserved by tunneling electron micros copy (TEM -Philips model CM120). Room temperature \nmagnetization measurements were done via vibrating sample magnetometer (Meghnatis \nDaghigh  Kavir  Co.). Current -voltage  (I-V) measurements were done by two probe method using Keithley \n(model 2450) as sourcemeter.  \n \n2.3. Magnetoimpedance  (MI)  Measurement  \nAmorphous Co -based ribbons  Co68.15Fe4.35Si12.5B15 (1 mm width, 40 mm length and ~ 20 µm thickness) is \nprepared by a conventional melt -spinning technique. The Gr/Ni composite with different molarities of 0.05, \n0.075 and 0.1 M of NiCl 2 was drop coated on the two surface  sides  of the ribbon  at room temperature. \nEvaporation of water and nanoparticle solution was required befo re the data acquisition process  start. To \nmeasure the MI response of  the samples, an external magnetic field produced by a solenoid was applied \nalong the ribbon axis and the impedance was measured by means of the  four-point probe method. The ac \ncurrent passed through the longitudinal direction of the ribbon with different frequenc ies was supplied by \na function generator (GPS -2125), with a 50 Ω resistor in the circuit . The impedance was evaluated by \nmeasuring the voltage and current across t he sample using a digital oscilloscope (GPS -1102B). The MI \nratio is defined as  \n𝑀𝐼%=𝑍(𝐻𝑒)−𝑍(𝐻𝑒𝑚𝑎𝑥)\n𝑍(𝐻𝑒𝑚𝑎𝑥)×100; (2) \nwhere Z refers to the impedance as a function of the external field 𝐻𝑒. The 𝐻𝑒max is the maximum field \napplied to the samples during  the MI measurement.  \n \n \n \n 6 \n 3. RESULT AND DISCUSSION   \nXRD analysis was employed to confirm the crystalline structures  of Gr/Ni composite s as shown in Figure  \n2a. The dominant  diffraction peak of the initial graphite  foil at  θ =  6.31°  nd cates an  nter ayer d-spacing \nof 3.38 Å, while in the final samples the (002) diffraction peak is appeared at 26. 16° with an interlayer d-\nspacing of 3. 40 Å. There are some  diffraction peaks that belong to Ni(OH) 2. One broad  diffraction peak at \naround  θ=13.45° and another one at  θ = 33. ° represent the (001) and (100) p anes of the β -phase, \nrespectively (JCPDS No: 001 -1047). Wh  e (110) and (111) p anes of the α -phase appear  with broad \nd ffract on peak at  θ = 36 -39.3° (JCP DS No: 06 -0075)  36. Other two d ffract on peaks w th  θ = 59° and \n6 .1° represent (110) and (111) p anes of β -phase Ni(OH) 2. The diffraction peaks for Ni were also observed \nat  θ = 44.6°  and 52° which  correspond to (111)  and (200) planes , respectively  (JCPDS No: 001 -1260) , \nindicating the presence of face centered cubic (fcc) Ni with lattice constant of 3.51 Å. By increasing the \nconcentration of NiCl 2, less relative intensity of Ni peaks can be seen. The average size of the deposited Ni \nand Ni(OH) 2 crystals were calculated based on Scherrer equation  and found to be  43 and 32 nm, \nrespectively . Values are close to those observed in TEM images that will be explained later.   \nTo elucidate the interaction of Ni nanoparticles (Ni NPs)  with the G r sheets, F TIR spectra were recorded \nand analyzed. FTIR is a useful technique to confirm that the nanoparticles are anchored to G r surface, as \nhas been shown by several groups  37,38. Figure  2b shows the FTIR spectra of the graphite  foil and produced \nsample s. It shows a number of oxygen functionalities and this may be because of the residual functional \ngroups in the Gr 39. The absorption peaks at ~1633 cm‐1, ~1550  cm‐1 and ~1375 cm‐1 should be assigned to \nstretch ng v brat ons (υ) of  =O. The presence of O -H is confirmed by the strong and broad band at 3446 \ncm-1. The intensity of this peak is stronger for the synthesized Gr-Ni structures  than the initial graphite foil \nwhich shows the effect of aqoueos soloution of experiment on generation of hydroxyl elements.  The \nabsorption peak at 1053 cm-1 shou d be attr buted to stretch ng v brat ons (υ) of  -O-C 40 that disappears \nafter exfolation in Gr-Ni structures . The bands at 2925  cm‐1 and 2852 cm-1 are assigned to the asymmetric \nand symmetric vibrations of C -H, respec tively  41,42. Comparing the FTIR spectra of graphite foil with the \nfinal sample , the spectrum of Gr-Ni structures   clearly exhibite  a considerable redshift (particularly to the \nmajor vibration of C=C and –OH bonds) in the FTIR peaks, which may be because of the Ni bonded to the \nGr layers 43. The absorption bands at 651, 563 and 426 cm-1 are attributed to the NiO  and n (Ni –OH) \nvibratio n 43. 7 \n  \nFigure 2. (a) X-Ray Diffraction (XRD)  and (b) FTIR  spectra  of 0.1, 0.075 and 0.05 M samples and graphite foil . \n \nHigh -resolution XPS spectra  is used to probe the chemical compositions of the 0.05 and 0.1 M Gr/Ni \nsamples (Figure  3a-d). The C 1s and Ni 2p peaks are decomposed into multi -peaks analyzing with \ndistribution of C –C/C–O/C=O and Ni –O/ Ni –OH bonds.  Peak positions and FWHM values are presented \nin Table 1. The peaks related to Ni 2p show presence of Ni(OH) 2 and Ni -oxides and no Ni.  But Ni peaks \nwere observed in the XRD measurements  which  reconfirms the presence of Ni compo unds . This contrary \nis expected  as the XPS is able to probe the surface  of the materials  only. Also, rather than the oxides at the \nsurface of the Ni, there are some oth er covering compounds which cause the peak related to the Ni does not \nappear in the spectrum.  \n10 20 30 40 50 60 70 80#\n# \n \nPos. [°2TH.] Graphite# \n \n 0.1 M******(002)\n#(004)\n#\n (111)(110)(200)(111)(101)(110)(100 )(001)Intensity (a. u.)\n 0.075 M*^**^# Graphite#\n##\n**  \n \n 0.05 M* Ni(OH2)^ Ni*****\n*^\n^(a)\n1000 2000 3000 40000.60.70.80.91.01.1\n  Transmittance (%)\nWavenumber (cm-1) Graphite\n 0.05 M\n 0.075 M\n 0.1 M\n3446 (O-H)2852 (C-H)\n2925 (C-H)1633 (C=O)1550 (C=C)1458 (C=OH)1375  (C=O)1053  (C-O-C)651 (Ni-O)563 (Ni-O)426 (Ni-OH)(b)8 \n  \nFigure 3. High -resolution XPS spectra of C 1s and Ni 2p core level of  Gr/Ni sample  of (a, b) 0.05 M  and  \n(c, d) 0.1 M . \n \nTable 1 . Peak positions and FWHM values from high -resolution XPS spectra of C 1s and Ni 2p core level of  0.05 and \n0.1 M  Gr/Ni sample .  \n \n \nThe area ratios allow calculating relative proportions of atoms in each binding for a given element. The \nconcentration ratio of  Ni:C was estimat ed using the following relation  for 0.05 and 0.1 M  samples:  \n𝐶Ni\nCC=INi\nSNi×SC\nIC                                                      (3) \n890 880 870 860 850Ni2+Ni2+\nNi3+Ni3+Ni2+ Intensity (a.u.)\nBinding Energy (eV) Ni(OH) 2\n Ni2O3\n NiO\n Satellite peakNi2+\n0.05 M(b)\n292 290 288 286 284 282 280\n  Intensity (a.u.)\nBinding Energy (eV) C-Al\n C=C\n C=O\n C-O\n0.05 M(a)\n292 290 288 286 284 282 280\n  Intensity (a.u.)\nBinding Energy (eV) C-Al\n C-O\n C=O\n C=C\n0.1 M(c)\n890 880 870 860 850Ni3+Ni3+Ni2+\nNi2+Ni2+\n  Intensity (a.u.)\nBinding Energy (eV) Ni(OH)2\n Ni2O3\n Satellite peak\n NiONi2+\n0.1 M(d)\n)+3Ni(2/1p2Ni )+2Ni(2/1p2Ni )+3Ni(2/3p2Ni )+2Ni(2/3p2Ni sample\nFWHM ( eV) B.E. (eV) FWHM ( eV) B.E. (eV) FWHM ( eV) B.E. (eV) FWHM ( eV) B.E. (eV)  \n1.7 875.1 1.8-1.9 873.6 -871.9 2.7 857.4 2.3 -1.9 855.9 -853.7 0.05 M\n1.6 875.9 2.2 -1.8 873.9 -872.4 2.7 857.4 2.3 -2 856.4 -854 M0.1 C1s(C=O) C1s(C−O/C−Cl) C1s(C=C) C1s(C−Al) sample\nFWHM ( eV) B.E. (eV) FWHM ( eV) B.E. (eV) FWHM ( eV) B.E. (eV) FWHM ( eV) B.E. (eV)  \n4.1 287.6 1.7 285.9 1.3 284.8 2 282.5 0.05 M\n4.1 287.3 1.7 285.7 1.6 284.8 2.6 282.6 M0.1 9 \n where C, I and S are the elemental concentrations, XPS peak area and corresponding sensitivity factors. \nThe relative surface ratio of 𝐶Ni\nCC for 0.1 M sample is 0.36 which increases to 0.51 for 0.05 M sample. These \nresults indicate that Ni was enriched on the surface of carbon. Also 𝐶Ni\nCO for 0.1 M sample is 0.71 which \nincreases to 0.85 for 0.05 M sample and CC\nCO for 0.05 M sample  is 1.66 and increases to 1.93 for 0.1 M \nsample.  \nTEM images of samples are shown in Figure  4. Hexagonal structures appearing in Figure  4a-c are presumed \nto be related to Ni(OH) 2 nano -crysta s, wh ch the r crysta   zat on at β -hexagonal phase  was identified from \nour XRD data. This Ni -based nano -crystals are separated by dashed lines in Figure  4c. They have also \ndifferent alignments like rolled up structure s, which are shown by an arrow in Figure  4c that occurs with \nmore contents with increasing molarities, i.e. 0.1 M sample. Comparing all features dictates small thickness \nof such crystal, as they look to have thin planar geometry. In  Figure  4b, d we see crumpled Gr and nano -\ncrystals stick to it for 0.75 M sample. The nano -crystals (Ni, Ni(OH) 2, Ni-oxides) are randomly distributed \non Gr sheets which are supposed to be the reason of their superparamagnetic -like response  (presented at \ncontinue) . TEM results confirm the size of Ni-based nano -crystals obtained by XRD data.  The brightness \nof Gr sheets appearing in this image shows their small thickness. The image in Figure  4d shows micron \nsize formation of Gr sheets which is  in agreement with DLS results.  DLS measurements show ed the size \nof particles  to be  about 1.  μm. SAED pattern in Figure  4e, measured for a randomly selected Gr sheet  of \nthe 0.05 M sample , represents the polycrystalline nature of the Gr.  \n \nFigure 4. TEM images of (a) 0.1 M, (b) 0.075 M, (c) 0.05 M samples  and (d) micron boundary size of Gr sheets  and \n(e) SAED pattern  of the 0.05 M sample.  \n(a) (b) (c)\n(d)\n (e)\n(c)10 \n  \nTo investigate the effect of the molarity of NiCl 2 on the magnetic properties of the Gr/Ni composite, the \nmagnetic hysteresis loops of the 0.05, 0.075 and 0.1 M samples  were recorded at room temperature.  The \nVSM results shown in Figure  5a, represent superparamagnetic nature of the samples . By decreasing the \nmolarity of NiCl 2 the magnetization of the product increases (see inset of Figure  5a ). The coercivity for all \nof the samples was very small , indicating that the majority of the MNPs are in a superparamagnetic state.  \nXRD results showed the lower consentration of Ni in samples synthesized by higher NiCl 2 molarities. The \nprobability of the formation of Ni or Ni(OH) 2 depends on the consentration of OH- at the cathode. It means \nthat the more the molarity of the NiCl 2 results in more generation rate of  OH- and therefore the more ratio \nof Ni(OH) 2 to Ni  would be gained . Consecuently, by controlling the molarity of the solution, the \nconsentration of ferromagnetic Ni and so the magnetization will be controlled. The XPS results are \ncompared with magnetizatio n which indicates that the concentration of Ni to Ni (OH) 2 is more in 0. 05 M \nsample and caused  a higher magnetization. This also confirms that the conductivity of 0.05 M sample is \nmore than that of 0.1 M sample which has been seen by our I -V experiment.  For I -V measurements, silver \npaste coated pellets of the samples were used. I-V plots are linear, showing conductive behavior of all \nsamples  (Figure  5b). Resistivity of samples was determined by linear fitting  of I-V plots. Resistivity  \nincreases from 0.05  M sample to 0.1  M sample which is shown in the inset of Figure  5b. According to the \naforementioned growth mechanism , we can interpret that as the molarity of NiCl 2 in solution increases, \nthere is more OH- generation on the cathode and more hydroxide elements in Gr, and hence more resistivity \nin samples. XPS and XRD data show more hydroxide densities and lower Ni elements for samples prepared \nfrom solutions with higher molarities of NiCl 2. \n \nFigure 5. (a) VSM magnetization plots of 0.05 M, 0.075 M and 0.1 M samples. Inset shows the saturation \nmagnetization of samples versus concentration. (b) Current -Voltage ( I-V) curve  measurement  for 0.05 M, 0.075 M \nand 0.1 M samples  and inset shows the resistivity ( 𝜌) of samples  versus concentration . \n-10 -5 0 5 10-1.0-0.50.00.51.0\n0.050 0.075 0.1000.40.50.60.70.80.9\n  Magnetization (emu/g)\nApplied Field (kOe) 0.1 M\n 0.075 M\n 0.05 M(a)\nMs(emu/g)\nConcentration (M)\n  \n-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3-0.15-0.10-0.050.000.050.100.15\n0.050 0.075 0.1002468\n  I (A)\nV (v) 0.1 M\n 0.075 M\n 0.05 M(b)\nohm.cm )\nConcentration (M)\n  11 \n 3.1. Magnetic Gr/Ni Detection by Magnetoimpedance Measurement : Testing of Biosensor   \nIn the biosensor sector, the detection of functionalized magnetic nanoparticles has become a current active \nresearch issue  22,44. Besides high sensitivities, the nanoparticle detector should also display low power \nconsumption, small size,  quick response,  stability of operation parameters, resistance to aggressive medium,  \nand low -cost. Different sensing techniques are employed for the  nanoparticle detection: anisotropic \nmagnetoresistance  45, spin-valves  46, giant magnetoresistance  47 and magnetoimpedance (MI)  48,49. Among \nthem, biosensors based on the so -called MI effect have been proposed as an alternative procedure for the \ndetection of magnetic nanoparticles.   \nTo evaluate the potential applications in sensing devices , we study  the magnetic field and frequency \ndependences of MI response of a Co -based  ferromagnetic ribbon and used this prototype as a backg round \nin order to determine the effects of the stray fields introduced by the magnetic Gr/Ni nanoparticle s. By \napplying an ac charge current with frequency f to the samples , we investigate how the impedance of the \nribbon  changes as a function of the extern al magnetic  field. We perform field sweep impedance \nmeasurements at an arbitrary frequency of f = 10 MHz for ribbon (as -cast) and ribbon  drop coated  by Gr/Ni \nnanoparticle with three different molar ratios of 0.05, 0.075 and 0.1 , and  a current of I = 66 mA applied to \nthe samples . Schematic illustration of the measurement setup for the MI response and the structural features \nof the  MI based  sensor  is presented in Figure  6a. The lower the molarity of the samples , the higher the \ncontent of the Ni is in the sample. Each of the Ni nanoparticles can produce the stray field on the surface \nof the ribbon  and thereby  affect  the magnetic anisotropy and permeability of the ribbon . As can be seen \nfrom Figure  6b, the MI ratio for the bare ribbon ha s the smallest value and increase s for the drop -coated  \nsamples. The maximum values of the MI ratio are 201%, 218%, 238% and 27 5% for the bare ribbon  and \nthe ribbon with the  drop-coated  Gr/Ni  composite with molar ratios of 0.1, 0.075 and  0.05, respectively.  The \ninset of Figure  6b shows an enlarged portion of the low -field MI curves . One can see that  the anisotropy \nfield is 1.5 Oe for the bare ribbon while for the drop -coated  sample , the amount of the anisotropy field \ndecreases and reach es to 1 Oe.  When the Ni nanoparticles are affected by the external field , they can \nproduce a measurable  stray field.   As a result , presence of the Gr/Ni composite  leads to  a sizable increase \nin the MI  ratio near  the anisotropy field and the displacement of the MI  curve.  This finding  is of practical \nimportance, as the Gr/Ni  composite  can be better used as a magnetic biomarker for applications in  medical \ndiagnosis.  Specially in the cases which Gr plays the role of bio marking 50. The increa se in the MI ratio due \nto the  presence of the Ni nanoparticles  or the Gr/Ni  nanocomposite  can be explained by considering the \ndisturbance of  the applied dc longitudinal and ac transverse fields caused by  the presence of the stray  fields \nof the Ni nanoparticles on the  surface of the ribbon.  In order to evaluate difference between the MI ratio of \nthe bare ribbon and Gr/Ni composite  covered  ribbon , we measured the MI  at different frequencies  ranging 12 \n from 1  MHz to 15  MHz . The maximum MI ratio of all s amples versus frequency is plotted in Figure  6c. It \nis noted in Figure  6c that for all investigated samples, with increasing frequency, the maximum MI ratio \nfirst increases, reaches a maximum at a particular frequency ( 10 MHz), and then decreases for higher \nfrequencies. This trend can be interpreted by considering the relative contributions of DW motion and \nmoment rotation to the transverse magnetic permeability and hence to the MI  20,23,51. Note that as frequency \nincreases well above 100 kHz , the cont ribution of DW motion is damped due to the presence of the eddy \ncurrent and moment rotation becomes dominant  20. Thus, the MI ratio decreases at high frequencies.  \n \n \nFigure 6. (a) Schematic illustration of the measurement setup for the MI response and the structural conditions in the \nMI sensor when ribbon is coated with  Gr/Ni composites . The influence of the Gr /Ni layer on the MI is related to stray \nfields induced by magnetic Ni nanoparticles. The stray fields change the magnetization distribution in the ma gnetic \nribbon  and affect the permeability and the MI effect.  (b) The MI ratio as a function of applied magnetic field for bare \nribbon and ribbon drop coated by Gr/Ni nanoparticle with three different molar ratios. The inset shows an enlarged \nportion of the  low-field MI curves.  (c) Frequency dependence of the impedance response of all samples . (d) Magnetic \nhysteresis loops of bare ribbon and ribbon drop coated by Gr/Ni nanocomposite with molar ratios of 0.05 and 0.1 . \n5 10 15150200250300\n  MI(%)\nf (MHz) Bare ribbon\n 0.05 M\n 0.075 M\n 0.1 M(c)(a)\n-40 0 40-60060Magnetization(emu/g)\nApplied Field(Oe) Bare ribbon\n 0.1 M\n 0.05 M\n  \n(d)13 \n Finally,  to investigate the effect of the drop -coated Gr/Ni composite on the magnetic properties of the \nribbons, conventional VSM  method  was used to obtain the magnetic hysteresis loops . Figure  6d shows the \nmagnetic hysteresis loops of bare ribbon and ribbon drop coated by Gr/Ni composite with molar ratios of \n0.05 and 0.1 . It can be seen that the loops are all thin and narrow, and magnetization was saturated at a \nsmall applied field, indicating their soft ferromagnetic characteristics.  According to the Figure  6d, after \ncoating, the saturation field decreases and the differential  magnetic permeability of the sample which is \nproportional to the magnetization slope has increased dictating a weak field sensitive and magnetic ally \nsofter state.  In order to und erstand the obtained experimental results, we proposed and developed the \nfollowing electromagnetic model.   \n \n4. MODEL FOR IMPEDANCE OF RIBBON/GrNi STRUCTURES  \nIn order to describe the experimental results , we propose a model for the MI response of the ribbon/ GrNi \nstructures . MI response can be found by means of a solution of Maxwell equations for the electromagnetic \nfields and Landau –Lifshitz –Gilbert  (LLG)  equation for the magnetization dynamics  with appropriate \nboundary conditions at the interfaces bet ween the layers and outer surfaces . The influence  of the GrNi  layer \non the MI is related to stray fi elds induced by magnetic Ni nanoparticles. The stray fields change the \nmagnetiza tion distribution in the magnetic ribbon  and affect the permeability and MI effect. To describe \nqualitatively the infl uence of stray fields on the MI, we assume that  the GrNi  layer generates a spatially \nuniform effective field 𝐻𝑠 in the film. The value of 𝐻𝑠 is assumed to be proportional to the concent ration \nof MNPs in the GrNi, since the GrNi  saturation magnetization increases linearly ( see Figure  5a) with the \nconcentration of nanoparticles. It is assumed that the effective stray field 𝐻𝑠 has the opposite direction with \nrespect to the magnetization vector in the GrNi  layer.  Also, w e assume that the values of the permeability \nin the magnetic layer are determined by the magnetization rotation only  because the domain -wall motion is \nstrongly damped  at sufficiently high frequencies  (above 100 kHz ) 20. \nA schematic of the ribbon/ GrNi structures and the coordinate system used for the analysis  is shown in \nFigure  7. There are three reg ons ‘‘0,’’ ‘‘1,’’ and ‘‘ ’’  denoting  the magnetic ribbon  layer, the GrNi layer , \nand air.  The film structure having length 𝑙 and width 𝑤<𝑙 and consi sts of thickness  𝑡0 and 𝑡1 for the ribbon  \nand Gr /Ni layer , respectively . The structure  is subjected to an alternating driving field  𝐸=𝐸0𝑒−𝑖𝜔𝑡, and \nan external magnetic field 𝐻��� is parallel to the long side of the sample (y -axis).  It is assumed that the film \nlength and width are much  larger  than its thickness. Introducing the scalar and vector potentials, 𝜑 and 𝑨, \nMaxwe  ’s equat ons can be expressed as ( 𝛁∙𝑨=0 is used ) \n 14 \n 𝛁2𝑨𝑗(𝑟,𝑡)=𝜇𝑗𝜎𝑗[𝛁𝜑𝑗(𝑟,𝑡)+𝑑\n𝑑𝑡𝑨𝑗(𝑟,𝑡)]  \n𝛁2𝜑𝑗(𝑟,𝑡)=0 (4) \nwhere 𝑗=0, 1 corresponds to reg ons ‘‘0’’ and ‘‘1’’ and 𝜎𝑗 , 𝜇𝑗 corresponds to conductivity and permeability \nof the magnetic ribbon for 𝑗=0 and Gr/Ni layer for 𝑗=1. The general solutions for the vector potential in \nthe three regions can be expressed as 52 \n \nFigure 7. Schematic of the ribbon/ GrNi structures and the coordinate system . \n \n \n𝐴0=𝑖𝐸0\n𝜔[𝐵0cosh(𝛼0𝑧)+𝐶0sinh(𝛼0𝑧)−1] (5) \n𝐴1=𝑖𝐸0\n𝜔[𝐵1cosh(𝛼1𝑧)+𝐶1sinh(𝛼1𝑧)−1] (6) \n𝐴2=𝑖𝐸0\n𝜔𝐵2[𝑙\n𝑤ln(𝑟+𝑤\n𝑟−𝑤)−2𝑧\n𝑤arctan(𝑤\n2𝑦𝑙\n𝑟)+ln(𝑙+𝑟\n√𝑤2+4𝑧2)] (7) \n \nWhere , 𝑟=(𝑤2+4𝑧2+𝑙2)1\n2 , 𝛼𝑘=(1+𝑖)(𝜔𝜇𝑗𝜎𝑗\n2)1\n2 and due to symmetry along the 𝑍 axis, 𝐴0(𝑍)=\n 𝐴0(−𝑍), the constant of 𝐶0 is zero. The boundary conditions allow one to find the constants 𝐵0, 𝐵1, 𝐶1, \nand 𝐵2 in Equations ( 5)–(7) and describe completely the distribution of the vector potential  (see Appendix) . \nWhen the  potential distribution is obtained, since  the magnitude of the dri ving electric field is constant,  the \nimpedance can be obtained as the proportionality factor between the voltage and the total current in the \ndevice :  \n \n𝑍=𝑙𝐸0\n2𝑤(∫𝐽0(𝑧)𝑡02⁄\n0𝑑𝑧+∫𝐽1(𝑧)𝑑𝑧𝑡02⁄ 𝑡1\n𝑡02⁄)−1  \n \n(8) \n \n \nwhere,  \n \n𝐽0=𝐸0𝜎0𝐵0cosh (𝛼0𝑧) (9) \n𝐽1=𝐸0𝜎1[𝐵1cosh(𝛼1𝑧)+𝐶1sinh (𝛼1𝑧)] (10) \n \nFinally, we obtain the impedance Z of the film with the Gr/Ni layer:  \n15 \n  \n𝑍=𝑙\n2𝑤1\n𝛾′cosh(𝛼1𝑡0\n2)cosh (𝛼1𝑡)[(tanh(𝛼1𝑡)−tanh(𝛼1𝑡0\n2)) (𝛾+𝛽𝛾′tanh(𝛼0𝑡0\n2))+(1−tanh(𝛼1𝑡0\n2) tanh (𝛼1𝑡))(𝛾′+𝛽𝛾tanh(𝛼0𝑡0\n2))]\n2𝜎0\n𝛼0tanh(𝛼0𝑡0\n2)+2𝜎1\n𝛼1[sinh(𝛼1𝑡1)+𝛽tanh (𝛼0𝑡0\n2)cosh(𝛼1𝑡1)−𝛽tanh(𝛼0𝑡0\n2)]  \n(11) \n \nWhere , 𝑡=𝑡0\n2+𝑡1, 𝛾=1+𝑙𝑛2𝑙\n𝑤 , 𝛾′=𝜋\n𝑤𝜇1\n𝜇2𝛼1 and 𝛽=(𝜇1𝜎0\n𝜇0𝜎1)1\n2. \n \n \n4.1. Effect of Gr /Ni Layer on Ribbon Permeability  \n    \nThe MI response of the ribbon  is controlled by the transverse magnetic permeability. The transverse \npermeability depends on many factors, such as the domain  structure, anisotropy axes distribution, mode of \nthe magnetization variation, and so on. Since in the experiment the current fre quencies are sufficiently high,  \nthe value  of the transverse permeability in the ferromagnetic layers is  governed by the magnetization \nrotation . This approximation is valid at sufficiently high frequencies  (>100 KHz) , when the domain -wall \nmotion is  damped  20. We also suppose that the ferromagnetic layers have in -plane uniaxial anisotropy and \nthe direction of the anisotropy axis is close to the transverse one.  \nThe magnetization distribution in the ferromagnetic layers can be found by minimizing the free  energy. \nTaking into account the effective stray field , 𝐻𝑠, the minimi zation procedure results in the following \nequation for the equilibrium magnetization angle , 𝜃: \n \n𝐻𝑎sin(𝜃−𝜓)cos(𝜃−𝜓)−𝐻𝑠sin(𝜃−𝜑)− 𝐻𝑒𝑐𝑜𝑠𝜃  =0 (12) \n \nHere , 𝐻𝑎 is the anisotropy field in the ferromagnetic layers and 𝜓 is the deviation angle of the anisotropy \naxis from the transverse direction.   \nIn the magnetic susceptibility model, magnetization dynamics is governed by the Landau - Lifshitz -Gilbert \nequation, given by  \n \n𝜕𝒎\n𝜕𝑡=−𝛾𝒎×𝑯𝒆𝒇𝒇+𝛼𝒎×𝜕𝒎\n𝜕𝑡 (13) \n \nwhere 𝒎 is vector magnetization, 𝑯𝒆𝒇𝒇 is the effective field, 𝛾 is the electron gyromagnetic ratio and α is \nthe Gilbert damping constant. The solution of the linearized  LLG  equation results in the following \nexpression for the  transverse permeability  in the ribbon layer  53: \n \n𝜇0=1+𝜔𝑚[𝜔𝑚+𝜔1−𝑖𝛼𝜔]𝑠𝑖𝑛2𝜃\n[𝜔𝑚+𝜔1−𝑖𝛼𝜔][𝜔2−𝑖𝛼𝜔]−𝜔2 (14) \n \n \nHere 𝜔𝑚 =4π𝛾𝑀𝑠, 𝑀𝑠 is the saturation magnetization of the ribbon , 𝜔 = πf is the angular frequency, and   \n 16 \n 𝜔1=𝛾[𝐻𝑎cos2(𝜃−𝜓)−𝐻𝑠cos(𝜃−𝜑)+ 𝐻𝑒𝑠𝑖𝑛𝜃]  (15) \n𝜔2=𝛾[𝐻𝑎cos2(𝜃−𝜓)−𝐻𝑠cos(𝜃−𝜑)+ 𝐻𝑒𝑠𝑖𝑛𝜃] (16) \n \nThus, the MI response in the ribbon with a Gr/Ni layer can be calculated  as follow s. The first step is the \ndetermination of the transverse permeability in the ferromagnetic layer by using Eq . (14). The second step \nis substitution of  Eq. 1 4 into Eq. 1 1 and then the calculation of the impedance Z of the film with the Gr/Ni \nlayer. When the impedance Z  is found, the MI response  of the ribbon  with Gr/Ni layer  can be obtained by \nmeans of Equation ( 2). \n \n4.2. MI Response of  Ribbon/ GrNi Structures: Comparison of Experimental and Modelling Results  \n \nFigure  8a shows the f e d dependence of the MI rat o ΔZ/Z ca cu ated at 10 MHz for the ribbon  without \nGr/Ni layer  and the ribbon  with Gr/Ni layer  for different values of the effective stray field , 𝐻𝑠. Note that \nthe results are presented only for the range of pos itive fields, since the MI ratio is symmetric with respect \nto the sign of the external field. The effective stray field increases with the concentration of MNPs in the \nGr/Ni layer  due to the growth of the Gr/Ni layer  saturation magnetization. As a result, the MI ratio shifts \ntoward low external field s with increas ing  𝐻𝑠 and exhibit s a dependence similar to the one observed in the \nexperiment.   \nTo analyze the variation of the MI effect, let us introduce the impedance  field sensitivity, which is defined \nas follows  \n𝑆=|∆𝑍|\n|∆𝐻𝑒|=𝑍(𝐻𝑒=−𝐻𝑎)−𝑍(𝐻𝑒=0)\n𝐻𝑎 (17) \n \nFigure  8b presents the frequency dependence of the impedance sensitivity calculated by means of Eq. ( 17) \nfor different values of the effective stray field , 𝐻𝑠. The field sensitivity increases at relatively low \nfrequencies and attains  a peak at f= 20 MHz. With a growth of the effective stray field , 𝐻𝑠, the position of \nthe highest sensitivity shifts to lower frequencies.   \nThe proposed  model  allows one to describ e qualitatively  the main features of the experimental results  of \nthe MI response of  a ribbon with a Gr/Ni layer.  However, some experimental results cannot be explained \nin the framework of the model.  It is demonstrated that the contribution of the stray fields induced by MNPs \nin the Gr/Ni layer leads to a shift toward low external field s with an increase of 𝐻𝑠. It should be noted that \nthe model does not describe the  essential increase of the MI respo nse with an increase in the concentration \nof MNPs in the Gr/Ni layer. The disagreement between theoretical and experimental results may be \nattributed to  the fact that  in the proposed model , the contribution of surface  modification , i.e. roughness , is \nnot considered.  There are many experimental  works that demonstrated the fact that reduction of roughness  \ncan lead to increase of the MI respons e 27,28,31,33. The increase of MI response in the magnetic layer deposited 17 \n on ribbon has been already explained  according to modifications of the ribbon surface and the closure of \nmagnetic flux paths of deposited ribbons with magnetic materials . Note also, that the simplified presentation \nof the stray fiel ds created by the MNPs by means of the effective fi eld, 𝐻𝑠 qualitatively describes the effect \nof the Gr/Ni layer on the MI of the ribbon . Therefore, to estimate the exact value of 𝐻𝑠, an approximate \ndistribution of the stray fields should be found by means of a numerical solution for the magnetostatic \nequation.  \n \nFigure  8. (a) The f e d dependence of the MI rat o ΔZ/Z ca cu ated at f = 𝜔\n2𝜋 =10 MHz for the ribbon without GrNi \nlayer and the ribbon with GrNi layer for different values of the effective stray field 𝐻𝑠. (b) The frequency dependence \nof the  impedance sensitivity calculated by means of Eq. ( 17) for different values of 𝐻𝑠. Parameters used for \ncalculations are  𝑤=8𝑚𝑚 , 𝑙=4𝑐𝑚, 𝑡0=20𝜇𝑚, 𝑡1=10𝜇𝑚, 𝜎0=5.81×1071\n𝛺𝑚 , 𝜎1=1031\n𝛺𝑚, 𝑀𝑠=800 Oe, \n𝜃=0.75𝜋, 𝜑=−0.1𝜋, 𝜓=0.48𝜋, 𝛼=0.1, 𝐻𝑎=4 Oe. \n \n5. CONCLUSION  \nIn summary, the present study demonstrates that electrochemically exfoliated Gr is a promising method for \nfabricat ing magnetic Gr sheets /Ni-nano -crystal composite s in high  quality.  By changing the molarity of a \nNiCl 2 aqueous solution, different magnetic strengths of the final composite  material  were gained.  Different \nmolarities of 0.05, 0.075 and 0.1 M of NiCl 2 for fabrication of the samples results in different magnetization \nof the material. The higher molarity resul ts in lower magnetization . In addition to the focus on the novel \nfabrication of materials, we further investigate the application of the synthesized materials in MI sensors \nwith observed enhanced ratio and sensitivity when applied atop a n MI sensor. The ex perimental observation \nof the MI response of our sensor shows higher MI% for the sensor coated with the Gr/Ni sample with a \nhigher magnetization. A corresponding MI model was developed on the basis of the solution of linearized \nMaxwell equations and Landau -Lifshitz equation for the magnetization dynamics in order to understand \n0 5 10 15 20 25050100150200250MI (%)\nHe (Oe) Hs=0Oe\n Hs=0.1Oe\n Hs=0.2Oe\n Hs=0.3Oe(a)\n  \n0 20 40 60 80 100012345Sensitivity (%/Oe)\nf (MHz) Hs=0 Oe\n Hs=0.1 Oe\n Hs=0.2 Oe\n Hs=0.3 Oe(b)\n  18 \n the origin of the behavior of the MI response in the presence  of Gr/Ni composite.  The results can be \nconsidered for future developments of MI based sensors and biosensors.   \n \nAppendix  \nBoundary conditions require 𝐴 and 1\n𝜇𝑑𝐴\n𝑑𝑧 be continuous:  \n \n𝑨𝟎=𝑨𝟏 \n  \n𝑧=𝑡0\n2 \n𝜇1𝑑\n𝑑𝑧𝑨𝟎=𝜇0𝑑\n𝑑𝑧𝑨𝟏  \n \n \n𝑨𝟏=𝑨𝟐  \n \n𝑧=𝑡0\n2+𝑡1 \n𝜇2𝑑\n𝑑𝑧𝑨𝟏=𝜇1𝑑\n𝑑𝑧𝑨𝟐 (A1)  \n \nBy substitution of Eq. ( 5-7) into Eq. ( A1) with approximation of 𝑨𝟐 for 𝑤<<𝑙: \n𝐵0cosh(𝑥0)−𝐵1cosh(𝑦0)−𝐶1sinh(𝑦0)=0 \n𝐵0𝛽sinh(𝑥0)−𝐵1sinh(𝑦0)−𝐶1cosh(𝑦0)=0 \n𝐵1cosh(𝑧0)+𝐶1sinh(𝑧0)−𝐵2𝛾=1 \n𝐵1sinh(𝑧0)+𝐶1cosh(𝑧0)+𝐵2𝛾′=0 (A2) \n \nwhere  \n𝑥0=𝛼0𝑡0\n2 𝑦0=𝛼1𝑡0\n2 𝑧0=𝛼1𝑡 \n𝑡=𝑡0\n2+𝑡1 𝛾=1+𝑙𝑛2𝑙\n𝑤 𝛾′=𝜋\n𝑤𝜇1\n𝜇2𝛼1 \n \nSolving for the coefficients we obtain:  \n𝐵0=1\n𝐸 \n𝐵1=1\n𝐸(cosh(𝑥0)cosh(𝑦0)−𝛽sinh(𝑥0)sinh (𝑦0)) \n𝐶1=1\n𝐸(𝛽sinh(𝑥0) cosh(𝑦0)−sinh(𝑦0) cosh(𝑥0)) \n𝐸= 1\n𝛾′cosh(𝑥0)cosh(𝑦0)cosh(𝑧0)[(tanh(𝑧0)−tanh(𝑦0)) (𝛾+𝛽𝛾′tanh(𝑥0))+(1−tanh(𝑦0) tanh (𝑧0))(𝛾′+𝛽𝛾tanh(𝑥0))]        (A3)  \n \n \n \n \n 19 \n AUTHOR INFORMATION  \nCorresponding Authors  \n*M.R.H. e -mail: mrh.hajiali67@gmail.com \n*M.M. e -mail: m-mohseni@sbu.ac.ir \n \nORCID  \nMohammadreza Hajiali:  https://orcid.org/0000 -0001 -5068 -4437  \nMajid Mohseni:  https://orcid.org/0000 -0001 -5626 -533X  \nAuthor Contributions  \nThe manuscript was writt en through contributions of all authors. 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Giant Magneto -Impedance in Co -Rich \nAmorphous Wires and Films. IEEE Trans. Magn.  1995 , 31 (2), 1249 –1260. \nhttps://doi.org/10.1109/20.364815.  \n "    },    {        "title": "1908.00926v2.An_accelerating_approach_of_designing_ferromagnetic_materials_via_machine_learning_modeling_of_magnetic_ground_state_and_Curie_temperature.pdf",        "content": "An accelerating approach of designing ferromagnetic \nmaterials via machine lea rning modeling of magnetic ground \nstate and Curie temperature  \nT. Long, N. M. Fortunato, Yixuan Zhang, O. Gutfleisch, and H. Zhang*  \n \nInstitute of Materials Science, Technical University of Darmstadt, Darmstadt 64287, \nGermany  \n \n*Corresponding to Hongbin Zhang, hzhang@tmm.tu -darmstadt.de  \nAbstract  \nMagnetic materials have a plethora of applicatio ns ranging from informatics to energy harvesting \nand conversion. However, such functionalities are limited by the magnetic ordering temperature. \nIn this work, we performed machine learning on the magnetic ground state and the Curie \ntemperature (T C), using generic chemical and crystal structural descriptors. Based on a database of \n2805 known intermetallic compounds, a random forest model is trained to classify ferromagnetic \nand antiferromagnetic compounds and to do regression on the T C for the ferromagnets. The \nresulting accuracy is about 86% for classification and 92% for regression (with a mean absolute \nerror of 58K). Composition based features are sufficient for both classification and regression, \nwhereas structural descriptors improve the performance. Fin ally, we predict the magnetic ordering \nand T C for all the intermetallic magnetic materials in the Materials Project. Our work paves the \nway to accelerate the development of magnetic materials for technological applications.  \nIntroduction  \nThe continued growth of the global population has raised issues about sustainability and energy \nfuture, demanding improved efficiency of electricity production and consumption. Magnetic \nmaterials have a wide spectrum of applications, particularly in effici ent energy harvesting, \nconversion, and utilization.1,2 Specifically, permanent magnets (PMs) are the key components for \nthe energy related technologies, such as conventional generators, e -mobility, automatization and \nrefrigeration.3,4 Currently, two classes of PMs, namely, the ferrites and AlNiCo, and the high \nperformance PMs based on Nd -Fe-B and Sm -Co are widely used, with a gap in bet ween to be \nfilled by novel PMs, ideally those without critical elements such as heavy rare earths. Moreover, \nFM materials have been widely applied in spintronics, such as sensing, memory and logic, \nwhereas the emerging antiferromagnetic (AFM) spintronics h ave recently drawn intense \nattention.4,5 Two fundamental properties desired for promising candidate magnetic materials are a ferromagnetic (FM) ground state with strong magnetization and a high Curie temperature (T C) \nwhich governs the temperature range of functioning. These properties are also important for \nmagnetic refrigeration which promises enhanced energy efficiency over the conventional cooling \ntechnologies.6 \n \nAlthough T C is readily experimentally measurable, synthesis and optimization of real materials are \ntime-consuming and mostly done based on trial and error. Thus, the development of a \nmethodology to accelerate the developm ent of magnetic materials with a theoretical pre -screening \nis of natural interest. Typical theoretical approaches to evaluate T C rely on the parameterization of \ndensity functional theory (DFT) electronic structure to construct a Heisenberg Hamiltonian, whi ch \ncan be solved via atomistic Monte Carlo simulation. This approach fails even for elemental metal \nlike Co and Ni, due to the strong itinerant nature of magnetism therein.7,8 Moreover, DFT is not \nsufficient in describing the strongly correlated 4f electrons in rare -earths,9 while the orbital \ndependent functional (e.g., DFT+U) treatment is often chosen to fit to experiments. The \nstate-of-the-art DFT plus dynamical mean field theory (DMFT) method can be applied to tackle \nthe electronic correlation problem but is numerically expensive. Whereas the T C evalu ated for bcc \nFe based on DFT+DMFT is 50% off the experimental value.10,11 Therefore, there is a great \nimpetus for a predictive approach to obtain T C, which is applicable to compounds with arbitrary \ncompositions and crystal structures.  \n \nMachine learning is an emerging tool in materials science, being applied successfully to model the \nthermodynamic stability,12 band gap,13 elastic properties,14 inter-atomic potentials15 and in \npredicting potential high temperature superconductors.16 However, regression model s to predict \nordering temperature of magnetic materials have only been reported in a limited scope while \nclassification models to distinguish AFM and FM are absent in literature to the best of our \nknowledge. Sanvito et al. trained a linear regression model  over 40 intermetallic Heusler alloys \n(with experimental T C), and made predictions for another 20 compounds. By validating \nexperiments, they discovered Co 2MnTi with a remarkably high T C of 900K.17 Dam et al. focused \non selecting the best features for predicting T C of binary 3d -4f intermetallic compounds by \napplying Gaussian kernel regression on 108 compounds. The add -one-in test accuracy can reach \nabove 95% when only eight descriptors are used, with the rare -earth concentration being the most \nrelevant.18  \n \nIn this work, we develop a FM/AFM classification model along with a regression model to predict \nthe T C for intermetallic FM compounds, using the random forest (RF) method. These models are \nthen used to identif y the magnetic ground state of 5183 magnetic intermetallic compounds from \nthe Materials Project database and to predict the T C of those classified as FM. It is demonstrated \nthat our machine learning framework is efficient and predictive, and can be used to  accelerate the \nscreening for FM compounds which are promising for spintronics and permanent magnets \napplications.  Results  \nData  \nUsing the AtomWork database,19 1749 FM and 1056 AFM inter -metallic compounds are collected, \nwhere oxides, sulfites, chlorides, and fluorides having been excluded, along with compounds \nwithout either of Cr, Mn, Fe, Co, and Ni atoms, which are the typical magnetic atoms in tran sition \nmetal based intermetallic magnetic materials. The corresponding crystal structures are collected \nfrom AtomWork and Inorganic Crystal Structure Database (ICSD).20 For compounds with \nmultiple magnetic phase transitions, the critical temperature is defined as the magnetic transition \ntemperature from a disordered paramagnetic state to an ordered FM state. In this way, materials \nwith the first -order magnet o-volume, magneto -structural, and temperature dependent \nspin-reorientation transitions are excluded for the current work.  \n \nThe distribution of experimental FM ordering temperatures is shown in Fig. 1(A). It is dominated \nby compounds with low T C, with a max imum value around 1400 K, e.g., 1410 K and 1388 K for \nCo in face -centered -cubic (space group 225) and hexagonal -closed -pack (space group 194) \nstructures, respectively. The element resolved T C distributions are highlighted in Fig. 1(B -D) for \nFe-, Co-, and M n-based compounds, whereas the T C for Ni -based compounds are mostly in the \nlow-temperature range (Fig. S1(A)) and there are limited number of Cr -based ferromagnetic \nmaterials (Fig. S1(B)). It is clear that all compounds with T C higher than 1200K are Co -based (Fig. \n1(D)), the Fe -based compounds (Fig. 1(C)) consist of 1/3 of the database with T C normally \ndistributed around 600K, while the T C of Mn -based compounds (Fig. 1(B)) are mostly found at \nrelative low temperature (with a peak at 300 K) range. Thus, the Fe- and Co -based compounds are \noptimal for high temperature applications, and in the room temperature range all three classes are \ninteresting.  \n \nMoreover, the distribution of experimental AFM ordering Neel temperatures (T N) is shown in Fig. \nS2(A), where T N of most compounds are less than 100K. The element resolved T N distributions  in \nFig. S2(B -F) indicate  that Fe - and Mn - based compounds are more suitable for high temperature \napplication. It is noted that there are many more AFM compounds such as oxides, thu s the \ncollection of AFM intermetallic compounds in this work serves only for classification and we save \nthe regression of T N for future study. We have also collected 5193 magnetic intermetallic \ncompounds from Materials Project,21 in order to make predictions by applying the machine \nlearning models for magnetic ground state classification and T C regression, as discussed in detail \nbelow.   \nFig. 1.  Distribution of T C for the FM database.  (A) Histogram of T C for 1749 FM materials in \nthe database. Green, blue, cyan, gray and yellow represents Ni, Co, Fe, Mn and Cr based \ncompounds, respectively. (B) Histogram of Fe based FM compound categorized by T C. (C) \nHistogram of Co based FM compound categorized by T C. (D) Histogram of Mn based FM \ncompound categorized by T C. The bin size is fixed to be 50K.  \nDescriptors  \nThe Materials Agnostic Platform for Informatics and Exploration (MAGPIE)22 proposed by Ward \net al. is used to obtain the chemical descriptors, including 4 categories: stoichiometric attributes, \nelement properties statistic, electron structure attributes and ionic compound attributes. In this \nwork, we regroup them in the follow ing 5 classes, namely, norm (Lp norms of the fractions), \nmagnetic moment of the constituting elements, atomic number, valence electrons and other \nchemical descriptors. (Please check details in Methods and Table S1 and S2) These descriptors are \ncollectively  labeled as CHEM. As for structural descriptors (labeled as STR), Smooth Overlap of \nAtomic Positions (SOAP) is used to describe the local crystalline environment such as \ncoordination and distance between atoms. 23 Space group number is considered as a structural \ndescriptor as well. In total, 139 (25) CHEM (STR) descriptors for each compound are used. In \norder to get a better understanding of the relative feature importance and the underlying physical \npicture, we co nsidered two models with variations of the included features labeled as \nCHEM+STR and CHEM.  \nClassification  \nWhile the prediction of whether a material is magnetic or not can be straightforwardly done in \nDFT, the question of being AFM or FM is more complex. F or instance, the AFM ground state is \nmostly set aside in the Materials Project, despite the thermodynamic stability and electronic \nproperties hinge on the magnetic configurations. This is because that the number of possible AFM \nstates, especially after con sidering non -collinear magnetic configurations,24 would make \nhigh-throughput calculations intractable.25 \n \nTo enable the prediction of the magnetic ordering in a computationally inexpensive fashion, we \nperform a classification of the AFM or FM ground state. The training set consisted of 90% of the \ndatabase and 10% are used for validation. The quality of the classification model is judged in \nterms of the following statistic metrics.26 By looking at the positives (tp), false positives ( fp), false \nnegatives (fn) and true negatives (tn), the accuracy, precision, recall and F1 can be evaluated as \nfollowing:  \ntp tnaccuracytp tn fp fn  \n \ntpprecisiontp fp\n \ntprecalltp fn\n \n12precision recallFprecision recall\n \nwith true being FM and fal se AFM. The accuracy represents the overall quality of the prediction, \nthe precision is the proportion of those correctly classified as FM within all classified as FM, the \nrecall is the proportion of those correctly identified as FM with all known to be FM , and the F1 \nbridges the recall and precision metrics, denoting if there is a bias towards classifying one label. \nThe c onfusion matrix (CM) is a table that represents the instances in a predicted class versus the \nones in the actual class, as shown in Fig. 2(A), together with the resulting metrics for 10 \ncross -validation sets plotted in Fig. 2(B).  \n \nObviously, the best accuracy for classification is 86%, that is, 88.8% FM and 82.4% AFM \ncompounds are correctly classified. It is achieved by taking all chemical and structural features as \ndescriptors, i.e., the CHEM+STR model. This combined with an F1 score of 89% (Fig. 2(B)) indicates good predictability, with a slight bias towards predicting FM which might be due to the \nunbalanced number of FM/AFM compounds in t he database. By performing 10 -fold cross \nvalidation (Fig. 2(B)), the average accuracy is about 82%, meaning that the RF model has neither \noverfitting nor biased sampling.  \n \nInterestingly, in the CHEM+STR model, the descriptor group valence electrons is selected as the \nmost important feature, contributing 47% of feature importance, while the second most important \ndescriptor group, SOAP, only constitutes 15%, followed by the at omic number and magnetic \nmoment of the constituting elements. The contribution of valence electrons demonstrates that \nintrinsic properties of element have the strongest significance in predicting the magnetic ordering. \nTo confirm this, the CHEM model is co nsidered, which eliminates an explicit description of the \nlocal crystalline environment, leading to an accuracy of 82% and an F1 score of 86% (Table S3) \nwhere the order of importance for the chemical features remains the same, reinforcing their \nimportance relative to each other. Excluding the structural descriptor only results in a 4% drop in \naccuracy, indicating that the local environment does not affect the magnetic ordering directly, but \nexerts influence on the valence electrons of atoms and disturbs the  magnetic ordering. However, \nthe necessity of considering interaction between atoms in magnetic ordering cannot be \noverlooked.  \n \nFig. 2.  Performance of the classification model.  (A): Confusion matrix of FM and AFM \nclassification test set of 281 compounds. (B): Statistical metrics for 10 -folder cross validation.  \n \nRegression  \nTurning now to the regression of T C, which is done using the RF with 90%/10% partition for \ntraining/valida tion of the 1749 FM compounds, the best R2 obtained from the validation is as high \nas 92% using the CHEM+STR descriptors, indicating very good agreement between the \nexperimental and predicted values. As shown in Fig. 3(A), the corresponding mean absolute e rror \n(MAE) is 58 K. The agreement has to be weighed against the variation of T C in experiments due to \ndifferences in composition, synthesis and measurement techniques, which contributes to the error. \nIt is noted that, compared with those obtained based on DFT calculations, the machine learned \nvalues are more accurate. 7–9 \n \nThe valence electron features are still assigned with the highest importance of 41%, while that for \nthe magnetic moment of the constituting elements increases to 25% corresponding to the linear \nrelationship between magnetic moment  per atom with the total magnetization. Interestingly, the \nimportance of SOAP drops to only 9%, indicating that the local crystalline environment has less \nimportance when FM ordering is determined. When compared with the CHEM descriptors, the \neffect of inc luding the crystal structure is again marginal but noticeable. For instance, using only \nthe CHEM descriptors results in a R2 of 90% and an MAE of 60K (Table S4). Nevertheless, the \nCHEM only model must fail for edge cases, where different phases/volumes of the same \ncompound have different magnetic orderings or T C. For instance, Fe 3Nb with space group number \n225 has a predicted T C of 939K while the predicted T C is only 373.67K with space group number \n139 using CHEM+STR descriptors. However, Heusler alloys wit h magneto -structural transitions \nlike Ni 2MnGa exhibit the same predicted T C in both cubic and tetragonal structures, probably due \nto the lack of data in training, e.g., only 32 compounds in the  database have isomers.  \n \nFurthermore, in order to test the con vergence with respect to the feature space, we performed \nactive learning (AL), starting with 10% of the training data and take automatically 10% more \nsamples data which are the most outliers in the rest training set. In this way, the feature space that \nhas not been covered by the previous training set will be included, enabling to obtain higher \naccuracy with less data.27 As shown in Fig. S3, using only 50% of the data, the resulting accuracy \nbased on AL is already comparable with the standard model. Such training accelera ting is even \nsignificant when approximately 10% data selected by AL achieves an R2 of 84% compared with \nrandom selection only at 72%. Therefore, covering larger region of the feature space will make the \nregression modeling more robust. In addition, our dat abase is not guaranteed to be completed, the \nAL algorithm will make it possible to take further experimental cases into consideration.  \n \n  \nFig. 3. Regression model performance and feature importance.  (A) Predicted vs. experiment \nTC for test set in general regression model. (B) 10-folder cross validation and pie chart of feature \nimportance.  \n \nPrediction  \nUsing the model based on CHEM+STR, we performed AFM/FM classification of the 5193 \nintermetal lic compounds from Materials Project, leading to 2884 (2309) FM (AFM) (see Data file \nS1 in the Supplementary). Fig. 4 shows the predicted T C for the classified FM compounds, ranging \nup to 1280 K. Obviously, the compounds with T C higher than 600 K are again  mostly Co and Fe \nbased, and the element -wise distribution is comparable to that of our training set as shown in Fig. \nS4. The compounds with predict T C > 1100 K are listed in Table 1, which are all Co -based. It is \nfound that the predicted T C is in good agr eement with the values from the literature, which have \nbeen collected after obtaining the prediction. For instance, the largest difference is around 130K \nfor Co 17Tb2, where the predicted value is about 1283.7K while the experiment value is 1150K 28; \nwhereas the smallest difference is about 10 K for Co 17PrYb with experimental (predicted) T C \nbeing 1178K29 (1167.2 K). We note the total computational time for the classification and \nregression of these compounds is a few seconds, making it trivially inexpensive compared with \nthe computational effort that would be required by DFT.  \n \n \n \n \n \n \n \n \n \n \nTable 1.  List of potential high T C (>1100K) FM compounds by this work.  \nMP-id Formula  Predicted T C (K) Real T C (K) Space Group No.  \nmp-1072089  Co Around 1283.67  - 227 \nmp-669382  Co Around 1283.67  - 186 \nmp-1193227  Co Around 1283.67  - 136 \nmp-1096987  Co9Fe Around 1283.67  - 123 \nmp-1201816  Co17Gd2 Around 1283.67  120030 194 \nmp-1204082  Co17Lu2 Around 1283.67  119231 194 \nmp-1195194  Co17Np2 Around 1283.67  - 194 \nmp-568820  Co17Pu2 Around 1283.67  - 194 \nmp-1094061  Co12Sm Around 1283.67  - 139 \nmp-16932  Co17Th2 Around 1283.67  - 166 \nmp-1199370  Co17Tb2 Around 1283.67  115028 194 \nmp-1196360  Co17Tm 2 Around 1283.67  117028 194 \nmp-1219785  Co17PrSm  Around 1167.17  120032 160 \nmp-1219295  Co17GdSm  Around 1167.17  120032 160 \nmp-1220026  Co17ErPr Around 1167.17  116729 160 \nmp-1200096  Co17Sm 2 Around 1167.17  115028 194 \nmp-1215870  Co17PrYb  Around 1167.17  117829 194 \nmp-1199900  Co17Yb2 Around 1167.17  117629 194 \nmp-1216133  Co33Yb4Zr Around 1167.17  117533 156 \nmp-1215883  Co34Pr3Yb Around 1167.17  117329 8 \nmp-1219047  Co17SmY  Around 1167.17  120032 160 \nmp-356 Co17Nd2 Around 1167.17  112534 166 \nmp-1224958  Co33La4Ta Around 1167.17  - 8 \nmp-1226612  CeCo 17Y Around 1167.17  - 160 \nmp-1105621  Co17Pr2 Around 1167.17  - 166 \nmp-1220026  Co17ErPr Around 1167.17  - 160 \n \n  \nFig. 4.  Histogram of T C prediction.  (A) Histogram of predicted T C for 2884 FM materials in the \ndatabase. Green, blue, cyan, gray and yellow represents Ni, Co, Fe, Mn and Cr based compounds, \nrespectively. (B) The same as (A) but with predicted T C higher than 600K.  \nDiscussion  \nIt is demonstrated that mac hine learning using the RF algorithm is able to distinguish materials \nwith FM and AFM ordering, and further predict the T C of FM compounds. This solves two critical \nproblems in designing magnetic materials. For classification, the accuracy reaches 86% (82% ) \nusing chemistry plus structure (chemistry only) as descriptors. This outperforms the DFT \ncalculations25, which are applied on a selected set of compounds. For the resulting FM compounds, \nthe magnetization can be straightforwardly evaluated using DFT. Furthermore, the T C can be \naccurately modeled with R2  92% and MAE about 58K. This enables us to reduce the number of \ncandidates for further characterization. For instance, the magnetocrystalline anisotropy can be \nevaluated in a high throughput way,35 which sets an upper limit for the coercivity. Thus, the \nmachine learning model devel oped in this work in conjunction with DFT enables us to get all \nthree essential intrinsic magnetic properties evaluated.  \n \nSince machine learning is able to capture the mechanism behind magnetic ordering from the \nstatistical point of view, one interesting q uestion is to apply the same modeling on AFM \ncompounds to predict the Neel temperature. As the descriptors we used are robust, as suggested by \ncomparable accuracy with different sets of descriptors, we suspect that our methods are applicable \nto predict the  Neel temperature of AFM compounds as well. However, the AFM magnetic ground \nstates are not uniquely defined, which requires additional development based on either machine \nlearning modeling or high throughput DFT calculations.  \n \nIn conclusion, we have devel oped a robust machine learning framework which allows screening of \nthe magnetic ground state and Curie temperature of FM compounds. This paves the way to \ndevelop FM materials with systematic characterization of the intrinsic magnetic properties, with \nthe h elp of further high throughput DFT calculations.  \nAcknowledgement  \nThe authors are grateful to Qiang Gao, Chen Shen and Ilias Samathrakis  for useful discussion and \nsuggestions. The authors gratefully acknowledge computational time on the Lichtenberg High \nPerformance Supercomputer. Teng Long thanks the financial support from the China Scholarship \nCouncil. Nuno M. Fortunato thanks the financi al support from European Research Council.  \nNote  \nDuring the preparation of the manuscript, we noticed that Nelson and Sanvito did a similar work \n―Predicting the Curie temperature of ferromagnets using machine learning‖ (arXiv:1906:08534). \nThey focused on t he preselected FM compounds without classification, where the accuracy is \nabout 88% with a MAE of 50K. Consistent with our observation, it is concluded that only \nchemistry is required to model the Curie temperature.  \nReference  \n1. Gutfleisch, O. et al.  Magnetic Materials and Devices for the 21st Century: Stronger, Lighter, and More \nEnergy Efficient. Advanced Materials  23, 821 –842 (2011).  \n2. Skokov, K. P. &  Gutfleisch, O. Heavy rare earth free, free rare earth and rare earth free magnets - Vision and \nreality. Scripta Materialia  154, 289 –294 (2018).  \n3. Sander, D. et al.  The 2017 Magnetism Roadmap. J. Phys. D: Appl. Phys.  50, 363001 (2017).  \n4. 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Magnetic characteristics and lattice constants of some pseudobinary intermetallic compounds of the type R2T17. physica status solidi (a)  23, K15 –K18 (1974).  \n33. Hirosawa, S. & Wallace, W. E. Effect of Substitution of Zr and Pr on magnetic properties of R2Co17(R=Er,  \nYb). Journal of Magnetism and Magnetic Materials  30, 238 –242 (1982).  \n34. Andreev, A. V. et al.  High -field magnetization study of a Tm2Co17 single crystal. Physical Review B  81, \n134429 (2010).  \n35. Drebov, N. et al.  Ab initio  screening methodology applied t o the search for new permanent magnetic \nmaterials. New Journal of Physics  15, 125023 (2013).  \n "    },    {        "title": "1909.00527v5.Magnetically_induced_enhanced_coarsening_in_thin_films.pdf",        "content": "Magnetically induced/enhanced coarsening in thin \flms\nR. Backofen1and A. Voigt1, 2\n1Institute of Scienti\fc Computing, Technische Universit at Dresden, 01062 Dresden, Germany\n2Dresden Center for Computational Materials Science (DCMS), 01062 Dresden, Germany\nExternal magnetic \felds in\ruence the microstructure of polycrystalline materials. We explore the\nin\ruence of strong external magnetic \felds on the long time scaling of grain size during coarsen-\ning in thin \flms with an extended phase-\feld-crystal model. Additionally, the change of various\ngeometrical and topological properties is studied. In a situation which leads to stagnation, an ap-\nplied external magnetic \feld can induce further grain growth. The induced driving force due to\nthe magnetic anisotropy de\fnes the magnetic in\ruence of the external magnetic \feld. Di\u000berent\nscaling regimes are identi\fed dependent on the magnetization. At the beginning, the scaling expo-\nnent increases with the strength of the magnetization. Later, when the texture becomes dominated\nby grains preferably aligned with the external magnetic \feld, the scaling exponent becomes inde-\npendent of the strength of the magnetization or stagnation occurs. We discuss how the magnetic\nin\ruence change the e\u000bect of retarding or pinning forces, which are known to in\ruence the scaling\nexponent. We further study the in\ruence of the magnetic \feld on the grain size distribution (GSD),\nnext neighbor distribution (NND) as well as grain shape and orientation. If possible, we compare\nour predictions with experimental \fndings.\nI. INTRODUCTION\nGrain boundaries in polycrystalline materials are of\nparamount importance to various \felds of science and\nengineering. They have been intensively studied theo-\nretically and experimentally over decades. Quantitative\ncomparison of geometrical and topological properties be-\ntween theory or simulation and experimental data are\nstill unsatisfactory in general. Progress have been made\nfor nanocrystalline thin metallic \flms. Geometric and\ntopological characteristics of the grain structure can be\nshown to be universal and independent of many experi-\nmental conditions [1]. A phase \feld crystal (PFC) model\n[2, 3], which considers the essential atomic details but\noperates on di\u000busive time scales, was able to reproduce\nthe universal grain size distribution and showed similar\nscaling properties and stagnation as in the experiments\n[4]. This is in contrast with more classical Mullins-like\nmodels, which only consider the evolution of the con-\ntinuous grain boundary network [5]. Theoretical pre-\ndictions and simulations for this type of models lead to\nself-similar structures and coarsening laws for the aver-\nage grain size of the form t\u000b, with a scaling exponent\n\u000b= 1 in the original setting [6]. These models have been\nextended by including retarding and pinning forces for\ngrain boundary movement [7{9] and grain rotation [10{\n12]. The modi\fcations can explain the smaller scaling\nexponents in experiments and stagnation. However, also\nthese modi\fcations are unable to reproduce the univer-\nsal grain size distribution (GSD). A detailed comparison\nbetween these models with PFC simulations [4] and ex-\nperiments in Ref. [1] can be found in Ref. [13]. With the\nachieved agreement for various geometrical and topolog-\nical properties it is now time to use the PFC model as a\npredictive tool to control grain growth in thin \flms under\nthe in\ruence of external \felds.\nExternal magnetic \felds during processing in\ruence\ngrain growth and as such have been proposed as an ad-ditional degree of freedom to control the grain structure,\nsee [14, 15] for reviews. The PFC model has been ex-\ntended to include magnetic interactions in Refs. [16, 17]\nand was used in Ref. [18] to explain the complex inter-\nactions between magnetic \felds and solid-state matter\ntransport. An applied magnetic \feld in\ruences the tex-\nture during coarsening due to the anisotropic magnetic\nproperties of the single grains. Grains with their easy\naxis aligned to the external \feld are energetically pre-\nferred. They grow preferably at the expense of the other\ngrains. The mobility of grain boundaries in this model is\nfound to be anisotropic with respect to the applied mag-\nnetic \feld. Magnetostriction is naturally included in the\nextended PFC model. All these e\u000bects already change\ntexture on small time scales. In this paper we analyze\nthe long time scaling behavior and various geometrical\nand topological properties in grain growth under the in-\n\ruence of a strong external magnetic \feld.\nThe paper is organized as follows: We \frst review the\nunderlying PFC model, the physical setting and the con-\nsidered numerical approach. Then we consider the coars-\nening regime and analyze various geometrical and topo-\nlogical measures. Finally, we discuss the results, explain\nour \fndings and draw conclusions.\nII. MODEL AND NUMERICAL APPROACH\nThe model in Refs. [16{18] combines the rescaled num-\nber density 'of the original PFC model [2, 3] with a\nmean \feld approximation for the averaged magnetization\nm. The total energy,\nF[';m] =Z\nfPFC(') +!Bfm(m) +!Bfc(';m) dr;\nconsists of contributions related to local ordering in the\ncrystal,fPFC('), and the local magnetization, fm(m).\nThe magnetic anisotropy is included by coupling densityarXiv:1909.00527v5  [cond-mat.mtrl-sci]  2 Mar 20202\n\feld and magnetization in the last term, fc(';m).!B\nis a parameter to control the in\ruence of the magnetic\nenergy.\nAn extended PFC model (XPFC) is chosen in order to\nde\fne the crystal structure [19],\nfPFC(') =1\n2'(r)2\u0000t\n6'(r)3+v\n12'(r)4\n\u00001\n2'(r)Z\nC2(r\u0000r0)'(r0) dr0:\nThe magnetization is governed by\nfm(m) =W2\n0\n2(r\u0001m)2+rmm2\n2+\rmm4\n4\u0000m\u0001B+B2\n2;\nwherermand\rmcontrol the magnitude of magnetization\nandW2\n0the energy due to inhomogenities of magnetiza-\ntion [16, 18]. The magnetic anisotropy is modeled by\ncoupling the density wave with magnetization [16],\nfc(';m) =\u0000!m'2m2\n2\u00002X\nj=1\u000b2j\n2j(m\u0001r')2j:\nIn order to maximize the anisotropy, as in [18], a\nsquare ordering of the crystal is preferred, which is re-\nalized within the XPFC formulation for fPFC('), see\n[19, 20]. The correlation function C2is approximated\nin k-space as the envelope of a set of Gaussians and\nwith peaks chosen by the primary k-vectors de\fning the\ncrystal structure. For a square symmetry a minimum of\ntwo peaks is needed, bC2(k) = max ( bC2;0(k);bC2;1(k)) and\nbC2;i(k) =Aiexp [(ki\u0000k)2=(2\u00182\ni)]. The e\u000bect of temper-\nature on the elastic properties is seen in the width of the\npeaks and modeled by \u0018i.Aiis a Debye-Waller factor\ncontrolling the height of the peaks.\nMagnetization in an isotropic and homogenous mate-\nrial is modeled by fm(m). The last two terms describe\nthe interaction of the magnetization with an external\nand a self-induced magnetic \feld, BextandBind, respec-\ntively. The magnetic \feld is de\fned as B=Bext+Bind,\nwhere Bindis de\fned with help of the vector potential:\nBind=r\u0002Aandr2A=\u0000r\u0002 m.\nThe magnetic anisotropy of the material is due to the\ncrystalline structure of the material. Thus, the magneti-\nzation has to depend on the local structure represented\nby'and vice versa. The \frst term in fc(';m), changes\nthe ferromagnetic transition in the magnetic free energy.\nOn average '2is larger in the crystal than in the homo-\ngeneous phase. Thus, !mandrmcan be chosen to realize\na paramagnetic homogeneous phase and a ferromagnetic\ncrystal. The second term depends on average on the rel-\native orientation of the crystalline structure with respect\nto the magnetization.\nThe number density 'evolves according to conserved\ndynamics and magnetization according to non-conserved\ndynamics,\n@'\n@t=Mnr2\u000eF[';m]\n\u000e';@mi\n@t=\u0000Mm\u000eF[';m]\n\u000emi(1)tvMn\u0016'k0=1\u00181=2A0;1!B\u000b2;4\n1110.05\u0000\n2\u0019;p\n22\u0019\u0001\n(1;1)(1;1)1(-0.001, 0)\nTABLE I. Modeling parameters. The parameters are inspired\nby [20] and chosen to maximize the energetic di\u000berence be-\ntween square and triangular phase.\ni= 1;2, respectively. However, in the limit of strong ex-\nternal magnetic \felds, Bext, the magnetization, m, can\nassumed to be homogeneous in the crystal. As shown\nin Ref. [18] the magnetization becomes perfectly aligned\nwith the external magnetic \feld and independent of the\nrelative orientation of the crystal. For paramagnetic or\nferromagnetic materials near the Curie temperature, the\nmagnitude of the magnetization m=jmjdependents on\nthe magnitude of the external magnetic \feld Bext. In\nthis limitfm(m) is constant and does not in\ruence the\ndynamics. Furthermore, we are only concerned with the\ncrystal phase and assume !m= 0. The remaining param-\neters are chosen as in Table I and lead to a minimization\nof energy if the magnetization is aligned with the h1 1i-\ndirections of the crystal, the easy axis. The hard axis\nare theh1 0i-directions. Thus, a preferably or perfectly\naligned single crystal has a h1 1i-direction aligned with\nthe external magnetic \feld. Due to the direct relation\nbetween Bextandm, only the evolution equation for '\nremains and reads:\n@'\n@t=Mnr22\n4'\u0000t\n2'2+v\n3'3(2)\n\u0000Z\nC2(r\u0000r0)'(r0) dr0\n+!Br2X\nj=1\u000b2j(mr')2j\u00001m3\n5;\nwhere mis considered as a parameter. Increasing m\nleads to increasing anisotropy and magnetostriction [18].\nThe external magnetic \feld Bextand thus mis assumed\nto be parallel to the thin \flm. Thus, in this limit of\nstrong external magnetic \felds we can use mto vary\nthe strength and direction of the in\ruence of the ex-\nternal magnetic \feld on the thin \flm. The magnitude\nofmis varied between [0; 0.8] and varies the magnetic\nanisotropy.\nIn order to increase numerical stability, short wave-\nlength in the solutions of the density are gradually\ndamped in k-space by adding \u000010\u00006(2k1\u0000k)2to Ck(k).\nThe evolution equation is solved semi-implicitly in time\nwith a pseudo-spectral method. For numerical details we\nrefer to Refs. [21, 22]. The reduced model Eq. (2) is nu-\nmerically more stable and less costly compared to the full\nmodel (1). The timestep may be increased by an order of\nmagnitude. Thus, coarsening simulations for large times\nbecome feasible.\nHere, the thin \flm is modeled by a two dimensional\nslab perpendicular to the \flm height. The crystalline or-3\nder is de\fned by the density wave, '. The external mag-\nnetic \feld is assumed parallel to the \flm and induces a\nhomogeneous magnetization. The magnetic driving force\nin the model is controlled by the magnitude of the mag-\nnetic moments.\nWe choose a parameter set, which shows stagnation\nin coarsening to include the e\u000bect of retarding forces and\nre\rect the experimental \fndings. The simulation domain\nhas sizeL2= 819:22. The mean distance of density peaks\nis one and is resolved by ten grid points, (dx=0.1). Thus,\nthe whole systems consists of 6 :7\u0001105density peaks, rep-\nresenting particles. A time step of dt=0.1 was used.\nIII. COARSENING\nEquation (2), is used to model magnetic assisted an-\nnealing of thin \flms. The texture of the polycrystalline\nstructure is monitored during annealing in order to ex-\ntract geometrical and topological properties over time\nand compare them for di\u000berent magnitudes m. To gen-\nerate an appropriate initial condition we set m= 0, start\nwith a randomly perturbed density \feld ', and solve eq.\n(2) until we reach a polycrystalline structure with small\ncrystallites with square symmetry. The perturbation is a\nrandom distortion at every grid point. The small wave-\nlength perturbations are smoothed rapidly by the evo-\nlution equation, but long wavelength perturbation act\nas nucleation centers. Thus, at random positions grains\nwith random orientation begin to grow until they touch\nand from a network of grain boundaries. After impinge-\nment we got about 1,600 randomly oriented grains. This\ncon\fguration is used as initial condition for all simula-\ntions.\nA. Scaling\nFigure 1 shows the evolution of the mean grain area,\nhAi. Coarsening leads to an increase of the mean grain\narea over time. The coarsening is enhanced by increasing\nthe magnetization and, thus, the magnetic anisotropy.\nWe identify scaling regimes by a power law, hAi/t\u000b,\nwith a scaling exponent \u000b. In all cases a \frst scal-\ning regime Fig. 1(B) is reached after an initial phase\nFig. 1(A). Without magnetization a scaling exponent of\n\u000b= 1=3 is observed. Increasing the magnetic in\ruence\nincreases the scaling exponent. The maximum scaling\nexponent\u000b= 1 is achieved for m= 0:8. However, this\nscaling regime ends. For small magnetic in\ruence below\nsome threshold, it turns into stagnation, Fig. 1(C). Above\nthis threshold, here m\u00150:5, the scaling becomes inde-\npendent of magnetic interaction and we observe \u000b= 1=3,\nFig. 1(D).\nIt has been shown before that without magnetic driv-\ning force the scaling exponent depend on initial condi-\ntions and modeling parameters [4]. This also remains\nif magnetic driving forces are included. The identi\fed\nFIG. 1. Long time evolution of mean grain area for di\u000ber-\nent magnetization. Four di\u000berent regimes are identi\fed: (A)\ntowards scaling, an initial phase; (B) dependent scaling, a\nmagnetically enhanced scaling regime with the scaling expo-\nnent depending on m; (C) towards stagnation, a regime which\nis only present without or with low magnetic \felds; and (D)\nindependent scaling, a regime reached at late times, with a\nscaling exponent independent of magnetic anisotropy. mis\nvaried between [0; 0.8] and models the strength of magnetic\nin\ruence and anisotropy.\nregimes (A), (B), (C) and (D) thus also depend on initial\nconditions and modeling parameters.\nWithout magnetic driving force the texture becomes\nself similar during coarsening [1, 4]. This is not the case\nfor magnetically enhanced coarsening due to grain selec-\ntion. In the following we analyze texture evolution during\ncoarsening in detail in order to understand the change of\nthe scaling behavior.\nB. Orientation selection\nThe magnetic driving force leads to preferable growth\nof grains, which are preferably aligned with respect to the\nexternal magnetic \feld. Figure 2 shows typical orienta-\ntion distributions and how they evolve over time depen-\ndent on the magnetic in\ruence. The color represents the\nlocal crystal orientation, \u0012. A preferably aligned crystal\ncorresponds to \u0012= 0 and, due to symmetry, the \u0012varies\nin the range [\u00000:25\u0019;0:25\u0019].\nThe initial orientation distribution is constructed with-\nout magnetization. Thus, it is homogeneous, Figs. 2(a)\nand (b). There is no preferred orientation for the grains.\nWithout a magnetic driving force, m= 0, the orientation\ndistribution stays homogeneous, Figs. 2(c)-2(e). With\na magnetic driving force this changes and well aligned\ngrains grow preferentially, Figs. 2(f)- 2(k). Grains with\n\u0012\u00190 (green) grow at the expense of the other grains\n(blue, red). As already quanti\fed in Fig. 1, the en-\nhanced grain growth with increasing mcan be seen also\nby larger grain sizes for increasing m, Fig. 2(d), 2(g) and4\nFIG. 2. Grain structure during annealing. The color represents the local orientation of the easy axis with respect to the\nexternal magnetic \feld. The area fraction is shown as function of orientation for the initial and \fnal con\fgurations for di\u000berent\nmagnetic \felds m. The times for the snapshots for m=0, 0.5 and 0.7 are (9 \u0001103;2:7\u0001104), (1\u0001103;1:1\u0001104) and (4:1\u0001103;1:6\u0001104),\nrespectively.\n2(j). However, we are here interested in the orientation\ndistribution, which becomes sharply peaked at \u0012= 0,\nFig. 2(e), 2(h) and 2(k). The e\u000bect increases with in-\ncreasing magnetic driving force, as already analyzed for\nthe full model (1) in Ref. [18].\nFIG. 3. Mean magnetic force during coarsening for di\u000berent\napplied magnetic \felds m.\nThe narrowing in orientation distribution has an ef-\nfect on the total impact of the external magnetic \feld.\nAs it reduces the mean orientation di\u000berence of adja-\ncent grains it also reduces the mean magnetic driving\nforce. To measure this e\u000bect we de\fne the mean mag-\nnetic driving force as the average energy di\u000berence due to\nmagnetic anisotropy with respect to a perfectly alignedcrystal. Figure 3 shows this quantity over time. Initially\nthe mean magnetic driving force strongly depends on the\nstrength of the magnetic \feld. Large mlead to large\nmagnetic anisotropy and, thus, large magnetic driving\nforces. However, over time the mean magnetic driving\nforce decreases as the mean orientation deviation from a\nperfectly aligned crystal decreases due to grain selection.\nThe strength of this e\u000bect correlates with the strength of\nthe magnetic \feld. At large times, the mean magnetic\ndriving force falls below a threshold. This large time be-\nhavior correlates with the independent scaling regime in\nFig. 1(D), which occurs, when the mean magnetic driv-\ning force falls below \u00190:7\u000110\u00004. The time this threshold\nis reached depends on mand is indicated by the dashed\n(red) line in Fig. 1. Thus, orientation selection induced\nby the external magnetic \feld over time decreases the\nin\ruence of the magnetic \feld, which explains the tran-\nsition to the independent linear scaling (D) in Fig. 1.\nIn the case of stagnation, m < 0:5, the mean magnetic\ndriving force never exceeds the de\fned threshold.5\na)\n b)\n c)\nFIG. 4. Log-normal distribution parameters exp( \u0016) (a) and\u001b(b) over time and GSD (c) for \fnal averaged values for m\nbetween [0; 0.8]. The data for m= 0 correspond with [4] and the experimentally found universal GSD in [1].\nC. Grain Size Distribution\nThe external magnetic \feld does not only change the\norientation distribution but also the grain size distri-\nbution (GSD). Without external magnetic \felds it was\nshown in [4] that the coarsening becomes self similar and\nthe GSD is well described by a log-normal distribution:\n(p\n2\u0019\u001bx)\u00001exp (\u0000(logx\u0000\u0016)2\n2\u001b2), where x is the scaled ra-\ndiusR\nhRi. We calculate the GSD for all coarsening sim-\nulations and \ft log-normal distributions to our results.\nIn Figs. 4(a) and 4(b) the two values de\fning the log-\nnormal distribution, exp( \u0016) and\u001bare shown over time.\nDuring the dependent (magnetically enhanced) scaling,\nFig. 1(B), exp( \u0016) and\u001bchange: exp( \u0016) decreases, while\n\u001bincreases. Thus, the GSD is not constant over time\nand, thus, the coarsening is not self similar. Only within\nthe independent scaling regime and towards stagnation,\nFigs. 1(C) and 1(D), the GSD becomes stationary on av-\nerage. Thus, self similar growth is achieved.\nAs the number of grains is drastically decreased within\nthis regime the GSD statistics become more and more\nnoisy. Fluctuations in the GSD approximation increase\nfor larger times and higher magnetic in\ruence. In or-\nder to compare the GSD for di\u000berent external magnetic\n\felds in the limit of large times, we average exp( \u0016) and\u001b\nfor large times and use the averaged value to reconstruct\nthe log-normal distribution, see Fig. 4(c). Large exter-\nnal magnetic \felds, m > 0:5 shift the maximum of the\nGSD towards smaller sizes. However, the tail becomes\nwider. Thus, the number of large grains with respect to\nthe average grain size is increased. For smaller external\nmagnetic \felds, m < 0:5 the tendency is the same but\nthe di\u000berence is minor.\nD. Grain coordination and shape\nVarious other geometrical and topological measures\nhave been considered to de\fne the grain structure. The\nnext neighbor distribution (NND) or coordination num-\nber of grains counts the number of neighboring grains.\nThe shape of grains can be quanti\fed by approximating\nevery grain by an ellipse. The ratio of the axis of theellipse then measure the elongation of grains. This leads\nto the axis ratio distribution (ARD). Elongated grains\nmy have preferred direction of elongation. This is mea-\nsured here by the angle of the small axis with the external\nmagnetic \feld and lead to a small axis orientation distri-\nbution (SAOD).\nWe concentrate on large times for which the coarsen-\ning is self similar. Figure 5(a) shows the NND, which is\nalso \ftted by a log-normal distribution. With increasing\nexternal magnetic \feld the distribution broadens and the\nmaximum is shifted to smaller values. This can already\nbe related to the faster growth, which leads to larger\ngrains and thus also an increased di\u000berence in grain size.\nClassical empirical laws for topological properties in grain\nstructures, such as the Lewis' law and the Aboav-Weair's\nlaw, see Ref. [23] for a review, show a linear relation\nbetween the coordination number and the area of the\ngrains and postulate that grains with high (low) coor-\ndination number are surrounded by small (large) grains,\nrespectively. These e\u000bects are further enhanced by the\nelongation of grains, which lead to more neighbors. Addi-\ntionally, small grains between elongated grains have less\nneighbors.\nThe ARD can also be approximated by a log-normal\ndistribution, Fig. 5(b). With increasing magnetic\nanisotropy the ratio increases and more and more elon-\ngated grains are present. The orientation of the elonga-\ntion is correlated with the external magnetic \feld. In\nFig. 5(c) the orientation distribution of the small axes\nwith the direction of the external magnetic \feld (SAOD)\nis shown. Here the distribution is approximated by a\ncosine. The elongated grains become more and more ori-\nented perpendicular to the external magnetic \feld.\nIV. DISCUSSION\nClassical Mullins-like models for grain growth predict\nself similar growth and a scaling law hAi/t\u000bwith a\nscaling exponent \u000b= 1 [6]. This also does not change if\nexternal magnetic \felds are introduced as an additional\ndriving force. In contrast to our simulation, see Fig. 1,\nno in\ruence of the scaling behavior is observed. Even\nthough the texture depends on strength and direction of6\na)\n b)\n c)\nFIG. 5. Log-normal description for next neighbor distribution (NND) (a), the smoothed distribution should of course be\ninterpreted in a discrete setting, axis ratio distribution (ARD) (b) and cosine description for small axis orientation distribution\n(SAOD) (c), obtained from late time coarsening regime. mis varied between [0; 0.8]. Note: the NND has only discrete values\nand isrepresented by a smooth density distribution to show the in\ruence on m.\nthe external magnetic \feld [24{29]. In these simulations\nthe increase of growth of well aligned grains is leveled by\nthe decrease of growth of not well aligned grains. Thus,\nthe scaling exponent is predicted to be independent of\nthe additional driving force. In these models, smaller\nexponents and stagnation of grain growth, as observed\nin experiments [1], can only be achieved by introducing\nadditional retarding or pinning forces.\nWithin the considered PFC model triple point and ori-\nentational pinning are naturally present, which is one\nreason for the observed lower scaling exponent and the\nstagnation [4]. External magnetic \felds introduce an ad-\nditional driving force to the system. If large enough they\ncan overcome the retarding forces and enhance growth.\nThis explains the dependent growth regime with scaling\nexponents depending on the applied magnetic \feld. If\nthe magnetic driving force is large enough all retarding\nforces are overcome and an exponent of \u000b= 1 is reached.\nGrain growth under an applied magnetic \feld leads\nto preferable growth of well aligned grains. It is this\ngrain selection which decreased the mean magnetic driv-\ning force over time. If the texture is dominated by well\naligned grains, the magnetic driving force is no longer\na function of the applied \feld but is limited by the tex-\nture, see Fig. 3. Only parts of the retarding forces can be\novercome and the scaling exponent becomes independent\nof the magnetic interaction. Turning o\u000b the magnetic\n\feld in this regime of well aligned grains leads to stagna-\ntion. It can only be speculated about the origin of this\nretarding forces and the mechanism they are overcome\nby the magnetic \feld. However, crystalline defects and\nelastic properties are known to be modi\fed by the local\nmagnetization [18] and lead to magnetization dependent\nmobilities. The same mechanism may also open new re-\naction paths for defect movement which might remove\nthe retarding forces.\nIn the case of small magnetic \feld the coarsening stag-\nnates. In this regime the magnetic driving force is not\nlarge enough to overcome the retarding force responsible\nfor stagnation.\nWithin the independent scaling regime self similar\ngrowth is observed which allows to compute various geo-\nFIG. 6. GSD of Zr sheet after annealing with and without\nmagnetic \felds of 19 T. Data is extracted from Fig. 14 in\nRef. [30]. The mean grain size hRiis 10\u0016m for the sample\nannealed without magnetic \feld and 18 \u0016m for the sample\nannealed with magnetic \feld.\nmetrical and topological properties of the grain structure.\nTheir dependence on the magnitude of the applied mag-\nnetic \feld has been analyzed. The considered grain size\ndistribution (GSD), next neighbor distribution (NND)\nand axis ratio distribution (ARD) broaden with increas-\ning magnetic anisotropy, leading to larger grains, more\ngrains with very few and many neighbors, and more elon-\ngated grains, see Figs. 4 and 5. The shift in the NND to\nsmaller coordination number has also been reported for\nsimulations based on Mullins type models [26].\nEven though, texture control by magnetic \felds is\nof increasing interest [14] there are not much data on\nthe in\ruence of magnetic \felds on GSD in thin \flms\navailable. In [30] the texture and grain size evolution\nof thin Zr sheets annealed with and without magnetic\n\felds at di\u000berent temperatures are studied. Increasing\ntemperature and applying external magnetic \felds lead\nto increasing mean size of the grains. The orientation of\nthe \fnal grains are in\ruenced by the magnetic \feld and\nthe orientation distribution becomes peaked at favorable\norientations. The same tendency as predicted by our\nsimulations, Fig. 2. In Fig. 6 the GSD are compared for\nthese samples after annealing with and without magnetic\n\feld. The magnetic \feld shifts the peak of the GSD\ntowards smaller values leading to an increase of relatively\nsmall grains and relatively large grains. The GSD also7\nwidens and the tail is increased by the magnetic \feld.\nAlso these details in the evolution follow qualitatively\nour simulation results, Fig. 4. But we are not aware of\nan experimental study showing the increased elongation\nof the grains perpendicular to the external magnetic \feld.\nV. CONCLUSION\nWe studied magnetically enhanced coarsening with an\nextended PFC model. The external magnetic \feld is as-\nsumed to be strong enough to prescribe the magnetiza-\ntion of the thin \flm. That is, the magnetization is con-\nstant and perfectly aligned with the external magnetic\n\feld. The anisotropy of the magnetic properties of the\ncrystal lead to a magnetic driving force. Well aligned\ncrystals grow at the expense of not well aligned crystals.\nAdditionally, magnetostriction leads to deformation of\ncrystal and defect structures.\nThe magnetic driving force leads to grain selection\nand a texture dominated by well aligned grains. As the\namount of similar oriented grains increase, the mean ori-\nentation di\u000berence between grains decreases. Thus, the\nmean magnetic driving force also decreases with time due\nto texture change. The scaling exponent becomes inde-\npendent for large times and for large enough magneti-\nzation. Stagnation and variation of scaling exponents is\ndue to retarding and pinning forces for grain boundarymovement. There are two mechanisms in magnetically\nenhanced coarsening, which change the e\u000bect of retard-\ning forces. Firstly, the magnetic driving force helps to\novercome the retarding forces during coarsening. This\nexplains the scaling regime dependent on the magnetic\nanisotropy. Secondly, the change of structure of the crys-\ntal due to magnetostriction can decrease the energy bar-\nriers representing the retarding force. Then the driving\nforce due to minimization of grain boundary energy may\nbecome large enough to overcome the retarding forces.\nThis could explain the independent scaling regime.\nBut not only the scaling changes, characteristic geo-\nmetric and topological properties are also in\ruenced by\nthe applied magnetic \feld. At least for GSD and NND\nexperiments show the same tendency as predicted by our\nsimulations.\nACKNOWLEDGMENTS\nA.V. and R.B. acknowledge support by the German\nResearch Foundation (DFG) under Grant No. Vo899/21\nin SPP1959. We further gratefully acknowledge the\nGauss Centre for Supercomputing e.V. 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In this introductory article we briefly review the key properties\nand functionalities of various magnetic semiconductor fam ilies, including europium chalcogenides,\nchromium spinels, dilute magnetic semiconductors, dilute ferromagnetic semiconductors and insu-\nlators, mentioning also sources of non-uniformities in the magnetization distribution, accounting\nfor an apparent high Curie temperature ferromagnetism in ma ny systems. Our survey is carried\nout from today’s perspective of ferromagnetic and antiferr omagnetic spintronics as well as of the\nemerging fields of magnetic topological materials and atomi cally thin 2D layers.\nKey words: Magnetic and dilute magnetic semiconduc-\ntors; topological materials; 2D systems\nDOI: PACC: 7550P\nMagnetic semiconductors\nThe discovery of ferromagnetism in some europium\nchalcogenides and chromium spinels a half of the cen-\ntury ago[1]came as a surprise, since insulators typically\nshowed antiferromagnetic or ferrimagnetic spin ordering,\ndrivenbyasuperexchangeinteraction,whereasferromag-\nnetism was considered a domain of metals. However, the\nGoodenough-Kanamori-Andersonrules indicate in which\ncases the superexchange can lead to ferromagnetic short-\nrange coupling between localized spins. This mechanism\naccounts e.g. for Curie temperature TC= 130K in\nCdCr2Se4[2]. In the case of EuO and EuS, the antiferro-\nmagnetic superexchange is overcompensated by a direct\nf–dferromagnetic exchange, which results in TC= 68K\nand 16K, respectively[3].\nSoon after their discovery, magnetic semiconductors\nwere found to exhibit outstanding properties, including\n∗dietl@ifpan.edu.pl\n†alberta.bonanni@jku.at\n‡ohno@riec.tohoku.ac.jpcolossal magnetoresistance and magnetooptical effects,\nassigned rightly to the exchange interaction of band car-\nriers and localized spins. This sd–fcoupling results in\na giant spin splitting of bands below TC, as well as in\nmagnetization fluctuations near TCgenerated by band\ncarriers[4, 5]. Due to the presence of a sizable sd–fex-\nchangeinteraction,byelectrondoping(oxygenvacancies,\nEu substituted by Gd) the magnitude of TCcan be en-\nhanced by about 50K in EuO, in agreement with the\nRuderman-Kittel-Kasuya-Yosida (RKKY) theory[3].\nMore recently, EuS and related ferromagnetic insula-\ntors have been proposed as functional overlayersthat can\nlead to novel topological properties by exchange split-\nting of interfacial bands viaa ferromagnetic proximity\neffect[6]. At the same time, HgCr 2Se4(TC= 110K) ap-\npears to be a Weyl semimetal[7]. Furthermore, antiferro-\nmagnetic EuTe constitutes a test-bench for the emerging\nfield of antiferromagnetic spintronics[8].\nAnother class of materials that attract considerable\nattention are van der Waals magnetic semiconductors\nstudied down to the atomically thin limit[9]. A com-\npetition between ferromagnetic and antiferromagnetic\nsuperexchange accounts for the magnetic properties of\nCrI3and related systems as a function of the number\nof layers, electric field, and strain[9, 10]. In particular,\nmonolayers are ferromagnetic in the case of these com-\npounds in which, similarly to 2D (Cd,Mn)Te[11], the low-\ntemperature 2D spin order is stabilized by the largemag-\nnitude of the uniaxial magnetic anisotropy[12]. On the2\nother hand, TMPX 3, where TM is a transition metal and\nXiseitherS orSe, areantiferromagneticsemiconductors.\nTheirpropertiesasafunction ofthe layernumberareun-\nder investigations[9].\nDilute magnetic semiconductors\nThis material family was initially named semimag-\nnetic semiconductors[13], as it comprises standard semi-\nconductors doped with magnetic impurities that are ran-\ndomly distributed, electrically inactive, and do not show\nany long-range spatial spin ordering[14, 15]. Typical rep-\nresentatives are (Cd,Mn)Te, (Zn,Co)O, and (Pb,Eu)S.\nHere, the short-range antiferromagnetic superexchange\nbetween Mn[16, 17]and heavier 3 dtransition metal ions\n(e.g. Co) or long-range dipolar interactions between rare\nearth magnetic moments[18]lead to spin-glass freezing\natTf<1K forx <0.1. Magnetooptical and quan-\ntum transport experiments have allowed to reveal and\nquantify the influence of sp–dcoupling upon the exci-\nton, polariton, and Landau level spectra, quantum Hall\neffects, one-electron and many-body quantum localiza-\ntion, universal conductance fluctuations, and carrier spin\ndynamics in dilute magnetic semiconductor systems of\nvarious dimensionalities realized in bulk, 2D quantum\nstructures, nanowires, and quantum dots[15, 19–21], as\nwell as made it possible to optically detect electrical spin\ninjection[22]. Conversely, the dynamics of localized spins\nhas been probed by the Faraday effect[23]or quantum\nnoise spectroscopy[24]. The physics of bound magnetic\npolarons, single electrons interacting with spins local-\nized within the confining potential of shallow impurities\nor quantum dots, has been advanced in dilute magnetic\nsemiconductors[25]. While the majority of experimental\nresultscan be explained within virtual crystaland molec-\nularfieldapproximations,strongcouplingeffectsshowup\nin the case of oxides and nitrides doped with transition\nmetal impurities[26, 27].\nMost of the end binary compounds, for instance MnSe\nand EuTe, are insulating antiferromagnets and, there-\nfore, attract attention from the point of view of anti-\nferromagnetic spintronics. Another ultimate limit of di-\nlutemagneticsemiconductorsisrepresentedbyqubitsys-\ntems consisting of single magnetic ions in single quantum\ndots[28, 29].\np-type dilute ferromagnetic semiconductors\nIn these systems, a high density of delocalized or\nweakly localized holes leads to long-range ferromag-\nnetic interactions between transition metal cations,\nwhich dominate over the antiferromagnetic superex-\nchange, and are well described by the p–dZener\nmodel[30, 31], more universal compared to RKKY-type of\napproaches. The flagship example of this material fam-\nily is (Ga,Mn)As in which Mn ions introduce spins andholes to the valence band[32], but to this category be-\nlong also other types of magnetically doped p-type com-\npounds, in which holes originate from point defects, like\n(Pb,Sn,Mn)Te[33], or from shallow acceptor impurities,\ne.g., (Cd,Mn)Te/(Cd,Mg)Te:N[11]and (Zn,Mn)Te:N[34],\nrather than from Mn ions. The reported magnitudes of\nCurie temperature TCreach 200K in (Ga,Mn)As[35–37],\n(Ge,Mn)Te[38, 39], and (K,Ba)(Zn,Mn) 2As2[40]with less\nthan 10% of Mn cations, xeff<0.1, measured by satura-\ntion magnetization in moderate fields, µ0H/lessorsimilar5T. The\nvalues of Curie temperatures achieved in p-type dilute\nferromagnetic semiconductors are to be contrasted with\nTC≈0.16K in n-Zn 1−xMnxO:Al with x= 0.03[41]. Such\nlow values of TCin n-type systems reflect the relatively\nsmall magnitudes of both s–dexchange integral and den-\nsity of states. Higher TCvalues are observed only in spe-\ncific situations, for instance, at the crossings of electron\nLandau levels under quantum Hall effect conditions[42].\nGround breaking spintronic functionalities have been\ndemonstrated and theoretically described for (Ga,Mn)As\nand related systems[31, 43]. They rely on the strong p–\ndcoupling between localized spins and hole carriers, as\nwell as on sizable spin-orbit interactions in p-like or-\nbitals formingthe valence band ororiginatingfrom inver-\nsion asymmetry of the host crystal structure. Numerous\nfunctionalities of (Ga,Mn)As and other p-type dilute fer-\nromagnetic semiconductors (electrical spin injection[44],\nmagnetization control by an electric field[45–47], current-\ninduced domain-wall motion[48, 49], anisotropic tunneling\nmagnetoresistance[50, 51], and spin-orbit torque[52, 53])\nhave triggered the spread of spintronic research over\nvirtually all materials families. In particular, they are\nnow explored in multilayers of transition metals support-\ning ferromagnetism above room temperature[54]. At the\nsametime, the searchforhigh TCferromagneticsemicon-\nductors led to the discoveryof Fe-based superconductors,\nwhereas theoretical approaches to the anomalous Hall ef-\nfect in (Ga,Mn)As in terms of the Berry phase[55, 56]and\nstudies of spin-orbit torque in the same compound[52, 53]\nhave paved the way to spintronics of topological[57, 58]\nand antiferromagnetic systems[59].\nDilute ferromagnetic insulators\nAs mentioned, the Goodenough-Kanamori-Anderson\nrules indicates in which cases superexchange can lead\nto ferromagnetic short-range coupling between localized\nspins. According to experimental and theoretical stud-\nies, suchamechanismoperatesforMn3+ionsinGaNand\naccounts for TCvalues reaching about 13K at x≈10%\nin semi-insulating wurtzite Ga 1−xMnxN[60], an accom-\nplishment preceded by a long series of material develop-\nment stages[61, 62]. Since there is no competing antiferro-\nmagnetic interactions and due to high atom density, an\nunusually high magnitude of magnetization for the dilu-\ntionx= 0.1 is observed in Ga 1−xMnxN[62]. In contrast\nto random antiferromagnets (such as II-VI dilute mag-3\nnetic semiconductors), there is no frustration in the case\nof ferromagnetic spin-spin interactions. Furthermore, ac-\ncording to the tight-binding theory, the interaction en-\nergy decays exponentially with the spin-spin distance[60].\nAccordingly, the dependence of TConxcannot be ex-\nplained within the mean-filed approximation, but cor-\nroborates the percolation theory[63], and can be quanti-\ntatively described by combiningtight-binding theory and\nMonte Carlo simulations[60].\nThe significant inversion symmetry breaking specific\nto wurtzite semiconductors and the insulating charac-\nter of the system allow controlling the magnetization\nby an electric field viaa piezoelectromagnetic coupling\nin (Ga,Mn)N. As shown experimentally and confirmed\ntheoretically[64], the inverse piezoelectric effect changes\nthe magnitude of the single-ion magnetic anisotropy spe-\ncific to Mn3+ions in GaN, and thus the magnitude of\nmagnetization.\nDilute magnetic topological materials\nIn these systems[57, 58], there are novel consequencesof\nmagnetization-inducedgiant p–dexchangespin-splittings\nof bulk and topological boundary states. In particular,\nin thin layers of 3D topological insulators, this split-\nting turns hybridized topological surface states into chi-\nral edge states. Striking consequences of this trans-\nformation include: (i) the precise quantization of the\nHall conductance σxy=e2/hdemonstrated for thin\nlayers of ferromagnetic (Bi,Sb,Cr) 2Te3at millikelvin\ntemperatures[65], as predicted theoretically[66]; (ii) the\nefficient magnetization switching by spin-locked electric\ncurrents in these ferromagnets[67]. In the case of 3D\nDirac materials with magnetic impurities coupled by\nantiferromagnetic superexchange, such as (Cd,Mn) 3As2\nor strained (Hg,Mn)Te, the magnetic field-induced for-\nmation of Weyl semimetals with non-zero topological\ncharges has been predicted[68].\nFurthermore, inverted band ordering specific to topo-\nlogical matter enhances the role of the long-range inter-\nband Bloembergen-Rowlandcontributionto spin-spin in-\nteractions, resulting in higher spin-glass freezing temper-\naturesTfintopologicalsemimetals, suchas(Cd,Mn) 3As2\nand (Hg,Mn)Te, compared to topologically trivial II-\nVI Mn-based dilute magnetic semiconductors[17]. This\nspin-spin exchange mechanism, taken into consideration\nwithin the p–dZener model[69]and named Van Vleck’s\nparamagnetism, was proposed to lead to ferromagnetism\nin topological insulators of bismuth/antimony chalco-\ngenides containing V, Cr, or Fe[66], a proposal discussed\nearlier in the context of zero-gap (Hg,Mn)Te[70]. How-\never, according to ab initio studies and corresponding\nexperimental data, the superexchange in the case of V\nand Cr, and the RKKY coupling in Mn-doped films\nappear to contribute significantly to ferromagnetism in\ntetradymite V 2-VI3compounds[71]. It appears that the\nco-existence of ferromagnetic superexchange with ferro-magnetic Bloembergen-Rowlandand RKKYinteractions\naccounts for TCas high as 250K in (Cr xSb1−x)2Te3with\nx= 0.44[72]. Ferromagnetic coupling between magnetic\nimpuritiesmediatedbycarriersresidinginsurfaceoredge\nDirac cones is rather weak for realistic values of spatial\nextension of these states into the bulk[31].\nHeterogeneities in magnetic semiconductors\nIt becomes increasingly clear that spatial inhomo-\ngeneities in the magnetization distribution account for\na number of key properties of magnetic semiconductors,\nincludingthemagnitudeof TC. Wediscussherethreedis-\ntinct mechanisms accounting for the non-uniform distri-\nbution of magnetization and describe their consequences.\n(1) According to the percolation theory applicable to\ndilute systems with short-range spin-spin interactions,\ncritical temperature Tccorresponds to the formation of a\npercolative cluster that incorporates typically about 16%\nof the spins in the 3D case. This cluster extends over all\nspins at T= 0, implying that at T >0 ferromagnetism\nco-exists with superparamagnetismproduced by not per-\ncolative clusters[73]. Presumably this effect accounts for\nthermal instabilities of the quantum anomalous Hall ef-\nfect even at T≪TC, and shifts below 100 mK the op-\neration range of the potential resistance standards based\non this phenomenon[74, 75].\n(2) A specific feature of conducting magnetic semi-\nconductors is the interplay of carrier-mediated ferromag-\nnetism with carrier localization, which results in spa-\ntialfluctuationsofmagnetizationandsuperparamagnetic\nsignatures generated by the critical fluctuations in the\ncarrier density of states in the vicinity of the metal-to-\ninsulator transition. This effect has been extensively dis-\ncussed in the context of both ferromagnetic[5, 76]and di-\nlute ferromagnetic semiconductors[77–79].\n(3) The magnetic properties discussed above have con-\ncerned systems with a random distribution of cation-\nsubstitutional magnetic impurities. A major challenge\nin dilute magnetic materials is the crucial dependence\nof their properties on the spatial distribution of mag-\nnetic ions and their position in the crystal lattice. These\nnanoscale structural characteristics depend, in turn, on\nthe growth and processing protocols, as well as on dop-\ning with shallow impurities[80, 81]. By employing a range\nof photon, electron, and particle beam methods, with\nstructural, chemical, and spin resolution down to the\nnanoscale, it has become possible to correlate the sur-\nprising magnetic properties with the spatial arrange-\nment and with the electronic configuration of the mag-\nnetic constituent[82, 83]. In particular, the aggregation\nof magnetic cations which are either introduced delib-\nerately or present due to contamination, accounts for\nthe high TCvalues observed in a number of semiconduc-\ntorsandoxides[84, 85]. The symmetryloweringassociated\nwith this aggregation is also responsible for remarkable\nmagnetic and magnetotransport anisotropy properties of4\n(Ga,Mn)As[86]and (In,Fe)As[87].\nOutlook\nSemiconductors and insulators doped with magnetic\nimpurities, in addition to showing fascinating physics,\nhave well established applications as semi-insulating sub-\nstrates and layers, and as building-blocks for solid state\nlasers, optical insulators, and detectors of high energy\nphotons. More recent works have revealed surprising\nmaterials science features, such as a dependence of the\nspatial distribution of magnetic impurities on the Fermi\nlevel position. At the same time, studies of dilute fer-\nromagnetic semiconductors have led to discoveries of\nbreakthrough functionalities, like for instance spin-orbit\ntorque and magnetization manipulation by an electric\nfield, which are on the way of recasting the concept of\ncomputer hardware. 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Jerry Qib*, Ruike Zhaoa* \n \naDepartment of Mechanical and Aerospace Engineering, The Ohio State University, \nColumbus, OH, 43210, USA  \nbThe George W. Woodruff School of Mechanical Engineering, Georgia Institute of \nTechnology, Atlanta, GA 30332, USA  \ncGeorgia Tech Research Institute, Atlanta, GA 30332, USA  \ndDepartment of Physics, The Ohio State University, Columbus, OH, 43210, USA  \n \n§ These authors made equal contribution to this work.  \n*Corresponding authors. Email: qih@me.gatech.edu ; zhao.2885@osu.edu  \n \n \nShape -programmable  soft materials that exhibit integrated  multifunctional  \nshape manipulation s, including reprogramm able, untethered , fast, and reversible \nshape transformation  and locking, are highly desirable for a plethora of \napplications, including soft robot ics, morphing structures , and biomedical devices . \nDespite recent progress , it remains challenging to achieve  multiple shape \nmanipulations in one material system . Here, we report a novel magnet ic shape \nmemory polymer composite to achieve this . The composite consists  of two types of \nmagnetic particles in an amorphous shape memory polymer  matrix . The matrix \nsoftens v ia magnetic inductive heating of low -coercivity particles , and high -\nremanence particle s with reprogram mable  magnetization profile s drive  the rapid \nand reversible shape change under actuation magnetic field s. Once cooled, t he \nactuated shape can be locked. Additionally , varying the particle loadings for \nheating  enables  sequential actuation. Th e integrated multi functional shape  2 manipulation s are further exploit ed for applications including soft magnetic \ngripper s with large grabbing force , sequential logic for computing , and \nreconfigurable antenna s. \n \nKeyword:  shape memory polymers, soft active materials, magnetic soft material , soft \nrobotics, soft material computing    3 Introduction  \nShape programmable soft materials that exhibit integrated multi functional  shape \nmanipulations, including reprogrammable, untethered , fast, and reversible shape \ntransformation  and locking,  in response to external stimul i, such as  heat, light, or \nmagnetic field1-5, are highly desirable for a plethora of applications, including soft \nrobotics6, actuators7-9, deployable devices10,11, and biomedical devices6,12-15. A wide \nrange of novel materials  have been developed in the past, including liquid crystals \nelastomer s16,17, hydrogel s18, magnet ic soft materials6,19, and shape memory polymer s \n(SMP s)1,20-22. Magnetic soft materials composed of magnetic particles in a soft polymer \nmatrix have drawn great interest  recently  due to their untethered control for shape \nchange23,24, motion6,7,25, and tunable mechanical propertie s26. Among them , hard-\nmagnetic soft material s utilize high-remanence, high-coercivity  magnetic particles , \nsuch as neodymium -iron-boron (NdFeB) , to achieve  complex programmable shape \nchanges6,19,27-29. Under  an applied magnetic field , these particles  with programmed \ndomains  exert micro -torques , leading to a large  macrosc opic shape change . However, \nmaintain ing the actuated shape need s a constant ly applied magnetic field, which is  \nenergy inefficient.  In many practical applications, such as soft robotic grippers30,31 and \nmorphing antenna s32,33, it is highly desirable that the actuated shape can be locked so \nthat the material can fulfil l certain functions without the constant presence of an \nexternal field.  \nSMPs can be  programmed  and fixed into a temporary shape and then recover the \noriginal shape under external stimul i, such as heat or light 34,35. Typically, a thermally   4 triggered SMP uses a transition  temperature ( Ttran), such as glass transition temperature \n(Tg), for the shape memory effect . In a shape memory cycle, an SMP is programmed to \na temporary shape by  an external force at a temperature above Ttran followed by cooling  \nand unloading . The SMP recover s its original shape at temperature s above Ttran by direct \nheating  or inductive heating36,37. \nIn this article , motivated by the advantages of hard -magnetic soft material s and \nSMP s, we report a novel magnet ic shape memory polymer  (M-SMP)  with integrated \nreprogrammable , untethered , fast, and reversible actuation and shape locking. The M -\nSMP  is composed of two types  of magnetic particles (Fe3O4 and NdFeB) in an \namorphous  SMP  matrix . The Fe3O4 particle s enable indu ctive heating under a high \nfrequency alternating current (AC) magnetic field  and thus are employed for shape \nlocking and unlocking . The NdFeB particles are magn etized and remagne tized with \npredetermined magn etization profi les for programmable  actuation . We demonstrate that \nthe integrated multifunctional  shape manipulation s offered by M -SMPs can be \nexploited for a wide range of novel applications, including soft  grippers  for heavy loads , \nsequential logic circuit s for digital computing , and reconfigurable morphing antenna s.   \n \nResults  \nDesign and characterization  \nTo demonstrate the concept, we fabricate an ac rylate-based amorphous SMP  with \nembedded NdFeB micro particles and Fe 3O4 micro particles (Methods , Figs. S1-S3, \nTable S1  & S2). Before  use, the M-SMP  is magnetized to have a desired magnetic  5 profile under an impulse mag netic field (~1.5  T). Fig. 1a  shows the working mechanism \nby using an M -SMP cantilever  with a magnetization polarity  along its longitud inal \ndirection. A t room temperature, the cantilever is stiff and cannot deform  under an \nactuation magnetic field ( Ba). When a n AC magnetic field ( Bh) is applied, the inductive \nheating of the Fe3O4 particles heats the M -SMP above its Tg, and the modulus of the M-\nSMP drops significantly. Then , a small Ba can bend the cantilever . By alternating  Ba \nbetween  up (+) and down ( -) directions  at this moment , fast transforming between \nupward and downward bending can be easily achieved. Upon removal of Bh, the \nbending  shape can be locked without further applying Ba once the temperature of the \nM-SMP drops below its Tg. Moreover, t he magn etization profi le of the M-SMP can be \nrepro grammed for different shape transformation by remagnetization . For example, \nremagnetizing the beam when it is mechanically locked in a folding shape will change \nthe actuation shape to folding under the same Ba (bottom row of Fig. 1a ). \nNeat SMP and M -SMP  samples  are prepared to characterize  their \nthermo mechanical properties . Fig. 1b  shows the thermomechanical properties of the \nneat SMP and the M -SMP P15 -15, where the two numbers represent the volume \nfractions of Fe 3O4 and NdFeB  particles , respectively. The storage modulus of P15 -15 \ndecreases from 4.6 GPa to 3.0 MPa when the temperature T increases from 20oC to \n100oC. Tg, measured as the temperature at the peak of the tan  curve, is ~56 °C for the \nneat SMP, and ~58oC for P15 -15 (Fig. S4 ). The Young’s  modulus of the M-SMP at \nhigh-temperature increases linearly with the increasing particle load ing (Fig. 1c). Fig. \n1d shows  the strain, stress , and temperature as function s of time during the shape  6 memory test of P15-15. When P15 -15 is programmed  at 85 °C, it has the  shape fixity \nand shape recovery ratio s of 87.8% and 87.2%, respectively  (Fig. S5).  \n   The Fe 3O4 particles , due to their low coercivity , can be easily magnetized and \ndemagnetized under a small high frequency AC magnetic field , leading  to a magnetic \nhysteresis loss for inductive heating.  In contrast, the NdFeB particles , due to their  high \ncoercivity , can retain high remnant magnetization for magnetic actuation  (Fig. S6 & \nS7). Note that the NdFeB  particles start to be demagnetized  when the temperature is \nabove ~150oC (Fig. S8). Therefore , the temperature for shape unlocking and actuation \nshould be limited to below 150oC.  \n \nFast transforming and shape locking  \nHere, we experimentally demonstrate the remote fast transforming and shape locking \nof the M -SMP , which can be used as  a soft robotic gripper . The experimental setup for \nM-SMP heating and actuation consists of two types of coils  (Fig. 2a ): a pair of  \nelectromagnetic  coils generate Ba for actuation ; a solenoid  provide s Bh for inductive \nheating. A n M-SMP (P15 -15) cantilever  is fabricated with magnetization along its \nlongitudinal directio n in such a way that  the beam  will tend to bend u nder a vertical \nmagnetic field (Fig. 2b ). To heat and actuate the beam , we use Bh = 40 mT at 60 kHz \nand Ba = 30 mT, respectively . The magnetic field profiles for Ba and Bh, as well as the \nmeasured cantilever displacement versus time , are shown in Fig. 2c . The a pplication of  \nBh gradually increases the temperature and the deflection of the M -SMP \n(Supplementary Video  S1). Here, we alternate Ba at 0.25 Hz to show the reversible  7 fast transforming . Upon removal of Bh at 30  s, the temperature drops by air cooling and \nthe modulus of the M-SMP increases dramatically  (Fig. 1b). The bending shape  can \nthen be locked without further application of  Ba. Fig. 2d shows the M-SMP  cantilever \ncarrying a weight (23 g) that is 64 times heavier than its own weight (0.36  g).  \n   Soft robotic gripper s are intensively research ed due to their capability of adapting \ntheir morphology to grab  objects . However, the low-stiffness nature of soft materials  \nsignificantly limits the actuation force , making most soft robotic gripper s incapable of  \ngrabbing heavy object s. Taking M-SMPs ’ advantage of shape locking,  we demonstrate \na soft robotic gripper  that grabs  an object  much heavier than its own weight.  Fig. 2e  \nshows the design and magnetization directions of a four-arm gripper (Fig. S9 ). By \napplying Bh and a positive Ba (upward) , the gripper  softens and opens up for grabbing. \nUpon switching Ba to negative, the gripper  conforms to the lead ball . At this moment, \nthe ball slips if the gripper is lifted (Fig. 2f ). However, the gripper can be locked into \nthe actuated shape and provide a large grabbing force w hen we remove  Bh and cool \ndown the material . As demonstrated in Fig. 2g , the stiffened gripper can effectively lift \nthe lead ball without any external stimulation ( Supplementary Video S2 ). The weig ht \nof the lead ball is 23  g, which is 49 times heavier than the gripper (0.47  g).  \n \nSequential actuation  \nThe s equential  shape transformation of an object in a predefined sequence can enable a \nmaterial or system to fulfill multiple functions25,38. Here, we show that the sequential \nactuation of an M -SMP system  can be ac hieved by designing and actuating material  8 regions with  different Fe3O4 loading s for different resultant heating temperature s and \nstiffness es under the same applied Bh. We prepare three M-SMP s with the same \ndimension containing the same amount of NdFeB (15  vol%) but different amount s of \nFe3O4 (5 vol%, 15  vol% , and 25  vol% , named as P5 -15, P15 -15, and P25 -15, \nrespectively ). Fig. 3a  shows the mechanical and heating characterizations of the three \nM-SMP s under the same Bh (Methods , Fig. S10 ). To reach the temperature (around \n50oC) at which  the M -SMPs become reasonably soft to deform  under Ba, it takes 5  s, \n11 s, and 35  s for P25-15, P15 -15, and P5 -15, respectively .  \nBased on the mechanism of sequential actuation , we design a flower -like structure \nmade of M-SMP petals using  P5-15 and P25 -15 to demonstrate the programm able \nsequential motion (Fig. 3b ). The P5 -15 petals are designed to be longer than the P25 -\n15 ones, and t he magnetization is along the outward radial direction for all petals (Fig. \nS11). Fig. 3c  shows Bh (red) and Ba (black) profiles as function s of time. The deflections \nof P5 -15 and P25 -15 petals , defined as the vertical displacements of the endpoints, are \nplotted as black and blue curves in Fig. 3c , with  the sequential shape change  illustrated \nin Fig. 3d . Upon the application of Bh and a negative Ba, the P25-15 petal s soften and \nstart to bend first due to the large heating  power . During this time, the P5 -15 petal s are \nheated slowly and remain straight due to their lower temperature and high stiffness. \nWith increasing heating time, the P5 -15 petals  start to soften and bend at 18  s and are \neventually  (at 32  s) fully actuated to lift the entire flower . After r emoving  Bh and cooling \nthe flower down  to room temperature, all petals are locked in their deformed  shape. \nFast transforming feature of M -SMP s is also demonstrated by switch ing the magnetic  9 field direction during the actuation process  (Supplementary Video  S3). \nSupplementary Video  S4 shows a flower blooming -inspired  sequential  shape -\ntransformation of an M-SMP system using  P5-15, P15 -15, P25 -15.   \n \nSequential actuation for digital computing  \nSoft active materials and structures have recently been explored for programmable \nmechanical computing due to its capability of integrating actuation and computing in \nsoft bodies  for potential applications in self -sensing  of autonomous soft robots39,40, \nnonlinear dynamics -enabled non conventional computing41, and mechanical logic \ncircuits42-44. Taking M-SMPs ’ advantage s of reversible actuation and shape locking, we \ndemonstrate that M -SMP s can be used to design  a sequential logic device , the D-latch , \nfor storing one bit of information, which can be readily extended to a memory with \narbitrary bits.  The truth table for the D-latch logic is shown in Fig. 3e: when the input \nE is 1, the output Q keeps the same value as the input D; when the input E is 0, the \noutput Q stays latched and is independent of the input D. We achieve this D-latch logic \nutilizing the controlled actuation of an M-SMP beam  switch ( Fig. 3f & 3g). The \nmagnetic  fields Bh and Ba work as inputs and the LED serves as the indicator of the  \noutput.  The time -dependent actuation /locking  of M -SMPs  is interpreted as an RC delay \ncircuit  between Bh and the D -latch, where the heating /cooling time of the M -SMP is \nregarded as the charging /discharging time of a capacitor  (Fig. S12 ). When Bh is on and \nthe beam  is unlocked  (T>T g, E=1), the downward Ba (D=1) or upward Ba (D=0) \ndetermines whether the circuit is closed  or open , leading to the on ( Q=1) or off ( Q=0)  10 state of the LED. When Bh is off and the beam  is locked  (T<T g, E=0), Ba is no longer \ncapable of actuating  the beam  and, consequently , cannot change  the status of the  LED. \nIn other  words, the previous state of Q is stored in the system . \nWith the M-SMP -enabled D-latch system, we next design a sequential digital logic \ncircuit as a three -bit memory , which contains  three  M-SMP beams  (P5-15, P15 -15, and \nP25-15) and three LEDs  shown in Fig. 3h  (Methods , Fig. S13 ). Fig. 3i  shows the three -\nstep logic for this three -bit memory, with E1, E2 and E3 representing the input E for P5-\n15, P15 -15, and P25 -15, respectively. During the operation, h eating for 28  s unlocks all \nM-SMPs ( E1, E2, E3=1); heating for 12 s unlocks  P5-15 and P25 -15 (E1=0, E2, E3=1); \nheating for 6  s only unlocks P25-15 (E1, E2=0, E3=1). Followed by  cooling and actuation  \n(changing D) , the M -SMP switches can lock their shapes and retain the output status . \nFig. 3j  shows the original state and output  states for the three M -SMP switches \nindicated by the LEDs. In the first step,  unlock ing all M -SMPs ( E1, E2, E3=1) with D=1 \nchanges the memory state from 0 -0-0 to 1 -1-1. After cooling and locking  (E1, E2, E3=0), \nwe next  unlock P 15-15 and P25 -15 (E1=0, E2, E3=1) and switch D to 0, which change s \nthe memory state from 1 -1-1 to 1 -0-0. In the third step  after locking ( E1, E2, E3=0), we \nonly unlock P25 -15 (E1, E2=0, E3=1) and switch D to 1  to finally change the memory \nstate to 1 -0-1 (Supplementary Video  S5). This example demonstrate s that b y \ncontrolling the two inputs  Bh (E) and Ba (D), we can erase and rewrite the information \nin the memory  (Table S3  & S4). Theoretically, a n electronic device with n -bit memory \ncan be realized with n M-SMPs with varying particle loadings. In this way, 2n states can \nbe achieved and stored with n steps by manipulating  two inputs. Additionally , we can  11 tune the NdFeB particle loading and Tg to provide more design flexibility for more \ncomplex computing system s using M -SMP s. \n \nReprogrammable morphing radiofrequency antennas  \nThe ability to change the antenna shape on the fly provides the capability to either \nremotely deploy an antenna45,46 or reconfigure its functionality47-49. Here, w e \ndemonstrate two morphing radio -frequency (RF) antennas that can rapidly , reversibly \ntransform between on -demand  shapes . The shape locking of M -SMP s allows  the \nantennas to retain the ir actuated , functional shape s without the need for a constant \napplication of  external stimulation . Fig. 4a  shows the design of a cantilever -based \nmorphing monopole antenna  (48 mm long ). It can be reprogrammed  to different \nmagnetization profiles  to transform into different shapes  for deployable and \nreconfigurable antennas . Being  magnet ized al ong its longitudinal direction , gravity \ndrive s the cantilever to bend  down (Down shape)  upon heating . The Down shape can \nbe actuated to the Up shape under  Ba = 20 mT ( Fig. 4b). Fig. 4 c shows the antenna \nworks as a deployable monopole antenna  due to its  poor impedance  (S11 larger than  the \nacceptable value, -10 dB45,47) in the Down shape  but good S 11 value  with a resonant \nfrequency of 0.95  GHz in the Up shape . Moreover, this deployable antenna can be \naltered to a reconfigurable antenna  by reprog ramming the M -SMP’s  magnetization \nprofile . Here, the same cantilever is remagnetized to have a sinusoidal shape with a \nheight of 24  mm under Ba = 80 mT (Fig. 4b, Supplementary Video  S6, Fig. S14). Fig. \n4c shows the resonant frequency of this antenna shifts  from  0.95 GHz (Up shape) to  12 1.25 GHz (sinusoidal shape ), representing  a 32% change , with good agreement \nachieved between the simulation and experimental results. The radiation pattern \nsimulations and polar plot s are similar  for all these configurations  (Fig. S14 ), which is \nbenefi cial as a reconfigurable antenna.   \nUtilizing M -SMP’s advantages of shape transformation and locking, t he on-demand \nshape transformation from a planar state to a 3D structure can also be achieved. Here  \nwe design a tapered helical antenna to achieve frequency reconfigurability. The antenna \nis composed of a thin M -SMP substrate with printed conductive silver wir e on its \nsurface  (Fig. 4d ). The M-SMP substrate  is magnetized in a stretched , spring -like \nconfiguration ( Fig. S15) to realize the pop -up actuation with programmable heights and \nconfigurations under a controlled vertical Ba (Fig. 4e , Supplementary Video S7). The \nsimulation and experimental results in  Fig. 4f show that the resonant frequenc y of the \nantenna  can be readily tuned between 2.15 GHz and 3.26 GHz. T he simulated  radiation \npatterns at resonance with similar profile s shown  in Fig. 4g  indicate that the operating \ndirection of the antenna remains constant, which is desirable for antenna applications . \nDue to the shape locking capability offered by the M-SMP, the reconfigured antenna \ncan retain the new shape with out assistance from the external field, which lowers the \noverall energy requirements. Using M -SMPs as a substrate material for a remote ly \ncontrolled reconfigurable antenna is thus advantageous over the mechanically \nprogrammed antenna33 and conventional magnetic -responsive antenna50.  \n \nConclusion   13 In summary, the reported magnetic  shape memory polymer integrates reprogrammable , \nuntethered, fast,  and reversible shape transformation and shape locking into one system . \nUtilizing  two types of embedded magnetic particles for inductive heating and actuation, \nthe material can be effectively unlocked and locked for energy -efficient operations and \nfunctions  as soft robotic grippers, sequential actuation  devices , digital l ogic circuit s, \nand deploy able/reconfigurable antenna s. With recent advances in  simulation  tools and \n3D/4D printing  for design optimization and fabrication  of complex structures , these \ndemonstrations  suggest that  the novel M -SMP can serve as a material platform for a \nwide range of applications, including biomedical devices for minimum invasive surgery , \nactive metamaterials,  morphological computing, autonomou s soft robots, and \nreconfigurable , flexible electronics, etc. 14 Methods  \nPreparation of the M -SMPs   \nOur neat SMP is an acrylate -based amorphous polymer. The resin contains aliphatic \nurethane diacrylate (Ebecryl 8807, Allnex, GA), 2 -Phenoxyethanol acrylate (Allnex, \nGA), isobornyl acrylate (Sigma -Aldrich, St.  Louis, MO), and isodecyl acrylate \n(Sigma -Aldrich, St. Louis, MO) with a weight ratio of 0.7:60.2:30. 1:9. A thermally -\ninduced radical initiator (2,2' -Azobis(2 -methylpropionitrile), 0.7 w%) is added for \nthermal curing. Additionally , 2 wt% of fumed silica with an average size of 0.2 -0.3 μm \n(Sigma -Aldrich, St. Louis, MO) and 0.4  wt% of 2,2' -Azobis(2 -methylpr opionitrile) are \nadded to ensure good mixing of the matrix resin with the magnetic particles. The \ncomposite is prepared by adding predetermined amounts of Fe 3O4 (0-25 vol%) (particle \nsize of 30  m, Alpha Chemicals, MO, USA) and NdFeB magnetic particles (15 vol %) \n(average particle size of 25  m, Magnequench, Singapore) in the matrix resin. The M -\nSMP composite is denoted as Px -y with x and y representing  the Fe3O4 and NdFeB \nvolume fraction s, respectively . The mixture is manually mixed, degassed under vacuum, \nand then sandwiched between two glass slides with different spacings  for thermal \ncuring. The thermal curing is conducted by precuring at 80oC for 4  h and postcuring at \n120oC for 0.5 h. The cured composite films are magnetized and remagnetized under  \nimpulse magnetic fields ( ~1.5 T for first magnetization and 5.5 T for remagnetization) \ngenerated by an in -house built impulse magnetizer.  \nElectromagnetic coil system for actuation  and inductive heating  \nWe use a pair of in -house built electromagnetic coils with coil spacing  of 74 mm for \nactuating. The two coils are connected in series and powered by a custom \nprogrammable power supply with up to 15 A output current. The coils can ge nerate a \ncentral magnetic field with a ratio of 7 mT/A. A water -cooled solenoid is connected to \nan LH -15A high -frequency induction heater to generate an AC magnetic field with a \nfrequency of 60  kHz and a magnetic field ranging from 10  mT to 60  mT. \nPhysical properties characterization  \nUniaxial tension tests are conducted on a dynamic mechanical analysis (DMA) tester \n(DMA 850 , TA Instruments, New Castle, DE) at various temperatures. The thin film \nsamples (dimension: about 20  mm × 3 mm × 0.6 mm) are str etched at a strain rate of \n0.2/min. The dynamic thermomechanical properties are measured on the DMA tester. \nThe strain is oscillated at a frequency of 1  Hz with a peak -to-peak amplitude of 0.1%. \nThe temperature is ramped from 0oC to 120oC at the rate of 3oC/min. The shape memory \ntests are carried out on the DMA tester in the uniaxial tensile mode with controlled \nforce. The thermal imaging video (Video S 1) and temperature profiles (Fig. 2c  & Fig. \n3a) are recorded using a thermal imaging camera (Seek Thermal , Inc., Santa Barbara, \nCA). The dimension of M -SMPs used for the temperature profiles in Fig. 3 a is 10 mm \n× 10 mm × 1 mm.  \nCantilever experiments  \nThe M -SMP film is cut into a strip with a length of 35 mm and width of 4.5 mm. Two \nacrylic pieces (15 mm  × 7 mm × 2 mm) are used to clamp one end of the M -SMP strip \nto create a cantilever with a length of 20  mm.    15 Gripper experiments  \nTwo M -SMP strips (47 mm  × 5 mm  × 0.5mm ) are cut and glued together to form a \ncross shape. The dimension of each arm is 21 mm x  5 mm. The four -arm gripper is \nsoftened  and mechanically deformed to a grasp  shape  (diameter: 15  mm). The gripper \nis then cooled down and magnetized along the direction shown in Fig. S9. After \nmagnetization, a glass  rod holder  is glued to the central part of the grippe r. \nFlower -like structure experiments  \nThe flower -like structure has two type s of petals, one is P5-15 and one is P25-15. The \ndimensions of P5 -15 and P25 -15 petals are shown in Fig. S11. The petals are cut with \nthe assist of a crylic molds. The individual petals are magnetized along the outward \nradial direction . The central part is 3D -printed using  a commercial rigid resin using a  \nFormlabs Form2 3D printer (Formlabs, Somerville, MA, USA). The petals are then \nglued to the central part.  \nSequential logic circuit experiments  \nAll beams used as switches in the sequential logic circuit have the dimension of 20 mm \n× 5 mm × 0.8mm . Each beam is fixed at one end to the printed circuit. Silver paste \n(Dupont ME603)  is uniformly painted on the bottom surface of the beams and cured at \n80°C for 20 min. The LED leads, the fixed end of beams, and the copper wires for \nconnecting the power supply are all attached to the printed circuit using silver paste. \nFinally, the asse mbled circuit is cured at 80°C for another 20 min.  \nSingle cantilever -based antenna experiments  \nThe M -SMP film is cut into a strip with dimension of 50 mm × 10 mm  × 0.25 mm . The \ndesigned silver band  has a width of 6  mm and a length of 50  mm. Silver paste is painted \non one side of the strip and cured at 80°C for 20 min. One end of the cantilever -based \nantenna sample is glued to a 3D -printed PLA base.  \nHelical antenna experiments  \nThe helical antenna is fabricated with a 3D -printed PV A mold us ing an Ultimaker S5. \nThe mold is filled with the M -SMP resin mixture and sandwiched between two glass \nslides for thermal curing.  The curing reaction is conducted by pr ecuring at 80oC for 4 \nh and post -treatment at 120oC for 0.5 h. The PVA mold is then dissolved using water. \nThe cured sample  is heated until soft, deformed to the shape as shown in Fig. S15 , and \nmagnetized along the height direction.  \nAntenna simulation and measurements  \nThe antenna is transformed and locked to the expected actuated shape and is fed by a \n50 Ω coaxial probe. The antenna ’s return loss (S 11) is measured using a Vector Network \nAnalyzer (VNA). In all experiments, the antenna is connected to a 50  Ω SMA connector \non a 300  mm x 300 mm aluminum ground plane. After achieving th e desired antenna \nshape using Bh and Ba, the feed pin of the SMA connector is connected to the conductive \nsilver lines on the antenna, exciting the antenna for measurements. The bandwidths of \ninterest during the measurement are from 0.5  GHz to 2  GHz for the cantilever -based  \nantenna s and 2  GHz to 4  GHz for the helical antenna. All antenna simulations are \nconducted using ANSYS Electromagnetic Suite V19.10 HFSS.    16 References  \n1 Lendlein, A. & Kelch, S. Shape -memory polymers. Angew Chem Int Edit  41, 2034 -2057 (2002).  \n2 Jeon, S. -J., Hauser, A. W. & Hayward, R. C. Shape -Morphing Materials from Stimuli -Responsive \nHydrogel Hybrids. Accounts of Chemical Research  50, 161 -169 (2017).  \n3 Liu, Y ., Genzer, J. & Dickey, M. D. “2D or not 2D”: Shape -programming polymer sheets. 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V. in 2015 IEEE International Symposium on \nAntennas and Propagation & USNC/URSI National Radio Science Meeting.   2215 -2216 (IEEE).  \n46 Costantine, J., Tawk, Y ., Christodoulou, C. G., Banik, J. & Lane, S. CubeSat deployable antenna \nusing bistable composite tape -springs. IEEE Antennas and Wireless Propagation Letters  11, \n285-288 (2012).  \n47 Kovitz, J. M., Rajagopalan, H. & Rahmat -Samii, Y . Design and implementation of broadband \nMEMS RHCP/LHCP reconfigurable arrays using rotated E -shaped patch elements. IEEE \nTransactions on Antennas and Propagation  63, 2497 -2507 (2015).  \n48 Oliveri, G., Werner, D. H. & Massa, A. Reconfigu rable electromagnetics through \nmetamaterials —A review. Proceedings of the IEEE  103, 1034 -1056 (2015).  \n49 Bernhard, J. T. Reconfigurable antennas. Synthesis lectures on antennas  2, 1-66 (2007).  \n50 Langer, J. -C., Zou, J., Liu, C. & Bernhard, J. T. Micromachi ned reconfigurable out -of-plane \nmicrostrip patch antenna using plastic deformation magnetic actuation. IEEE Microwave and \nWireless Components Letters  13, 120 -122 (2003).  \n \nAcknowledgement  \nR.Z. acknowledges the Haythornthwaite Foundation Research Initiati on Grant and the \nOSU Emergent Materials Research Grant. H.J.Q acknowledges the support of an \nAFOSR grant (FA9550 -19-1-0151; Dr. B. -L. “Les” Lee, Program Manager). F.Y .Y \nacknowledges the support from US Department of Energy under Grant No. DE -\nSC0001304. J.M.K . acknowledges the Georgia Tech Research Institute’s indepe ndent \nresearch and development HIVE program under the direction of Mr. Benjamin Riley. \nThe authors would like to thank Dr. Ken Allen and Dr. Todd Lee for their review of this \narticle.  \n \nAuthor Contributions  \nR.Z. and H.J.Q. conceived the concept and designe d the study.  X.K.  and Q.Z. developed \nthe material s. Q.Z. and F.Y . Y. characterized the magnetic properties of the material s. \nX.K., S.W., and S.M.M. characterized the thermomechanical properties and shape \nmemory performance of the material s. Q.Z. and S.W. designed the magnetic actuation \nexperiments and Q.Z., S.W., and RD. Z. conducted these experiments. J.W., J.M.K., \nand Q.Z. designed morphing antennas. J.W. and J.M.K. conducted the antenna \nsimulations and experiments. All the authors read t he manuscript.  \n \nCompeting Financial Interests  \nThe authors declare no competing financial interests.  \n   19 Figures and Figure Captions  \n \n \nFigure 1. Schematics and properties of magnetic shape memory polymers (M -\nSMPs). (a) Working mechanism of M -SMPs. (b) Storage modulus and tan δ versu s \ntemperature for the neat SMP and P15 -15 (M -SMP with 15  vol% Fe 3O4 and 15  vol% \nNdFeB). (c) Effect of NdFeB and Fe 3O4 particle loadings on the Young’s modulus of \nthe M -SMP at 85oC. (d) Shape memory performance of P15 -15 (black dashed line: \nstress; blue solid line: strain; red dotted line: temperature).  \n \n \n 20  \nFigure 2. Fast -transforming and shape locking of M-SMPs via super imposed \nmagnetic fields. (a) Experimental setup for the super imposed magnetic fields: the two \nparallel electric coils are used to generate the actuation magnetic field, Ba; the solenoid \ncoil in the middle is used to generate the heating magnetic field, Bh. Scale bar: 15 mm. \n(b) Cantilever bending and shape locking. Scale bar: 5 mm. (c) Magnetic field profiles \nof Ba and Bh and beam deflection and temperature with respect to time. The gradient \nbackground color illustrates the time -dependent temperature change with  the scale bar \non the side. (d) Locked bending beam carrying a weight (23g) 64 times heavier than its \nown weight (0.36g). (e) Design and magnetization profile of a fo ur-arm M -SMP gripper \n(0.47g). (f -g) M-SMP  gripper lifting  a lead ball (23g) without (f) and with (g) shape \nlocking . Scale bar: 5 mm.  \n \n 21  \nFigure 3. Sequential actuation of M -SMPs and its application as digital logic \ncircuits . (a) Temperature and corresponding Young’s moduli of three M -SMPs \ncontaining different Fe 3O4 loadings. (b) Design of a flower -like structure using P5 -15 \nand P25 -15 M -SMPs. (c) Magnetic field profiles ( Ba and Bh) and deflection of the \nsequentially actuated M -SMPs with respect t o time. (d) Sequential shape transforming \nand shape locking. (e) Truth table for a D-latch. (f) Schematic of an M-SMP D -latch \nlogic with two magnetic fields ( Ba and Bh) serving as input and LED state as output. (g) \nRelationship between Bh and the enabled input E of the D -latch. (h) Design of the \nsequential logic circuit using M -SMPs with different Fe 3O4 loadings (P5 -15, P15 -15, \nand P25 -15). (i) Magnetic control for a se quential logic circuit with three steps and \ntunable outputs. (j) LED indications for four different output states. Scale bars in (d) \nand (j): 5 mm.   \n \n 22  \nFigure 4. Application of M -SMP s for morphing antennas. (a) Schematic of a single -\ncantilever monopole an tenna. (b) Cantilever antenna with t wo different magnetization \nprofiles by reprogramming . (c) Experimental (solid lines) and simulation (dashed lines) \nresults of the S 11 spectrum. (d)  Schematic and magnetization profile of a reconfigurable \nhelical antenna.  (e) Actuation of the helical antenna under different Ba. (f) Experimental \n(solid lines) and simulation (dashed lines) results of S 11 band for the reconfigurable \nhelical antenna at di fferent heights. (g) 2D polar plot of the simulated radiation patterns \nof the helical antenna at different heights.  Scale bars in ( b) and ( e): 5 mm.  \n \n"    },    {        "title": "1910.01218v1.Magnetic_Scattering_Chapter.pdf",        "content": "1 \n Magnetic Scattering  \n \nJeffrey W. Lynn1 and Bernhard Keimer2 \n1NIST Center for Neutron Research  \nNational Institute of Standards and Technology  \nGaithersburg, MD 20899- 6102 (USA)  \n  \n2Max -Planck Institute  for Solid State Research  \nHeisenbergstrasse 1  \nD-70569 Stuttgart (Germany)  \n \nin  \n \nHandbook of Magnetism  \nedited by Michael Coey and Stuart Parkin  \n  2 \n  \nContents  \nI. Introduction  ......................................................................................................................................... 3 \nII. Magnetic Neutron Diffraction Technique  ..................................................................................... 3 \n a.  Polarized Neutron Techniques  \n b.  Polarized Neutron Reflectometry  \nIII. Resonant Magnetic X -ray Diffraction Technique  ........................................................................ 8 \nIV.       Dynamics  ........................................................................................................................................ 11 \n          a.  Inelastic Neutron Scattering Technique  .................................................................................... 11 \n         b.  Resonant Inelastic X -ray Scattering Technique  ........................................................................ 12 \nV.     Magnetic Diffraction Examples with Neutrons  .......................................................................... 13 \nVI. Magnetic Diffraction Examples with X -rays  .............................................................................. 16 \nVII.  Spin Dynamics with Neutrons  ...................................................................................................... 18 \nVIII.     Spin Dynamics with RIXS ............................................................................................................ 20 \nIX. Facilities  and online information  .................................................................................................  21 \nX. Summary and Future Directions  ..................................................................................................... 22 \nXI. Acknowledgments  ......................................................................................................................... 22 \nReferences  \n \n  3 \n I. Introduction \nScattering measurements provide essential information about the intrinsic electronic \ninteractions in magnetic material s.  Traditionally neutron scattering has been the unique magnetic \nscattering tool,  [1] [2] but that situation has changed recently following  remarkable advances in \nresonant x -ray scattering techniques.  [3] Both techniques collect data as functions of the energy \nand the momentum transf erred from the spin system to the neutron or photon beam. The resulting \nfive-dimensional data sets serve as powerful probes of magnetic materials. Elastic scattering \nelucidates the magnetic configuration, direction of the spins, symmetry of the magnetic state, \nspatial distribution of the magnetization density, and dependence of the order parameter on \nthermodynamic fields such as temperature, pressure, magnetic and electric fields.  Inelastic \nscattering determines the energies of the fundamental excitations , which can be used to elucidate  \nthe nature and strength of the exchange interactions.   \nThe information provided by x -ray and neutron scattering is  largely complementary.  [4] For \ninstance, resonant elastic x -ray scattering can be used to measure magneti c order parameters in an \nelement -specific manner  by tuning the x -ray energy to an electronic transition , and t he enormous \nphoton flux at modern synchrotron facilities allows  measurement s of very small samples and films \nas thin  as a single unit cell . [5] [6] The magnetic dynamics of such samples can be explored with \nResonant Inelastic x-ray Scattering (RIXS),  [7] and pump- probe techniques open new avenues of \nresearch into non -equilibrium phenomena.  Elastic neutron scattering, on the other hand, provides \nquantitative information about the magnitude  of magnetic order parameters that is difficult to \nobtain with x-rays, and the energy resolution offered by inelastic magnetic neutron scattering is \norders -of-magnitude finer  than can be currently achieved in RIXS experiments .  Both techniques \ncan also measure the lattice structure and dynamics, as reviewed elsewhere in this volume , and \nthose cross section s can be uniquely distinguished from magnetic scattering by polarization \ntechniques. [8, 9]    \nIn this chapter, we describe how to employ neutron and x -ray scattering to explore the \nmagnetism of materials, paying particular attention to the complementarity of both techniques.   \n \nII. Magnetic Neutron Diffraction Technique  \n \nMagnetic neutron scattering originates from the neutron’s magnetic dipole moment . As a spin -\n½ particle , the neutron carries a magnetic dipole moment of - 1.913 nuclear magnetons that \ninteracts with the unpaired electrons in the sample, either through the dipole moment associated \nwith an electron’s spin or via the orbital motion of the elec tron.  The strength of this magnetic \ninteraction is comparable to the neutron- nuclear interaction . The  magnetic scattering cross -section  \nreveal s the magnetic structure and dynamics of materials over wide ranges of length scale and \nenergy.  Magnetic neutron scattering plays a central role in determining and understanding the \nmicroscopic properties of a vast variety of magnetic systems  – from the fundamental nature, \nsymmetry, and dynamics of magnetically ordered materials to elucidating the magnetic \ncharacteristics essential in technological applications.   4 \n  \nOne traditional role of magnetic neutron scattering has been the measurement of magnetic \nBragg intensities in the magnetically ordered regime.  Such measurements can be used to \ndetermine the spatial arrangement and directions of the atomic magnetic moments, the atomic \nmagnetization density of the individual atoms in the material, and the value of the ordered moments \nas a function of external  parameters such as temperature, pressure, and applied  magne tic or electric \nfields.  These types of measurements can be carried out on single crystals, powders, thin films, and \nartificially grown multilayers, and often the information collected can be obtained by no other \nexperimental technique.  For magnetic pheno mena that occur over length scales that are large \ncompared to atomic distances, the technique of magnetic Small Angle Neutron Scattering (SANS) \ncan be applied.  This is an ideal technique to explore domain structures, long wavelength \noscillatory magnetic s tates, vortex structures in superconductors, skyrmions, nanomagnets, and \nother spatial variations of the magnetization density on length scales from 1 to 1000 nm.  Another \nspecialized technique is neutron reflectometry, which can be used to investigate the  magnetization \nprofile in the near -surface regime of single crystals, as well as the magnetization density of thin \nfilms and multilayers.  This particular technique has enjoyed dramatic growth during the last \ndecade or so  due to the rapid advancement of at omic -layer  deposition capabilities.  \n \nThe cross -section for magnetic Bragg scattering can be written as  [10]  \n() ()22\n22\n)(2hkl M B hkl M FAmceC I g ggθγM\n\n\n=        (1) \n \nwhere IM is the integrated intensity for the magnetic Bragg reflection located at the reciprocal \nlattice vector g hkl, the neutron- electron coupling constant in parentheses is − 0.27× 10-12 cm, C is an \ninstrumental constant which includes the resolution of the measur ement, A(θB) is an angular factor \nwhich depends on the method of measurement (sample angular rotation, θ :2θ scan, etc.), and M g \nis the multiplicity of the reflection (for a powder sample).  The magnetic structure factor F M(g) is \ngiven in the general case by  [1] [11]   \n \n() ()ej jW\njN\njig\nhkl M e F−\n=⋅\n\n× × =∑ gg g grˆ ˆ\n1M (2) \n \nwhere gˆ is a unit vector in the direction of the reciprocal lattice vector g hkl, Mj(ghkl) is the vector \nform factor of the jth ion located at rj in the unit cell, Wj is the Debye -Waller factor that accounts \nfor the thermal vibrations of the jth ion, and the sum is over all (magnetic) atoms in the unit cell.  \nThe triple cross product originates from the vector nature of the dipole -dipole interaction of the \nneutron with the electron.  A quantitative calculation of M j(g) in the general case involves \nevaluating matrix elements of the form ±+±⋅O SeiRg2 , where S  is the electron  spin operator, O  \nis the symmetrized orbital operator introduced by [12] , and ± represents the angular momentum \nstate.  This can be quite a complicated angular -momentum computation involving all the electron 5 \n orbitals in the unit cell, but has the simple result that only the components of the magnetic moment \nthat are perpendicular to g hkl (or more generally the wave vector K ) contribute to the scattering.  \nOften the atomic spin density is collinear , by which we mean that at each point in the spatial extent \nof the electron’s probability distribution, the atomic magnetization density points in the same \ndirection .  In this case the direction of M j(g) does not depend on g, and the form factor is just a \nscalar function, f (g), which is simply related to the Fourier transform of the magnetization density.  \nThe free -ion form factors have bee n tabulated for essentially all the magnetic elements  [see, for \nexample, https://www.ill.eu/sites/ccsl/ffacts/ffachtml.html ].  Note that for x -ray scattering the \nform factor for charge scattering corresponds to the Fourier transform of the total charge density \nof all the electrons, while in the magnetic neutron case it is the transform of the “magnetic” \nelectrons  only, which are the electrons whose spins are unpaired.  Recalling t hat a Fourier \ntransform inverts the relative size of objects, the magnetic form factor typically decreases much \nmore rapidly with |ghkl| than for the case of x -ray charge scattering  since the unpaired electrons are \nusually the outermost ones of the ion.  T his dependence of the scattering intensity on f (g) is a \nconvenient way to distinguish magnetic cross -sections from nuclear cross -sections, where the \nequivalent of the form factor is just a constant since the nucleus ( ≈10-5 Å) looks like a point particle \nto a thermal/cold neutron ( see below the nuclear coherent scattering amplitude b in Eq.  (6)).  \nIf in addition to the magnetization density being collinear, the magnetic moments in the \nordered state point along a unique direct ion (i.e. the magnetic structure is a ferromagnet, or a \nsimple + -  + - type antiferromagnet), then the square of the magnetic structure factor simplifies to  \n \n \n() ()22^^21∑−⋅−=⋅\nji\nj M ee f FjW\njz\njj\ngrg\ng g µηη  (3) \n \nwhere η^ denotes the (common) direction of the ordered moments and ηj the sign of the moment \n(±1), µz\nj is the average value of the ordered moment in thermodynamic equilibrium at ( T, H, P, \n…), and the orientation factor 2^^\n1⋅−ηg  represents an average over all possible domains.  If \nthe magnetic moments are the same type, then this expression further simplifies to  \n \n() ()2\n22 2^^21 ∑−⋅⋅−=\njW i\njz\nM ee f Fj jrgg g g η µη     .      (4) \n \nWe see from these expressions that neutrons can be used to determin e several important quantities; \nthe location of magnetic atoms in the unit cell and the spatial distribution of their magnetic \nelectrons;  the dependence of < µz> on temperature, field, pressure, or other thermodynamic \nvariables, which is directly related to the order parameter for the phase transition (e.g. the sublattice 6 \n magnetization).  Often the preferred magnetic axis η^ can also be determ ined from the relative \nintensities.  Finally, the scattering can be put on an absolute scale by internal comparison with the \nnuclear Bragg intensities I N from the same sample, given by  \n \n() ()2)( g g FN B N A C I θτM=          (5) \nwith  \n()2\n2∑−⋅=\njW i\nj N eeb Fj jrgg     .          (6) \nHere b j is the coherent nuclear scattering amplitude for the jth atom in the unit cell, and the sum is \nover al l atoms in the unit cell.  Typically  the nuclear structure is known accurately and F N can be \ncalculated, whereby the saturated value of the magnetic moment in Bohr magnetons can be \nobtained. \n \n There are several ways that magnetic Bragg scattering can be distinguished from the \nnuclear scattering from the structure.  Above the magnetic order ing temperature all Bragg peaks \nare nuclear (structural) in origin, while as the temperature drops below the ordering temperature \nthe intensities of the magnetic Bragg peaks rapidly develop, and for unpolarized neutrons the \nnuclear and magnetic intensities  simply add.  If these new Bragg peaks occur at positions that are \ndistinct from the nuclear reflections, then it is straightforward to distinguish magnetic from nuclear \nscattering.  In the case of a ferromagnet, however, or for some antiferromagnets which contain two \nor more magnetic atoms in the chemical unit cell, these Bragg peaks can occur at the same position.  \nOne standard technique for identifying the magnetic Bragg scattering is to make one diffraction \nmeasurement in the paramagnetic state well abo ve the ordering temperature, and another in the \nordered state at the lowest temperature possible, and then subtract the two sets of data.  In the \nparamagnetic state the (free ion) diffuse magnetic scattering is given by [5,6]   \n \n()Kfp Ieff ParamceC2 22\n22\n2 32\n\n\n\n=γ        (7) \n \nwhere peff is the effective magnetic moment (= g[J(J+1)]1/2 for a free ion).  This is a magnetic \nincoherent cross -section, and the only angular dependence is through the magnetic form factor \nf(K).  Hence this scattering looks like “background”.  There is a sum rule on the magnetic scattering \nin the system, though, and in the ordered state  this diffuse scattering shifts into the coherent \nmagnetic Bragg peaks  and magnetic excitations .  A subtraction of the high temperature data (Eq. \n(7)) from the data obtained at low temperature (Eq. (1)) will then yield the magnetic  Bragg peaks, \non top of a deficit (negative) of scattering away from the Bragg peaks due to the disappearance of \nthe diffuse paramagnetic scattering in the ordered state.  On the other hand, all the nuclear cross -\nsections usually do not change significantl y with temperature  (apart from the Debye -Waller factor \ne-2W), and hence drop out in the subtraction.  A related subtraction technique is to apply a large 7 \n magnetic field in the paramagnetic state, to induce a net (ferromagnetic -like) moment.  The zero \nfield (nuclear) diffraction pattern can then be subtracted from the high -field pattern to obtain the \ninduced- moment diffraction pattern. \n \na. Polarized Neutron  Technique s \n \n When the neutron beam that impinges on a sample has a well -defined polarization state, \nthen the nuclear and magnetic scattering that originates from the sample interferes coherently, in \ncontrast to being separate cross -sections like Eq. (1) and Eq. (5) whe re magnetic and nuclear \nintensities just add.  Polarized neutron diffraction measurements with polarization analysis of the \nscattered neutrons can be used to establish unambiguously which peaks are magnetic, which are \nnuclear, and more generally to separat e the magnetic and nuclear scattering at Bragg positions \nwhere there are both nuclear and magnetic contributions. The standard polarization analysis \ntechnique is straightforward in principle [9]  [13].  Nuclear coherent Bragg scattering never causes \na reversal, or spin -flip, of the neutron spin direction upon scattering.  Thus the nuclear peaks will \nonly be observed in the non- spin- flip scattering geometry.  We denote this configuration as (+ +), \nwhere the incident spin of the neutron is ‘ up’ spin and remains in the up state after scattering.  Non -\nspin- flip scattering also occurs if the incident neutron is in the ‘ down’  state, and remains in the \ndown state after scatter ing (denoted ( − −)).  The magnetic cross -sections, on the other hand, depend \non the relative orientation of the neutron polarization P  and the reciprocal lattice vector g.  In the \nconfiguration where P ⊥g, typically half the magnetic Bragg scattering involves a reversal of the \nneutron spin (denoted by ( − +) or (+ − )), and half does not ; the details depend on the specific \nHamiltonian describing the magnetism.  Thus for an isotropic Heisenberg -type model the magnetic \ncontribution to the reflec tion consists of the spin- flip (− +) and non -spin- flip (+ +) intensities of  \nequal intensity.  For the case where P g, all the magnetic scattering is spin -flip.  Hence for a pure \nmagnetic Bragg reflection where (S x,Sy,Sz) are active, the spin- flip scattering should be twice as \nstrong as for the P ⊥g configuration. \n The arrangement of having P g or P⊥g provides an experimental simplification and hence \ndata that are straightforward to interpret.  More generally, however, P  and g can have any relative \nangle.  This more general technique of neutron polarimetry is more difficult to realize \nexperimentally an d can complicate the interpretation of  the data, but can provide additional details \nabout the magnetic structure that cannot be obtained otherwise . [14]    \nb. Polarized Neutron Reflectometry  \n \nIf neutrons are incident on a surface at (very small) grazing angles the scattering can be \ncast in the form of a neutron ‘optical potential’ , analogous to photons in optical fibers.  For most \nmaterials the waveleng th-dependent index of refraction for neutr ons (and x -rays) , n, is slightly less \nthen unity, so that at suffi ciently small angles of incidence the scattering can be described by the \none-dimensional Schrödinger equation and the neutrons undergo total external reflection —the \nbasis for neutron guides .  For a simple material with a net magnetization , interference between \nnuclear and magnetic scattering leads to the following expression for  n [10]  [15]  : 8 \n  2/1\n22 2\n21\n\n\n\n\n\n\n\n\n\n\n\n±−=±µγ\nπλ\nmcebN n        (8) \nwhere N is the number density of the material and < µ> is the average moment.  The magnetic form \nfactor is unity since we are scattering at very small angles.  Note that Nb  is the nuclear scattering \nlength density for the material, and the magnetic term is the magnetic scattering length density.  \nThe critical angle below which we have mirror reflection is given by  \n2/1\n22 22/1\n22 2\n2 2arcsin\n\n\n\n\n\n\n\n\n\n\n\n± ≅\n\n\n\n\n\n\n\n\n\n\n\n± = µγ\nπλµγ\nπλθmcebNmcebNC     (9) \nwhere ± denotes the two polarization states of the neutron.  Above the critical angle the neutrons  \npenetrate the surface, and Fourier transforming the scattering provides a quantitative measure of \nthe structural profile and magnetic  profile of the material .  For thin films and multilayers  the layers, \nsubstrate, and front and back surfaces produce interference effects that provide a standard and very \npowerful technique for determining the properties of a wide variety of magnetic materials. [16] \n[17] \n \nIII. Resonant M agnetic X -ray Diffraction  Technique  \n \nMagnetic x -ray scattering was first demonstrated off resonance, that is, with photons that were not \ntuned to any absorption edge of the material under study.  However, the non- resonant magnetic x -\nray scattering cross section is so small that this technique is not useful for magnetic structure \ndetermination.  Magnetic x -ray scattering has only risen to prominence when synchrotron radiation \nenabled experim ents with photons tuned to x -ray absorption edges, where the resonant cross \nsection can be enhanced by several orders of magnitude.  [5] [6] The enhancement is greatest when \nthe partially occupied valence shell is reached by an electric dipole- allowed transition, that is, at  \nthe L2,3-absorption edges of transition metals  with valence d- electrons , and at the M 4,5-absorption \nedges of lanthanides or  actinides  with valence f -electrons . Magnetic x -ray scattering is then \nactivated by the strong core- hole spin- orbit coupling in the intermediate state , prior to reemission \nof the photon. \n \nFrom an instrumental perspective , one can group magnetic x -ray scattering experiments into three \ncategories, depending on the photon energy E required to reach the respective absorption edges, \nnamely soft ( E < 1 keV), intermediate (1 ≤ E ≤ 5 keV) and hard ( E > 5 keV). Whereas soft x -ray \nexperiments use gratings to monochromate the synchrotron radiation, intermediate and hard x -ray \nexperiments are performed with single -crystal monochromators. Because of air absorption, soft \nand intermediate x -ray experiments are carried out under vacuum conditions. The soft and \nintermediate x -ray ranges comprise the L -edges of 3d (4d) metals and the M -edges of 4f - (5f-) \nelectron systems, respectively. Experiments at the dipole -active L -edges of 5d metals are carried \nout with hard x -rays, as are experiments at the K -absorption edges of d -electron systems and L -\nabsorption edges of  f-electron systems where the resonant en hancement of the magnetic cross \nsection is weaker.  9 \n  \nUnlike neutron scattering, resonant magnetic x -ray scattering experiments require photons with a \nspecific energy , so that only the direction and not the magnitude of the photon momentum  is \nadjustable . Momentum conservation yields kinematic constraints that are particularly severe for \nsoft x -ray experiments on the important class of 3d metal compounds , where simple \nantiferro magnetic Bragg reflections characteristic of a doubled crystallographic unit cell cannot be \nreached in many cases (Fig. 1 [7]). Magnetic order with larger periodicities (and correspondingly \nshorter reciprocal lattice vectors) can be studied by resonant x -ray diffraction, but dynamical \ndiffraction effects can be important (see the example below). For resonant x -ray diffraction with \nintermediate and hard x -rays (Fig. 1), these constraints do not apply.  \n \nIn contrast to magnetic neutron scattering  which is generally straightforward to interpret , a \ncomplete quantitative calculation of the magnetic x -ray scattering cross section requires numerical \nelectronic structure calculations that describe the many -body correlations in the intermediate state. \nIn many cases, however, one is interested in the magnetic moment orientation, which can be \nextracted from the dependence of the scattered intens ity on the photon polarization without \nreference to such calculations. In spherical symmetry, the scattering tensor can be expressed in the \nfollowing way:  [18]  \n ()()⋅−⋅⋅ +⋅× +⋅ =∗ ∗ ∗ ∗\no i j o j i j o i o i j M M E M E E EF εε εεσεεσεεσ31)( )( )( )()2( )1( )0( (10)  \nwhere Mj is the magnetization vector of the ion j, εi and εo are the polarization vectors of the \nincoming and outgoing photons, and σ(0), σ(1), and σ(2) are proportional to the x -ray absorption \n(XAS), x -ray magnetic circular dichroism (XMCD), and x -ray magnetic linear dichroism (XMLD) \ntensors , respectively.  Additional terms arise from the crystal field, but they tend to be small for \ncollinear spin structures , as long as M  points along a high- symmetry direction of the crystal lattice. \n[18] \nTo separate magnetic scattering from charge scattering  (first term in Eq. 10) , magnetic x -ray \nscattering experiments can be carried out in crossed linear polarization. With the caveats \nmentioned above, the  intensity of a magnetic Bragg reflection  of a collinear antiferromagnet  at the \nreciprocal lattice vector g  can then be written as  \n2\n)1()(∑ ⋅× =∗ ⋅\njj o i jrigM E e Ijεεσ         (11) \nwhere the summation r uns over the magnetic unit cell . To determine the spin structure of a given \nmaterial, one commonly uses the so- called “azimuthal scan” where the momentum transfer g is \nkept fixed, and the sample is rotated such that the orientation of  M varies relative to the photon \npolarization vectors. In this way, simple  spin structure s can be determined based on  a single Bragg \nreflection.   10 \n Even for simple spin structures, however, it is important to keep in mind that the spectral functions  \nσ(E) are tensors with properties that may be strongly influenced by the symmetry of the crystal \nlattice.  If the site symmetry is tetragonal, for instance, the XAS spectra for light polarized in the \nxy-plane and along the z -axis, σ(0)xy  and σ(0)z, are generally different – a phenomenon known as \n“natural linear dichroism”. σ(1) and σ(2) are also generally anisotropic.  \nThe deviations from spherical symmetry are particularly prominent in situations where orbital \norder is present. An elementary example is  the Cu2+ ion with electron configuration 3d9 (i.e., a \nsingle hole in the d- electron shell).  [18] Materials based on Cu2+ usually exhibit Jahn- Teller \ndistortions that lift the degeneracy between d- orbitals of x2-y2 and 3z2-r2 symmetry. The lobes of \nthese orbitals are extended in the xy- plane and along the z -axis, respectively . For instance, t he \ncuprate high-temperatur e superconductors  exhibit a tetragonal structure with hole s in the x2-y2 \norbital. In this case, the electric dipole selection rules prohibit excitation of a 2p core electron into \nthe valence shell with z -polarized light, so that σ(0)z = 0 whereas σ(0)xy ≠ 0. The selection rule  \ncompletely changes  the az imuthal scans,  as observed  in resonant elastic scattering experiments on \ncopper -oxide compounds . [19] This example illustrates the important influence of orbital order on \nazimuthal scans in magnetic x -ray scattering. Proper consideration of the crystal symmetry is \nespecially important for experiments performed with polarized incident light, but without \npolarization analysis of the  scattered beam, because magnetic and charge scattering may then both \ncontribute to the detected signal.  \nThe photon energy dependence of the scattering tensor σ(E) contains a lot of additional \ninformation , some of which can be extracted without extensive model calculations.  In particular, \nthe large enhancement of the scattering intensity  at the absorption edges of magnetic metal atoms \ngives rise to the element sensitivity of magnetic x -ray scattering, which is particularly useful for \nmultinary co mpounds and for magnetic multilayers with different magnetic species. In principle, \nresonant magnetic x -ray scattering is also sensitive to the valence state of metal ions, which can \nbe inferred from the maximum of σ (E). Resonant scattering experiments on mixed -valent \ncompounds have indeed been reported.  [20]  However, the analysis and quantitative interpretation \nof such experiments require careful consideration of the multiplets in the intermediate state.  \nIn the discussion so far, we have not considered the spin- orbit coupling in the valence shell, which \nis generally weak for 3d metal compounds. In 4f  and 5f electron systems, however, the spin- orbit \ncoupling is so strong that it dominates the interatomic exchange interactions, so that models of \nsuch compounds are ba sed on firmly locked spin and orbital angular momenta. In 4d and 5d \nelectron systems, on the other hand, the intra -atomic spin -orbit coupling turns out to be comparable \nto other important energy scales including the on -site Coulomb interactions and the inter -atomic \nexchange coupling. Comparative magnetic x -ray diffraction experiments at the L 2 and L 3 \nabsorption edges have recently proven to be a powerful probe of the s pin-orbit compos ition of the \nground state wave function in such materials.  [19]  \n \n 11 \n IV. Dynamics  \na. Inelastic Neutron Scattering Technique  \n \nNeutrons can also scatter inelastical ly, to reveal the magnetic fluctuation spectrum of a material \nover wide ranges of energy ( ≈10-8→1 eV) and over the entire Brillouin zone. Neutron scattering \nplays a truly unique role in that it is the only technique that can directly determine the complete \nmagnetic excitation spectrum, whether it is in the form of the dispersion relations for spin wave \nexcitations, wave -vector and energy dependence of critical fluctuations, crystal field excitations, \nmagnetic excitons, or moment/valence fluctuations.  In the present overview we will discuss some \nof these possibilities.  \n \nAs an example, consider identical spins S localized on a simple cubic lattice, with a coupling given \nby -JSi⋅Sj where J is the Heisenberg  exchange interaction between neighbors separated by the \ndistance a.  The collective excit ations are magnons [ref. Chap ter on Spin Waves] .  If we have J>0  \nso that the lowest energy configuration is where the spins are parallel (a ferromagnet), then the \nmagnon dispersion along the edge of the cube (the [100] direction) is given by  \n \nE(q)= 8 JS[sin2(qa/2)]   .    .    (12) \n \nAt each wave vector q  a neutron can either create a magnon at ( q, E) with a concomitant change \nof momentum and loss of energy  of the neutron, or conversely destroy a magnon with a gain in \nenergy.  The observed change in momentum and energy for the neutron can then be used to map \nthe magnon dispersion relation.  Neutron scattering is particularly well suited for such inelastic \nscattering studies since neutrons typically have energies that are comparable to the energies of \nexcitations in the solid, and theref ore the neutron energy changes are large and easily measured.   \nAdditional information about the nature of the excitations can be obtained by polarized inelastic \nneutron scattering techniques, which are finding increasing use.  The cross section for spin w ave \nscattering from a simple Heisenberg ferromagnet is given by [1]  [13]  [9] \n \n()( )()()GqKq\nGq,q − +\n\n\n= ∑\n\n\n\nΩ±\n   δδπ γσ\nEE nVS\nkkgfmce\ndd\n21212\n2')(2 223\n22\n22\n \n \n× \n\n\n\n••••+^ ^ ^2^ ^\n2 1 ηK KPηK     (13) \n \nwhere n q is the Bose thermal population factor and ^\nη is a unit vector in the direction of the spins.  \nGenerally s pin wave scattering is represented by the familiar raising and lowering operators S± = \nSx ± iSy, which cause a reversal of the neutron spin when the magnon is created or destroyed.  \nThese “spin -flip” cross -sections are denoted by (+ − ) and (− +).  If the neutron polarization P  is 12 \n parallel to the momentum transfer K , PK, then the spin angular momentum is conserved (as there \nis no orbital contribution in this case).  In this experimental geometry, Eq. (13) shows us that we \ncan only create a spin wave in the ( − +) configuration, which at the same time causes the total \nmagnetization of the sample to decrease by one unit (1 µ B for a spin -only system ).  Alternatively, \nwe can destroy a spin wave only in the (+ − ) configuration, while increasing the magnetization by \none unit.  This gives us a unique way to unambiguously identify the spin wave scattering, and \npolarized beam techniques in general can be used to distinguish magnetic from nuclear scattering \nin a manner similar to the case of Bragg scattering.  \nFinally, we note that the magnetic Bragg scattering is comparable in strength to the overall \nmagnetic inelastic scattering.  However, all the Bragg scattering is located at a single point in \nreciprocal space, while the inelastic scattering  is distributed throughout the three dimensional \nBrillouin zone.  Hence when actually making inelastic measurements to determine the dispersion \nof the excitations one can only observe a small portion of the dispersion surface at any one time, \nand thus the observed inelastic scattering  is typically two to three orders of magnitude less intense \nthan the Bragg peaks.  Consequently , these are much more time consuming measurements, and \nlarger samples are needed to offset the reduction in intensity.  Of course, a successful determination \nof the dispersion relations yields a complete determination of the fundamental magnetic \ninteractions in the solid.  \n \nb. Resonant Inelastic X -ray Scattering Techniqu e \nThe mechanism underlying magnetic resonant inelastic x -ray scattering (RIXS) is analogous to the \none for resonant elastic scattering discussed in Section III and depicted in Fig. 1. A photon tuned \nto a dipol e-allowed transition promotes a core electron into the pa rtially occupied valence s hell. In \nthe intermediate state, the core- hole spin- orbit coupling induces an electronic  spin- flip, so that  the \nre-emitted photon leaves a magnetically excited state behind. Single magnetic excitations are then \nobservable in crossed polarization, analogous  to elastic magnetic scattering (Eq. 10 ). [7] In this \nsense, the relationship between elastic and inelastic resonant x -ray scattering is analogous to the \none between elastic and inelastic neutron  scattering.  Another useful analogy is optical Raman \nscattering, where single magnetic excitations at q = 0 can be activated by the spin -orbit coupling \nin the intermediate state [21]  which is, however, usually much weaker than the core -hole spin-\norbit coupling in RIXS. A more com mon Raman scattering experiment  addresses  bi-magnon \nexcitations that do not involve an electronic spin- flip. Such experiments are also possible with \nRIXS in parallel polarization geometry. As in optical Raman scattering, however, they only \ndetermine the Brillouin -zone averaged spectrum of magnetic excitations. The unique advantage of \nsingle -magnon RIXS is that the full magnon dispersion can be determined even for single crystal s \nof micrometer dimensions , or for  atomically thin films and heterostructures.  \nFrom an instrumental perspective, RIXS experiments on magnetic excitations  are challenging \nbecause the energy of the photons required to induce the atomic dipole  transition ( E = 0.4- 1 keV \nfor 3d metal L -edges) l argely  exceeds the typical energy of magnons in solids. A breakthrough \nwas achieved in 2009, when the resolving power of soft x -ray RIXS instrumentation passed the \nthreshold of E/ΔE ≈ 10000. This enabled the first RIXS observation of  high-energy magnons in \nundoped layered cuprates , which exhibit an exc eptionally large bandwidth of ≈300 meV. [22] 13 \n Shortly thereafter, high -energy paramagnons were also observed by RIXS in superconducting \ncuprates [23] [24] and in iron -based high- temperature superconductors at the Fe L 2,3 edges.  [25]  \nKinematical constraints analogous to those in resonant elastic scattering restrict these experiments \nto a fraction of the Brillouin zone that does not include the magnetic ordering wave vectors  of the \nrespective parent compounds. The kinematical constraint s are even more severe for RIXS \nexperiments of bi -magnon excitations in metal oxides at the oxygen K -edge (1 s-2p, 0.5 eV).  [26]  \nParallel advances in RIXS instrumentation for hard x -rays allowed the observation of single \nmagnons in antiferromagnetically ordered iridium oxides with 5d electron systems . [27] The larger \nresonance energies of the 2p-5d transition, with  correspondingly larger photon wave vectors , allow \nthe detection of magnons over the entire Brillouin zone. Instrumentation for RIXS at the L -\nabsorption edges of 4d metals and M -edges of actinides at intermediate photon energies (2.5 ≤ E \n≤ 5 eV) ha s only recently been developed . [28]  \nIn contrast to inelastic neutron scattering, the theoretical description of RIXS is still under \ndevelopment, and several open questions are actively debated  in the literature. These include the \nseparation of spin excitations from orbital excitations in mul ti-orbital systems, and from charge \nexcitations in metallic systems. This challenge is particularly severe in the iron pnictides, which \nare metals with multiple Fermi surfaces originating from different Fe d- orbitals. A complete \nresolution of this problem will likely require a transition to full polarization analysis in RIXS, so \nthat the  different excitation channels  can be separated completely . The first experiments using \nRIXS polarimeters have already been reported.  [29] Another open issue is the influence of the \ncore- hole potential in the RIXS intermediate state of the valence electron system in metallic \nsystems , where the core- hole lifetime may be comparable to intrinsic time scale s of the valence \nelectrons .  \n \nV.  Magnetic Diffraction Examples with Neutrons  \n \nAs an example of magnetic powder diffraction, the scattering from a sample of Na 5/8MnO 2 is \nshown in Fig. 2  [30].  This material exhibits Mn3+ and Mn4+ charge stripes and vacancy ordering \nof the Na  subsystem , which results in a rather complicated low -temperature magnetic structure \nthat can be determined from this pattern.  Of course, Rietveld refinement s for the crystallographic \nstructure can be performed from the full patterns at both high and low  temperatures to determine \nthe full crystal structure;  lattice parameters, atomic positions in the unit cell, site occupancies, etc. , \nas well as the value of the ordered moment.  The inset shows the temperature  dependence of the \nmagnetic peak intensity, which we see from Eq. (4) is the square of the sublattice magnetization —\nthe order parameter of the magnetic phase transition .  Note that we can identify the magnetic \nscattering through its temperature dependence,  as mag netic Bragg peaks vanish above the Néel \ntemperature where long range magnetic order occurs.  Note also that  the magnetic intensities \nbecome weak at high scattering angles as f (g) falls off with increasing scattering angle.  14 \n A more elegant way to iden tify magnetic scattering is to employ the neutron polarization \ntechnique , particularly if the material has a crystallographic rearrangement or distortion associated \nwith the magnetic transition.  It is more involved and time -consuming experimentally, but yields \nan unambiguous identification and separation of magnetic and nuclear Bragg peaks.  Figure 3  \nshows the polarized beam results for two peaks of polycrystalline YBa 2Fe3O8. [31]  The top section \nof the figure shows the data for the P ⊥g configuration. The peak on the left has the identical \nintensity for both spin- flip and non- spin- flip scattering, and hence we conclude that this scattering \nis purely magnetic in origin.  The peak on the right has strong intensity for (+ +), while the inte nsity \nfor (- +) is smaller by the instrumental flipping ratio.  Hence this peak is a pure nuclear reflection.  \nThe center row shows the same peaks for the  P||g configuration, while the bottom row shows the \nsubtraction of the P ⊥g spin-flip scattering from the P||g spin- flip scattering.  In this subtraction \nprocedure instrumental background, as well as all nuclear scattering cross sections, cancel, \nisolating the magnetic scattering.  We see that there is magnetic intensity only for the low angle \nposition, whi le no intensity survives for the peak on the right, unambiguously establishing that the \none peak is purely magnetic and the other purely nuclear.  These data also demonstrate that all \nthree components of the angular momentum contribute to the magnetic scat tering. This simple \nexample demons trates how the technique works;  obviously it plays a more critical role in cases \nwhere it is not clear from other means what is the origin of the peaks, such as in regimes where \nthe magnetic and nuclear peaks overlap, or i n situations where the magnetic transition is \naccompanied by a structural distortion  where the structural peaks change significantly in intensity . \nWhen investigating the magnetic structures of new materials, it is generally best to first carry \nout powder diffraction experiments to establish the basic properties of the magnetic structure, \nassuming of course that the ordered moment is large enough to observe the magnetic Bragg peaks.  \nOnce the basics are established, on the other hand, measurements on a singl e crystal can provide \nmuch  higher quality and more detailed information about the magnetic properties.  Figure 4  shows \na map of the scattering intensity in the ( h,k,0) scattering plane at 22 K for a single crystal of the \nmultiferroic Co 3TeO 6, which orders antiferromagnetically at 26 K. [32]  The crystal structure is \nmonoclinic, and we see four satellite magnetic peaks around each (integer) structural peak, \nindicating that the initial magnetic structure is incommensurate in both the h and k (and l as well, \nit turns out [33] ) directions .  With further decrease of temperature a series of additional transitions \nare observed , detail that would be difficult to determine with a powder.  At lower temperature , \nseparate commensurate peaks develop, then there is a lock -in transition along k  that includes a \nferroelectric order parameter, and then finally a transition into the ground state with both \ncommensurate magnetic order and incommensurate order along h, k , and l . [33] [34]  \nThe magnetic superconductor ErNi 2B2C goes superconducting at T C = 11 K, and then develops \nincommensurate antiferro magnetic order below T M= 6 K as shown in Fig. 5 . [35]  The wave vector \nfor the ordering is (h,0,0) with h ≈ 0.55, with the spin direction transverse, along (0,y ,0).  Initially \nthe magnetic order exhibits a simple sinusoidal spin -density -wave (SDW) that is transversely \npolarized, as shown in the  bottom of the  figure.  As the amplitude of the SDW increases, third, \nfifth, and higher -order wave vector peaks develop as the wave squares  up.  This is the expected 15 \n behavior since for localized moments entropy mandates that a simple spin density wave cannot be \nthe ground state magnetic structure.  \nFor any SDW  structure, only odd- order peaks will have non- zero intensity due t o time-reversal \nsymmetry , because on average the net magnetization is zero.  Below 2.3 K we see that a new set \nof even-order peaks is found along the ( h,0,0) direction of ErNi 2B2C.  One possibility is that the \neven -order peaks are due to a structural distortion , a charge- density wave (CDW)  that follows the \nSDW  due to a magnetoelastic interaction.  Hence  the even -order peaks would be structural peaks \nand the odd- order peaks magnetic.  In t he present material, however, a net magnetization develops \nin the superconducting state  in the magnetic ground state , so that the even -order peaks could be \nstructural, magnetic, or both. To establish the nature of these peaks unambiguously  polarized \nneutron diffraction was used , as shown in Fig. 6.  The data are measured in the ( h,0,l) scattering \nplane, with k  then perpendicular to the scattering plane.  For P ||g the spins are perpendicular to the \nscattering plane and hence perpendicular to P and then the magnetic scattering is all spin -flip.  \nNote that the polarization dependence of the cross sections is quite different than the YBa 2Fe3O8 \nexample above, emphasizing that the spin- flip and non- spin- flip magnetic cross sections depend \non the details of the magnetic structure.  The structural scattering is always non -spin- flip.  The data \nshow that both odd- order and even -order  are purely magnetic  in this system .   \nFor antiferromagnets there is no net magnetization produced by the magnetic ordering.  When \nthe sublattice magnetizations are not compensated and there is a net magnetization, on the other \nhand, the superconductivity must respond to and try to screen this magnetization.  If the internally \ngenerated field is below HC1 then the supercurrents will exactly compensate the net magnetization \nand the total field will be zero.  If the field exceeds H C2 then the superconductivity will be \nextinguished as happens in ma terials such as ErRh 4B4 and HoMo 6S8. [36]  Between these two  \ncases , vortices are expected to be spontaneously generated, and this possibility can be i nvestigated \nwith SANS.  Figure 7  shows SANS data from  a single crystal of ErNi 2B2C. [37]  The inset presents \nthe image on the two -dimensional SANS detector, where K =0 is in the center.  We see the expected  \nhexagonal pattern of scattering from the vortex lattice.  Below the ferromagnetic transition \nadditional vortices spontaneously form due to the internally generated magnetic field, which adds \nto the applied field. To accommodate the additional vortices the y rearrange themselves with a \nsmaller lattice parameter for the vortex lattice, which is reflected by the peak of the vortex \nscattering moving to larger K . [38]  \nThe above examples demonstrate scattering from long range magnetic  order where the \nmagnetic diffraction consists of resolution -limited Bragg peaks .  But that is not always the case , \nand some of the best examples occur where competing magnetic interactions lead to frust ration \nand suppress the order or  prevent it completely.  Arguably the best example of  a frustrated lattice \noccurs in the cubic rare -earth  (R) pyrochlore (R2Ti2O7) systems  where the R  ions occupy corner -\nsharing tetrahedra. [39]  For R = Ho, Dy, for example, the single -ion anisotropy restricts the \nmoments to point along diagonal [111] directions, along lines that intersect the center of each \ntetrahedron.  The ground state turns out to be with two of the moments pointing into each \ntetrah edron and two pointing out.  But you don’t know which two are in and which two are out, \nexactly like the hydrogen bonding in ice where two H  move into the oxygen in the center of the 16 \n tetrahedron and bond and two move out , resulting in a macroscopic degener acy that violates the \nthird law of thermodynamics.  The first measurement of the ground state correlations was carried \nout for Ho 2Ti2O7, where the observed scattering  from the correlated moments  agreed quite well \nwith simulations.  [40]  An interesting simplification occurs for a field applied along the [111] \ndirection, which isolates the layers and forms  two-dimensional ‘kagom é spin- ice’.  The scattering \nfor this case is shown in Fig. 8 for Dy 2Ti2O7, which shows the broad distributions of diffuse \nmagnetic scattering that are in excellent agreement with Monte Carlo simulations.  [41]  \nThe ground state properties are not the only remarkable property of spin- ice, as the magnetic \nexcitations are equally fascinating.  Theory showed that these excitations, which simply consist of \nflipping one of the spins in a tetrahedron so that you have three pointing out and one pointing in \n(and in the adjacent tetrahedron three point in and one out), correspond to the creation of a \nmagnetic monopole and anti -monopole. [42]  The subsequent motion of these particles is  governed \nby the Coulomb Hamiltonian for magnetic charges, and this scenario was subsequently confirmed \nby neutron scattering measurements.  [43]  [41]  [44] \nAdvances in  thin film de position methods  have facilitated the synthesis of complex \nheterostructures with atomic layer accuracy, which has enabled investigators to control the \nmagnetic properties by tailoring the exchange interactions within and between layers.  These \ncapabilities combined with advance s in experimental reflectometry techniques  have made neutron \nscattering an essenti al tool to elucidate the atomic depth profile and magnetization density of thin \nfilms and multilayers.  An interesting example is the multilayer oxide heterostructure consisting \nof the (approximately cubic) antiferromagnets  LaMnO 3 and SrMnO 3, grown on a Sr TiO 3 substrate.  \nThe structural indices of refraction for these two materials are almost identical, rendering the \nstructural scattering practically invisible.   Occasionally an extra layer of LaMnO 3 was deposited \nto dope the interface, which produced an eff ective composition of La 0.44Sr0.56MnO 3, which is in \nthe ferromagnetic regime.  Figure 9  shows the non- spin- flip polarized neutron reflectivity data in \nthe two polarization states, R++ and R--, that are sensitive to the ferromagnetism.  The resulting \nmagnetic depth profile reveals that the magnetic modulation is quite large, varying from 0.7 µB to \n2.2 µB, and that its  period corresponds precisely to  the LMO superlattice structure.  [45]  High angle \ndiffraction data on the epitaxial multilayer confirmed the canted modulated spin structure of the \nsuperlattice.  \n \nVI. Magnetic Diffraction Examples with X -rays \n \nAs an example of resonant magnetic x -ray scattering, we first highlight experiments on the \nantiferromagnet Sr 2IrO 4 with hard x -rays tuned to the Ir L 2,3 edges  [46]. The crystal structure of \nSr2IrO 4 is composed of IrO 2 square lattices , closely similar to La 2CuO4, the parent compound of a \nprominent family of high -temperature superconductors.  Prior to the x -ray experiments, m agnetic \nsusceptibility measurements had suggested antiferromagnetic order with a Néel temperature of  \n240 K, but neutron diffraction experiments had proven difficult because of the large neutron \nabsorption cross section of Ir, and because large single crystals could not be grown. The hard x -17 \n ray data on a crystal of sub- millimeter dimensions show multiple magnetic Bragg reflections that \ncan be analyzed by refining the Bragg intensities according to Eq. (11) in a manner entirely \nanalogous to m agnetic neutron diffraction. The analysis revealed a canted  antiferromagnetic \nstructure in the IrO 2 planes , with alternating stacking in the direction perpendicular to the planes.  \n \nThe photon energy dependence of the resonant magnetic x -ray scattering cross section yields \nadditional information about the magnetic ground state of Sr 2IrO 4 that would be difficult to obtain \nwith neutron diffraction, even under ideal conditions. The Ir va lence electrons occupy 5d orbitals \nof xy, xz, and yz  symmetry. For materials with 3d valence electrons, the crystal field lifts the \ndegeneracy between these orbitals and quenches the orbital magnetization. In the 5d electron shell, \nhowever, the strong intra -atomic spin -orbit coupling can generate  complex admixtures of these \norbitals in the ground- state wave function, which correspond to a nonzero orbital magnetic  \nmoment. This, in turn, affects the matrix elements for the photon- induced transitions from the spin-\norbit split 2p  shell into the 5d shell such that the diffraction intensities at the L 2 and L 3 edges ( 2p1/2-\n5d and 2p3/2-5d, respectively) can become different. The strong disparity of the diffraction \nintensities observed experimentally (Fig. 10 ) [46] indicates that the orbital magnetization is largely \nunquenched, and t hat the spin and orbital components of the magnetic order parameter in Sr 2IrO 4 \nare of comparable magnitude. Similar observations have been made for other iridates. M odels of \nmagnetism in the iridates are therefore commonly expressed in terms of the total angular \nmomentum, J eff =S+L . For Sr 2IrO 4, Jeff = ½ in the ground state.  \n \nThe large resonant scattering cross section, combined with the hi gh photon flux at synchrotron \nbeamlines and the focusing capability of advance d x-ray instrumentation, allow magnetic x -ray \nscattering experiments with beam dimensions well below typical magnetic domain sizes. Figure \n11 provides an example of such an experiment  on the layered antiferromagnet La 0.96Sr2.04Mn 2O7, \nwhich comprises alternately stacked sheets of ferromagnetically aligned Mn spins  [47]. The (001) \nmagnetic Bragg reflection of this spin array can be reached with photons tuned to the Mn L 3-edge.  \nThe data shown in Fig. 11 were taken with a beam of 3 00 nm diameter. T hey reveal that the \ndiffracted intensity varies on a characteristi c length scale of several microns. A detailed analysis \nshows that the intensity variation results from domains with different spin directions, which \ndiffract photons with different scattering amplitude due to the photon polarization dependence of \nthe scatt ering cross section (Eq . (11)). In another study, domains with different helicities in a spiral \nmagnet were imaged by resonant diffraction with circularly polarized x -rays. The spatial resolution \nand imaging capabilities of magnetic x -ray scattering methods are expected to dev elop rapidl y \nwith the advent of coherent x -ray beams at fourth -generation synchrotron sources.  \n \nIn analogy  to neutron reflectometry, polarized magnetic x -ray reflectometry has recently \ndeveloped into a powerful, element -sensitive probe of complex oxide thin films, heterostructures , \nand superlattices . As an example, we discuss  resonant x-ray diffraction data on R NiO 3-based films \nand superlattices (where R denote s a lanthanide atom). R NiO 3 perovskites exhibit  a Mott metal-\ninsulator transition as a function of the radius of the R  cation, which modulates the Ni -O-Ni bond \nangle. Recent work has shown that the metal -insulator transition can also be controlled by epitaxial 18 \n strain and by spatial confinement of the c onduction electron system. Antiferromagnetism with \nordering vector g  = (¼, ¼, ¼) develops in the Mott -insulating phase. Fig. 12( top) shows azimuthal \nscans at the corresponding magnetic Bragg reflection taken with photons tuned to the Ni L 3-edge  \n[48]. The data analysis demonstrates that the magnetic order is non -collinear, with Ni spins \nforming a spiral propagati ng along the (111) direction of the perovskite unit cell. The polarization \nplane of the spiral can be contr olled by epitaxial strain.   \n \nFig. 12(bottom ) shows a contour map of the resonant scattering intensity from a  LaNiO 3-LaAlO 3 \nsuperlattice as a function of the azimuthal angle and the momentum transfer perpendicular to  the \nsuperlattice  plane.  [49] Strong modifications of the azimuthal- angle dependence of the intensity \noccur particularly under grazing- incidence or grazing -exit conditions, where t he incident or \nscattered beams are strongly refracted at the external and internal interfaces of the superlattice. \nThese data illustrate the possibly important influence of dynamical effects in resonant soft x -ray \ndiffraction from thin -film structures, which go beyond the kinematic approxi mation usually \nemployed in the analysis of such data.  \n \nVery recently, x -ray free -electron lasers have enabled time- resolved resonant magnetic diffraction \nexperiments capable of imaging the real -time dynamics of magnetic order  under non- equilibrium \nconditions . As an illustration of th is emerging capability, Fig.  13 shows the time evolution of the \ng = (¼, ¼, ¼) antiferromagnetic Bragg peak of a NdNiO 3 film following a THz pump pulse exciting \nan infrared -active phonon mode  of the LaAlO 3 substrate  [50].  As the phonon propagates from the \nsubstrate through the film , it obliterates the antiferromagnetic order in its wake on a picosecond \ntime scale. The mechanism underlying this “non -thermal  melting” phenomenon may involve \ntransient distortions of the NiO 6 octahedra, which weaken  the magnetic exchange interactions \nbetween Ni spins.  \n \nVII. Spin Dynamics with Neutrons  \nThere are many types of magnetic excitations and fluctuations that can be measured  with \nneutron scattering techniques, such as magnons, spinons, critical fluctuations, crystal field \nexcitations, magnetic excitons, and moment/valence fluctuations.  We start with classic magnons \nin an isotropic ferromagnet , where  the excitations are gapless and the dispersion relation  is given \nby Eq. (12).  Figure 14 (left) shows a measurement for La 0.67Ca0.33MnO 3, which is a colossal \nmagnetoresistive (CMR ) material.  [51]  The data reveal two magnon peaks  at a given  wave vector , \none in energy gain where the neutron destroys a magnon and gains energy, and one in energy loss  \nwhere a magnon is created .  This is a small q  (long wavelength) excitation, and in fact this sample \nis polycrystalline rather than single crystal, and the data were collected around the (0,0,0) \nreciprocal lattice position.  Such measurements are restricted in wave vector and energy, and are \nonly viable for isotropic ferromagnets; otherwise the excitation s fall outside the accessible \nexperimental window  dictated by momentum and energy conservation.  If there is a question of \nwhether these excitations are magnons or phonons, t he polarized  beam technique can be employed \nas shown in Fig. 14(right)  for the prototypical isotropic ferromagnet  amorphous Fe 86B14. [52].  \nThese data were taken with the neutron polarization P  parallel to the momentum transfer K  (PK).  \nIn this configuration magnons require the neutron spin direction to reverse (spin- flip), while 19 \n phonons can only be observed in the non- spin- flip configuration.  For magnons  we should be able \nto create a spin wave only in the ( − +) configuration  where t he incident neutron moment is \nantiparallel to the magnetization; the scattered neutron moment is then parallel to the \nmagnetization direction , and the magnetization is decreased by one unit by the creation of the \nmagnon.  On the energy gain side the proces s is reversed and we destroy a magnon  only in the (+ \n−) configuration.  This is prec isely what we see in the data; for the ( − +) configuration the spin \nwaves can only be observed for neutron energy loss scattering (E > 0), while for the (+ − ) \nconfiguration spin waves can only be observed in neutron energy gain (E < 0).  This behavior of \nthe scattering uniquely identifies  these excitations  as magnons . \nExpanding the sine in Eq. (12) we see that the small- q dispersion relation can be written as E sw \n= D(T)q2, where D  is the spin wave “stiffness” constant.  The general form of the spin wave \ndispersion relation is the same for all isotropic ferromagnets, a requirement of the (assumed) \nperfect rotational symmetry  of the magnetic system, while the numerical value of D  depends on \nthe details of the magnetic interactions and the nature of the magnetism.  The small- q dispersion \nrelation can be readily measured , as shown in Fig. 15(left)  for a single crystal of La 0.85Sr0.15MnO 3, \nand D(T) obtained. [53]  The effect of  temperature is to soften the average exchange interaction as \nthe magnetization decreases, and hence the magnons renormalize to lower energies with increasing \ntemperature as also  shown Fig. 15.  With single crystals the dispersion curves can be determine d \nin different directions and throughout the Brillouin zone , as shown in Fig. 15(right)  for a number \nof perovskite CMR systems. [54]  Such measurements enable to determine in detail all  exchange \ninteractions, rather than just the long wavelength (average) behavior.  Any gap(s) in the excitation \nspectrum can also be directly measured.  \nIn addition to the magnon energies, t he lifetimes  of the excitations  can also be determined by \nextracting the intrinsic widths of the excitations, both in the ground state for itinerant electron \nsystems, and as a function of temperature.   An example of the li newidths in the ground state are \nshown for La 0.85Sr0.15MnO 3 in Fig. 16. [53]  In the simplest localized -spin model negligible  \nintrinsic spin wave linewidths would be expected at low temperatures, while we see here that the \nobserved linewidths are substantial  at all measured wave vectors and highly anisotropic, indicating \nthat an itinerant electron type of model is a more appropr iate description for this system.   In \nparticular , the linewidths become very large at large wave vectors .  These substantial linewidths \nare easy to measure with conventional instrumentation.  I nsulating magnets, on the other hand, \ngenerally have much sm aller linewidths and require much higher instr umental resolution to \nmeasure.  Figure 16(right)  shows the measured linewidths for the prototype insulating \nantiferromagnet  Rb2MnF 4. [55]  Here the spin- echo triple -axis technique has been employed , \nwhich has extraordinarily good (µeV) resolution.  The theoretically calculated linewidths from \nspin- wave th eory are shown by the solid curves at a series of temperatures, and are in quantitative \nagreement with the data.   \nOne area where neutron scattering has played an essential role is elucidating the spin dynamics \nof the high temperature superconductors, first for the copper  oxide systems  [56] and more recently \nfor the iron -based superconductors. [57]  The magnetic excitations in these classes of materials \nextend to quite high energies —as high as ≈0.5 eV —making the measurements particularly \nchallenging since the magnetic form factor requires that the magnitude of K  must be kept small, \nnecessitating  quite high incident ene rgy neutrons.  These requirement s are well matched to the \ntime-of-flight capabilities of spallation neutron facilities where high energy neutrons are plentiful.  20 \n To illustrate the basic technique, consider the excitations from BaFe 2As2, which is one of the \nantiferromagnetic ‘parent’ materials of the iron -based superconductors.  The antiferromagnetic \nordering temperature T N = 138 K, which corresponds to a thermal energy of  just ≈12 meV  (1 meV \n 11.605 K) .  Yet  we see from Fig. 17  that the magnons extend up to 200 meV, an order -of-\nmagnitude  higher energies than the ordering temperature represents , indicating that the system has \na substantial component of low -dimensional character . [58]  The in-plane dispersion relations are \nalso quite anisotropi c, even though the orthorhombic distortion away from tetragonal symmetry \n(that accompanies the magnetic order) is small.  Another very interesting aspect of the magnetic \nexcitations is that they have quite large linewidths at high energies, indicating that the magnetic \nelectrons are itinerant in nature.  Indeed,  the iron d-bands where the magnetism originates cross \nthe Fermi energy —the definition of itineracy.   \nOur final neutron example concerns the spin dynamics of one -dimensional  (1D)  magnets , \nwhich  (together with  2D magnets) have played a special role  in developing a fundamental \nunderstanding of quantum magnetic systems .  This is because they are theoretically  more tractable \nand therefore enable a deeper comparison with experiment .  They also entail the emergence of new \ntypes of cooperative states and their associated excitations.  Arguably the most  interesting case is \nfor the spin one -half antiferromagnet chai n, where quantum effects are maximal, represented by \nmaterials such as KCuF 3 [59] and CuSO 4⋅5D 2O [60] which have enjoyed a long and int eresting \nhistory of  investigations.   The ground state turns out to be an entangled macroscopic singlet , but \nwhere the two -spin correlation function decay s only algebraically, rendering long  lengths of the \nchain to be correlated antiferromagnetically.  The fundamental excitations of such a 1D system are \nspinons  in the se (isolated)  spin chain s, which can be considered to a first approximation as moving \ndomain wall s.  Measurements of the dynamic structure factor for CuSO 4⋅5D 2O are shown in Fig. \n18. [60]  Spinons carry fractional spin, and hence these fractionalized excitations can only be \ncreated in pairs in the scattering process .  Thus  the lower energy part of the spectrum  corresponds \nto two -spinon excitations  and has t he appearance of a  simple  antifer romagnetic spin wave \ndispersion relation.  However, only 71 % of the spectral weight is contained in this two -spinon \ncomponent , with essentially  all the remainder bein g accounted for by the four -spinon contribution.  \nPrecise calculations of the dynamic str ucture factor for two -spinon and four -spinon scattering are \nalso show n in Fig. 18, which account for essentially the entire measured spectral weight , and are \nin excellent agreement with the measurements.  [60]    \n \nVIII.  Spin Dynamics with RIXS  \nThe set of materials investigated by high -resolution RIXS is thus far limited to magnets with \ncharacteristic exchange interactions of the order of 100 meV. A milestone was set by early \nexperiments on La2CuO 4, the antiferromagnetic, Mott- insulating e nd member of a family  of high-\ntemperature superconductors, which exhibit s an exceptionally large magnon bandwidth of ≈ 300 \nmeV. A RIXS spectrometer with  energy resolution of ΔE ≈ 100 meV proved to be  capable of \nseparating  these excitations from the elastic line over a substantial fraction  of the Brillouin zone  \n(Fig. 19) . [23, 22]  Comparison with prior inelastic neutron scattering data on the same materials \ndemonstrated that the  RIXS excitation features  indeed originate from single antiferromagnetic \nmagnons.  21 \n RIXS experiments have also revealed the persistence of high -energy paramagnon excitations in \nhighly doped, superconducting cuprates. Based on the polarization dependence of t he scattering \ncross section  at specific scattering geometries , they can be separated from charge excitations , as \nshown in Fig. 20 for YBa 2Cu3O6+x. [29]  The measurements are complementary to inelastic neutron \nscattering experiments, which have much higher energy resolution and can therefore access spin \nexcitations with energies from 1 -100 meV, comparable to the superconducting energy gap. The \nRIXS measurements, on the other hand, are more sensitive to high -energy excitations , which can \nalso be investigated with high energy neutrons from spallation sources. The photon energy \ndependence of the RIXS intensity yields additional insight into the nature of these excitations. \nWhereas the spin excitation energy is indep endent of photon energy, as expected for collective \nmodes, the spectral weight of the charge excitations shifts upon detuning the photon energy away \nfrom the L -edge resonance, signaling a broad excitation continuum. This supports models that treat \ncollecti ve spin excitations as mediators of unconventional superconductivity.   \nHard x -ray RIXS experiments on the layered iridates have revealed magnon dispersions \nremarkably similar to those of the cuprates  – a finding that has fueled predictions of \nunconventional superconductivity in the iridates. In addition to the usual low -energ y magnon \nbranches emanating from the antiferromagnetic Bragg reflections , these experiments have also \nrevealed weakly dispersive  “spin -orbit exciton” modes corresponding to spin excitations from the \nJeff = ½ ground state into the J eff = 3/2 excited state (Fig. 21). [27] Since the dispersion of these \nmodes is controlled by the combination of the intra -atomic spin -orbit coupling, the crystalline \nelectric field, and the inter -atomic exchange interactions, RIXS experiments are an incisive probe \nof the low -energy electronic structure of these materials.   \nFinally, to illustrate  the diversity of inelastic x-ray scattering methods applied to magnetism, we \nhighlight results of an  x-ray emission spectroscopy study of i ron arsenide superconductors  of \ncomposition Ca 1-xRxFe2As2 (where R = rare earth).  [61] The goal of this experiment was to \nelucidate the origin of a pressure -induced structural phase transition from an antiferromagnetic to \na nonmagnetic state that is associated with a large volume reduction.  [62] [63] To measure the \nlocal magnetic moment of the Fe  ions independent of any interatomic correlations , x-ray photons \nwere  tuned to the Fe K -absorption edge (1s -3d), and the spectrum of emitted x -rays was monitored \naround the  dipole -active  Kβ emission line ( 2p-1s). A local moment on the Fe site induces a splitting \nof this line (inset of Fig. 22) whose size depends on the moment amplitude. These experiments led \nto the discovery of a pressure induced spin- state transition  from a high- spin to a low -spin \nconfiguration of the Fe atoms. The lower volume of the low -spin Fe atoms  explains the vol ume \ncollapse in the nonmagnetic phase  at high pressures.  \n \n \nIX. Facilities  and Online Information  \n A list of current neutron scattering facilities around the world can be found at \n(http://e n.wikipedia.org/wiki/Neutron_research_facility ). Numerical values of the free- ion 22 \n magnetic form factors for neutrons can be obtained at \nhttps://www.ill.eu/sites/ccsl/ffacts/ffachtml.html .  Values of the coherent nuclear scattering \namplitudes and other nuclear cross -sections can be found at http://www.ncnr.nist.gov/resources/n-\nlengths/ . \n \n A list of current x -ray scattering facilities can be found at  \n(http://en.wikipedia.org/wiki/Lis t_of_synchrotron_radiation_facilities ).   \n \nValues for characteristic x -ray energies and a guide to the literature on x -ray form factors can be \nfound at http://xdb.lbl.gov/ . \n \nEnergy Units:  Traditionally  magnetic excitations are  quoted in units of meV but sometimes \nauthors use THz, particularly for phonons in older literature.  Raman and IR experimenters often \nuse cm-1.  1 meV   0.24180 THz   8.0655 cm-1  11.605 K. \n \nFor a wavelength λ  = 1.54 Å the photon energy is 8.05 keV, the electron energy is 63.4 eV, and \nfor a neutron the energy is 34.5 meV. \n \nX. Summary and Future Directions  \nIn this review we have discussed the basic characteristics of magnetic neutron and x -ray \nscattering and provided a number of experimental examples of how these techniques can be \nemployed.  Neutron scattering is a rather mature technique which has the advantage of being  a \nweakly interacting probe that does not affect the properties of the sample.  The source of neutrons \nhas traditionally been steady state reactor based facilities, but this has now been complemented by \nthe newer, pulsed spallation neutron source facilities  which can offer higher peak flux than steady -\nstate reactors .  Both types of sources have many different types of spectrometers that enable \nmagnetic investigations over many orders -of-magnitude in both spatial and time domains.  In \naddition to new sources and new types of sources, many of the advancements in ne utron technique s \nover the years ha ve com e from developments in how to tailor and manipulate neutrons, vast arrays \nof detectors, and the software to analyze and visualize the data , and this progress continues \nunabated .  New sources and new instrumentation currently are being planned and developed, with \nthe anticipation that measurement capabilities will be greatly increased together with an increased \nquantity and scale of data acquired.  \n Resonant x -ray scattering is a much newer technique, with high brightness that allows \nmeasurements of small bulk samples, thin films, and multilayers.  It also has the advantage of \nbeing element specific as the resonance is tuned to an absorption edge.  Tremendous progress in \nmeasurement capabilities has been realized in the last few years, both with magnetic diffr action \nand magnetic inelastic scattering.  In contrast to neutron scattering which is on a solid theoretical \nfoundation, the theoretical understanding and interpretation of magnetic x -ray scattering is \nundergoing considerable development, which should lead to improved interpretation of \nexperimental data and exciting new capabilities.  \nThe future of both techniques is brilliant, pun intended. 23 \n  \nAcknowledgments  \nWe thank our many colleagues who have collaborated with us on our own projects, where \nwe have taken a number  of the examples for our own convenience and familiarity with the \nsystems.   \n \nReferences  \n \n[1]  S. W. Lovesey, Theory of neutr on scattering from condensed matter, Oxford: Clarendon Press - \nOxford, 1984.  \n[2]  J. W. Lynn, \"Magnetic Neutron Scattering,\" J. Applied Physics, vol. 75, p. 6806, 1994.  \n[3]  M. Blume, \"Magnetic Scattering of X -rays,\" J. Appl. Phys., vol. 57, p. 3615, 1985.  \n[4]  E. Balcar and S. 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(bottom) Reciprocal lattice of \northorhombic perovskite antiferromagnets, with structural (black) and magnetic (red) Bragg \nreflections. Circles indicate the maximal coverage of RIXS with photons at the Cu (green) \nand Ru (blue) L 2,3-edges.  (top panel adapted from [7], © American Physical Society 2011).  30 \n  \nFigure 2 .  Magnetic diffraction pattern for Na 5/8MnO 2, obtained by subtracti ng the \ncrystallographic diffraction pattern obtained at 100 K, above the antiferromagnetic phase \ntransition, from the data at 2.5 K in the magnetic ground state.  The structural scattering cancels \nin the subtraction if there is no significant change when the  sample magnetically orders.  The \ninset shows the temperature dependence of the intensity of the strongest magnetic peak, and \nreveals a transition temperature of ≈60 K ( adapted from  [30] , ©Spinger Nature 2014).  \n31 \n  \nFigure 3.  Polarized neutron diffraction on polycrystalline YBa 2Fe3O8.  The top portion of the \nfigure is for P⊥g, where the open circles show the non- spin- flip scattering and the filled circles \nare in the spin -flip configuration.  The low angle peak has equal intensity for both cross sections, \nand thus is identified as a pure magnetic reflection, while the ratio of the (+ +) to ( - +) scattering \nfor the high angle peak is just the instrumental flipping ratio.  Hence this is a pure nuclear \nreflection.  The center portion of the figure is for P||g, and the bottom portion is the subtraction \nof the spin- flip data for the P ⊥g configuration from the spin- flip data for P||g.  Note that in the \nsubtraction procedure all background and nuclear cross sections cancel, thereby  isolating the \nmagnetic scattering.  (reprinted by  permission from  [31] , © American Physical Society 1992 ). \n32 \n  \nFigure 4 .  Neutron diffraction intensity map observed in the ( h, k, 0) scattering plane of a single \ncrystal of the multiferroic Co 3TeO 6.  The temperature is 22 K, just below the antiferromagnetic \nphase transition at T N=26 K.  The nuclear Bragg peaks at integer positions are accompanied by \nfour satellite magnetic reflections, indicating the development of incommensurate (ICM) \nmagnetic order.  Note that the ordering wave vector is incommensurate in both h and k .  No \nenerg y analyzer was used for these measurements so that the data are energy -integrated, and \nthere is clear diffuse scattering surrounding the ICM peaks at this temperature originating from  \ninelastic magnetic excitations ( adapted from  [32] , © American Physical Society 2012 ). \n \n33 \n  \n \n \nFigure 5 .  (top) Unpolarized neutron diffraction measurements along the ( h,0,0) direction at 1.3 \nK, 2.4 K, and 4.58 K  of a single crystal of ErNi 2B2C.  At 10 K no peaks are observed in this \nwave vector range.  The data have been offset along the intensity axis for clarity. Above the \nweak ferromagnetic transition at 2.3 K the fundamental incommensurate peak is observed at \nh=0.55, along with higher odd- order harmonics.  Below the ferromagnetic transition a new set of \neven -order harmonics develops, indicated by the arrows.  (bottom) Schematic of the i nitial \ntransversely polarized spin- density -wave, and ground state square -wave ( adapted from  [35] , © \nAmerican Physical Society 2001 ). \n \n34 \n  \nFig. 6.  Polarized neutron diffraction measurements on a single crystal of ErNi 2B2C showing \nboth the odd order (5th) and even- order (16th) harmonics for the P||g configuration.  The solid \ncircles ( - , +) and solid triangles (+ , -) are spin -flip scattering, while the open circles (+ , +) and \nopen triangles ( - , -) are non -spin- flip scattering.  The data demonstrate that both types of \nreflections are magnetic in or igin, with the moment direction along the b axis  (adapted from  [35] , \n© American Physical Society 2001 ). \n \n35 \n  \n \nFigure 7 .  Radially averaged small angle neutron scattering i ntensity of the vortex scattering in \nErNi 2B2C vs. wave vector K  at 85  mT, above and below the weak ferromagnetic transition.  The \nshift in the peak position demonstrates that additional vortices spontaneously form as the \nmacroscopic magnetization develops at low temperatures.  The temperature dependence shows \nthat this spontaneous vortex formation is directly related to the weak ferromagnetic transition.  \nThe inset shows vortex Bragg peaks on the two- dimensional SANS detector;  K  = 0 is in the \ncenter.  (adapted from  [37] ). \n  \n36 \n  \nFigure 8.  (A ) Neutron measurements of the diffuse magnetic scattering in the kagomé spin- ice \ncompound Dy 2Ti2O7 at T=0.43 K and B=0.5 T.  The sharp structural Bragg peaks, such as (2,-\n2,0), are contained within one pixel and have been removed from the plot.  (C ) Monte Carlo \nsimulations of the expected scattering in this  kagomé spin- ice state.   The overall features are in \nexcellent agreement with the data ( adapted from  [41] , © The Physical Society of Japan 2009 ). \n  \n37 \n  \nFigure 9 .  (a) Non -spin- flip polarized neutron reflectivity data R++ (red) and R- -(blue) on a \nLaMnO 3/SrMnO 3 multilayer, measured in a 675 mT  field at 120 K.  The inset shows a schematic \nof the superlattice.  (b) Magnetic depth profile  determined  by the fit to the data.  Location  of the \nLaMnO3 (pink)  and SrMnO3 (green)  layers  are shown.  (c)  Spin- flip intensity,  showing  the \nantiferromagnetic peak and satellite peak.   Inset  shows the non- spin- flip scattering in the same \nrange ( adapted from  [45] , © American Physical Society 2011 ). \n38 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 1 0.  Photon energy dependence of the (1, 0, 22) magnetic Bragg reflection at the  L3-\n(left) and L2-edges  (right) of Sr 2IrO 4. The black lines show the x -ray absorption spectra for \ncomparison ( reprinted with permission from  [46] , © American Associati on for the \nAdvancement of Science 2009). 39 \n  \n \n \n \n \n \n \n  \nFigure 11.  Map of the resonant elastic x -ray scattering intensity at the (0, 0, 1) magnetic \nBragg reflection of La 0.96Sr2.04Mn 2O7 at the Mn L 3-edge. The data indicate domains where \nthe Mn spins point in different directions in the MnO 2 layers ( reprinted with  permission \nfrom  [47] , © American Physical Society 2013).  40 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 12. (top left) Ni L 3-edge scans through the (¼, ¼, ¼) magnetic Bragg reflection of \nLaNiO 3-LaAlO 3 superlattices with different numbers of consecutive unit cells. The absence \nof the magnetic Bragg peak in superlattices with 3 or more LaNiO 3 layers indicates that the \nmagnetic order in the 2x2 superlattice is induced by spatial confinement of the conduction \nelectrons. ( reprinted with  permission from  [48]) (top right) Azimuthal angle dependence of \nthe (¼, ¼, ¼) magn etic Bragg peak of nickelate thin films and superlattices with simulations \nthat rule out collinear (CM) and favor non- collinear (NCM) magnetism. (bottom) Simulated \ncontour map of the scattering intensity of 2x2 LaNiO 3-LaAlO 3 superlattice as functions of \nazimuthal angle and momentum transfer perpendicular the to superlattice plane, \ndemonstrating the importance of dynamical diffraction effects ( reprinted with  permission \nfrom [49] , © American Physical Society 2016).  41 \n  \n  \nFigure 13.  (a) Schematic illustration of the demagnetization proce ss of a NdNiO 3 film \ntriggered by a coherently excited photon of the LaAlO 3 substrate. (b) Depth profile of the  \n(¼, ¼, ¼) resonant magnetic Bragg peak  intensity at different time delays between the \nphonon pump pulse and the resonant x -ray diffraction probe measurement. ( reprinted with  \npermission from  [50] , ©Springer Nature 2015).  42 \n  \nFigure 14 .  Spin waves in isotropic ferromagnets.  (left) Energy scan at a wave vector q  of 0.07 \nÅ-1 for La 0.7Ca0.3MnO 3, (published with permission from  [51] , © American Physical Society \n1996)  showing the spin waves in energy gain (E<0) and energy loss (E>0).  (Right) polarized \nbeam energy scan on the Fe86B14 amorphous ferromagnet at a fixed wave vector of 0.09 Å-1, with \nthe neutron polarization parallel to q .  In this configuration spin angular  momentum is conserved, \nand the neutron can only create an excitation (E>0) if its moment is initially antiparallel to the \nmagnetization, and can only destroy a spin wave (E<0) when its moment is parallel  (reprinted \nwith permission from  [52] , © American Institute of Physics 1996) . \n  \n43 \n  \n \n \n \nFigure 15 .  (a) Low energy spin wave dispersion relations at two different temperatures.  The \ndispersion relation follows a quadratic dependence expected for a ferromagnet, which defines the \nspin stiffness D , and no significant gap in the excitation spectrum is observed indicating an \nisotropic system.  D (T) is shown in (b), which follows a power law behavior as the Curie \ntemperature is approached  (reprinted with  permission from  [53] , © American Physical Society \n1998).  (right)  Spin wave dispersion relations for a series of colossal magnetoresistive \nperovskite oxides  (reprinted with  permission from  [54] , © American Physical Society 2006 ). \n  \n44 \n  \n \nFigure 16 .  (left) Intrinsic spin wave linewidths for the ground state magnetic excitations in \nLa0.85Sr0.15MnO 3.  The linewidths are quite anisotropic, and are significant at small wave vectors \nbut become very large at large q ( reprinted with  permission from  [53] , © American Physical \nSociety 1998).   (Right) magnon linewidths as a function of temperature for a series of q’s in the \ninsulating antiferromagnet Rb 2MnF 4, measured using the high resolution spin- echo triple -axis \ntechnique  (reprinted with  permission from  [55] , © American Physical Society 2006 ).  The solid \ncurves are calculations using spin wave theory.  \n  \n45 \n  \n \n \n \nFig. 17.  Left:  Constant -energy cuts of the magnetic excitations in BaFe 2As2 at a series of \nenergies.  The solid curves are the fits to the spin wave model.  (Right) Spin wave dispersion \nalong the (1, K ) direction as determined by energy and Q cuts of the raw data.  The solid line is a \nHeisenberg model calculation using anisotropic exchange couplings SJ 1a = 59.2 ± 2.0, SJ 1b = \n−9.2 ± 1.2, SJ 2 = 13.6 ± 1.0, SJc = 1.8 ± 0.3 meV determined by fitting the full cross section. \nThe dotted line is a Heisenberg model calculation assuming isotropic exchange coupling SJ 1a = \nSJ1b = 18.3 ± 1.4, SJ 2 = 28.7 ± 0.5, and SJc = 1.8 meV  (adapted from  [58] , © American \nPhysical Society 2011 ). \n \n46 \n  \n \nFig. 18.  Intensity color maps of the experimental inelastic neutron scattering spectrum measured \nalong the Cu chain  in CuSO 4⋅5D 2O are shown in the left, compared with the theoretical two - and \nfour-spinon dynamic structure factor  (reprinted with  permission from  [60] , © Springer Nature \n2013) . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n47 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 19.  (left) RIXS profile of La 2CuO 4 taken at the Cu L 3 edge with ≈100 meV energy \nresolution. The spectrum can be decomposed into elastic (A), single magnon (B), multiple \nmagnon (C) and optical phonon (D) components . The inset shows the x -ray absorption \nspectrum  near the Cu L 3 edge, the arr ow marks the energy of the incident photons. (right) \nSingle magnon dispersion determined by RIXS (blue dots), compared to inelastic  neutron \nscattering data ( dashed line)  (reprinted with  permission from  [23] , © American Physical \nSociety 2010).  48 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 20.  Photon energy dependence of the RIXS intensity ((a),(b) ) for undoped \nantiferromagnetic YBa 2Cu3O6.1 and ((c), (d) ) superconducting  Ca-substituted YBa 2Cu3O7 in \npolarization geometries that predominantly select spin ( a),(c) and charge (b),(d) excitations. \nThe horizontal dashed lines highlight the energy i ndependence of the magnetic pe ak \nposition, while the dashed green line is a guide to the e ye underlining the fluorescence \nbehavior of the continuum of charge excitations from the doped holes  (reprinted with  \npermission from  [29] , © American Physical Society 2015).  49 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 21.  (top)  The spin –orbital level scheme of Sr 2IrO 4. The spin -orbit coupling λ splits \nthe d -electron manifold into J eff=1/2 and 3/2 multiplets. The crystal field Δ lifts the \ndegeneracy of the J eff=3/2  multiplet. O range (blue) colors in the images of the orbitals \nrepre sent spin up (down) projections. (bottom) Dispersion of magnons and spin- orbit \nexcitons (marked with QP for “quasiparticle”) extracted from RIXS data at the Ir L -edge \n(reprinted with  permission from  [27] , © Springer Nature 2014).  50 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 22.  Fe K β emission spectra of Ca 1-xRxFe2As2 with R = Nd, and difference spectrum \nwith FeCrAs where Fe is in a nonmagnetic spin -0 state. The difference spectrum indicates a \nsplitting of the emission line due to a local magnetic moment on the Fe site (inset) \n(reprinted with  permission from  [61] , © American Physical Society 2013).  "    },    {        "title": "1910.03425v1.Magnetic_2D_materials_and_heterostructures.pdf",        "content": "1 \n Magnetic 2D materials and heterostructures  \nM. Gibertini 1,2 , M. Koperski 3 ,4 , A. F. Morpurgo 1 ,5 , K. S. Novoselov 3,4  \n1Department of Quantum Matter Physics, University of G eneva, CH-1211 Geneva, Switzerland \n2National Centre for Computational Design and Discove ry of Novel Materials (MARVEL), École \nPolytechnique Fédérale de Lausanne, CH-1015 Lausann e, Switzerland \n3School of Physics and Astronomy, University of Manch ester, Oxford Road, Manchester, M13 9PL, UK \n4National Graphene Institute, University of Manchest er, Oxford Road, Manchester, M13 9PL, UK \n5Group of Applied Physics, University of Geneva, CH-12 11 Geneva, Switzerland \n \nThe family of 2D materials grows day by day, drasti cally expanding the scope of possible \nphenomena to be explored in two dimensions, as well  as the possible van der Waals \nheterostructures that one can create. Such 2D mater ials currently cover a vast range of properties. \nUntil recently, this family has been missing one cr ucial member – 2D magnets. The situation has \nchanged over the last two years with the introducti on of a variety of atomically-thin magnetic \ncrystals. Here we will discuss the difference betwe en magnetic states in 2D materials and in bulk \ncrystals and present an overview of the 2D magnets that have been explored recently. We will \nfocus, in particular, on the case of the two most s tudied systems – semiconducting CrI 3 and \nmetallic Fe 3GeTe 2 – and illustrate the physical phenomena that have been observed. Special \nattention will be given to the range of novel van d er Waals heterostructures that became possible \nwith the appearance of 2D magnets, offering new per spectives in this rapidly expanding field. \n  2 \n Introduction \nThe family of two-dimensional (2D) crystals has bee n growing with an impressive speed over the last \nfew years, with many new systems have already been introduced 1-5, while many more have been \npredicted and await discovery 6-9. The range of properties covered by such materials  is now \nextremely large: metals, semimetals, topological in sulators, semiconductors and insulators – all are \npresent within this class of crystals 2,10 . The scope of correlation phenomena hosted in thes e systems \nis also rather broad, and includes superconductivit y, charge density waves, Mott insulators, etc . \nMore often than not, one atom thick crystals posses s properties that are strikingly different from \nthose of their three dimensional (3D) parent compou nds: graphene is a zero gap semiconductor 11 , \nwhereas graphite is a semimetal with a band overlap ; many monolayers of transition metal \ndichalcogenides (TMDC) in 2H phase are direct band gap semiconductors 12 , whereas in their bulk \nform (and even in their bilayers) the band gap is i ndirect. Combinations of such materials with \ndifferent properties have already led to the creati on of unique heterostructures, allowed \ninvestigation of new physical effects, as well as t he creation of novel devices 2,13,14 .  \nA conspicuously missing member of the family of 2D materials are magnetic atomically-thin \ncrystals 15 . Until recently 2D magnetism remained an elusive d ream and such crystals were not \navailable for experimental work. At the same time, the opportunities which would be opened by 2D \nmagnets are huge, ranging from the possibility of e xploring – or even exploiting – a plethora of \ndifferent 2D magnetic states to that of controlling  the magnetic properties, e.g. with an external \nelectric field. Furthermore, using such materials a s parts of van der Waals heterostructures is likely  \nto disclose new exciting directions. For instance, stacking different magnetic crystals together –or \nrestacking the same 2D crystals with different orie ntation– might result in a different magnetic order  \nor in new physical phenomena.  \nSomehow, the absence of magnetic 2D materials was s urprising, because layered, van-der-Waals-\nbonded, magnetic crystals have been available to us  for a long time 16,17 . It should be possible to \nexfoliate them down to one atomically thin unit and  potentially grow them on substrate as individual \nmonolayers. Indeed, some theoretical efforts had be en done to demonstrate that such crystals \nwould be stable and exhibit a finite critical tempe rature 18-23 Tc. \nIt was only recently, however, that a number of 2D magnets have been obtained experimentally 24 . \nThis breakthrough was immediately followed by inten se experimental and theoretical activities to \nstart exploring the field of 2D magnetism. The full  arsenal of techniques developed for van der Waals \nheterostructures was applied to the study of such c rystals and a number of specific effects have \nbeen observed: from renormalization of Tc to the control of magnetic ordering by gating. Her e we \ndiscuss which particular physical phenomena can man ifest themselves differently in 2D (as \ncompared to 3D) and summarize the current experimen tal situation in the field. \nFrom bulk to 2D magnetism \nTheoretical consideration  \nThe hallmark of magnetism is the existence of an or dered arrangement of magnetic moments over \nmacroscopic length scales, with a spontaneous break ing of time-reversal symmetry. This is typically \ndriven by the interaction between the neighbouring spins (exchange coupling) that tends to favour \nspecific relative orientations between them. At zer o temperature ( T), this local order can extend \nover macroscopic length scales. With increasing T, thermal fluctuations tend to misalign magnetic \nmoments in neighbouring regions, so that long-range  order is destroyed above Tc. Indeed, whether \nor not a system undergoes a phase transition at a f inite Tc depends on the effectiveness of thermal 3 \n fluctuations, which is governed by few general para meters irrespective of specific details of the \nsystem. In particular, dimensionality plays an esse ntial role in determining the impact of thermal  \nfluctuations on the critical behaviour of  many-body systems. \n \nIn a 3D system, a magnetic phase transition can alw ays occur at a finite temperature, while in the \none-dimensional case long-range order is possible  only at T = 0 25 . Being at the border between \nthese two extremes, the situation in 2D is more com plex. In this case, the existence of magnetic \nlong-range order at any finite temperature cruciall y depends on the number n of relevant spin \ncomponents, usually called spin dimensionality (Fig . 1), and determined by the physical parameters \nof the system (e.g., the presence and strength of m agnetic anisotropy, see below). \nFocusing for definiteness on finite-range exchange interactions, the Mermin-Wagner-Hohenberg \ntheorem 26,27  states that thermal fluctuations destroy long-rang e magnetic order in 2D systems at any \nfinite temperature when the spin dimensionality is three (isotropic Heisenberg model 28 ). This is due \nto the fact that the continuous symmetry of the iso tropic model leads to gapless long-wavelength \nexcitations (spin waves) that have a finite density  of states in 2D and can thus be easily excited at \nany finite temperature, with detrimental effects on  magnetic order. On the contrary, the exact \nsolution by Onsager 29  of the 2D Ising(-Lenz) model 30,31  shows that a phase transition to a \nmagnetically ordered phase occurs at Tc > 0 when n = 1. In this case, the anisotropy of the system, \nwhich favours a specific spin component, opens a ga p in the spin-wave spectrum thus suppressing \nthe effect of thermal fluctuations.  \nFor planar 2D magnets ( n=2), conveniently described by the so-called XY mod el, there is no \nconventional transition to long-range order, althou gh the susceptibility diverges below a finite \ntemperature. Berezinskii 32 , Kosterlitz and Thouless 33  pointed out that this divergence is associated \nwith the onset of topological order, characterized by an algebraic decay of spin correlations and by \nthe presence of bound pairs of vortex and antivorte x arrangements of spins. Thus, below the so- \nFigure 1 . Role of spin dimensionality and evolution of Tc. a, A spin dimensionality n=1 means that the \nsystem has a strong uniaxial anisotropy and the spi ns point in either one of the two possible orientat ions \n(“up” or “down”) along a given direction. The syste m behaves effectively as if it had only a single sp in \ncomponent along the easy axis and the underlying sp in Hamiltonian for localized spins is called Ising model. \nThe case n=2 corresponds to an easy-plane anisotropy that fav ours the spins to lie in a given plane, \nalthough the orientation within the plane is comple tely unconstrained. The spins can be thus considere d to \nhave effectively only two components (associated wi th the two in-plane directions), being successfully  \ndescribed within the so-called XY model.  Note, tha t in this case χ→∞  for T<TKT . Finally, for isotropic \nsystems n=3 and there is no constraint on the direction of t he spins. The underlying spin Hamiltonian in this \ncase is the isotropic Heisenberg model. b, TC (normalised to bulk Tc3D  for the particular material) as a \nfunction of the number of layers. Data for CrI 3 are adapted from 47 , for Fe 3GeTe 2 (red curve) – from 59 , for \nFe 3GeTe 2 (orange curve) – from 55 , for CrGeTe 3 – from 48 , for FePS 3 – from 42   \n4 \n called Kosterlitz-Thouless temperature TKT , quasi-long-range magnetic order is established an d the \nexistence of a finite order parameter is suppressed  only marginally with the system size. \nIn the absence of truly 2D magnetic crystals, such gamut of critical phenomena has been \nexperimentally validated over the last thirty years  in magnetic thin films grown on a substrate 34,35  or \nin 3D layered transition metal compounds 16,17 , which can be viewed as a stack of weakly-coupled 2D \nmagnetic layers. From these studies it clearly emer ges that the theoretical scenarios presented \nabove in terms of different spin dimensionalities c an be considered as idealized limiting cases of a \nmore realistic situation where spins have three com ponents, but magnetic interactions restrict them \nto point mainly either along a given direction or i n a specific plane.  \nA simple and yet successful strategy that allows in terpolating between the above scenarios and to \ndescribe more realistic situations relies on the us e of a generalized Heisenberg spin Hamiltonian: \n  \u0001 = −\u0004\n\u0005∑ \u0007 \t \n \u000b∙ \n\r+  Λ \u0010\u000b\u0011\u0010\r\u0011\u0012 〈\u000b,\r〉 −∑\u0016 \u0017\u0010\u000b\u0011\u0018\u0005\u000b    (1) \n  \nHere J is the exchange coupling between spins \n\u000b and \n\r on neighbouring sites (favouring either \nferromagnetic, J>0, or antiferromagnetic, J<0, order), while A and Λ are the so-called on-site and \ninter-site (or exchange) magnetic anisotropies, res pectively. This model captures the essential \nfeatures of most experimental systems 17 , and recovers in specific limits the idealized the oretical \nscenarios illustrated above. In particular, the iso tropic Heisenberg model ( n=3) corresponds to the \nabsence of magnetic anisotropy ( \u0016 ≈ 0  and Λ ≈ 0 ), while the Ising ( n=1) or XY ( n=2) model can be \nrecovered in the limit of strong easy-axis or easy- plane anisotropy (e.g. \u0016 → + − ⁄∞, respectively). \nThis model can effectively take into account some s ubtle effects, as in the case of dipole-dipole \ninteractions that can be partially described in ter ms of a renormalization of the on-site anisotropy A. \nStill, in some cases more sophisticated models are needed, which include either further neighbours \nor different kinds of magnetic coupling, such as Dz yaloshinsky–Moriya36,37 , Kitaev 38,39 , or higher order \n(e.g. bi-quadratic) interactions.  \nA wide range of physical phenomena are expected dep ending on the values of \t, Λ, and  \u0016 or on the \npresence of additional magnetic interactions. The k nowledge collected in the 1990s on magnetic \nlayered van-der-Waals-bonded compounds, combined wi th the recent development on the \nproduction of monolayers by exfoliation or differen t growth methods, is now disclosing new \nopportunities. First, the rich variety of magnetic 2D materials that can be isolated offers the \npossibility to study model spin Hamiltonians in a b road range of parameter regimes, including \nregimes that have been remained so far unexplored. In addition, the generalised Heisenberg \nHamiltonian (1) stems from a course-graining of the  electronic properties, obtained by integrating \nout all degrees of freedom but spin. A change in th e electronic structure, therefore, is expected to \nresult in a modification of the effective spin Hami ltonian describing the system. This suggests that \nthe substantial versatility of 2D materials and the ir sensitivity to external manipulations – gating, \nstrain, coupling to other 2D materials in van der W aals heterostructures– provide unprecedented \nopportunities to further expand or tune the range o f model parameters.  \nWhereas the approach based on the effective Hamilto nian in Eq. (1) is well suited to investigate the \nequilibrium configuration of 2D magnetic materials,  it is far less ideal to understand the interplay o f \nthe magnetic states with the opto-electronic proper ties of atomically thin crystals, i.e. one of the \ndomains in which magnetic 2D materials are anticipa ted to show new interesting physical \nphenomena and reveal unexpected results. The point is that the “effective” nature of the interaction \nhides the microscopic origin of the magnetic intera ction parameters. Magnetic exchange integrals of 5 \n the same magnitude could originate from direct exch ange between nearby spins or from indirect \nmechanisms, such as super- or double- exchange, tha t are typically mediated by intermediate states \ninvolving non-magnetic atoms. Whereas the differenc e may be immaterial to determine the \nmagnetic state in a system of given thickness, it i s crucial to understand the evolution of the \nmagnetic interactions upon reducing thickness. Dire ct exchange of two orbitals that are rather \nstrongly localized in real space is unlikely to dep end strongly on thickness. Conversely, indirect \nexchange mechanisms, being sensitive to the energy difference with the intermediate states and to \nthe corresponding hopping energy, can be drasticall y altered as a result of the change in band \nstructure that commonly happens in 2D materials whe n the thickness is reduced. Proper capturing \nof the nature of the coupling between the magnetic configuration and the single-electron \nwavefunctions is also clearly essential for our und erstanding of the influence of magnetism on such \nprocesses as tunnelling, inter-band radiative trans ition, and electron transport (and their evolution \nupon varying thickness). Early experiments (see bel ow) are indeed showing that these processes are \nof great interest in 2D magnetic semiconductors, bu t very little is currently understood theoretically . \nThere is plenty of room for theoretical work aiming  at establishing the key concepts needed to \nproperly capture the effect of the magnetic state o n the semiconducting properties of magnetic 2D \nmaterials. \nFirst efforts towards the isolation of magnetic 2D materials were carried out late in 2016 with the \nexfoliation of mono- and few-layers of NiPS 3 40 , FePS 3 41-43 , and CrSiTe 3 44 . These compounds display \nmagnetic order in bulk form 45,46 , and Raman measurements provide –albeit only indir ectly– evidence \nthat magnetism is preserved in thin crystals. The m ajor breakthrough occurred in 2017, with the first \nclear experimental confirmation of magnetism in ato mically-thin CrI 3 47  and CrGeTe 3 48  down to the \nmono- and bi-layer limit, respectively.  \nThese ground-breaking experiments have now sparked an intense activity on atomically-thin \nmagnetic crystals 49 . Apart from rare exceptions 50-52 , most efforts are focusing on the exfoliation of \nlayered compounds that are known to be magnetic in their bulk form. Nonetheless, from the \ndiscussion above it should be clear that the inheri tance of magnetism from 3D in the bulk to 2D in \nindividual monolayers is not guaranteed a priori, a nd crucially depends on the magnetic anisotropy \nof the system. For instance, materials with isotrop ic exchange interactions can be magnetic in their \n3D form, but thermal fluctuations strongly suppress  the critical temperature as we reduce their \nthickness, until magnetism is completely destroyed in the 2D limit. This is the case of CrGeTe 3, for \nwhich the critical temperature goes to zero as we a pproach the monolayer limit 48 . The application of \neven a very small external magnetic field B introduces an anisotropy in the system, opening up  a gap \nin the spin-wave spectrum and thus allowing to have  a non-zero magnetization at finite \ntemperature. For CrGeTe 3, the authors of Ref. 48  introduce an effective critical temperature that i n \nbilayers is already half the bulk Tc (61 K) even just at B = 0.065 T. On the contrary, when a material \nhas already an intrinsic anisotropy, stemming, for instance, from a strong spin-orbit coupling 53,54  \n(magneto-crystalline anisotropy), magnetism is supp osed to survive also in the monolayer limit (with \nthe dimensional crossover from 3D to 2D leading to a different critical temperature, Fig. 1b). This is  \nthe case of CrI 3 and Fe 3GeTe 2, which will be discussed in details below. In addi tion to the spin-orbit \ncoupling, other effects (e.g. magnetostatic dipole- dipole interactions) can contribute to a small \nmagnetic anisotropy, yet sufficient to stabilize lo ng-range magnetic order in 2D. 6 \n Given the crucially different properties of 3D and 2D systems, a fundamental question that arises \nwhen exfoliating magnetic layered compounds concern s the critical thickness below which a \nmaterial can be considered as 2D. Indeed, the inter -layer exchange coupling JL, although typically \nvery small, leads normally to a 3D critical behavio ur of several quantities in sufficiently thick \nsamples. One can then take the dimensional crossove r in the critical exponents from the values \nexpected in 3D to those in 2D (see Table 1) to mark  the transition from a 3D behaviour to a truly 2D \ncharacter 17,34 . This is for instance what was done 55  for Fe 3GeTe 2. Another fundamental quantity to \nmonitor is Tc, which we expect  to evolve from its bulk value Tc3D  to its 2D value Tc2D  as the number of \nlayers is reduced. Theory predicts 56,57  that the critical temperature for a N-layer structure, Tc(N), \nshould approach Tc3D  for large N as \n  T c(N)/ Tc3D = 1 - ( C/N)λ   (2) \nwhere C is a non-universal constant and λ = 1/ν is related  to the critical exponent ν of the bulk 3D \nsystem (and thus depends on the spin dimensionality ; see Table 1). For small N, instead, the critical \ntemperature should increase linearly with the numbe r of layers 58 . This prediction describes very well \nthe behaviour of Fe 3GeTe 2 where the critical temperature has been measured o ver a wide range of \nsample thicknesses 59,60 .  \nDespite having rather general validity, all conside rations here above do not exclude the possibility \nthat particular compounds may exhibit a different b ehaviour due to a variety of specific \nmechanisms. For example, the situation can be easil y imagined in which the evolution of the band \nstructure in a metallic layered material leads to a  change in the density of states (DoS) at the Fermi  \nlevel, which coincidentally exhibits a large peak w hen the material thickness is reduced to the atomic  \nlevel. Such a peak in the DoS may enhance the sensi tivity to interaction effects, and lead to magnetic  \norder in agreement with the Stoner criterion 61 . Indeed, there are cases in which a ferromagnetic \nstate has been reported in monolayers but not in th e bulk, as first experiments on VSe 2 appear to Mode l β \n\u001e\u0017\u001f<\u001f!\u0018\n∝|$|% γ \n&∝|$|'( ν \n)∝|$|'* δ \n\u001e\u0017\u001f!\u0018∝|+|\u0004/- Tc \n2D Ising  1/8  7/4  1 15  2\t[01ln \u00071+√2\u0012] ⁄  Ref 29  \n3D Ising  0.3265  1.237  0.630  4.789  \u001f7\u00058 \u00171+9\u0017\t:\t⁄\u0018;/<)  Ref 109  \n3D XY  0.348  1.318  0.672  4.787  \u001f=> +9|\t|/\u0017ln \u0017|\t\t:⁄|\u0018\u0018\u0005  Ref 110   \n3D Heisenberg  0.369  1.396  0.711  4.783  9|\t|/ln \u0017|\t\t:|⁄ \u0018 Ref 111,112  \nMean field  0.5  1 0.5  3 ? \t\u00171+?:?⁄\t:\t⁄\u001801⁄    Ref 113  \n \nTable 1. Critical behaviour from 2D to 3D.  Critical exponents governing the behaviour of magn etisation ( M), susceptibility \n(&), and correlation length ( )) as a function of the reduced temperature, $ = \u001f \u001f !⁄ − 1, or the magnetic field B, for different \nmodels in 2D and 3D. Not all the critical exponents  are independent, as they are related by scaling re lationships 114  such as \n@ = A \u0017B − 1\u0018  and 2A + @ = C D , where d is the space dimensionality. While the critical ex ponents for the 2D Ising model \nand mean-field theory can be determined analyticall y, for the other models they are results of numeric al calculations 115 . In \n2D, the critical exponents can be rigorously define d only for the Ising model, which is the only one d isplaying a standard \nphase transition to long-range order at finite Tc. Since long-range order in the 2D XY model is supp ressed only in the \nthermodynamic limit, finite-size samples can sustai n a non-vanishing magnetisation and its critical be haviour has been \nexperimentally observed (e.g. in Rb 2CrCl 4 116 ), with a corresponding critical exponent 117  A ≈ 0.23 . For each model an \nexpression for the critical temperature Tc is also provided ( C is a numerical coefficient and z and zL are the intra- and inter-\nlayer coordination numbers, respectively), assuming  in the 3D case a layered structure with | \t:\t⁄| ≪ 1  (see Box 1). In the \ncase of the 2D Ising model, the argument of the log arithm depends on the lattice type, and the reporte d expression refers \nto a square lattice.  7 \n indicate 50 . It seems a realistic possibility that the occurre nce of magnetism in this case is a \nconsequence of the specific monolayer band structur e having an enhanced DOS at the Fermi level 19 . \nIt will be nonetheless important to see whether thi s very interesting result will be reproduced \nexperimentally, and hopefully extended to more 2D f erromagnetic metals. \nCurrently a good dozen 2D magnetic crystals have be en studied, either obtained by exfoliating 3D \nparent layered magnets (see Box 1) or by growing th em directly on a substrate 50-52 . Due to the \nlimited space available, it is impossible to review  all of them here, and we will focus on the two mos t \nstudied systems - CrI 3 and Fe 3GeTe 2. Since CrI 3 is a semiconductor and Fe 3GeTe 2 is a metal – such a \nselection of materials allows us to overview the wh ole breadth of the techniques used. We will also \nfocus on the most common features of the magnetic m aterials, so that much of the information \npresented can be applied to other materials as well . \nBox 1: Examples  of magnetic configurations of vdW layered ma terials  \nThe strong geometrical anisotropy of vdW layered cr ystals –which leads to a significant difference in magnitude \nbetween intra-layer ( J) and inter-layer ( JL) exchange coupling– also manifests itself in diffe rent kinds of inter-\nlayer spin alignment found in different material sy stems. Here we illustrate different examples, along  with \nexperimentally-reported materials realisations. Sem iconducting and metallic compounds are distinguishe d \nwith different colours; the scheme does not illustr ate the magnetic anisotropy of the different materi als (for \ninstance in the bottom right quadrant, CrI 3 and CrCl 3 are both layered antiferromagnets with all spins \nferromagnetically order in each plane, but whereas in CrI 3 the layer magnetization points perpendicular to th e \nplane, in CrCl 3 it lies in the plane) \n Interlayer  \n Antiferromagnetic  Ferromagnetic   \nFerromagnetic J<0; JL>0  J>0; JL>0  Intralayer    \nCoPS 3 Ref 105  \nMnPS 3 Ref 106,107,119-121  \nNiPS 3 Ref 40,45,122   CrI 3 (bulk) Ref 5,65  \nCrBr 3 (bulk, few layers) Ref 67,94,123  \nCr 2Ge 2Te 6 Ref 48,124  Fe 3GeTe 2 Ref 55,125  \nVSe 2 Ref 10,19,50  Antiferromagnetic \n J<0; JL<0  J>0; JL<0  \n  \nFePS 3 Ref 42,126  \nMnPSe 3 Ref 127   CrI 3 (few layers) Ref 47  \nCrCl 3 (bulk) Ref 128   \nGreen – semiconductor materials; orange - metallic  \n8 \n  \nCrI 3 \nSemiconducting layered van der Waals materials that  undergo low-temperature magnetic transitions \noften exhibit different forms of antiferromagnetic order. Nevertheless, transitions to ferromagnetic \nstates have also been reported in many cases and, q uite naturally, early studies of 2D magnetism on \nexfoliated crystals have relied on compounds with r elatively large bulk critical temperature, such as \nCrI 35,23,47,62-64. \n \nExperimental investigations of bulk CrI 3 and other trihalides date back to sixties 65-71 , but only recently \nMcGuire and collaborators have reported a well-docu mented study of the temperature-dependent \nstructural properties and magnetic response of this  material 5. Magnetization and magnetic \nsusceptibility measurements show that bulk CrI 3 is a strongly anisotropic ferromagnet below the \nCurie temperature ( Tc=61K), with its easy axis pointing perpendicular to  the layers, and a saturation \nmagnetization consistent with a S=3/2 state of the Cr atoms. The behaviour observed is typical of a \nvery soft ferromagnet, with the formation of magnet ic domains causing the remnant magnetization \nto vanish and the absence of magnetic hysteresis. C lear evidence for a second magnetic phase  \nFig. 2. Magneto-optical Kerr effect for thin films of CrI 3. The evolution of magnetization with the \napplication of an external magnetic field in out-of -plane configuration may be traced by measuring the  Kerr \nrotation angle. Data for mono-, bi- and trilayer fi lms of CrI 3, illustrated in panels a, b and c, unravel the \nthickness-dependent behaviour of magnetic order in such structures. Samples with an odd number of \nlayers display a hysteresis loop with a coercive fi eld ∼0.13 T, indicative of a ferromagnetic state. The bi layer \nis characterised by ferromagnetic order within the individual layers, with antiferromagnetic coupling \nbetween the layers, giving rise at low field to a v anishing Kerr rotation angle. At a field  ∼±0.65 T a spin-flip \nprocess occurs in one of the layers, so that the ma gnetization of both layers points in the same direc tion. \nAdapted from 47 . \n9 \n transition at T ~ 50K is found in low-field magnetization measurem ents with B applied parallel to the \nlayers, but neither its nature nor the details of t he ensuing magnetic state are currently \nunderstood 72 . This is worth pointing out, because this second t ransition appears to be related to the \nunexpected behaviour observed in atomically thin cr ystals (see below). \nThe first low-temperature magneto-optical Kerr effe ct experiments (see Box 2) demonstrating the \npersistence of magnetism in atomically-thin CrI 3 crystals were reported in 2017 47 . The \nmeasurements show an unexpected systematics in the magnetic response of CrI 3 mono-, bi-, and \ntrilayers, Fig. 2. In mono- and trilayers, a hyster etic Kerr rotation switching at low B (~0.1 T) and \nsaturating at higher B is observed upon sweeping the magnetic field appli ed perpendicularly to the Box 2: The Kerr effect and its application to atomi cally -thin materials  \nThe magneto-optical Kerr effect (MOKE) 129  is an example of circular birefringence –i.e. a di fferent response in \nreflection or transmission of a dielectric medium d epending on the polarisation state of the incoming photon, \nwhich is induced by the presence of magnetic order in the reflecting material. In the case of a specim en \nexhibiting magnetization along the vertical z-axis (polar MOKE), an additional phase difference will arise \nbetween HI and H' circularly polarised photons travelling along the z-axis upon reflection from the surface. \nFor linearly polarised light, this translates into a rotation of the polarisation direction upon refle ction by an \nangle JK, which is known as the Kerr rotation (panel a). In  the simple case of a bulk specimen, one can consid er \nonly single reflection processes, and the Kerr rota tion angle is directly proportional to the magnetiz ation of the \nsample ( LK∝ M ). However, when adopting MOKE to assess the magnet isation of atomically thin films of \nlayered materials, several difficulties arise. Firs t, the ultimate thinness of 2D magnetic materials a nd their \nincorporation into van der Waals heterostructures r esults in a more complex dependence LK\u0017M\u0018, which, to a \nfirst approximation, can be interpreted in terms of  multiple reflections 47 . Moreover, such macroscopic \napproach does not consider the character of the ele ctronic structure of the magnetic material (e. g. e xistence of \nexcitons), which may alter the state of the reflect ed photons via additional thickness-dependent contr ibutions \nto the dielectric functions, leading to, e. g., a h ighly dispersive nature of MOKE signal (see panels b and c for \nKerr rotation measurements of the same CrI 3 monolayer for 633 nm and 780 nm lasers). Finally, and even more \nimportantly, the description that links the Kerr ro tation to the magnetization of the sample is based on \nMaxwell’s equations for macroscopic dielectric medi a, which is not necessarily always correct for atom ically-\nthin materials. Nonetheless, at present MOKE (toget her with other polarisation-based optical technique s \nsuitable for probing magnetisation, such as magneti c circular dichroism arising from unequal absorptio n \ncoefficients for photons characterised by opposite circular polarisation) has been very effective in r evealing the \nmagnetic properties of 2D materials. \nPanels (b) and (c) are adapted from 40,47 . \n \n10 \n CrI 3 layers. In bilayers, however, the Kerr angle vanis hes at low field, and only when sweeping the \nfield over a larger range ( B ~ 0.65 T) a sharp, non-hysteretic jump to a state with finite Kerr rotation \noccurs. \nConventional theory describing magneto-optics of ma croscopic media enables a coherent \ninterpretation of these observations. In the measur ement configuration, the magnetic field \ndependence of the Kerr angle indicates that in mono - and tri-layers a finite magnetization \nperpendicular to the layers is present at B=0, switching hysteretically under an applied field . Mono- \nand tri-layers are therefore strongly anisotropic f erromagnets with an easy axis perpendicular to the \nCrI 3 layers, as expected from the bulk. The same logic,  however, leads to the conclusion that the \nmagnetization of CrI 3 bilayers vanishes at B=0, and only appears in the presence of a sufficien tly \nlarge applied field. This behaviour suggests that t he constituent monolayers have equal and opposite \nmagnetization at B=0, so that the total magnetization vanishes. Only when a sufficiently large \nexternal field is applied, the magnetic moments in the two layers reorient and point in the same \ndirection, resulting in a finite total magnetizatio n. \nThe outlined scenario implies that the microscopic interlayer exchange coupling in CrI 3 is \nantiferromagnetic, since only then the low-field bi layer magnetization vanishes. Mono- and trilayers \n(and more in general all multilayers formed by an o dd number of layers) are ferromagnetic because \nthe magnetization on one of the layers remains unco mpensated. Such a possibility is internally \nconsistent and agrees with the data, but it is uncl ear how it can be reconciled with the ferromagnetic  \nstate of bulk crystals that seemingly requires the interlayer exchange coupling to be ferromagnetic. \nIndeed, ab-initio  calculations based on the actual low-temperature b ulk CrI 3 structure predict the \ninterlayer exchange coupling to be ferromagnetic 5,62 . It has nevertheless been noticed that the same \ncalculations performed using the high-temperature c rystal structure (i.e., the structure stable above \na structural phase transition 5 occurring at T ~ 200K) give an antiferromagnetic exchange interla yer \ncoupling 73-77 . This finding has led to the suggestion that the s tructure of thin CrI 3 multilayers may \ncorrespond to the high-temperature bulk CrI 3 structure and not to the low temperature one. So f ar, \nno experimental information is available to support  this idea, although clear evidence for a change in  \nmagnetic state associated with a structural transit ion has been recently reported 78  in few-layer CrI 3 . \nIt remains therefore an open question how (and on w hich length scale) the crossover from \nantiferromagnetic to ferromagnetic interlayer coupl ing occurs as the thickness CrI 3 is increased from \nmonolayer to bulk. \nFe 3GeTe 2 \nAnother popular 2D ferromagnet is Fe 3GeTe 2. Unlike CrI 3, Fe 3GeTe 2 is a metal with carrier \nconcentration that has been reported to vary from 1 .2×10 19  cm -3 in samples grown by molecular \nbeam epitaxy 51  (with a carrier mobility of ~50 cm 2/V·s) to ~10 21  cm -3 (approximately 8×10 13  cm -2 per \nlayer) in crystals grown using the chemical vapour transport method 79 . Alongside Kerr \nmeasurements, the anomalous Hall effect has been us ed to study magnetism 51 . Micromechanical \ncleavage of this material is difficult using conven tional methods and, in order to increase the \nchances to obtain monolayer flakes, exfoliation was  performed on freshly-evaporated gold 55  or on \nAl 2O359 . Similarly to CrI 3, the strong spin-orbit coupling results in a magne to-crystalline anisotropy \nalong the c-axis (perpendicular to the planes), wit h a ferromagnetic arrangement of spins within \neach layer. Although most reports suggest ferromagn etic arrangements between the layers 59 , \nmagnetic force microscopy indicates more complex ar rangements with an antiferromagnetic phase \ndominating or coexisting with the ferromagnetic one80 . 11 \n When studying the magnetization as a function of th ickness, two transitions have been observed. \nOne transition can be seen through the shape of the  hysteresis loops. For thin samples (the \ndefinition of “thin” depends on the experiment: it corresponds to less than 100 nm in Ref 60  and less \nthan 15 nm in Ref 55 ) no domain structure is reported, leading to sharp  and perfectly rectangular \nhysteresis loops, whereas for thicker samples labyr inthine domains appear above a critical \ntemperature and result in a more complex structure of the hysteresis loop. \nAnother transition can be seen in very thin samples  through the deviation of the dependence of Tc \non the number of layers from the expected scaling b ehaviour in equation (1). For samples above ~5 \nlayers, the thickness dependence follows equation ( 1), with the constant C corresponding to \napproximately 2-3 layers 59,60  and λ of the order of 1.7 – close to what is expec ted for 3D Heisenberg \nferromagnetism 59,60 . Deviations from this behaviour occur for thinner samples, which has been taken \nas an indication for a transition from 3D to 2D beh aviour59 . More evidence for such 3D to 2D \ntransition comes from the careful study of the crit ical behaviour of the magnetization as a function \nof temperature. For samples above 5nm in thickness the magnetization dependence on temperature \nis well described by the power law M(T)= M(0)(1- T/Tc)β with β in the range of 0.25-0.27. For thinner \nsamples the critical exponent gradually decreases, reaching 0.14 for a monolayer 55 , close to the \nvalue expected for the 2D Ising model (see Table 1) . \nMagnetic circular dichroism 55  and anomalous Hall effect 59  measurements have shown that \nmagnetism survives down to the monolayer limit, wit h Tc dropping down to 130K and 20K, \nrespectively. At this moment of time it is difficul t to establish the origin of this discrepancy, beca use \nmonolayers of this material are difficult to produc e, and thus rather unconventional fabrication \nmethods are being used 59  which might lead to changes of the properties of t he resulting devices \n(either due to introduction of defects or due to th e unintentional doping). For instance, slight \nchanges in the iron content are known to affect sig nificantly the critical temperature of the bulk 81 . \nAlso, the monolayers studied in Ref. 55  were in direct contact with a gold substrate, wher eas those \nstudied in Ref. 59 were not, so that substantial charge transfer coul d have taken place in the first case \nand not in the second. As we know that Tc is very sensitive to the electron density - this d ifference \nmay have had a large influence. \nGating of atomically thin magnetic crystals \nElectrostatic control of magnetism in gated structu res is of great interest for fundamental reasons \nand for its potential for future device application s 82 . In thin films of conducting materials, gate-\ncontrolled magnetism has been demonstrated over the  last two decades, first in magnetically doped \nsemiconductors and subsequently in conventional fer romagnetic metals 83-86 . Gating typically causes \nchanges in the critical temperature and in the coer cive field, which are driven by the accumulated \ncharge carriers 59 . The possibility to influence electrostatically th e magnetic properties of so-called \nmultiferroics, i.e. systems in which the magnetizat ion and the electrical polarization are intrinsical ly \ncoupled, is also well-established 87,88 . In this case, changes in magnetization are driven  by the applied \nelectric field and not by the carrier density, a ph enomenon normally referred to as magnetoelectric \neffect.  \nEstablishing the possibility to gate-tune the magne tic properties of atomically thin crystals has been  \na prime goal of experiments on virtually all magnet ic 2D materials investigated so far. For CrI 3 and \nFe 3GeTe 2, different experiments have been performed to show  the influence of gating on \nmagnetism. For semiconducting CrI 3, work has relied on MOKE measurements in singly an d double \ngated devices, whereas for metallic Fe 3GeTe 2 the anomalous Hall resistance has been measured. 12 \n  \nWork on Fe 3GeTe 2 done at Fudan University illustrates the possibili ty to control the Curie \ntemperature of atomically thin layers over an unexp ectedly large range 59 . This was shown by \nmeasuring the anomalous Hall resistance of trilayer  Fe 3GeTe 2 devices gated with a lithium-based \nelectrolyte (note, however, that there is a possibi lity that apart of pure electrostatics, there are a lso \neffects due to intercalation).  At zero gate voltag e, the trilayer Tc determined from the \ndisappearance of the hysteresis in the Rxy (B) curve was found to be Tc ~ 100 K, much lower than the \nbulk value. Upon applying a positive gate voltage, however, Tc was found to vary in a non-monotonic \nway, reaching values in excess of 300 K. Interestin gly, the evolution of the coercive field Hc extracted \nfrom fixed-temperature gate-dependent measurements parallels that of the critical temperature, \nwith Hc and Tc exhibiting maxima and minima at the same gate volt age values, Fig. 3a,b. The \nphenomenon was attributed to the shift in the Fermi  energy EF of Fe 3GeTe 2 –with a concomitant \nchange in DoS at EF- and used to interpret the data in terms of Stoner ’s criterion for itinerant \nferromagnetism 61 . Whereas the experimental observations are certain ly tantalizing and the \nproposed interpretation plausible, a more comprehen sive characterization of the experimental \nsystem as a function of gate voltage will be needed  to establish definite conclusions. For instance, i t \nwill be essential to measure the gate-induced charg e density, to understand whether or not lithium  \nFig. 3. Doping and gate control of layered magnets.  a, b Measurements of the anomalous Hall effect in \ntrilayer Fe 3GeTe 2 demonstrate the possibility to tune the Curie temp erature with ionic gating. a, The \nresulting phase diagram shows that the transition f rom the ferromagnetic to the paramagnetic state \nstrongly depends on the gate voltage, and that the Curie temperature may be increased as high as up to  \nroom temperature. b, The coercive field characterising Rxy  hysteresis loops at low temperatures also \ndisplays a strong dependence on the ionic gate volt age, which correlates with the dependence of Tc. c, d \nDouble gated structures allow for separate investig ation of the doping-related and electric-field-rela ted \nchanges in the magnetic state of bilayer CrI 3. c, The magnetization evolution, traced by measuremen ts of \nmagnetic circular dichroism (MCD), is presented for  different doping levels. Electron/hole doping \nlowers/increases the value of magnetic field requir ed to break the interlayer antiferromagnetic coupli ng. \nd, The three panels present the impact of an electri c field on the magnetic state of CrI 3 bilayer. The major \neffect arising from the presence of the electric fi eld is the opening of a hysteresis loop centred aro und \nzero magnetic field, indicative of a non-zero magne tization in the antiferromagnetic phase. Panels a and b \nare reproduced from 59 , panel c - from 90 , and panel d - from 118 . \n13 \n intercalates, and to consider other effects that li thium may have besides causing charge transfer \n(e.g., on the Fe 3GeTe 2 crystal structure).  \nGate-dependent magnetism in CrI 3 has been investigated by means of magneto-optics ( i.e., gate \ndependent Kerr effect measurements) in structures w ith conventional solid-state gates. The most \ninteresting results have been obtained on double-ga ted bilayer CrI 3 devices, i.e., structures in which \nan hBN-encapsulated CrI 3 bilayer connected to graphene electrodes is sandwi ched between two \nconducting materials acting as gates. Different inf ormation is obtained when biasing the two gates \nwith the same or opposite polarity. If the voltage polarity is opposite, the CrI 3 bilayer can be kept at \nzero total accumulated charge (the charge accumulat ed by one gate is removed by the other) while \nimposing a perpendicular electric field of varying strength, proportional to the difference of applied  \nvoltages. Alternatively, if voltages of the same po larity are applied, the perpendicular electric fiel d \ncan be made to vanish, so that the only effect of g ating is to accumulate charge. Double-gated \ndevices, therefore, allow the influence of an appli ed perpendicular electric field and of accumulated \ncharge carriers to be discriminated experimentally.  \nThe application of a perpendicular electric field a nd the accumulation of charge have strikingly \ndifferent effects, which can again be monitored by Kerr effect measurements. Doping of CrI 3 crystals \n(through biasing the two gates with voltages of the  same polarity) causes a progressive reduction in \nthe value of magnetic field (Fig. 3c) that is neede d to switch the CrI 3 from the antiferromagnetic \nstate stable at low field, into a ferromagnetic sta te stable at high field (i.e., the state in which t he \nmagnetization of the two CrI 3 monolayers have the same orientation). The same ph enomenon has \nbeen reported by two independent teams of researche rs, albeit the magnitude of the observed \nreduction in switching field differ significantly i n the two cases: one team reported 89  the switching \nfield to decrease by about 30%, whereas the other f ound the switching field can be made to vanish \naltogether 90 . This is a quite remarkable finding, as it implies  that charge accumulation actually turns \nbilayer CrI 3 from an antiferromagnetic into a ferromagnetic sta te at B=0. \nThe application of a perpendicular electric field r esults in a distinctly different modification of th e \nmagnetic properties. As mentioned earlier, in an ex tended interval of applied magnetic field around \nB=0 (up to values B∼ ± 0.5-0.7 T, depending on the device) the total ma gnetization of bilayer CrI 3 \nvanishes, due to the antiparallel orientation of th e spin in the adjacent layers, and thus, no Kerr \nrotation is observed. When applying a perpendicular  electric field ( Ez), however, a finite and \nhysteretic Kerr rotation appears in this same low m agnetic field range, with the Kerr angle that \nincreases linearly with Ez, Fig. 3d. If interpreted in terms of conventional magneto-optics theory, this \nobservation implies that the applied electric field  induces a finite magnetization Mz∝Ez, and thus the \npresence of a strong magneto-electric effect in bil ayer CrI 3.  \nThe overall consistency of the qualitative aspects of the results reported in different experiments \ngives confidence in the robustness of the observed phenomena. Much remains to be understood, \nhowever, about the precise conditions of the experi ments and the underlying microscopic physics. \nFor instance, it is currently unclear how large is the accumulated charge density when the two gates \nare biased with the same voltage polarity and – sin ce the Fermi level is inside the band-gap of CrI 3– \nwhich electronic states are occupied by the accumul ated charge. Some caution is also desirable \nwhen interpreting the effect of the applied electri c field, because theory predicts 91  that when the \ncombination of inversion and time-reversal symmetry  is broken (which is the case in CrI 3 bilayers in \nthe presence of a perpendicular electric field) Ker r rotation can occur in an antiferromagnet with \nvanishing magnetization. Hence, the observation of finite Kerr rotation does not always imply that a 14 \n finite magnetization is present and a direct measur ement of the electric field induced magnetization \nwould be desirable.  \nHeterostructures \nThe use of magnetic 2D materials as part of van der  Waals heterostructures pursues two targets. \nOne is expanding the functionality of the heterostr uctures available for experiments and for future \napplications. The other is to use phenomena origina ting from proximity to other 2D crystals as a tool \nto study the magnetic properties of the magnetic la yers themselves. \nMagnetic tunnel junctions \nDue to the absence of broken covalent bonds, the su rfaces of layered van der Waals bonded \nmaterials normally possess high-quality electronic properties.  This enables one to realise magnetic \ntunnel junctions by sandwiching thin films of insul ating or semiconductor materials (like hBN or \nTMDC) between two layers of metallic ferromagnetic crystals, as has been done recently with \nFe 3GeTe 292 . A pronounced spin-valve effect has been observed in Fe 3GeTe 2/hBN/Fe 3GeTe 2 junctions \ndue to different coercive field of the top and bott om electrodes (because of different \ndemagnetisation factors for the two electrodes with  different shape), Fig. 4a. The observed 160% \nmagnetoresistance corresponds to a 66% spin polariz ation at the Fermi energy, i.e., to the presence \nof 83% and 17 % of majority and minority spins, res pectively. These experiments also confirmed that \nthe magnetic properties of the surface of Fe 3GeTe 2 crystals are very similar to those of the bulk.  \nA less conventional type of magnetic tunnel junctio ns can be realized by using exfoliated insulating \nmagnetic 2D materials as tunnel barriers, in combin ation with non-magnetic electrodes (typically \ngraphene multilayers). Two basic mechanisms leading  to magnetoresistance in such structures have \nbeen identified. One is magnon-assisted tunnelling:  at sufficiently high-bias (typically 10 mV or \nhigher) additional inelastic tunnelling channels op en up, in which tunnelling electrons emit another \nquasiparticle, thus increasing the differential con duction, Fig. 4b. In the case of non-magnetic \nbarriers (and for magnetic barriers above Tc) such effects are dominated by phonon emission 93 . \nHowever, for T<Tc electrons can also emit magnons, identifiable by t he shift of their energy upon \napplying magnetic field 94,95 . \nMuch larger effects can originate from switching of  the magnetic state in the barrier, as observed in \nCrI 3. Due to the antiferromagnetic coupling of the magn etization in the adjacent layers, CrI 3 tunnel \nbarriers exhibit negative magnetoresistance in exce ss of 10000% 72,73,96 , occurring in a series of sharp \njumps, Fig. 4c-e. The phenomenon can be understood by considering that in the antiferromagnetic \nstate each next layer acts as a tunnel barrier for electrons with the opposite spin. In the \nantiferromagnetic state, therefore, the tunnelling resistance is high for both spins. In the \nferromagnetic state, when the magnetization in all layers is aligned, the barrier is high for the \nminority spins and much lower for the majority spin s, which reduces the overall resistance of the \nstructure. Since the different layers in the multil ayer CrI 3 switch their magnetization at different \nmagnetic field, multiple jumps occur, each correspo nding –in the simplest scenario– to one layer \nreverting its magnetisation, as it can be inferred by comparing the tunnelling magnetoresistance to \nthe magnetic field evolution of the MOKE measured o n the same devices. \nFurthermore, thin multilayers CrI 3 with the same overall magnetisation, but different  magnetisation \nsequence of the individual monolayers (e.g., ↑↓↑↑ a nd ↑↑↓↑ in a tetralayer) can exhibit \ndifferent magnetoresistance 97,98 . In practice, even for thick CrI 3, usually two magnetoresistance \njumps at approximately 0,85 and 1.9 T are found to dominate the switching virtually in all devices, \nup to crystals exceeding 10 nm (in which tunnelling  occurs in the Fowler-Nordheim regime under the 15 \n application of high bias 73 ). The reason why these two jumps in magnetoresista nce are so much larger \nand more robust than all the others remains to be u nderstood in detail. Still, it is noteworthy that \nthe values of the magnetic field at which these mos t prominent jumps occur correspond precisely \nthe values in which the exchange field induced by a  20-nm CrI 3 crystal into a WSe 2 monolayer is also \nseen to switch (see below). \n \nCurrent works are exploring different aspects of Cr I 3 tunnel barriers to realize transistors with \nmagnetic functionality. These include, for example,  the dependence of the tunnelling \nmagnetoresistance on the position of the Fermi leve l in the graphene contacts and the effect of the \naccumulated charge on the magnetic state of CrI 396-98 . Investigations are also attempting to \nunderstand how the magnetic behaviour crosses over from that of exfoliated crystals to that of the  \nFig. 4.  Heterostructures with layered magnetic materials  a, Ferromagnetic layered metals may be \nemployed as magnetic contacts, enabling fabrication  of devices acting as spin filters. The data presen ted \nshow the magnetic field evolution of resistance for  a Fe 3GeTe 2/BN/Fe 3GeTe 2 junction which displays a spin-\nvalve behaviour with magnetoresistance of up to 160 % arising from a difference in conductance when the  \nmagnetization of the two Fe 3GeTe 2 electrodes is parallel or antiparallel.  b, Differential tunnelling \nconductance for Graphite/CrI 3/Graphite junctions measured as a function of bias voltage and magnetic \nfield applied along the magnetization easy axis of CrI 3 reveals low-energy resonances exhibiting a linear \ndependence on the magnetic field, consistently with  the expected evolution of magnon states in CrI 3. The \nemergence of such inelastic tunnelling process prov ides a way to explore low-energy collective states in \nlayered magnetic materials. c-e, Vertical transport experiments in tunnel junction s based on thin layers of \nCrI 3. The impact of magnetic field on the tunnelling curr ent of 2-, 4- and ~10-layers CrI 3 devices is \npresented in panels c, d and e, respectively. The emerging current/resistivity ju mps at certain values of \nmagnetic field correspond to changes in the magneti c state of CrI 3. For the bilayer device, the sharp \nresistivity step arises from breaking the interlaye r exchange coupling. The 4-layers device exhibits a  more \ncomplex evolution of the tunnelling current, which may be attributed to layer-by-layer spin-flip proce sses, \nleading to the appearance of multiple steps. In the  10-layers sample, two major jumps of resistance ar ise. \nPanel a is reproduced from 65,92 , b - from 95 , c and d - from 96  and e - from 45,73 . \n16 \n bulk, a highly non-trivial question since the behav iour of exfoliated CrI 3 crystals that are 10-20 nm –\nseemingly thick enough to be considered as bulk– is  similar to that of atomically-thin crystals and no t \nto that of truly macroscopic samples.  \nHeterostructures with semiconducting TMDCs \nProximity between magnetic 2D crystals with other l ayered materials may serve as a very sensitive \ntool to study the details of the magnetic structure . The first investigations demonstrating \npronounced magnetic phenomena at van der Waals inte rfaces were focused on CrI 3/WSe 2 \nstructures 63 . In such devices, the low-temperature photolumines cence of WSe 2 reveal zero-field \nZeeman-type splitting of excitonic resonances. The effect is due to an interfacial exchange energy \nequivalent to the presence of an effective 13T magn etic field. The fact that the effect disappears \nabove the bulk Curie temperature of CrI 3 confirms the origin of the phenomena. These first \nexperiments also probed the evolution of the exchan ge field with applied perpendicular magnetic \nfield, which was found to be more complex than expe cted for a conventional magnet. One of the \nmost pronounced features in the photoluminescence m easurements is a sharp jump at B ~ 0.85 T, \nindicative of a field-induced change in the magneti c state of the system. The position of the jump \ncorresponds well to those observed in the tunnellin g experiments, calling for further investigation of  \nthe phenomena. \nOutlook \nThe field of 2D magnetic materials has just started , but the experimental results already reported \nallow us to catch the first glimpse of a vast unexp lored domain. Based on the work done until now, \nwe can identify different questions that require ur gent attention. \nBroadening the experimental investigations to a muc h wider spectrum of compounds is urgently \nneeded (many candidate materials are available, wit h a clear predominance of layered \nsemiconducting systems). This would allow us to bet ter structure our knowledge of the behaviour of \nthe layered ferromagnetic materials, and to underst and microscopically how this behaviour is \ndetermined by the strength on inter- and intra-laye r exchange interactions, uniaxial anisotropy, etc . \nIt will be instructive, for instance, to find more semiconductors with an out-of-plane layer-\nantiferromagnetic spin configuration, and check if giant tunnelling magnetoresistance always occurs \nas it does in CrI 3. It is not obvious that it should be so, because t he coupling between the spin \nconfiguration and the electronic states mediating t unnelling need not be the same in different \ncompounds. Indeed, the description of magnetic 2D m aterials requires the introduction of a variety \nof microscopic physical parameters, such as in-plan e and out-of-plane magnitude of exchange \ninteractions, different spin-orbit coupling terms a nd their strength, etc. How these parameters \ndetermine the magnetic configuration of the system and influence the single-electron states \nresponsible for the opto-electronic response of dif ferent 2D magnetic materials is far from being \nwell-understood. Comparing systematically the behav iour of materials with different microscopic \nparameters is the only way to the progress. These c onsiderations should be extended to van der \nWaals interfaces and heterostructures of different 2D magnetic materials, whose investigations \nenable atomically thin crystals to be probed in uni que ways and have a great potential for \nunexpected discoveries. \nEnlarging the family of magnetic monolayers entails  also the possibility to provide experimental \nevidence for long-standing theoretical predictions that still lack validation in truly 2D systems. Thi s \nincludes the realization of exotic states of matter  like intrinsic Chern insulators 99  and quantum spin \nliquids 100 , which have currently been observed only in non-st oichiometric magnetically-doped 17 \n materials 101  or in 3D layered crystals not yet exfoliated into monolayers 102 , respectively. Moreover, \nstrong Dzyaloshinskii-Moriya interactions could sta bilize topological spin configurations known as \nskyrmions that, in addition to their fundamental in terest, could provide a new paradigm for high-\ndensity, low-power data storage 103 . \nFuture experiments will additionally require signif icant technical development and new ideas to \nprobe magnetism quantitatively on very small length  scales and in the conditions of very large \ndemagnetisation fields. This is certainly needed, f or instance, for the investigation of \nantiferromagnetic 2D materials, with adjacent spins  in a same layer pointing in alternating \ndirections. A number of these compounds have been s tudied as bulk cystals 104,105  –MnPS 3, FePS 3, \nCoPS 3 – but only some of them have been exfoliated down to monolayer 106-108 .  In the bulk, \nantiferromagnetic order is probed by neutron diffra ction experiments, which – due to the weak \ninteraction of neutrons with matter – requires the use of very large bulk crystals. For \nantiferromagnetic atomically thin crystals new stra tegies will have to be followed. The use of Raman \nspectroscopy 42  to detect antiferromagnetic ordering in thin (down  to monolayer) crystals of FePS 3 \nillustrates an idea that has been pursued recently.  We anticipate that in the future different types o f \nscanning probes will play an important role. 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Magnetic behavior and spin-lattice coupling in cle avable van der Waals \nlayered CrCl 3 crystals. Phys. Rev. Mater.  1, 014001, doi:10.1103/PhysRevMaterials.1.014001 \n(2017). \n129 Kerr, J. XLIII. On rotation of the plane of pol arization by reflection from the pole of a magnet. \nThe London, Edinburgh, and Dublin Philosophical Mag azine and Journal of Science  3, 321-\n343, doi:10.1080/14786447708639245 (1877). \n \nAcknowledgements.  \nWe would like to acknowledge useful discussions wit h (in alphabetical order): I. Gutiérrez-Lezama, H. \nHenck, G. Long, N. Ubrig, Z. Wang. This work was su pported by the EU Graphene Flagship Program \n(K.S.N. and A.F.M.), European Research Council Syne rgy Grant Hetero2D (K.S.N.), the Royal Society, \nEngineering and Physical Research Council (EPSRC UK , K.S.N), US Army Research Office (W911NF-16-\n1-0279, K.S.N. and M. K.), Swiss National Science F oundation Ambizione grant program (M.G.) and \nDivision II (A.F.M.). \nCompeting interests \nThe authors declare no competing interests.  "    },    {        "title": "1911.00894v2.Spin_and_Magnetism_in_2D_Materials.pdf",        "content": "J. Phys. D: Appl. Phys. ## (2020) ######   Topical Review  \nSpin and Magnetism in 2D Materials  \nRoland K. Kawakami, The Ohio State University  \n \nStatus .   Two-dimensional (2D) materials provide a uni que platform for  spintronics and magnetism , where the atomic thinness \nof the layers leads to  strong tunability  via electrostatic gates , as discussed by S. O. Valenzuela in the 2017 Magnetism \nRoadmap . Various types of  2D materials contribute distinct spin -dependent properties  (Fig. 1) : graphene pr ovides excellent \nspin transport [ 1], transition metal dichalcogenides (TMD Cs: MX 2, with M = Mo, W and X = S, Se ) provide strong spin -orbit \ncoupling and valley -select ive optical transitions  [2] and 2D  magnets provide  non-volatile storage  and capabilities for  spin \nfiltering, injection , and detection [ 3]. By combining these materials  in stacked van der Waals (vdW) heterostructures, their \nvarious  properties are  integrated within a single s tructure. Beyond the simple addition of functionalities, quantum mechanical \ninteractions across interfaces produce spin proximity effects where properties of  2D layer s are altered by imprinting  \ncharacteristics of neighboring layers. These properties enable  potential applications in efficient non-volatile memory , spin -\nbased logic, and spin -dependent optoelectronics . \n \n \n Graphene exhibits the longest  room temperatur e spin diffusion length (~30 µm) of any material , but the weak spi n-\norbit coupling has limited its  capabilitie s for spin -charge conversion and  electrical manipulation of spin  [1]. Stacking a TMD C \nlayer onto graphene imparts a proximity spin -orbit coupling, which was most convincingly demonstrated through spin \nprecession experiments on  MoSe 2/graphene and WS 2/graphene spin valves  [1]. Fig. 2a  [4] shows oblique spin precession \nmeasurements on a WS 2/graphene spin valve, where the dependence of the spin signal on the B -field angle displays a highly \nnon-linear dependence (green data). This indicates a much lon ger spin lifetime for out -of-plane spins vs. in -plane spins, which \nis a smoking -gun indicator of proximity spin -orbit coupling in graphene induced by the WS 2. Subsequently, proximity spin -\norbit coupling in TMD C/graphene heterostructure s were  used to demons trate spin -charge conversion by spin Hall  and Rashba -\nEdelstein effect s, using spin p recession to avoid spurious signals  [1]. Meanwhile, e lectrical control of spin transport and spin \nrelaxation was demonstrated . Control of spin transport by electric gates was achieved using a graphene spin valve  with MoS 2 \non top [1]. Fig. 2b  [5] shows the increase of  conductivity (black curve) of the n -type MoS 2 with gate voltage ( Vg). This \nincreases the spin absorption from graphene to MoS 2, which shunts away spin cur rent from the graphene, eventually leading \nto zero spin current (RNL = 0) for V g > 15 V . In addition, e lectric control of spin relaxation was achieved in gated bilayer \ngraphene, surprisingly without the need for  proximity spin -orbit coupling [1]. Applying a perpendicular electric field opens \nup a band gap and the intrinsic spin -orbit splitting, though small (~24 µ eV), produces an out -of-plane spin -orbit field to \nstrongly increase the out -of-plane spin lifetime while decreasing the in -plane spin lifetime.  This was identified through obliqu e \nspin precession measurements on bilayer graphene (Fig. 2c  [6]), using a  measurement  geometry similar to  Fig. 2a.  \n Monolayer TMD Cs are direct gap semiconductors with spin -valley coupled states in the K and K’ valleys , where  \ncircularly polarized light excites  a particular valley (Fig. 1)  [2]. Optical pump -probe measure ments  established spin-valley  \nlifetimes of a few microseconds in p -type monolayer WSe 2 [7,8]. In addition, o ptical generation of spin -valley  polarization in \nmonolayer TMD Cs has  been used for injecting spins into neighboring graphene layers , which serves  as a buildi ng block for \n2D optospintronics [ 1]. As shown in Figure 2d [9] circularly -polarized light generates spin -valley polarization in monolayer \nMoS 2, which transfers into graphene, subsequently precess es in a transverse B -field, and is detected by a ferromagnetic \nelectrode. The observation of an anti -symmetric Hanle curve (blue) that flips for opposite detector magnetization (grey) \nprovides convincing evidence for this.  \n The most recent class of 2D materials for spintronics are the monolayer and few -layer vdW magnets. Intrinsic \nferromagnetism was observed in exfoliated CrI 3, CrGeTe 3, and Fe 3GeTe 2 by magneto -optic Kerr effect (MOKE)  below room \ntemperature  [3]. Room temperature intri nsic ferromagnetism was reported in ep itaxial VSe 2 and MnSe 2, as well as Fe 3GeTe 2 \nmodified by patterning or  ionic liquid gating [ 3]. The us e of 2D magnets for spintronics  (see Section 6) was demonstrated in \nrecent experiments. Electric control of magnetic interlayer coupling was realized in bilayer CrI3, which has a split hysteresis \nloop indic ating antiferromagnetic coupling  [3]. As shown in Figure 2e  [10], the variation of the splitting with applied electric \nfield indic ates real -time control of the antiferromagnetic  coupling. Vertical transpor t through insulating bilayer CrI 3 produces \na large tunne ling magnetoresistance  (> 10,000%) due to spin filtering effects  [3]. Figure 2f  [11] shows  the tunneling current \nas a function of applied  magnetic  field, showing larger (smaller) current in the parallel (antiparallel) magnetization state. More \ntraditional metal/barrier/metal magnetic tunnel junc tions (MTJs) were realized in Fe 3GeTe 2/hBN/Fe 3GeTe 2 with TMR of \n160% [3].  Also relevant for spintronic memory, spin -orbit -torque (see Section 4) was observed in Fe 3GeTe 2/Pt [12,13 ]. \nCurrent and future challenges.    While 2D magnets exhibit a range of interesting magnetoelectronic p henomena , these have \nonly been observed at low temperatures . So far, none of the room temperature 2D ferromagnets have exhibited high remanence \nor ability to integrate into heterostructures. Thus, continued materials development is needed to simultaneously increase Curie \nFigure 1.  2D materials for spintronic heterostructures . J. Phys. D: Appl. Phys. ## (2020) ######   Topical Review  \ntemperature, magnetic remanence, material integration capability, and air -stability. A recent advance along these lines is a Fe -\nrich version of Fe 3GeTe 2, namely Fe 5GeTe 2 [14] which exhibits ferromagnetic  order near room temperature.  Further work on \ndeveloping new room temperature 2D magnets with improved characteristics is an important  challenge.  \nFor applications in magneto electronic memor ies, one of the challenges of the field is to reduce the critical current \ndensity needed for magnetic switching. Three viable approaches are s pin-orbit -torque in FM/heavy metal bilayers  (see Section \n4), spin -transfer -torque in FM/barrier/FM MTJs  (see Section 6 ), and voltage controlled magneti sm. 2D magnets are attractive \nin this regard, as the strong covalent bonding of the atomic sheets enables low magnet ic volume by scaling down to  atomic \nlayers. Reported values of critical current densities for spin -orbit -torque switching in  initial studies is  ~1011 A/m2 [12,13 ], \nwhich is  promising . Further development with a lternative heavy metal layers such as WTe 2, Bi2Se3 and other  vdW materials \nwith high spin-orbit coupling sh ould improve device performance.  Strong elec trostatic gating effects are a hallmark  of 2D \nmaterials, which will likely  maximize effects such as voltage -controlled magnetic anisotropy ( VCMA), which is a candidate \nfor low power dynamic magnetization switching  [15]. Combinations of VCMA and spin -torque c ould enable ultra -efficient \nmagnetization switching. For higher switching speed, antiferromagnetic materials such as MnPS 3 and other layered \ntrichalcogenides could provide fast switching due to high magnetic resonance frequencies, which is a general motiva tion fo r \nantiferromagnetic spintronics.  \n \nFigure 2.  (a)  Oblique spin precession measurements of TMDC/graphene spin valves, demonstrating proximity spin -orbit coupling through \nobservation of spin lifetime anisotropy. Adapted from [ 4]. (b) Two-dimensional field -effect spin switch composed of MoS 2 on graphene spin \nvalve. Adapted from [ 5]. (c) Oblique spin precession measurements of dual -gated bilayer graphene spin valves, demonstrating electric control \nof spin lifetime anisotropy.  Adapted from [6]. (d) Opto -valleytronic spin injection from MoS 2 into graphene . Adapted from [9]. (e) Electric \nfield control of antiferromagnetic interlayer coupling  in bilayer CrI 3. Adapted from [10]. (f) Giant  spin-filtering tunne ling magnetoresistance  \nin vertical transport across bilayer CrI 3. Adapted from [11]. \n \nFor multifunctional spintro nics, a crucial issue is understanding and optimizing spin proximity effects in \nheterostructures of  graphene, TMD Cs, and 2D magnets . Proximity spin -orbit coupling has been observed i n TMD C/graphene  \nand proximity exchange fields have been observed in TMD C/FM insulator and graphene/ FM insulator systems  [1]. Future \nchallenges include  the control of such proximity effects by electric gates and by twist angle between the layers. The \nramifications of such proximity effects are i n four areas: electr ically -controlled spin switches , efficient  magnetization \nswitching by spin -orbit torque  (see Section 4) , optospintronics and opto magnetic switching  (see Section 5) , and the realization \nof topological states such as  the quantum anomalous Hall effect  (QAHE) .  \n \nAdvances in science and technology to meet challenges.    While exfoliated films are good for fundamental science, epitaxial  \nfilms are needed for a manufacturable technology. Various forms of  chemical vapour deposition  have  been useful for growth \nof graphene and TMD Cs, while m olecular beam epitaxy has been useful for growth of 2D magnets and TMD Cs. Optimizing \nsuch materials and c ontrolling interface quality is crucial in many contexts. To maximize spin proximi ty effects , it is important  \nto employ  methods for achieving  clean interfaces , such as the stacking of 2D materials inside gloveboxes or under vacuum . \nFor many air -sensitive 2D conductors and magnets, stacking inside a glovebox is essential. Electrical spin  injection into \ngraphene requires injection across tunnel barriers , which continue s to advance .  \nThe use of advanced microscopies and spectroscopies that can image the magnetic order and electronic structure \nwill be important for developing new 2D magnets and spintronic heterostructures. Spin -polarized scanning tunneling \nmicroscopy can image magnetism with atomic resolution to correlate the atomic -scale structure with the magnetic ordering , \nas discussed by D. Sander in the 2017 Magnetism Roadmap .  NV diamon d microscopy can probe the local magnetic field of \nburied layers with high spatial resolution. Second harmonic generation is a nonlinear optical probe that is sensitive to \nsymmetry -breaking, which therefore probes the layer stacking and antiferromagnetic o rder. Micron and nanometer -scale angle -\nresolved photoemission spectroscopy (micro/nanoARPES) enables spatial mapping of electronic band structure, which will be \nimportant for the development of 2D magnets, topological edge states, and spintronic devices.  \nConcluding remarks.    The study of spin and m agnetism in  vdW heterostructures is in its early stages and progressing rapidly, \nas exemplified by  the recent emergence of spin proximity effects and 2D magnets. The development of electrically -tunable, \nmultifun ctional spintronic devices  will rely on coupled advances in synthesis, assembly, and measure ment  and will take \nadvantage of the unique properties of 2D materials.  \n \nReferences  \n \n1. Avsar, A., Ochoa, H., Guinea, F., Ozyilmaz, B., van Wees, B. J. & Vera -Marun, I. J. 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Recent Progress in the Voltage -\nControlled Magnetic Anisotropy Effect and the Challenges Faced in Developing Voltage -Torque MRAM. \nMicromachines (Basel)  10, 327 (2019).  \n "    },    {        "title": "1911.02881v1.Additive_manufactured_isotropic_NdFeB_magnets_by_stereolithography__fused_filament_fabrication__and_selective_laser_sintering.pdf",        "content": "Additive manufactured isotropic NdFeB magnets by stereolithography,\nfused filament fabrication, and selective laser sintering\nChristian Hubera,b,∗, Gerald Mitteramskoglerc, Michael Goertlerd, Iulian Telibane, Martin Groenefelde,\nDieter Suessa,b\naPhysics of Functional Materials, University of Vienna, 1090 Vienna, Austria\nbChristian Doppler Laboratory for Advanced Magnetic Sensing and Materials, 1090 Vienna, Austria\ncIncus GmbH, 1220 Vienna, Austria\ndInstitute for Surface Technologies and Photonics, Joanneum Research Forschungsgesellschaft GmbH, 8712 Niklasdorf,\nAustria\neMagnetfabrik Bonn GmbH, 53119 Bonn, Germany\nAbstract\nMagnetic isotropic NdFeB powder is processed by the following additive manufacturing methods: (i) stere-\nolithography (SLA), (ii) fused filament fabrication (FFF), and (iii) selective laser sintering (SLS). For the\nfirst time, a stereolithography based method is used to 3D print hard magnetic materials. FFF and SLA\nuse a polymer matrix material as binder, SLS sinters the powder directly. All methods use the same hard\nmagnetic NdFeB powder material. Complex magnets with small feature sizes in a superior surface quality\ncan be printed with SLA. The magnetic properties for the processed samples are investigated and compared.\nSLA can print magnets with a remanence of 388 mT and a coercivity of 0 .923 T. A complex magnetic design\nfor speed wheel sensing applications is presented and printed with all methods.\nKeywords:\nMagnets, Photopolymerization, Powder Bed Fusion, Material Extrusion, NdFeB\n1. Introduction\nAdditive manufacturing (AM) offers a new era of\npossibilities for magnetic materials and advanced\nmagnetic sensing applications. AM methods cre-\nating solid structures layer-by-layer from a form-\nless or form-neutral feedstock by means of chemical\nor thermal processes. This leads to several advan-\ntages like: design freedom, net shape capabilities,\nwaste reduction, minimum lead times for prototyp-\ning, compared to traditional manufacturing meth-\nods like sintering of full-dense magnets or injection-\nmolding of polymer-bonded magnets.\nFused filament fabrication (FFF) or fused depo-\nsition modeling (FDM) is a well-known and widely\nused AM method to print thermoplastic materi-\nals. It uses a wire-shaped thermoplastic filament as\nbuilding material. The filament is feed to a mov-\nable extruder where it heated up above its softening\n∗huber-c@univie.ac.atpoint. The molten material is pressed out of the\nextruder nozzle and builds the structure layer-by-\nlayer on the already printed and solidified layer [1].\nA sketch of the FFF method is shown in Fig. 1(a).\nBy mixing magnetic soft- or hard magnetic materi-\nals into the thermoplastic binder, FFF can be also\nused to 3D print polymer-bonded magnets with fill-\ning ratios up to 90 wt.%. [2, 3, 4, 5, 6, 7, 8, 9].\nA big disadvantage of polymer-bonded permanent\nmagnets is their lowered maximum energy prod-\nuct (BH)maxcompared to sintered magnets due to\ntheir plastic matrix material.\nTo maximize the performance of permanent mag-\nnets, the (BH)maxmust be increased. Powder bed\nfusion (PBF) processes does not need a matrix ma-\nterial. It completely melts the metallic powder\nby the aid of a high-power laser or electron beam\nsource [10]. To optimize the printing process and\nthe quality of the prints, powders with a spherical\nmorphology are preferred [11]. Fig. 1(b) shows a\nsketch of the printing process. This means theo-\nPreprint submitted to arXiv.org November 11, 2019arXiv:1911.02881v1  [physics.app-ph]  7 Nov 2019retically, that fully dense magnets can be printed.\nHowever, the rapid liquefaction and cooling rates\nof the material at the localized heat source, influ-\nences the microstructure (size of the grains and the\ncomposition of the grain boundaries), and there-\nfore the magnetic properties of the printed struc-\ntures [12, 13, 14]. The optimization of the mag-\nnetic properties of PBF processed magnets is an\nactive research field. The PBF process can be di-\nvided into the heating source. Following commonly\nused PBF printing techniques are used to investi-\ngate the capability to print hard or soft magnets:\nelectron beam melting (EBM) [15], selective laser\nmelting (SLM) [16, 17, 18, 19, 20, 21], and selective\nlaser sintering (SLS). SLS does not completely melt\neach powder layer but sinter the particles to retain\ntheir original microstructure. As a second step, the\ncoercivity of the SLS printed samples can be sub-\nstantially increased by a grain boundary infiltration\nmethod [22].\nStereolithography (SLA) was the first commer-\ncially available AM technology. The layers of the\nsliced computer model is scanned by a visible or\nultraviolet (UV) light to cure the photosensitive\nresin selectively for each cross-section, or a digi-\ntal light processing (DLP) engine projects and cure\nthe whole image of every layer. After each fin-\nished layer, the workpiece is lowered by one-layer\nthickness. Then, the resin sweeps across the cross-\nsection of the partly finished object, and coating\nit with a new layer of fresh resin. This layer is\nscanned and cured-on the previous hardened layer.\nFig. 1(c) shows the principle of SLA. SLA of soft\nmagnetic materials with a very low filler content of\nonly 30 wt.% is described in [23]. Up to now, no\npublication about SLA of hard magnetic materials\nexists.\nThis publication deals with SLA of magnetic\nisotropic powder in a photo reactive resin. The res-\nolution and quality of the 3D printed permanent\nmagnetic samples are superior. Fig. 1(d) shows\na 3D printed magnetic St. Stephen’s Cathedral,\nVienna with a minimum feature size of the model\nof 0.1 mm and a layer height of 60 µm. Further-\nmore, the same magnetic isotropic powder is used\nto print polymer-bonded magnets with FFF and\nsintered magnets with SLS. All advantages and dis-\nadvantages of each method are discussed in detail.\nComplex magnets are printed and their magnetic\nproperties are investigated and compared.\nFigure 1: Different used additive manufacturing (AM) meth-\nods. (a) fused filament fabrication (FFF). (b) selective laser\nsintering (SLS). (c) stereolithography (SLA). (d) 3D printed\nmagnetic St. Stephen’s Cathedral, Vienna by SLA.\n2. Materials & Methods\nWe are using a commercial isotropic NdFeB pow-\nder (MQP-S-11-9 supplied by Magnequench Corpo-\nration) for all three presented AM methods. This\npowder has a spherical morphology with a powder\nsize distribution of d 50of 38 µm and the tap den-\nsity exhibits 61 % of the materials full density. A\nscanning electron microscopy (SEM) image of the\npowder can be seen in Fig. 2(a). Its main field of ap-\nplication is the manufacturing of bonded magnets,\nparticularly by injection molding or extrusion. The\npowder particles have nano-sized NdFeB grains,\nit have a uniaxial magnetocrystalline anisotropy\nwhich are random orientated. This leads to mag-\nnetic isotropic behavior of the bulk magnet. This\npowder is produced by a gas atomization process\nand a followed heat treatment. The chemical com-\nposition states Nd 7.5Pr0.7Fe75.4Co2.5B8.8Zr2.6Ti2.5\n(at.%) [20].\nFor the FFF of bonded magnets, a conventional\nend-user 3D printer Builder from Code P is used.\nWe are using a prefabricated compound (Neofer ®\n25/60p) from Magnetfabrik Bonn GmbH. It con-\nsists of 89 wt.% MQP-S powder inside a PA11 ma-\ntrix. To get the wire-shaped filaments for the 3D\nprinter extruder, the Neofer ®25/60p compound\ngranules are extruded at the University of Leoben\nwith a Leistritz ZSE 18 HPe-48D twin-screw ex-\ntruder. The extrusion temperature is 260◦C, and\nthe hot filament is hauled off and cooled by a cooled\nconveyor belt. The diameter of 1 .75 mm and tol-\nerances of the filament are controlled by a Sikora\nLaser Series 2000 diameter-measuring system. The\n2Builder 3D printer can build structures with a max-\nimum size of 220 ×210×165 mm3(L×W×H).\nThe layer height resolution can be varied between\n0.05 and 0.5 mm. Printing speed ranges from 10\nto 80 mm/s, traveling speed ranges from 10 to\n200 mm/s. To avoid clogging of the nozzle due\nto the height filler content, the minimum nozzle\nsize diameter is 0 .4 mm. This large nozzle diam-\neter defines the minimum feature size of the prints.\nThe printing temperature for the PA11 compound\nis 260◦C. For a better adhesion of the first layer,\nthe printing bed is heated up to 80◦C.\nFor sample fabrication with the SLS system,\na commercial Farsoon FS121M LPBF-machine is\nused. It is equipped with a continuous wave 200 W\nYb-fibre laser with a wavelength of 1 .07µm and\na spot size of 0 .1 mm. It has a build space of\n120×120×100 mm3(L×W×H). The printing of\nthe MQP-S powder is performed under Ar atmo-\nsphere with oxygen content below 0 .1 %. A layer\nthickness of 100 µm, and the powder recoating was\ndone with a carbon fiber brush. All specimens were\nprinted without support structures directly onto a\nsteel substrate plate to ensure proper heat dissipa-\ntion. The laser power Pis varied between 20 W\nand 100 W, and the scan speed vis varied between\n50 mm/s and 2000 mm/s to find the optimal print-\ning parameters. The line energy Eline=P/v is a\nconvenient printing parameter. For sintering of the\nMQP-S powder, line energies between 0 .03 J/mm\nand 0.07 J/mm at 40 W, and a hatch spacing hof\n0.14 mm is practicable.\nIncus GmbH developed an industrial vat pho-\ntopolymerisation process called Lithography-based\nMetal Manufacturing (LMM). The LMM machine\nis based on a top-down SLA principle. The liquid\nphoto-reactive feedstock is polymerized from above\nby a high-performance projection unit (Fig. 1(c)).\nThe building platform with the submerged parts\nis lowered, layer-by-layer, according to the chosen\nlayer thickness. For this study, a layer thickness of\n60µm is used. After the curing of a layer, the wiper\nblade applies a fresh film of feedstock. The size of\nthe building platform is 75 ×43 mm2and the resolu-\ntions in the xandydirections are 40 µm each. The\nprinting time of a single layer is 35 s, which results\nin a build speed of 6 mm/h in z-direction (about\n20 cm3/h in volume). A photo-reactive feedstock is\nprepared, based on commercially available di- and\npolyfunctional methacrylates (60 wt.%). The reac-\ntive components included an initiation system and\na proprietary photoinitiator, which absorbs lightin the wavelengths emitted by the projector. A\nsolid loading of MQP-S powder up to 92 wt.% is\nachieved. The binder components and the mag-\nnetic powder were added in a mixing cup and ho-\nmogeneously dispersed via centrifugal mixing. The\nself-supporting function of the material facilitates\nthe volume-optimized placement of different parts\non a single building platform without the need for\nadditional support structures.\n3. Results & discussion\nThe focus in this paper is the discussion of the\nmagnetic properties of the different used AM meth-\nods. To test the magnetic properties, cubes with\na dimensions of 5 ×5×5 mm3are printed with\nthe above described printing parameters and tech-\nniques. No post-processing of the printed samples\nis performed. SEM images of the surface and the\nlayer structure are presented in Fig. 2(b)-(d). The\nsample printed by FFF shows the rawest surface\n(Fig. 2(b)). Fig. 2(b) indicating a partly densi-\nfied SLS sample while several cracks can still be\nseen in the microstructure. The surface of the SLA\nsample shows the best quality of all three methods\n(Fig. 2(d)).\nVolumetric mass density /rho1is measured with with\na hydrostatic balance (Mettler Toledo, AG204DR)\nbased on the Archimedes principle. The filling\nfraction of the MQP-S powder inside the polymer\nmatrix for the FFF and the SLA printed samples\nis measured with by thermogravimetric analysis\n(TGA) (Tab. 1). The density for the FFF sam-\nple is around 20 % lower compared to the theo-\nretical value ( /rho1= 4.35 g/cm3). SLS shows a den-\nsity that is in the same range as the tap density\nof the powder ( /rho1= 4.3 g/cm3). This shows that\nthe MPQ-S powder is sintered without complete\nmelting of the material. SLA has the highest vol-\numetric mass density of the investigated printing\ntechniques. For the measurement of the magnetic\nhysteresis curve and the magnetic properties of the\nsamples, a permagraph (magnetic closed loop mea-\nsurement) from Magnet-Physik Dr. Steingroever\nGmbH with a JH 15-1 pick-up coil is used. The\nmagnetic hysteresis curved are shown in Fig. 3, and\nthe magnetic properties are summarized in Tab. 1.\nThe coercivity of the SLS sample is around 25 %\nlower compared to the data sheet value of the pow-\nder. This is a result of the inhomogeneous mi-\n3100 µm(a)\n1 layer 1 layer\n100 µm(b)\n(d) (c)\n50 µm\n1 layer50 µm\n50 µm\n100 µm100 µmFigure 2: All presented AM methods use the same isotropic\nNdFeB powder (MQP-S-11-9, Magnequench). (a) SEM im-\nage of the initial MQP-S powder. SEM images of the sur-\nfaces of magnetic samples, printed with: (b) FFF, (c) SLS,\n(d) SLA.\ncrostructure, in particular the grain size distribu-\ntion [22].\nTable 1: Properties of the isotropic NdFeB powder (MQP-\nS-11-9 from Magnequench Corporation.) and the samples\nprinted with the different AM methods. wf...filling mass\nfraction,/rho1... volumetric mass density, Br...residual Induc-\ntion, andµ0Hcj...intrinsic coercivity.\nsamplewf\n(wt.%)/rho1\n(g/cm3)Br\n(mT)µ0Hcj\n(T)\npowder – 7.43 746 0 .880\nFFF 89 3.57 344 0 .918\nSLS 100 4.47 436 0 .653\nSLA 92 4.83 388 0 .923\nThe capabilities of the different presented AM\nmethods are discussed on a magnetic speed wheel\nsensing system. Such high precision sensor systems\nare embedded in many applications, especially in\nautomotive application, e.g. in anti-blocking sys-\ntem (ABS) or engine management systems [24]. A\npossible design of such speed sensors consist of a\nmagnetic field sensor, e.g. Hall effect or giant mag-\nnetoresistance (GMR) sensor, a permanent mag-\nnet which provide a bias field and a soft magnetic\n−1.6−1.2−0.8−0.40.0 0.4 0.8 1.2 1.6\nmagnetic field strength µ0Hint(T)−700−500−300−100100300500700polarization J(mT)\nFDM\nSLS\nSLAFigure 3: Hysteresis loops of the different printing methods.\nwheel. Normally, the magnet is underneath the sen-\nsor (back-bias magnet) and the rotating soft mag-\nnetic wheel modulates the magnetic field of the\nback-bias magnet. The rotational velocity of the\nwheel is direct proportional to the modulation of\nthe field. Fig. 4(a) shows a sketch of a possible\nwheel speed sensing arrangement.\n(a)\ngear\nwheel\nback-bias\nmagnetmagnetic\nfield sensor(b)\n1mm(T)\nFigure 4: Magnetic wheel speed sensing. (a) Principle of\nthe magnetic speed sensing. A permanent magnet is un-\nderneath the magnetic field sensor (back-bias magnet). A\nsoft magnetic gear periodically modulates the bias field of\nthe magnet. (b) Special back-bias magnet design for giant\nmagnetoresistance (GMR) sensors.\nIf a GMR sensor is designated to detect the field\nmodulation, some special magnetic design criteria\nmust be considered. GMR sensors are in-plane sen-\nsitive and the linear range is very small [25]. This\nmeans that the back-bias magnet must have very\nlow magnetic in-plane field components. This can\nbe achieved by a specific design of the magnet.\nFig. 4(b) shows the cross-section of a well-known ge-\nometry that minimizes the components of the mag-\nnetic stray field Binxandydirection in a wide\nrange along the x-axisrx. In this case, an accurate\nfield distribution is more important than a maxi-\nmum field. Prototyping of such complex magnetic\ndesigns is one of the biggest advantage of AM meth-\nods.\n4Starting from the design as described above, a\nback-bias magnet for speed wheel sensing is 3D\nprinted with: (i) FFF, (ii) SLS, and (iii) SLA.\nThe overall size of the magnet is 7 ×5×5.5 mm3\n(L×W×H). After the printing process, the magnet\nis magnetized in an electromagnet with a maximum\nmagnetic flux density of 1 .9 T in permanent opera-\ntion mode. Fig. 5 shows a line scan of the magnetic\nflux density B, 2.5 mm above the pyramide tip (T)\nfor all three AM methods. The magnetic flux den-\nsity is measured with a Hall probe and the FFF\n3D printer as described in [2]. The magnet printed\nby FFF has the weakest flux density Bzbecause of\nthe smaller remanence Brcompared to the other\nAM methods. However, all three methods show a\nminimum stray field BxandByalong thex-axis.\nA picture of the printed magnets is illustrated in\nFig. 6. It is clearly visible that SLA produces the\ngeometrical most accurate prints.\n−4−2 0 2 4\nx-axis rx(mm)−100−50050100magnetic flux density Bi(mT)\nFDMBx\nFDMBy\nFDMBzSLSBx\nSLSBy\nSLSBzSLABx\nSLABy\nSLABz\nFigure 5: Line scan of the magnetic flux density B, 2.5 mm\nabove the pyramid tip (T).\n(mm) 10 20(a) (b) (c)\nFigure 6: Picture of the back-bias magnets printed by: (a)\nFFF, (b) SLA, (c) SLS.\n4. Conclusion\nAM of magnetic materials with different meth-\nods and materials is an active research field. Manygroups use the hard magnetic isotropic NdFeB\nMQP-S powder due to the spherical morphology\nand robustness against corrosion.\nNevertheless, this publication describes SLA of\nhard magnetic materials for the first time. Even\nmore, the SLA method is compared to polymer-\nbonded magnets printed with FFF and sintered\nmagnets printed with SLA. Magnets printed with\nSLA show the best magnetic performance and a\nvery high surface quality compared to samples\nprinted with FFF or SLS. The modification of the\nmicrostructure of the powder during the SLS pro-\ncess is the reason for its lower magnetic performance\ncompared to the other methods. FFF is the most\naffordable and simplest way to print magnets, but\ndue to the large nozzle diameter, the accuracy of\nthe physical dimensions is limited. Additionally,\nthe lower volumetric mass density compared to the\ntheoretical value is a reason for the lower remanence\nof the printed magnets.\nIn summary, it can be said that the MQP-S pow-\nder perfectly meets the requirements of the SLA\nprinting process. We can see a huge potential for\nthe manufacturing of complex magnetic designs in\na superior quality.\nAcknowledgment\nThe support from CD-Laboratory AMSEN (fi-\nnanced by the Austrian Federal Ministry of Econ-\nomy, Family and Youth, the National Foundation\nfor Research, Technology and Development) is ac-\nknowledged.\nReferences\n[1] N. Guo, M. C. Leu, Additive manufacturing: tech-\nnology, applications and research needs, Frontiers of\nMechanical Engineering 8 (2013) 215–243.\n[2] C. Huber, C. Abert, F. Bruckner, M. Groenefeld,\nO. Muthsam, S. Schuschnigg, K. Sirak, R. Thanhoffer,\nI. Teliban, C. Vogler, R. Windl, D. 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Dhanasekhar1,2, Praveen Chaddah3†, D Chandrasekhar Kakarla4,5, H. D. Yang4,5, Z. H. \nYang4, B. H. Chen6, Y. C. Chung6 and A. K. Das1 \n1Department of Physics, Indian Ins titute of Technology, Kharagpur 721302, West Bengal, India  \n2Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India  \n3UGC -DAE Consortium for Scientific  Research, Khandwa Road, Indore 452001, Madhya Pradesh , India  \n4 Department of Physics, National Sun Yat -Sen University, Kaohsiung 804, Taiwan  \n5Center of Crystal Research, National Sun Yat -Sen University , Kaohsiung 804, Taiwan  \n6National Synchrotron Radiation Research Center, Hsinchu, 30076, Taiwan  \n \n*tiwariarchana9@phy.iitkgp.ernet.in  \n† Since retired;  chaddah.praveen@gmail.com  \n \nABSTRACT  \nThe magnetic glassy state is a fascinating phenomenon, which  results from the kinetic arrest of the first or der \nmagnetic phase transition . Interesting properties, such as metastable magnetization and nonequilibrium magnetic \nphases, are naturally developed  in the magnetic glassy state. Here,  we report magnetic glass property in the low \nspin state of Co3+ in EuBaCo 2O5+δ (δ = 0.47) cobaltite  at low temperature ( T < 60 K) . The measurements of \nmagnetization under t he cooling and heating in  unequal fields , magnetization relaxation and thermal  cycling of \nmagnetization show the kinetic arrest of low magnetization state below 60 K. The kinetically arrested low \ntemperature  magnetic phase is further supported  through the  study of  isothermal magnetic entropy , which show s \nthe significant entropy change . The present results will open a new window to search the  microscopic  relation \nbetween the spin state transitions and the kinetic arrest  induced magnetic glassy  phenomena in complex \nmaterials.  \n \nKeywords:  first order phase transition , kinetic arrest , magnetic glass, magnetic entropy  \n \n \n \n \n \n 2 \n 1. Introduction   \nThe first-order magnetic phase transition (FOMPT) in magnetic materials shows several fascinating phenomena , \nsuch as giant magnetocaloric effect , colossal ( or giant) magnetoresis tance , magnetic shape memory effect , etc [1–\n5]. In recent years , the coexistence  of magnetic phases  (or multiple magnetic phases)  is observed  in some magnetic \nmaterials below the FOMPT. The phase coexistence arises from the kinetic arrest of the supercooled, metastable, \nand nonequilibrium magnetic phase in the background of a stable equilibrium magnetic phase. Under certain \nconditions , the nonequilibrium supercooled phase  is transformed into the stable equilibrium phase  and the  \nnonequilibrium phase is named as magnetic glass (M G) [5–12]. The origin of MG phenomenon is different from \nthe conventional spin glass (SG) and the reentrant spin glass (RSG)  and is observed i n a variety of magnetic \nsystems [13,14] . In the majority of MG materials, it is observed that the FO MPT is accompanied with \nthermomagnetic irreversibility  (TMI) and this  TMI can be understood from the appearance of phase coexistence \nand metastability  [4–13].     \n \nFurther, the kinetically arrested FOMPT is coupled with a structural change  in most of the MG materials.  For \nexample s, the high temperature ferromagnetic ( FM) to low temperature antiferromagnetic ( AFM ) transition in \nchemically doped -CeFe 2 alloys [6,15,16 ] involves cubic to rhombohedral structural changes, whereas the high \ntemperature AFM to low temperature FM transition in Gd 5Ge4 [17] is associated with the orthorhombic  structural \ntransition from the  Sm 5Ge4-type to orthorhombic Gd 5Si4-type. On the other  hand , the formation of MG is also \nreported in the layered YBaCo 2O5+δ (δ = 0.50) cobaltite with 5% Ca substitution at Y and Ba sites  [10]. In \ncomplement to the Ca substitution  studies , we have shown  that the electron  doped  YBaCo 2O5.5-δ (δ = 0.13)  \ncobaltite  also showed  the presence of MG state at low temperature ( T) [18]. However, magneto structural coupling \nhardly plays a role in both the cobaltites and the MG phenomenon is  explained considering the local phase \nseparation of FM and AFM clusters.   \n \nOur recent studies  [19] on layered EuBaCo 2O5+δ (δ = 0.47) (EBCO) cobaltite showed  metal to insulator transition \nat 355 K  (TMIT), paramagnet ( PM) to FM transition at 267 K  (TC), FM to AFM1 transition at 240 K  (TN1) and \nAFM1 –AFM2 transition at 204 K  (TN2). The FM to AFM transition  show ed the  significant  first order \nthermomagnetic hysteresis effect  and further, we showed that the AFM p hases were  shifted to low temperature \nunder  the external magnetic field . More importantly , the octahedral Co3+ showed the high spin (HS) to intermediate \nspin (IS) state transition at TMIT and this IS state gradually  converted  to the low spin (LS) state  below  155 K.  \n  \nIn this article, we report the kinetic arrest of magnetization at low temperature in the presence of the magnetic \nfield (H) and a glasslike dynamical response  in EuBaCo 2O5+δ (δ = 0.47) (EBCO) cobaltite.  The non-equilibrium \nmagnetic glassy state in EBCO also contributes a significant entropy change at low temperature.  The observed \nMG phenomena in this cobaltite differs  from the other magnetic glassy materials  because  the various high \ntemperature magnetic phases (FM, AFM1 and AFM2) in EBCO ar e strongly coupled with the Co3+ spin state \ntransitions and the kinetic arrest is observed at the low temperature, where LS (S=0) state of Co3+ presents.  Finally , \nwe show  that the kinetic arrest of FOMPT in EBCO is not associated with a structural transition , and the other \nvarious  possibilities  that can elucidate the origin of kinetic arrest and MG phenomena are discussed.  \n 3 \n 2. Experiment details   \nThe polycrystalline EuBaCo 2O5+δ (δ = 0.47) cobaltite w as prepared by conventional solid -state reaction method \nand the details can be found in Ref.  19. The temperature -dependent structural analyse s were  performed  using the \nsynchrotron  X-ray powder diffraction (SXR PD) at TPS -09A beam line at the National Synchrot ron Radiation \nResearch Centre,  Hsinchu, Taiwan. Finely grounded powder sample was loaded in 0.3  mm glass capillary. The \ndiffraction patterns were measured in the range of 2θ = 1.8 –100° with a step size of Δ( 2θ) = 0.00375° with the \nincident wave length of 0.61992 Å. The magnetization  versus temperature, i.e. M  (T) measurements w ere carried \nout using Physical Property Measurement System (PPMS), Cryogenic Limited, UK. A C magnetization , time \ndependence of  magnetization i.e. M  (t) and temperature cycling magnetization measurements were performed \nusing Magnetic Property Measurement System (MPMS SQUID VSM), Quantum Design, USA.  \n3. Results  \n3.1. Ac magnetization studies  \nTo confirm the various dc M  transitions and to find out whether presence of any conventional spin glass signature  \nin EBCO, a frequency dependent ac  M measurements  under an ac field of 1 Oe were performed and the obtained \nreal part of χ𝑎𝑐′ (T) is shown in figure  1. The various dc M transitions  [19] also appear  in χ'( T) and those transitions \nare frequency independent in the measured frequency range. The χ𝑎𝑐′ (T) curves show a drop below 60 K, which \nmatches with the zero field cooled ( ZFC) M behavior  (see figure  2(b)). Further, to ruled out the spin or cluster \nglass like behavior below 60 K, we performed a memory effect in χ𝑎𝑐′ (T) and the result is shown in the inset of \nfigure  1. In these measurements, the sample was cooled to the base  temperature of 5 K while administering halt \nat 25 K for t w = 3 hours. After reaching 5 K, the sample was warmed to 380 K and χ𝑎𝑐′ (T) was recorded during \nthe heating cycle and the result is shown in the inset of figure  1. The data taken  after the halt is referred as  χ𝑎𝑐′ ℎ𝑎𝑙𝑡  \nand referred as  χ𝑎𝑐′ 𝑟𝑒𝑓 without the halt time. The difference between the   χ𝑎𝑐′ ℎ𝑎𝑙𝑡− χ𝑎𝑐′ 𝑟𝑒𝑓 (≈ Δ χ𝑎𝑐′) is also \nshown in the inset of figure  1. In most of the spin glass or cluster glass the  Δχ𝑎𝑐′  showed  dip kind of signatures at \nthe halt temperatures  [20] and in our present case the Δ χ𝑎𝑐′ have  not show n any such signatur es at 25 K. The \nfrequency independent χ𝑎𝑐′ (T) peaks in combined with Δ χ𝑎𝑐′ result ruled out the SG and cluster glass (CG) like \nfeatures below 60 K. The observed ac magnetization results are in line with the single crystal studies of \nGdBaCo 2O5+x, where it is showed that the conventional spin -glass behavior is not poss ible for any x and suggests \nthat all magnetic irreversibilities are interrelated to the spin rearrangement  within a spin -ordered state  [21].  \n3.2. Kinetic arrest and metastability  \nThe detailed  ac M studies show the absence of conventional glass  like feature in the EBCO  and the detaile d dc M \nstudies show  the signatures  of phases coexistence  [19]. To identify the kinetic arrest -induced phase coexistence, \nBanerjee et al. proposed a special magnetic measurement protocol, i.e., cooling and heatin g in unequal fields  \n(“CHUF”)  [22,23 ]. Following this protocol, the EBCO  sample was  cooled  from  the 380 to 10 K under various \ncooling magnetic fields (H cool), i.e. 0 , 1, 2, 3, 5 and 7 T respectively  and at 10 K  the H cool is changed  to measuring \nmagnetic field  (Hmeasure ) of 2 T and M was reordered on warming  cycle (see figure  2(a)). For T > TN1 the M \nbehavior of EBCO under the CHUF protocol remains the same irrespective of the H cool. For T < TN1, the M curves 4 \n show strong dispersion and more interesting behavior is observed below 60 K. For H cool = 0 T, the M of the sample \nshows an initial in crease from 5 -60 K and becomes f lat. Whereas in case of cooling under 1  T and measuring at  2 \nT the M increases up to 35  K and becomes flat or plateau is seen . While for H cool = H measure  the M follows the same \npath without showing any initial rise. For H cool > H measure the initial increase in M is absent and shows slightly \ndecrease in M with increasing the  Hcool. This result clearly suggests that the low temperature M is metastable and \nis kinetically arrested while cooling. Further, the kinetically arrested M state is de -arrested and  reaches a stable \nmagnetization above 60  K. We note that the de -arrest takes place  at low T with increasing  the coo ling H . This is \nfurther supported by the ZFC measured at  different low  fields , where the initial increase in M is shifted to low T \nwith increasing the H  as shown in the figure  2(b). The H-T phase diagram along with kinetic arrest  band obtained  \nfrom the ZFC  magnetization is shown in the figure  2(d).      \nThe M relaxation measurements at 20 K are measured under 2 T for  the ZFC and the field cooled ( FC) cases  on \nEBCO provide further evidence for the  metastability  of the lower M state. In ZFC measurement, the EBCO sample \nwas cooled from 380 K to 20  K in zero field ; then the sample  was held at T = 20 K for a time t w = 300 s;  after then \nH was applied and M was recorded as a function of time. In FC measurement the sample was cooled from 380 K \nto 20 K in presence of an applied magnetic field ( H); then the sample was held at T = 20 K for a time t w = 300 s;  \nafter which the H was removed and the M was recorded as a function of time. The ZFC magnetization of EBCO  \nis increased  from 0.102 to 0.1 10 µB/f.u. and the corresponding  change in M is ∼ 7.8 % (figure  2(c)) . On the other \nside, the FC magnetization  is decreased  from 0.176 to 0.172 µB/f.u. and the corresponding change in M is ∼ 2.2 \n% (figure  2(c)).  At 20 K the initial FC m agnetization is almost double of the ZFC magnetization, which  indicates  \nthat the FC magnetization  of EBCO  is close to the equilibrium state, whereas  the ZFC magnetization has very low \nvalue  and is metastable and relaxes fast.  \nTo understand  the metastable nature of the low temperature ZFC magnetization  of EBCO, we have further  \nmeasured the effect of T cycling on M applying H of 3 T and the obtained  results a long with the field cooled \ncooling ( FCC ) and field cooled warming ( FCW ) magnetization curves  are shown in figure  3. For better \nunderstanding, the reference ZFC magnetization of the EBCO obtained under  H of 3 T is also shown in the inset \nof Fig. 3 . In ZFC temperature cycling, initially the sample was cooled  from 380 K to 10 K and at 10 K, the \nmagnetic field of 3 T was applied and M w as recorded while increasing as well as  decreasing the temperature in \nstep of 10 K from 10  K to 120 K. In these measurements, the M value increases  continuously from that of the \nprevious one and above 60 K the ZFC magnetization matches with the FCW magnetization. This observation is \nvery similar to that of the reported one for Gd 5Ge4 [24]. This clearly su ggests that, at low temperature , even at a \nfixed H, the thermal cycling is able to convert some of the supercooled metastable low M phase to equilibrium \nhigher M phase. This de -arrest of the metastable st ate is occurred  over a range of temperature  which  is identified \nas a ‘kinetic arrest band’. Similar result is obtained under the ZFC T cycling at H = 5 T , which implies that the \nobserved metastable nature persists even at high H (see inset of figure  3). The temperature over which this de -\narrest is occurred or the ‘kinetic arrest band’  is shifted  to lower temperature  at larger H.  \n3.3. Magnetic entropy studies  \nTo further investigate the MG property  and the nature of the various magnetic phases , the magnetocaloric (MC) \nmeasurement  was performed on EBCO . Usually, most of the MC stud ies are devoted to understand  the usefulness  5 \n of the materials  for refrigeration applications; however, this technique can also be us ed to probe the H induced \nmagnetic phase transitions. The T variation of isothermal magnetic entropy change (∆S M) has been calculated \nfrom the isothermal M curves using Maxwell’s equation  \n|∆𝑆𝑀|= 𝜇𝑜∫ (𝜕𝑀\n𝜕𝑇)\n𝐻𝐻𝑚𝑎𝑥\n0 𝑑𝐻 \nThe above relation states that the change in entropy of any material is  directly associated with the first derivative \nof M with respect to T, which makes it  more sensitive to probe the small change  in M. In most of the studies, the \nT variation of entropy change is represented with -∆SM vs. T curve, which shows a positive peak in the vicinity of \nthe FM transition and a negative peak at the onset of the AFM ordering.  \n \nThe obtained -∆SM as a function of T for different ΔH  for the EBCO  is shown in figure  4. We have  notice d a \npositive symmetric peak in -∆SM curves at TC and a crossover from positive to negative  in -∆SM curves at TN1, \nwhich is hi ghlighted in the inset of figure  4. While at TN2 the -∆SM curves show  the broad asymmetric negative \npeak. More importantly , the positive to negative crossover  and the position of the maximum in -∆SM curve s at TN1 \nand TN2 are found to be shifted towards the low T with increasing H, which is correlated  with the M (T) behaviour \n[19]. On the other side the peak position of -∆SM curve at TC is insensitive to H. The -∆SM curves show the  \nmaximum value of 0.67 J/kg K ( -0.35 J/kg K) at TC (TN2) for the maximum H of 7 T. Interestingly , -∆SM increase s \nlinearly with H  when T decreases  for T < 60 K and at  10 K it is found to be -0.61 J/kg K for H  = 7 T. The total \nmagnetic entropy change due to the spin only is related to   ∆𝑆𝑀 𝛼 ln(2S+1), where S is the total spin. According \nto this relation one would expect ∆ SM ((IS); S =1) > ∆ SM ((LS); S =0). The obtained ∆ SM values at 7 T are far \nbelow than the theoretical spin only values. However, the observed magnitude of ∆ SM gives the additional \ninformation about the Co3+ spin state transitions . In the vicinity of AFM2 , the observed |∆SM| at 7 T is almost half \nof the FM phase. As discussed in ref . 19 that in  AFM2 phase, the IS states of Co3+ is gradually converted to LS \nstate and the complete  LS state of Co3+ would be expect ed at low  T. At low T one would expect that |∆SM| will \ntend to be  constant (if one considers the contribution of pyramidal Co3+ IS state). On contrary |∆SM| is increase d \nwith H below 60 K , where the magnetic glassy phase is observed. The magnetic entropy studies clearly show that \nthe arrested low T magnetic phase contrib utes significant entropy  change . \n \n4. Discussion  and concluding  remarks  \nThe comprehensive measurements  of M  on EBCO provide  a macroscopic evidence of the kinetic arrest and  the \nformation of magnetic glassy state below 60 K.  As discussed in the introduction , the magneto -structural transitions \nare the major ingredients that drive the FOMPT and cause the kinetic arrest in most of the magnetic glassy \nmaterials.  To understand the effect of structural changes in  various magnetic transitions , we have reordered the T \nvariation of  synchrotron X-ray diffraction (SXR PD) on EBCO sample in the temperature range of  99 - 400 K. \nThe SXR PD patterns match with the other members of RBaCo 2O5+x (x ≈ 0.5) cobaltites  [25–29] in which the  \ncrystal symmetry was found to be invariant, i.e. orthorhombic ( Pmmm ) in the whole investigated T range.  figure s \n5(a & c) show  the SXR PD pattern of selected 2θ range for the 400 K and 99 K respectively. figure  5(b) shows the \ncontour map of intensity variation for the selected 2θ range  with respect to T. The main three intensity  reflections , \n(120), (022) and (102) , corresponding to the Pmmm  space group have  been shifted to the higher angle with 6 \n decrease of T. However, this trend was found to be dramatic in the vicinity of TMIT (350 K -310 K), where \nreflections (120) and (102) are shifted towards  the lower angle  and (022) is shifted to the higher angle.  Figure  5(d) \nshows the T dependence of the lattice constants of EBCO,  where  the lattice parameters show  unusual  variation  in \nthe vicinity of TMIT and HS to IS  transition of Co3+ is observed. With decreasing of  T in the range of  360-300 K , \nboth a & c parameters show negative drops  and at the same T, b parameter show s positive jump. Further decreasing \nof T, such anomalous change is not observed but gradual decrease is found in the lattice parameters. The \nanomalous changes in lattice constants close to TMIT matches with  the results  of the latent heat  measured  by \ndifferential scanning calorimetry (DSC ), which indicates  that the transition is of  first-order type  [19]. This overall \nbehavio r of the present EBCO matches with the other members of this family  where  the first order structural \nchanges are coupled with the metal to insulator transition (TMIT) and also with the spin state transitions of Co3+.  \n \nFurther , the thermal studies on different cobaltites of this family show that the PM –FM transitio n is second order \nin nature and  FM–AFM1 is the fir st order magnetic transition  [30]. Although the  zero field SXR PD studies do not \nshow any first order signatures at FM–AFM1 transition ( TN1=240 K ) in EBCO but  the M ( T) and M (H) studies \nshow that there is significant thermal hysteresis , which confirms  that the TN1 is the first order in nature.  For \ninstance, metamagnetic transi tions are  also observed in the EBCO, and more importantly , the metamagnetic \ntransitions are shifted to the high field with decreasing T and are found to be incomplete at low T up to 7 T [19,31 ]. \nSuch behavior has been observed in a variety of magnetic materials, and in these materials, the kinetic arrest arises  \ndue to the incomple te first -order phase transition  [8,23,32 ].  \n \nCompare d to the existing magnetic glassy materials , the MG state in EBCO seems unusual . As discussed above \nthat the firs t order structural changes seem  almost decoupled with the magnetic transitions and more importantly \nthe magnetic active ion i.e. Co3+ shows T variation of the spin state transitions. In EBCO, at TMIT the octahedral \nCo3+ shows HS to IS state transition and with decreasing T the IS of Co3+ is gradually transformed into LS state \n[19]. The muon -spin relaxation measurements on the RBaCo 2O5.5 cobaltites showed that the AFM1 and AFM2 \nphases have different types of spin s tate order (SSO)  [33]. The AFM1 state  is having two types of SSO, while \nAFM2 phase is having up to four different types of SSO  structures . In both AFM1 and A FM2 phases , the IS Co3+ \nions on pyramidal sites show AFM coupling, while the neighboring octahedral Co3+ holds either IS or LS state. \nAlthough  the AFM1 and AFM2 phases having the different types of the SSO structures , but the self -energies of \nthese SSO structures are almost same and cause s the microscopic phase separation at FM -AFM 1 transitions  [33]. \nFurther, it was suggested that these two AFM phases were  independent of each other and develop ed as a  well \nseparated phases with decreasing  T. This  has been further confirmed experimentally in our previous studies of \nEBCO cobaltite  through the M ( T) and M (H)  studies  [19].  \n \nHere, we further argue that the identification of  non-equilibrium magnetic phase  at low T in the EBCO is very \ndifficult ; because  the octahedral  HS Co3+ (S=2) sites  are transformed into the diamagnetic LS Co3+ (S = 0) with \nthe decreasing  T, and the pyramidal IS Co3+ sites have an AFM coupling. One would expect a very low magnetic \nmoment at low T. As shown in figure  2 (a), the change of t he magnetic moment between 60 -10 K in the arrested  \nT region is very small,  i.e., 0.2 μ B/f.u. In most of the reported magnetic glassy materials, the higher T magnetic \nphase is generally arrested and shows a non -equilibrium glassy signature at low T. Ideally, in EBCO, the high T 7 \n magnetic phase, i.e. ferri  or ferromagnetic phase has to be kinetically arrested at low T in the background of the \nAFM phase and would expect  to show a non -equilibrium glassy signature. In contrast , the obtained kinetic arrest \nband (figure  2(d)) at different magnetic fields matches  with the ideal FM ground states  (the high T magnetic phase \nis AFM)  [34]. Further, the small kinetically arrested magnetic moment at low T may be due to several factors, \nsuch as partially arrested high T ferro or ferrimagnetic phase, or incomplete first order metamagnetic phases or \nfrom one of the arrested SSO ordered AFM phase. These signatures clearly indicate  the complex behavior  of the \narrested phase.  \n \nFinally, it is worth to  compare the MG state of the present EBCO with the other available reports of this cobaltite \nfamily. The MG state is  reported in the YBaCo 2O5+δ (δ = 0.50) cobaltite with 5% Ca substitution at Y and Ba sites  \n[10] and also in the electron doped YBaCo 2O5.5-δ (δ = 0.13)  [18] cobaltite. Although the MG state i s observed  for \nall samples  at low T, but the high T magnetic state is very different in the case of EBCO as compared to the both \nY cobaltites. In Y cobaltites, the AFM2 phase, field induced multip le metamagnetic transitions and the signatures \nof octahedral  Co3+ IS to LS state transition are not observed. The evaluation of phase separation between the FM \nand AFM clusters in the  doped Y cobaltites causes the MG state at low T. It is noted that the phase separation in \nthe case of undoped YBaCo 2O5+δ (δ = 0.50)  is minimal  and sufficiently significant in case of EBaCo 2O5+δ (δ = \n0.47)  [19]. This difference  clearly indicates  that the microscopic origin of MG state at low T in the present EBCO \nis different from the doped Y cobaltites, where the intrinsic magnetic phase separation in EBCO is coupled  with \nthe octahedral Co3+ spin state transitions.  \n \nIn conclusions , we have firmly established the formation of the magnetic glassy state in EuBaCo 2O5+δ (δ=0.47) \ncobaltite  through  various magnetization measurements . The nonequilibrium magnetic glassy state is observed at \nthe low spin state of Co3+, where the arrested phase has very low magnetization. The isothermal magnetic entropy \nstudies also show  the large entropy change at the  kinetically arrested phase. Though t his study  demonstrates a \npossible mutual coupling among the magnetic phase se paration, spin state transition  and the magnetic glassy state, \nbut the microscopic origin of the kinetic arrest that has  led to the glassy dynamics in EBCO is still lacking  and \nchallenging. Further, the observed MG state in this cobaltite family is generic, not limited to the above studied \nmaterials but also can be extended to various chemical substitutions  as well as for the nanostructured thin films. \nThe s tructural and magnetic imaging techniques in the presence of magnetic field may play a key role in \nunderstanding the microscopic relationship between the kinetic arrest and the supercooling phase [35–38]. Such \nstudies should be initiated to a large extent to  investigate the microscopic relationship between  kinetic arrest  and \nthe spin state transition driven magnetic phase separation in these cobaltites.  \n \n \nAcknowledgements  \nThe authors of IIT Kharagpur acknowledge  IIT Kharagpur  funded VSM SQUID magnetometer at central research \nfacility and cryogenic physical property measurement system at department of physics. 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Matter 25 496011  \n[37]    Pallavi Kushwaha, Archana Lakhani, R. Rawat, A. B anerjee, and P. Chaddah  2009  Phys. Rev. B  79 132402             \n[38]    X. F. Miao, Y. Mitsui, A. Iulian Dugulan, L. Caron, N. V. Thang, P. Manuel, K. Koyama, K. Takahashi,  \n           N. H. van Dijk, and E. Bruck 2016 Phys. Rev. B 94  094426  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 10 \n Figure captions:  \nFigure  1. (Color online) In -phase ac -magnetization measured at different frequencies as a funct ion of temperature  \nfor EBCO sample . Insets show the different magnetic transitions and memory effect.  \n \nFigure  2. (Color online) Temperature dependent magnetization obtained  for EBCO sample  (a) during warming \nafter being cooled in different magnetic fields and (b) in ZFC mode under various applied fields. (c) Magnetization \nversus  time plot for EBCO at 20 K along ZFC and FC path in the presence of H = 2 T. (d)  Schematics of kinetic \narrest band diagram as the  corresponding ZFC magnetization curves for the case with high magnetization ground \nstate.  \n \nFigure  3. (Color online) Magnetization versus temperature plots obtained in ZFC, FCC, and FCW modes in an \napplied field of 30 kOe  for EBCO sample . In the ZFC mode the sample is subjected to thermal cycling at various \ntemperatures and the detailed measured protocol is described in the corresponding text. The top inset shows the \nZFC mode thermal cycling of magnetization  in an applied field of 5 0 kOe . The lower inset shows the ZFC \nmagnetization without thermal cycling, i.e. normal ZFC magnetization.  \n \nFigure  4. (Color online) Plot of the change in magnetic entropy ( -∆SM) for 1 T field interval as a function of \ntemperature  for EBCO sample calculated using the thermodynamic Maxwell relation. Insets show the expanded \nview of -∆SM around the TN1 (240 K).  \n \nFigure  5. (Color online)  Selected portion of the SXRPD  patterns (2  = 12.87 °-13.24 °) at (a) 400 K and (c)  99 K \nrespectively. (b)  Contour map of  three main reflection s for the selected 2  (= 12.87 °-13.24 °) for  the temperature  \nrange of 400 - 99 K ; the intensity variation is denoted  by the colour profile and the reflections  are indexed on the \nbasis of  orthorhombic ( Pmmm , ap x 2a p x 2ap) lattice . (d) Temperature dependence of lattice constants of \nEuBaCo 2O5.47. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 11 \n Figures:  \n \n \n \n \n \n \n \n \n \n \n \n \nFigure  1. \n \n     \n \n \n  \n \n \n \n \n \n \nFigure  2. \n12 \n  \n \n \n \n \n \n \n \n \n \n \n \n                   Figure  3. \n \n \n \n \n \n \n \n \n \n \n \n \n     \n          \n                 Figure  4.    \n \n13 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure  5. \n"    },    {        "title": "1911.03470v1.Magnetic_i_MXene__a_new_class_of_multifunctional_two_dimensional_materials.pdf",        "content": "arXiv:1911.03470v1  [cond-mat.mes-hall]  8 Nov 2019Magnetic i-MXene: a new class of multifunctional two-dimen sional materials\nQiang Gao and Hongbin Zhang∗\nInstitute of Materials Science, Technische Universit ¨at Darmstadt, 64287 Darmstadt, Germany\n(Dated: November 12, 2019)\nBased on density functional theory calculations, we invest igated the two-dimensional in-plane\nordered MXene (i-MXenes), focusing particularly on the mag netic properties. It is observed that\nrobust two-dimensional magnetism can be achieved by alloyi ng nonmagnetic MXene with magnetic\ntransition metal atoms. Moreover, both the magnetic ground states and the magnetocrystalline\nanisotropy energy of the i-MXenes can be effectively manipul ated by strain, indicating strong piezo-\nmagnetic effect. Further studies on the transport propertie s reveal that i-MXenes provide an inter-\nesting playground to realize large thermoelectric respons e, antiferromagnetic topological insulator,\nandspin-gapless semiconductors. Thus, i-MXenesare anewc lass ofmultifunctional two-dimensional\nmagnetic materials which are promising for future spintron ic applications.\nI. INTRODUCTION\nThe MAX compounds with a general chemical for-\nmula of M n+1AXn(M: early transition metal; A: main\ngroup element; X: C or N; n: integer up to 6)\nhave drawn intensive attention recently, due to poten-\ntial applications as structural, electrical, and tribologi-\ncal materials.1The corresponding two-dimensional (2D)\nnanosheets M n+1Xn, dubbed as MXene, can be ob-\ntained by etching the A elements away, e.g., Ti3C2from\nTi3AlC2.2This defines a new class of 2D materials be-\nyond graphene,3with a number of materials reported\nshowing multi-functionality covering energy, spintronic,\nnano-electronic, and topological applications.4,5For in-\nstance, theoretical calculations predicted that Ti 3C2is\na high performance anode material for lithium ion bat-\nteries,6which was confirmed later by experiments.7Al-\nthough most MXenes are metallic due to the partially\noccupied d-shells of the transition metal atoms, function-\nalization can be applied to open up a finite band gap\nthus further tailor their properties. For instance, passi-\nvated MXene Mo 2MC2(M = Ti, Zr and Hf) with the O 2\ngroup are found to be robust quantum spin Hall insula-\ntors with band gaps from 0.1 eV to 0.2 eV.8Particularly,\nthe magnetic properties of MXenes deserve further inves-\ntigation, driven by the discovery of 2D magnetic mono-\nlayers such as CrI 3and Cr 2Ge2Te6.9–11Gaoet al.12pre-\ndicted that Ti 2C and Ti 2N are nearly half-metals, which\ncan be tuned into spin-gapless semiconductor (SGS) un-\nder biaxial strain. Theoretical calculations also demon-\nstrated that Mn 2N with functional groups O, OH and F\ncan be half-metals with high Curie temperatures.13How-\never, high-throughput density functional theory (DFT)\ncalculationsrevealthat it isdifficult to obtainMAXcom-\npounds with Fe, Co, and Ni,14,15leading to a challenge\nto realize magnetic MXenes.\nRecently, the in-plane ordered MAX (i-MAX) com-\npounds with aformulaof(M 2/3M′\n1/3)2AXhavebeen syn-\nthesized by substituting 1 /3 foreign transition metal or\nrare earth element M′for M in M 2AX compounds.16,17\nBased on theoretical studies for the i-MAX compounds\n(Mo2/3M′\n1/3)2AC (M′= Sc, Y; A = Al, Ga, In, Si,Ge,In), it was found that the stable conditions for an i-\nMAX are: (1) the significant difference between the\natomic radii for the dopant metal M′and parent M; (2)\nsmall A atoms.18For the synthesized Cr-bases i-MAX\ncompounds, DFT calculations suggest (Cr 2/3Zr1/3)2AlC\nis stable in an anti-ferromagnetic spin configuration,19\nwhile (Cr 2/3M′\n1/3)2AlC (M′= Sc and Y) are probably\nstable in a mixed state due to small energy difference be-\ntweentheanti-ferromagneticandferromagneticphases.20\nThis suggests that various magnetic states are possible\nin the i-MAX compounds, and thus interesting for the\ncorresponding i-MXenes in the monolayer limit. On the\nother hand, the i-MXenes can also be obtained from i-\nMAX by etching away the A-element atoms, such as the\nordered i-MXene W 4/3C and Nb 4/3C.21,22\nIn this work, we performed systematic density func-\ntionaltheorycalculationstoinvestigatethe magneticand\nelectronic properties of i-MXenes with a general chemi-\ncal formula (M 2/3M′\n1/3)2X, focusing particularly on the\ncases where M′is magnetic. It is observed that robust\nmagnetism can be induced, and there exists significant\nmagneto-structural coupling, leading to tunable mag-\nnetic ground state and magnetic anisotropy by strain.\nMoreover, our calculations suggest that the i-MXenes\nhost fascinating transport properties, such as large See-\nbeck effect, antiferromagnetic topological insulators, and\nspin-gapless semiconductors.\nII. COMPUTATIONAL METHOD\nTo maintain reasonable computational effort, we con-\nsidered11knownnonmagneticMXeneastheparentcom-\npounds, namely, Sc 2C, V2C, Mo 2C, Nb 2C, Ta2C, Ti2C,\nZr2C, Hf2C, Ti2N, Zr2N, and Hf 2N.23The dopant ele-\nment M′is chosen to be one of the transition metal el-\nements except Tc, resulting in 319 compounds with the\ngeneral chemical formula (M 2/3M′1/3)2X. As sketched in\nFig. 1, the symmetry for the hexagonal MXenes will be\nlowered after substituting the dopants, leading to a rect-\nangular structure. In our calculations, both the hexago-\nnal and rectangular structures are considered in order to2\nunderstand the effect of strain. To determine the mag-\nnetic ground state, we consider non-magnetic (NM), fer-\nromagnetic (FM), interlayer antiferromagnetic (AFM),\nand intralayer AFM configurations for the magnetic M′\nsublattice (Fig. S2 in the Supplementary material).\nOur DFT calculations are performed in an automated\nway using the in-house developed high-throughput en-\nvironment,24–27which is interfaced to Vienna ab initio\nSimulation Package (VASP)28,29and full-potential local-\norbital minimum-basis code (FPLO).30,31The exchange-\ncorrelationfunctional in the generalized gradientapprox-\nimation (GGA) is applied, as parameterized by Perdew,\nBurke, and Ernzerhof (PBE).32To guarantee good con-\nvergence, the plane-wave energy cutoff and k-mesh den-\nsity are set as 500 eV and 60 ˚A−1, respectively. The\ncalculations to obtain optimized structures and vari-\nous magnetic configurations are carried out using VASP,\nwhile the electronic structure and physical properties are\nobtained using the FPLO and WIEN2k33codes, as de-\ntailed in our previous work.24,25,34\nFIG. 1: [a] Top view for i-MXene (M 2/3M′\n1/3)2X in both\nhexagonal lattice (red) and rectangular lattice (green). S ide\nviews for i-MXene in hexagonal [b] and rectangular [c] lat-\ntices. The ”-A” and ”-B” denote the atom above or below\nthe central layer of B in i-MXene. The original pure MXene\ncrystallizes in a hexagonal lattice. After inducing 1/3 for eign\ntransition metal dopant, the hexagonal symmetry is broken\nand tilted to the rectangular lattice.\nIII. RESULTS AND DISCUSSIONS\nAs summarized in Fig. 2, among the 319 i-MXenes\nin the rectangular geometry, 257 compounds are non-\nmagnetic, and the remaining 62 cases are magnetic (with\ntotal magnetic moments greater than 0.2 µB/f.u. in the\nferromagnetic spin configuration), where 36 (26) being\nwith ferromagnetic (antiferromagnetic) ground states.\nFurthermore, for the compounds in the imposed hexag-\nonal geometry, the magnetic M′sublattice forms a tri-\nangular lattice, which is frustrated and thus can lead\nto in-plane noncollinear magnetic structure. We found\nthat there are 13 i-MXenes with AFM intralayer ex-\nchange coupling between the moments on the M′sites(cf. TableS1inSupplementarymaterial), indicatingpos-\nsible 2D noncollinear magnetic states. Such compounds\nwill be saved for detailed investigation in the future and\nwithin the current work we focus only on the collinear\nmagnetic configurations.\nFIG. 2: Classification of the magnetic ground states for\nthe 319 i-MXene, where ”NM to FM” and ”AFM to FM”\nmark the compounds whose ground state changes by impos-\ning hexagonal geometry.\nInterestingly, comparing the magnetic ground states\nfor i-MXenes in the rectangular and hexagonal geome-\ntries, there are 6 compounds whose magnetic configura-\ntions can be changed (Fig. 2), e.g., 3 NM cases become\nFM, and 3 AFM compounds change to FM. For instance,\n(Zr2/3Ti1/3)2N changes from NM (in rectangular lattice)\nto FM (in hexagonal lattice) states with a total magnetic\nmoment of 0.95 µB/f.u.. The same NM →FM transi-\ntion occurs for (Ti 2/3Ru1/3)2C and (Zr 2/3Cu1/3)2C, with\nmagnetic moments of 0.29 and 0.30 µB/f.u., respectively.\nThe underlying mechanism can be understood based on\nthedensityofstates(DOS) ofthe Zr-5dorbitalsasshown\nin Fig. S4 in the in the Supplementary material. Based\non the Stoner model, the criterion for a stable ferromag-\nnet is\nUN(EF)>1 (1)\nwhere U is the exchange integral and N(EF) is the\nDOS at Fermi level in the paramagnetic state. For the\nrectangular lattice, the Fermi level is located around a\npeak of DOS. In the imposed hexagonal structure, the\nDOS at Fermi level is significantly enhanced from 5.2\nstates/eV/u.c. to 6.0 states/eV/u.c. (Fig. S4 in the in\nthe Supplementary material), resulting in the instability\nof becoming ferromagnetic. The same behavior is ob-\nserved also for (Ti 2/3Ru1/3)2C and (Zr 2/3Ru1/3)2C (cf.\nFig. S4 in the Supplementary material).\nFor the compounds with the AFM →FM transition,\nsuch as (Hf 2/3Fe1/3)2C and (Ti 2/3V1/3)2C, the magnetic\nmoments of the Fe (V) atoms change from 1.09 (0.58) µB3\nin the rectangular lattice to as large as 1.79 (0.64) µBin\nthe hexagonal lattice, respectively. For (Hf 2/3Fe1/3)2C,\nthe magnetic moments of Fe atoms areenhanced by 64%,\nwhich can be attributed to the enhanced exchange split-\nting (cf. Fig. S5 in the Supplementary). It also turns out\nto be a gapless semiconductor, which will be discussed\nin detail later. On the other hand, we have shown the\nband structures for (Ti 2/3V1/3)2C in Fig. S6 in the in the\nSupplementary material in both rectangular (Rect.) and\nhexagonal (Hex.) lattices together with the anomalous\nhall conductivity (AHC) for (Ti 2/3V1/3)2C in the hexag-\nonal lattice. In the rectangular lattice, (Ti 2/3V1/3)2C is\nAFM with zero AHC. Obviously, for the FM state in the\nhexagonal lattice there exists a finite AHC of -160 S/cm\nat the Fermi level. In this sense, strain can be applied to\ntune the topological transport properties, leading to the\npiezospintronic effect as discussed in the bulk materials\nas well.35,36\nItisnotedthatthelatticedeformationfromtherectan-\ngular to the hexagonal lattices is of marginal magnitude.\nFor instance, the strains along the a and b directions for\n(Ti2/3Fe1/3)2C are just about 0.14% and 0.18%, where\nthe energy difference between rectangular and hexago-\nnal lattices is as small as 1.2 meV/atom. Thus, it is\nstraighforwardtotailorthe i-MXenesfromlowsymmetry\nrectangular lattice to the hexagonal lattice with higher\nsymmetry. On the other hand, it has been previously\nreported that the piezomagnetic effect can be realized in\npure MXene M 2C (M = Hf, Nb, Sc, Ta, Ti, V, and Zr)\nby applying biaxial strain.37In this regard, we suspect\nthat the piezomagnetic effect is dramaticallyenhanced in\ni-MXene as manifested by the five aforementioned cases.\nAs the hexagonal lattice with high symmetry is more in-\nterestingand easyto obtain, wewill focus onthe physical\nproperties of such systems in the remaining part of this\nwork.\nFor two-dimensional magnets, according to the\nMermin-Wagner theorem, there exists no long range or-\ndering if there is continuous symmetry for the order pa-\nrameters.38In this regard, to stabilize 2D magnets at fi-\nnite temperature, magnetocrystalline anisotropy energy\n(MAE) is essential, which breaks the rotational symme-\ntry of Heisenberg moments.39The MAEs for the mag-\nnetic i-MXenes are evaluated using the force theorem:40\nMAE =/summationdisplay\ni∈occ.(ǫ[100]\ni−ǫ[001]\ni) (2)\nwhereǫ[001]\niandǫ[100]\nidenote the energy eigenvalues of\nthei−thbandforthe magnetizationalong[001]and[100]\ndirections, respectively.\nTable I lists the 5 i-MXenes with a MAE larger than\n0.5meV/f.u. amongallthe magnetici-MXenes(TableS2\nin Supplementary). It is noticed that (Hf 2/3Fe1/3)2C\nhas the largest MAE of 1.39 meV/f.u., favoring out-of-\nplane magnetization direction. In addition, the MAEs\nof (Zr2/3Fe1/3)2C and (Ti 2/3Fe1/3)2C are 0.74 meV/f.u.\nand 0.03 meV/f.u., respectively. Such a trend of increas-\ning MAE as X varying from Ti, Zr and Hf can be at-tributed to the variation of the strength of the atomic\nspin-orbit coupling (SOC) as 11.2 meV (Ti), 42.1 meV\n(Zr)and126.5meV(Hf).41,42Furthermore,theenhanced\nMAE of (Zr 2/3Fe1/3)2C and (Hf 2/3Fe1/3)2C is originated\nfrom the trigonal crystal fields which lead to strongly\nSOC coupled bands around the Fermi energy, as mani-\nfested by the orbital projected band structures (Fig. S9\nin the supplementary material). As shown in the or-\nbital projected band structures (Fig. S9 in the supple-\nmentary material), for (Hf 2/3Fe1/3)2C, the Fe-3d xyand\nFe-3dx2−y2bands are split by 73 meV along the K-Γ\nsection. While for (Zr 2/3Fe1/3)2C, the Fe-3d xyand Fe-\n3dx2−y2bandsaresplitby40meValongtheM-Ksection.\nAs the d xyand dx2−y2orbitals are strongly coupled by\nSOC,suchaspecial electronicstructureleadsto the large\nMAEs. Such a mechanism is in good correspondence to\nthe giant MAE realized in artificial Fe atoms adsorbed\non III-V nitride thin films with local trigonal symmetry\nas well.43\nTo obtain the Curie temperature, we took\n(Hf2/3Fe1/3)2C as an example and evaluated the\nexchange parameters between magnetic Fe atoms by\nmapping the DFT total energies to the Heisenberg\nmodel:\nH=−1\n2/summationdisplay\ni/negationslash=jJijSi·Sj (3)\nwhereJijis the exchange parameter for the local mo-\nments on the iandjsites,Si/jmarksthe on-sitespin op-\nerator. Considering three magnetic configurations, i.e.,\nFM, AFM- α, and AFM- β(cf. Fig. S2 for spin configu-\nrations AFM- βand AFM- α), the energy differences can\nbe formulated in terms of the interlayer ( Jinter) and in-\ntralayer ( Jintra) exchange parameters (cf. Fig. S3 in the\nsupplementary material for JintraandJinter):\nEFM−EAFM−α=−JinterS2, (4)\nEFM−EAFM−β=−8JintraS2. (5)\nHere we use the square of local spin moment length\nto represent the square of the spin operator. The re-\nsulting exchange coupling parameters for (Hf 2/3Fe1/3)2C\nare: J inter= 33.94 meV and J intra= 0.70 meV. Simi-\nlarly, we obtained the exchange coupling parameters for\nthe other ferromagnetic cases such as (Hf 2/3Cr1/3)2N,\n(Ta2/3Fe1/3)2C,(Ti2/3Hf1/3)2N,and(Zr 2/3Fe1/3)2C(Ta-\nble I). Obviously, the value of interlayer exchange cou-\npling is much larger than that of the intralayer coupling\nfor all the listed compounds. Due to the dramatic large\nmagnetic anisotropy, we can use the 2D Ising model to\nestimate the Curie temperature44,45\nTC=2Jinter\nkBln(1+√\n2). (6)\nwith the results listed in Table I. Surprisingly, the Curie\ntemperatures for (Hf 2/3Fe1/3)2C and (Hf 2/3Cr1/3)2N are4\nTABLE I: The basic information of the magnetic i-MXene candi dates in hexagonal lattice with an out-of-plane MAE larger\nthan 0.5 meV/f.u., including the MAE in unit of meV/f.u., the magnetic moment per magnetic atom in unit of µB, magnetic\norder, the exchange coupling parameters and the main magnet ic atom.\nCompound MAE Magnetic Magnetic Magnetic JinterJintraTC\nmeV/f.u. Moment order atom meV meV K\n(Ta2/3Fe1/3)2C 0.86 1.82 AFM Fe -10.05 0.56 -\n(Zr2/3Fe1/3)2C 0.74 1.71 FM Fe 10.16 -2.24 267.54\n(Hf2/3Fe1/3)2C 1.39 1.79 FM Fe 33.94 0.70 893.67\n(Hf2/3Cr1/3)2N 0.76 1.01 FM Cr 13.06 0.20 343.90\n(Ti2/3Hf1/3)2N 0.71 0.30 FM Ti 7.22 0.97 190.11\neven above the room temperature. It is noted that\nfor the recently synthesized 2D magnet CrI 3with a\nCurie temperature of 45 K, its MAE is about 1.71\nmeV/f.u.9,10,39,46Obviously, the out-of-plane MAEs of\n(Hf2/3Fe1/3)2C and (Hf 2/3Cr1/3)2N are almost the same\nas that of CrI 3. Futhermore, for CrI 3the interlayer and\nintralayer exchange parameters are 11.64 meV and 2.37\nmeV.47So, the exchange parameters for (Hf 2/3Fe1/3)2C\nand (Hf 2/3Cr1/3)2N are larger than that of CrI 3. In this\npoint of view, we suspect the i-MXene (Hf 2/3Fe1/3)2C\nand (Hf 2/3Cr1/3)2N being promising 2D magnets with\nhigh Curie temperature.\nIV. ELECTRONIC PROPERTIES\nA. Thermoelectric properties of semiconductors\n[a]\n[b]\nFIG. 3: Band structures and Seebeck coefficients for\n(Sc2/3Cd1/3)2C [a] and (Sc 2/3Hg1/3)2C [b]. The horizontal\ndashed lines denote the Fermi level.\nPreviously, it has been reported that the 2D semicon-ductorshavevery largethermoelectric power.48,49Forin-\nstance, SnSe monolayers are regarded as promising ther-\nmoelectric material, e.g., the Seebeck coefficient is in-\ncreased from 160 µV/K to 300 µV/K when temperature\nis increased from 300 K to 700 K.48Although most i-\nMXenestendtobemetallic, thereexistninenonmagnetic\nsemiconductors i.e., (Sc2/3X1/3)2C (X = Au, Cu, Ir, Ni,\nZn, Cd, Hg), (Hf 2/3Ir1/3)2C and (Ti 2/3Au1/3)2C. Among\nthem, (Sc 2/3Cd1/3)2C and (Sc 2/3Hg1/3)2C are large gap\nsemiconductors (with finite band gap larger than 0.4 eV\nand less than 1.0 eV), which can be good candidate ther-\nmoelectric materials. The band structures of these two\nsemiconductors in the hexagonal lattice are displayed in\nFig. 3 togetherwith the Seebeck coefficients as a function\nof chemical potential. The band gaps of (Sc 2/3Cd1/3)2C\nand (Sc 2/3Hg1/3)2C are as large as 0.41 and 0.79 eV,\nrespectively. Furthermore, the valence band maximum\n(VBM) and conduction band minimum (CBM) are flat\naround the Fermi level. Such a behavior implies that\nboth compounds have the potential to realize large See-\nbeck effect. To confirmthis, the Seebeck coefficientshave\nbeen evaluated for such two compounds, as shown in\nFig. 3. The peak of Seebeck coefficients around Fermi\nlevel for (Sc 2/3Cd1/3)2C ((Sc 2/3Hg1/3)2C) are remark-\nably as high as 2100 (3000) µV/K at 100 K, 820 (1200)\nµV/K at 300 K, and 520 (700) µV/K at 500 K. The en-\nhanced Seebeckcoefficients can be attributed to the large\nderivative of the DOS with respect to the energy and\nhence the flat bands above and below the Fermi energy.\nIt isnotedthat the densityofstatesforthesetwosystems\nare very comparable for the hexagonal and rectangular\nlattices, leadingto negligiblechangesin the Seebeckcoef-\nficients(cf. Fig.S7andS8inSupplementary). Therefore,\nwe suspect that (Sc 2/3Cd1/3)2C and (Sc 2/3Hg1/3)2C are\ngood candidates as 2D thermoelectric materials.\nB. Topological properties\nAs discussed above, most i-MXenes are metallic due\nto the partially filled d-shells of the transition metal\natoms. However, motivated by the reported nontriv-\nial topological state in some functionalized MXenes,8\nwe studied also the topological nature of the i-MXene\n(Ta2/3Fe1/3)2C with a tiny band gap of 1.34 meV. Sur-\nprisingly, (Ta 2/3Fe1/3)2C with a collinear AFM ground5\n[a]\n[b]\nFIG. 4: Band structures [a] and edge states [b] for AFM topo-\nlogical insulator (Ta 2/3Fe1/3)2C. When SOC is turned off, the\nband structures of spin up channel overlap with that of the\nspin down channel. So we only show band structure in one\nspin channel. In both [a] and [b], the dashed line denote the\nFermi level.\nstate hosts a nontrivial state, as shown in the bulk and\nsurface electronic structures in Fig. 4. Without consid-\nering spin-orbit coupling (SOC), the VBM and CBM al-\nmost cross each other at the Fermi energy. According to\nthefatbandanalysis(Fig.S9andS10inSupplementary),\nthese two bands are mainly contributed by the Fe-3d yz\nand Fe-3d zxorbital character, which is strongly coupled\nby SOC. Turning on SOC, a remarkable indirect gap of\n75.0 meV is opened, leading to band inversion and thus\nthe occurrence of the topological nontrivial state. This is\nclearly confirmed by our explicit calculation of the edge\nstatesofthe corresponding1Dribbons with a width of70\nunits, as shown in Fig. 4(b). Furthermore, the nontrivial\nedge state is protected by the combination of the time\nreversal ( T) and space inversion ( P) symmetries ( T P),\nas demonstrated for CuMnAs in Ref. [50].\nC. Spin-gapless Semoconductors\nIt is reported that MXene Ti 2C can be a spin-gapless\nsemiconductor (SGS) under 2% strain.12We found two\nSGS candidates among i-MXenes in the hexagonal geom-\netry,e.g., (Hf2/3Fe1/3)2C and (Zr 2/3Fe1/3)2C with total\nmagnetic moments of 1.79 and 0.74 µB/f.u., respectively.\nSuch two SGSs are not within the classification of four[a]\n[b]\nFIG. 5: The band structures (left) and anomalous Hall con-\nductivity (right) for (Hf 2/3Fe1/3)2C [a] and (Zr 2/3Fe1/3)2C\n[b]. The horizontal dashed lines denote the Fermi energies.\ntypes SGSs defined in our previous work,34because the\nVBM and CBM have both spin characters as shown in\nFig. 5. For such tow SGSs at Fermi level, in the spin\nup channel the CBM and VBM touches each other di-\nrectly between K−Γ (in A region), while in the spin\ndown channel there is only a small gap between for the\nCBM and VBM between K−M(in region B). That is\nthe VBM and CBM of spin up and down channels can\nbe roughly seen at the same energy level. When SOC is\nswitched on, the touching bands open a local band gap,\nsuggesting a topologically nontrivial state. This can be\nmanifested by explicit evaluation of the anomalous Hall\nconductivity. Forinstance, itisobservedthatthe anoma-\nlous Hall conductivity is finite around the Fermi energy,\nwhich is almost quantized to 3 and 5 e2/hwith tiny band\ngaps of 3.8 meV and 0.3 meV for (Hf 2/3Fe1/3)2C and\n(Zr2/3Fe1/3)2C, respectively. That is, such i-MXenes are\nChern insulators with a nontrivial Chern number of 3\nand 5, though the band gaps are small.\nV. CONCLUSION\nIn conclusion, we have done a systematic study on\nthe magnetic and electronic properties of i-MXene com-\npounds, whichprovideaninterestingplaygroundformul-\ntifunctional applications. The magnetic ground states\nfor i-MXene are investigated for both rectangular and\nhexagonal lattices, where slight strain can be applied to\ntune the magnetic ground state. Due to the underly-\ning crystal fields, the magnetocrystalline anisotropy of i-\nMXenecanbesignificantlyenhanced, i.e., thereare7sys-6\ntems with amagneticcrystallineanisotropyenergylarger\nthan 0.5 meV/f.u.. Furthermore, investigation on the\nelectronic properties reveals that i-MXene can host fas-\ncinating transport properties, including significant ther-\nmoelectric effects, antiferromagnetic topological insula-\ntor state, and spin gapless semiconductors. Our calcu-\nlations suggest that i-MXene is a class of 2D materials\nwhich are promising for future applications, calling for\nfurther experimental exploration.ACKNOWLEDGMENTS\nQiangGaothanksthefinancialsupportfromtheChina\nScholarshipCouncil. 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It resembles  more like an eight figure or else joint double quantum -dots \ninstead of the classical, spherical more  uniform field of a bar magnet. T his actual subatomic quantum magnetic field of an \nelectron at rest,  is called Quantum Magnet or else a Magneton. It is today verified experimentally by qu antum magnetic \nfield imaging methods  and sensors like SQUID scanning magnetic microscopy  of ferromangets  and also seen in Bose -\nEinstein Condensat es (BEC) quantum ferrrofluids experiments.  Normally, a macroscale bar magnet should behave like a \nrelative giant Quantum Magnet with identical magnetic dipole field  imprint since all of its individua l magnetons  \ncollectively  inside the material, dipole moments are uniformly aligned forming the total net field of the magnet. However \ndue to Quantum Decoherence (QDE) phenomenon at the macroscale and macroscopic  magnetic field imaging sensors \nlimitations  which cannot  pickup these rapid quantum magnetization fluctuations, this field is masked and not visible at \nthe macroscale.  By using the relative inexpensive submicron resolution Ferrolens quantum magnetic  optical \nsuperparam agnetic thin film sensor for field real time imaging  and method we have researched in our previous \npublications, we can actually make this net magneton field visible on macroscale magnets. We call this net total field \nherein, Quantum Field of Magnet (QFM) differentiating  it  therefore  from the field of the  single subatomic magneton  thus \nquantum magnet.  Additionally, the unique potential of the Ferrolens device to display also the magnetic flux lines of this \nmacroscopically projected giant Magenton gives us the opportunity  for the firs t time to study the individual magnetic flux \nlines geometrical pattern that o f a single subatomic magneton. W e describe this particular magnetic flux of the magneton \nobserved, qua ntum magnetic flux.  Therefore a n astonishing novel observation has been made  that the Quantum  \nMagnetic Field of the Magnet -Magneton (QFM) consists of a dipole vortex shaped magnetic flux geometrical pattern \nresponsible for creating the classical macroscopic N -S field of magnetism as a tension field between the two polar \nquantum fl ux vortices North and South poles. A physical interpretation  of this quantum decoherence mechanism \nobserved is analyzed  and presented and conclusions made showing physical evidence of the quantum origin irrotational \nand therefore conservative property of m agnetism and also demonstrating that ultimately magnetism at the quantum \nlevel is an energy dipole vortex phenomenon.  ©2019  The Authors. Published by Elsevier B.V. This is an open access article \nunder the CC BY -NC-ND license ( http://creativecommons.org/licenses/by -nc-nd/4.0/ ). \n \nKeywords:  quantum decoherence; quantum magnet; magneton ; ferromagnets; quantum optics; magneto optics; \nsuperparamagnetic thin films; ferrolens; electromagnetism;fundamentals physics ;conservative fields.  \n \n http://dx.doi.org/10.1016/j.rinp.2019.102793  \n1. INTRODUCTION  \nA novel magnetic field physical sensor, quantum magnetic optic device is used \nto demonstrate and validate for the first time a correlation between the quantum \nnet vortex field existing on macroscopic ferromagnets d escribed in our previous \nwork [1] and the familiar classical N-S macroscop ic magnetic field  imprint \nobtained, showing that macroscopic magnetism to be a Quantum Decoherence \n(QDE) effect. Therefore providing us a potentially important link between the \nquantum and macrocosm and possible enhancing our understanding over our \nphysica l world and magnetism in general. Similar quantum field images were \npreviously reported independently in Bose -Einstein condensate quantum \nferrofluids [21]  (i.e. ferrofluid close to absolute zero temperature)  and SQUID \nmagnetic microscopy of ferromagnets [2 5], but never a valid correlation and \nobservation was made with macroscopic magnetism and describing an existing \nquantum decoherence mechanism responsible for the transition from the \nquantum net magnetic field , thus quantun magnet  to the macroscopic  classical .   \nThis thin film magnetic optic lens can detect and display in real time \nthis otherwise unobservable quantum field with conventional \nmacroscopic sensors. A condensed soft quantum matter thin film, magnetic \nfield sensor like the ferrolens, gives  us a holographic imprint of the actual \ncausality  quantum magnet field  of permanent magnets  that more condensed \nmacroscopic solid matter  sensors are not able to show due to quantum \ndecoherence phenomena  limitations . This is not due an intrinsic \nproperty o f the thin film ferrolens as we have established  in our \nprevious two publications referenced in this article but exclusively externally induced by the magnetic field under observation. This net effect quantum vortex \nfield we experimentally discovered in e very macro dipole magnet, is potentially  \nthe cause and responsible for the creation of its macroscopic classical axial field \nimprint we are all familiar with shown for example by macro field sensors like \nthe iron filings experiment. The same or similar to the field vortices  observed  in \nBose -Einstein condensate quantum ferrofluids.  \nThe fact that this same quantum vortex field can be observed  by a lens (i.e. \nFerrolens) also at the macroscopic level and shown existing as a net result in \nevery macro ferromag net or electromagnet is remarkable and potentially \nimportant . The ferrolens acts like a quantum microscope and our method gives \nus potentially a unique opportunity for the first time to map the magnetic flux \ngeometry and dynamics of  the single elementary M agneton -Quantum Magnet \nthus the stationary magnetic field of a single electron . The whole magnet acts like \na giant macroscopic stationary  magneton when observed through  the Ferrolens.  \n2. METHOD   \nA F errolens1 quantum imaging device [1][2][13] , \n(https://tinyurl.com/y2cgp59x ), was used  and its similar to aqueous \nmagnetic film, magnetophotonic properties also independently \nreported by others [3,4]. This superparamagnetic [5–7] thin film optical lens \nexhibits minimal magnetic quantum decoherence [8–12] and has a sub -micron \nspatial resolution of the magnetic field under observation. The magnetic object \n                                                                                 \n1 https://tinyurl.com/y4np83fn  under investigation is placed above  or under the ferrolens at the center either \nwith its pole facing the lens, polar -field (https://ti nyurl.com/yctntnjc ) view or \nat its side, side -field (https://tinyurl.com/y26yuru5 ) view although other \nconfigurations are also possible. The ferrolens can be activated using \nartificial lighting either by a light emitting diode (LED ) ring around the perimeter \nof the lens or a single light source.  In the case of the LED lighting, a wire -\nframe holographic pattern of the quantum field of the magnet is shown whereas if single light source is used, a cloud pattern of th e field is \ndisplayed. Many types of materials of permanent magnets of different \nshapes were examined, a small sample we are presenting herein, \nrangin g from neodymium Nd magnets, ferrite  magnets  as well \nelectromagnets, all showing the same QFM vortex field geometry as \nshown below in fig.1 . \n3. RESULTS AND DISCUSSION  \n \nFig. 1. Quantum vortex flux of the field of various magnets as shown by the quantum magneto optic device ferrolens in real time. A RGB light ring was used on the \nperimeter of the lens showing the wire -frame individual quantum flux lines, of the field. (a) side f ield view of cube magnet under the ferrolens. North and South Poles are \nindicated and the domain wall of the magnet shown by the arrow. The two polar back to back vortices  are clearly shown joint at the domain wall (b) polar field view \ninside the quantum v ortex of a pole of cube magnet. Enormous holographic depth of the field shown by the ferrolens. A white light LED ring was us ed during the \nexperiment since it produces the highest sensitivity for the ferrolens. The pole of the magnet is under the ferrolens  facing it. The individual quantum flux lines shown are \nnot criss -crossing but overlapping in 3D Euclidian space holographically shown by the lens. The ferrolens is magnetically transparent t o both axial poles of the magnet \ntherefore both poles of the cube  magnet are projected simultaneously at its 2D surface  [13] (c) A Halbach array under the ferrolens. Normally when magnet is under the \nferrolens its quantum flux lines are extending outside and above the ferrolens surface before of course they all curl back toward s the magnet’s poles. However in this \nphotograph taken, the two poles of the center magnet in the array shown are magnetically conf ined by the quantum polar vortices of the other magnets in the array left \nand right. Thus, vortex ring, torus, are formed on the two poles of the center magnet. The middle column at the center is the  domain wall region of the dipole magnet.  \nFrom the sample experimental  material taken of fig.1. above, the \nquantum vortex shaped magnetic flux existing in  macro magnetic \ndipoles  and Quantum Magnet  is demonstrated and revealed for the \nfirst time by a quantum magneto optic devi ce. Specifically, in fig.1(c) \nthe North and South polar quantum flux forming perfect vortex rings \nor torus, is representing best this new Quantum Field of Magnets \n(QFM) geometry shown by the experiments and in compliance with \nthe Maxwell equations [14] which demand zero divergence ∇⋅B = 0 (1) and full curl for magnetic fields . It is also remarkable how much \nevident with the ferrolens is,  in fig. 1(a)(c), the domain wall or else \nreferred as Bloch domain wall [15][16] region of the magnets. A region \nno more than 100 nm wide quantum effect, with the domain wall itself \nbeing a few atoms thick which is magnetically and spatially magnified \nby the lens and shown. Classically, in the domain wall region of a \nmagnetic dipole field, the transition in polarity N -S or vice versa occurs. \nIn the QFM vortex flux of a dipole magnet the domain wall is the joint \nbetween its two quantum flux N -S polar vortices which are axially \nconnected and have a counter geometrical spi n as shown in fig.1.  \n \nFig. 2. Physical mechanism for explaining classical macroscopic E -field of macro \nmagnetic dipoles with quantum vortex M -field (QFM) causality effect observed \nwith the ferrolens.  Fig.2(c) [17].  This discovered,  QFM  or else Quantum Magnet,  vortex flux could \nalso provide a possible physical mechanism for explaining the \nmacroscopic field imprint of magnetic dipoles as shown in fig.2. \nThe effect is similar as described by vortex hydrodynamics [18–\n20], quantum Bose -Einstein condensate ferrofluids [21]  and \ngeneral vortex model theory [22,23]  very often encountered in \nnature from the quantum scale to the macro world [13] . The \nquantum vortex flux we call, M -field shown in fig.2(a) with red \nand fig. 2(b) as displ ayed in real time by the ferrolens, is \npossible responsible and the cause for the macroscopic axial \nflux we call E -field, fig.2(a) with blue and fig.2(c), we usually \nobserve with macro field imaging sensors which have usually a \nmm scale size and are theref ore susceptible to quantum \ndecoherence phenomena.  On the other hand ferrolens has very little \nquantum decoherence since it is using a nano scale imaging sensor (i.e. \n10nm in average Fe3O4 magnetite nanoparticles).  \n \n \nFig. 3. Quantum Decoherence (QDE) effect mechanism responsible for the \ntransmutation of the dipole net quantum field present on magnets (see thick \ncolored lines shown in the ferrolens )  into the classical macroscopic field imprint \n(see tension  field formed , brown iron filings thin lines, between the two black holes \nN-S poles of the magnet) .   \nThe total QFM of a dipole magnet with the combined E -M fields is \ndisplayed in fig. 2(d) and fig.3 using overlaid transparent images from \nthe experiments to best demonstrate th is causality effect of the two \nquantum polar vortices we call M-field  as illustrated in fig.2a, acting like \ndrain s generating therefore the familiar, axial N-S classical macroscopic \nfield we call E-field as illustrated in fig.2a, between the two poles  (i.e. \nsee the two black holes in fig.2d and fig.3) . \nIn fig.4 we see a sample of the single light source experiments we \ncontacted with the ferrolens. A single LED was  used as artificial lighting to \nactivate the lens and was placed 1cm under the ferrolens at it s center.  The \nmagnets used in the experiments were placed on top of the ferrolens. \nAll observations and results obtained are in real time without any kind \nof processing. In this experimental configuration with a single light, a \ncloud pattern of the QFM -Magneton  flux is displayed instead of the \nwire -frame flux pattern we get when using a LED ring. This allows us \nto observe more clearly and in a concise way the outer shell and \ngeneral 360 ° outline geometry of the QFM flux in magnetic dipoles  \nwhich is actu ally representing the field of a single magneton thus the \nQuantum Magnet . All experimental results obtained and a small sample is \nshown above in fig. 4a -d, demonstrate the same QFM geometry, independent of \nshape or material of magnet (i.e. neodymium or fer rite) used in the experiments.  \nThus, a hemispherical field as subsequence of the quantum vortex field extending \nand curling in 3D Euclidian space. The two polar fields are joint axially at the \ndomain wall forming the final spherical nature of  magnetic fiel ds. However as \nshown the two distinct N -S polar quantum vortices are also kept \nmarginally separated by the domain wall.  \n \n \nFig. 4. Single light source ferrolens experiments and correlation of the total QFM -Magneton field  of magnetic dipoles with classical vortex theory and Modons. (a) \nCylindrical magnet placed on its side (side -field view) above the ferrolens with single white LED one cm under the lens at the center. Its two hemispherical polar fields  \noutli ne is shown , magneton field . (b) Same experiment but with a yellow light LED. Look at inner theta θ pattern , magneton field . The outer perimeter is the rim of the lens. \nThe hemispherical polar quantum field outline geometry is clearly again demonstrated wi th the domain wall joining at the middle and at the same time separating the \ntwo fields. (c) Same pattern seen on spherical magnet with the 2D Bloch domain wall disk  at the middle. Upper part of the photograph is the North Pole and at the lower \nthe South P ole. (d) Closer look at the domain wall of the spherical magnet. We can clearly see the separation of the polar quantum field s. (e) A modon. Counter spinning \nwhirlpools [13] in a pool joining their vortices underwater as shown (i.e. food coloring was used to make the effect visible) . These vortices are joint togethe r but at the same \ntime keep their  distance. (f) Velocity (flux) graphs of the modon [24]. Striking resemblance with the QFM of dipole magnets shown previously in (a)(b)(c) & (d). (g) Nested \ncounter -torus hemispherical geometry of the total Quantum Field of Magnets (QFM) illustration. All of its geometrical fi eld features are indicated including both of its flux \nfield modalities, polar vortex flux and axial flux. The two polar quantum fields are joint at the domain wall in the middle a s indicated. (h) Dipolar field of ferromagnet \nexperimentally shown by another  quantum sensor, scanning SQUID microscopy  [25]. \nSurprisingly, we can observe the same field geometry and dipole \nbehavior, in water modons [24,26]  of  two unaxially counter spinning \nwhirlpools [13] as shown in fig. 4(e)(f). These whirls are joint and hold \ntogether underwater by the vortex made visible using food coloring \nresembling thus the domain wall of macro magnetic dipoles. Also, in \nthe same time the whirlpool p air keeps its distance from each other \navoiding merging. The same exact behavior which evidently is shown \nin all the sample QFM experiments, fig.1 -4. Similar  dipolar field pattern  \nwe can see also independendly  reported , from ferromagnets  scanning \nSQUID magnetic microscopy, a quantum sensor, fig 4(h)  [25]. \nNotice the striking resemblance of the analytic modon velocity (flux) \ngraph in fig. 4f [24]. In fig. 4(g) we illustrate conclusively the observed \ntotal QFM geometry of magnetic dipoles as shown by the ferrolens and \nother experiments. A nested double counter torus joint hemispheres \nfield geometry constituting a sphere. This field geometry is repeating in shells like an onion  and extending fractally from center outwards in \n3D Euclidian space. The two counter geometry North and South poles \nquantum vortex fields are indicated as well as the axial macroscopic \nfield making up the final perpendicular EM fields of a dipole magnet \nwith its  domain wall essent ially a 2D disk, in the middle of the sphere \nas illustrated in fig4(g).  \nA special macroscopic ferrolens at fig. 5(a), was constructed to \ndemonstrate our point about quantum decoherence effect being \nresponsible for reverting the quantum vortex field of a ma gnet QFM   \nshown previously in fig. 1 to fig. 4,  to its familiar axial  macroscopic E -\nfield imprint at fig. 5(b). The Fe iron  field sensor particles we used are \n40 μm in size, thus, x4,000 larger than the ones used in a normal \nquantum ferrolens, 10nm in s ize. The Fe iron particles are suspended \nin mineral oil carrier fluid and have a volume concentration percentage \ncompared to the carrier fluid of about 30%. In a normal ferrolens this \nsame percentage is no more than 0.75%. In fig. 5(b) a cylindrical Nd \nmag net is placed above the macro ferrolens and its N-S magnetic field \nis displayed . \n \n \nFig. 5. Quantum decoherence effect demonstrated. (a) A 40 μm Fe iron particles \nmacroscopic ferrolens was made. The particles are suspended inside the lens in a \nmineral oil thin film. The particle size is here 4,000 times larger than the 10nm \nparticle size used in a normal ferrolens. (b) Macroscopic familiar field imprint of a \ncylindrical magnet shown  by the above macro ferrolens due to quantum \ndeco herence effect of the actual net quantum vortex field present on the magnet \npreviously displayed in fig.1,2,3&4 , [13] . \nIn the above macrosc opic ferrolens experiment [13] shown in fig. 5, \nquantum decoherence rate δ [9,10,27]  of the 40 μm iron Fe macr o \nsensor particles inside the colloidal carrier fluid, about 200 μm in \nthickness encapsulated film, can be actually estimated by calculating \nthe Brownian relaxation time τΒ [28,29]  of the  magnetic particles \nwhich is dominant at this relative macroscale size:  \n \n \n         \n                                                  (2) \n \n    \n                                                       (3)  \nIn equation (2), VB , ηο and k are the particle volume, the dynamic \nviscosity of the carrier fluid and the Boltzmann constant approximately \nat 1.38X10-23 respectively, with T being the temperature set at 300K, \nroom temperature for the purpose of this calculation. The spherical \nparticle volume size VB  in equation (3) is calculated for a radius R=D/2 \nat 20 μm with the particles surfactant thickness d, being R » d and \ntherefore ca n be dismissed. The other specified value used in the \ncalculation is ηο=2.4 cP thus, 2.4X10-3 Kg/m/s for the carrier fluid \nmineral oil dynamic viscosity.  \n \nThe quantum decoherence rate is estimated therefore to:  \n \n                                                               (4) \n \nIndicating that our open system  ferrolens, is interacting strongly with \nthe environment and behaving macroscopic, slowing down to a nearly \nhold, quantum fluctuations. Also since τΒ  relaxation time is larger than \nexperiment observation times, the ferrolens is now ferromagnetic as \ntheory predicts [29]. \nIn contrast,  a superparamagnetic ferrolens with for example 10nm \nin diameter size magnetite Fe3O4 nanoparticles, would re sult due to its \ndominant Néel relaxation time τ Ν over Brownian motion, at a 10-9 s \ntime [28] and therefore to a δ  109 Hz, suggesting rapid magnetization \nfluctuations  and that it is quantum act ive. \n \n3.1 The irrotational net Qua ntum Field of \nMagnets (QFM) as actual cause for the cases \nof conservation in classical static \nmacroscopic magnetic fields  \n \nIn the ideal case of an undisturbed  isolated by any external \ninfluence say for example a permanent magnet left alone and \nhaving a time invariant static field  for a simply connected \ndomain (i.e. without any discontinuation in the field)  where \nthere is no free electrons current present or an y other external \ncharge introduced, has a zero curl \n0 H (5), and can be \ndescribed  by the negative gradient  of its  scalar potential, \nH\n(6) and we also know that there is no work done and \nenergy consumed by th e magnet since there is no extern al \ncharge introduced in field and in general there is no energy lost \nand the field is dormant . Although the above case does not \nclassify as a vector force field, in the context  of quantum \nmechanics since the magnet has aligned orbiting unbounded \nelectrons interacting in its matter we can extent the definition \nof conservative fields for this case and say that, all three \nequivalent conditions described  above are meet and the field  is \ntherefore described as conservative  [30] .  \nAt the moment we  introduce an external charge in the field of \nthe magnet (i.e. real world condition since our magnet is an \nopen system and interacting with the environment ), it becomes \nnow a vector force field and in general except special cases, the \nmagnetic field of our  permanent magnet  is non -conservative in \nthe presence of currents or time -varying electric fields  by \ndefinition of the three criteria  we described before and their \nequivalence proof meaning that when any one criterion  holds \nthere other two also hold and t he field is described as \nconservative.  \nA conservative field should have a closed line integral (or curl) \nof ze ro [31] . Maxwell's fourth equation (Ampere's law) can be \nwritten here as:   \n \n0 0 0t  EBJ\n  (7) \nso we can see this will equal zero only in certain cases.  \n \nMagnetic  force is also only conservative in special cases. The \nforce due to an electromagnetic field is written  \n \nqq  F E v B   (8) \n \nFor this to be conservative then  \n0 F  and \n \n \n( ). qq     F E v B  (9) \n \nBut from Faraday's law we know that  \n,t BE\n   so, \n \n( ) ( ) ( ) ( ) q q q q qt         BF v B B v B v v B\n \nFrom Gauss’s  law for magnetism  \n0 B always, and  for a \nsingle charge introduced \n/ ( ) 0.t     vr  \n \nFurthermore,   \n( ) ( ) 0t    B v B r\n,  so,  \n \nx y zqt x t y t z t                B B B BF\n \n \n \ndqdt BF   (10)  \n \nand the force is only conservative in the case of stationary static \nmagnetic fields which also includes the case of our permanent \nmagnet in our example used for this analysis.  \n \nAlso we know that in the case of the single external  charge \nintroduced in the field of a permanent magnet, zero net work \n(W) is done by the force when moving a particle through a \ntrajectory that s tarts and end s in the same point closed loop or \nelse the work done is path -indepedent  (i.e. not necessarily at \nthe same point in the z -axis in 3D space, solenoid  case ) meaning \nequal amount of potential energy is converted to kinetic energy \nand vice versa and there is n o energy loss in the system.   \nd0\nCW F r  \n   (11)  \nBy the equivalence  proof we already even with one criterion  \nmeet as we shown before, we should characterize the magnet  \ninteracting with the single charge as a conservative vector force \nfield however in this case surprisingly  the third criterion  does \nnot hold  (12)  and the equivalence  proof is therefore broken  \n \nF\n   (12)  \n \nsince the force cannot  be described as the negative gradient of \nany potential Φ we know. This anomaly directly implies that \nthe single charge interacting magnetic field is not a \nconservative field but as a matter of fact not even a vector force \nfield which  is a contradiction of all we know and established \nabout Electromagnetism.  \n \nA different approach and analysis  is needed to establish  that \nthe static magnetic field of our magnet in our example, \ninteracting with a charge is actual a conservative field thus \npath -independent, assuming that there is no work done as all \nexperiments in the literature show and the equiv alence  only \napparently does not hold for the third criterion  as mentioned \nabove in equation (12) simply because of our not complete \n100% yet k nowledge about Electromagnetism and not because \nthe vector force field is not really conservative.  \n \nActually this is not the only case in physics where a path -\ndependent  thus non -conserv ative field can have zero curl . So \nwe must be very carefully with the equivalence  proof of the \nthree criteria across different force fields and zero curl does not \nnecessarily imply cons ervative although the opposite is always \ntrue thus, non zero curl fields cannot  be conservative.  Most \nvelocity -dependent forces, such as friction, do not satisfy any of \nthe three conditions,  and therefore are  non -conservative  [32] . \n \nHowever the case we described of the static magnetic field \ninteracting with a single charge, it  is the only case in our \nknowledge that fulfills  both of the zero curl and zero net work \ndone crit eria and fails in the third \nF\n  (13) which is the \nmost important and ,   in the context of the  gradient theorem  \n[33] , it is an exclusive criterion  and condition, thus, a  vector \nfield  F is conservative if and only if it has a potential function  f  \nwith  F=∇f  in general .  Therefore, if you are given a potential \nfunction  f or if you can find one, and that potential function is \ndefined everywhere, then there is nothing more to prove. You \nknow that  F is a conservative vector field, and you don't need to \nworry about the other tests we mention here. Likewise, if you \ncan demonstrate that it is impossible to find a function  f that \nsatisfies  F=∇f, then you can similarly conclude that  F is non -\nconserv ative, or path -dependent.  As we stated before we do not \nyet know  such a function potential to prove conservation and \ntherefore another research lead and approach must be \ninvestigated.  \nSo far our analysis using Maxwell theory for electromagnetism \nresulted that we cannot  conclude that a static magnetic force \nvector field interacting with a single charge is conservative but \nwithout however rejecting the possibility . \n Another large obstacle we have in characterizing the force field \nin our case and example as conservative by investigating  \nalternative theories and an alysis  in the literature  [34–37], is the \nconclusion coming directly from  the gradient theorem and \nvector calculus in general  saying that all conservative  vector \nfields thus path -indep endent meaning that in our case,  that the \nwork done in moving a particle  charge  inside our magnetic \nfield between two points is independent of the path taken  and \nequivalently, if a par ticle travels in a closed loop the net work \ndone by a conservative force field is zero , are also irrotational \nin 3D space and therefore  must have a vanishing curl  (e.g. a \nvortex)  within a simply connected domain . \n \nThis last we can use as an exclusive criterion  for conserva tive \nvector fields and all macroscopic magnetic fiel ds according to \nMaxwell and experiments have nothing even near a vanishing \ncurl  even more are not irrotational by any means.  Therefore, \nusing this criterion  no classical macroscopic magnetic field can \never be characterized as conservative.  \n \nHowever,  our res earch presented herein shows an  underlying \nhidden to the macro world  net quantum dipole vortex field \nexisting in every macro magnet as the actual causality field for \ngenerating the classical macroscopic magnetic field as a tensor \nfunction of the two polar quantum magnetic N-S polar vortices \n(fig. 3).  This quantum magnetic field of magnets (QFM) is not \npresent and visible at the macroscopic level due to Quantum \nDecoherence (QDE) phenomenon and mechanism.  \n \nThis is a crucial finding  that shows the actual vortex nature of \nmagn etism in origin and suggesting  that static magnetic fields \nat the quantum level are irrotational fields and therefore \nconclusively proving to be  conservative.  \n \nAlso notice, that unlike 2D space where a hole in the domain \narea of the field is not a simply  connected domain which is a \nsub condition  for the irrotational field being conservative as \nshown above, in 3D space a domain can have a hole in the \norigin center and still be a simply connected domain as long \nthis hole does not pass all the way through th e domain.  This is \nalso shown in all our experiments an observations made with \nthe ferrolens   of the field of permanent dipole magnets . The two \nN-S polar quantum vortex holes are not connecting and \ntherefore the shown dipole vortex field is irrotational in a \nsimply connected 3D domain of the total field of the magnet \nfig.4g  and therefore conservative.  \n \n3.2 The logarithmic spiral arms of the QFM  \nFrom vor tex theory [18] we know that a free vortex thus an \nirrotational vortex (i.e. most common case of vortices occurring  \nin nature), has a vanishing curl therefore  a zero curl \n0 B\n and all of its individual particle trajectories are \nlogarithmic spirals.  Also since it has a vanishing curl its \ntrajectory progressively  as the distance r  0 from the vortex  \norigin (i.e. eye of vortex) diminishes, becomes a cirlce therefore \na non -zero curl or else called the pole of the vortex.   \nFig. 6. (a) A logarithmic spiral  (b) The QFM dipole vortex field of a magnet shown \nby our experiments using the Ferrolens same as fig.2b & fig. 3 but this time the \nnegative of the photograph was used to select two spiral arm segments (s ee \nwhite dotted lines) of the QFM flux for numerical analysis . The xy Cartesian  \ncoordinates of these spiral segments  were extracted using  xy image digitizing \ntechnique.  (c) A Left arm spiral segment of the QFM, used for numerical analysis . \nUp is its xy li near scale plot, down is its xy logarithmic scale plot  (d) Right arm \nspiral segment and its corresponding data plots.  \nThe experimental data of fig. 3 dipole vortex photograph was \nused in the numerical analysis by extracting the xy Cartesian  \ncoordinates  of two QFM flux lines segments using image \ndigitization technique and the corresponding xy plots of these \nwere drawn to decide if these spiral arms are logarithmic.  \nIn a logarithmic spiral  its polar equation is , \n     \nbr ae                                                   (13)  \nwhere r is the distance from the origin {0,0} of the vortex, θ is \nthe angle from the x -axis, and a and b are arbitrary constants.  \n \n \n \n \nAlso its xy Cartesian  coordinates are equal to,  \n \ncos cosbx r a e\n   (14) \n \nsin sinby r a e\n                                          (15) \n \nand the change of its radius  is, \n \nb drabe brd\n\n   (16) \n \nIn fig.6c and fig. 6d we can see the plots derived from the \nnumerical analysis of the two spiral arms of the QFM vortex \nfield shown b y the Ferrolens. The first plots of each column   are \ndrawn with linear scale xy axes (see links for the extracted \nnumerical values  https://tinyurl.com/y3nekkub  and \nhttps://tinyurl.com/yyojfz57 ), the second plots below on each \ncolumn  are the same plot numerical data  but now the xy axes \nused have logarithmic scale  (https://tinyurl.com/yyln6jsn  and \nhttps://tinyurl.com/y554k3nt  ). We see that in the second case \nwhere the logarithmic  scale axes are used, the plots give  \nexponential functions and therefore the original spiral ar ms of \nthe QFM of the magnet shown by the Fe rrolens are logarithmic \nspirals allowing also a minute  optical distortion depending on \nthe view angle from the lens  (see log -log verification plots at  \nhttps://tinyurl.com/yxh8r9cg  , https://tinyurl.com/y5eq7mo3  \naccordingly) . \n \n3.3  Quantum Magnet  \n \nA novel observation has been made that the collective net Quantum \nMagnetic Field of the Magnet -Magneton (QFM) consists of a dipole \nvortex shaped magnetic flux geometrical pattern responsible for \ncreating the classical macroscopic N -S field of magnetism as a tension \nfield between the two polar quantum flux vortices North and South \npoles. A physical interpretation o f this quantum d ecoherence \nmechanism observed was  analyzed and  presented, showing physical \nevidence of the quantum origin irrotational and therefore conservative \nproperty of magnetism and also demonstrating that ultimately \nmagnetism at the quantum level is  an energy dipole vortex \nphenomenon  (i.e. we known that magnetic flux lines are made up of \nvirtual photons flow) . \nIn this point it is important to explain that the authors here by the \ndata and the analysis presented, never reported or implied that the \nmagn etic flux geometrical vortices of permanent magnets shown on \nthe ferrolens are a result of topological defects, giant skyrmions, thus \ninduced by the superparamagnetic thin film hermitically sealed inside \nthe lens. But are quite accurately representations o f the actual net \nQuantum Magnet  field (QFM) we discovered present in all \nmacroscale permanent magnets [1]. To our knowledge, there were \nnever reported by the literature ever existing cm in size, stable \nmagnetic  skyrmions [39] of any type superfluidic or superconductive \nlet alone at room temperatures. Therefore the quantum magnetic field \nimprint on the ferrolens is exclusively induced by the external macro \nmagnet under observation. Additionally, the 25 -50 μm th ick depending \nthe ferrolens construction, thin film colloidal magnetite Fe 3O4 \nferrofluid -kerosene mixture used inside the ferrolens and sealed to \nprevent evaporation, is non-magnetohydrodynamic , meaning it is an \nelectrical insulator and the nanoparticles h ave antistatic coating [1]  [2]. \n  \nFig. 7 (a ) Synthetic Magnetic Ring Array and its quantized vortex pole at the \ncenter shown on a ferrolens (b) The 12 perimeter magnetic plates used in the \narray, quantized flux shown by the ferrolens (c) Surface map B -field strength of \nthe ring array measured with a 3-axis magnetometer.  \nIn order to demonstrate herein that the ferrolens is driven \nexclusively by the external magnetic field, in fig. 7(a), a special magnetic \nring array [1] was constructed, a magnetic flux twister, consisting of \ntwelve 10mm X 1mm square magnet plates inserted vertically in a 3D \nprinted PVC frame. The combined quantized magnetic flux of the \ntwelve magnets in the ring array, produces a circular type magnetic \nmoment clockwise or anticlockwise depending the magnets placement \nbut because the m agnets have a skewed angle to each other inside the \nring it results to an elliptical vortexing magnetic moment with a \nvanishing curl and inducing to the ferrolens nanoparticles a \nprogressively incremental angular velocity  which its magnetic imprint \nis accurately picked up by the ferrolens as shown (i.e. see  the field  at the \ncenter of the ring array which is placed on top of the ferrolens) . In fig. \n7(b) we see the quantized flux of the twelve magnets ring placed under \nthe f errolens and because their very small 1mm thickness and skew \nangles, the magnetic flux is forced to curl and wave around and in \nbetween the magnets in the ring almost like a sinusoidal wave \nfunction. In fig. 7(c) we see the magnetic field B intensity surfa ce map \nof the synthetic magnetic ring array as measured with 3 -axis \nmagnetometer , placed 1cm away from the face of the ring array \n(measured values found in the link https://tinyurl.com/yypmhglv ). \nEach hump on the perimeter is a magnetic plate of the ring with the net \nfield single Pole of the ring formed at the center. The elliptical \nperimeter and downward slope of the surface map indicating the \nvortexing external magnetic field generated by the synthetic magnetic \nring array prototype.    \n \nFig. 8 (a) Nineteen, 5mm sphere magnets triangular lattice (b) Twenty -one 5mm \nsphere magnets triangular lattice (c) & (d) Their corresponding quantized fields \nand magnetic flux vortices as shown by the ferrolens. The triangular ph ases \nformed, of the quantized magnetic field lattices are also shown. Some dust and \nreflection photographic artifacts are present.    \nSo far, we have demonstrated and analyzed the dipole Quantum \nMagnet  Field  (QFM)  of macro permanent magnets shown by the \nferrolens, a quadrupole face part from a Halbach magnetic array and \nthe quantized field of a twelve magnets synthetic ring array, magnetic \nflux twister.  \nIn fig. 8 experiment, we placed different triangular lattices \nconfigurations of 5mm in diameter small sphere dipole permanent \nmagnets under the ferrolens with a black cupboard paper inserted in \nbetween for increased contrast of the magnetic field imprint. Notice, \nthat the net quantized field of the lattices induced, is relative small in \nstrength, near the sensitivity limit of the used ferrolens in the \nexperiment. This requires at least minimum field strength of 15 mT to \nstart to display.  Therefore the phot ographs taken of the field shown by \nthe ferrolens are very fade and were software enhanced for increased \ncontrast. The displayed, see fig. 8(c)&(d),  in real time by the ferrolens, \nartificially induced by the magnetic lattices, macroscopic -near \nmicroscopic quantized vortices and their triangular field phases \ninteractions, demonstrates the capability of the lens to pick up and \ndisplay macroscopic externally induced quantized magnetic flux  and \nvortices. Similar to the generated quantized vortex lattices obser ved in \nBECs [21]  [39]. However, we have to stress out here that,  this quantum \nmagnetic optic device must not in any way assumed of  being capable \nand responsible, for the generation of these stable quant ized vortices \ndue topological defects in the medium since it does not exhibit any \nsuperfluidic nor superconductive properties at room temperatures as \nexplained before. In other words, the Ferrolens device can show \nmacroscopic quantized magnetism and vortic es but cannot generate \nthem. The ferrolens optic display device is an electric insulator at all \ntemperatures therefore not a superconductor. In addition we tested \nrigorously for superfluidic behavior concerning quantization of \nmagnetic vortices generation at different values of induced angular \nmomentum [21]  [39] without any observable effect.  \nThe device remains neutral and optical isotropic unless disturbed \nby an external magnetic field (see link https://tinyurl.com/y26yuru5 ) and \nreturns to its previous isotropic idle condition instantly after the \nmagnet is removed. The encapsulated thin film of ferrofluid inside the \nferrolens in this state, does not flow, but exists i n a balanced state of \nequilibrium no matter what position the cell is oriented.  The \nnanoparticles inside the ferrolens do not settle with gravity.  Moreover, when a magnet is fastened at the ferrolens surface the magnetic field \ndisplayed is unaffected by an y motion of the ferrolens device.   \nTherefore, from all the experimental results and analysis \npresented herein and our previous research [1][2], we come to a \nconclusion and deduction by elimination process, that the ferrolens is \nprobing and accurately displaying the actual net collective Quantum \nMagnet Field (QFM) present  in every macroscopic size magnet  and \nother wise masked at the macroscale and macroscopic field sensors  \ndue Quantum Decoherence (QDE) phenomenon and also recognize \nthat the whole phenomenon warrants further investigation in the \nfuture.  \n \nFig.  9 (a) Forced orientation  magnetic array (b) Field shown with ferrolens.  \n \nFinally, because the field images obtained in fig. 8 from the freely \noriented 5mm sphere magnet lattices are at the limit of this ferrolens \nsensitivity, we used in this particular experiment and in order to \ndemonstrate the actual field analytical display capability of the \nferrolens device, we constructed a forced polarity orientation, \nmagnetic array shown in fig. 9(a) using much stronger sixteen \nNeodymium N42 10mmX4mm disk magnets. The magnetic arrays \nwas pla ced under the ferrolens in contact and a black cupboard paper \nwas inserted in between to mask the view of  the magnets and make \nsure that only the field is displayed and also increase the contrast.  \nThe field real time display of this magnetic array by the ferrolens is \nshown in fig. 9(b) and the orientation of the individual magnetic \nmoments N -S in the array is indicated in fig. 9(a). Each vortex, black \nhole, shown is a Pole of a magnet in the array and each dipole vortex \n(i.e. vortices pair) two joint hemis pheres is a magnet. A close inspection \nof fig. 9b reveals even the individual separated magnetic flux lines \ngeometrically vortexing around the Poles. We can also observe the \ndomain walls triangular phases formed, see grid of white flux lines, due \nthe inter action of the magnets inside the array. Once again the \nQuantum Magnet Field (QFM)  emerges here for each single magnet \nin the array as we demonstrated previously in fig. 4a&b for the dipole \nmagnet, thus the two hemispherical polar fields or polar vortices, \naxially joint at the domain wall in the middle.  \nThe four gray spots in the center forming a square are field \ncancelation areas in the magnetic array formed by antiparallel pairing \nof the fields of the magnets in these areas.  \nLast but not least, we observe  the four corner Poles in the \nmagnetic array to be opened up vortices as expected since \nthese particular Poles are very little to none interacting with \nthe rest of the magnets in the array and therefore are not \nmagnetically confined.   \n \n4. CONCLUSIONS  \n \nWe have  herein introduced, presented and analyzed by \nexperimental evidence and theoretical analysis, the existence of a \nsecond hidden quantum vortex flux residing in every macro magnetic \ndipole (i.e. we experimented only with macro magnets and not with any \nmicros copic or quantum sized magnetic dipoles)  besides the \nmacroscopic axial flux observed in magnetic dipoles. This second \nquantum vortex field is masked from the macrocosm and macroscopic \nsensors due to quantum decoherence (QDE) and is showing up only \nwhen a q uantum magnetic field imaging device is used with nano sized \nsensors. In a way we can say that the ferrolens is the quantum version \nof the classical Faraday’s iron filings experiment.  \n  Therefore we made the novel observation  that the collective net \nQFM -Magneton field consists of two quantum magnetic flux vortices \ngeometr ical patterns, joint back to back , and with each vortex residing \non each pole of the Quantum Magnet  as shown in fig.2a  & fig.3  and in \ntheir outline form in fig.4(a)(b) , present in every m acro magnetic \ndipole besides its  macroscopic classi cal N -S axial flux field.  \n A valid correlation was made between our experimental results and \nanalysis with SQUID magnetic microscopy,  quantum Bose -Einstein \ncondensate ferrofluids, general vortex theory and  specifically \nhydrodynamics modon dipole phenomenon occurring in nature, with \nstriking similar effects with our observations with the ferrolens of \nmagnetic dipoles  and also latest quantum dots research for the \nmagnetic model of the electron [38].   \nThis newly experimentally observed  vortex structure QFM in dipole \nmagnets and Quantum Magnet found is complying to the Maxwell \nequation  (1) (i.e. ∇⋅B = 0  ) for zero divergence  and exhibi ts full curl \nwhen observed with the ferrolens . Nevertheless, it could also \npotentially provide a mechanism as shown, to explain the macroscopic \nclassical magnetic field imprint of magnets as a subsequent quantum \ndecoherence effect and prove therefore the actual we believe, \nultimately vorte x nature of magnetism we have researched by \nanalyzing the individual magnetic flux lines geometry  of the magneton \nobserved with the ferrolens and concluding that magnetism is a dipole \nenergy vortex phenomenon  (i.e. magnetic flux lines are made up of \nvirtual photons  flow ).  \nIn fig. 5 shown experiments, we demonstrated clearly that quantum \ndecoherence is taking place here and is responsible for reverting the \nreported by us masked, quantum vortex field QFM of macro dipole \nmagnets to the familiar macroscopic N-S field. Also, theoretical \ncalculations were made to confirm that quantum decoherence is the \ncause for this phenomenon.  \nAlso, a numerical analysis  of the experimental data was carried out \nto confirm that the QFM vortices shown by the Ferrolens are natural \nlogarithmic spirals thus free irrotational vortices.  \nAdditionally , a theoretical analysis was presented  show ing that in \nspecial cases where a magnetic force vector field is to be considered as \nconservative it must be necessarily irrotational with a vanishin g curl \ntherefore confirming our findings  about the actual quantum origin, \nelementary dipole vortex shaped  field and nature of magnetism.  Besides the experiments carried out with normal dipole macroscale \nmagnets, in section 3.3 we demonstrated and analyzed also \nexperimental results obtained from magnetic arrays and lattices and \nconcluded that these results, vortices shown on the Poles of magnets, \nare not due topological defects of the ferrolens medium, quantum field \nimaging sensor or other phenomena we previ ously investigated [1]. \nThus, superfluid or superconductive magnetic quantized vortices \nshown usually in Bose -Einstein Condensates, but exclusively due to the \nQuantum Magnet  collective net aligned magnetic moments and \ncollective field (see fig. 4a&b)  of the unpaired electrons i nside the \nmagnet material  projected into the Ferrolens quantum magnetic optic \ndevice and successfully displayed. The collective quantum field of the \nmagnet resembles closely to the stationary intrinsic magnetic dipole \nfield of the single electron called Q uantum Magnet or else Magneton.  \nTherefore, the ferrolens acts like a quantum optic microscope and \nour method presents to us potentially a unique opportunity for the first \ntime to map the magnetic flux geometry (see fig. 1, fig. 2&3 and fig. 4g)  \nand dynami cs of the single elementary Magneton -Quantum Magnet \nthus the stationary magnetic field of a single electron.  In essence what \nwe see with the F errolens of the field of a macroscopic permanent \nmagnet is, we look directly at the field  of a giant sized statio nary \nmagneton.   \n This macroscale projected quantum phenomenon reported first \nhere we believe warrants further investigation in the near future with \nthe potential of new discoveries and new physics.  \n     \nAcknowledgements.  The authors acknowledge their colleagues, Dr. Michalis \nTatarakis, Dr. Manolis Maravelakis, and Dr. Stelios Kouridakis Technological \nEducational Institute of Crete Academic staff members, for their support. Also Mr. \nMichael Snyder MSc, Timm Vanderelli inde pendent researcher s USA and Dr. \nAlberto Tufaile University of São Paulo Brazil for their support.  \nDisclosures.  All authors contributed equally to the work. The authors declare \nthat there are no conflicts of interest related to this article.  \n  \nAppendix : Supplementary material  to this article can be found online at  \nhttps://tinyurl.com/y3je49tb  [13].  \n \n \nGraphical Abstract : Quantum magnetic vortex field (QFM) of a neodymium ring \nmagnet shown by the ferrolens in real time. The field on the perimeter of the ring \nmagnet extends and curls on the outside holographically shown by the ferrolens \nwith the field i nside the ring magnetically confined and forming a torus.  \n \nReferences  \n \n[1] E. 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Liepmann, Elements of Gas Dynamics., in: Dover \nPublications, 2013: pp. 194 –196. \nhttps://books.google.gr/books/about/Elements_of_Gas_Dynami\ncs.html?id=IWrCAgAAQBAJ&redir_esc=y (acces sed July 3, 2019).  \n[37] D.J. Acheson, Elementary fluid dynamics, Clarendon Press, 1990. \nhttps://books.google.gr/books/about/Elementary_Fluid_Dynami\ncs.html?id=GgC69 -WUTs0C&redir_esc=y (accessed July 3, 2019).  \n[38] P. Stano, C. -H. Hsu, L.C. Camenzind, L. Yu,  D. Zumb hl, D. Loss, \nOrbital effects of a strong in -plane magnetic field on a gate -\ndefined quantum dot, Phys. Rev. B. 99 (2019). \ndoi:10.1103/PhysRevB.99.085308.  \n[39] L.S. Leslie, A. Hansen, K.C. Wright, B.M. Deutsch, N.P. Bigelow, \nCreation and detectio n of Skyrmions in a Bose -Einstein \ncondensate, Phys. Rev. Lett. 103 (2009). \ndoi:10.1103/PhysRevLett.103.250401.  \n \nPlease cite this article as: E. Markoulakis, A. Konstantaras, \nJ.Chatzakis, R. Iyer, E. Antonidakis, Real time observation of a \nstationary mag neton,  Results in Physics, 2019 , doi: \nhttps://doi.org/10.1016/j.rinp.2019.102793  "    },    {        "title": "1911.12268v1.Synthesis__magnetization_and_heat_capacity_of_triangular_lattice_materials_NaErSe__2__and_KErSe__2_.pdf",        "content": "arXiv:1911.12268v1  [cond-mat.mtrl-sci]  27 Nov 2019Synthesis, magnetization and heat capacity of triangular l attice materials NaErSe 2\nand KErSe 2\nJie Xing,1Liurukara D. Sanjeewa,1Jungsoo Kim,2William R. Meier,1Andrew F.\nMay,1Qiang Zheng,1Radu Custelcean,3G. R. Stewart,2and Athena S. Sefat1\n1Materials Science and Technology Division, Oak Ridge Natio nal Laboratory, Oak Ridge, Tennessee 37831, USA\n2Department of Physics, University of Florida, Gainesville , Florida 32611, USA\n3Chemical Sciences Division, Oak Ridge National Laboratory , Oak Ridge, Tennessee 37831, USA\n(Dated: November 28, 2019)\nIn this paper we report the synthesis, magnetization and hea t capacity of the frustrated magnets\nAErSe2(A=Na,K) which contain perfect triangular lattices of Er3+. The magnetization data sug-\ngests no long-range magnetic order exists in AErSe2(A=Na,K), which is consistent with the heat\ncapacity measurements. Large anisotropy is observed betwe en the magnetization within the ab\nplane and along the caxis of both compounds. When the magnetic field is applied alo ngabplane,\nanomalies are observed at 1.8 µBin NaErSe 2at 0.2 T and 2.1 µBin KErSe 2at 0.18 T. Unlike\nNaErSe 2, a plateau-like field-induced metamagnetic transition is o bserved for H /bardblcbelow 1 K in\nKErSe 2. Two broad peaks are observed in the heat capacity below 10 K i ndicating possible crystal\nelectric field(CEF) effects and magnetic entropy released un der different magnetic fields. All results\nindicate that AErSe2are strongly anisotropic, frustrated magnets with field-in duced transition at\nlow temperature. The lack of signatures for long-range magn etic order implies that these materials\nare candidates for hosting a quantum spin liquid ground stat e.\nI. INTRODUCTION\nGeometrically frustrated magnets with many novel\nproperties in the low temperature region have been stud-\nied for many years[1]. One intriguing phenomenon is the\nmagnetization plateau. An ordered ground state of 120◦\nbetween each spin was proposed in two-dimensional spin\n1/2 triangular lattice Heisenberg antiferromagnet (TL-\nHAF) [2]. Due to the degeneration of the energy in the\nquantum system, the magnetization under the magnetic\nfield is predicted with a plateau at 1/3 of the saturation\nmagnetization in TLHAF [3–5]. On the experimental\nside, the 1/3 plateau is found in triangular lattices ma-\nterials with varied spin number from 1/2 to 3/2, such as\nCs2CuBr4, CuFeO 2, RbFe(MoO 4)2and Ba 3CoSb2O9[6–\n9]. Furthermore,besidestheTLHAF,strongspin-lattices\ncoupling could induce a 1/2 magnetization plateau in the\ngeometrical frustrated system[10, 11].\nAnother important topic relevant to geometrically\nfrustratedmagnetsisthe quantumspinliquid(QSL). The\nQSL state is a highly entangled quantum state lead-\ning to the fractionization of spin degrees of freedom[12].\nRecently, Majorana fermions were also proposed in the\nquantum spin liquid system [13]. Until now several com-\npoundswith S=1/2areproposedasthe QSLcandidates,\nsuch as honeycomb iridates A2IrO3(A=Na,Li,H 3Li,Cu),\nκ-(BEDT-TTF) 2Cu2(CN)3, EtMe 3Sb[Pd(dmit) 2]2, and\nRuCl3[14–28].\nRecently, both theoretical and experimental results in-\ndicate the magnetic rare earth ions located in the ge-\nometrically frustrated lattices (e.g., triangular lattice)\nmay also form QSL states [29, 30]. Rare earth ions in\nconfigurations with an odd number of 4 felectrons sup-\nport Kramer doublets which can be treated as an effec-\ntive spin Jeff=1/2. Compared to transition metals, thespin-orbital coupling in the rare earth system is much\nstronger and highly anisotropic exchange couplings are\nexpected [31]. YbMgGaO 4with the YbFe 2O4structure\nis studied as rare earth QSL candidate. The Yb3+ions\nconstruct a triangular layer and are octahedrally coordi-\nnated by O2−ions. The heat capacity, thermal conduc-\ntivity, neutronscatteringandmuonspinrelaxationreveal\na possible gapless QSL ground state [32–39]. However,\nthe existence of Ga/Mg disorders may drive the system\nto another state [32, 36, 37, 40–43]. Replacing Yb by\nEr also reveals possible QSL behavior[44, 45]. This en-\ncourages research on rare earth materials with the ideal\nfrustrated triangular structures. Very recently a classic\nsystemARECh 2(A=Alkali metal, RE=Rare earth ele-\nments,Ch= chalcogens) with perfect triangular lattices\nof rare earth ions were proposed as QSL candidates [49–\n53]. Comparingto the other frustrated lattice structures,\nthis family with less ions and simple triangular struc-\nture suggests less possibilities of impurities or disorders.\nNo structural or magnetic transition was found in the\nNaYbCh2(Ch=O,S,Se) down to 50 mK from heat capac-\nity and magnetization [49]. Single crystal studies out-\nline the strongly anisotropic, quasi 2D magnetism and\nJeff=1/2 state in NaYbS 2[50]. Specific heat data con-\nsistent with a gapless QSL state was found in NaYbO 2\nand magnetic field promotes a quantum phase transi-\ntion above 2 T [51–53]. Besides these possible QSL\nstate studies, an anisotropic spin of the rare earth ions\nin a crystal electric field has been proposed by theoret-\nical calculations[46–48]. This system is a large family\nwith diversity originating from rare earth, alkali or tran-\nsition metal and chalcogen, which could support more\nnovel properties of multiple exchange couplings and crys-\ntal field effects. From this aspect, this system is a good\nplatformtostudyfrustratedmagnetismofrareearthions2\nTABLE I. The crystal structure of the NaErSe 2and KErSe 2phase at 300K. Both samples are belong to the same space group\nR3m.\ncomponent instrument a(˚A) c(˚A) z(Se) R\nNaErSe 2single crystal XRD 4.0784(4) 20.746(3) 0.25694(1) 0.03\nNaErSe 2powder XRD 4.0876(1) 20.7980(6) 0.2562(1) 0.08\nKErSe 2 single crystal XRD 4.1466(1) 22.743(5) 0.2647(1) 0.02\nKErSe 2 powder XRD 4.1470(1) 22.7662(8) 0.2649(1) 0.07\non a triangular lattice.\nUntil now all the measurements in this ARECh2sys-\ntem focused on the A=Na and RE=Yb3+[49–53]. The\nfruitful resultspromptedustoundertakeaninvestigation\nof the physical properties in AErSe2(A=Na, K) systems.\nIn this paper, we report the synthesis of Na/KErSe 2sin-\ngle crystals and survey the magnetization and heat ca-\npacityinthissystem. BesidesthepriorreportedNaErSe 2\nandNaLuSe 2, wesynthesizebothpowderandsinglecrys-\ntal of new KEr(Lu)Se 2. Good crystallinity is confirmed\nby the powder X-ray diffraction (XRD) and single crys-\ntallinesXRD. Nosite vacanciesorimpurities arefound in\nthe crystal. No magnetic transitionis found down to 0.42\nK in the magnetization and heat capacity characteriza-\ntion for small applied magnetic fields. Large anisotropy\nis exposed between H ∝bardblaband H∝bardblc. A tiny slope change\nin the isothermal magnetization is found only for H ∝bardblab\nat low temperature. The moment where this occurs is\n1.8µBand 2.1 µBin NaErSe 2and KErSe 2. KErSe 2\nshows a clear plateau-like metamagnetic transition with\nm=2.3µBat H∝bardblcbelow 1 K. The heat capacity shows\ntwo broad features without any λanomaly above 0.4 K\nat 0 T. All these physical properties indicate the absence\nof long range order and existence of short-range order in\nthese materials.\nII. MATERIALS AND METHODS\nTransparent millimeter-scale AErSe2andALuSe2sin-\ngle crystals were synthesized by the two-step method.\nPictures of the crystals are shown in Fig.1(a-b). At first\nNa chunks (Alfa Aesar 99.9%), K chunks (Alfa Aesar\n99.9%), Er chunks (Alfa Aesar 99.99%) and Se chunks\n(Alfa Aesar99.99%)weremixed by the stoichiometricra-\ntioA:Er(Lu):Se=1.1:1:2. The alkali element excess 10%\ndue to the slight reaction with the inner wall of the sil-\nica tube. The whole mixture was loaded into a carbon\ncrucible and sealed in the silica tube under vacuum. The\nampules were slowly heated to 220◦C and held for 24\nh. Then we heated them up to 900◦C and held for 3\ndays. After the first reaction, the samples cooled down\nto room temperature over 24 h. The reacted compounds\nwere ground and rinsed with deionized water and ace-\ntone. This process gets rid of the multiple binary phases\nand excess alkali metal. The pure AErSe2powders are\nconfirmed by powder XRD, as shown in Fig. 1(c). The\nsecond step is the flux synthesis. We used NaCl and KClas the flux for the NaEr(Lu)Se 2and KEr(Lu)Se 2, respec-\ntively. The mass ratio between the flux and the powder\nis 10:1. The mixture was sealed in the silica tube under\nvacuum. The ampoules were heated up to 850◦C and\nheld for 2 weeks. The compounds were taken out after\nthe fast furnace cooling and leached out in the deion-\nized water and acetone. The sizable, plate-shape single\ncrystals were found and the thickness is related to the\ndwell time at the second step. No impurities peaks and\nsharp (00l) manifest good quality for both compounds,\nas shown in the Fig. 1(c). The slow cooling rate do not\nexhibit obvious relation with the size of the crystals.\nMagnetic properties were measured in Quantum De-\nsign (QD) Magnetic Properties Measurement System\n(MPMS3) with iHe3 option. Due to the shape of these\ncrystals, we stacked several crystals along cwith mass\n3 mg and used them in the regular magnetic measure-\nment above 2 K. Single crystals of AErSe2(A= Na and\nK) around 0.5-1 mg with as-grown (00l) surfaces were\nused in the3He option of the MPMS3. Demagnetization\nfactors are considered as N=1 in the isothermal mag-\nnetization at H ∝bardblc. Temperature dependent heat capac-\nity was measured in the QD Physical Properties Mea-\nsurement System (PPMS) with3He option using the re-\nlaxation technique at different applied fields along the\nc-axis. Single crystal X-ray diffraction(XRD) measure-\nment wasperformed by BrukerApex single-crystalX-ray\ndiffractometer. Powder XRD was done by a PANalyti-\ncal X’pert Pro diffractometer equipped with an incident\nbeam monochrometer (Cu K α1radiation) at room tem-\nperature.\nIII. RESULTS AND DISCUSSION\nFig.1(d) shows the schematic structure of AErSe2\n(A=Na, K with space group R¯3m). The perfect triangu-\nlar layers of magnetic Er3+ions are isolated by the non-\nmagnetic Na/K layers suggesting anisotropic and frus-\ntrated magnetism in this system. The crystal structure\nparameters of AErSe2(A=Na, K) obtained from single\ncrystal XRD and powder XRD are shown in Table 1.\nBoth methods yield similar results from the refinements.\nIn this paper we use the powder parameters for discus-\nsion, avoiding the possible effect of the stress in the sin-\ngle crystals. Due to the large radius of K+, the dis-\ntances between each layer are 6.932 ˚A in NaErSe 2and\n7.588˚A in KErSe 2. The distance between the near-3\nFIG. 1. (a-b) The crystal picture of AErSe2(A= Na and K) against 1 mm scale. (c) XRD pattern of AErSe2(A= Na and\nK) powder and crystals on (00l) direction. The red lines show the refinement result of the two compounds. (d) The schematic\nstructure of R¯3m AErSe2(A = Na and K). The red edge-sharing distorted ErSe 6octahedra construct the triangular layer\nbetween the Alkali ions. (e) The Se-Er-Se angle of distorted ErSe6octahedra. (f) The ideal triangular Er3+layer in the ab\nplane. The nearest neighbour Er3+ions are connected by two Se ions.\nest neighbor Er3+ions also extended from 4.087 ˚A to\n4.147˚A by replacing K+from Na+. Although the frus-\ntrated interactions exist in the triangular lattice, the\ninterlayer interaction may induce the magnetic transi-\ntionatthe low-temperatureregion[9]. Therefore,KErSe 2\ncould show more two-dimensional charactor correspond-\ning to the larger c/aratio in the structure. In addition,\ndue to the 8% larger c/aratio in KErSe 2, the distortion\nof ErSe 6octahedra is stronger than the NaErSe 2. The\nSe-Er-Se angle αandβin Fig.1(e) change from 91.51◦\nand 88.49◦in NaErSe 2to 93.13◦and 86.87◦in KErSe 2.\nThe change in local crystalline environment inspires us\nto investigatehowthe propertiesofthese two compounds\ncompare.\nFig. 2(a-b) presents the temperature dependence of\nmagnetic susceptibility χ=M/Hand inverse magnetic\nsusceptibility from 2 K to 350 K for the powder and sin-\ngle crystals. The anisotropic behaviour appears below 40\nK. The magnetization of the single crystals along H∝bardblab\nandH∝bardblcis well matched with the result of the powder\nabove 40 K. No visible features are found above 2 K sug-\ngesting there are no magnetic or structural transitions.\nThe inset shows the AC susceptibility of NaErSe 2and\nKErSe 2from 0.4 K to 1.8 K. No hint of spin freezing orlong-range magnetic order appears above 0.4 K. These\nresults all suggest that the perfect triangular lattice with\nEr3+isstableatlowtemperaturewhichisconsistentwith\nother Yb compounds in this system[49–53]. No structure\ntransition or distortion in Er3+imply that this system\nis a steady platform to investigate novel magnetic prop-\nerties in the frustrated system. The Curie-Weiss fitting\nof the powder above 200 K yield a Curie-Weiss temper-\natureθCW=-10.9 K and effective moment µeff=9.5µB\nin NaErSe 2andθCW=-8 K,µeff=9.5µBin KErSe 2,\nrespectively. The effective moments of these two com-\npounds match the theoretical value of 9.6 µB/Er3+for\nfree ions. The negative value of θCWindicates the domi-\nnantantiferromagneticinteractionbetweenthe Er3+ions\nin the triangular lattice. Curie-Weiss fitting is also ap-\nplied in both samples from 10 K to 30 K: θCW=-4.3 K,\nµeff= 9.4µBfor NaErSe 2andθCW=-3.8 K, µeff= 9.4\nµBfor NaErSe 2. The change of θCWmay be caused by\nthe thermal population of CEF levels.\nFig. 2(c)-(f) present the temperature dependence of\nmagneticsusceptibilityunderdifferentmagneticfieldsfor\nNaErSe 2and KErSe 2. When the temperature decreases\nbelow 10 K, M(T)deviates from the Curie-Weiss behav-\nior. The relation observed in this region is M∝T−0.4.4\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s49/s50/s51\n/s49/s49/s48\n/s49 /s49/s48/s49/s50/s51/s49/s49/s48\n/s49 /s49/s48/s48/s46/s53/s49/s49/s46/s53/s50/s99/s32 /s40/s101/s109/s117/s32/s109/s111/s108/s45/s49\n/s32/s79/s101/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s32/s112/s111/s119/s100/s101/s114\n/s32/s72/s124/s124 /s97/s98\n/s32/s72/s124/s124 /s99/s40/s97/s41/s78/s97/s69/s114/s83/s101\n/s50\n/s48/s49/s50/s51/s52\n/s49/s47 /s99 /s32/s40/s49/s48/s32/s109/s111/s108/s32/s79/s101/s32/s101/s109/s117/s45/s49\n/s41\n/s40/s99/s41/s40/s98/s41\n/s53/s32/s84/s52/s32/s84/s51/s32/s84/s50/s32/s84/s49/s32/s84/s99/s32 /s40/s101/s109/s117/s32/s109/s111/s108/s45/s49\n/s32/s79/s101/s45/s49\n/s41/s48/s46/s50/s32/s84\n/s48/s46/s53/s32/s84/s72/s124/s124 /s97/s98\n/s55/s32/s84\n/s40/s101/s41\n/s72/s124/s124 /s99/s48/s46/s49/s32/s84/s99/s32 /s40/s101/s109/s117/s32/s109/s111/s108/s45/s49\n/s32/s79/s101/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s48/s46/s53/s32/s84\n/s49/s32/s84\n/s50/s32/s84\n/s51/s32/s84\n/s52/s32/s84\n/s53/s32/s84\n/s55/s32/s84/s54/s32/s84/s40/s100/s41\n/s72/s124/s124 /s97/s98/s48/s46/s48/s49/s32/s84/s99/s32 /s40/s101/s109/s117/s32/s109/s111/s108/s45/s49\n/s32/s79/s101/s45/s49\n/s41/s48/s46/s49/s32/s84\n/s48/s46/s53/s32/s84\n/s49/s32/s84\n/s50/s32/s84\n/s51/s32/s84\n/s52/s32/s84\n/s53/s32/s84\n/s55/s32/s84/s54/s32/s84\n/s40/s102/s41\n/s72/s124/s124 /s99/s99 /s32/s40/s101/s109/s117/s32/s109/s111/s108/s45/s49\n/s32/s79/s101/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s48/s46/s48/s49/s32/s84\n/s48/s46/s49/s32/s84\n/s48/s46/s53/s32/s84\n/s49/s32/s84\n/s50/s32/s84\n/s51/s32/s84\n/s52/s32/s84\n/s53/s32/s84/s84/s42\n/s55/s32/s84/s54/s32/s84/s49/s46/s48 /s49/s46/s53/s48/s46/s52/s49/s48/s48/s48/s46/s52/s49/s48/s50\n/s55/s32/s84\n/s50 /s52/s48/s46/s52/s53/s48/s46/s53/s48/s48/s46/s53/s53\n/s52/s32/s84/s84/s42/s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s52/s53/s54/s99 /s39/s32/s40/s101/s109/s117/s32/s109/s111/s108/s45/s49\n/s32/s79/s101/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s55/s53/s55/s32/s72/s122\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s49/s50/s51/s75/s69/s114/s83/s101\n/s50\n/s32/s112/s111/s119/s100/s101/s114\n/s32/s72/s124/s124/s97/s98\n/s32/s72/s124/s124/s99/s99 /s32/s40/s101/s109/s117/s32/s109/s111/s108/s45/s49\n/s32/s79/s101/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s49/s50/s99 /s39/s32/s40/s101/s109/s117/s32/s109/s111/s108/s45/s49\n/s32/s79/s101/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s55/s53/s55/s32/s72/s122\n/s48/s49/s50/s51/s52\n/s49/s47 /s99 /s32/s40/s49/s48/s32/s109/s111/s108/s32/s79/s101/s32/s101/s109/s117/s45/s49\n/s41\nFIG. 2. (a-b) Temperature dependence of magnetic suscepti-\nbility and inversed magnetic susceptibility from 2 K to 350 K\nfor AErSe 2powder and crystal along abplane and caxis. The\ninset show the ac susceptibility from 0.4K to 1.8K for AErSe 2.\n(b)-(f) Temperature dependence of magnetic susptibility i n\nNaErSe 2and KErSe 2crystal with H /bardblabor H/bardblc. The inset\nshows the temperature dependence of magnetic susceptibili ty\nat high magnet field. T* is the temperature of the maximum\nin each magnetic field. The light green dash lines are the\nfitting from CEF at 0.2 T in (c) and 0.1 T in (d-f).\nIt is consistent with that recently observed in Os 0.55Cl2,\nwhich displays many physical properties similar to those\nobserved in quantum spin liquid candidates[54]. It worth\nnoting that there is no significant long-range magnetic\ntransition above 0.4 K from low magnetic field DC sus-\nceptibility and AC susceptibility in zeroapplied DC field.\nThe temperature-independent regions under different\nmagnetic fields are found in these two compounds with\nH∝bardblab. However, when the magnetic field is along the c\naxis, the magnetic susceptibility reaches a maximum and\nthen decreases with temperature. We choose the start-\ning point of d χ/dT(T)=0 as T* forH∝bardblc. The subtle\nhump feature for H∝bardblcin NaErSe 2at 7 T is shown in\nthe inset of Fig. 2(e). The magnitude of the ∆ χis only\naround 0.02% and is challenging to distinguish at the low\nmagnetic field. The similar but remarkable situation also\nappears in KErSe 2. The magnitude of ∆ χis 7% which\n/s53/s48/s32/s75/s32/s72/s124/s124 /s97/s98 /s32\n/s53/s48/s32/s75/s32/s72/s124/s124 /s99 /s32/s50/s48/s32/s75/s32/s72/s124/s124 /s99/s50/s48/s32/s75/s32/s72/s124/s124 /s97/s98/s72/s124/s124 /s99/s72/s124/s124 /s97/s98\n/s50/s32/s75/s77/s111/s109/s101/s110/s116/s32/s40 /s109\n/s66/s47/s69/s114/s51/s43\n/s41\n/s72/s32/s40/s84/s41/s48/s46/s52/s50/s32/s75/s78/s97/s69/s114/s83/s101\n/s50\n/s50/s32/s75/s48/s46/s52/s50/s32/s75\n/s40/s97/s41/s50/s32/s75\n/s40/s98/s41/s53/s48/s32/s75/s32/s72/s124/s124 /s97/s98 /s32\n/s53/s48/s32/s75/s32/s72/s124/s124 /s99 /s32/s49/s48/s32/s75/s32/s72/s124/s124 /s99/s49/s48/s32/s75/s32/s72/s124/s124 /s97/s98\n/s72/s124/s124 /s99/s72/s124/s124 /s97/s98\n/s75/s69/s114/s83/s101\n/s50/s77/s111/s109/s101/s110/s116/s32/s40 /s109\n/s66/s47/s69/s114/s51/s43\n/s41\n/s72/s32/s40/s84/s41/s48/s46/s52/s50/s32/s75\n/s50/s32/s75/s48/s46/s52/s50/s32/s75\n/s40/s99/s41\n/s48/s46/s56/s32/s75/s100/s77/s47/s100/s72/s32/s40 /s49/s48/s45/s51\n/s109\n/s66/s32/s84/s45/s49\n/s41\n/s72/s32/s40/s84/s41/s48/s46/s52/s50/s32/s75/s48/s46/s52/s50/s32/s75\n/s48/s46/s56/s32/s75\n/s72/s124/s124 /s97/s98/s48/s46/s52/s50/s32/s75\n/s72/s124/s124 /s99\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48/s46/s52/s50/s32/s75\n/s72/s124/s124 /s97/s98\n/s78/s97/s69/s114/s83/s101\n/s50\n/s100/s77/s47/s100/s72/s32/s40 /s49/s48/s45/s51\n/s109\n/s66/s32/s84/s45/s49\n/s41\n/s72/s32/s40/s84/s41/s75/s69/s114/s83/s101\n/s50/s72/s124/s124 /s99\n/s72/s124/s124 /s97/s98/s40/s102/s41\n/s40/s101/s41/s40/s100/s41\n/s48/s46/s56/s32/s75/s48/s46/s52/s50/s32/s75\n/s48/s46/s56/s32/s75\n/s75/s69/s114/s83/s101\n/s50/s78/s97/s69/s114/s83/s101\n/s50/s99/s44\n/s32/s40/s101/s109/s117/s32/s109/s111/s108/s45/s49\n/s32/s79/s101/s45/s49\n/s41\n/s72/s32/s40/s84/s41/s48/s46/s52/s50/s32/s75\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s49/s48/s50/s48\n/s72/s124/s124 /s97/s98/s99/s44\n/s32/s40/s101/s109/s117/s32/s109/s111/s108/s45/s49\n/s32/s79/s101/s45/s49\n/s41\n/s72/s32/s40/s84/s41/s48 /s50 /s52 /s54/s48/s52/s56/s109/s111/s109/s101/s110/s116/s32/s40 /s109\n/s66/s47/s69/s114/s51/s43\n/s41\n/s72/s32/s40/s84/s41/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s50/s52/s100/s77/s47/s100/s72/s32\n/s40/s49/s48/s45/s52\n/s109\n/s66/s32/s84/s45/s49\n/s41\n/s72/s32/s40/s84/s41\n/s48/s46/s52/s50/s32/s75\n/s72/s124/s124 /s99/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s50/s52/s100/s77/s47/s100/s72/s32\n/s40/s49/s48/s45/s52\n/s109\n/s66/s32/s84/s45/s49\n/s41\n/s72/s32/s40/s84/s41\nFIG. 3. (a) Isothermal magnetization of NaErSe 2up to 7 T\nfrom 0.42 K to 50 K at H/bardblabandH/bardblc. The pink dash lines\nshow the fitting from the CEF. (b) Isothermal magnetization\nof KErSe 2up to 7 T from 0.42 K to 50 K at H/bardblabandH/bardblc.\nThe pink dash lines show the fitting from the CEF at 0.42\nK. (c-d) Derivation of isothermal magnetization of NaErSe 2\nand KErSe 2from 0 T to 1 T at H/bardblab. Inset: Derivation of\nisothermal magnetization of NaErSe 2andKErSe 2from 0T to\n1 T along H/bardblcat 0.42 K. (e-f) AC susceptibility of NaErSe 2\nand KErSe 2. Inset: The isothermal magnetization of KErSe 2.\nThe dash line indicates the moment and field range of the\nplateau.\nis much larger than NaErSe 2. This maximum in χis ob-\nserved as high as 3 K and is present for fields between\napproximately 4 and 6 T. The origin of this feature may\nbe the population of the CEF states, long-range order\nor short-range magnetic interaction between Er3+ions.\nWe could rule out the long-range magnetic order since\nnoλanomaly is found in the same temperature range in\nspecific heat measurements (Fig. 4). The mixture of the\nCEF states with magnetic fields in KErSe 2could poten-\ntially contribute to the broad peak in the temperature\ndependence of magnetization.\nThe isothermal magnetization is sensitive to the tran-\nsition induced by the magnetic field. We measured5\nthe isothermal magnetization from 0.42 K to 50 K for\nNaErSe 2and KErSe 2up to 7 T, as shown in Fig. 3 (a)\nand Fig. 3 (b). The magnetic moments are not saturated\nat 0.42 K and 7 T for either compound and orientation.\nLarge anisotropy is found in both of them below 50 K. If\nwe linear extrapolated high field data to 0 T, we could\ngetstronganisotropymomentsforNaErSe 2:mab=5.6µB\nandmc=2.1µB. The anisotropic values are lower than\nthe expected value of Er3+which may be affected by the\nCEF. A subtle change of slope is found at the low mag-\nnetic field around 0.2 T presenting obviously by dM/dH\nin Fig. 3(c). This subtle feature appearsonlybelow 0.6K\nwiththemoment1.8 µB, whichiscloseto1/3of mab=5.6\nµBin NaErSe 2. The similar subtle hump is also found in\nthe KErSe 2when H∝bardblab, as shown in Fig. 3(d) as well.\nThe moment around 2.1 µBis even close to 1/3 of the\nextended moment mab=6.2µBin KErSe 2. Becausethere\nis no sublet kink near the same magnetic fields at H ∝bardblcin\nthe inset of Fig. 3(c-d), we can exclude the influence of\nthe impurities or other isotropic impacts. The possible\nphase transition or population of CEF states could in-\nduce the slope change. To confirmed this feature, we also\nmeasured AC susceptibility at 757 Hz below 1 K along H\n∝bardblabandH∝bardblc. The similar broad peaks are located only\natH∝bardblabas shown in Fig. 3(e-f). The magnetic fields of\nthe maximum are almost identical with the DC magne-\ntization in Fig. 3(c). Besides these features at H∝bardblab, an\napparent plateau with upturn at high field is recognized\nonly in the KErSe 2atH∝bardblc, as shown in Fig. 3(b). The\nfield rangeof the plateau is 2 T to 4.5 T indicating by the\nverticalgreenlinesin theinset ofFig.3(f). Thelinearex-\ntended dash line indicates 2.3 µB. The length of plateau\ndecreases as increasing the temperature and finally fade\nout to curved behavior at 2K. It is impressive to find\ndifferent features along two directions implying the large\nanisotropic CEF state of Er3+and possible short-range\ninteraction between Er3+. In addition, it is worth noting\nthat the compound is not long magnetic ordering in this\ntemperature range even with the field induced plateau-\nlike transition. This behavior is rare in the frustrated\nsystem.\nTo investigate the magnetic entropy at low temper-\nature, we measure the heat capacity of NaErSe 2and\nKErSe 2downtothe0.4K,asshowninFig.4(a). Thereis\nnodiscernible λanomalydownto0.4Kwhichagreeswith\nthe magnetizationmeasurement. We synthesize nonmag-\nnetic compounds NaLuSe 2and KLuSe 2to estimate the\nphonon contribution, which are also presented in Fig.\n4(a). The magnetic entropy release from high temper-\nature around 50 K in both compounds. This temper-\nature is much higher than the Yb-112 compounds in-\ndicating the strong CEF with Er3+. Two broad peaks\nare observed in both samples from 0.4 K to 10 K. The\nmulti-level Schottky or the short-range magnetic order\nmay cause these peaks in the heat capacity. The low-\ntemperature peak moves to high temperature with the\nmagnetic field. Due to the low-temperature peak over-\nlaps with the high-temperature peak above 3 T, it is/s49 /s49/s48 /s49/s48/s48/s49/s69/s45/s52/s48/s46/s48/s48/s49/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48/s49/s48/s48/s49/s48/s48/s48\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s53/s49/s48\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s53/s49/s48\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s53/s49/s48/s49/s53/s67/s32/s40/s74/s32/s109/s111/s108/s45/s49\n/s32/s75/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s32/s78/s97/s69/s114/s83/s101/s50 /s32 /s32/s75/s69/s114/s83/s101/s50\n/s32/s78/s97/s76/s117/s83/s101/s50 /s32 /s32/s75/s76/s117/s83/s101/s50\n/s32/s78/s97/s40/s67/s69/s70/s41/s32 /s32/s75/s40/s67/s69/s70/s41\n/s48/s32/s84/s40/s97/s41/s40/s98/s41\n/s32/s48/s32/s84/s32 /s32/s49/s32/s84\n/s32/s51/s32/s84/s32 /s32/s53/s32/s84\n/s32/s55/s32/s84/s67/s32/s40/s74/s32/s109/s111/s108/s45/s49\n/s32/s75/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s78/s97/s69/s114/s83/s101\n/s50\n/s75/s69/s114/s83/s101\n/s50/s40/s99/s41/s67/s32/s40/s74/s32/s109/s111/s108/s45/s49\n/s32/s75/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s32/s48/s32/s84/s32 /s32/s49/s32/s84\n/s32/s51/s32/s84/s32 /s32/s53/s32/s84\n/s32/s55/s32/s84/s48/s32/s84/s40/s100/s41/s83/s32/s40/s74/s32/s109/s111/s108/s45/s49\n/s32/s75/s45/s49\n/s41\n/s84/s32/s40/s75/s41/s32/s78/s97/s69/s114/s83/s101\n/s50\n/s32/s75/s69/s114/s83/s101\n/s50/s48 /s49/s48 /s50/s48/s48/s52/s56/s48/s46/s51 /s48/s46/s54 /s48/s46/s57/s48/s53/s49/s48/s67/s47/s84/s40/s49/s48/s45/s50\n/s74/s32/s109/s111/s108/s45/s49\n/s32/s75/s45/s50\n/s41\n/s84/s32/s40/s75/s41\n/s48/s46/s51 /s48/s46/s54 /s48/s46/s57/s48/s53/s49/s48/s67/s47/s84/s32/s40/s49/s48/s45/s50\n/s74/s32/s109/s111/s108/s45/s49\n/s32/s75/s45/s50\n/s41\n/s84/s32/s40/s75/s41\nFIG. 4. (a) The temperature dependence of heat capacity of\nNaErSe 2, KErSe 2, NaLuSe 2, KLuSe 2and CEF fitting from\n0.4Kto150K.TheinsetpresenttheheatcapacityofNaErSe 2\nand KErSe 2below 20 K. (b) Temperature dependence of C of\nNaErSe 2under different magnetic fields. The inset shows the\ntiny peaks at 3 T, 5 T, 7 T. (c) Temperature dependence of\nC of KErSe 2under different magnetic fields. The inset show\nthe tiny peaks at 3 T, 5 T, 7 T. (d) Temperature dependence\nof the magnetic entropy of NaErSe 2and KErSe 2.\nhard to distinguish these two peaks. The combined\nhigh-temperature peak does not change the magnitude,\nwhich strongly suggests the Schottky effects from the\nCEF above 3 T. However, the magnitude of the peak\nchanges below 3 T and the high temperature peak shift\nslightly to low temperature at 1 T. These features in-\ndicate the entropy changes across 3 T by the possible\nmixtured CEF states or short range effects. The larger\nmagnetic field( >3 T) could induce a tiny anomaly below\n0.8 K, as shown in the inset of Fig. 4(b-c). The tem-\nperature of this tiny transitions increases with enhancing\nthe magnetic field which violates antiferromagnetic tran-\nsition. This feature indicates the possible subtle interac-\ntion of CEF states with different fields. At last, we want\nto estimate the entropy release in NaErSe 2and KErSe 2\nat 0 T. The entropy from 0.4 K to 200 K is close to 13 J\nmol−1K−1in both compounds. This value is larger than\nthe Rln2 and much smaller than the expected 23 J mol−1\nK−1forEr3+. This indicateslargeentropyreleasesbelow\n0.4 K or possible high energy CEF exits.\nWe use a CEF model to fit the magnetization vs tem-\nperature and field data to examine the role of single ion\neffects in these two systems. CEF Hamiltonian of D3d\nsymmetry is HCEF\nD3d=B0\n2ˆO0\n2+B0\n4ˆO0\n4+B3\n4ˆO3\n4+B0\n6ˆO0\n6+\nB3\n6ˆO3\n6+B6\n6ˆO6\n6. This point symmetry splits the J=15/2\nstates into the 3 Γ+\n45doublets and 5 Γ+\n6doublets. The6\nfitted curves show good agreements with the magnitude\nof the M(H) data and the high temperature data. The\nground state doublet is reliably Γ+\n6with the Γ+\n45as the\nfirst excited state around 10-25 K for both samples. The\npoor agreements at low fields or low temperature reveal-\ningthattheCEFmodelisnotsufficientalone. Thefitting\nshows a plateau feature could be induced by the mixture\nof CEF states with magnetic field. Considering the two\nbroadpeaksoverlapandformSchottky-likepeakbetween\n1 T and 3 T in the heat capacity, the plateau for H ∝bardblc in\nKErSe 2is suggestedas the metamagnetictransition from\nCEF. The short-range interaction between Er3+ions or\nlow-lying CEF levels could cause the anomalies at low\nfield for H ∝bardblab and the deviation below 2 K. A detailed\nstudyofCEFisexpectedtobedoneinneutronscattering\nmeasurement.\nIV. CONCLUSION\nIn conclusion, we present the experimental results\nof magnetization and heat capacity measurement on\nAErSe2(A=Na and K) single crystals. Large anisotropy\nis found below 40 K along two directions H∝bardblabandH∝bardblc.\nThe isothermal magnetization shows a subtle kink fea-\nture only H∝bardblabat low temperature in both compounds.\nKErSe 2also shows a wide plateau-like metamagnetic\ntransition for H∝bardblc. The heat capacity show two broad\npeaks below 10 K at zero field. All results indicate\nthatAErSe2are strongly anisotropic, frustrated magnetswith field-induced transitions at low temperatures. No\nlong-range magnetic order implies that these materials\nare candidates for hosting a quantum spin liquid ground\nstate.\nV. ACKNOWLEDGEMENTS\nThe research is supported by the U.S. Department of\nEnergy (DOE), Office of Science, Basic Energy Sciences\n(BES), Materials Science and Engineering Division. The\nX-ray diffraction analysis by RC was supported by the\nUS Department of Energy, Office of Science, Basic En-\nergy Sciences, Chemical Sciences, Geosciences, and Bio-\nsciences Division. Work at Florida by J. S. Kim and G.\nR. S. supported by the US Department of Energy, Basic\nEnergy Sciences, contract no. DE-FG02-86ER45268\nNotice: This manuscript has been authored by UT-\nBattelle, LLC under ContractNo. DE-AC05-00OR22725\nwith the U.S. Department of Energy. The United\nStates Government retains and the publisher, by ac-\ncepting the article for publication, acknowledges that\nthe United States Government retains a non-exclusive,\npaid-up, irrevocable, world-wide license to publish or re-\nproduce the published form of this manuscript, or al-\nlow others to do so, for United States Government pur-\nposes. 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We find that the\nmodel induces multifarious magnetic orders depending on th e Fermi wavenumber and gives a good\nstarting point for understanding the experimental observa tion. In addition, we predict the presence\nof an undiscovered novel magnetic order called cuboc order a t large Fermi wavenumber region.\nPACS numbers: Valid PACS appear here\nRecent experimental studies on Tsai-type quasicrys-\ntals including rare-earth ions [1, 2] have successively pro-\nvided startling discoveries of novel phenomena: valence\nfluctuation [3, 4], quantum criticality [5, 6] and super-\nconductivity [7, 8]. These phenomena are induced by\nstrongly-correlated electrons originating from rare-earth\nions, particularly Yb and Ce. The physics behind the\nphenomena might be similar to that of heavy-fermion\nmaterials. However, crucial roles of quasi-periodicity in\nthese phenomena still remains an open question in spit\nof recent theoretical efforts [9–13].\nTsai-type quasicrystalline approximants, which have\nthe same local structure as the quasicrystals but keep\nthe translational symmetry [14], have also attracted\nmuch attentions due to experimental discovery of vari-\nous magnetic orders [15], e.g., ferromagnetism, antifer-\nromagnetism, and spin-glass-like magnetism, both in bi-\nnary [16–20] and ternary [21–28] compounds. This is\nin contrast to quasicrystal where there is no report on\nmagnetic order so far [29–37]. In the approximants, the\nmagnetic moments located on rare-earth ions can inter-\nact each other via the Rudderman-Kittel-Kasuya-Yosida\n(RKKY) interaction [38–40]. Atomic composition in the\napproximants is a controlling parameter of the Fermi\nwavenumber kF, which changes spacial extension of the\nRKKY interaction. Therefore, an idea that the RKKY\ninteraction as a function of kFis crucial for understand-\ningthevariousmagneticordershasbeenputforward[15].\nHowever, the crystal structure of Tsai-type quasicrys-\ntals/approximantsis too complicated to theoretically an-\nalyze the magnetic behaviors. Actually, magnetic orders\nin three dimensional quasicrystals/approximants remain\nunclear from theoretical viewpoints, contrary to pioneer-\ningworksonmagnetisminlow-dimensionalquasiperiodic\nsystems [41–49].\nAu-Al-Gd system is one of 1/1 Tsai-type quasicrys-\ntalline approximants Au-X-RE (X=Al, Si; RE=Gd, Tb,\nHo, Dy) showing magnetic orders [22, 26, 50]. With\n∗sugimoto.takanori@rs.tus.ac.jpincreasing Au concentration, i.e., decreasing kF, mag-\nnetism in Au-Al-Gd changes from spin glass to ferromag-\nnetism and to antiferromagnetism [24, 26]. Since single-\nionanisotropydue tospin-orbitcouplingandcrystalfield\nis weak on a Gd ion, the Au-Al-Gd system is a good ma-\nterial to investigate interplay of RKKY interactions and\nthecomplicatedstructureinthe Tsai-typeapproximants.\nIn this paper, we theoretically investigate magnetic\nstates in Au-Al-Gd based on the classical approximation\nof localized quantum spins on Gd ions. The approxima-\ntion is justified by the fact that (i) magnetic moment on\nGd ions has been estimated to be µeff= 8.74µBfrom\nmagnetic susceptibility [24, 26] (cf. the effective moment\nof a single Gd3+ion is 7.94µB), which is large enough\nto be approximated as a classical spin, and (ii) magnetic\npropertiesinaTsai-type1/1approximantAu-Si-Tbwith\nthe same crystal structure as Au-Al-Gd but with strong\nsingle-ion anisotropy [20, 27] has been described well by\nthe classical approximation [50]. Calculating possible\nmagnetic orders in a simple model with both Gd ions\nin the Tsai-type 1/1 approximant structure and RKKY\ninteraction, we confirm a good qualitative accordance\nwith the experimental changeof magnetic orders[24, 26].\nIn addition, we predict an undiscovered magnetic order\ncalled cuboc order at large kFregion.\nA unit cell in the 1/1 approximant includes two Tsai\nclusters. Gd ions occupy icosahedral vertices of the Tsai\ncluster. The lattice of Gd ions corresponds to body-\ncentered cubic (bcc) of icosahedrons (see Fig. 1), and\nthe localized magnetic moments are located on the Gd\nions. Thus, there are 24 spins ( ns= 24) in the unit cell.\nThe minimum model Hamiltonian with the RKKY in-\nteraction only is given by\nH=−/summationdisplay\n|r−r′|<RcJ|r−r′|Sr·Sr′, (1)\nwhere the exchange energy between two spins, Srat\nrandSr′atr′, is given by J|r−r′|=Jf(2kF|r−r′|)\n(J >0) with the function of Friedel oscillation f(x) =\n(−xcosx+ sinx)/x4. In this work, we use a classi-\ncal approximation of quantum spin and thus Sris ex-\npressed as a three-dimensional normalized vector with2\nFIG. 1. Magnetic lattice of the Au-Al-Gd alloy. Red balls\ndenote Gd ions on vertices of icosahedron. The icosahedra\ncompose bcc structure.\n|Sr|= 1. In the 1/1 approximant Au-Al-Gd, the mag-\nnetic phases successively change with changing the Au\nconcentration [24, 26]. Here we assume that kFis de-\ntermined by the electron density nassociated with the\nAu concentration via kF= (3π2n)1/3. In the Hamilto-\nnian (1), we introduce a cutoff range Rcof the RKKY\ninteraction for simplicity of calculation.\nWe perform numerical calculation with the classi-\ncal Monte-Carlo (MC) method to obtain spin config-\nurations at zero temperature. In this calculation, we\nuse single-update heat-bath method combined with the\nover-relaxation technique and the temperature-exchange\nmethod. The system size is set to Nc= 8×8×8 unit\ncells, corresponding to Ns=Ncns= 12288 spins, with\nperiodic boundary condition. We take the number of\nreplicasNR= 200,thenumberofMCstepsforrelaxation\nNMC= 2400,andthelowesttemperature TM/J= 1.0−7.\nAfter performingthe MC simulation, weupdate the state\nuntil the energy converges at T= 0 to obtain the ground\nstate. We use Rc= 50˚A, which is larger than three times\nas long as the lattice unit a= 14.7˚A [51].kFis changed\nfrom 1.28˚A−1to 1.61˚A−1, where the experimentally de-\ntermined kFin Au-Al-Gd is included [24, 26].\nTo classify the spin configurations of the ground state\nusing our method, we consider commensurability defined\nby\nC=1\nns/summationdisplay\ni/bardbl/angbracketleftSi/angbracketright/bardbl (2)\nwith the averaged magnitude of spins /angbracketleftSi/angbracketrightover all unit\ncells:/angbracketleftSi/angbracketright=N−1\nc/summationtext\njSi,j, whereSi,jrepresents the i-\nth spin in the j-th unit cell. If the spin configuration is\ninvariant with respect to translation of the lattice, theF A A Cb IC IC \n1.0\n0.01.0\n0.01.0\n0.01.0\n0.0\n1.3 1.4 1.5 1.6\nFIG. 2. Phase diagram of the Hamiltonian (1) at zero tem-\nperature. Four panels represent commesurability C, ferro-\nmagnetic order OF, antiferromagnetic order OA, and cuboc\norderOCb, as a function of the Fermi wavenumber kF. Color-\nshaded regions denote commensurate phases, where only one\nof the order parameters OF,OA,OCbis finite: the ferromag-\nnetc (F) phase in red region, the antiferromagnetc (A) phase\nin blue region, the cuboc (Cb) phase in green region. The\ngray region corresponds to the incommensurate (IC) phase.\ncommensurability equals to unity. It should be noted\nthat the commensurability is zero in the case of two sub-\nlattice configuration; e.g., a spin in a unit cell is directed\nopposite to the corresponding spin in neighboring unit\ncells. Thus, this quantity is a measure of ferroic char-\nacter in the spin configuration. The top panel in Fig. 2\nshows the commensulabirity Cas a function of kF. The\nvalue of Calternates between 1 and 0 from kF= 1.28˚A\nto 1.61˚A, that is, magnetic state switches from commen-\nsurate to incommensurate states and vise versa several\ntimes in this region.\nTo clarify the commensurate state in detail, we con-\nsider ferromagnetic and antiferromagnetic order param-\neters defined by\nOF=1\nns/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/summationdisplay\ni/angbracketleftSi/angbracketright/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble, (3)\nOA=1\nns/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftBigg/summationdisplay\ni∈CCI/angbracketleftSi/angbracketright−/summationdisplay\ni∈BCI/angbracketleftSi/angbracketright/parenrightBigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble,(4)\nrespectively, where CCI (BCI) represents the set of posi-\ntions on the cubic-cornered icosahedron (body-centered\nicasahedron) in the unit cell. The second and third pan-\nels in Fig. 2 show these order parameters. We find that\nthese order parameters are equal to 0 or 1 in the whole\nregion.OF= 1 represents perfect ferromagnetism, while\nOA= 1 represents an antiferromagnetic phase with N´ eel3\n(a) (b)\nFIG. 3. Cuboc order. (a) An example of spin configuration\nwithOCb= 1. Each plane has four spins denoted by the\nsame color, e.g., the first to fourth spins belong to the xy\nplane. Neighboring spins in the same plane exhibit 90 degree\norder. (b) Spin configuration observed in the MC calculation\nwithkF= 1.60˚A−1. The spin configurations (a) and (b) are\nequivalent under a global O(3) rotation.\norderwhere spins in an icosahedronexhibit ferromagnetc\norder while spins located in a neighboring icosahedron\nhave the opposite direction. Antiferromagnetic phases\nare located in narrow regions in Fig. 2.\nIn Fig. 2, there is a commensurate region of neither\nferromagnetic nor antiferromagnetic states above kF=\n1.55˚A−1. In the bottom panel of Fig. 2, we successfully\nidentify this magnetic phase with so-called cubocorder\nusing a corresponding order parameter [52–54] defined\nby\nOCb=|/angbracketleftKxy/angbracketright·/angbracketleftKyz/angbracketright×/angbracketleftKzx/angbracketright| (5)\nwith the averaged vector chirality /angbracketleftKα/angbracketrightof neighboring\nspins in the α=xy,xz, andzxplanes given by\n/angbracketleftKα/angbracketright=1\n8/summationdisplay\n/angbracketlefti,i′/angbracketright∈αplane/angbracketleftSi/angbracketright×/angbracketleftSi′/angbracketright,(6)\nwhere the summation runs over all neighboring bonds\n/angbracketlefti,i′/angbracketrightin theαplane of icosahedron; e.g., /angbracketlefti,i′/angbracketright=/angbracketleft1,2/angbracketright,\n/angbracketleft2,3/angbracketright,/angbracketleft3,4/angbracketright, and/angbracketleft4,1/angbracketrightfor thexyplane in Fig. 3. Note\nthat there are two icosahedra in the unit cell, leading to\nthe normalization factor 1 /8 in (6). Supposing a per-\nfect 90 degree order represented in Fig. 3(a), we obtain\nOCb= 1. The cuboc order is invariant under global O(3)\nrotationofspin configuration. Forinstance, the spin con-\nfiguration in Fig. 3(b) obtained by the MC simulation\nwithkF= 1.60˚A−1coincides with the perfect 90 degree\norder in Fig. 3(a) by a proper global O(3) rotation.\nTo confirm phase boundaries determined by the order\nparameters, we calculate the total energy Eas a function\nofkFand its derivative d E/dkFas shown in Fig. 4(a).\nWe findclearjumps ind E/dkFattwoantiferromagnetic-\nincommensurate boundaries with kF= 1.29˚A−1and1.25\nr0.251.3 1.6Y10 r3\n0.01.0 2.0r2.01.0Y10 r31.3 1.60\nr25 (a)\n(b)\nA F A Cb IC IC \nExp. Region\nFIG. 4. (a) Total energy of the ground state versus the Fermi\nwavenumber kFand its derivative d E/dkF. In the incom-\nmensurate (IC) phases, black triangles denote points where\nthe derivative jumps/cusps indicating the the first-/secon d-\norder phase transition. Other anomalies of the derivative\ncoincide with the phase boundaries obtained in Fig. 2. (b)\nCurie-Weisse temperature θCWas a function of kF. The inset\nrepresents θCWin a wider region of kF, where shaded area is\nfromkF= 1.25˚A−1to 1.65˚A−1. The vertical dotted lines\ndenote the position of triangles in (a). The ferromagnetic ( F),\nantiferromagnetic (A), cuboc (Cb), and IC phases are taken\nfrom Fig. 2. The notation ‘Exp. Region’ indicates an region\nexperimentally investigated [24, 26], which is estimated b y\nelectron density in the Fermi gas approximation.\n1.535˚A−1and at a antiferromagnetic-cuboc bound-\nary with kF= 1.55˚A−1, indicating the first-order\nphase transition. At two incommensurate-ferromagnetic\nboundaries with kF= 1.315˚A−1and 1.41˚A−1, there\nis anomaly like cusp, corresponding to the second-order\nphase transition. In addition to the expected anomaly\nat the phase boundaries determined by the order pa-\nrameters, we find several anomalies in d E/dkFwithin\nthe incommensurate phases in Fig. 4(a) as denoted by\nblack triangles. These anomalies suggest the presence of\ninternal magnetic structures inside the incommensurate\nphase, which have not been detected by the order param-\neters examined above. The detailed study on the internal\nstructures remains as a future work.4\nIn order to make clear the physical origin of magnetic\nphases, we plot the Curie-Weisse temperature θCWin\nFig. 4(b), which is determined by the sum of total ex-\nchangeenergies θCW= (3Ns)−1/summationtext\n|r−r′|<RcJ|r−r′|based\non high-temperature expansion formalism. The inset in\nFig. 4(b) shows an alternation of θCWbetween positive\nandnegativevaluesinthewiderrangeof kF,comingfrom\noscillating nature of the RKKY interaction. We find that\na ferromagnetic phase appears when θCWis positively\nlarge as expected. On the contrary, antiferromagnetic\nphases emerge near θCW∼=0. This indicates that the\nantiferromagnetic state composed of ferromagnetic order\ninside an icosahedral cluster and antiferromagnetic spin\narrangementbetweentheneighboringclustersisachieved\nwhen the magnitude of intra-cluster ferromagnetic inter-\nactions is comparable to that of inter-cluster antiferro-\nmagnetic interactions. The cuboc phase also appears\nwhenθCW∼=0, indicating that ferromagnetic and anti-\nferromagnetic interactions are totally canceled out each\nother but there is strong frustration as evidenced from\n90 degree order. The incommensurate phase also needs\nstrong frustration, where partial antiferromagetic inter-\nactions may play a key role.\nLet us compare the calculated phases with experi-\nmentally observed ones in the Au-Al-Gd approximant.\nThe experimental data have shown the change of phases\nfromantiferromagnetictoferromagneticandtospin-glass\nphases with increasing kF[24, 26]. The region of experi-\nmentally determined kFbased on the Fermi gas approx-\nimation is denoted in Fig. 4(b) by “Exp. Region”. In\nthe region, however, our calculated phases are only fer-\nromagnetic and incommensurate phases. Therefore, we\nhave two inconsistencies: (i) antiferromagnetic phase ob-\nserved in the experiment does not exist in the region and\n(ii) spin glass phase is not obtained in our calculation.\nWe speculate that the inconsistencies could be ex-\nplainedbytheeffectofrandomnessduetochemicaldisor-\nders and defects in real Au-Al-Gd alloys. First, the ran-\ndomness will shorten the mean free path of conduction\nelectrons, andthusthecutoffrangeofthe RKKYinterac-\ntion,Rc, would be shorter than the assumed Rc= 50˚A.\nThe shortened cutoff range suppresses absolute value of\nthe Curie-Weisse temperature θCWand shifts the peak\nofθCWatkF= 1.35˚A−1in Fig. 4(b) to larger kF\n(not shown). The change of θCWcan narrow the fer-\nromagnetic phase and may shift it to larger kF, accom-\npanied by the shift of antiferromagnetic phase around\nkF= 1.30˚A−1possibly to the estimated experimentalregion. This will explain the inconsistency (i). The ran-\ndomness may also explain the inconsistency (ii) about\nthe absence of spin glass state in our result. In gen-\neral, spin glass state is induced by not only frustration\nbut also randomness. Since the incommensurate state in\nour phase diagram is expected to originate from strong\nfrustration as discussed above, including randomness in\nour model would bring a spin glass behavior in coopera-\ntion with strong frustration in a certain range of incom-\nmensurate phase. This is one of possible explanations of\nthe inconsistency (ii). However, if so, we unfortunately\nmeet another problem that experimentally observed θCW\nis negative in the spin glass phase [26] in contrast to the\ncalculated positive θCWin our incommensurate phase.\nIn order to resolve this new problem, we may need treat\nanisotropy of the crystal field around Gd ions, inhomo-\ngeneity ofchemical disorders, and magnetic dipole-dipole\ninteractions as precise as possible. This remains as an\nimportant problem to be solved in the future.\nIn summary, we have theoretically investigated mag-\nnetic phases in the Tsai-type quasicrystalline approxi-\nmant Au-Al-Gd alloy using a minimal magnetic model\nincluding the RKKY interaction only. The change of\nAu/Al concentration in the alloy is assumed to corre-\nspond to the change of the Fermi wavenumber in our\nmodel. Classical MC calculations have been performed\nto obtain spin configurations at zero temperature. We\nhave found a successive change among antiferromag-\nnetic, ferromagnetic, and incommesurate phases. At\nlarge Fermi wavevector, to which real Au-Al-Gd alloy\nhave not reached yet, we have discovered a cuboc phase.\nRecently such a cuboc order has attracted much atten-\ntion as an exotic magnetic order experimentally observed\nin kagome spin compounds [55, 56] and as an origin of\ntheoretically predicted anomalous magnetic behavior in\na spin tube [57, 58]. Therefore the 1/1 Tsai-type ap-\nproximant Au-Al-Gd alloy is one of possible candidates\nfor new magnetic phenomena associated with the cuboc\norder.\nWe would like to thank R. Tamura, T. J. Sato and\nT. Hiroto for fruitful discussions. This work was partly\nsupported by Challenging Research Exploratory (Grant\nNo. JP17K18764), Grant-in-Aid for Scientific Research\non Innovative Areas (Grant No. JP19H05821). Numeri-\ncal computation in this work was carried out on the Su-\npercomputer Center at Institute for Solid State Physics,\nUniversity of Tokyo and the supercomputers at JAEA.\n[1] A. P. Tsai, A. Niikura, A. Inoue, T. Masumoto, Y.\nNishida, K. Tsuda, and M. Tanaka, Philos. Mag. Lett.\n70, 169 (1994).\n[2] A. P. Tsai, J. Q. Guo, E. Abe, H. Takakura, and T. J.\nSato, Nature (London) 408, 537 (2000).\n[3] D. Kawana, T. Watanuki, A. Machida, T. Shobu, K.Aoki, andA. P. Tsai, Phys. Rev.B 81, 220202(R) (2010).\n[4] T. Watanuki, S. Kashimoto, D. Kawana, T. Yamazaki,\nA. Machida, Y. 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Jpn.\n86, 114701 (2017)."    },    {        "title": "1912.07374v1.3D_printing_of_polymer_bonded_anisotropic_magnets_in_an_external_magnetic_field_and_by_a_modified_production_process.pdf",        "content": "3D printing of polymer-bonded anisotropic magnets in an external magnetic field and by a modified\nproduction process\nKlaus Sonnleitner,1, 2,\u0003Christian Huber,1, 2Iulian Teliban,3Spomenka Kobe,4Boris Saje,5Daniel\nKagerbauer,6Michael Reissner,7Christian Lengauer,8Martin Groenefeld,3and Dieter Suess1, 2\n1Physics of Functional Materials, University of Vienna, 1090 Vienna, Austria\n2Christian Doppler Laboratory for Advanced Magnetic Sensing and Materials, 1090 Vienna, Austria\n3Magnetfabrik Bonn GmbH, 53119 Bonn, Germany\n4Department of Nanostructured Materials, Jožef Stefan Institute, 1000 Ljubljana, Slovenia\n5Kolektor Magnet Technology GmbH, 45356 Essen, Germany\n6Atominstitut, TU Wien, 1020 Vienna, Austria\n7Low Temperature Facilities, TU Wien, 1040 Vienna, Austria\n8Department of Mineralogy and Crystallography, University of Vienna, 1090 Vienna, Austria\n(Dated: December 17, 2019)\nThe possibility of producing polymer-bonded magnets with the aid of additive processes, such as 3D printing,\nopens up a multitude of new areas of application. Almost any structures and prototypes can be produced cost-\neffectively in small quantities. Extending the 3D printing process allows the manufacturing of anisotropic\nmagnetic structures by aligning the magnetic easy axis of ferromagnetic particles inside a paste-like compound\nmaterial along an external magnetic field. This is achieved by two different approaches: First, the magnetic field\nfor aligning the particles is provided by a permanent magnet. Secondly, the 3D printing process itselfs generates\nan anisotropic behavior of the structures. An inexpensive and customizable end-user fused filament fabrication\n3D printer is used to print the magnetic samples. The magnetical properties of different magnetic anisotropic Sr\nferrite and SmFeN materials are investigated and discussed.\nFused Filament Fabrication (FFF) is a well established ad-\nditive manufacturing (3D printing) process using thermoplas-\ntic materials [1, 2]. Key advantages of FFF are: It is an af-\nfordable, fast and end-user friendly manufacturing process.\nThis work deals with the 3D printing of polymer-bonded and\nmagnetic anisotropic magnets. Polymer-bonded magnets are\ncomposed of magnetic powder and a polymer matrix. There-\nfore, the maximum energy product (BH)maxof bonded mag-\nnets is limited compared to sintered magnets. Nevertheless,\nmany applications need an accurate and complex magnetic\nfield distribution instead of highest field strengths, for ex-\nample in sensor and electric drive technology [3]. Polymer-\nbonded magnets can be produced from magnetic powder that\nis magnetically isotropic or anisotropic. If a high energy prod-\nuct(BH)maxof polymer-bonded magnets is not the most im-\nportant parameter, magnetic isotropic powder is preferred be-\ncause it is associated with lower costs and more flexibility.\nOn the other hand, anisotropic magnets can only be magne-\ntized in one certain direction, which can be restrictive with re-\nspect to shape. For producing polymer-bonded magnets, mag-\nnetic powder is mixed with a binder such as a thermoplastic\npolymer. The resulted compound can be used for injection\nmolding, compression and for extrusion. For the FFF process,\nthermoplastic or compounds are extruded into wire-shaped fil-\naments. Due to the high filler content of the compound, the\nviscosity increases which can lead to filling and flowing prob-\nlems. Polyamides such as PA6, PA11 and PA12 offer a good\ncombination of viscosity, bonding and mechanical properties\nsuch as a high tensile- and impact strength [4, 5]. For 3D\nprinting, PA6 is more difficult to process because the water\n\u0003klaus.sonnleitner@univie.ac.atabsorption capacity is significantly higher compared to PA12:\nPA6 absorbs approximately 1 :13 wt%, while PA12 absorbs\n0:15 wt% [6]. To prevent printing errors, PA6 compounds\nhave to be dried at around 90\u000eC for 8 h.\nCertain applications of magnetic designs have complex re-\nquirements on the magnetic field distribution, for example\nhighest possible magnetic field homogeneity in a certain vol-\nume. 3D printing of isotropic NdFeB magnets with focus on\ntailoring the external stray field by topology optimisation of\nthe magnet was successfully implemented. [7–9]. With Big\nArea Additive Manufacturing (BAAM) isotropic NdFeB mag-\nnets were manufactured [10], where a special heat and extrud-\ning unit is needed and only large structures can be realized.\nGoal of this work is to produce magnetic anisotropic polymer-\nbonded magnets with the help of two different approaches by\nan end user 3D printer: Printing directly on the surface of a\nstrong permanent magnet and investigating how the printing\nprocess itself affects the magnetic anisotropy of the sample.\nThree different magnetic anisotropic compounds are inves-\ntigated:\n1. Strontium Hexaferrite inside a PA6 matrix\n(Sproxr10/20p), fill grade: 49 vol%,\n2. Strontium Hexaferrite inside a PA12 matrix\n(Sproxr11/22p), fill grade: 53 vol%,\n3. Sm 2Fe17N3inside a PA12 matrix, fill grade: 44 vol%.\nThe two Sproxrcompounds are prefabricated by the Mag-\nnetfabrik Bonn. Sm 2Fe17N3and the thermoplastic PA12 are\ncompounded with twin screw extruder at the University of\nLeoben, Austria[11]. The advantage of these ferrite powders\nis that they are inexpensive but have a lower (BH)maxcom-\npared to rare earth materials. The volume filler constant forarXiv:1912.07374v1  [cond-mat.mtrl-sci]  16 Dec 20192\nthese materials is determined by the Loss on Ignition (LOI),\nwhereby the sample is heated to 1100\u000eC and the plastic is\nevaporated[12].\nPictures generated with a Scanning Electron Microscope\n(SEM) reveal information of the microstructure of the ma-\nterials. The Strontium Hexaferrite grains are platelets with\n6\u00022\u00022 µm3(L\u0002W\u0002H)and with the basic magnetic prop-\nerties of Br=196mT and with HcJ=183kA =m, Fig. 1 (b) and\n(c). Sm 2Fe17N3particles are approximately spherical with a\ndiameter of 3 to 4 µm with Br=1:31T and HcJ=889kA =m,\nFig. 1 (a).\nFIG. 1: Images of a Scanning Electron Microscope: (a)\nSm2Fe17N3+PA12, (b) Sproxr10/20p and (c)\nSproxr11/22p.\nBy applying an external magnetic field with varying\nstrength during the printing process, the minimum alignment\nfield to orient the magnetic particles can be determined. The\nSproxrand Sm 2Fe17N3grains are particles with uniaxial\nanisotropy and show one magnetic easy axis. In our case, we\nare using a permanent magnet to generate an external field\nduring the printing process. The used magnets provide fields\nof approximately 550 mT for a 40 \u000240\u000240 mm3(L\u0002W\u0002H)\nNdFeB magnet in the center and 150 mT for a plate cylinder\nwith a diameter of 25 mm and a thickness of 3 mm. The cal-\nculation of the flux density along the symmetry axis of a cube\nand a cylinder magnet is performed analytically [13]. The dis-\ntance between the permanent magnet and the printed sample\nis varied. The distances are chosen to create a Bzof 200, 150\nand 100 mT. As the thermoplastic matrix melts, the particles\ncan orient itself under the external field to form an anisotropic\nmagnet. It is necessary to print the magnets as close as possi-\nble to the center of the permanent magnet since the field gra-\ndients increase with increasing distance from the symmetry\naxis. As a consequence a force acts on the magnetic particle\nwhich leads to deflection and a unfavorable magnetization of\nthe sample. The advantage of this method is that the printer\ndoes not require any modifications, only the permanent mag-\nnet has to be attached to the printing bed. According to the\nStoner-Wohlfarth model the remanent magnetization mrand\nthe coercivity hcare reduced by a factor of 0 :5 for an assem-bly of randomly oriented particles in comparison to a fully\naligned structure [14]. This means mr=ms=0:5 for an assem-\nbly of not aligned, single domain Stoner-Wohlfarth particles\nandmr=ms=1 for an assembly of parallel to the easy axis\naligned, single domain particles, where msis the saturation\nmagnetization. The hysteresis loop is measured in the Vibrat-\ning Sample Magnetometer (VSM) and reveals how good the\nparticles in the matrix are aligned. The samples examined are\ncubes and are printed with a modified FFF printer. Cubes with\nthe size 10\u000210\u00025 mm3(L\u0002W\u0002H)are printed and then\nprocessed to a smaller format: 3 \u00023\u00023 mm3(L\u0002W\u0002H).\nThis was necessary because printing the sample directly is\ndifficult and the VSM supports only geometries smaller than\n4\u00024\u00024 mm3(L\u0002W\u0002H)where the dipole approximation\nis still valid. The used printer, Velleman’s K8200, allows\nthe exchange of components, since the printer software Mar-\nlin is open source [15] and temperatures up to 350\u000eC can\nbe reached by replacing the stock extruder with E3D’s Titan\nAero. The viscosity of the material can be reduced by increas-\ning the printing temperatures, which has a positive effect on\nprocessing. The optimal print parameters for the three mate-\nrials are shown in Tab. I.\nTABLE I: Best parameter settings for printer and Slic3r[16],\nempirically found.\nParameter Value\nExtruder temp 300\u000eC\nLayer height 0 :25 mm\nFill density 100 %\nFill pattern rectilinear\nPrinter speed 15 to 20 mms\u00001\nBuild platform painter’s tape\nBed temperature 40\u000eC\nFig. 2 shows the hysteresis loops of all three materials in\nhard and easy axis, with samples printed at different external\nmagnetic flux densities m0Hext=100, 150 and 200 mT and\nTab. II lists the corresponding ratio mr=mswith the remanence\nmrand the saturation magnetization ms. The internal field\nm0Hint=m0Hext\u0000Nmis displayed on the x-axis, assuming\na demagnetization factor of N=1=3 and mis the magnetiza-\ntion. This applies to a perfect cube and for real cubes small\ndeviations are to be expected [17].\nTABLE II: The ratio mr=msin magnetic easy axis of samples\nprinted at various magnetic flux densities.\nmr=ms 200 mT 150 mT 100 mT\nSprox 10/20p 0.88 0.76 -\nSprox 11/22p 0.94 0.42 0.4\nSm2Fe17N3+PA12 - 0.75 0.7\nFor both Sproxrmaterials an external flux density of\n200 mT is sufficient for aligning the particles because of their\nlower coercive field. However, 150 mT is the highest exter-\nnal field in which Sm 2Fe17N3+PA12 can be printed, which\nis not enough for alignment. Larger flux densities destroy the3\n−101\nSprox\n10/20pm/mmax\n−4−3−2−101234Sm2Fe17N3\n+ PA12\nµ0Hint(T)\n−4−3−2−101234−101\nSprox\n11/22p\nµ0Hint(T)m/mmaxhard axis easy axis\n200 mT\n150 mT\n100 mT\nFIG. 2: Hysteresis loops in hard- and easy axis of three\nsamples measured with the VSM, where Hintis the internal\nfield, considering a demagnetization factor of N=1=3. The\nsamples are aligned with different magnitudes of the external\nfield.\nshape of the sample due to the larger remanence magnetiza-\ntion of the material compared to Sproxr. Also the density\nand therefore the remanence of the printed sample is reduced\ncompared to the raw compound, Tab. III. All values in the\ntable, measured and from the data sheet, refer to anisotropic\nsamples.\nTABLE III: Br1,HcJ1andr1measured properties of printed\nanisotropic sample, Br2,HcJ2andr2according to datasheet.\nFor Sm 2Fe17N3+PA12 the printed anisotropic ( Br1) and\nprinted isotropic ( Br2) samples are compared.\nSprox 10/20p Sprox 11/22p Sm2Fe17N3+PA12\nBr1(mT) 201 220 308\nHcJ1(kA=m) 162 281 565\nBr2(mT) 222 225 389a\nHcJ2(kA=m) 207 239 899\nr1(gcm\u00003) 2.861 2.962 3.404\nr2(gcm\u00003) 3.2 3.2 3.796b\nBr1=Br2 0.91 0.98 0.79\nrr1=rr2 0.89 0.91 0.9\nacalculated\nbdensity measurement of compound, no data sheet available\nA further functional principle is being investigated in which\nthe printing direction is observed in relation to the orientation\nof the particles, the so-called \"flow anisotropy\". The magnetic\nhard and easy axis depend on the geometry: The magnetic\neasy axis of the Sproxrparticles is perpendicular to the long\nside of their cuboid geometry. The orientation of the parti-\ncles is shown with a light microscope in reflected light mode.\nFig. 3 (a) shows an isotropic sample and Fig. 3 (b) a sample\nthat is printed under an external field of 545 mT. By com-\nparing the images in Fig. 3 one can see the alignment of theparticles in the external magnetic field in vertical direction.\nFIG. 3: Reflected light microscope images: (a) Isotropic\nSproxr11/22p (b) aligned anisotropic particles of the printed\nmagnet in an external magnetic field of 545 mT.\nThe cuboid structure of Sproxrallows mechanical orien-\ntation over the printing direction whereas the spherical struc-\nture of SmFeN remains unaffected. Fig. 4 (a) shows how this\nidea is realized: When a cuboid is printed consisting only of\nperimeters, areas arise where print direction is the same over\nthe whole volume of the sample. For soft magnetic materials,\nthe influence of the 3D printing process was investigated by\nPatton et al. [18].\nHysteresis loops with the applied field parallel to and or-\nthogonal to the print direction were performed in the VSM\nwith the 3\u00023\u00023 mm3(L\u0002W\u0002H)cubes. Fig. 4 (b) shows\nthe hysteresis loops of the different samples. The SmFeN\nsample shows no difference between the two measured direc-\ntions, whereas for the Sprox samples a difference is visible.\nThis is a result of the Sprox aligned in the printing direction\nand no alignement of the SmFeN sample. The values of the\nnormalized remanance mr=msare shown in Tab. IV.\nTABLE IV: The ratio mr=msin parallel and perpendicular\ndirection of print - movement.\nmr=msk- direction?-direction\nSprox 10/20p 0.58 0.39\nSprox 11/22p 0.62 0.4\nSmFeN 0.66 0.67\nThis paper presents two methods for aligning the easy axis\nof magnetic particles during 3D printing. A ferromagnetic\npowder inside a thermoplastic matrix is processed. The de-\nscribed techniques allow the printing of samples with a higher\nremanence Brthan isotropic powders. Compared to the pre-\nsented approaches, providing an external alignment field pro-\nduces better results in terms of mr=msbut the observed flow\nanisotropy shows, that the print process itself has an influ-\nence on the alignment of the particles. The results show that4\nFIG. 4: (a) Every vertical shell of the dark blue sample\n(10\u000210\u00026 mm3(L\u0002W\u0002H)) is printed in direction of the\nlight blue arrows consisting only of perimeters (walls). The\nsample for the VSM measurement (orange) is cut out of the\ndark blue sample: all layers including infill and thus the\nwhole volume (3\u00023\u00023 mm3(L\u0002W\u0002H)) of the resulting\ncube are printed in the same direction. (b) Hysteresis loops in\ndirection of printing and orthogonal to print direction of three\nsamples measured with the VSM, where Hintis the internal\nfield.a magnetic field Bzof approximately 200 mT is sufficient for\naligning the the hardmagnetic Strontium Hexaferrite particles\nin the Sproxrcompound materials. For SmFeN the thresh-\nold value stays unknown, since printing above external fields\nbigger than 150 mT is not possible right now. An improved\ncooling could contribute to the increase of the magnetic field\nduring printing SmFeN. As a next step the results motivate the\ndevelopment of a customized 3D printer capable of aligning\nthe particles arbitrarily during printing by applying a variable\nexternal magnetic field of 200 mT in the desired direction for\nthe alignment of the particle. After successfully implement-\ning this design, complex structures with special magnetic ca-\npabilities should be printable. This would be a breakthrough\nin the development and manufacturing of magnets with vary-\ning local magnetization direction that can not be produced by\nfabrication techniques that are state of the art.\nThe support of the CD-Labor AMSEN (financed by the\nFederal Ministry of Economics, Family and Youth, the Na-\ntional Foundation for Research, Technology and Develop-\nment) is acknowledged. The SEM sample preparation and\nimage generation is performed at the Faculty Center for Nano\nStructure Research at the University of Vienna, Austria.\n[1] I. Zein, D. W. Hutmacher, K. C. Tan, and S. H. Teoh, Bioma-\nterials 23, 1169 (2002).\n[2] S. A. Tofail, E. P. Koumoulos, A. Bandyopadhyay, S. Bose,\nL. O’Donoghue, and C. Charitidis, Materials today 21, 22\n(2018).\n[3] K. Elian and H. Theuss, in Proceedings of the 5th Electron-\nics System-integration Technology Conference (ESTC) (IEEE,\n2014) pp. 1–5.\n[4] J. Xiao, J. U. Otaigbe, and D. C. Jiles, Journal of magnetism\nand magnetic materials 218, 60 (2000).\n[5] G. Salmoria, J. Leite, L. Vieira, A. Pires, and C. Roesler, Poly-\nmer Testing 31, 411 (2012).\n[6] J. J. Rajesh, J. Bijwe, B. Venkataraman, and U. Tewari, Journal\nof materials science 37, 5107 (2002).\n[7] C. Huber, C. Abert, F. Bruckner, M. Groenefeld,\nS. Schuschnigg, I. Teliban, C. V ogler, G. Wautischer, R. Windl,\nand D. Suess, Scientific reports 7, 9419 (2017).[8] C. Huber, C. Abert, F. Bruckner, M. Groenefeld, O. Muthsam,\nS. Schuschnigg, K. Sirak, R. Thanhoffer, I. Teliban, C. V ogler,\net al. , Applied Physics Letters 109, 162401 (2016).\n[9] C. Huber, C. Abert, F. Bruckner, C. Pfaff, J. Kriwet, M. Groene-\nfeld, I. Teliban, C. V ogler, and D. Suess, Journal of Applied\nPhysics 122, 053904 (2017).\n[10] L. Li, A. Tirado, I. Nlebedim, O. Rios, B. Post, V . Kunc,\nR. Lowden, E. Lara-Curzio, R. Fredette, J. Ormerod, et al. , Sci-\nentific reports 6, 36212 (2016).\n[11] M. Kracalik, S. Laske, M. Gschweitl, W. Friesenbichler, and\nG. R. Langecker, Journal of Applied Polymer Science 113,\n1422 (2009).\n[12] O. Heiri, A. F. Lotter, and G. Lemcke, Journal of paleolimnol-\nogy25, 101 (2001).\n[13] J. Camacho and V . Sosa, Revista mexicana de física E 59, 8\n(2013).\n[14] C. Tannous and J. Gieraltowski, European journal of physics5\n29, 475 (2008).\n[15] S. Moore, P. Armstrong, T. McDonald, and M. Yampolskiy, in\n2016 Resilience Week (RWS) (IEEE, 2016) pp. 46–51.\n[16] A. Ranellucci, Canessa, E., Fonda, C., and Zennaro, ed.,“Low-\ncost 3D Printing for Science, Education, and Sustainable De-velopment , 75 (2013).\n[17] A. Aharoni, Journal of applied physics 83, 3432 (1998).\n[18] M. V . Patton, P. Ryan, T. Calascione, N. Fischer, A. Morgen-\nstern, N. Stenger, and B. B. Nelson-Cheeseman, Additive Man-\nufacturing 27, 482 (2019)."    },    {        "title": "1912.12477v1.Manipulating_magnetic_fields_in_inaccessible_regions_by_negative_magnetic_permeability.pdf",        "content": "Manipulating magnetic fields in inaccessible regions by negative magnetic\npermeability\nRosa Mach-Batlle,1, 2Mark G. Bason,3Nuria Del-Valle,1and Jordi Prat-Camps4,∗\n1Departament de Fisica, Universitat Autonoma de Barcelona, Bellaterra 08193, Spain\n2Center for Biomolecular Nanotechnologies, Istituto Italiano di Tecnologia,\nVia Barsanti 14, 73010 Arnesano (LE), Italy\n3Department of Physics and Astronomy, University of Sussex,\nFalmer, Brighton BN1 9QH, United Kingdom\n4INTERACT Lab, School of Engineering and Informatics,\nUniversity of Sussex, Brighton BN1 9RH, United Kingdom\nThe control of magnetic fields underpins a wide variety of cutting-edge technologies, ranging\nfrom the guiding of micro-robots for medical applications to the trapping of atoms for quantum\nexperiments. Relying on different arrangements of coils, these technologies demand advances in the\nmanipulation of magnetic fields at a distance from the sources that generate them. Here, we present a\nstrategy for remotely shaping magnetic fields based on active magnetic metamaterials with negative\npermeability and we theoretically and experimentally demonstrate the possibility of emulating and\ncancelling magnetic sources at distance. Our strategy provides magnetic field focusing deep into\nphysically inaccessible volumes where neither sources of magnetic field nor magnetic materials can\nbe placed.\nMagnetism is a physical phenomenon that is found\nat the basis of a wide range of technologies, from\nconventional motors and generators to sophisticated\nbiomedical techniques and quantum technologies. In\nthe last decade, the advent of metamaterials has en-\nabled the manipulation of magnetic fields in unprece-\ndented ways, leading to magnetic cloaks that can\nmake objects magnetically undetectable [1, 2], mag-\nnetic hoses that can guide the magnetic field [3], and\neven magnetic wormholes that can connect two points\nthrough an undetectable path [4]. However, the de-\ngree of control one can exert over magnetic fields is\nrestrained by some fundamental limitations. One of\nthe most striking examples is the impossibility of gen-\nerating local magnetic field maxima in free space, as\nstated by Earnshaw’s theorem [5, 6]. This strong lim-\nitation prevents the stable magnetostatic trapping of\nparamagnetic and ferromagnetic materials, limits the\nability to concentrate magnetic fields, and makes it\nimpossible to exactly create and cancel magnetic field\nsources, such as magnets or electric wires, at a dis-\ntance.\nHere we propose a strategy for manipulating mag-\nnetic fields remotely that makes it possible to over-\ncome these stringent limitations. In particular, we\ndemonstrate the possibility of emulating the field of a\nmagnetic source (a straight current wire) at a distance\nfrom the structure that generates such field. Our\napproach is based on negative-permeability magnetic\nmaterials, which have recently been shown to exhibit\nappealing unconventional properties [7]. Such mate-\nrials do not naturally occur [8], but they can be effec-\ntively mimicked as active metamaterials consisting of\nprecise arrangements of currents [9]. The emulation of\nmagnetic sources remotely is not only scientifically in-\ntriguing but could also have deep impacts on several\nmagnetic field-based, cutting-edge technologies. For\nexample, it could improve the guidance of magneticmicrorobots and functional nano-particles for medical\napplications [10, 11], enhance techniques like transcra-\nnial magnetic stimulation [12] by providing strong and\nspatially-localized magnetic fields at target locations,\nopen new venues in atom trapping, or provide rad-\nical new ways of shielding undesired magnetic fields\nremotely.\nFigure 1: A magnetic replicator. Artistic sketch of the\ndevice. A long wire is surrounded by a magnetic replicator\ngenerating a replica of the wire outside it.arXiv:1912.12477v1  [physics.class-ph]  28 Dec 20192\nBased on transformation optics (see the Supplemen-\ntary Material), we demonstrate that a long cylindri-\ncal shell with relative magnetic permeability µ=−1\nacts analogously to a perfect lens for electromagnetic\nwaves [13, 14] in the magnetostatic limit. When this\nnegative-permeability shell surrounds a magnetic field\nsource the field distribution outside the shell cor-\nresponds to the field created by another magnetic\nsource, a replica source, which can appear at a dis-\ntance from its external surface, i.e. in free space (see\nFig. 1). Even though materials with negative perme-\nability do not exist, a cylindrical shell with µ=−1 can\nbe mimicked by two surface current densities, KM1(ϕ)\nandKM2(ϕ), flowing on its internal and external sur-\nfaces, respectively [7, 9]. Our goal now is to demon-\nstrate that these current densities can lead to the cre-\nation of a magnetic source at a distance. For the sake\nof generality, we focus on the creation of a wire; any\nother field source with translational symmetry along\nthe device’s axis can be regarded as the superposition\nof the field created by a combination of wires.\nWe solve the magnetostatic Maxwell’s equations\nconsidering a long (along z) cylindrical shell with\nµ=−1, internal radius R1and external radius R2\nwhich we coin a magnetic replicator . It surrounds\na long (along z) straight wire of current Iplaced at\n(x,y) = (d,0). The solutions show that the magnetic\nfield in the space beyond the replicator corresponds to\nthe field created by a replica wire of current Iplaced at\n(x,y) = (d/prime,0), whered/prime=dR2\n2/R2\n1. When the replica\nwire appears in empty space, d/prime>R 2, the field solu-\ntion in the region ρ∈(R2,d/prime), diverges [15, 16] (where\nρis the standard cylindrical radial coordinate). This\nannular region of divergent field solution makes it pos-\nsible to have a divergence of the field at the position\nof the replica wire without having a local field maxi-\nmum in empty space, thus ensuring that Earnshaw’s\ntheorem is obeyed (see the Supplementary Material)\nFrom the field distribution, we obtain the current\ndensities required to emulate a wire at a distance.\nBoth KM1(ϕ) and KM2(ϕ) are found as an infinite se-\nries but, while KM1(ϕ) is always convergent, KM2(ϕ)\nonly converges when the replica wire appears in the\ndevice volume ( d/prime≤R2). Therefore, the creation of\na replica wire in empty space ( d/prime> R 2) would re-\nquire infinite currents at the external boundary of the\ndevice. However, one can still approximate well the\nfield created by a wire at a distance by truncating the\nsummation in KM2(ϕ) up tonTterms. The higher the\nnumber of terms nT, the more the external field distri-\nbution resembles the field created by a wire (compare\nFigs. 2A-C with D). As shown in Fig. 2E, the spatial\nfocusing of the Bycomponent is greatly improved by\nadding more terms. The Bxcomponent also converges\nto the distribution created by a wire by increasing nT\n(see Fig. 2F).\nThe inspection of the role of the two current den-\nsities KM1(ϕ) and KM2(ϕ) leads to a simplification\nFigure 2: Numerical calculations of a magnetic replica-\ntor. (A)-(C) Plots of the normalized y-component of\nthe magnetic induction ByR2/(µ0I) along the plane XY\nfor a long (along z) straight wire carrying a current I\nplaced at (x,y) = (3/8R2,0) surrounded by the magneti-\nzation currents of a magnetic replicator of internal radius\nR1= 1/2R2, external radius R2and relative magnetic\npermeability µ=−1 centered at the origin of coordi-\nnates. The magnetization current density KM2is trun-\ncated to (A) nT= 5 terms, (B) nT= 10 terms, and\n(C)nT= 20 terms. (D) Calculation of the field created\nby a wire with current Iplaced at the position of the\nreplica, (x,y) = (3/2R2,0). Line plots of the normalized\ny-component (E) and x-component (F) of the magnetic in-\nduction along the dashed line in (A)-(D) ( x= 7/4R2) for\nthe casesnT= 5,nT= 10,nT= 20, and for the replica\nwire.\nof our device. The creation of a straight replica wire\ncarrying a current Iat a distance d/primefrom the shell\ncenter only requires a centered wire of current Iand\nthe surface current density\nKnT(ϕ) =nT/summationdisplay\nn=1I\nπR2/parenleftbiggd/prime\nR2/parenrightbiggn\ncos(nϕ)z (1)\nfed at the cylindrical surface ρ=R2(see details in the\nSupplementary Material). Therefore the information\non the explicit values of dandR1are irrelevant to our\nexperimental implementation.\nFor the experimental emulation of a current-\ncarrying wire at a distance, we constructed a cylindri-\ncal magnetic replicator of external radius R2= 40 mm\nand height 400 mm. We set the replica wire current\ntoI=−0.5A (the negative sign indicates the current3\nflows in the negative z−direction) and its position to\nd/prime= 60 mm. We truncated the summation in Eq.\n(1) up tonT= 10. The resulting continuous sheet\ncurrent density was converted into a discrete set of 20\nstraight current wires (current values and exact posi-\ntion of the wires can be found in the Supplementary\nMaterial). Figure 3A shows the field distribution ob-\ntained with numerical calculations when considering\nthis discrete set of wires surrounding a centered wire\nof currentI=−0.5 A.\nThe magnetic field was measured in the rectangu-\nlar region delimited in Fig. 3A using two miniature\nfluxgate magnetometers. The experimental results\nmatch very well with the results calculated numeri-\ncally for the metamaterial device, which confirms that\nthe magnetization currents of a shell with negative\npermeability can be used to create magnetic sources\nat a distance (Figs. 3B and 3C).\nThe remote creation of magnetic sources provides\na new strategy for focusing magnetic fields in inac-\nFigure 3: Calculations and measurements of a magnetic\nreplicator. (A) Plot of the calculated field |By|created by\nthe 21 currents (orange dots) forming the replicator with\nR2= 40mm and d/prime= 60mm. The calculation assumes the\nwires have a length (in z) of 400mm. The shaded rectangle\nindicates the measured area. (B) |By|field calculated at\nthe positions of the measurements. (C) Measured field\n|By|. Standard errors associated with the measurements\ncan be found in the Supplementary Material; values are\nsmaller than 0 .025µT everywhere.cessible volumes. Imagine a strong focusing of mag-\nnetic fields is required in a region where magnetic field\nsources cannot be placed, e.g. inside the body of a\npatient. One can place a magnetic replicator in the\naccessible volume (outside the body of that patient)\nin order to emulate the field of an image wire carrying\na currentIclose to the region where the focusing is\nrequired. Let us assume that the inaccessible region\nisx>x INand that the surface of the magnetic repli-\ncator is placed just at the border of the inaccessible\nregion (the device is centered at the origin of coordi-\nnates and its radius is R2=xIN). The magnetic field\ndistribution along different y−lines in the inaccessi-\nble region obtained using this strategy shows a much\nsharper peak than the field distribution that would be\nobtained by simply placing a wire with current Iat\nx=xIN(Fig. 4A). The gradient of the field, ∂By/∂y\nis thus greatly increased.\nMoreover, we experimentally demonstrated that the\nfield created by a magnetic source can be remotely\ncancelled by a magnetic replicator. Together with our\nmetamaterial device, we placed a straight wire of cur-\nrentI= 0.5 A in the position of the replica wire\n(which we call the “target wire”). The superposition\nof the field generated by the wire and that generated\nby the metamaterial cancel out. The experimental\nmeasurements give a very low magnetic field strength\nin the region ρ>d/prime(Fig. 4B). This demonstrates that\nmagnetic sources placed in inaccessible regions (like\nthe interior of a wall, for example) can be cancelled\nremotely, without the need for surrounding them with\nmagnetic cloaks.\nThe magnetic replicator we propose, similar to\nparity-time perfect lenses for electromagnetic waves\n[17, 18], does not require the design of bulk metama-\nterials with cumbersome magnetic permeability dis-\ntributions; it can be realized by a precise arrange-\nment of current-carrying electric wires. Thus, some\nof the issues associated with magnetic metamaterials,\nsuch as the non-linearity of their constituents [19], are\navoided.\nEven though our results have been demonstrated\nfor static magnetic fields, this same device could work\nfor low-frequency AC fields [20]. For time-dependent\nmagnetic fields with associated wavelengths larger or\ncomparable to the size of the device (for which the in-\nduced electric fields can be neglected), one could feed\nAC currents to the same active metamaterial. Thus,\nour experimental setup would maintain its function-\nality up to frequencies around 100MHz.\nThe emulation of a magnetic source at distance is\nvalid everywhere in space except in the circular re-\ngion delimited by the position of the replica source\nand can be made as exact as desired at the expense\nof increased complexity and power requirements. Ac-\ncording to Eq. (1), the higher the chosen number of\ntermsnT, the larger the current density that has to be\nfed to the metamaterial shell. Since the summation4\nFigure 4: Focusing of magnetic field and cancellation of\nthe magnetic field created by a wire. (A) Plots of −By\nalong different parallel lines at x/prime= 9.1,10.1 and 12.1mm\n(pink-circles, purple-squares, and yellow-triangles, respec-\ntively). Measurements are shown in symbols and the cor-\nresponding calculations in solid lines. Dashed lines are\ncalculations of the field created by a single finite wire car-\nrying a current I=−0.5A located at x/prime=−20mm,y/prime= 0.\n(B) Measurements (line y/prime= 0mm) of the field created by\na wire located at the position of the image wire (”target\nwire”) and of the field created by the magnetic replicator\nand the target wire (”cancellation”). Error bars (standard\nerrors) are smaller than the symbol size. Insets show the\nmeasured|B|for the target wire and the cancellation con-\nfigurations. The black grid shows the measurement points;\nthe color surface is obtained as a linear interpolation be-\ntween them. Standard errors are smaller than 0 .02µT.\ninKnTis dominated by its last term, the required\nnumber of wires to emulate the shell is 2 nT. Not only\nits number but also the current each of them carries\nincreases with nT; the current carried by the wire at\n(x,y) = (R2,0), for example, increases with nTas\n(d/prime/R2)nT. Therefore, the currents and the required\npower to create a source at a distance rapidly increase\nwith the distance of the target source ( d/prime/R2) as well\nas with the total number of terms in KnT. In scenarios\nwhere high field accuracy and strength are demanded,\nhigh-T csuperconductors could be used to create the\ncircuits forming the metamaterial.\nIn spite of considering translational symmetryalong thez−direction throughout the article, these\nsame ideas could be applied to emulate a 3D magnetic\nsource, like a point magnetic dipole. In that case, one\ncould consider a spherical shell with negative perme-\nability and calculate the corresponding magnetization\ncurrents, which would likely result in cumbersome\ninhomogeneous distributions of surface and volume\ncurrent densities.\nResults presented here open a new pathway for con-\ntrolling magnetic fields at a distance, with poten-\ntial technological applications. For example, a wide\nvariety of microrobots and functional micro/nano-\nparticles are moved and actuated by means of mag-\nnetic fields [10, 21–24]. They can perform drug trans-\nport and controlled drug release [11], intraocular reti-\nnal procedures [22], or even stem cell transplanta-\ntion [25]. However, the rapid drop off of field strength\nwith target depth within the body is acknowledged\nto pose severe limitations to the clinical development\nof some of them [10, 11]. Another example is tran-\nscranial magnetic stimulation (TMS), which uses mag-\nnetic fields to modulate the neural activity of patients\nwith different pathology [12]. In spite of its success,\nTMS suffers from limited focality, lacking the ability\nto stimulate specific regions [12]. Our results could\nbenefit both technologies by enabling the precise spa-\ntial targeting of magnetic fields at the required depth\ninside the body. In actual applications, though, one\nshould take into account that the region between the\nmetamaterial and the replica image would experience\nstrong magnetic fields.\nAnother area of application is in atom trapping.\nDepending on the atom’s state, they can be trapped\nin magnetic field minima (low-field seekers) or max-\nima (high-field seekers). Since local maxima are for-\nbidden by Earnshaw’s theorem, high-field seekers are\ntypically trapped in the saddle point of a magnetic po-\ntential that oscillates in time [26]. However, these dy-\nnamic magnetic traps are shallow compared to traps\nfor low-field seekers [27, 28]. By emulating a magnetic\nsource at distance, one would be able to generate mag-\nnetic potential landscapes with higher gradients at the\ndesired target position resulting in tighter traps.\nIn conclusion, our results demonstrate that a shell\nwith negative permeability can emulate and cancel\nmagnetic sources at distance. This ability to manipu-\nlate magnetic fields remotely will enable both the ad-\nvancement of existing technology and potentially new\napplications using magnetic fields.\nAcknowledgments\nJPC is funded by a Leverhulme Trust Early Career\nfellowship (ECF-2018-447). We thank P. Maurer, I.\nHughes and C. Navau for their feedback on the article.5\n∗Electronic address: j.prat.camps@gmail.com\n[1] S. Narayana, Y. Sato, Advanced Materials 24, 71\n(2012).\n[2] F. G¨ om¨ ory, et al. ,Science (New York, N.Y.) 335,\n1466 (2012).\n[3] C. Navau, J. Prat-Camps, O. Romero-Isart, J. Cirac,\nA. Sanchez, Physical Review Letters 112, 253901\n(2014).\n[4] J. Prat-Camps, C. Navau, A. Sanchez, Scientific Re-\nports 5, 12488 (2015).\n[5] W. H. Wing, Progress in Quantum Electronics 8, 181\n(1984).\n[6] W. Ketterle, D. E. Pritchard, Applied Physics B Pho-\ntophysics and Laser Chemistry 54, 403 (1992).\n[7] R. Mach-Batlle, et al. ,Physical Review Applied 9,\n034007 (2018).\n[8] O. Dolgov, D. Kirzhnits, V. Losyakov, Solid State\nCommunications 46, 147 (1983).\n[9] R. Mach-Batlle, et al. ,Physical Review B 96, 094422\n(2017).\n[10] M. Sitti, Nature 458, 1121 (2009).\n[11] J. Estelrich, E. Escribano, J. Queralt, M. Busquets,\nInternational journal of molecular sciences 16, 8070\n(2015).\n[12] T. Wagner, A. Valero-Cabre, A. Pascual-Leone, An-\nnual Review of Biomedical Engineering 9, 527 (2007).\n[13] J. B. Pendry, S. A. Ramakrishna, Journal of Physics:\nCondensed Matter 14, 8463 (2002).[14] J. Pendry, Optics Express 11, 755 (2003).\n[15] S. Anantha Ramakrishna, J. B. Pendry, Physical Re-\nview B 69, 115115 (2004).\n[16] G. W. Milton, N. A. P. Nicorovici, R. C. Mcphe-\ndran, V. A. Podolskiy, Proceedings of the Royal So-\nciety A: Mathematical, Physical and Engineering Sci-\nences 461, 3999 (2005).\n[17] R. Fleury, D. L. Sounas, A. Al` u, Physical Review Let-\nters113(2014).\n[18] F. Monticone, C. A. Valagiannopoulos, A. Al` u, Phys-\nical Review X 6(2016).\n[19] J. M. Coey, Magnetism and magnetic materials\n(Cambridge university press, 2010).\n[20] J. Prat-Camps, C. Navau, A. Sanchez, Advanced Ma-\nterials 28, 4898 (2016).\n[21] R. Dreyfus, et al. ,Nature 437, 862 (2005).\n[22] M. P. Kummer, et al. ,IEEE Transactions on Robotics\n26, 1006 (2010).\n[23] W. Hu, G. Z. Lum, M. Mastrangeli, M. Sitti, Nature\n554, 81 (2018).\n[24] Y. Kim, H. Yuk, R. Zhao, S. A. Chester, X. Zhao,\nNature 558, 274 (2018).\n[25] S. Jeon, et al. ,Science Robotics 4, eaav4317 (2019).\n[26] E. A. Cornell, C. Monroe, C. E. Wieman, Physical\nreview letters 67, 2439 (1991).\n[27] J. Fort´ agh, C. Zimmermann, Reviews of Modern\nPhysics 79, 235 (2007).\n[28] E. A. Hinds, I. G. Hughes, Journal of Physics D: Ap-\nplied Physics 32, R119 (1999).Supplementary Material for ‘Manipulating magnetic fields in\ninaccessible regions by negative magnetic permeability’\nRosa Mach-Batlle, Mark G. Bason, Nuria Del-Valle, Jordi Prat-Camps\nContents\n1 A negative-permeability cylindrical shell by transformation optics 2\n1.1 Space transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2\n1.2 Analytic expressions of the magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.3 Transformation of the magnetic field created by an inner wire . . . . . . . . . . . . . . . . . . . . 3\n2 A negative-permeability cylindrical shell by Maxwell equations 5\n2.1 Analytic derivation of the magnetic vector potential for a negative-permeability shell surrounding\na wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n2.2 Mimicking the magnetic replicator with current distributions . . . . . . . . . . . . . . . . . . . . 6\n2.3 Design of an active metasurface that emulates a wire at a distance . . . . . . . . . . . . . . . . . 8\n3 Materials and Methods 10\n3.1 Design and construction of the magnetic replicator . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n3.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n3.2.1 Numerical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\n3.2.2 Measurements of the field created by the magnetic replicator . . . . . . . . . . . . . . . . 12\n3.2.3 Measurements of the cancellation of the field created by a wire . . . . . . . . . . . . . . . 15\n3.2.4 Measurements of the field created by the individual circuits . . . . . . . . . . . . . . . . . 17\n1arXiv:1912.12477v1  [physics.class-ph]  28 Dec 20191 A negative-permeability cylindrical shell by transformation optics\n1.1 Space transformation\nOur goal is to derive the properties of a cylindrical shell with negative permeability that acts on the magnetic\nfield in the same way as a perfect lens acts on the electromagnetic waves (18, 31, 32). For internal magnetic\nfields, i. e. when a magnetic source is located inside the hole of the cylindrical shell, this shell must transform\nthe space as illustrated in Fig. S1.\nFigure S1: Sketches of (a) the undistorted original space and (b) the transformed space for magnetic field\nsources located inside the shell.\nConsidering a cylindrical shell infinitely long along the z−direction with internal radius R1and external\nradiusR2and a cylinder of radius R0< R 1, the space is transformed as follows. The space in the region\nR0<ρ<∞is expanded through\n\n\nρ/prime=/parenleftBig\nR2\nR0/parenrightBig\nρ,\nϕ/prime=ϕ, ρ∈(R0,∞)\nz/prime=z,(S1)\nwhereρ,ϕandzare the coordinates in the original physical space and ρ/prime,ϕ/primeandz/primeare the coordinates in the\ntransformed virtual space. Simultaneously, the space R0≤ρ≤R1is folded (arrows in Fig. S1b) as\n\n\nρ/prime=R1/parenleftBig\nρ\nR1/parenrightBigk\n,\nϕ/prime=ϕ, ρ∈[R0,R1]\nz/prime=z,(S2)\nwherekis a negative parameter ranging from 0−to−∞. To guarantee the continuity of the space at ρ/prime=R2,\nEqs. (S1) and (S2) give the following relation between kandR0,\nR0=R1(R2/R1)1/k. (S3)\nTransformation optics theory can be applied to obtain the permeability resulting in the presented space\ntransformation. The permeability tensors are\nµ/prime=\nµρρµρϕµρz\nµϕρµϕϕµϕz\nµzρµzϕµzz\n=\n1 0 0\n0 1 0\n0 0/parenleftBig\nR2\nR1/parenrightBig2/k−2\n (S4)\nin the external region ρ/prime>R 2,\nµ/prime=\nk0 0\n0 1/k 0\n0 01\nk/parenleftBig\nρ/prime\nR1/parenrightBig2/k−2\n (S5)\n2in the cylindrical shell region R1≤ρ/prime≤R2, andµ/prime= 1 in the internal region, ρ/prime<R 1, where the space is not\ntransformed.\nBecause we are considering static magnetic fields and in magnetostatics the magnetic and the electric fields\nare decoupled, the permittivity εof the medium does not play any role in the shaping of magnetic fields. If\none assumes translational symmetry along the z−axis, thez−component of the magnetic field is zero and only\nthe left-upper 2x2 minors of the permeability tensors have physical relevance. In this case, Eq. (S4) shows that\na negative-permeability shell, different from the perfect lenses proposals for electromagnetic waves (18,31,32),\ndoes not require magnetic material neither inside the hole nor in the external region. According to Eq. (S5),\nthe relative magnetic permeability of a negative-permeability cylindrical shell is homogeneous and anisotropic,\nwith radial and angular components µρρ=kandµϕϕ= 1/k, respectively. Since kwas defined as a negative\nparameter, there are infinite possible anisotropic shells, all with negative values of both µρρandµϕϕ. All these\ncylindrical shells exhibit slightly different properties, since they result from different space transformations [Eqs.\n(S1)-(S3)].\nSimilar space transformations demonstrate that this anisotropic long cylindrical shell with negative radial\nand angular components of the relative magnetic permeability kand 1/k, respectively, transforms externally\napplied magnetic fields in an analogous way [32].\n1.2 Analytic expressions of the magnetic field\nOne can apply the transformation optics theory to find out how a negative-permeability shell transforms a\nmagnetic field distribution everywhere in space. The analytic expression of the magnetic field when the space\nis transformed according to Eqs. (S1) and (S2) is\nH/prime(ρ/prime,ϕ/prime) =H(ρ/prime,ϕ/prime), ρ/prime∈[0,R1) (S6)\n\n\nH/prime\nρ(ρ/prime,ϕ/prime) =1\nk/parenleftBig\nρ/prime\nR1/parenrightBig1/k−1\nHρ/parenleftbigg\nR1/parenleftBig\nρ/prime\nR1/parenrightBig1/k\n,ϕ/prime/parenrightbigg\n,\nH/prime\nϕ(ρ/prime,ϕ/prime) =/parenleftBig\nρ/prime\nR1/parenrightBig1/k−1\nHϕ/parenleftbigg\nR1/parenleftBig\nρ/prime\nR1/parenrightBig1/k\n,ϕ/prime/parenrightbigg\n, ρ/prime∈[R1,R2](S7)\nH/prime(ρ/prime,ϕ/prime) =/parenleftbiggR2\nR1/parenrightbigg1/k−1\nH/parenleftBigg/parenleftbiggR2\nR1/parenrightbigg1/k−1\nρ/prime,ϕ/prime/parenrightBigg\n. ρ/prime∈(R2,∞) (S8)\nEquation (S6) shows that the field inside the hole, ρ/prime∈[0,R1), is not modified by the presence of the shell, while\nEqs. (S7) and (S8) indicate that the field distribution in the shell volume, ρ/prime∈[R1,R2], and in the external\nregion,ρ/prime∈(R2,∞), are transformed by the shell and depend on the magnetic field source and on the shell\nradii and parameter k.\n1.3 Transformation of the magnetic field created by an inner wire\nHere we use the general expressions in Eqs. (S6)-(S8) to derive how a negative-permeability shell transforms\nthe magnetic field generated by a long straight wire of dc current Iplaced inside its hole, at a distance dfrom\nthe center of the shell (see Fig. S2). Other magnetic field distributions with translational symmetry along the\nz−direction, such as the field created by a long magnetic dipole or a long magnetic quadrupole, can be regarded\nas the superposition of the field created by several wires. For example, the field created by two antiparallel\nwires of current Iseparated by a small distance d→0 approximates well the field created by a dipolar source\nwith dipolar magnetic moment per unit length m=Id. Therefore, the transformation of the field created by a\nwire can be used to understand how negative-permeability shells transform these other field distributions.\nBecause the shell does not transform the field inside its hole, the field in the region ρ<R 1corresponds to\nthe field of the original wire. In the exterior of the shell, the field distribution is found to be equivalent to the\nfield of a replica wire of current Ilocated at\ndEXT\ni=/parenleftbiggR2\nR1/parenrightbigg1−1/k\nd. (S9)\nIf the original wire is placed outside the folding region ( d<R 0, as in Fig. S2a), the replica wire appears in the\nshell region dEXT\ni<R 2. In contrast, if the original wire is placed in the folding region ( R0<d<R 1, as in Fig.\nS2b), the replica wire appears at a distance from the external surface of the shell, at dEXT\ni> R 2. However,\nthe magnetostatic Maxwell equations ∇×H=Jfand∇·B= 0 indicate that magnetic field lines can only\nemanate from actual magnetic sources. This makes it impossible to achieve the same field distribution as if\n3Figure S2: Sketches of two long (along z) cylindrical negative-permeability shells (blue regions) with internal\nradiusR1and external radius R2surrounding a long straight wire of current I(black dot) placed at a distance\ndfrom the center of the shell. According to the space transformation in Eqs. (S1) and (S2), the shell effectively\nfolds the space in the region ρ∈[R0,R1]. The radius R0is plotted as a dotted line. In (a) the wire is located\nin the non-folding region ( d<R 0), while in (b) it is located in the folding region ( R0<d<R 1).\na source was located in empty space (at dEXT\ni> R 2) if no real source is actually placed at dEXT\ni. Therefore,\nthe transformation optics results for the cases in which the folding region contains magnetic sources require\nthe consideration of actual sources at the replica location. Perfect lenses for electromagnetic waves derived by\ntransformation optics show similar limitations [31].\nIn the shell region, the magnetic field can only be written in terms of replica wires when the permeability of\nthe shell is isotropic, with µρρ=µϕϕ=−1. In this case, the magnetic field Hin the shell region corresponds\nto the superposition of the field of a replica wire of current Ilocated at the center of the shell hole plus another\none of current−Iplaced atdSHE\ni=R2\n1/d. Becauseµ=−1 and B=µµ0H, the magnetic induction Bin the\nshell is that of a replica wire of current −Ilocatedρ= 0 plus that of another one of current Iplaced atdSHE\ni.\n42 A negative-permeability cylindrical shell by Maxwell equations\n2.1 Analytic derivation of the magnetic vector potential for a negative-permeability\nshell surrounding a wire\nConsider a cylindrical shell of internal radius R1and external radius R2with linear, homogeneous and isotropic\nrelative magnetic permeability µ=−1, which we coin a magnetic replicator . It surrounds a long straight wire\nof currentIdisplaced a distance dfrom the center of the hole, as illustrated in Fig. S2.\nThe magnetic field everywhere in space can be derived from the magnetic vector potential AasB=∇×A.\nDue to the symmetry of the field created by the wire, the magnetic vector potential is found to be A=A(ρ,ϕ)z.\nFrom the magnetostatic Maxwell equation ∇×H=Jfand the constitutive relation B=µµ0H, one finds that\n(in the Coulomb Gauge; i.e. ∇·A= 0) the magnetic vector potential must fulfill\n∇2A(ρ,ϕ) = 0, (S10)\ninside the hole (INT), in the magnetic replicator volume (MR), and in the exterior of the shell (EXT) while\n∇2A(ρ,ϕ) =−µ0Jf, (S11)\nin the region occupied by the wire, which we consider to be a point.\nThe solution of Eq. (S10) can be written as,\nAINT(ρ,ϕ) =Aw(ρ,ϕ) +∞/summationdisplay\nn=1anρncos(nϕ) (S12)\nAMR(ρ,ϕ) =b0ln(ρ) +∞/summationdisplay\nn=1/parenleftbigg\nbnρn+cn\nρn/parenrightbigg\ncos(nϕ), (S13)\nAEXT(ρ,ϕ) =d0ln(ρ) +∞/summationdisplay\nn=1dn\nρncos(nϕ), (S14)\nwhereAwis the magnetic vector potential due to a non-centered long straight wire and we have taken into\naccount that the magnetic field the shell creates must be finite at ρ→0 and tend to zero at ρ→∞ . In order\nto obtain the magnetic vector potential coefficients it is convenient to write the magnetic vector potential of a\nwireAwas a Fourier series, which reads as\nAw(ρ,ϕ) =\n\n−µ0I\n2πln(ρ) +/summationtext∞\nn=1µ0I\n2πn/parenleftBig\nd\nρ/parenrightBign\ncos(nϕ), ρ>d\n−µ0I\n2πln(d) +/summationtext∞\nn=1µ0I\n2πn/parenleftbigρ\nd/parenrightbigncos(nϕ). ρ≤d(S15)\nThe coefficients b0,d0,an,bn,cn, anddn, obtained by imposing that the radial component of the magnetic\ninduction and the angular component of the magnetic field must be continuous at the material boundaries\nρ=R1andρ=R2, are\nb0=µ0I\n2π, (S16)\nd0=−µ0I\n2π, (S17)\nan= 0, (S18)\nbn=µ0I\n2πn/parenleftbiggd\nR2\n1/parenrightbiggn\n, (S19)\ncn= 0, (S20)\ndn=µ0I\n2πn/parenleftbiggdR2\n2\nR2\n1/parenrightbiggn\n. (S21)\nSome general properties of the magnetic replicator result from the magnetic vector potential solutions. First,\nbecausean= 0, the shell does not distort the magnetic field created by the wire inside its hole (compare Figs.\nS3b and f to Figs. S3a and e, respectively). Second, when d≤R2\n1/R2, the magnetic vector potential in the\nexternal region converges to the potential of a replica current Ilocated atdEXT\ni=d(R2/R1)2(compare Figs.\n5S3b and c). If d>R2\n1/R2, the replica current appears outside the shell ( dEXT\ni>R 2) andAEXT(ρ,ϕ) converges\nto the potential of the replica wire not in the whole external region but only in the region ρ > dEXT\ni; in the\nregionρ∈(R2,dEXT\ni)AEXT(ρ,ϕ) diverges (compare Figs. S3f and g). Third, when d≤R2\n1/R2, the magnetic\nvector potential in the shell region converges to the potential of a replica wire of current −Iplaced atρ= 0\nplus the potential of a second replica wire of current Iplaced atdSHE\ni=R2\n1/d(outside the shell, dSHE\ni>R 2). If\nd>R2\n1/R2, the second image appears in the shell volume ( R1<dSHE\ni<R 2). In this case, AMR(ρ,ϕ) converges\nto the potential of the two replica wires in the region ρ∈(R1,dSHE\ni) and diverges in the region ρ∈(dSHE\ni,R2).\nFigure S3: In panels (a)-(c) and (e)-(g), numerical calculations of the normalized magnetic induction ByR1/(µ0I)\na straight wire of current I(sketched as a black dot) creates in different configurations. In (a) the wire is placed\nat (x,y)= (d, 0). In (b) the wire in (a) is surrounded by a cylindrical magnetic replicator with radii R1= 3d\nandR2= 2R1and relative magnetic permeability µ=−1 centered at the origin of coordinates. In (c) the\nwire is located at (4 d, 0). In (e) the wire is placed at (2 d, 0). In (f) the wire in (e) is surrounded by the same\nshell as in (b). In (g) the wire is placed at (8 d, 0). The white annular regions in (f) indicate that the field\ndiverges. Panels (d) and (h) illustrate the creation of replicas achieved by the magnetic replicator in (b) and (f),\nrespectively. The replica wire sketched in red in (d) and (h) gives the field distribution in the external region,\nwhile the replica wires sketched in blue give the field distribution in the shell volume.\n2.2 Mimicking the magnetic replicator with current distributions\nThe magnetic replicator requires a negative value of the magnetic permeability. Materials with negative perme-\nabilities do not naturally occur, but their behaviour can be effectively mimicked by substituting the hypothetical\nmaterial with negative permeability by its magnetization currents (12). Here we derive the magnetization cur-\nrents of a cylindrical magnetic replicator with permeability µ=−1 surrounding a long straight wire, as sketched\nin Fig. S2. Because the material is homogeneous and isotropic, the volume magnetization currents are zero,\nJM(ρ,ϕ) =∇×M(ρ,ϕ) = 0. The surface magnetization current densities KM1(ϕ) and KM2(ϕ), flowing at\nρ=R1andρ=R2respectively, are found from the material magnetization as,\nKM1(ϕ) =M(R1,ϕ)×n1=Mϕ(R1,ϕ)z=2\nµ0Bϕ(R1,ϕ)z=−2\nµ0∂AMR(ρ,ϕ)\n∂ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nρ=R1z, (S22)\n6KM2(ϕ) =M(R2,ϕ)×n2=−Mϕ(R2,ϕ)z=−2\nµ0Bϕ(R1,ϕ)z=2\nµ0∂AMR(ρ,ϕ)\n∂ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nρ=R2z, (S23)\nwhere n1andn2are the unity vectors perpendicular to the surfaces ρ=R1andρ=R2, respectively, pointing\noutwards the material surfaces; n1=−ρρρandn2=ρρρ. From Eqs. (S13) and (S19), the surface magnetization\ncurrent densities are obtained as\nKM1(ϕ) =/parenleftBigg\n−I\nπR1−∞/summationdisplay\nn=1I\nπR1/parenleftbiggd\nR1/parenrightbiggn\ncos(nϕ)/parenrightBigg\nz, (S24)\nKM2(ϕ) =/parenleftBigg\nI\nπR2+∞/summationdisplay\nn=1I\nπR2/parenleftbiggdR2\nR2\n1/parenrightbiggn\ncos(nϕ)/parenrightBigg\nz. (S25)\nThe sum in the current density KM1is always convergent and gives\nKM1(ϕ) =/parenleftbigg\n−I\nπR1+I\nπR1d[d−R1cos(ϕ)]\nR2\n1+d2−2dR1cos(ϕ)/parenrightbigg\nz. (S26)\nIn contrast, the current density KM2only converges when d<R2\n1/R2. In this case,\nKM2(ϕ) =/parenleftbiggI\nπR2−I\nπd[dR2−R2\n1cos(ϕ)]\nR4\n1+d2R2\n2−2dR2\n1R2cos(ϕ)/parenrightbigg\nz. (S27)\nWhen the summation in Eq. (S25) is not convergent one can still approximately obtain the same field\ndistribution as that resulting from a magnetic replicator with permeability µ=−1 by truncating the external\ncurrent density summation up to nTterms,\nKnT\nM2(ϕ) =/parenleftBigg\nI\nπR2+nT/summationdisplay\nn=1I\nπR2/parenleftbiggdR2\nR2\n1/parenrightbiggn\ncos(nϕ)/parenrightBigg\nz. (S28)\nAs shown in Fig. 2 of the main text, the higher the number of terms nT, the more the external field\ndistribution resembles the field of the replica wire.\nWe can now analyse what is the field distribution that the current densities KM1(ϕ) and KM2(ϕ) create in\nthe external region. On the one hand, KM1(ϕ) cancels the field created by the original wire and creates the\nsame field distribution as that of a centered wire of current −I. On the other hand, KM2(ϕ) cancels the field\ncreated by a centered wire of current −Iand creates the field of the replica wire. In this way when the original\nwire is surrounded by the two current densities KM1(ϕ) and KM2(ϕ) the field in the external region corresponds\nto the field created by a replica wire.\nIf the goal is to create the field of a wire in the external region regardless of the field in the volume occupied\nby the shell, the original wire and the current density KM1(ϕ) can be substituted by a centered wire of current\n−I; this centered wire together with KM2(ϕ) still create the field of a replica wire. Actually, as shown in Fig.\nS4, the currents can be simplified even further. Since the first term in the summation of KM2(ϕ) creates the\nsame field distribution in the external region as a centered wire of current 2 I, the field of the replica wire can\nbe obtained by a centered wire of current Isurrounded by the surface current density\nKnT(ϕ) =nT/summationdisplay\nn=1I\nπR2/parenleftbiggdR2\nR2\n1/parenrightbiggn\ncos(nϕ)z, (S29)\nwhich can be written in terms of the position of the replica wire as\nKnT(ϕ) =nT/summationdisplay\nn=1I\nπR2/parenleftbiggdEXT\ni\nR2/parenrightbiggn\ncos(nϕ)z. (S30)\n7Figure S4: Normalized vertical component of the magnetic induction ByR2/(µ0I) for (a) a wire of current I\nplaced atx=dsurrounded by the sheet current densities in Eqs. (S26) and (S28) with R1= 4/3d,R2= 2R1,\nandnT= 20, (b) a centered wire of current Isurrounded by the current density in Eq. (S30) with R2= 8/3d,\ndEXT\ni= 4dandnT= 20, and (c) a wire of current Iplaced atx= 4d.\n2.3 Design of an active metasurface that emulates a wire at a distance\nConsider a magnetic replicator that is designed to create a target source at a distance, i.e. the chosen parameters\nfulfillR2\n1/R2<d<R 1<R 2<dEXT\ni. For a practical realization of such a device, the continuous surface current\ndensity in Eq. (S30) must be converted into a discrete set of straight wires constituting an active metamaterial\nor metasurface. The current of each of the wires can be calculated according to\nIm=/integraldisplay(θm+1+θm)/2\n(θm+θm−1)/2KnT(ϕ)R2dϕ, (S31)\nwhereθmis the angular positions of each wire.\nIf one chooses to place a wire at the positions where the function KnT(ϕ) has a relative minimum or a\nmaximum, the metasurface must consists of 2 nTstraight wires, as the summation in Eq. (S30) is dominated\nby its last term. Therefore, the creation of a replica wire of current Irequires 2nTstraight wires of current Im\n(m= 1,2,3...2nT) plus a centered wire of current I2nT+1=I.\nAccording to Eq. (S30), the higher the chosen number of terms nT, the larger the current density. The\nnumber of straight wires and the current Imof each of the wires also increases with nT. For example, the\ncurrent of the wire that is placed at ( x,y) = (R2,0), which is the wire that is placed closer to the replica source\nand the one that must carry the largest current, increases with nTas/parenleftbig\ndEXT\ni/R2/parenrightbignT, approximately. Because\nthe currents required to create a source at a distance increase with nT, the power dissipated by the metasurface,\nwhich can be written as the summation of the power dissipated by each of the wires,\nPT=2nT+1/summationdisplay\nm=1I2\nmRm, (S32)\nalso increases with nT.\nWe can now compare the power dissipated by the metamaterial magnetic replicator that creates the field\nof a replica wire at a distance, PT, for different nTto the power that would be dissipated by an actual replica\nwire,P0. Assume a long straight wire of current I2nT+1=Iplaced at the origin of coordinates surrounded\nby a cylindrical metasurface of radius R2consisting of different sets of long straight wires of current Im. The\ncurrentsImare calculated from Eqs. (S30) and (S35) assuming dEXT\ni= 1.5R2and 5, 10 and 20 nTterms.\nConsider first that we have 2 nTlong straight wires connected to the same voltage source, which provides a\nconstant voltage Vm=V. We assume that the resistance of each wire can be tuned so that each of them carries\nthe appropriate current, Im. The total power dissipated by the central wire and the 2 nTwires in the cylindrical\nsurface normalized to the power that would be dissipated by the replica wire ( P0=IV) is\nPT\nP0=1\nIV2nT+1/summationdisplay\nm=1ImVm=2nT+1/summationdisplay\nm=1Im\nI. (S33)\n8The values of PT/P0obtained for nT= 5,nT= 10, andnT= 20 are 11.13, 84.59 and 4877.69, respectively. The\nhigher the number of terms that are considered (and therefore, the more the field resembles that of a source\nappearing at a distance), the larger the power needed to feed the metamaterial.\nConsider now that we followed a different strategy for feeding the current to all the wires. Imagine we\nhad a single wire of current Icand that we used this wire to achieve all the currents Imin the experiment by\nadjusting the number of turns at each angular position. In this way, the dissipated power normalized to the\npower dissipated by the replica wire ( P0=IV=I2R) would be calculated as\nPT\nP0=1\nI2R2nT+1/summationdisplay\nm=1I2\nmRm=2nT+1/summationdisplay\nm=1I3\nm\nI3, (S34)\ntaking into account that the resistance Rmis proportional to the number of turns of the wire of current Ic\nrequired for achieving Im. For the cases nT= 5,nT= 10, andnT= 20, the obtained values of PT/P0are 45.73,\n4531.72, and 1 .93·108, respectively, which show the strong dependence of the power dissipated on the number\nof termsnTthat is considered and also on the chosen experimental method for feeding the currents.\n93 Materials and Methods\n3.1 Design and construction of the magnetic replicator\nWe designed and built a cylindrical magnetic replicator that emulates a wire at a certain distance from the\nshell. In particular, our replicating shell has an external radius R2= 40mm and generates a replica current at\ndEXT\ni = 60mm (see Fig. S5c). Since the replica wire appears outside the shell, the required KM2would not\nconverge. For this reason, we truncate the summation in Eq. (S30) up to nT= 10. The resulting continuous\nsheet current density is converted into a discrete set of 20 straight current wires. The angular position of these\nwires is chosen to be at the positions where the function KnT(ϕ) has a relative minimum or a maximum. The\ncurrent each of these wires carries is given by\nIm=/integraldisplay(θm+1+θm)/2\n(θm+θm−1)/2KnT(ϕ)R2dϕ, (S35)\nwhereθmare the angular positions of each wire given in Table 1. The current of each wire ( Im) is also given.\nThese values are obtained assuming the replica wire carries a current I=−0.5A (the negative sign indicates\nthe current flows in the negative z-direction). The sheet current density KM1converges to a finite number. As\ndiscussed in the previous section, if we are interested in the field distribution outside the shell only, this current\ndensity can be replaced by a single wire placed at the center of the cylindrical replicator. This central wire is\nreferred as m= 21 in Table 1.\nTable 1 indicates that 12 different values of current are needed. We used only 4 independent circuits, each\nof them fed with a different value of current, to generate all the values. We combined different number of turns\nof each circuit to produce the required effective current as described in Table 1 (negative values indicate turns\nFigure S5: (a) 3D render of the experimental setup. Red, yellow, and black parts represent different 3D-printed\nsupport structures to position the wires (shown in orange-copper color) at the desired locations. The scan area\nis shown in translucent blue. The inset shows the two vertices that were used to determine the relative position\nof the magnetic replicator with respect to the sensors. (b) Pictures of the experimental device, showing the\nconnections between different turns (left) and the actual experimental setup. (c) Sketch indicating the position\nof the different wires and the definition of the angle θ.\n10in the negative direction of the z axis). Circuits 1, 2 and 3 were fed with 2 .5, 0.25, and 0.025A, respectively,\nand were combined to generate the current for the 20 external wires. Circuit 4 carried a current of 0.5A and\nfed the central wire ( m=21) only. Circuits 1, 2, and 3 consisted of single enameled copper wires of 1.59, 0.50\nand 0.22mm of diameter, respectively. Circuit 4 used a single AWG wire of 1.5mm of diameter.\nThe actual experimental magnetic replicator was designed to have a length (along the z direction) of 400mm.\nIn order the position all the wires at the appropriate positions, a non-magnetic plastic structure was designed\nand 3D-printed (using a FlashForge Creator Pro printer and polylactic acid [PLA] filament). This cylindrical\nstructure (shown in yellow and black in Fig. S5a-c) had grooves at the appropriate angular positions, where\nthe different turns where positioned and glued. Each of the 4 circuits was winded continuously, as can be\nseen in Fig. S5b. The central wire was guided through the central hole of the cylinder using specially-designed\n3D-printed parts placed at the top and the bottom of the cylinder. In a similar way, a wire was placed at the\nposition of the replica wire. This ”target” wire was only fed in some parts of the experiment to demonstrate, for\nexample, the cancellation properties of our magnetic replicator. Finally, a 3D-printed support structure (shown\nin red in Fig. S5a and c) held the structure upright and clamped it to the table. Two reference points of this\nstructure (see inset in Fig. S5a) were used to calibrate the relative position of the whole structure with respect\nto the magnetic sensors. For this point on, the origin of coordinates is set at the position of the target wire as\ndepicted in Fig. S5a. All the measurements and simulations shown next follow this convention.\nwire number,\nmθm\n(deg)nominal current\nIm(A)turns circuit 1\n(2.5A)turns circuit 2\n(0.25A)turns circuit 3\n(0.025A)generated\ncurrent (A)\n1 0.00 -6.493 -3 4 0 -6.500\n2 20.41 4.630 2 -2 5 4.625\n3 39.53 -2.819 -1 -1 -3 -2.825\n4 57.68 2.159 1 -1 -4 2.150\n5 75.46 -1.662 -1 3 4 -1.650\n6 93.03 1.478 1 -4 -1 1.475\n7 110.51 -1.251 0 -5 0 -1.250\n8 127.92 1.206 0 5 -2 1.200\n9 145.30 -1.080 0 -4 -3 -1.075\n10 162.65 1.101 0 4 4 1.100\n11 180.00 -1.031 0 -4 -1 -1.025\n12 197.35 1.101 0 4 4 1.100\n13 214.70 -1.080 0 -4 -3 -1.075\n14 232.08 1.206 0 5 -2 1.200\n15 249.49 -1.251 0 -5 0 -1.250\n16 266.97 1.478 0 6 -1 1.475\n17 284.54 -1.662 -1 3 4 -1.650\n18 302.32 2.159 1 -1 -4 2.15\n19 320.47 -2.819 -1 -1 -3 -2.825\n20 339.59 4.630 2 -2 5 4.625\n21 - -0.500 - - - -0.5\nTable 1: Parameters for each of the wires forming the magnetic replicator. The angular position of each wire,\nθm, follows the convention defined in Fig. S5c. Positive values of current indicate the current is flowing in the\nz-direction as defined in the figure. Wire m= 21 corresponds to the central wire.\n3.2 Measurements\nThe x and y components of the magnetic field were measured using two miniature fluxgate magnetometers\n(DRV425) whose sensor size is 4x4 mm and noise floor is around 1 nT/√\nHz. The two modules were placed one\non top of the other, with a vertical gap of approximately 1 mm between them, and were oriented perpendicularly\nto measure the two components of the field. No cross-talk was measurable for the range of conditions presented.\nA 3D-printed support structured held the two sensors at the right position. This structure was in turn attached\nto a plastic support arm which was mounted on two perpendicular manual linear stages. The error associated\nto the x,y coordinates of the sensors given by the linear stages was ∆ x,y= 0.1mm.\nThe two evaluation modules were connected to a 16-bit analogue to digital converter (Labjack T7) from\nwhich two voltages, VxandVy, were obtained corresponding to the two sensors.\nTo perform a measurement of the field, we first recorded the background magnetic field for a period of 5\nseconds at a sample rate of 132 Hz and calculated the mean value and the standard error of the measurements,\nVOFF\na and ∆VOFF\na, respectively ( a={x,y}). This was followed by measurements with the same settings\n11in which the appropriate currents were on, obtaining VON\naand ∆VON\na. The magnetic field created by the\ncurrents is Va=VON\na−VOFF\na and the associated standard error ∆ Va=/radicalbig\n(VONa)2+ (VOFFa)2. These two sets\nof recordings were repeated five times. To obtain a best estimate of the voltages, these five sets of values ( Va,i,\n∆Va,i,i={1,2,3,4,5}) were combined using the individual variances as weighting [33].\nThe sensors were calibrated in a known field, which was cross-checked with a precision fluxgate (Stefan\nMayer Fluxmaster). The linearity was within 0.1%.\n3.2.1 Numerical calculations\nIn order to validate our measurements, we performed numerical calculations of the field created by different\ndistributions of finite wires. We numerically integrated the 1D Biot-Savart equation to obtain the magnetic\nfield at different positions of interest. The wires were assumed to be single uni-dimensional vertical current lines\nextending from z=−200mm toz= 200mm (the connections between wires at the top and bottom ends of the\nstructure were omitted). The current assigned to each wire corresponded to the ”generated current” value in\nTable 1. The positioning of the wires in the xy plane was done by taking into account the measured position of\nthe two reference points of the structure (see sect. 3.1). The coordinates of these two points were used to find\nthe center of the replicator and its relative rotation. Based on this information and on the nominal angles of\nthe wires,θm, we calculated the final x and y coordinates of each wire.\n3.2.2 Measurements of the field created by the magnetic replicator\nWe measured the magnetic field created by the magnetic replicator by feeding the 21 wires. The field was\nmeasured along the middle plane ( z= 0 in our coordinates system). Magnetic field values, Bi, and their\ncorresponding standard errors, ∆ Bi, were obtained as discussed in Sec. 3.2. Results are shown in Figs. S6 and\nS7. The corresponding numerical calculations are shown in Fig. S8. We used the same color scales, ranges, and\npositions to make the grids directly comparable.\nFigure S9 shows 1D plots of the measured fields (in symbols) and their corresponding calculations (solid\nlines). Fields are plotted as a function of y for different values of x corresponding to different parallel lines.\nFigure S6: (a) Measured Bx. (b) Standard error corresponding to the measurements and (c) histogram of the\nerrors.\n12Figure S7: (a) Measured By. (b) Standard error corresponding to the measurements and (c) histogram of the\nerrors.\nFigure S8: Numerical simulations of (a) Bxand (b)Byobtained using Biot-Savart.\n13Figure S9: Measured Bx(a) andBy(b) along lines of constant x (in symbols). Solid lines are the corresponding\nnumerical calculations.\n143.2.3 Measurements of the cancellation of the field created by a wire\nWe measured the ability the magnetic replicator has to cancel the magnetic field created by a wire placed at\nthe position of the replica wire. For this experiment we first measured the field created by a wire located at\nthe position of the replica wire carrying I= 0.5A. Results are shown on the left column of Fig. S10 (”target\nwire”). We then fed the same wire together with the 21 wires forming the magnetic replicator. Results are\nshown on the right column of Fig. S10 (”cancellation”) and demonstrate a great reduction of the magnetic field.\nA 1D measurement along the y= 0 line is shown in Fig. S11, confirming the cancellation is achieved up to large\ndistances. These results show how the magnetic replicator can be used to cancel the field created by a long\nwire.\nThe two measured magnetic field components were combined to plot the absolute value of the magnetic field\n(|B|=/radicalBig\nB2x+B2y) for the ”target wire” and ”cancellation” cases. Results are shown in Fig. S12.\nFigure S10: Measured field components when the target wire is fed [left, panels (a) and (c)] and when the\nmetamaterial and the target wire are fed [right, panels (b) and (d)]. The largest standard error obtained in\nthese measurements is 0 .02µT in all four panels.\n15Figure S11: Measured field component Bywhen the target wire is fed (blue) and when the magnetic replicator\nand the target wire are fed (magenta).\nFigure S12: Measured absolute value of the field ( |B|=/radicalBig\nB2x+B2y) when (a) the target wire is fed and (b) when\nthe magnetic replicator and the target wire are fed. The largest standard error obtained in these measurements\nis 0.02µT in the two panels. For each position, the coordinates corresponding to the BxandBymeasurements\nwhere slightly different due to small differences in the position of the sensors. To generate this plot, we averaged\nthe x and y coordinates.\n163.2.4 Measurements of the field created by the individual circuits\nWe performed measurements of the field created by the 4 individual circuits forming the magnetic replicator.\nResults are shown in symbols together with their corresponding numerical calculations in Fig. S13. BxandBy\ncomponents were measured along two different lines. Since the two magnetic sensors measuring the different\ncomponents where located at slightly different positions, the coordinates of the measuring lines for BxandBy\ndo not match exactly.\nMeasurements show that the 4 circuits create magnetic field distributions similar to the calculated values.\nThe agreement is better for the y-components. To explain the potential sources of disagreement between\nmeasurements and calculations one first needs to consider that the calculations assumed a single straight wire\nlocated at the nominal position for each of the 21 wires forming the replicator. In the actual setup, several\nturns of wire were used in the 20 wire positions. The different circuits were winded one after the other, so\nthat different turns were located at slightly different radial distances from the center of the structure. Another\nrelevant source of error is the inaccurate verticality of the whole structure, which has a small base (a diameter\nof 80mm) compared to its height (of 400mm). Finally, the wires that supplied the current to the magnetic\nreplicator could be responsible for creating undesired fields in the measuring area. The wires were not shielded\n(and we did not use coaxial wiring) and they were just moved as far as possible from the setup to minimize\ntheir effect.\nFigure S13: Measured (in symbols) and corresponding simulations of the fields created by the 4 independent\ncircuits measured along the specified lines (the origin of corrdinates is set to the position of the target wire).\nThe current supplied to the circuits was IC1= 2.5A,IC2= 0.25A,IC3= 0.05A,IC4= 0.5A.\n17References\n[29] M. Yan, W. Yan, M. Qiu, Cylindrical superlens by a coordinate transformation. Phys. Rev. B 78, 125113\n(2008).\n[30] H. Chen, C. T. Chan and P. Sheng, Transformation optics and metamaterials. Nature Materials 9, 387396\n(2010).\n[31] L. Bergamin, and A. Favaro, Negative index of refraction, perfect lenses and transformation optics - some\nwords of caution. 2010 URSI International Symposium on Electromagnetic Theory,760-763 (2010).\n[32] C. Navau, J. Prat-Camps, and A. Sanchez Magnetic Energy Harvesting and Concentration at a Distance\nby Transformation Optics. Phys. Rev. Lett., 109(2012).\n[33] Mandel, John. The statistical analysis of experimental data. Courier Corporation, 2012.\n18"    },    {        "title": "2001.03153v1.Two_dimensional_magnetic_nanoelectromechanical_resonators.pdf",        "content": " 1 Two-dimensional magnetic nanoelectromechanical resonators \n \nShengwei Jiang1, Hongchao Xie1, Jie Shan1,2,3*, Kin Fai Mak1,2,3*  \n \n1Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA. \n2School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA \n3Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY, USA \nEmail: jie.shan@cornell.edu ; kinfai.mak@cornell.edu  \nThese authors contributed equally: Shengwei Jiang, Hongchao Xie. \n \n \nTwo-dimensional (2D) layered materials possess outstanding mechanical, electronic \nand optical properties, making them ideal materials for nanoelectromechanical \napplications 1,2. The recent discovery of 2D magnetic materials 3,4,5,6,7,8 has promised \na new class of magnetically active nanoelectromechanical systems (NEMS). Here we \ndemonstrate resonators made of 2D CrI 3, whose mechanical resonances depend on \nthe magnetic state of the material. We quantify the underlining effects of exchange \nand anisotropy magnetostriction by measuring the field dependence of the \nresonance frequency under a magnetic field parallel and perpendicular to the easy \naxis, respectively. Furthermore, we show efficient strain tuning of magnetism in 2D \nCrI3 as a result of the inverse magnetostrictive effect using the NEMS platform. Our \nresults establish the basis for mechanical detection of magnetism and magnetic \nphase transitions in 2D layered magnetic materials. The new magnetic NEMS may \nalso find applications in magnetic actuation and sensing. \n \nTwo-dimensional (2D) layered magnetic materials are attractive building blocks for \nnanoelectromechanical systems  (NEMS): while they share high stiffness and strength and \nlow mass with other 2D materials, they are magnetically active 1,2,3,6,7,8. Among the large \nclass of newly emerged 2D layered magnetic materials, of particular interest is few-layer \nCrI3, whose magnetic ground state consists of antiferromagnetically coupled \nferromagnetic (FM) monolayers with out-of-plane easy axis 3,5. The interlayer exchange \ninteraction is relatively weak, a magnetic field on the order of 0.5 T in the out-of-plane ( 𝒛ො) \ndirection can induce spin-flip transition in bilayer CrI 3. Remarkable phenomena and \ndevice concepts based on detecting and controlling the interlayer magnetic state have \nbeen recently demonstrated, including spin-filter giant magnetoresistance 9 , 10 ,11 ,12, \nmagnetic switching by electric field 13 or electrostatic doping 14 , 15 , 16 ,17, and spin \ntransistors 18. The coupling between the magnetic and mechanical properties in \natomically thin materials, the basis for 2D magnetic NEMS, however, remains elusive \nalthough NEMS made of thicker magnetic materials 19 or coated with FM metals 20,21 \nhave been studied.  \n \nHere we demonstrate magnetic NEMS resonators made of 2D CrI 3 drumhead membranes \nand investigate magnetostriction and its inverse effect in the material (Fig. 1a). We focus  2 on even-layer CrI 3 for maximum effect. We encapsulate CrI 3 (an air sensitive insulator) \nwithin two stable 2D materials, few-layer graphene (below) and monolayer WSe 2 (above) \n(Fig. 1b). In addition to protect CrI 3 from degradation under ambient conditions, few-\nlayer graphene acts as a conducting electrode, while monolayer WSe 2 as a strain gauge as \nwe discuss below. The 2D heterostructure was first assembled and then transferred on \nprefabricated circular microtrenches (2 – 3 μm in radius) with patterned Au electrodes \nand Si back gate. The optical image of a sample device is shown in Fig. 1c. We drive \nmechanical resonances in the membrane electrically in the linear regime using a small r.f. \ngate voltage, which is superposed on a DC gate voltage Vg that applies static tension to \nthe membrane by pulling it towards the gate. We detect the mechanical resonances \ninterferometrically at 633 nm using a network analyzer. We also measure the out-of-\nplane magnetization of the membrane using magnetic circular dichroism (MCD) at 633 \nnm. Unless otherwise specified, all measurements were performed at 4 K in a helium \nexchange gas environment (<10-5 Torr). (See Methods for details on device fabrication \nand measurement techniques.)  \n \nFigure 1d shows the fundamental resonance mode of a bilayer CrI 3 membrane at Vg = 0. \nIt has a Lorentzian lineshape (solid line) with a peak frequency f around 25.5 MHz and a \nwidth around 10 kHz, corresponding to a quality factor of about 2500. The resonance \nfrequency scales inversely with the drumhead radius R, and is determined by stress on the \nmembrane 𝜎 and its 2D mass density  𝜌. Figure 1e shows gate tuning of the resonance \nfrequency, which is well described by the continuum model (assuming zero bending \nstiffness) with fully clamped boundary: 𝑓=క\nଶగோටఙ\nఘ (red solid line) . Here 𝜉 ≈2.405 is the \nfirst root of the zeroth order Bessel function and the stress is obtained by the second \nderivative of the total energy (elastic plus electrostatic) of the membrane with respect to \nits vertical displacement. Near Vg = 0, the frequency is set by the built-in stress 𝜎, \nwhereas at higher Vg the gate-induced stress 3𝑌𝜖(𝑉), which is determined by the \neffective 2D Young’s modulus  𝑌 and the gate-induced strain 𝜖(𝑉), causes the \nfrequency to increase roughly as 𝑉ସ. This behavior is fully consistent with other 2D \nNEMS such as graphene and MoS 2 [Ref. 22,23,24]. By calibrating 𝜖(𝑉) from the exciton \npeak shift of the WSe 2 layer (63 meV per % of biaxial strain 25), we determine the basic \nparameters of the resonator 𝜎≈ 0.5 N/m, 𝑌≈600 N/m and 𝜌≈ 3×10ିହ kg/m2 \n[Ref. 22,23,24] (see Methods).  \n \nWe investigate the effect of magnetic field on CrI 3 resonators at Vg = 0. The \nnanomechanical resonance of the bilayer CrI 3 device is measured while the field is swept \nfrom 1 T to -1 T back to 1 T along 𝒛ො (Fig. 2a). The resonance frequency 𝑓 is independent \nof field except an abrupt redshift of ~ 0.06% when the field magnitude exceeds ~ 0.5 T. \nThe linewidth and amplitude basically do not change throughout the entire field range \n(Supplementary Fig. S1). We correlate the field dependence of 𝑓 (Fig. 2b) with the \nmembrane MCD (Fig. 2c). The measured MCD is consistent with the reported results for  3 bilayer CrI 3 [Ref. 14]. It shows that the membrane is antiferromagnetic (AF) under small \nfields (the finite MCD signal is caused by layer asymmetry in the heterostructure) and \nundergoes a first-order AF-FM spin-flip transition (with hysteresis) around ±0.5 T. The \nmechanical resonance frequency is thus correlated with the sample’s magnetic state with \n𝑓ி>𝑓ிெ, where 𝑓ி and 𝑓ிெ denote the resonance frequency in the AF and FM state, \nrespectively. In contrast to MCD, 𝑓 is insensitive to the magnetization direction. \n \nSimilar behavior is observed in all CrI 3 resonators that were investigated in this study. \nFigure 2d - 2f shows the corresponding result for a 6-layer CrI 3 resonator under an out-\nof-plane field up to 2.3 T. A second 6-layer device is shown in Supplementary Fig. S2. \nCompared to bilayer CrI 3, there are now two spin-flip transitions around 0.9 T and 1.8 T, \nwhich correspond to spin flip in the surface layer and the interior layers, respectively 9,18. \nThe resonance frequency redshifts at each spin-flip transition. The total redshift from 𝑓ி \n(~ 0 T) to 𝑓ிெ (>1.8 T) is about 0.23%. This is nearly 5 times larger than in bilayer CrI 3.  \n \nIn contrast, the behavior of the resonators under an in-plane magnetic field is distinct. \nFigure 3a shows the nanomechanical resonance of the same 6-layer CrI 3 device as in Fig. \n2d - 2f while the magnetic field is swept from 8 T to -8 T to 8 T along the in-plane \ndirection. The field dependence of 𝑓 is shown in Fig. 3b after subtraction of a small linear \ndrift due to initial stress relaxation and/or slow temperature drift. Sharp transition and \nhysteresis that are characteristic for the first-order spin-flip transition are now absent. \nInstead, 𝑓 redshifts smoothly with increasing field magnitude and saturates beyond ~ 6 T. \nThis behavior is correlated with the spin-canting effect observed in 2D CrI 3 [Ref. 9,18]. \nBecause of the large out-of-plane magnetic anisotropy, the spins under an in-plane field \nare canted continuously from 𝒛ො until reaching the saturation field, beyond which they are \nfully aligned with the in-plane field. The total change in 𝑓 under the in-plane field (~ \n0.13%) is about half of the value under the out-of-plane field for the 6-layer CrI 3 \nresonator. \n \nThe resonance frequency of the NEMS at Vg = 0 is determined by the built-in stress 𝜎. \nThe observed correlation between the resonance frequency and the magnetic state of 2D \nCrI3 suggests that magnetostriction is a result of competition between minimizing the \nelastic energy and the internal magnetic interactions. Other effects such as arisen from the \nmagnetostatic energy cannot explain the experimental observations (see Methods). The \nelastic energy of the membrane per unit area can be expressed as 𝑈=ଷ\nଶ𝑌𝜖ଶ [Ref. \n22,23,24]. Here the effective 2D Young’s modulus 𝑌 is dominated by that of few-layer \ngraphene because of its much higher mechanical stiffness and is independent of magnetic \nfield (Methods); strain 𝜖= (𝑎−𝑎)𝑎⁄ is defined as the fractional change of the in-\nplane lattice constant 𝑎 that conforms to fixed boundary from its equilibrium value 𝑎. If \nwe neglect the strain dependence of the demagnetization energy and leave out the \nintralayer exchange interactions, which do not play a role in the interlayer metamagnetic  4 transition, the part of free energy that is strain dependent can be expressed for bilayer \nCrI3 to the lowest relevant order as  \n \n𝐹=𝐽ୄ൫𝑺𝒕∙𝑺𝒃൯+𝐾ቀห𝑺𝒕×𝒛ොหଶ+ห𝑺𝒃×𝒛ොหଶቁ.   (1) \n \nHere 𝑺𝒕 and 𝑺𝒃 denote the spin unit vector of the top and bottom CrI 3 layers, respectively. \nThe two terms describe, respectively, the interlayer exchange coupling with  energy per \nunit area 𝐽ୄ (> 0) and anisotropy with effective energy per unit area per layer  𝐾 (> 0). \nFor simplicity we only consider the effect of bond length change on 𝐽ୄ and 𝐾. The \nequilibrium lattice constant of the strained membrane in each magnetic state can be found \nby minimizing the total free energy ( 𝑈+𝐹) with respect to 𝑎 or equivalently strain 𝜖, \nfrom which stress ( 𝜎=𝑌𝜖) and mechanical resonance frequency can be evaluated 26 \n(see Methods).  \n \nFor spin-flip transitions, the anisotropy energy is not relevant because spins are along the \neasy axis. Minimizing ( 𝑈+𝐹) for bilayer CrI 3 yields a change in the strain level \n(−ଶ\nଷడ఼\nడఢ) and a fractional change in the resonance frequency ( −ଵ\nଷఙబడ఼\nడఢ) for the AF-\nFM transition. We estimate డ఼\nడఢ𝐴௨≈ −3 meV, where 𝐴௨ (≈ 0.47 nm2) is the unit cell \narea, from the experimental resonance frequency shift and the measured built-in stress 𝜎.  \nThe negative sign indicates that strain weakens AF ordering . We also estimate the \nsaturation magnetostriction |ଶ\nଷడ఼\nడఢ| ~ 10-6 for the bilayer CrI 3 heterostructure. We \nnote that bilayer CrI 3 by itself is expected to have a much higher saturation \nmagnetostriction because of its much smaller Young’s modulus compared to graphene. \n \nIn 6-layer CrI 3 there are 5 interfaces. The nearly 5-time stronger magnetostrictive \nresponse observed in experiment can be understood in terms of a 5-time stronger total \ninterlayer exchange energy 5𝐽ୄ൫𝑺𝒕∙𝑺𝒃൯ and similar elastic energy compared to the \nbilayer CrI 3 device. Under an in-plane field, both interlayer exchange and anisotropy \ncontributions to the free energy need to be considered and a similar energy minimization \nscheme can be applied to yield −ଵ\nଷఙబቀ5డ఼\nడఢ+6డ\nడఢቁ for the fractional resonance \nfrequency shift. The observed ~ 50% smaller frequency shift in Fig. 3b compared to Fig. \n2e suggests that డ\nడఢ ≈ −ଵ\nଶడ఼\nడఢ in the 6-layer CrI 3 device.  \n \nFinally, we use the NEMS platform to demonstrate strain tuning of magnetism through \nthe inverse magnetostrictive effect. We probe the spin-flip transition in the same bilayer \nCrI3 device as in Fig. 2a-c at different Vg’s by MCD. The application of Vg can potentially \ninduce both tension by an electrostatic force and electrostatic doping to the suspended \nmembrane. Figure 4a and 4b compare the behavior of a suspended and substrate-\nsupported region of the membrane. Only the MCD data near the spin-flip transition for  5 the positive sweeping direction of the field is shown for clarity. For the substrate-\nsupported region, only the electrostatic doping effect is relevant. The spin-flip transition \nfield 𝐻 varies linearly with Vg (red symbols, Fig. 4c), which is in good agreement with \nprevious studies 14,15. In contrast, the behavior of the suspended region is symmetric \nabout Vg = 0; 𝐻 decreases nonlinearly with magnitude of Vg (blue symbols, Fig. 4c). We \nconclude from the comparison that electrostatic doping into suspended CrI 3 is negligible \nand the change in 𝐻 is caused primarily by strain.  \n \nFigure 4d shows 𝐻 as a function of gate-induced strain calibrated from the exciton peak \nshift in the WSe 2 layer. It decreases linearly with strain by as much as ~ 32 mT. \nQuantitatively, 𝐻 can be related to the interlayer exchange coupling 𝐽ୄ through 𝜇𝐻=\n఼ାఓబெబమ/ௗ\nெబ [Ref. 14], where 𝜇, 𝑀 and 𝑑 denote the vacuum permeability, the saturation \nmagnetization per CrI 3 monolayer, and the interlayer distance in bilayer CrI 3, respectively \n(see Methods). We obtain డ఼\nడఢ𝐴௨≈ −5 meV from the slope in Fig. 4d and 𝑀 by \nassuming that each Cr3+ cation carries a magnetic moment of 3 𝜇 (Bohr magneton). The \nvalue agrees well with that estimated from the mechanical resonance measurement. The \ndiscrepancy is largely due to uncertainty in the resonator parameters (such as initial stress \nand mass). Nevertheless, the good agreement between two independent measurements \nillustrates the importance of interlayer exchange magnetostriction in 2D CrI 3.  \n \nIn conclusion, we have demonstrated a new type of magnetostrictive NEMS based on 2D \nCrI3. An inverse magnetostrictive effect has also been demonstrated, which allows \ncontinuous strain tuning of the internal magnetic interactions.  Our results have put in \nplace the groundwork for potential applications of these devices, including magnetic \nactuation and sensing, as well as a general detection scheme based on mechanical \nresonances for emerging magnetic states and phase transitions in 2D layered magnetic \nmaterials. \n \n \nMethods \nDevice fabrication \nWe used the layer-by-layer dry transfer method to fabricate drumhead resonators made of \natomically thin CrI 3 fully encapsulated by few-layer graphene and single-layer WSe 2 as \nshown in Fig. 1b. Atomically thin flakes of CrI 3, graphene and WSe 2 were mechanically \nexfoliated onto silicon substrates with a 285-nm oxide layer  from the corresponding bulk \nsynthetic crystals (from HQ Graphene). A polymer stamp made of a thin layer of \npolycarbonate (PC) on polydimethylsiloxane (PDMS) was used to pick up the desired \nflakes one by one to form the heterostructure. The complete heterostructure was first \nreleased onto a new PDMS substrate so that the residual PC film on the sample can be \nremoved by dissolving it in N-Methyl-2-pyrrolidone. The sample was then transferred \nonto a circular microtrench of 2 – 3 𝜇m in radius and 600 nm in depth on a silicon \nsubstrate with prepatterned Ti/Au electrodes . A small amount of PDMS residual was left  6 untreated. CrI 3 was handled inside a nitrogen-filled glovebox with oxygen and water less \nthan 1part per million (ppm) to avoid degradation. The thickness of atomically thin \nmaterials was first estimated by their optical reflection contrast and then measured by \natomic force microscopy (AFM). The thickness of CrI 3 flakes was further verified by the \nmagnetization versus out-of-plane magnetic field measurement. The thickness of WSe 2 \nflakes was verified by optical reflection spectroscopy.  \n \nMechanical resonance detection  \nAn optical interferometric technique was applied to detect the out-of-plane displacement \nof the resonators 22,24,27 (Fig. 1a). The resonators were mounted in a closed-cycle cryostat \n(Attocube, attoDry1000). The output of a HeNe laser at 632.8 nm was focused onto the \ncenter of the suspended membrane using a high numerical aperture (N.A. = 0.8) objective. \nThe beam size on the device was on the order of 1 μm and the total power was kept \nbelow 1 μW to minimize the laser heating effect. The reflected beam was collected by the \nsame objective and detected by a fast photodetector. Motion of the membrane was \nactuated capacitively by applying a small r.f. gate voltage (~ 1 mV) between the \nmembrane and the back gate. As the membrane moves in the out-of-plane direction, the \noptical cavity formed between the membrane and the trenched substrate modulates the \ndevice reflectance. The amplitude of the motion as a function of driving frequency was \nmeasured by a network analyzer (Agilent E5061A), which both provided the r.f. voltage \nand measured the fast photodetector response. The amplitude of the motion reaches its \nmaximum when the r.f. frequency matches the natural frequency of the resonator. \n \nMagnetic circular dichroism (MCD) measurements  \nMCD was employed to characterize the membranes’ magnetic properties. A nearly \nidentical optical setup as the one used for the mechanical resonance detection was \nemployed. The polarization of the optical excitation at 632.8 nm  was modulated between \nleft and right circular polarization by a photoelastic modulator at 50.1kHz. Both the a.c. \nand d.c. component of the reflected beam were detected by a photodiode in combination \nwith a lock-in amplifier and multimeter, respectively. MCD was determined as the ratio \nof the a.c. and d.c. component.   \n \nStrain calibration  \nGate-induced strain in the suspended membranes was determined by using monolayer \nWSe2 as a strain gauge assuming no relative sliding between the constituent 2D layers. It \nrelies on the fact that the fundamental exciton resonance energy in monolayer WSe 2 \nredshifts linearly with strain (63 meV/%) for a relatively large rage of biaxial strain 25. \nThe fundamental exciton energy in monolayer WSe 2 was determined as a function of Vg \nby optical reflection spectroscopy. In these measurements, broadband radiation from a \nsingle-mode fiber-coupled halogen lamp was employed. The collected radiation was \ndetected by a spectrometer equipped with a charge-coupled-device (CCD) camera. The \nexcitation power on the device was kept well below 0.1 µW to minimize the laser heating \neffect. The exciton resonance energy and the calibrated gate-induced strain as a function \nof Vg is shown in Supplementary Fig. S3.   7  \nCharacterization of resonator parameters   \nBy minimizing the sum of the elastic energy and the electrostatic energy with respect to \nstrain (i.e. డ\nడఢቂଷ\nଶ𝑌𝜖ଶ−ଵ\nଶ𝐶𝑉ଶቃ= 0), we obtain the gate-induced strain 𝜖൫𝑉൯≈\nଵ\nଽቀோ\nቁଶ\nቀఌబమ\nఙబቁଶ\n∝𝑉ସ for 𝜖൫𝑉൯≪𝜖 (𝜖 is the built-in strain).  Here 𝐶 is the back gate \ncapacitance, which is strain or gate dependent because of the gate-induced vertical \ndisplacement of the membrane, 𝐷 is the vertical separation between the membrane and \nthe back gate at Vg = 0, R is the drumhead radius, and 𝜀 is the vacuum permittivity. The \nbuilt-in stress 𝜎 can be obtained from the slope of 𝜖 as a function of 𝑉ସ (Supplementary \nFig. S3c) . The effective Young’s modulus 𝑌 and the 2D mass density 𝜌 of the \nmembrane can be obtained by fitting the experimental gate dependence of the resonance \nfrequency 𝑓=క\nଶగோටఙ\nఘ with 𝜎=ோమ\nସడమ\nడ௭మቂଷ\nଶ𝑌𝜖ଶ−ଵ\nଶ𝐶𝑉ଶቃ, where 𝑧 is the vertical \ndisplacement [Ref. 28,29,30]. The extracted values for bilayer CrI 3 resonator 1 are 𝜎≈ 0.5 \n𝑁𝑚ିଵ, 𝜌≈ 3×10ିହ 𝑘𝑔𝑚ିଶ, and 𝑌≈ 600 𝑁𝑚ିଵ. The mass density is 1.9 times of \nthe expected mass density of the heterostructure presumably due to the presence of \npolymer residues and other adsorbates on the membrane. The effective Young’s modulus \nis also consistent with the reported values. We estimate the effective Young’s modulus of \n2D heterostructures as the total contribution of constitute layers, 𝑌=∑𝑛𝑌ଶ, , where \n𝑛 and 𝑌ଶ, are the layer number and the 2D elastic stiffness of the i-th material per layer. \nWe obtain 𝑌≈ 1200 𝑁𝑚ିଵ for bilayer device 1 using 𝑌ଶ,= 340 𝑁𝑚ିଵ, 𝑌ଶ,ூయ=\n25 𝑁𝑚ିଵand 𝑌ଶ,ௐௌమ= 112 𝑁𝑚ିଵ reported for the constitute 2D materials [Ref. \n31,32,33,34]. The discrepancy may come from the presence of polymer residues and/or \nwrinkles on the membrane. Most devices studied in this work consist of 3 layers of \ngraphene, 1 layer WSe 2, and 2-6 layers of CrI 3. 𝑌 is therefore dominated by the \ncontribution of graphene.  \n \nMechanisms for mechanical resonance shift in 2D CrI 3 under a magnetic field  \nIn the main text, we have assigned the competition between the internal magnetic \ninteractions and elastic energy as the major mechanism for the observed mechanical \nresonance shift in 2D CrI 3 resonators under a magnetic field at Vg = 0. Other effects could \npotentially also give rise to a mechanical resonance shift in 2D CrI 3 under a magnetic \nfield. One such possibility is the magnetostatic pressure from the gradient of the \nmagnetostatic energy in the 𝒛ො direction, 𝑴∙𝜕𝑩\n𝜕𝑧. Here 𝑴 is the sample magnetization and \n𝑩 is the magnetic field at the sample. However, for a nanometer thick sample with a \nlateral size of a few microns the gradient is expected to be negligible. This is supported \nby the absence of field dependence for the mechanical resonance up to high field except \nat the spin-flip transitions (Supplementary Fig. S4).  \n \nModel for exchange magnetostriction in bilayer CrI 3 \nWe consider bilayer CrI 3 under an out-of-plane magnetic field 𝜇𝐻. The free energy per \nunit area for a free membrane in the AF and FM state in the zero-temperature limit can be \nexpressed as   8  \n𝐹ி= 2𝐹−𝐽ୄ,                                                  (S1) \n𝐹ிெ= 2𝐹+𝐽ୄ−2𝜇𝑀ቀ𝐻−ெబ\n௧ቁ.                     (S2) \n \nHere 𝐹, 𝑀, 𝐽ୄ and 𝑡 denote the free energy of each monolayer, saturation magnetization \nof each monolayer, interlayer exchange interaction, and interlayer distance, respectively. \nThe spin-flip field 𝜇𝐻 can be evaluated by requiring 𝐹ி=𝐹ிெ [Ref. 13]. The \nequilibrium lattice constant in the two interlayer magnetic states differs slightly in free \nmembranes. In the presence of fixed boundary, strain is developed with an elastic energy \nof 𝑈=ଷ\nଶ𝑌𝜖ଶ, where 𝜖= (𝑎−𝑎)𝑎⁄ is the fractional change of the in-plane lattice \nconstant 𝑎 (which is fixed by the boundary) from its equilibrium value 𝑎. A new \nequilibrium configuration with lattice constants 𝑎ி and 𝑎ிெ is reached for the two \nmagnetic states to minimize the corresponding total free energy ( 𝑈+𝐹) with respect to \n𝑎. Strain in the AF and FM states 𝜖ி and 𝜖ிெ can thus be found by solving the \nfollowing equations: \n \n𝑎డ(ାிಲಷ)\nడబ= 2డிబ\nడఢ−డ఼\nడఢ−3𝑌𝜖ி= 0.                               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Additional data are available from the corresponding authors \nupon request. \n   11Figures and figure captions \n \nFig. 1: 2D CrI 3 nanoelectromechanical resonators. a, Schematic of the measurement \nsystem. The resonator (suspended 2D membrane on a Si trench of depth D) is actuated by \nan r.f. voltage from a vector network analyzer (VNA) through a bias tee. A DC voltage \nVg is superimposed to apply static tension to the membrane. The motion is detected \ninterferometrically by a HeNe laser, which is focused onto the center of the resonator. BS: \nbeam splitter; PD: photodetector. b, Schematic of a bilayer CrI 3 resonator with AF CrI 3 \nencapsulated by few-layer graphene and monolayer WSe 2. Filled spheres and arrows \ndenote Cr atoms and spins in the top and bottom CrI 3 layer. c, Optical microscope image \nof a bilayer CrI 3 device suspended over a circular trench. Dashed line shows the \nboundary of the CrI 3 flake. Scale bar is 4 μm. d, Fundamental mechanical resonance \n(symbols) of bilayer CrI 3 resonator 1 (radius 2 μm) and a Lorentzian fit of the resonance \nspectrum (solid line). e, Gate dependence of the measured resonance frequency (symbols) \nand fit to the continuum model (solid line) with 𝜎≈0.5 𝑁𝑚ିଵ, 𝜌 ≈ 3×\n10ିହ 𝑘𝑔𝑚ିଶ  (1.9 times the mass density of the membrane) and 𝑌≈ 600 𝑁𝑚ିଵ . \n \n 12 \nFig. 2: Mechanical detection of the spin-flip transition in 2D CrI 3. a, Normalized \nvibration amplitude of bilayer CrI 3 resonator 1 vs. driving frequency under an out of-\nplane magnetic field ( 𝜇𝐻ୄ) that sweeps from 1 T to -1 T to 1 T. b, Resonance \nfrequency  𝑓 extracted from a as a function of magnetic field. c, MCD of the membrane as \na function of magnetic field. The red and blue lines in b and c correspond to the \nmeasurement for the positive and negative sweeping directions of the field, respectively. \nd-f, Same measurements as in a-c for 6-layer CrI 3 resonator 1 (radius 3 μm).   \n \n 13 \nFig. 3: Mechanical detection of spin canting in 2D CrI 3. a, Normalized vibration \namplitude vs. driving frequency under an in-plane magnetic field ( 𝜇𝐻||) that sweeps \nfrom 8 T to -8 T to 8 T for 6-layer CrI 3 resonator 1. b, Resonance frequency  𝑓 extracted \nfrom a as a function of magnetic field. Red and blue lines correspond to the measurement \nfor the positive and negative sweeping directions of the field, respectively. \n \n \n \n \n \n \n \n \n 14 \nFig. 4: Strain tuning of the spin-flip transition in 2D CrI 3. a, b, Normalized MCD as a \nfunction of out-of-plane magnetic field ( 𝜇𝐻ୄ) that sweeps from - 1 T to 1 T (only - 0.6 T \nto 0.6 T is shown here for clarity) at different Vg’s for a suspended ( a) and substrate-\nsupported region ( b) of bilayer CrI 3 resonator 1. c, Spin-flip transition field as a function \nof gate voltage for the suspended (blue) and substrate-supported (red) region of the \nmembrane. The lines are a guide to the eye. d, Spin-flip transition field as a function of \ngate-induced strain (symbols). The solid line is a linear fit.  \n \n  \n 15Supplementary figures and figure captions \n \nSupplementary Fig. S1: Mechanical resonance of 2D CrI 3 under an out-of-plane \nmagnetic field. a, b,  Field dependence of the amplitude ( a) and linewidth ( b) of the \nfundamental resonance of bilayer CrI 3 resonator 1. The field dependence of the resonance \nfrequency is shown in Fig. 2b. The red and blue symbols correspond to the measurement \nfor the positive and negative sweeping directions of the field, respectively. c, Field \ndependence of the reflectance at 633 nm normalized by the reflectance at zero field. \n \n 16  \nSupplementary Fig. S2: Results for 6-layer CrI 3 resonator 2 under an out-of-plane \nmagnetic field. a , Normalized vibration amplitude vs. driving frequency under an out of-\nplane magnetic field ( 𝜇𝐻ୄ) that sweeps from 2.3 T to -2.3 T to 2.3 T at  𝑉 = 0 V. b, \nResonance frequency  𝑓 extracted from a as a function of magnetic field. c, MCD of the \nmembrane as a function of magnetic field at 𝑉= 0 𝑉. The red and blue lines in b, c \ncorrespond to the measurement for the positive and negative sweeping directions of the \nfield, respectively. The radius of the drumhead is 3 μm. \n \n \n \n \n \n \n \n \n 17 \n  \nSupplementary Fig. S3: Strain calibration and determination of resonator \nparameters.  a, Reflection contrast of bilayer CrI 3 resonator 1 from 1.6 – 1.9 eV as a \nfunction of gate voltage. The main feature is a dip (around 1.75 eV at 𝑉= 0 𝑉), which \ncorresponds to the fundamental exciton resonance of monolayer WSe 2. The feature \nredshifts slightly with 𝑉 up to about 40 V followed by a much larger redshift with further \nincrease of 𝑉. b, Representative spectra at selected 𝑉. c, Exciton resonance energy \nextracted from a (left axis) and gate-induced strain calibrated from the exciton resonance \nenergy (right axis) as a function of 𝑉ସ for 𝑉 up to 39 V. The solid line is a linear fit from \nthe slope of which t he built-in stress 𝜎 was determined. \n \n \n \n \n 18 \nSupplementary Fig. S4: 2D CrI 3 resonators under high out-of-plane fields. \nNormalized vibration amplitude vs. driving frequency under an out of-plane magnetic \nfield up to 5 T for 6-layer CrI 3 resonator 1. The resonance frequency, amplitude and \nlinewidth basically do not change up to 5 T except at the spin-flip transitions.  \n \n \n"    },    {        "title": "2001.04044v1.Gate_tunable_spin_waves_in_antiferromagnetic_atomic_bilayers.pdf",        "content": " 1 Gate -tunable spin wave s in antiferromagnet ic atomic bilayers  \n \nXiao -Xiao Zhang1,2, Lizhong Li3, Daniel W eber4, Joshua Goldberger4, Kin Fai Mak1,3,5*, \nJie Shan1,3,5* \n \n1Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY 14853, USA  \n2Department of Physics, University of Florida  \n3School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA  \n4Department of Chemistry and Biochemistry , Ohio State University  \n5Laboratory of Atomic and Solid State Physics, Cornell Unive rsity, Ithaca, NY 14853, USA.  \n \n \nThe emergence of two -dimensional (2D) layered magnetic materials has open ed an exciting \nplayground for both fundamental studies of magnetism in 2D and exploration s of spin -\nbased applications  1-4. Remarkable properties , including spin filtering  in magnetic tunnel \njunctions  and gate control  of magnetic states , have recently been demonstrated  in 2D \nmagnetic materials  5-12. While these studies focu s on the static properties , dynamic \nmagnetic properties  such as excitation  and control of spin waves  have remain ed elusive. \nHere we excite spin waves and probe their dynamics in antiferromagnetic CrI 3 bilayers by \nemploying an ultrafast optical pump/ magneto -optical Kerr probe technique . We identify \nsub-terahert z magnetic resonances under an in -plane magnetic field , from which we \ndetermine the anisotropy and interlayer  exchange field s and the spin damping rates . We \nfurther show tuning of antiferromagnetic resonances by tens of gigahertz  through  \nelectrostatic gating . Our results shed light on magnetic excitations and spin dy namics in 2D \nmagnetic materials, and dem onstrate their unique potential for applications in ultrafast \ndata storage and  processing.  \n \nSpin waves, first predicted by F. Bloch in 1929, are propagating disturbance s in magnetic \nordering  in a magnetic material  13. The quanta of spin waves  are called magnons . The rich s pin-\nwave phenomena in magneti c materials have attracted fundamental  interest  and impacted on \ntechnology of telecommunication  systems , radars, and potentially  also low-power information \ntransmission and processing due to their decoupling from charge current  14,15. The main magnetic \nmaterial s of interest  have so far  been f erromagnets (FM) . The operation speed of FM-based \ndevice s is typically in  the GHz range , which is limited by  the ferromagnetic resonance (zero-\nmomentum resonance ) frequency . One of the major attractions of antiferromagnets (AFM ), a \nclass of much more common magnetic materials,  is the prospect of high -speed operation. The \nantiferromagnetic resonance s are in the frequency range of as high as THz due to the spin-\nsublattice exchange  16. The AFMs , however,  are difficult to access due to the absence of \nmacroscopic  magnetization.  \n \nThe recent discovery of t wo-dimensional (2D) layered magnetic materials  17-19, particularly A-\ntype AFMs  such as bilayer CrI 3 that are made of two antiferromagnetically coupled \nferromagnetic monolayers  17, present s new opportun ities to unlock the properties of AFMs. With \nfully uncompensated ferromagnetic surfaces , the magnetic state  can be easily accessed  and \ncontrolled  20. The van de r Waals nature allows their convenient integration into heterostructure s \nwith high -quality interface s 21. And t he atomic thickness  allows  the application of strong electric  2 field and large electrostatic doping to control the properties of 2D magnet ic material s. Although \nrapid progress has been made  in both fundamental understanding and potential applications  1-12,22, \nthe s pin dynamics , including basic properties such as magnetic resonances and damping , have \nremained unexplored  in these materials . A major technical challenge arises from the smal l \namount of spins present in atomically thin  samples of typical lateral dimensions of a few microns. \nThis makes studies with conventional probes,  such as  neutron scattering and microwave \nabsorption  23,24, extremely d ifficult or impossible. Microwave absorption  measurements are  \nfurther hindered by the high antiferromagnetic resonance frequencies.  \n \nHere we investigate spin-wave excitations in bilayer CrI 3 using  the time-resolved magneto -\noptical Kerr effect (MOKE). The sample consists of a heterostructure of bilayer CrI 3 and \nmonolayer WSe 2, which  is encapsulated in two hexagonal boron nitride (hBN) thin layers for \nprotect ion of  air-sensitive CrI 3 (Fig. 1a). While monolayer CrI 3 is a ferromagnetic semiconductor \nwith out -of-plane anisotropy  below the Curie temperature  of about 40 K, bilayer CrI 3 is an AFM \nwith spins in the two ferromagnetic monolayer s anti-aligned  below the N éel temperature of \nabout 45 K 17. Monolayer WSe 2 is a direct gap non-magnetic semiconductor  with strong spin-\norbit interaction  25. It is believed to have a type -II band alignment with CrI 3 26 (Fig. 1 b). The \nintroduction of WSe 2 significantly enhances  optical absorption of the pump and hot carrier  \ninjection into CrI 3 for magnetic excit ations . As will be discussed below , WSe 2 also breaks the \nlayer symmetry in bilayer CrI 3 to enable the detection of different oscillation  mode s of spin \nwaves in the polar MOKE  geometry . Figure 1 c is the magnetization of bilayer CrI 3 as a function \nof out -of-plane magnetic field at 4 K probed by  magnetic circular dichroism (MCD)  at 1.8 eV. \nThe antiferromagnetic behavior  is fully consistent with the reported results  17. The small nonzero \nmagnetization near zero field is a manifestation of the broken layer symmetry. The sharp turn -on \nof the magnetization around 0.75 T corresponds to a spin -flip transition, which provides a \nmeasure of the interlayer exchange field HE.  \n \nA pulsed laser (200 -fs pulse duration) was employed for the time -resolved  measurements . The \nheterostructure was excited by a  light pulse centered  near the WSe 2 fundamental exciton \nresonance  energy  (1.73 eV), and the change in CrI 3 magnetization is probed  by a time -\nsynchronized pulse  at a lower energy ( 1.54 eV ). Both t he pump and probe were  linearly \npolarized and at normal incidence. The polarization  rotation of the reflected probe beam  locked \nto the modulation frequency of the pump  was detected. In this configuration  the MOKE signal is \nsensitive only to the out -of-plane m agnetization.  An in -plane magnetic field H|| was applied, \nwhich causes the magnetization  of both the top and bottom layer s to cant (Fig. 1d). The field \nrequired to rotate the ordered moments into the  in-plane direct ion, which is referred to as the \nsaturation field HS, has been reported to be near 3.8 T for bilayer CrI 3 at 2 K8. Unless otherwise \nspecified, all measurements were  performed at 1.7 K.  (See Methods for details on the sample \nfabrication and the time -resolved MOKE set up.) \n \nFigure  2a displays the time evolution of the pump -induced change in the MOKE signal of bilayer \nCrI 3 under  H|| ranging from 0 – 6 T. For all fields , the MOKE signal shows a sudden change at  \ntime zero, followed by a decay  on the scale of 10’s – 100’s ps . This  reflects the incoherent  \ndemagnetization  process , in which the magnetic order is disturbed instantaneously by the pump \npulse and is slowly reestablishe d. Oscillations in the MOKE sig nal that are also instantaneous  3 with the optical excitation become clearly observable with increasing magnetic field . The \namplitude, frequency and damping of these oscillations evolve systematically with H||.  \n \nFigure 2b is the fast Fourier transform (FFT)  of the oscillat ory part  of the time traces in Fig. 2a.  \nTwo examples are shown in Fig. 3a and 3b for H|| at 1. 5 and 3.75 T, respectively. The \nexponential decay  of the incoherent demagnetization dynamics has been  subtracted from the \nMOKE signal before performing  FFT. At low magnetic field , a resonance around 70 GHz is \nobserved. As H|| increases, it splits into two  resonances  with one that redshifts significantly  and \nthe other  that exhibits minimal shifts in frequency until 3.3 T. Above  this field , both resonance s \nblueshift with  increasing  H||. While t he low -energy mode quickly becomes too weak to be \nobserved , the amplitude of the high -energy mode  does not depend strongly on  field.  \n \nWe performed a careful analysis of the MOKE dynamics directly in the time domain , fittting the \noscillations  with two damped harmonic waves  (red lines , Fig. 3a, b ). The extracted resonance \nfrequenc ies, damping rate s and amplitudes as a function of H|| are summarized in Fig. 3 d, 3e and \nSupplementary Fig. S 7, respectively. We first focus on the resonance frequencies. The field \ndependence  of the resonance frequencies  shows two di stinct regimes . Below about  3.3 T, the two \nnearly degenerate modes (at small fields) both soften with increasing field , one slightly and the \nother nearly to zero frequency . Above 3.3 T , both modes show a linear increase  in frequency \nwith a slope equal to the electron gyromagnetic ratio 𝛾/2𝜋 ≈ 28 GHz/T . The latter is \ncharacteristic of a ferromagnetic resonance  under high fields .  \n \nThe observed magnetic -field dispersion of the resonances is indicative of  their magnon origin \nwith 3.3 T correspond ing to the saturation field HS in bilayer CrI 3. The two modes are the spin \nprecession eigenmodes  of the coupled top and bottom layer magnetizations under an in -plane \nfield (Fig. 3c) . Above the saturation field , the spins are aligned  along the applied field and the \nspin waves become ferromagnetic -like. This interpretation is further supported by the  \ntemperature dependence  of the resonances (Supplementary Fig. S3-5). Clear mode softening is \nobserved with increasing temperature and the resonance feature disappears near the Néel \ntemperature  of bilayer CrI 3. The microscopic mechanism for the observed ultrafast excitation of \nspin waves in bilayer CrI 3 is not fully understood. A plausible  process is the exciton generation \nin WSe 2 by the optical pump , followed by ultrafast exciton dissociation and charge transfer at the \nCrI 3-WSe 2 interface  26, and an impulsive perturbation to  the magneti c interactions  27,28 in CrI 3 by \nthe hot carriers . Detail s on the supporting experiments of this mechanism are provided in \nMethods.  \n \nWe model the field dependent spin dynamics using  the coupled Landau -Lifshitz -Gilbert (LLG) \nequations, which d escribe  precession  of antiferromagnetically coupled top and bottom layer \nmagnetization s under an in -plane field 𝐻∥ 29 (Details are provided in Methods ). The effective \nmagnetic field responsible for spin precession  in each layer includes contributions from t he \napplied field H||, intralayer anisotropy  field HA, and the interlayer exchange field HE. In the \nsimple case of negligible damping and symmetr ic top and bottom layers , the frequency of the \nprecession eigenmodes are found as 𝜔𝑇= 𝛾[ 𝐻𝐴( 2𝐻𝐸+𝐻𝐴) + 2𝐻𝐸−𝐻𝐴\n 2𝐻𝐸+𝐻𝐴𝐻||2]1\n2, 𝜔𝐿=\n 𝛾[ 𝐻𝐴( 2𝐻𝐸+𝐻𝐴)− 𝐻𝐴\n 2𝐻𝐸+𝐻𝐴𝐻||2]1\n2  (before saturation ); and 𝜔𝑇= 𝛾√𝐻||(𝐻||−𝐻𝐴), 𝜔𝐿= 4  𝛾√(𝐻||−2𝐻𝐸)(𝐻||−2𝐻𝐸−𝐻𝐴)  (after saturation).  As shown schematically in Fig. 3c, the low-\nenergy mode  corresponds to the longitudinal (with respect to 𝐻∥) mode  𝜔𝐿, which  has net \nmoment oscillations only along the applied field direction  (the y-axis). The high-energy mode \ncorresponds to the transverse (with respect to 𝐻∥) mode 𝜔𝑇, which has net moment oscillations \nin the x-z plane. The longitudinal mode 𝜔𝐿 drops to zero at the saturation field 𝐻𝑆 = 2𝐻𝐸+𝐻𝐴.  \n \nThe simple solution fits the experimental data well for the entire magnetic field range (dashed \nlines, Fig. 3 d) with HA ≈ 1.77 T and HE ≈ 0.76 T . The interlayer  exchange  HE is in good \nagreement with th e value from  the spin-flip transition  measurement  under an out -of-plane field  \n(Fig. 1 c). The intralayer anisotropy HA or the saturation field ( HS  ≈ 3.3 T) is slightly smaller than \nthe reported value  8, likely due to the different doping  levels  present in different samples (see \ngate dependence studies  below) . In contrast to the simple model, t he measured 𝜔𝐿 is always \nfinite likely due to the layer asymmetry in  bilayer CrI 3 (caused by coupling to monolayer WSe 2), \nas well as inhomogeneous broadening (see below). The layer asymmetry also allows the \nobservation of the low-frequency mode  in the polar MOKE geometry, which would otherwise \nhave zero out-of-plane magnetization . \n \nNext  we discuss the damping of the spin waves  in 2D CrI 3. Figure 3 e is the magnetic -field \ndependence of the normalized damping rate 2𝜋\n𝜔𝜏 for both the transverse and longitudinal modes. \nOverall, damping is substantially higher below and near the saturation field for both modes . In \naddition, da mping of the longitudinal mode is generally higher than the transverse mode.  The \nhigh damping  observed below and near HS is likely originat ed from inhomogeneous broadening \nof the magnetic resonances  and spin wave dephasing . In this regime, the resonance frequencies \nare strongly dependent on internal magnetic interactions , which are sensitive to local doping and \nstrain  within the 2D layers . For instance, a ±10 % variation in the  interlayer exchange field alone  \n(which is comparable to the typical inhomogeneity reported in bilayer CrI 3 3) can account for the \nobserved damping  of the transverse mode  at HS. Inhomogeneous broadening also explains the \nseemingly larger damp ing for the longitudinal mode  near HS, where ωL has a steep dependence \non HE and HA. Above HS, the resonance  frequencies are basically determined by the  applied field \nand inhomogeneous broadening becomes insignificant , especially in the high-field limit ( e.g. at 6 \nT). Other damping mechanisms such as interfacial damping and  spin-orbit coupling of the i odine \natom  could be come  relevant  here. However,  our experiment on few -layer CrI 3 in the high -field \nlimit  show s weak dependence of  (𝜏𝑇)−1 on layer number  (Supplementary Fig. S 6), suggesting \nthat interfacial damping is not  important . Future systematic studies are warranted to fully \nunderstand the microscopic damping mechanisms.  \n \nFinally we demonstrate control of the spin waves by electrostat ic gating  using a dual -gate device \n(Methods) . Figure 4 a shows the FFT amplitude spectra of coherent spin oscillations under a \nfixed magnetic field  of 2 T at different gate voltages . The resonance shifts continuously  from ~ \n80 GHz to ~ 55 GHz when the gate voltage is var ied from -13 V to + 13 V  (corresponding to \nfrom ‘hole doping ’ to ‘electron doping ’). Figure 4b shows t he entire magnetic -field dispersion  of \nthe transverse mode at  varying gate voltages  (the longitudinal mode  is not studied because of its \nsmall amplitude) . As in the  zero gating  case, the initial redshift of the mode  is followed by a \nblueshift with increasing  magnetic  field at all gate voltages . The turning point, which is \ndetermine d by the sat uration field HS, is tuned by about  1 T by gate voltage . Furthermore, while \nthe dispersion  of 𝜔𝑇 is nearly unchanged by gating  above HS, it is strongly modified below HS.  5 In this regime the resonance frequency decreases by as much as 40 % when the gate voltage is \nvaried from -13 V to +13 V .   \n \nThe observed magnetic -field dispersion  of 𝜔𝑇 at all gate voltages  can be described by the simple \nsolution of the LLG equations  discussed above  (inset of Fig. 4 b) with doping dependent \ninterlayer exchange HE and intralayer anisotropy  HA (Fig. 4 c). Both fields decrease linearly with \nincreasing gate voltage , with HA at a faster rate than HE. A similar doping dependence  for HE has \nbeen reported previously from the spin-flip transition  measurement  under an out -of-plane field  6. \nSuch doping dependences of the magnetic interactions  can be understood as a consequence of \ndoping dependent electron occupancy of the magnetic Cr3+ ions and their wavefunction overlap . \nBased on this picture, increasing electron density weakens the magnetic interactions, and in turn \nthe effective magnetic fields responsible for spin precession below HS. Above HS, the \nmagnetization is fully saturated in the in -plane direction and  the spin resonance frequency is \nalmost solely determined by the applied field H|| and is therefore doping i ndependent.  A \nquantitative description  of the experimental result , however, would require ab initio  calculations \nand is beyond the scope of th e current  study.  \n \nIn conclusion, we have demonstrated the generation and detection of spin waves in a prototype \n2D magnetic material  of bilayer CrI 3 with a time-resolved optical pump -probe method . The \nresults allow the characterization  of important parameters such as the internal magnetic \ninteraction s and damping. We have also demonstrated widely gate tuna ble magnetic resonances \nin this 2D magnetic system ，revealing  the potential  of using 2D AFMs  to achieve  local gate \ncontrol of spin dynamics  for reconfigurable ultrafast spin-based devices  30,31.  \n \n \nMethods   \nSample and device fabrication  \nThe measured sample is a stack of 2D materials composed of (from top to bottom) few -layer \ngraphite, hBN , monolayer WSe 2, bilayer CrI 3, hBN, and few -layer graphite . The top and bottom \ngraphite/hBN pairs serve as gates.  An additional stripe of graphite is attached to the WSe 2 flake  \nfor grounding  and charge injection . The thickness of hBN layers is ~ 30 nm, and the graphite \nlayers , about  2-6 nm. Bulk crystals of hBN were purchased  from HQ graphene. Bulk CrI 3 \ncrystals were syn thesized by chemical vapor transport following methods described in previous \nreports32,33. These crystals crystallized into the C2/m space group with typical lattice constants \nof a=6.904Å,  b=11.899Å,  c=7.008Å and  β=108.74°, and Curie temperatures of 61 K. All \nlayer  material s were first exfoliated from their bulk crystals onto SiO 2/Si substrates and \nidentified by the ir color contrast under an optical microscope . The heterostructure was built by \nthe layer -by-layer d ry transfer technique  34. It was then released on to a substrate with pre -\npatterned gold electrodes, which contact the bottom gate, top gate, and grounding graphite flake.  \nThe steps involving CrI 3 before its full encapsulation in hBN layers were  performed inside a \nnitrogen -filled glovebox  because CrI 3 is air sensitive. In the gating experiment, equal top and \nbottom gate voltages were applied to the heterostructure and the gate voltage  shown in Fig. 4  \nwas the v oltage on each gate.  \n \nTime -resolved magneto -optical Kerr effect (MOKE)  and magnetic circular dichroism \n(MCD)   6 In the time-resolved MOKE setup, the probe beam  is the output of a Ti:Sapphire  oscillator  \n(Coherent Chameleon  with a repetition rate of 78 MHz  and pulse duration of 200 fs) centered at \n1.54 eV , and the p ump beam is the second harmonic of an optical parametric oscillator ( OPO ) \n(Coherent Chameleon compact OPO) output centered at 1.73 eV . The time delay between the \npump and probe pulses was controlled by a motorized linear delay stage. Both the pump and \nprobe beam  were  linearly polarized. The pump intensity was modulated at 100 kHz by a \ncombination of a half-wave photoelastic modulator (PEM ) and a linear polarizer whose \ntransmission axis is perpendicular to the original pump polarization. The pump and probe beam \nimpinge d on the sample at normal i ncidence. The reflected light was  first filtered to remove the \npump , passed through a half -wave Fr esnel rhomb  and a Wollaston prism , and detected by a pair \nof balanced photodiodes . The pump -induced change in Kerr rotation was determined  as the ratio \nof the intensity imbalance of the photodiodes obtained from a lock -in amplifier locked at the \npump modulation frequency and the intensity of each photodiode.   \n \nFor the MCD measurements, a single beam centered at  1.8 eV was used. The light beam  was \nmodulated at 50 kHz between the left and right circular  polarization  using a PEM . The reflected \nlight was focused onto a  photodiode . The MCD was determined as the ratio of the ac component  \nof the photodiode signal  measured by a lock -in amplifier at the polarization modulation \nfrequency  and the dc component of the photodiode signal measured by a voltmeter.  \n  \nFor a ll measurements samples were mounted in an optical cryostat (attoDry2100) with a base \ntemperature of 1.7 K and a  superconducti ng solenoid magnet  up to 9 Tesla. For measurements \nunder an out -of-plane field, the sample was mounted horizontally  and light was focused onto the \nsample at normal incidence  by a microscope objective . For measurements under an  in-plane field, \nthe sample was mounted vertically and the light beam was  guided by a  mirror at  45° and focused \nonto the sample at normal incident with a lens.    \n \nLandau -Lifshitz -Gilbert (LLG) equations  \nWe model the field  dependent spin dynamics in antiferromagnetic bilayer CrI 3 using coupled \nLandau -Lifshitz -Gilbert (LLG) equations  29,  \n \n𝜕𝑴𝑖\n𝜕𝑡=−𝛾𝑴𝑖×𝑯𝑖𝑒𝑓𝑓+𝛼\n𝑀𝑆𝑴𝑖×𝜕𝑴𝑖\n𝜕𝑡.  (1) \n \nwhere i = 1, 2. In Eqn. 1  𝑴𝑖 is the magnetization  of the top or bottom layer  (which are assumed \nto have  an equal magnitude 𝑀𝑆), 𝛾/2𝜋 ≈ 28 GHz/T is the electron gyromagnetic ratio , 𝛼 is the \ndimensionless damping factor , and 𝑯𝑖𝑒𝑓𝑓 is the effective magnetic field in each layer that is \nresponsible for spin precession . In the absence of applied magnetic field, 𝑴1 and 𝑴2 are anti-\naligned along the easy axis  (z-axis) . When  an in -plane field  𝑯∥ (along the y -axis) is applied, 𝑴1 \nand 𝑴2 are tilted symmetrically towards the y -axis, before fully turned into the applied field \ndirection at  the saturation field  𝐻𝑆 = 2𝐻𝐸+𝐻𝐴. Here 𝐻𝐸 and 𝐻𝐴 are the interlayer exchange  and \nintralayer anisotropy field s, respectively . A schematic is shown in Fig . 3c. The effective field  \n𝑯1,2𝑒𝑓𝑓=𝑯∥−𝐻𝐸\n𝑀𝑆𝑴2,1+𝐻𝐴\n𝑀𝑆(𝑴1,2)𝑧𝒛̂ has contributions from the applied field, the interlayer \nexchange  field,  and the intralayer anisotropy field . We search for solution in the form of a \nharmonic wave  𝑒𝑖𝜔𝑡 with angular frequency  ω. For the simpl e case of zero damping (𝛼 = 0), two \neigen mode frequencies 𝜔𝑇 and 𝜔𝐿 are given in the main text.   7  \nIn case of finite  but weak  damping, we find the following transverse and longitudinal modes  \nafter simplifying the LLG equations : \n \nBefore saturation (𝐻||<𝐻S), \n \n𝜔𝑇2(1+𝛼2)−𝑖𝛼𝜔𝑇𝛾(𝜔𝑇02𝛾2⁄\n2𝐻𝐸+𝐻𝐴+2𝐻𝐸+𝐻𝐴)−𝜔𝑇02=0; \n𝜔𝐿2(1+𝛼2)−𝑖𝛼𝜔𝐿𝛾(𝜔𝐿02𝛾2⁄\n𝐻𝐴+𝐻𝐴)−𝜔𝐿02=0; \nAfter saturation (𝐻||>𝐻𝑆), \n \n𝜔𝑇2(1+𝛼2)−𝑖𝛼𝜔𝑇𝛾(2𝐻||−𝐻𝐴)−𝜔𝑇02=0; \n \n𝜔𝐿2(1+𝛼2)−𝑖𝛼𝜔𝐿𝛾(2𝐻||−4𝐻𝐸−𝐻𝐴)−𝜔𝐿02=0. \n \nHere 𝝎𝑻𝟎 and 𝝎𝑳𝟎 correspond to  the solution  at zero damping  (𝜶 = 0). In particular, when 𝜶 << \n1, the oscillation frequency ( the real part of the solution for  𝝎𝑻 and 𝝎𝑳) becomes  𝝎𝟎\n√𝟏+𝜶𝟐, where \n𝝎𝟎 is the undamped solution  for the two modes . Overall,  the eigenmode frequencies are reduce d \ndue to damping , and t he two mode s will no longer be degenerate  at 𝑯||=𝟎 taking into  account  \nof higher order corrections of  𝜶.  At low temperature, we found this correction insignificant for \nthe high-frequency branch , which has a larger oscillation  amplitude and was measured with a \nhigher precision . Fitting the experimental data with the damped LLG solution yield ed similar \nvalues for 𝑯𝑬 and 𝑯𝑨.  \n \nMechanism for ultrafast excitation of coherent magnons   \nWe have investigated t he mechanism for the o bserved ultrafast excitation of magnons in bilayer \nCrI 3. A plausible picture involves exciton generation  in WSe 2 by the optical pump , ultrafast \nexciton dissociation and charge transfer at the CrI 3-WSe 2 interface, and  an impulsive \nperturbation to the magnetic anisotropy and exchange  fields  in CrI 3 by the injected hot carriers . \nSeveral  control experiments  were performed to test this picture . Pump-probe measurement s were \nperformed on both monolayer WSe 2 and bilayer CrI 3 areas alone  (non-overlapped  regions in  the \nheterostructure) under the same experimental conditions . Negligible pump -induced MOKE \nsignal  was observed . In addition , measurement was done on the heterostructure at different pump \nenergies . The magnetic resonance frequencie s were found unchanged, but the amplitudes follow \nthe absorption spectrum of WSe 2 (Supplementary Fig. S 1). These two experiments show that \nmagnons are generated through  optical excitation of  excitons in WSe 2. It has been reported \nearlier that CrI 3-WSe 2 heterostructure s have a type -II band alignment , which can facilitate \nultrafast exciton dissociation and charge transfer  26. Next t he onset of coherent oscillations  is \ninstantaneous with optical  excitation  in our experiment . This exclud es lattice heating in CrI 3 as a \ndominant mechanism  for the generation of magnons, which typically takes a longer time  to build \nup. Moreover , the resonance amplitude is independent of the pump laser polarization \n(Supplementary Fig. S 2), indicating that hot carriers, rather than the angular momentum  of the \ncarriers , are responsib le for the excitation of magnons. Finally, as we show in the main text , the  8 magnetic anisotropy and exchange can be effectively altered by carrier doping in CrI 3. These \nexperiments are all consistent with the proposed mechanism of ultrafast excitations  of magnons \nin CrI 3-WSe 2 heterostructures.  \n \nTemperature dependence of magnon modes  \nWe have performed the optical pump/MOKE probe experiment in CrI 3-WSe 2 heterostructures at \ntemperature ranging from 1.7 K to 50 K. No obvious oscillations can be measured above 50  K \nwhen bilayer CrI 3 is close to its N éel temperature.  The results at 1.7 K are presented in the main \ntext. Supplementary Fig. S3 and S 4 show the c orresponding measurements and analysis for 25  K \nand 45  K, respectively. With increasing temperature , the magnon frequency decreases and the \nsaturation field (estimated from the minimum of the frequency dispersion) also decreases . A \nsystematic temperature dependence is shown in Supplementary Fig. S 5 for the high -frequency \nmode 𝜔𝑇 at a fixed in -plane field of  2 T. The frequency has a negligible temperature dependence \nwell below the Néel temperature ( < 20 K), and decreases rapidly w hen the temperature \napproaches the N éel temperature . \n \nAdditiona l measurements on few -layer CrI 3   \nWe have measured the magnetic response from a few-layer CrI 3 (6-8 layer ) sample . Because of \nthe larger MOKE signal and higher optical absorption in thicker samples , magnetic oscillations \ncan be measured without the enhancement from monolayer WSe 2. The results are shown in \nSupplementary Fig. S 6. The comparison of results from samples of different thickness es \nprovides insight into the origin of magnetic damping. 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Bilayer CrI 3 is outlined with a purple line , and monolayer WSe 2, a black line. \nScale bar is 5 𝜇m. b, Schematic of a type-II band alignment between monolayer WSe 2 and CrI 3. \nOptically excited exciton in WSe 2 is dissocated at the interface and electron is transferred to CrI 3 \n26. c, MCD of the heterostrucutre as a function of out -of-plane magnetic field  at 4 K . Hysteresis \nis observed for field swe eping along two opposing  directions. Insets are schematics of the \ncorresponding  magnetizations in the top and bottom layers of  blayer CrI 3. The dashed line s \nindicate the spin -flip transition around  0.75 T. d, Schematic of bilayer CrI 3 under an in -plane \nmagnetic field H||. Below the saturation field, the magnetizations of the top and bottom layer are \nsymmetrically canted towards the applied field direction.  \n \n \n 12  \nFigure  2 | Time -resolved magn on oscillations. a, Pump-induced Kerr rotation as a function of \npump -probe delay time in bilayer CrI 3 under different in -plane magnetic field s. The curves are \ndisplaced vertically for clarity.  b, FFT amplitude spectra of the time dependences shown in a \nafter the demagnetization dynamics  (exponential decay ) were remov ed. The spectra are vertically \ndisplaced for clarity.  \n \n \n \n \n \n \n \n \n \n \n \n 13  \nFigure 3 |  Magnon dispersion and damping. a, b, Pump -induced MOKE dynamics in bilayer \nCrI 3 under two representative in -plane fields of 1.5  T (a) and 3.75 T  (b). Grey lines  are \nexperiment after subtracting the demagnetization dynamics , and red lines , fits to two damped  \nharmonic oscillations .  c, Illustration of two spin wave eigen modes in an  AFM : the transverse \nmode ( left) and the longitudinal mode  (right) . The dashed line s indicate the equilibrium top and \nbottom layer magnetization M 1 and M 2, which are  titled symmetrically from the z-direction \ntowards  the applied field direction ( y-axis) . The ma gnetization s precess follow ing the green and \nblue arrows  in the order  1 through 4.  d, e, Oscillation frequencies  (d) and da mping rates  (e) of \nthe transverse and longitudinal modes extracted from the two harmonic oscillation fit as a \nfunction of in-plane magnetic field. The error bars  are the fit uncertainties . The vertical d otted \nlines indicate the in -plane saturation magnetic field. Dashed lines in d are fits to the LLG \nequations as described in the text.  \n \n 14  \nFigure 4 | Gate  tunab le magnon  frequency . a, FFT amplitude spectra of the magnon s as a \nfunction of gate voltage  under a fixed in -plane field of  2 T. The dashed line is a guide to the eye \nof the evolution of the resonance frequency with gate voltage  and triangle s indicat e the peak of \nthe resonance . b, Magnetic -field dispersion of the transverse mode  at different gate voltage s. The \ninset shows the fits of the experimental data to the LLG equations. The same colored line s (LLG \nequation) and symbols (experiment) deno te the same gate voltage. c, Anisotropy  field HA and \nexchange field  HE extracted from the fits in b at different gate voltages . Error bars are the \nstandard deviation from the fitting. Dashed lines are linear fits.  \n \n \n \n \n \n \n \n  \n 15 Supplementary figures  \nFigure S1 | Amplitude  of the s pin wave s under a  fixed in-plane magnetic field of 2 T as a \nfunction of pump wavelength. The dependence  resembles that of the excitonic resonance in \nmonolayer WSe 2. The spectral broadening arises  from the additional WSe 2 trion absorption and \nthe linewidth of the light pulses  (~ 5 nm in full width at half maximum  (FWHM )) employed in \nthe pump -probe measurement . \n \n \nFigure S2 | Spin wave dynamics under H|| = 2 T excited by optical pump  of different  \npolarization s. The red, orange and blue lines correspond to left circularly polarized, linear \npolarized , and right circularly polarized pump , respectively . The curves were  vertically shifted \nfor easy comparison. The oscillation  amplitude does not depend on the pump polarization (i.e. \nphoton angular momentum ).  \n \n 16  \n \nFigure S3 | Magnon oscillations at 25  K. a, Spin dynamics in bilayer CrI 3 under  different \nmagnetic field s. The curves were  vertically displaced for clarity . b, FFT amplitude spectr a of a. \nc, In-plane field dispersion of the two magnon modes  extracted from fitting the time -resolved \nMOKE signal with two harmonic oscillations . \n \n \n \n \n \n \n \n \n \n  \n 17  \nFigure S4 | Magnon oscillations at 45 K. Same as in Supplementary Fig. S3. Due to the weak \nsignal, we can only identify the transverse mode 𝜔𝑇.  \n  \n 18  \nFigure S5 | Temperature dependence of the transverse magnon mode frequency under  a fixed in -\nplane magnetic field of 2  T. All other experimental conditions are the same  as in Fig. S3 and S4 .  \n \n \nFigure  S6 | Pump -probe m easurements on few -layer CrI 3 at 1.7  K. a, Spin wave dynamics \nunder  different in -plane magnetic field s. b, The corresponding FFT amplitude spectrum of a. The \ndamping at 6T is estimated to be ~0.04, which is similar to bilayer CrI 3.  \n \n 19  \nFigure S 7 | Amplitude of the longitudinal and transverse magnon modes extracted from  the \ntime-resolved MOKE measurement ( Fig. 2 of the main text ).  \n \n  \n"    },    {        "title": "2001.06900v3.Disentangling_magnetic_and_grain_contrast_in_polycrystalline_FeGe_thin_films_using_four_dimensional_Lorentz_scanning_transmission_electron_microscopy.pdf",        "content": "1 \n Disentangling magnetic and grain contrast  in polycrystalline FeGe thin films using four-dimensional  \nLorentz scanning transmission electron microscopy  \n \nKayla X. Nguyen1,2*, Xiyue  S. Zhang2*, Emrah Turgut2,3, Michael C. Cao2, Jack Glaser2, Zhen Chen2, \nMatthew J. Stolt4, Celesta S. Chang5, Yu-Tsun Shao2, Song Jin4, Gregory D. Fuchs2,6, David A. Muller2,6 \n \n1. Department of Chemistry and Chemical Biology, Cornell University, Ithaca, NY 14853, USA  \n2. School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA  \n3. Department of Physics, Oklahoma State University, Stillwater, OK 74078, USA  \n4. Department of Chemistry, University of Wisconsin –Madison, Madison, Wisconsin 53706, USA  \n5. Department of Physics, Cornell University, Ithaca, NY 14853, USA  \n6. Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY 14853, USA  \n \n*These Authors Contributed Equally  \n \nAbstract  \n \nThe study of nanoscale chiral magnetic order in polycrystalline materials with a strong Dzyaloshinkii -\nMoriya interaction (DMI) is interesting for the observation of magnetic phenomena at grain boundaries and \ninterfaces. This is especially true for polycrys talline materials, which can be grown using scalable \ntechniques and whose scalability is promising for future device applications. One such material is sputter -\ndeposited B20 FeGe on Si, which has been actively investigated as the basis for low -power, high -density \nmagnetic memory technology in a scalable material platform.  Although conventional Lorentz electron \nmicroscopy provides the requisite spatial resolution to probe chiral magnetic textures in single -crystal FeGe, \nprobing the magnetism of sputtered B2 0 FeGe is more challenging because the sub -micron crystal grains \nadd confounding contrast. This is a more general problem for polycrystalline magnetic devices where \nscattering from grain boundaries tends to hide comparably weaker signals from magnetism.  W e address \nthe challenge of disentangling magnetic and grain contrast by applying 4 -dimensional Lorentz scanning \ntransmission electron microscopy using an electron microscope pixel array detector. Supported by \nanalytical and numerical models, we find that t he most important parameter for imaging magnetic materials \nwith polycrystalline grains is the ability for the detector to sustain large electron doses, where having a \nhigh-dynamic range detector becomes extremely important.  Despite the small grain size in  sputtered B20 \nFeGe on Si, using this approach we are still able to observe helicity switching of skyrmions and magnetic \nhelices across two adjacent grains as they thread through neighboring grains. We reproduce this effect using \nmicromagnetic simulati ons by assuming that the grains have distinct orientation and magnetic chirality and \nfind that magnetic helicity couples to crystal chirality.  Our methodology for imaging magnetic textures is \napplicable to other thin -film magnets used for spintronics and memory applications, where an \nunderstanding of how magnetic order is accommodated in polycrystalline materials is important.  \n \nI. Introduction  \n \nPower -efficient memory devices based on magnetic skymions  in sputtered, polycrystalline thin films \nare increasingly promising  [1-5]. Although the crystalline perfection afforded by the bulk synthesis of chiral \nmagnetic materials has enabled scientific understanding  [6-8], applications will require materials that a re \ngrown using scalable techniques such as sputtering.  The chiral magnetism themself  arises because of the \nDzyaloshinskii -Moriya interaction (DMI)  [9, 10]  that is present at specimen interfaces an d in the volume \nof noncentrosymmetric materials with broken inversion symmetry.  Here we focus our investigation on thin \nfilms of cubic B20 FeGe sputtered on Si. This material lacks crystalline inversion symmetry, and has \nenantiomers with left -handed and r ight-handed crystal chiralities, which are observed by Lorentz \nmicroscopy to couple to the magnetic helicity  in Mn 1-xFexGe [11]. However, in thin -film form, the magnetic \nstate is strongly modified by substrate -induced strain [12, 13]  and small grain size  relative to bulk crystals. 2 \n Our work seeks to answer relevant questions including: how does sub-micron grain size and the presence \nof many grain boundaries alter the nanoscale chiral magnetism in this material? Imaging the noncollinear \nspin textures in these polycrystalline films is a direct approach to understanding questions relevant to \napplications.  \n \nElectron microscopy has been widely used to investigate the real space magnetization profiles of chiral and \ntopological spin textures, e.g. helices and magnetic skyrmions  [11, 14, 15] .  Conventional Lorentz \ntransmission electron microscopy (LTEM) provides nanometer spatial resolution; however, the electron \nbeam must be defocused to obtain magnetic contrast causing strong Fresnel fringes at the grain boundaries \nand obscuring magnetic infor mation [6, 15 -18]. Another well -known method is off -axis electron \nholography, which utilize s a biprism to split the post -specimen diffracted electron beam into reference and \nimage waves, such that at the image plane, the two waves overlap forming the electron hologram  [19].  In \nthis method, the phase shift o f the electron beam can be measured and related back to the local \nelectromagnetic field in thin samples.  However, electron holography has a limited field of view, often \nunder 5 𝜇m, and the reference beam must pass through a hole in the sample. The field of view and resolution \nare coupled by the number of pixels on the detector, whereas for scanning transmission electron microscopy \n(STEM) methods the number of pixels recorded is adjustable and set by the dynamic range and noise on \nthe scan coils – usually a few picometers in modern instruments. Holography is also sensitive to artifacts \nfrom dynamical diffraction, nonlinearities with sample thickness and small changes in crystal orientation  \n[19-23].  In addition, all TEM methods require a thin (< 200 nm) sample, but this can be prepared from a \nbulk device in a few hours using a focused ion beam. For non -destructive imaging, magnetic force \nmicroscopy (MFM), a form of atomic force mi croscopy, utilizes a magnetic tip to scan across a magnetic \nsample, where the magnetic force between the tip and sample can be used to image magnetic structures \nsuch as skyrmions  [24].  For MFM, resolution is limited to ~10 nanometers  [25] compared to recent \nadvances in electron microscopy in field -free mode, demonstrating sub -Angstrom spatial resolution [26]. \n \nLorentz STEM (LSTEM) presents an alternative  [27, 28]  in the form of differential phase contrast \n(DPC) imaging, which uses a focused electron beam  thus avoiding information delocalization,  and has been \nshown to be more sensitive at measuring local magnetic fields than electron holography [29].  DPC imaging \nmeasures the deflection of a focused electron beam due to the electromagnetic field within the sample , \nwhich enables the study of the internal structure of magnetic skyrmions  [30, 31] . However, there are \nmultiple constraints to DPC imaging including: (1) limited dynamic range of the detector limits the \nsensitivity to detect the extremely small magnetic fields that  are expected in modern devices, (2) \nnonlinearities introduced by the signal normalization when the detector is not perfectly centered, and (3) \nchanging beam shape due to electron channeling, which gives rise to contrast reversals as the electron beam \nmove s through the sample  [32-34].  As a result, experiments using both LSTEM and DPC have mostly \nfocused on single -crystal samples that exhibit sparse grains and few defects, thus presenting a structurally \nuniform medium for imaging.  \n \nHere, we  use a version of LSTEM known as 4 -dimensional Lorentz scanning transmission electron \nmicroscopy (4 -D LSTEM) with an electron microscope pixel array detector (EMPAD) [35]. This technique  \ncan sense electromagnetic beam deflections by acquiring electron diffraction patterns at every x -y scan \nposition with a high k x-ky momentum resolution.  From the electron diffraction pattern, a center of mass \n(COM) signal can be extracted and used to quan titatively reconstruct the sample’s magnetic field [34]. The \nkey advantage of the EMPAD is their high dynamic range, which enables sensitive mapping of magnetic \nfields, (𝜇𝑇\n√𝐻𝑧), and captures simultaneously the crystalline and magnetic information without saturating the \ndetector.  From this, we distinguish magnetic field information from crystalline grain structure effects and \nprobe each effect independently.  In the following sections, we (1) highlight the fundamental ideas behind \ndetection of magnetic fields with high sensitivity, (2) show that the precision relies on the number of \nelectrons a detector ca n sustain, stressing the importance of a high -dynamic -range and high -speed detector, 3 \n and (3) demonstrate our technical approach to disentangling magnetic signal from the signal due to grain \ncontrast. Using this approach, we observe changes in the magnetic textures from the different DMI between \nadjacent grains.  \n \nII. Deflection of the Electron Beam due to Magnetic Fields  \n \nTo detect a magnetic field in Lorentz Microscopy, we consider a parallel electron beam source that is \nincident perpendicular to the plane of the specimen, along the z -axis.  Here, we ignore stray field effects, \nwhich gives rise to an unmeasurable z -component deflection and consider only the deflections due to the \nprojected in-plane magnetic field  [36, 37] . The equation for the electron beam deflection is then given as:  \n \n 𝛽(𝑥)=𝑒𝜆𝐵0𝑡\nℎ, (1) \n \nwhere B 0 is the local magnetic field, e is the electron charge, 𝜆 is the wavelength of the electron beam, t is \nthe sample thickness and h is Planck’s constant.  This deflection of the electron beam,  𝛽, can be measured \nby tracking the COM of the diffraction pattern as we scan the electron beam, which is related to the l ocal \nmagnetic field.  \n \nThe argument above presents the classical treatment of the electron beam deflection in a magnetic \nfield. On the length scale of electron microscopy, electron -specimen interactions and the beam shape should \nalso be treated quantum mec hanically.  For elastic scattering, the deflection of the electron beam by the \nlocal magnetic field is the change in momentum of the electron wave function, <𝑝⃗> [34], which is the \nCOM signal in our diffraction pattern.  Using Ehrenfest’s theorem  [38], the expectation value of the Lorentz \nforce in terms of  <𝑝⃗> is [34], \n \n𝑑⟨𝑝⃗⟩\n𝑑𝑡= ⟨[𝐻̂,𝑝̂]⟩=⟨𝑒(𝐸⃗⃗+𝑣⃗×𝐵⃗⃗ )⟩    (2) \n \nwhere 𝐻̂ is the Hamiltonian, e is the charge of the electron, 𝐸⃗⃗ is the electric field, 𝑣⃗ is the velocity at which \nthe electron is traveling and 𝐵⃗⃗ the magnetic field. Again, for elastic scattering the velocity remains constant \nso the rate of change of momentum with time can be mapped to the rate of change of momentum  with \nthickness into the sample. This <𝑝⃗> can be extracted from the intensity of the scattered wave function,  𝛹, \nas: \n \n 〈𝑝⃗⃗〉= ∫𝛹∗(𝑝⃗⃗)𝑝̂𝛹(𝑝⃗⃗)𝑑𝑝⃗⃗=∫𝑝̂|𝛹(𝑝⃗⃗)|2𝑑𝑝⃗⃗= ∫𝑝̂𝐼(𝑝⃗⃗)𝑑𝑝⃗⃗   (3) \n \nwhere I is the intensity of the diffraction pattern and 𝑝̂ is the momentum oper ator.  From equations 2 and 3, \na shift in <𝑝⃗>is perpendicular to the magnetic field and parallel to the electric field. Rewriting <𝑝⃗>in \nreal space, we see that it is also the probability current flow, ⟨𝑗⃗⟩, where the expectation value for ⟨𝑗⃗⟩ is \ncalculated for an evolving electron wave function, changing as it propagates through the specimen:  \n \n⟨𝑝⃗⃗⟩ =ℏ\n2𝑖∫[𝛹∗(𝑟⃗, 𝑟⃗𝑝)𝛻⃗⃗𝛹(𝑟⃗, 𝑟⃗𝑝)−𝛹(𝑟⃗, 𝑟⃗𝑝)𝛻⃗⃗𝛹∗(𝑟⃗, 𝑟⃗𝑝)]𝑑𝑟⃗=2𝑚⟨𝑗⃗⟩  (4). \n \nHere, r is the real space position at an arbitrary point. r p is the posit ion of the probe in real space, and m is \nthe mass of the electron. Although the equations above give us a direct relation for <𝑝⃗> to the \nelectromagnetic field, every signal from <𝑝⃗> results in a signal with contributions from the nuclear, core, \nvalence a nd external electric and magnetic fields. Disentangling these contributions based on their different \nspatial distributions is discussed in section V. Equations (2) -(4) describe the fundamental physics used \nwhen determining electromagnetic fields from a def lection of the electron beam. To experimentally 4 \n quantify the parameters that affect our signal sensitivity, we develop a framework to model and predict the \nsensitivity based on the parameters we can optimize in the electron microscope.  \n \nIII. Determining Magnetic Field Detection Sensitivity  \n \nThe magnetic field sensitivity that we can obtain using COM imaging depends on how well we can track \nthe deflection of the electron beam. To quantify this, we first present an analytical approach to determine \nthe sensi tivity of the magnetic field based on a model previously described by Chapman et al.  for a split \ndetector [37] (Appendix A). Here, Fig. 1 shows a schematic of a DPC detector from which the shift of \nscattered electron beam due to a magnetic field in the sample is detected [37].  \n \nFIG. 1. Schematic of signal deflection for a DPC detector used in LSTEM, where we show change in the  \nangular deflection of the electron beam due to the change in the Lorentz force as it is scanned across the \nsample from the left to right magnetic domain.  \n \nHere, the  deflection  sensitivity of DPC imaging has two dependencies: (1) 𝛼, the semi -convergence angle \nof the electron beam, which provides the limit in spatial resolution, and (2) n, the number of electrons or \ndose of the electron beam, which tells you how many electrons are needed to sustain a signal that is \nadequately above P oisson noise. These two variables are parameters that can be easily adjusted in the \nelectron microscope.  Using the analytical model in Appendix A, we observe the minimum deflection 𝛽𝐿 \nas: \n \n𝛽𝐿=𝜋\n4𝛼\n√𝑛                           (5). \n \nIn this analytical model, we only show how deflection sensitivity, hence field sensitivity, is dependent on \na DPC detector or a  detector with two pixels  in one dimension .  To calculate the field sensitivity of a \ndetector with more than two pixels, we move from an analytical to a numerical model to investigate how \ndeflection sensitivity is affected by n, the number of electrons that we use in our electron beam, and 𝑗, the \nnumber of pixels in our detector described in Appendix B. Doing so, we answer the question, do more \npixels in a detector lead to an improved detection sensitivity to fields? We find that as a general trend, the \npercentage error in locating the center of m ass of the beam, (𝛽𝐿𝛼⁄), scales as Poisson noise, 1√𝑛⁄ and the \n5 \n numerical pre -factor depends on the number of pixels.  \nIn our numerical model, we assume a one -dimensional detector that is entirely illuminated by an electron \nbeam represented as a top hat distribution with varying Poisson noise.  We used Poisson noise to simulate \nthe actual shape of the electron beam so that their shape is not a perfect uniform top hat probe. We chose \nPoisson noise because the EMPAD detector has a detector noise level of 0.007 primary e -/pixel  [35], while \na typical number of recorded primary electrons/pixel is ~100,000, giving a Poisson noise of √100 ,000 =\n316 e-.  In other words, the detector noise is about 50,000x smaller than the Poisson noise and can be safely \nneglected. Therefore, the noise of our data is dominated by the intrinsic Poisson noise determined from the \nnumber electrons of the incident beam. Given this, we can model the distortion of the shape as only being \nfrom Poisson noise. We convert this value to a deflection and a magnetic field where the statistical model \nis used to simulate the sensitivity of the magnetic field measurement.  For our numerical model, the detector \nnoise is assumed to be negligible compared to the signal detected.  \nWe vary the nu mber of pixels in our detector, the number of electrons used in our electron beam, and hence \nthe Poisson noise. We then calculate the COM of the electron beam on the detector over 2000 electron \nbeam configurations at each dose setting with random varying Poisson noise to find the standard deviation \nof the COM measurements from the statistics of 2000 samples . The standard deviation that we obtain is the \nestimated random  error from Poisson noise in our COM measurement, related to how prec isely we can \nextract the  signal from  COM. We define a calibration factor  to be 2𝛼\n(𝑗−1) in Ap pendix B , with 𝛼 being the \nsemi -convergence angle and 𝑗 being the number of detector pixels , so that by multiplying the standard \ndeviation in the unit of detector pixel with the calibration  factor in the unit of rad/pixel we can convert the \nCOM standard deviation to the minim um deflection angle of our electron beam, 𝛽. In our numerical model, \nwe choose 𝛼 to be 200 µrad because it corresponds to a spatial resolution of 7.6 nanometers for a 200 kV \nelectron beam, typical of the resolution needed to image magnetic skyrmions in single crystal FeGe.  \nBecause we assume a square electron beam shape for simplificat ion in the simulation, the associated \nminimum  detectable deflection angle  for a split detector  is  𝛽𝐿=𝛼\n √𝑛 (Appendix  B). This is a different \nnumerical prefactor from the round detector case of equation 5 . Our s imulation result s in the case of  a two-\npixel detector agree with the minimum deflection angle s calculated from the analytical model with a split \ndetector  in the version of square probe shape . From Fig. 2(a), we find that our deflection, 𝛽, is strongly \ndependent on the total number of electrons, and only weakly on the number of pixels in our detector, with \ndiminishing returns once more than 10 pixels per disk are added.   \n \nTo convert 𝛽 to magnetic field, we use a beam energy of 200 keV and a sample thickness of 100 nm [Fig. \n2(b)].  Here, our results show that if we want to observe magnetic fields with strength less than 1 milliTesla \n(mT), the number of electrons must be greater than 1 million. From Fig. 2(a) and 2( b), we find that a detector \nwith high dynamic range is more important for signal sensitivity than a detector with more pixels.  \nTraditional arguments for detectors with many pixels emphasize the importance of angular resolution, \nwhere more pixels in the de tector would give higher sensitivity in deflection detection.  Here, we show \nthrough our models that this is not necessarily the case for COM measurements.  We observe that even with \nlarger number of pixels, deflection sensitivity is still limited if a det ector cannot collect or sustain a high \nnumber of electrons integrated over all channels. While there is diminishing return in improving the SNR \nwith an increased number of pixels, for detectors with poor dynamic range, there sometimes might be a \nbenefit to  spreading the total dose over more pixels to avoid saturation, provided the dose/pixel is much \nlarger than the readout noise per pixel.  In summary, we find that we do not need many pixels on our detector \nto detect small fields. Instead, the most importan t thing is the ability of a detector to handle large total doses; \nhere, dose/pixel becomes important. For detecting very small deflections or fields, dynamic range is the \nlimiting factor.   \n 6 \n  \nFIG. 2. Numerical simulations using a 200 µrad  semi -convergence  angle a t a beam en ergy of 200 keV, and \nsample thickness of 100 nm. We show (a) the minimum measurable deflection  angle , (b) the smallest \nmeasurable magnetic field, as a function of electron dose and number of detector pixels , and (c) the \ndependence of mag netic field sensitivity (𝜇𝑇\n√𝐻𝑧) on beam current ( nA) and number of  detector  pixels . The \nwhite  lines represent the contours for deflection error (a), magnetic field (b) and B field sensitivity (c) as \n0.2, 0.5 (dotted lines) and 1 (solid lines) times powers of 10. From our plots, we observe that field resolution \nis more dependent on the number of el ectrons than the number of pixels on a detector, with diminishing \nreturns beyond 10 pixels.      \n \nTo measure these small deflections with high precision experimentally, we need cameras with higher \ndynamic ranges and faster read -out speeds than what is poss ible with scintillator -based technologies. The \nEMPAD presents one such a solution  [35]. Compared to traditional diffraction detectors such as CCDs, the \nEMPAD has a relatively fast acquisition at 0.86 ms  per frame and a high dynamic range of 30 bits; this \nmeans that the EMPAD can detect one to one million electrons without detector saturation at high signal to \nnoise.  Using the EMPAD, DC magnetometry using a COM approach can achieve a comparable sensitivi ty, \nalbeit at higher spatial resolution,  to other  high-sensitivity DC magnetic imaging techniques, such as \nscanned diamond nitrogen -vacancy centers that achieve sensitivities in the few 𝜇𝑇\n√𝐻𝑧 range using optically -\ndetected magnetic resonances [39-41]. AC modulation can improve the sensitivity in all cases.  Field \nsensitivity as a function of  beam current  is shown in Fig.2c. For LSTEM, the beam deflection comes from  \nsample in-plane magnetic field summed over projection, and the field sensitivity scales  inversely  with \nsquare root of electron beam current  and inversely  proportional with sample thickness  for thin samples  \n(Appendix A ). Sampl e thickness es greater than an electron mean free path  will suffer from electron  multiple \nscattering  out of the central beam  which  would  exponentially  atten uate the deflection signal , and the \ntradeoff between the two effects implies an optimal range of sample thickness . Peak field sensitivity, ~ 10 \nμT\n√Hz, can be obtained at a film thickness of around 100 nm, and dose rate limit of 10nA , which is a high, but \nstill-usable beam current for a cold field emission source . Here, using simulations, we have analyzed how \nto optimize the detection of extremely small magnetic field variations in the electron microscope, \nhighlighting the importance of a high -dynamic range diffraction detector such as the EMPAD.  \n \n \nIV. Experimental R esults  \nIn this section, we apply 4 -D LSTEM to study bulk single -crystal FeGe, a well -characterized material \nsystem  [6, 15, 42 -44], before proceeding to our  investigation of polycrystalline materials in section V. The \nquantitative comparison to a known material allows us to test the accuracy and precision of the LSTEM \nmeasurements. The sample preparation of single crystal FeGe was discussed previously in  [45-47] .  \nSamples are prepared using a focused ion beam and subsequently imaged at 300 keV with a 230 micro \nradian semi -convergence angle on a FEI -Titan. The beam current for imaging is 198pA and dwell time is \n1ms. The sample was cooled to  240 K using a Gatan cryo -holder and imaged under at 130 mT field \nperpendicular to the sample plane.  The objective lens was used to induce the 130 mT field to the sample. \n7 \n We calibrate the strength of the objective lens using Hall probe measurements , with a Hall probe mounted \nin a Protochips Aduro holder with electrical feedthroughs .  As the strength of the lens changes, we can \nrelate their strength to a value for the induced magnetic field on our Hall probe.   \nWe perform 4 -D LSTEM on single crystal B20 FeG e, a sample with a uniform background.  During \nscanning we record the full convergent beam electron diffraction (CBED) pattern at high -dynamic range \nand high speed [Fig. 3(a)].  The CBED pattern contains all of the magnetic and crystallographic information , \nincluding the Lorentz force, which is extracted from the beam COM. The sample’s magnetic induction field \nfor the x and y direction are shown in Fig. 3(b) and (c), respectively.  Using Fig. 3(b) and (c), we reconstruct \nthe magnetic field magnitude, which shows the skyrmion lattice [Fig. 3(d)]. Our in -focus 4 -D LSTEM \nimage of the skyrmion lattice agrees quantitatively with prior measurements using LTEM [15]and DPC  [30]. \nTo account for the thickness of the sample in our magnetic field measurements, we use electron energy loss \nspectrum (EELS) to map the inelastic mean free path, and  from this  we estimate the thickness to be about \n100nm. We confirmed the se estimat es locally from the fractional intensity remaining in the central beam \nof each EMPAD pattern using the Beer -Lambert law with an analytic estimate of  the elastic mean free \npath[48], and the two measurements  agree to within 10%. However, when the sample becomes too thick  \n(larger than a mean -free p ath, or strongly diffracting) , the intensity of the incident beam is reduced and \nmultiple scattering and dynamical diffraction dominates the signal. As a practical matter, the phase \napproximation breaks down  at about a mean free path (~120 -150 nm)  and it b ecomes difficult to distinguish \nseparate the  Lorentz deflection from  diffraction artifacts . \nTo estimate the noise in our measurements, we take a line profile of the magnetic field in Fig. 3(c) and \nperform a fit to our line.  Here, we obtain the root mean s quare (RMS) error by measuring the deviations in \nthe line profile of the magnetic field compared to our fit [Fig. 3(e)].  We find a root mean squared (RMS) \nerror of 3.6 mT and observe that we indeed have mT sensitivity in field.  \n \n \nFIG. 3. 4D Lorentz -STEM of single crystal B20 FeGe imaged at 240K with 130 mT applied magnetic field \non a FEI -Titan. (a) Schematic of magnetic imaging with EMPAD in Lorentz -STEM mode. We capture the \nfull scattered distribution in momentum space with the EM PAD at every scan position, from which we can \nalso reconstruct the magnetic field components in the (b) x and (c) y directions (in units of Tesla).  From \n8 \n (b) and (c), we obtain (d) the magnitude of the skyrmion field. We record a line profile from the magn etic \nfield deflection in y (c), and we compare it to their fit (e).  Here, we observed a root mean square (RMS) \nnoise of 3.6 mT.  White scale bar in (d) also applies to panels (b) and (c).   \n  \nAside from magnetic fields, the 4 -D LSTEM with the EMPAD also s imultaneously provides the \ninformation needed to generate images that are usually obtained from conventional techniques such as \nannular dark field, bright field, and specialized techniques such as quantitative measurements of thickness, \nstrain, tilt, polar ity, atomic fields, and the long -range electric fields. Although we can detect sensitive \nmagnetic fields, crystallographic scattering effects could dominate signals from real specimens, leading to \nartifacts or misinterpretation of magnetic signals.  For sp intronic devices, magnetic specimens are typically \nsputtered as thin films where grain sizes range from a couple of nanometers to microns.  One such material \nis sputter -deposited B20 FeGe on Si [3, 12, 39, 40, 49] . Probing the magnetism of sputtered B20 FeGe is \nextremely challenging because the sub -micron crystal grains exist on the same length scale as the \ncharacteristic length scale of the magnetic texture.  Therefore, disentangling magnetic effects from gr ain \ncontrast is extremely important. We present an approach to disentangle the signal of magnetic contrast from \ngrain contrast, allowing each to be probed independently.  This is enabled by the high dynamic range of the \nEMPAD, where multiple imaging modes can be used to extract and separate the two signals.  \n \n \nV.  Disentangling grain and magnetic contrast  \n \nAs a starting point, we must first consider how electron scattering is affected when both magnetic and grain \ncontrast are present. Previously, Chapman et al. [50] showed that by using an annular quadrant detector \ninstead of a solid quadrant detector, the unwanted grain contrast could be reduced,  and a smoother magnetic \ncontra st could be obtained by narrowing the collection angles of the detector. Kohl et al. [51] discussed \nhow using only a narrow range of annular cut -off angles relative to the direct electron beam acts as a low -\npass filter for the image. Kra jnak et al. [52] showed that magnetic contrast can also be enhanced by tracking \nonly the edge of the bright field disk, which also serves to suppress high -spatial -frequency, non -magnetic \nfeatures in very fine grain materials. This can be understood  in terms of Kohl and Majerts’s [51] analysis \nby recognizing that magnetic field signals are enhanced due to their slowly -varying  long-range potentials.  \nDiffraction contrast from grains and grain boundaries, on the other hand, are encod ed as short -range \npotentials, where this signal can be smaller than the probe size for a large electron beam [53]. Cao et al. \n[53] explored the asymptotic limits for both long -range potentials where the potential v aries slowly across \nthe probe shape, and short -range potentials where the potential is much smaller than the probe size, showing \nhow their signals are distributed differently in momentum space.  For magnetic imaging, where the probe \nis a few nanometers in size, magnetic fields are long -range with respect to the probe, and changes in grain \nstructure and contrast are short range. A short -range potential, such as the atomic arrangement at a grain \nboundary, changes intensity distribution inside the bright field  disk and do not deflect the disk boundaries. \nA long -range potential, such as a magnetic field in a ferromagnet, uniformly shifts the entire bright field \ndisk similar to the classical analysis of Ref. [36]. These different angular distributions are difficult to track \nwith only a quadrant detector.  A key aspect of this angular -separation technique is having a sufficient \nnumber of pixels on the detector to separate signals from long and shor t-range potentials, while recording \nsufficient signal from diffracted beams to track grain orientation simultaneously.   \nRecent work from  Wang et al. [54] has taken a different approach to en hancing  magnetic  contrast \nin polycrystalline FeGe by noting that equation 5 shows angular precision scales linearly with the \nconvergence angle. Thus higher sensitivity can be reached by choosing the smallest aperture (and hence the \nlargest probe) that can just resolve the magnetic features (~20 nm). However, at this limit, the magnetic \nfeatures are no longer much larger than the probe size, and instead of appearing as shifts of the diffraction \ndisk, the magnetic contrast is now present as features inside the disk thems elf.  Consequently, disk-edge \ntracking is no longer effective for separating magnetic and grain contrast  under th ose conditions .   Instead \nbandpass filtering becomes necessary.   9 \n  \nTo investigate long and short -range potentials and their effects on magnetism, we study B20 FeGe  (176 nm \nthickness) films sputtered on Si to test how the intensity distribution signal changes in the diffraction pattern \nusing the full 128x128 pixels on the EMPAD and their high dynamic range. We polished our sample in \nplan-view using a 3 degrees polish ing angle to remove the Si region at the tip of the sample, exposing a \nthin wedge of free -standing FeGe.  We use a FEI-Tecnai F20 at 200 keV with a Gatan double -tilt cryo -\ncooling holder to image our sample at 100 K in 4 -D LSTEM mode with the EMPAD, where w e collect a \ndiffraction pattern at each scan position.  We do not use the main objective lens for focusing because it \ninduces a 2 T field to the sample, which would fully saturate their magnetization and erase all chiral \nmagnetic texture. Instead, we use t he condenser lenses to focus the probe and we use the objective and \ncondenser mini lenses to either null the field on the sample or add a small external vertical field. In this \nexperiment, we chose a semi -convergence angle of 615 μrad giving us spatial res olution of 2.5 nm.  \n \nHere, we can explore the short and long -range potentials as intensity distributions in our diffraction pattern, \nwhere we can enhance the signal for magnetic fields [Fig. 4(i)] and decouple it from the grain structures \n[Fig.  4(j)]. To decouple the magnetic field from grain contrast, 1) we vary a virtual aperture on our \ndiffraction pattern and 2) for each aperture, we extract a COM signal, where the sampled angular \ndistribution of the scattered electron beam is limited by the aperture si zes [Fig. 4(a) -(c)].  Following Cao et \nal.’s [53] analysis, when the size of the aperture is less than the size of the incident diffracted disk, we treat \nthe COM signal as coming from only the short -range potentials such as grain boundaries in the sample.  \nWhen the aperture is larger than the size of the  incident beam, we treat the signal as coming from both the \nlong-range field (magnetic field) and the diffracted beams (grain structure). Here, we choose  virtual \ncollection  aperture sizes of 430 μrad, 700 μrad and 4.6 mrad [Fig. 4(a) –(c)], which correspond  to the radii \nof the apertures. At aperture sizes smaller than our semi -convergence angle, 615 μrad [Fig. 4(a)], we find \nthat the signal only comes from the short -range potentials, i.e. the grain contrast [Fig. 4(e)].  When we \nincrease our aperture size to  700 μrad [Fig. 4(b)], which is slightly larger than our semi -convergence angle, \nwe observe a signal from the magnetic field in the form of magnetic helices [Fig. 4(f)].  When we extend \nour aperture size further to 4.6 mrad [Fig. 4(c)], we observe both the  signal for the magnetic field and grain \ncontrast [Fig. 4(g)], similar to Fig. 4(f); however, there is an increased background from thermal diffuse \nscattering (including Kikuchi bands and multiple scattering) at higher scattering angles, as well as contras t \nmodulations from Bragg beams, reducing the signal to background ratio of the magnetic field signal than \nin Fig. 4(f).  From this, we see the that optimal outer angle for collecting the magnetic signal comes from \nchoosing an aperture angle that is slightl y larger than our semi -convergence angle.  \nTo suppress the short -ranged grain boundary and diffraction contrast, we exclude the interior of central disk \nfrom the analysis. Instead, we focus on the edge shift by evaluating a narrow annulus around the dis k edge \nby forming Fig. 4(h), which is the COM image when the aperture is slightly smaller than the incident disk \n[Fig. 4(e)] subtracted away from the COM image when the aperture is slightly larger than the incident disk \n[Fig. 4(f)]; schematic shown in Fig.  4(d). From the difference signal in Fig. 4(h), we can see that magnetic \ncontributions are enhanced, and grain contrast has been significantly reduced compared to Fig.4(e -g). It is \nworth pointing out that dynamical scattering from a relatively thick crysta lline sample also leads to intensity \nvariations in the center disk, which can also be significantly reduced by the subtraction scheme of Fig. 4(h). \nTherefore, our approach also improves field measurements for thicker samples because they reduce \ndistortions  from strong dynamical scattering.  Finally, we use  the COM -X and COM -Y from the decoupled \nmagnetic contrast and reconstruct a vector map showing magnetic helices [Fig. 4(j)], where we can plot the \namplitude and direction relative to the magnetic induction  color wheel, shown as the inset of Fig. 4(j) .   \nOn the other hand , we can investigate the grain contrast by suppressing the magnetic signal; here, we \nexclude the edge of the central disk by subtracting Fig. 4(f) from Fig. 4(g) such that we subtract away \nmagnetic effects and leave mostly grain contributions.  The resulting COM signal reflect the tilt of the 10 \n Ewald sphere and shifts of the Kikuchi bands, giving a qualitative sense of changes in grain orientation \nwith reduced sensitivity to thickness changes [ Fig. 4(j)].  When we compare Fig. 4 (i) and (j), we find \nmagnetic helical phases in Fig. 4(i) do not appear in our grain contrast image [Fig. 4(j)], where most of the \nmagnetization components have been subtracted away.  For finer grain sizes and overlappin g grains, our \ntechnique would still work so long as we can make our semi -convergence angle small enough to separate \nthe center disk from the rest of the diffracted disks.  When comparing the Fig. 4(i) and Fig. 4(j), we find \nchanges due to the electrostatic  potential; here, the yellow contrast in Fig. 4(i) comes from the thickness \ngradient of a wedge sample, which is a long -range electrostatic contribution to the Lorentz signal. In \naddition, charging artifacts such as two small pieces left over from polishin g can be seen on the right of Fig. \n4(i), sticking out from the edge of the sample. It is not present in the orientation map of Fig. 4(j) which was \ndesigned to reduce contributions from slowly varying thickness changes.  \nHaving established a method to resol ve nanoscale magnetic contrast in the presence of structural disorder, \nwe now discuss our observations of magnetic helices.  The orientation of helical magnetic states can be \ndefined in terms of a Q -vector – the vector about which the magnetic moment spira ls as a function of \nposition.  Thin -film FeGe (and other B20 magnetic materials) samples typically exhibit a helical state with \na Q-vector oriented out -of-plane, which has been verified by both neutron scattering experiments [13] and \nby the ferromagnetic resonance of the helical state  [12]. This tendency can be attributed to substrate -induced \ntensile strain in the film which produces anisotropy oriented in the sample plane [12, 30, 55 -58], consistent \nwith strain -dependent experiments in single -crystal FeGe  samples [56].  To see helices in an electron \nmicroscopy experiment, the Q -vector must lie in the plane.  In the case of out -of-plane Q -vectors, the \nmagnetization always remains in the sample plane as it spirals about the out -of-plane direction.  Th erefore, \nany deflection acquired by the electron beam as it passes through the upper part of the film will be \ncompensated by a deflection in the opposite direction in the lower part of the film.  In contrast, when the \nQ-vector lies in the plane, the beam d eflections are not canceled, and they vary with the lateral position \nleading to the stripe -like pattern seen in Fig. 4.  This both offers an opportunity to study the helical and \nskyrmion textures as they thread through different crystalline grains.  These topics are addressed in Sec. VI.  \n 11 \n  \nFIG. 4. 4D Lorentz -STEM of polycrystalline B20 FeGe thin film on silicon substrate imaged at 100K \nwith 0 mT applied magnetic field on a FEI -F20. Here, we optimize magnetic field contrast by varying \nthe collection apertur e size.  With a 615 μrad convergence semi -angle, w e calculate the COM in the x -\ndirection, using with virtual collection aperture sizes of (a) 430 µrad (b) 700 µrad and (c) 4.6 mrad.  Figures \n(e) – (g) show the x -component of the COM images corresponding to  apertures (a) – (c).  In (d), we use our \nsubtraction technique where we subtract the COM image when the aperture is slightly smaller than the \nbright field disk (a) from when the aperture is slightly larger than the bright field disk (b) so that we can \nobserve only shifts from the edge of the bright field disk (d).  Here, (h) corresponds to the COM image by \nsubtracting (e) from (f).  By exploiting the long -range potential due to the magnetic field through subtraction, \nwe can extract both COM -X and COM -Y and  turn them into vector components of the magnetic field, \nwhere we observed that grain contrast contributions have been greatly reduced.  We also perform our \nsubtraction technique to recover only the grain contrast by subtracting (f) from (g), where we take  their \nCOM -X and COM -Y to create a false colored vector map in (j), which shows only grain contrast image (as \na tilt of the Ewald Sphere). The color wheel plots the phase of the field or crystal orientation as the hue in \na continuous color scale, and the C OM amplitude is mapped to the brightness. We used COM -X and COM -\nY as vector components and represent them as an amplitude and phase, such that a change in direction of \nmagnetic induction [Fig. 4(i)] or grain orientation [Fig. 4(j)] can be shown.  White scale bar in (i) is 200 \nnm.  \n \n12 \n VI. Skyrmions and Helical Magnetic Phases at Grain Boundaries  \n \nThe ability to image structural features and magnetic effects at the same time has enabled us to study how \nskyrmions  and helical phases change along the boundaries of individual FeGe grains. Previous studies in \ncrystalline domains of alloyed Mn 1-xFexGe (x~0.7)  helimagnets using Lorentz -TEM have shown \ncorrelations between skyrmion helicity and crystal chirality with vary ing chemical compositions [11]. For \nchiral magnets made from thin -film sputtering with smaller polycrystalline granules, grain contrast and \nsubstrate effects have limited direct imaging of these chiral magnets due to image distortions caused by \nFresnel fringes in Lorentz -TEM, where  artifacts arising from grain contrast could be misinterpreted for \nmagnetic effects [16-18, 52] . In addition, structural effects from the grains in B20 FeGe were previously \nshown by electron back scattering diffraction (EBSD) to be twinned, which causes switching of crystal \nchirality [12], adding another layer of difficulty when decoupling magnetic from grain effects.  Therefore, \nto decouple all of these effects and observe the intrinsic magnetic properti es of the B20 FeGe magnets, we \nutilize 4 -D LSTEM with the EMPAD.  \n \nTo investigate the relationship between magnetic textures and crystal chirality of sputtered B20 FeGe on \nSi, we prepared a planview sample using a focused ion beam. The sample  is thinned at ~ 3° with respect to \nthe planar surface to create a gradient of Si thicknesses while milling, this also creates a thin wedge of free -\nstanding FeGe thin film in regions where the Si is completely milled away.  To account for the thickness \nof the sample in our magnetic field measurements, we use electron  energy loss spec troscopy (EELS)  to \nmeasure the thickness in terms of inelastic mean free path s [48]. From our results,  our region of interest \n[Fig. 5] is measured to be about 130nm. We then perform ed 4D Lorentz -STEM at 100K with a 340 𝜇rad \nsemi -convergence angle and 111 mT applied magnetic field on a Thermo Fisher Titan Themis at 300kV. \nTo decouple magnetic and grain effects, we utilize the technique described in section V. In this section, we \nwill d escribe our observations of skyrmion and helical phases in adjacent grains with opposite and similar \ncrystal chirality.   \n \nWe start by looking at how the skyrmion helic ity is affected in two adjacent grains with different crystal \nchirality [Fig. 5].  Here,  we find that the top grain shows anti -clockwise rotations of skyrmions and the \nbottom, clockwise rotations [Fig. 5(a)].  We examine the origin of these skyrmion rotations using annular \ndark field [15] where diffraction contrast from the contour of the two adjacent grains marks the grain \nboundary, shown as a blac k dashed line [Fig. 5(b)].  Using an atomic sized probe, we obtain atomic \nresolution images of the top and bottom grains (Fig. 5 (c -d)) at room temperature.  The two grains are \nviewed in project ion down the (111) zone axis, with corresponding simulations i n Fig. 5 (c -d). Structurally, \nthe differences in chirality can be seen in the tiny rotations of the purple atoms (Ge) that have clockwise \nrotations for the top grain (Fig. 5(c)) relative to the corner orange atoms (Fe and Ge in projection) and \nanticlockwis e rotations for the bottom grain (Fig. 5 (d)).  When compared to the simulations for right and \nleft-handed crystal chirality as inset of Fig. 5 c and d, the top grain exhibit s right -handed crystal chirality \n(Fig. 5(c)) and the bottom left -handed (Fig. 5(d) ).  This result was further confirmed by Kikuchi bands from \nthe top (Fig. 5(e)) and bottom (Fig. 5(f)) grains where the positions of the wide bands (marked by the yellow \narrows) and thin bands (marked by the blue arrows) are switched across the grain bound ary.  From these \nresults, we infer  as in Ref [11], that magnetic helici ty is coupled to the crystal chirality . \n 13 \n  \n \n \n     Fig. 5. The Skyrmion helicity is coupled to the crystal chirality of B20 -FeGe grains.  (a) 4D Lorentz -\nSTEM showing skyrmion helicity inverts on two adjacent grains, measured at 100K with 111 mT applied \nmagnetic field on a Thermo Fisher Titan Themis at 300kV. The same technique for plotting the vector fields \nof Fig. 4 is used here. The arrows indicate the direction of the local magnetic field. There is a helicity \ninversion across the grain boundary, as the t op grain shows anticlockwise skyrmion vortices and the bottom \nshows clockwise vortices. (b) Annular dark field image shows contrast from the two adjacent grains with a \ngrain boundary marked by a black dashed line. (c)& (d) are atomic resolution HAADF image s with (c) \nfrom the top grain and (d) from the bottom grain viewed down the [111] zone axis, showing the reversal of \ncrystal chirality, as illustrated by the colored atoms in the insets. The change in crystal chirality can be \ntracked by the tiny rotation o f the marked purple atoms(Ge) relative to the corner yellow atoms(Fe and Ge \nin projection), where it’s clockwise in (c) and anticlockwise in (d). This is also reflected in the Kikuchi \nbands of the convergent beam diffraction patterns of (e) and (f) taken f rom the top and bottom grains \nrespectively.  The positions of the wide bands(marked by yellow arrows) and thin bands(marked by blue \narrows) switch between the 2 grains, consistent with the change in crystal chirality.  \n \nTo test our inference, we study helical phases and skyrmions when adjacent grains have the same crystal \nchirality. Using the same technique described above, we find that the magnetic helical phases remained \nunchanged as they cross between two adjacent grains under  no applied magnetic field [Fig. 6(a)]. When we \nturned on the magnetic field to 111 mT, skyrmion rotations in both grains exhibit anti -clockwise rotations \neven at the grain boundary [Fig. 6(b)]. Here, annular dark field image shows contrast from two adjace nt \ngrains with a grain boundary marked by a dashed line [Fig. 6(c)].  At atomic resolution, annular dark field \nimage reveal that the twin boundary of both grains is projected down the [111] zone axis, where crystal \nchirality remains the same across the gra in boundary, albeit a 180 degree in -plane rotation [Fig. 6(d)], \nhighlighted as insets of Fig. 6(d) exhibiting right -handed crystal chirality. This is also consistent with the \nrelated Kikuchi diffraction patterns of Fig. 6 (e) and (f) taken from the top and  bottom grains respectively. \nHere, the positions of the wide (yellow arrows) and thin (blue arrows) Kikuchi bands are the same across \nthe grain boundary, showing that crystal chirality remains unchanged.  From our two experimental results \n(Fig. 5 and 6), w e believe that magnetism is coupled to the crystal chirality, such that the orientations of \nskyrmions and helical phases change with respect to the local crystal chirality.  \n \n \n14 \n  \n \nFig. 6. Skyrmion helicity is the same when adjacent grains have the same crys tal chirality:  (a) &(b) \n4D Lorentz -STEM showing the magnetic helicity remains unchanged for two adjacent grains in both (a) \nthe helical phase with no applied magnetic field, and (b) the skyrmion phase with 111 mT applied magnetic \nfield, measured on a Therm o Fisher Titan Themis at 300kV. Same technique for plotting vector field of Fig. \n4 is used here. The helicity stays the same across the grain boundary. (c) Annular dark field image shows \ncontrast from two adjacent grains with a grain boundary marked by a d ashed line. (d) is an atomic resolution \nHAADF image right from the twin boundary with both grains viewed down the [111] zone axis showing \nthe crystal chirality stays the same across the grain boundary. The zoomed -in insets of the images show \nboth crystal s tructures match with right -handed FeGe, other than a 180 degree in -plane rotation. It’s also \nconsistent with the related diffraction patterns of (e) and (f) taken from the top and bottom grains \nrespectively. The positions of the wide (yellow arrows) and th in (blue arrows) Kikuchi bands are the same \nacross the grain boundary, again showing the crystal chirality is unchanged.  \n \nTo further confirm our results, we investigate the effects of grains on the helical configuration by \nperforming micromagnetic simulati ons with Mumax3 [59]. We first randomly generate grains with 320 nm \naverage size using Voronoi tessellation [59] [Fig. 7(a)]. Next, we randomly assign either positive or \nnegative DMI coefficients to these grains as shown in Fig. 7(b). From magnetom etry measurements, we \nconfirm that our films have a small easy -plane uniaxial anisotropy, K u, approximately −3500 J/m3 [6, 12] . \nWe note that the K u of the sample may be altered in the process of preparing it for 4 -D LSTEM due to \nmechanical polishing and thickness variation. To account for this ef fect, we vary K u between 0 and −3000 \nJ/m3 in our simulations and obtained a very similar spin configuration. In Fig. 7, we show the simulation \nfor K u = -3000 J/m3. Particularly, we start the simulation by initializing the system into a random magnetic \nconfiguration and allow it to micromagnetically relax to their ground state. The spin configuration of this \nground state is shown in Fig. 7(c) as color -coded for the magnetization direction. When we applied at 0.3T \nfield to the white dashed region in Fig. 7( c), which has been magnified in Fig. 7(d), we observed skyrmions \nwith clockwise and anticlockwise rotations at the grain boundary.  The applied magnetic field is larger than \nthe one used in the experiment, because our micromagnetic simulations are performe d at zero temperature; \ntherefore, the skyrmion phase shifts to higher magnetic fields. Using micromagnetic simulations, we \nnoticed that the helical vectors shift and the skyrmion rotation directions change at the grain boundaries, \nwhere we observed both in -plane and out -of-plane components of the helical phases [Fig. 7(c)].  In our \nexperiments, only in -plane component of the spin configurations can be detected using shifts of the electron \nbeam.    Although the out -of-plane component cannot be detected exper imentally, unless we tilt the sample, \nwe find in both simulation and experiments that there are helical vector shifts and skyrmion rotation \ndirection changes at the grain boundaries, suggesting that the sign of the DMI is coupled to crystal \n15 \n orientation.  W e conclude that for FeGe on Si(111), the crystal grain orientation couples to the crystal \nchirality, and thus to the DMI.  \n \n \n \nFIG. 7. Micromagnetic simulations of disordered medium of FeGe  thin films. (a) Randomly generated \ngrains, in which the opposite sign of DMI coefficients are assigned (b). (c) The relaxed ground state spin \ntexture, where spatial shift of the helical vector is observed.  (d) Magnified region white dashed region in \n(c), where we applied a 0.3T field and observed skyrmions with clockwise and anticlockwise rotations at \nthe grain boundary.  The simulation window is 1024 x 1024 nm2.  Color wheel in (c) shows the orientation \nof the magnetization where there is both in -plane a nd out -of-plane magnetic components.  \n \n \n \nVII. Conclusion  \n \nIn this paper, we provide three themes: (1) we quantify the sensitivity of an electron beam deflection in \nLSTEM and their dependence on the number of electrons and pixels on the detector, (2) we find having a \nhigh dynamic range detector is essential when measuring small electromagnetic fields, and (3) we show \nhow we can effectively use all the scattering information from the  CBED pattern collected by the EMPAD \nto disentangle magnetic and grain contrasts from each other. Decoupling them together enables an approach \nfor efficient Lorentz imaging of magnetic samples simultaneously with grain structure, and investigations \nof chan ges in the magnetic field from crystal structures at the nanometer scale.  We believe that our \napproach is important in the characterization of real devices where small changes in grain structure can be \ncritical to device performance.   \n \nMore systematicall y, we find that the sensitivity in magnetic field detection is more dependent on the \nnumber of electron than on the number of pixels in our detector, where high dynamic range detectors are a \nnecessity when imaging µT -level magnetic fields.  Whereas, having  more pixels in our detector becomes \nimportant when we decouple signals from magnetic and grain contrast.  Here, we filter long -range \n(magnetic contrast) and short -range (grain contrast) potentials using our acquired CBED pattern for thin \nfilm sputtered B2 0 FeGe on Si. We observed helicity  changes in skyrmions where grain structures are of \nopposite crystal chirality and no change in helical phases and skyrmion rotations when the crystal chirality \nis the same, even when there is a 180 degree in -plane rotatio n of the crystal structure. By performing \nmicromagnetic simulations, we generate random grain orientations and signs of DMI.  From these \nsimulations, we observed that the DMI is coupled to the crystal chirality, which can dictate how the \nskyrmion rotation and helical vectors change when going in between grains.  Our technique can also be \napplied to other chiral magnets.  Here, understanding the nanoscale physical and magnetic properties of \nchiral magnetic materials is key to advancing next generation magnet ically -driven devices.  \n \nACKNOWLEDGMENTS:  Electron microscopy experiments of thin -film FeGe samples and equipment \nwere supported by the Cornell Center for Materials Research, through the National Science Foundation \n16 \n MRSEC program, award DMR 1719875.  Electro n microscopy measurements of single -crystal FeGe \nsamples were supported by DARPA under cooperative agreement D18AC00009.  Thin -film FeGe growth \nand micromagnetic modeling were supported by the Department of Energy Office of Science under grant \nNo. DE -SC001 2245.  Growth of single -crystal FeGe was made by M.J.S. and S. J., who are supported by \nNSF grant ECCS -1609585. M.J.S. also  acknowledges support from the NSF Graduate Research Fellowship \nProgram  grant number DGE -1256259.  \n \n \nAppendix A : Determining Magnetic Field Detection Sensitivity: Analytical Model  \n \nHow sensitive the measurement for magnetic field using center of mass (COM) imaging depends on how \nwell we can track the deflection of the electron beam. To quantify this, we follow an analytical approach to \ndetermine the accuracy of the magnetic field usi ng a model previously described by Chapman et al. [27, \n60]. In this analytical model, the deflection sensitivity of our signal has two dependencies: (1) 𝛼, the semi -\nconvergence angle of the electron beam, which provides the limit in spatial resolution, and (2) n, the number \nof electrons or dose of the electron beam, which corresponds to the number of electrons needed to sustain \na signal that is adequately above Poisson noise. From Chapman et al., assuming a round detector divided \ninto quadrants, the signal S is the difference in intensity between th e left and right halves of the  detector, \ndescribed as:  \n𝑆= 4𝛽𝐿\n𝜋𝛼 (A1) , \nwhere 𝛽𝐿 corresponds to the deflection angle of the electron beam from the local magnetic field of the \nsample.  Here, the dose n is  \n𝑛=𝐵𝜋2𝛼2𝐷2𝜏\n4𝑒 (A2),  \n \nwhere 𝑒 is the elec tron charge constant, B is the brightness of the electron gun, D is the beam diameter, and \n𝜏 is the acquisition time. The signal with respect to the dose is 𝑆∗n, and the noise due to Poisson statistics \nis √𝑛.  We calculate the signal to noise ratio (SNR) as 𝑠=𝑆∗√𝑛. To estimate the standard deviation of \nthe noise, we set s = 1, and rewrite the equation in terms of the minimum deflection angle,  𝛽𝐿, \n \n𝛽𝐿=𝜋𝛼\n4 √𝑛  (A3).  \n \nNote that Equation A3 is the minimum  deflection angle acquired via a normalized difference map of a split \ndetector when illuminated by a circular beam illumination .  In our approach,  we use a modified version for \na square beam illumination that match es with our simulations in Appendix B – the difference between the \ntwo approaches is a 20% correction factor, but the other scalings remain unchanged and the numerical \nsimulation is greatly accelerated . \n  \nWe also know the beam deflection angle is given as:  \n \n 𝛽𝐿=𝑒𝜆𝐵0𝑡\nℎ                                                         (A4), \n \nwhere B 0 is the local magnetic field, 𝜆 is the wavelength of the electron beam, t is the sample thickness and \nh is Planck’s constant.  We can therefore know the minimum detectable magnetic fie ld B0: \n \n𝐵0=𝜋ℎ𝛼\n 4e𝜆𝑡√𝑛                                                              (A5) . \n \nTo be able to have a direct field sensitivity comparison with other magnetic imaging techniques such as 17 \n superconducting quantum interference device (SQUID) and nitrogen vacancy (NV) centers [61], we have \nto normalize the time interval /bandwidth  for taking measurements on LSTEM  so that neither our sensitivity \nnor the corresponding dose is dependent of dwell time.  Normalizing out the  time interval we get the field \nsensitivity  𝐵𝑡 in terms 𝑇\n√𝐻𝑧 , as:  \n \n𝐵𝑡=𝐵0√𝜏=𝜋ℎ𝛼\n 4√𝑒𝜆𝑡√𝐼             (A6). \nWe can see from (A 6) that the field sensitivity scales inversely with square root of electron beam current \nand scales inversely proportional  with sample thickness for thin samples. The field sensitivity for our \nsquare beam has a numerical prefactor of 1  vs 𝜋\n4  for the round beam , a 20% difference  (Appendix B) .  \n \n \n \nAppendix B : Determining Magnetic Field Detection Sensitivity: Numerical Model  \n \nIn our numerical simulation, we assume a one -dimensional detector, represented as the vector form \nof 𝑣⃗. Our detector, 𝑣⃗, is then entirely illuminated by an electron beam represented by a top hat vector , 𝑔⃗  \nwhich has the same  size as our detector  vector . For a detector with j number of pixels,  𝑣⃗  is composed of j \nnumber of index es that starts from (−𝑗−1\n2) and have  an increas ing spacing of 1  so that the indexes are \ncentered about the origin . For example, for a two -pixel detector , 𝑣⃗=[−1\n2,1\n2]. The top hat vector has a \nuniform  distribution with the sum of components to be 1 , thus for a j-pixel detector  every component  of 𝑔⃗ \nis equal to  1\n𝑗. To convert our top hat distribution into the number of  electron s distribu ted on the detector , we \nmultiply the top hat vector, 𝑔⃗ with the electron dose , 𝑛 , as: 𝑛∗𝑔⃗=[𝑛\n𝑗,𝑛\n𝑗,…,𝑛\n𝑗]. However, real electron \nprobes have intensity variations , thus to reflect a real electron probe, we added Poisson noise on top of our \ntop hat  distribution. To get statistics from the Poisson noise, we generate Poisson  noise on  N number of \nsamples  which ha ve the same  electron dose distribution  on the same number of pixel  detector , and a sample \nof the electron probe with noise is represented as 𝑝⃗.   \n \nUsing our model, we calculate the COM, for the k -th sample  that  has an electron  beam  illumination  pk⃗⃗⃗⃗⃗, \nas μk: \n \n𝜇𝑘=𝑝𝑘⃗⃗⃗⃗⃗⃗∙𝑣⃗⃗\n𝑠𝑢𝑚 (𝑝𝑘⃗⃗⃗⃗⃗⃗)     (B1). \n \nHere, 𝜇𝑘 represents the COM on our detector, 𝑣⃗, in units of pixels. We find the mean, 𝜇, from all N \nsamples  as: \n \n𝜇= ∑ 𝜇𝑘# 𝑜𝑓 𝑠𝑎𝑚𝑝𝑙𝑒𝑠\n𝑘=1\n𝑁                   (B2). \n \nUsing Equation B1 and B 2, we calculate the standard deviation, 𝜎, as  \n \n𝜎=√1\n𝑁−1∑ (𝜇𝑘−𝜇)2𝑁\n𝑘=1     (B3). \n \n𝜎 is the noise level  of COM measurements , in the unit of pixels.  To convert the noise level to units of \ndeflection angle, we need to calibrate the pixel with our semi -convergence angle 𝛼. The calibration  factor  \nis different  for different number s of pixel s on our  detector , since  for the same convergence angle, as pixel 18 \n number increase s, the size of each pixel relative to the size of the electron beam is reduced . To counteract \nthis effect, we must calibrate  𝜎 so that our noise measurements  with different number s of pixel s on our  \ndetector  are consistent with the unit of  angle s (radians) . To figure out what the calibration factor  is for a j-\npixel detector, we calculate the COM  for a signal of 𝛽𝐿 deflection angle . The example for a 2 -pixel detector \ncase is shown in Fig . B1.  \n \nFIG. B 1. The shift of the probe intensity distribution on a two -pixel detector  assuming a square \nillumination for a semi -convergence angle 𝛼 and a small Lorentz deflection angle 𝛽𝐿. The associated \nminimum deflection angle  with electron dose n  is  𝛽𝐿=𝛼\n √𝑛. \n \nWhen there is a  small  𝛽𝐿 deflection  compared to 𝛼 on a j -pixel detector , the signal of probe shift will only \ncause intensity change on the first and the last detector pixel s. 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The\nfundamental role played by the spin degree of freedom was then revealed, even for a non-\nmagnetic system, whenever the spin-orbit coupling was present with some strength. Here\nwe demonstrate that similarly to electrons in the presence of spin-orbit coupling, the propa-\ngating vibrational modes are also accompanied by a well-defined magnetic oscillation even in\nnon-magnetic materials. Although this effect is illustrated by considering a single layer of the\nWSe 2dichalogenide, the phenomenon is completely general and valid for any non-magnetic\nmaterial with spin-orbit coupling. The emerging phonon-induced magnetic oscillation acts\nas an additional effective flipping mechanism for the electron spin and its implications in the\ntransport and scattering properties of the material are evident and profound.\nIntroduction\nIn materials science, the most conventional point of view is to assume that propagating vibrational\ncollective modes (phonons) are not associated with any magnetic property if the material itself is\nnon-magnetic. In a magnetic material, however, it is natural to expect a magnetic oscillation as-\nsociated to a phonon mode. Phonons are commonly understood as sinusoidal patterns of atomic\ndisplacements which couple to electron states by the scalar potential induced by these atomic dis-\nplacements. Electrons have a well defined spin-polarization under spin-orbit interaction, but when\ntime reversal symmetry applies the spin polarizations for opposite momenta cancel each other and\nthe material results to be non-magnetic. There is a clear evidence that the lattice thermal conduc-\ntivity of diamagnetic materials couples to external magnetic fields [1], which in principle might\nseem contrary to the idea that phonons do not have any associated magnetic property. Here we\nshow that similarly to electrons, phonons are also accompanied by an induced effective magnetic\noscillation when spin-orbit coupling is present even for non-magnetic materials.\n1arXiv:2001.08629v1  [cond-mat.other]  23 Jan 2020Results\nIn a solid with spin-orbit coupling, electron states are described by two-component spinor wave\nfunctions for each kpoint spanning the Brillouin zone (BZ), \tk;i(r) = \nu\"\nk;i(r)\nu#\nk;i(r)!\neikr. Each\nelectron state has an associated spin-polarization defined as the expectation value of the Pauli\nvector mk;i(r) =h\tk;ij\u001bj\tk;ii. Time reversed Kramers pairs at kand\u0000khave opposite spin-\npolarizations that exactly cancel out when integrated, which therefore implies no magnetism. How-\never, the electron spin-polarization is a crucial physical magnitude in many non-magnetic systems\nwith spin-orbit coupling, one of the most outstanding being probably its role in the topological\nproperties of matter.\nA phonon excitation consists in a sinusoidal displacement of atoms and an induced (almost)\nstatic response of the electron gas which tries to weaken or screen out the electric perturbation\ngenerated by these displacements. Therefore, the question of whether a phonon perturbation cre-\nates a magnetic oscillation could be suitably treated considering a generalized dielectric theory of\ndimension 4\u00024mixing the magnetic and electric components of the potential [2]. An alternative\nand more transparent way to see whether an overall magnetic property emerges is to consider the\neffect of the perturbation on each electron spinor wave function and then integrate over the BZ. The\nkey point is that if a phonon is excited with momentum qit couples differently with the electrons\natkand\u0000k, the result being that the spin-polarization of electrons with time reversed momenta do\nnot cancel each other. Under these conditions the balance of the electron spin-polarizations within\nthe BZ is broken and the BZ integral gives a finite value and, therefore, a net real-space magnetic\noscillation with the same wave number as the phonon q.\nLet us focus on a single frozen phonon-mode ( \u0017) with momentum qwhich produces a pertur-\nbation on both components of the periodic part of the electron spinor wave functions, \u000eq;\u0017u\u001b\nk;i(r),\nwhere\u000eq;\u0017denotes the self-consistent variation in the context of density-functional perturbation\ntheory [3, 4] and u\u001b\nk;i(r)is the periodic part of each spinor component. The unit cell periodic part\nof the amplitude of the frozen charge-spin density wave is obtained by integrating the contributions\nfrom all occupied electron states (see Supplementary Note 1)\n\u000e~n\u001b;\u001b0\nq;\u0017(r) =occX\nk;ihu\u001b0\nk;i(r)j\u000eq;\u0017u\u001b\nk;i(r)i: (1)\nTherefore,\u000e~n\u001b;\u001b0\nq;\u0017(r)eiqrwould represent the complete oscillation wave in real space. The absence\nof a magnetic component accompanying a phonon mode would require the off-diagonal elements\n(\u001b6=\u001b0) to be zero and that both diagonal components are equal to each other, which, in general,\nare conditions only fulfilled at the \u0000point ( q=0). Writing the 2\u00022charge-spin matrix of equa-\ntion (1) in terms of Pauli matrices allows to explicitly distinguish the electronic charge, \u000e~n0\nq;\u0017(r),\n2and the magnetic components, \u000e~Mq;\u0017(r) = (\u000e~n1\nq;\u0017(r);\u000e~n2\nq;\u0017(r);\u000e~n3\nq;\u0017(r)):\nn\n\u000e~n\u001b;\u001b0\nq;\u0017(r)o\n!\u000e~ nq;\u0017(r) =\u000e~n0\nq;\u0017(r)\u001b0+\u000e~n1\nq;\u0017(r)\u001b1+\u000e~n2\nq;\u0017(r)\u001b2+\u000e~n3\nq;\u0017(r)\u001b3\n=\u000e~n0\nq;\u0017(r)\u001b0+\u000e~Mq;\u0017(r)\u001b: (2)\nIn real space, the time-dependent charge-spin field is given by the real part of the above\nfrozen complex amplitudes when accounting for the classical motion of atoms. For a single phonon\nmode (q;\u0017)of energy!q;\u0017we have\n\u000enq;\u0017(r;t) = Reh\u0010\n\u000e~n0\nq;\u0017(r)\u001b0+\u000e~Mq;\u0017(r)\u001b\u0011\nei(qr\u0000!q;\u0017t)i\n:\n=\u000en0\nq;\u0017(r;t)\u001b0+\u000eMq;\u0017(r;t)\u001b;\nconcluding that the appearance of an induced spin-density (or magnetization indistinctly) (\u000e~Mq;\u0017(r))\nis completely general for crystals with spin-orbit coupling since the only requirement is a non-\ntrivial pattern of the spin-polarization within the BZ associated to the absence of inversion sym-\nmetry [5]. The phonon modes break the symmetry of the BZ in a way that the electron spin-\npolarization is modulated within the BZ producing a net spin accumulation. The similarity of\nthe phonon magnetism and the electron spin-polarization with time-reversal symmetry (no net\nmagnetism) is strengthened by the fact that time-reversed phonon momenta give strictly opposite\nmagnetization exactly in the same way as for electrons \u000eMq;\u0017(r;t) =\u0000\u000eM\u0000q;\u0017(r;t).\nThe above description of the spin-charge field induced by phonons is completely classical\nand focussed on a single phonon with a fixed momentum. Therefore, physically \u000eMq;\u0017(r;t)would\nbe the time dependent magnetization associated with a single coherent phonon mode. In general\nthe vector field defined by \u000eMq;\u0017(r;t)shows an interesting real-space and time dependent non-\ncollinear pattern, which depends also on the particular atomic displacements (polarization vectors)\nassociated with each phonon branch. Actually, it is the motion of the W atom, i.e. the atom for\nwhich the spin-orbit interaction is dominant, which determines the direction of the magnetization.\nFor instance, for q=Kand for a mode in which Se atoms rotate clockwise with opposite phase in\nthe plane of the surface ( x-yplane) and W atoms vibrate in the perpendicular direction ( z) to the\nsurface, a net circularly polarized induced magnetization appears in the surface plane around the\nW atoms (Figs. 1 a-cand Supplementary Movie 1). However, for the same phonon propagation\nvector q=K, if W atoms rotate clockwise around their equilibrium positions in the plane of the\nsurface, the net magnetization shows along the perpendicular direction to the plane (Figs. 2 a-c\nand Supplementary Movie 2). It is noteworthy that the magnitude of the induced magnetization is\nonly an order of magnitude smaller than in the induced (scalar) charge (Fig. 2 b). Since q=K0is\nthe time-reversed momentum of q=K, as mentioned earlier, the real-space magnetization should\nbe opposite in sign. For the first example given above but in the case in which q=K0, Se atoms\nrotate anticlockwise in the plane and the W vibrates perpendicular to the plane (Supplementary\nMovie 3). Taking a snapshot in time for which the atomic positions coincide with those in Fig. 1\ndemonstrates that the magnetization shows exactly opposite chirality, and therefore the relation\n3\u000eMK;\u0017(r;t) =\u0000\u000eM\u0000K;\u0017(r;t)is fulfilled (Supplementary Figure 1). Note that the propagation\nof the atomic displacements of W in the perpendicular direction to the plane along q=K(Fig. 1 b)\nis exactly the same as the one along q=K0when it is looked from right to left (Supplementary\nFigure 1 b). A similar situation occurs for the second example, where the chirality of the atomic\ndisplacements reverses when changing q=Kto its time-reversed value [6] which again gives a mag-\nnetization opposite in sign (Supplementary Movie 4) and which can be compared to that of Fig. 2\nwhen the atomic positions are frozen to be the same (Supplementary Figure 2). The non-collinear\ncharacter in space and time of the magnetization is also observed when the atomic displacements\nare all linear. For instance, for an acoustic phonon with vector q=Mat which Se atoms move\nalong qand W atoms vibrate in the perpendicular direction to the plane (Supplementary Movie\n5) a chiral magnetization pattern similar to that of Fig. 1 is found (Supplementary Figure 3). In-\nstead, when all atoms oscillate linearly in the plane at right angles to the propagation vector q=M\n(Supplementary Movie 6), the magnetization appears in the perpendicular direction to the plane as\nin Fig. 2 (Supplementary Figure 4). Note that for this acoustic mode when all atoms go through\ntheir equilibrium positions both, the induced charge and spin-density fields, disappear. All the\nmagnetization patterns show a periodicity in real space according to the wave number qof the\npropagation of the excited phonon, as they should. At this point it is worth mentioning that the\nmagnetic polarization of the electron gas as described in this manuscript does not have a relation\nwith the angular momentum of the atoms as described by Zhang et al. in [7]. In our theory linearly\npolarized phonons with null angular momentum give a finite and meaningful contribution to the\nmagnetization. It is therefore clear that the physics described in [7] is different and not connected\nto the spin response of the electron gas as described in the present work.\nThe size of the fluctuations of the real space unit-cell average of these oscillations gives an\norder of the magnitude of this effect, even though it does not capture all the detailed structure\nin real space. It is nevertheless physically meaningful, allowing to analyze the momentum and\nmode dependence at the same time, and making a connection with the possibility of experimental\ndetection as it will be shown shortly. More specifically, for a given phonon qthe root-mean-square\n(RMS) of the time dependence of the periodic part of this quantity reflects the overall amplitude\nof the spin-density associated to a single phonon mode. For each cartesian component \u000bwe have\n(see Supplementary Note 2):\n\u000eM\u000b\nq;\u0017=vuut*\u0012\nReZ\n\nd3r\n\nh\n\u000e~M\u000b\nq;\u0017(r)e\u0000i!q;\u0017ti\u00132+\nT: (3)\nThe above classical RMS amplitudes of the magnetization are directly connected to the charge-\ncharge, spin-charge and spin-spin components of the dynamic structure factor, S\u000b;\f(!;q+G)\n(see Supplementary Note 2), which is accessed by inelastic neutron scattering, inelastic X-Ray\nspectroscopy and spin-polarized electron energy loss spectroscopy [8]. As van Hove first pointed\nout [9], the dynamic structure factor is the space and time Fourier transform of the density-density\ncorrelation function h\u000e^n\u000b(r;t)\u000e^n\f(r0;0)iT. If we consider a 4-dimensional spin-charge quantized\n4field\n\u000e^n(r;t) =X\nq\u0017\u0000\nay\nq;\u0017ei!q;\u0017t\u000e~ n\u0003\nq;\u0017(r) +aq;\u0017e\u0000i!q;\u0017t\u000e~ nq;\u0017(r)\u0001\n; (4)\nwhereaq;\u0017anday\nq;\u0017are creation and annihilation operators for a phonon mode ( q;\u0017) with energy\n!q;\u0017(Fig. 3 a), then the dynamic structure factor can be written as\nS\u000b;\f(!;q+G) = [1 + fB(!q;\u0017)]\u000e~~n\u000b\nq;\u0017(G)\u000e~~n\f\u0003\nq;\u0017(G)\u000e(!\u0000!q;\u0017) (5)\n+fB(!q;\u0017)\u000e~~n\u000b\u0003\nq;\u0017(G)\u000e~~n\f\nq;\u0017(G)\u000e(!+!q;\u0017);\nwhere\u000e~~n\u000b\nq;\u0017(G)indicates the Fourier transform or crystal field components of the real-space com-\nplex amplitudes \u000e~n\u000b\nq;\u0017(r),fB(!q;\u0017)denotes phonon occupation numbers and where we ignore the\nDebye-Waller factor [10]. Taking the G=0components (unit-cell average), it is easily seen that\nthe classical RMS terms of equation (3) are proportional to the diagonal ( \u000b=\f) spectral con-\ntributions to equation (5) for individual phonons:\u0000\n\u000eM\u000b\nq;\u0017\u00012\u0018\u000e~~n\u000b\u0003\nq;\u0017(0)\u000e~~n\u000b\nq;\u0017(0). This helps to\nphysically interpret the RMS of the induced magnetization defined as above because the terms par-\nallel (q\n(\u000eMx\nq;\u0017)2+ (\u000eMy\nq;\u0017)2) and perpendicular ( \u000eMz\nq;\u0017) to the WSe 2layer shown in Fig. 3 band\nFig. 3 c, respectively, are basically the spin contributions to the structure factor connected to a given\nphonon mode (q;\u0017). Said in other words, Fig. 3 band Fig. 3 cmay be interpreted as the momen-\ntum/energy and phonon mode resolved contributions to the spin sector of the dynamic structure\nfactor depicted along the high symmetry lines of the surface Brillouin zone. The magnetic char-\nacter associated inherently to phonons as proposed in this work is therefore accessible by means\nof any experimental setup probing the spin components of the dynamical structure factor in the\nenergy ranges corresponding to phonons.\nDiscussion\nWe conclude that in any material with a non-trivial spin-pattern within the Brillouin zone, even\nif non-magnetic, phonon modes are connected inherently to a magnetic property analogous to the\nelectron spin-polarization and it can be stated quite generally that phonon modes are accompanied\nby an induced spin-density (magnetization) which is rich in real space details. It is also shown that\nthis magnetic modulation is only one order of magnitude weaker than the purely electrostatic (spin-\ndiagonal) terms. All the above physics is illustrated convincingly for WSe 2in a mode by mode\nanalysis where the real space and time dependence of the induced magnetization is revealed for the\nmost relevant modes. The implications are extensive and profound because phonons, which are\nnow intrinsically attached to an effective magnetic moment, should be understood as an additional\nspin-flip mechanism even for materials without a net magnetic moment. This means that the whole\nelectron-phonon physics is modulated in every system with spin-orbit coupling and, for instance,\neven electron backscattering events may be aided by the phonon magnetic moment. Experimental\ndetection of magnetic oscillations for coherent phonons should be done by ultrafast probes and\nour calculated details of these fields may indicate a detection strategy. However, we also show\n5that probing the spin components of the dynamical structure factor may be an alternative route to\nmeasure what we could name as the spin polarization of phonons.\nData availability The data that support the findings of this study are available from the corresponding\nauthor upon reasonable request.\n[1] Jin, H. et al. Phonon-induced diamagnetic force and its effects on the lattice thermal conduc-\ntivity. Nature Materials 14, 601 (2015).\n[2] Lafuente-Bartolome, J., Gurtubay, I. G. & Eiguren, A. Relativistic response and novel spin-\ncharge plasmon at the Tl/Si(111) surface. Phys. Rev. B 96, 035416 (2017).\n[3] Baroni, S., de Gironcoli, S., Dal Corso, A. & Giannozzi, P. Phonons and related crystal\nproperties from density-functional perturbation theory. Rev. Mod. Phys. 73, 515–562 (2001).\n[4] Dal Corso, A. Density functional perturbation theory for lattice dynamics with fully rela-\ntivistic ultrasoft pseudopotentials: Application to fcc-Pt and fcc-Au. Phys. Rev. B 76, 054308\n(2007).\n[5] LaShell, S., McDougall, B. A. & Jensen, E. Spin splitting of an Au(111) surface state band\nobserved with angle resolved photoelectron spectroscopy. Phys. Rev. Lett. 77, 3419–3422\n(1996).\n[6] Zhu, H. et al. Observation of chiral phonons. Science 359, 579–582 (2018).\n[7] Zhang, L. & Niu, Q. Angular momentum of phonons and the Einstein–de Haas effect. Phys.\nRev. Lett. 112, 085503 (2014).\n[8] Sturm, K. Dynamic Structure Factor: An Introduction. Zeitschrift fr Naturforschung A 48,\n233–242 (1993).\n[9] Van Hove, L. Correlations in space and time and Born approximation scattering in systems\nof interacting particles. Phys. Rev. 95, 249–262 (1954).\n[10] Jancovici, B. Infinite susceptibility without long-range order: The two-dimensional harmonic\n“solid”. Phys. Rev. Lett. 19, 20–22 (1967).\nAcknowledgements The authors acknowledge the Department of Education, Universities and Research of\nthe Basque Government and UPV/EHU (Grant No. IT756-13), the Spanish Ministry of Economy and Com-\npetitiveness MINECO (Grant No. FIS2016-75862-P) and the University of the Basque Country UPV/EHU\n(Grant No. GIU18/138) for financial support. Computer facilities were provided by the Donostia Interna-\ntional Physics Center (DIPC).\n6Author contributions A.E. conceived the ideas. A.I. contributed to the early stages of the calculations.\nI.G.G. and A.E. carried out the calculations, discussed the results and contributed to the writing of the\nmanuscript.\nCompeting interests The authors declare no competing interests.\n7a)b)z displacement of W along q=K\n−0.04−0.03−0.02−0.010.000.010.020.030.04z displacement of atoms ( a0)\nc)Figure 1: Induced magnetization for the second lowest energy acoustic phonon mode at q=K\ninvolving pure out of plane displacements of W atoms a) Real-space representation of the\nmagnetization in the plane of the W atoms for 4\u00024unit cells along the hexagonal axes of WSe 2\nfor the second lowest energy mode at q=K(in orange in Figs. 3 aand 3 b). In this mode\nthe W atoms (filled circles) displace along the perpendicular direction (see colour-bar) and the\nSe atoms above (filled triangle up) and below (filled triangle down) the W plane rotate clockwise\nwith opposite phase around their equilibrium positions (crosses) in their respective planes. The\ncoloured vector-field is proportional to the in-plane magnetization at each point in real space, with\nyellow/light (blue/dark) arrows representing the largest (smallest) values. These arrows as well\nas the displacements of the Se atoms have been scaled to make them visible. b)The coloured\narrows give the z displacement of the W atoms along the q=Kdirection (dotted magenta line in\npanel a)) according to the colour-bar. The dashed line describes the propagation of the vibration\nalong several unit-cells in real space. Note that K= [1=3;1=3;0]in crystal axes, and hence the\nperiodicity of the wave. c)Side view of the WSe 2formula-unit in the lower left corner unit-cell.\nThe names of the atoms display their displacements from the equilibrium positions, denoted as in\npanel a). This figure is a snapshot of the time evolution of the induced magnetization for this mode\n(Supplementary Movie 1).\n8a)\n−1.00−0.75−0.50−0.250.000.250.500.751.00\nMagnetization along z in the W plane ( 10−3µB)\nb)\n−4−3−2−10 1 2 3 4Charge fluctuation in the W plane ( 10−3au)\nc)Figure 2: Induced magnetization for the highest energy acoustic phonon mode at q=K\ninvolving in-plane displacements of the W atoms a) Real-space representation of the perpendic-\nular component of the magnetization in 4\u00024unit cells along the hexagonal axes of WSe 2for the\nhighest energy acoustic mode for q=K(represented in green in Figs. 3 aand 3 c). This mode is\ncomposed by clockwise rotations of the W atoms (circles) around their equilibrium positions and\nin-phase anticlockwise rotations of the Se atoms located above (triangle up) and below (triangle\ndown) the W plane. The colour code represents the magnetization in the perpendicular direction.\nThe displacements of the atoms have been scaled to make them visible. b)Induced electronic\ncharge for the same atomic configuration as in panel a). Note that the induced magnetization is\nonly an order of magnitude smaller than the induced (scalar) charge. c)Side view of the WSe 2\nformula-unit in the lower left corner unit-cell. The names of the atoms display their displace-\nments from the equilibrium positions, denoted as in panel a). This figure is a snapshot of the time\nevolution of the induced magnetization for this mode (Supplementary Movie 2).\n90510152025303540q, (meV) \na)\n024681012|Mx\nq,|2+|My\nq,|2(106B) \nb)\nM\n K M K'\n024681012|Mz\nq,|(106B) \nc)\nKMK/prime\nq=K q=K\nq=K q=K q=K q=KFigure 3: Mode and momentum resolved magnetization induced by lattice vibrations in\nmonolayer WSe 2a)Phonon energy spectrum along high symmetry lines in the surface Brillouin\nzone (inset). b)andc)Plane (x,y) and out-of-plane ( z) components of the magnitude of the unit-\ncell average magnetization for each phonon mode in a)and with the same colour convention. Insets\nshow the corresponding polarization vectors for q=K, the length of the arrows being proportional\nto the magnitude of the atomic displacements. Vertical arrows represent linear displacements in the\nperpendicular direction to the plane, and semicircular arrows show circular displacements of the\natoms around their equilibrium positions. The direction of the magnetization is determined by the\nmotion of the W atom. When W vibrates in the perpendicular direction of the plane, it induces a net\nmagnetization in the plane (panel b)). However, when W atoms move on the plane (with circular\npolarization for q=K), the induced magnetization appears in the perpendicular direction (panel\nc)). For the three middle modes in the phonon spectrum W atoms move significantly less, yielding a\nsmaller magnetization. Panels b)andc)can be interpreted as a frequency and momentum resolved\ndynamic structure factor of phonons.10"    },    {        "title": "2001.09774v1.Non_exponential_magnetic_relaxation_in_magnetic_nanoparticles_for_hyperthermia.pdf",        "content": "arXiv:2001.09774v1  [physics.app-ph]  27 Jan 2020Non-exponential magnetic relaxation in magnetic nanopart icles for hyperthermia\nI. Gresits,1, 2Gy. Thur´ oczy,3O. S´ agi,2S. Kollarics,2G. Cs˝ osz,2B. G.\nM´ arkus,2N. M. Nemes,4, 5M. Garc´ ıa Hern´ andez,4,5and F. Simon2, 6\n1Department of Non-Ionizing Radiation, National Public Hea lth Center, Budapest, Hungary\n2Department of Physics, Budapest University of Technology a nd Economics and MTA-BME\nLend¨ ulet Spintronics Research Group (PROSPIN), Po. Box 91 , H-1521 Budapest, Hungary\n3Department of Non-Ionizing Radiation, National Public Hea lth Institute, Budapest, Hungary\n4GFMC, Unidad Asociada ICMM-CSIC ”Laboratorio de Heteroest ructuras con Aplicaci´ on en Espintronica”,\nDepartamento de Fisica de Materiales Universidad Complute nse de Madrid, 28040\n5Instituto de Ciencia de Materiales de Madrid, 28049 Madrid, Spain\n6Laboratory of Physics of Complex Matter, ´Ecole Polytechnique F´ ed´ erale de Lausanne, Lausanne CH-1 015, Switzerland\nMagnetic nanoparticle based hyperthermia emerged as a pote ntial tool for treating malignant tumours. The\nefficiency of the method relies on the knowledge of magnetic p roperties of the samples; in particular, knowledge\nof the frequency dependent complex magnetic susceptibilit y is vital to optimize the irradiation conditions and to\nprovide feedback for material science developments. We stu dy the frequency-dependent magnetic susceptibility\nof an aqueous ferrite suspension for the first time using non- resonant and resonant radiofrequency reflectometry.\nWe identify the optimal measurement conditions using a stan dard solenoid coil, which is capable of providing\nthe complex magnetic susceptibility up to 150 MHz. The resul t matches those obtained from a radiofrequency\nresonator for a few discrete frequencies. The agreement bet ween the two different methods validates our ap-\nproach. Surprisingly, the dynamic magnetic susceptibilit y cannot be explained by an exponential magnetic\nrelaxation behavior even when we consider a particle size-d ependent distribution of the relaxation parameter.\nPACS numbers:\nIntroduction\nNanomagnetic hyperthermia, NMH,1–13is intensively stud-\nied due to its potential in tumor treatment. The prospective\nmethod involves the delivery of ferrite nanoparticles to th e ma-\nlignant tissue and a localized heating by an external radiof re-\nquency (RF) magnetic field affects the surrounding tissue on ly.\nThe key medical factors in the success of NMH4–6,14include\nthe affinity of tumour tissue to heating and the specificity of\nthe targeted delivery.\nConcerning the physics and material science challenges, i)\nthe efficiency of the heat delivery, ii) its accurate control and\niii) its precise characterization are the most important on es.\nConcerning the latter, various solutions exists which incl udes\nmodeling the exciting RF magnetic field with some knowledge\nabout the magnetic properties of the ferrite15–19, measure-\nment of the delivered heat from calorimetry15,17,20–22, or deter-\nmining the dissipated power by monitoring the quality facto r\nchange of a resonator in which the tissue is embedded23,24.\nAll three challenges are related to the accurate knowledge\nof the frequency-dependent complex magnetic susceptibili ty,\n/tildewideχ=χ′−iχ′′, of the nanomagnetic ferrite material. The dis-\nsipated power per unit volume, Pis proportional to the value\nofχ′′at the working frequency, ω, as:P= 0.5µ0ωχ′′H2\nAC,\nwhereµ0is the vacuum permeability, HACis the AC mag-\nnetic field strength. Although measurement of /tildewideχ(ω)is a well\nadvanced field due to e.g. the extensive filter or transformer\napplications10,25–34, we are not aware of any such attempts for\nnanomagnetic particles which are candidates for hyperther -\nmia.\nKnowledge of /tildewideχ(ω)would allow to determine the optimal\nworking frequency, which is crucial to avoid interference d ue\nto undesired heating of nearby tissue e.g. by eddy currents1,3,9.\nIn addition, an accurate characterization of /tildewideχ(ω)can provide\nan important feedback to material science to improve the fer -\nrite properties. Last but not least, measurement of /tildewideχ(ω)wouldallow for a better theoretical description of the high frequ ency\nmagnetic behavior of ferrites. Most reports suggest1,3,9,35that\na single relaxation time, τ, governs the frequency dependence\nof/tildewideχ(ω). The magnetic relaxation time, τ, is given by to\nthe Brown and N´ eel processes; these two processes describe\nthe magnetic relaxation due to the motion of the nanomag-\nnetic particle and the magnetization of the nanoparticle it -\nself (while the particle is stationary). When the two pro-\ncesses are uncorrelated, the magnetic relaxation time is gi ven\nas1/τ= 1/τB+1/τN, whereτBandτNare the respective re-\nlaxation times. These two relaxation types have very differ ent\nparticle size and temperature dependence, which would allo w\nfor a control of the dissipation. Nevertheless, the major op en\nquestions remain, i) whether the single exponential descri p-\ntion is valid, and ii) what the accurate frequency dependenc e\nof the magnetic susceptibility is.\nMotivated by these open questions, we study the frequency\ndependence of /tildewideχon a commercial ferrite suspension up to\n150 MHz. We used two types of methods: a broadband non-\nresonant one with a single solenoid combined with a network\nanalyzer and a radiofrequency resonator based approach. Th e\nlatter method yields the ratio of χ′′andχ′for a few discrete\nfrequencies. The two methods give a good agreement for\nthe frequency-dependent ratio of χ′′/χ′which validates both\nmeasurement techniques. We find that the data cannot be ex-\nplained by assuming that each magnetic nanoparticle follow s\na magnetic relaxation with a single exponent even when the\nparticle size distribution is taken into account. Our work n ot\nonly presents a viable set of methods for the characterizati on\nof/tildewideχbut it provides input to the theories aimed at describing\nthe magnetic relaxation in nanomagnetic particles and also a\nfeedback for future material science developments.2\nI. THEORETICAL BACKGROUND AND METHODS\nThe physically relevant quantity in hyperthermia is the\nimaginary part of the complex magnetic susceptibility, /tildewideχ, i.e.\nχ′′as the absorbed power is proportional to it. Although,\nwe recently developed a method to directly determine the ab-\nsorbed power during hyperthermia23, a method is desired to\ndetermine the full frequency dependence of /tildewideχ. This would\nnot only lead to finding the optimal irradiation frequency du r-\ning hyperthermia but it could also provide an important feed -\nback to materials development and for the understanding of\nthe physical phenomena behind the complex susceptibility i n\nferrite suspensions.\nThe generic form of the complex magnetic susceptibility of\na material reads:\n/tildewideχ(ω) =χ′(ω)−iχ′′(ω). (1)\nLinear response theory dictates that these can be transform ed\nto one another by a Hilbert transform36,37as:\nχ′(ω) =1\nπP/integraldisplay∞\n−∞χ′′(ω′)\nω′−ωdω′, (2)\nχ′′(ω) =−1\nπP/integraldisplay∞\n−∞χ′(ω′)\nω′−ωdω′, (3)\nwherePdenotes the principal value integral.\nWe note that we use a dimensionless volume susceptibility\n(invoking SI units) throughout. If a single relaxation proc ess is\npresent (similar to dielectric relaxation or to the Drude mo del\nof conduction, which yield /tildewideǫ(ω)and/tildewideσ(ω), respectively), the\ncomplex magnetic susceptibility takes the form:\nχ′(ω) =χ01\n1+ω2τ2, (4)\nχ′′(ω) =χ0ωτ\n1+ω2τ2, (5)\nwhereχ0is the static susceptibility.\nThe corresponding χ′andχ′′pairs can be constructed\nwhen multiple relaxation times are present in the descrip-\ntion of their frequency dependence. There is a general\nconsensus1,4,8,14,17,19,38–42although experiments are yet lack-\ning, that the single relaxation time description approxima tes\nwell the frequency dependence of the magnetic nanoparticle s.\nThe frequency dependence of χ′′is though to be described by\nthe relaxation time of the nanoparticles: 1/τ= 1/τN+1/τB,\nwhere the N´ eel and Brown relaxation times are related to the\nmotion of the magnetization with respect to the particles an d\nthe motion of the particle itself, respectively.\nWe used a commercial sample (Ferrotec EMG 705, nominal\ndiameter 10 nm) which contains aqueous suspensions of sin-\ngle domain magnetite (Fe 3O4) nanoparticles. We verified the\nmagnetic properties of the sample using static SQUID mag-\nnetometry; it showed the absence of a sizable magnetic hys-\nteresis (data shown in the Supplementary Information), whi ch\nproves that the material indeed contains magnetic mono-\ndomains.A. Measurements with non-resonant circuit\nAt frequencies below ∼5−10MHz the conventional\nmethods of measuring the current-voltage characteristics can\nbe used for which several commercial solutions exist. This\nmethod could e.g. yield the inductivity change for an induct or\nin which a ferrite sample is placed. However, above these fre -\nquencies the typical circuit size starts to become comparab le\nto the electromagnetic radiation wavelength thus wave effe cts\ncannot be neglected. The arising complications can be con-\nveniently handled with measurement of the Sparameters, i.e.\nthe reflection or transmission for the device under test.\nObtaining /tildewideχ(ω)is possible by perturbing the circuit prop-\nerties of some broadband antennas or waveguides while mon-\nitoring the corresponding Sparameters43(the reflected am-\nplitude,S11, and the transmitted one, S21) with a vector net-\nwork analyzer (VNA). We used two approaches: i) a droplet\nof the ferrite suspension on a coplanar waveguide (CPW) was\nmeasured and ii) about a 100 µl suspension was placed in a\nsolenoid. It is crucial in both cases to properly obtain the n ull\nmeasurement, i.e. to obtain the perturbation of the circuit due\nto the ferrite only. For the solenoid, we found that a sam-\nple holder filled with water gives no perturbation to the cir-\ncuit parameters as expected. In contrast, the CPW parameter s\nare strongly influenced by a droplet of distilled water whose\nquantity can be hardly controlled therefore performing the null\nmeasurement was impossible and as a result, the use of the\nCPW turned out to be impractical. Additional details about\nthe VNA measurements, including details of the failure with\nthe CPW based approach, are provided in the Supplementary\nInformation.\nIn the second approach, we used a conventional solenoid\n(shown in Fig. 2) made from 1 mm thick enameled copper\nwire, its inner diameter is 6 mm and it has a length of 23 mm\nwith 23 turns. The coil is soldered onto a semi-rigid copper R F\ncable that has a male SMA connector. Fig. 2. shows the equiv-\nalent circuit which was found to well explain the reflection c o-\nefficient in the DC-150 MHz frequency range (more precisely\nfrom 100 kHz which is the lowest limit of our VNA model\nRohde & Schwarz ZNB-20). The frequency dependence of\nthe wire re ce due to the skin-effect was also taken into ac-\ncount in the analysis. The parallel capacitor arises from th e\nparasitic self capacitance of the inductor and from the smal l\ncoaxial cable section. Further details about the validatio n of\nthe equivalent circuit (i.e. our fitting procedure) are prov ided\nin the Supplementary Information.\nThe frequency dependent complex reflection coefficient, Γ\n(same asS11this case), and Zof the studied circuit are related\nby44:\nΓ =Z−Z0\nZ+Z0, (6)\nwhereZ0is the 50 Ωwave impedance of the cables and Z\nis the complex, frequency dependent impedance of the non-\nresonant circuit. It can be inverted to yield Zas:Z=Z01+Γ\n1−Γ.\nThe admittance for the empty solenoid reads:\n1\nZempty=1\nR(ω)+iωL+iωC, (7)\nThe analysis yields fixed parameters for R(ω)andC, whereas\nthe effect of the sample is a perturbation of the inductivity :3\nVNA \nC\nCR\nL\nFIG. 1: Upper panel: photograph of the solenoid used in the no n-\nresonant susceptibility measurements. Lower panel: the eq uivalent\ncircuit model including a parasitic capacitor, Cdue to the small coax-\nial cable section and the self capacitance of the inductor. Rhas a\nfrequency dependence due to the skin-effect.\nFIG. 2: Upper panel: photograph of the solenoid used in the no n-\nresonant susceptibility measurements. Lower panel: the eq uivalent\ncircuit model including a parasitic capacitor, Cdue to the small coax-\nial cable section and the self capacitance of the inductor. Rhas a\nfrequency dependence due to the skin-effect.\nL→L(1+η/tildewideχ(ω)). We introduced the dimensionless filling\nfactor parameter, η, which is proportional to the volume of the\nsample per the volume of the solenoid, albeit does not equal t o\nthis exactly due to the presence of stray magnetic fields near\nthe ends of the solenoid. This parameter, η, also describes\nthat the susceptibility can only be determined up to a linear\nscaling constant with this type of measurement. In principl e,\nthe absolute value of /tildewideχ(ω)could be determined by calibrating\nthe result by a static susceptibility measurement (e.g. wit h a\nSQUID magnetometer) and by extrapolating the dynamic sus-\nceptibility to DC. It is however not possible with our presen t\nsetup asχshows a strong frequency dependence down to our\nlowest measurement frequency of 100 kHz.\nA straightforward calculation using Eq. (7) yields that\nη/tildewideχ(ω)can be obtained from the measurement of the admit-\ntance in the presence of the sample, 1/Zsample as:\nη/tildewideχ=/parenleftBig\n1\nZsample−iωC/parenrightBig−1\n−/parenleftBig\n1\nZempty−iωC/parenrightBig−1\niωL(8)\nB. Measurements with resonant circuit\nFig. 3 shows the block diagram of the resonant circuit mea-\nsurements which is the same as in the previous studies23,24.\nThis type of measurement is based on detecting the changesSource\nLR\nCT50 ΩHybrid\njunctionDetectorCM\nResonator\nFIG. 3: Left: Block diagram of the resonant measurement meth od.\nRight: The schematics of the resonator circuit. It has 2 vari able ca-\npacitors, the tuning ( CT) is used for setting the resonant frequency\nand the matching ( CM) is for setting the impedance of the resonator\nto 50Ωat resonant frequency. Detailed description is in Ref. 23\nin the resonator parameters, resonance frequency ω0and qual-\nity factor, Q. The presence of a magnetic material induces a\nchange in these parameters43–45as:\n∆ω0\nω0+i∆/parenleftbigg1\n2Q/parenrightbigg\n=−η/tildewideχ (9)\nHerein,∆ω0and∆/parenleftBig\n1\n2Q/parenrightBig\nare changes in resonator eigenfre-\nquency and the quality factor. The signs in Eq. (9) express\nthat in the presence of a paramagnetic material, the resonan ce\nfrequency downshifts (i.e. ∆ω0<0) and that it broadens (i.e.\n∆/parenleftBig\n1\n2Q/parenrightBig\n>0) when both χ′andχ′′are positive.\nThe resonator measurement has a high sensitivity to minute\namounts of samples43however its disadvantage is that its re-\nsult is limited to the resonance frequency only. Eq. (9) is\nremarkable, as it shows that the ratio ofχ′′andχ′can be\ndirectly determined at a given ω0(we use throughout the ap-\nproximation that Qis larger than 10, thus any change to ω0\ncan be considered to the first order only). Namely:\nχ′′\nχ′=−ω0∆/parenleftBig\n1\n2Q/parenrightBig\n∆ω0=−∆HWHM\n∆ω0(10)\nwhere we used that the half width at half maximum, HWHM\nis: HWHM =ω0/2Q. The broadening of the resonator profile\nmeans that ∆HWHM is positive.\nThis expression provides additional microscopic informa-\ntion when the magnetic susceptibility can be described by a\nsingle relaxation time:\n−∆HWHM\n∆ω0=χ′′\nχ′=ω0τ (11)\nE.g. when a measurement at 50 MHz returns −∆HWHM\n∆ω0= 1,\nwe then obtain directly a relaxation time of τ= 3ns. The right\nhand side of Eq. (11) can be also rewritten as ω0τ=f0/fc,\nwhere we introduced a characteristic frequency of the parti cle\nabsorption process, i.e. where χ′′has its maximum.\nThis description has an interesting consequence: it makes\nlittle sense to use tiny nanoparticles, i.e. to push τto an exces-\nsively short value (or fcto a too high value). The net absorbed4\npower reads: P∝fχ′′and when the full expression is sub-\nstituted into it, we obtain P∝f2\nfc/parenleftbigg\n1+f2\nf2c/parenrightbigg. This function is\nroughly linear with fbelowfcand saturates above it to a con-\nstant value. This means that an optimal irradiation frequen cy\nshould be at least as large as fc.\nII. RESULTS AND DISCUSSION\n/s45/s48/s46/s48/s54/s45/s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54\n/s32/s82/s101/s97/s108\n/s32/s73/s109 /s104/s99 /s39/s40 /s119 /s41/s32/s44/s32 /s104/s99 /s39/s39/s40 /s119 /s41 /s71\n/s115/s97/s109/s112/s108/s101/s45/s71\n/s101/s109/s112/s116/s121\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s32/s104/s99 /s39/s40 /s119 /s41\n/s32/s104/s99 /s39/s39/s40 /s119 /s41\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s40/s77/s72/s122/s41\nFIG. 4: The reflection coefficient for the solenoid with the sa mple\ninside, relative to the empty solenoid (upper panel). The re al and\nimaginary parts of the dynamic susceptibility as obtained f rom the\nreflection coefficients according to Eqs. (7) and (8). Note th at neither\ncomponent of /tildewideχfollows the expected Lorentzian forms.\nWe measured the reflection coefficient for the non-resonant\ncircuit,Γempty, i.e. for an empty solenoid in the 100 kHz-\n150 MHz range. The lower frequency limit value is set by\nvector network analyzer and values higher than 150 MHz are\nthought to be impractical due to water dielectric losses and\neddy current related losses in a physiological environment1–3.\nWe also measured the corresponding reflection coefficients\nwhen the ferrite suspension sample, Γsample and only distilled\nwater was inserted into it. The presence of the water refer-\nence does not give an appreciable change to Γ(data shown\nin the Supplementary Information) as expected. The differ-\nenceΓsample−Γempty is already sizeable and is shown in Fig.\n4. The dynamic susceptibility is obtained by first determini ng\nthe empty circuit parameters (details are given in the SM) an d\nthese fixed R,LandCare used together with Eqs. (7) and (8)\nto calculate /tildewideχ(ω). The result is shown in the lower panel of\nFig. 4.\nWe note that the use of Eq. (8) eliminates Rand we have\nalso checked that the result is little sensitive to about 10 %change in the value of LandC, therefore the result is robust\nand it does not depend much on the details of the measurement\ncircuit parameters. The ratio of the two components is parti c-\nularly insensitive to the parameters: Lcancels out formally\naccording to Eq. (8) but we also verified that a 20 % change\ninCleaves the ratio unaffected.\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s40/s77/s72/s122/s41/s99 /s39/s39/s47 /s99 /s39/s32/s110/s111/s110/s45/s114/s101/s115/s111/s110/s97/s110/s116\n/s32/s114/s101/s115/s111/s110/s97/s110/s116\nFIG. 5: The comparison of the ratios of χusing the non-resonant\nbroadband and the resonator based (at some discrete frequen cies) ap-\nproaches.\nFig. 5 shows the ratio of the two terms of the dynamics\nmagnetic susceptibility, χ′′(ω)/χ′(ω), as determined by the\nbroadband method and using the resonator based approach.\nThe latter data is presented for a few discrete frequencies.\nThe two kinds of data are surprisingly close to each other,\ngiven the quite different methods as these were obtained. Th is\nin fact validates both approaches and is a strong proof that\nwe are indeed capable of determining the complex magnetic\nsusceptibility of the ferrofluid sample up to a high frequenc y.\nOne expects that the signal to noise performance of the res-\nonator based approach is superior to that obtained with the\nnon-resonant method by the quality factor of the resonator46,\nwhich is about 100. In fact, the data shows just the opposite o f\nthat and the resonator based data point show a larger scatter ing\nthan the broadband approach. This indicates that the accura cy\nof the resonator method is limited by a systematic error, whi ch\nis most probably related to the inevitable retuning of the re s-\nonator and the reproducibility of the sample placement into\nthe resonator.\nThe experimentally observed dynamic susceptibility has\nimportant consequences for the practical application of hy per-\nthermia. Given that the net absorbed power: P∝fχ′′, it\nsuggests that a reasonably high frequency, f, should be used\nfor the irradiation, until other types of absorption, e.g. d ue to\neddy currents1–3, limit the operation.\nWe finally argue that the experimental observation cannot\nbe explained by an exponential magnetic relaxation either d ue\nto the rotation of magnetization (the Ne´ el relaxation) or d ue5\nto the rotation of the particle itself (the Brown relaxation ). In\nprinciple, both relaxation processes are particle size dep en-\ndent; in the nanometer particle size domain the Brown proces s\nprevails and it was calculated in Ref. 3 that for a particle di -\nameter of d= 10 nm we get τ= 300 ns (fc= 21 MHz) for\nd= 11 nm,τ= 2µs (fc= 3 MHz), and for d= 9 nm,\nτ= 50 ns (fc= 125 MHz). These frequencies would in prin-\nciple explain a significant χ′′(ω)in the1−100MHz range,\nsuch as we observe.\nHowever, a simple consideration reveals from Eqs. (4) and\n(5) that for a single exponential magnetic relaxation for ea ch\nmagnetic nanoparticle, the ratio of χ′′/χ′is astraight line as\na function of the frequency, which starts from the origin wit h\na slope depending on the distribution of the different τparam-\neters and particle sizes. Similarly, a single exponential r elax-\nation would always give a monotonously decreasing χ′(ω),\nirrespective of the particle size and τdistribution. Clearly,\nour experimental result contradicts both expectations: χ′′/χ′\nis not a straight line intersecting the origin and χ′(ω)signif-\nicantly increases rather than decreases above 20 MHz. We\ndo not have a consistent explanation for this unexpected, no n-\nexponential magnetic relaxation, which should motivate fu r-\nther experimental and theoretical efforts on ferrofluids. W e\ncan only speculate that a subtle interplay between the Ne´ el\nand Brown processes could cause this effect, whose explana-\ntion would eventually require the full solution of the equat ion\nof motion of the magnetic moment and the nanoparticles, such\nas it was attempted in Ref. 35.\nSummary\nIn summary, we studied the frequency-dependent dynamic\nmagnetic susceptibility of a commercially available ferro fluid.\nKnowledge of this quantity is important for i) determining\nthe optimal irradiation frequency in hyperthermia, ii) pro vid-\ning feedback for the material synthesis. We compare the re-\nsult of two fully independent approaches, one which is based\non measuring the broadband radiofrequency reflection from\na solenoid and the other, which is based on using radiofre-\nquency resonators. The two approaches give remarkably sim-\nilar results for the ratio of the imaginary and real parts of t he\nsusceptibility, which validates the approach. We observe a sur-\nprisingly non-exponential magnetic relaxation for the ens em-\nble of nanoparticles, which cannot be explained by the distr i-\nbution of the magnetic relaxation time in the nanoparticles .\nAcknowledgements\nThe authors are grateful to G. F¨ ul¨ op, P. Makk, and Sz.\nCsonka for the possibility of the VNA measurements and for\nthe technical assistance. Jose L. Martinez is gratefully ac -\nknowledged for the contribution to the SQUID measurements.\nSupport by the National Research, Development and Innova-\ntion Office of Hungary (NKFIH) Grant Nrs. K119442, 2017-\n1.2.1-NKP-2017-00001, and VKSZ-14-1-2015-0151 and by\nthe BME Nanonotechnology FIKP grant of EMMI (BME\nFIKP-NAT) are acknowledged. The authors also acknowledge\nthe COST CA 17115 MyWA VE action.6\n1Q. Pankhurst, J. Connolly, S. Jones, and J. Dobson, Journal o f\nPhysics D-Applied Physics 36, R167 (2003).\n2M. Kallumadil, M. Tada, T. Nakagawa, M. Abe, P. Southern, and\nQ. A. 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Kostylev, Physica E Low-Dimensional\nSystems and Nanostructures 69, 253 (2015).7\nAppendix A: Magnetic properties of the sample\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s109/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s32/s40/s101/s109/s117/s41\n/s45/s48/s46/s48/s53/s48 /s45/s48/s46/s48/s50/s53 /s48/s46/s48/s48/s48 /s48/s46/s48/s50/s53 /s48/s46/s48/s53/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s109/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s32/s40/s101/s109/s117/s41\n/s109\n/s48/s72 /s32/s40/s84/s41/s72/s121 /s115/s116/s101/s114/s101/s115/s105/s115/s61/s32/s50/s32/s109/s84\nFIG. 6: The magnetic moment of the sample, m, versus the mag-\nnetic field strength, µ0H, curve for the ferrite particle suspension.\nThe absence of a sizeable magnetic hysteresis indicates tha t this is a\nmonodomain sample. The estimate for the maximum hysteresis value\nis about 2 mT.\nThe magnetic moment versus the magnetic field strength,\nµ0H, is shown in Fig. 6 as measured with a SQUID magne-\ntometer. Notably, the sample magnetism shows a saturation\nabove 0.1 T, however it has a very small hysteresis of about\n2 mT. Common hard, multidomain ferromagnetic materials,\nwhich saturate is small magnetic fields, usually display a si g-\nnificant hysteresis. Our observation agrees with the expect ed\nbehavior of the sample, i.e. that it consists of mono-domain\nnanoparticle, which can easily align with the external mag-\nnetic field.\nAppendix B: Details of the non-resonant susceptibility\nmeasurement\nWe discuss herein how the solenoid based broadband sus-\nceptibility measurement can be performed. We first prove tha t\nthe equivalent circuit, presented in the main text, provide s an\naccurate description. The reflectivity data is shown in Fig. 7.\nWe obtain a perfect fit (i.e. the measured and fitted curves\noverlap) when we consider the equivalent circuit in the main/s45/s49/s48/s49/s82/s101/s40 /s71 /s41\n/s32/s77/s101/s97/s115/s117/s114/s101/s100\n/s32/s77/s111/s100/s101/s108/s105/s110/s103/s44/s32 /s82 /s40/s119 /s41/s44/s32 /s76 /s44/s32 /s67\n/s32/s77/s111/s100/s101/s108/s105/s110/s103/s44/s32 /s82 /s40/s119 /s41/s44/s32 /s76\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s45/s49/s48/s49\n/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s40/s77/s72/s122/s41/s73/s109/s32/s40 /s71 /s41\nFIG. 7: The reflection coefficient, Γ, and its modelling with vari-\nous equivalent circuit assumptions. The best fit is obtained when the\nequivalent circuit containing a frequency dependent wire r esistance\n(with a small DC value) was considered in addition to an induc tor\nand a capacitor. The absence of the capacitor does not give an appro-\npriate fit (dotted red curve). We note that a constant wire res istance,\nhowever with an unphysically large value, gives also an appr opriate\nfit.\ntext with parameters RDC= 15.5(2) mΩ,L= 0.62(1)µH,\nandC= 4.65(1) pF. This fit also considered the frequency\ndependency of the coil resistance due to the skin effect, who se\nDC value is RDC= 13 mΩ. We also performed the fit without\nconsidering the skin effect, which gave an unrealistically large\nRDC= 150 mΩwhile the fit being seemingly proper. A fit\nwithout considering a capacitor does not give a proper fit (do t-\nted curve in the figure): its major limitation is that it canno t\nreproduce the zero crossing of Γ, i.e. a resonant behavior in\nthe impedance of the circuit. As a result, we conclude that th e\nequivalent circuit in the main text provides a proper descri p-\ntion of the measurement circuit and that the fitted parameter s\ncan be used to obtain the complex susceptibility of the sampl e,\nas we described in the main text.\nFig. 8. demonstrates that the presence of the sample gives\nrise to a significant change in the reflection coefficient, Γ,\nwhereas the reflection is only slightly affected by the prese nce\nof the water (maximum Γchange is about 0.2 % below 150\nMHz) and its effect is limited to frequencies above 150 MHz.\nProbably, the inevitably present stray electric fields (due to the\nparasitic capacitance of the solenoid) interact with the wa ter\ndielectric, which results in this effect. The stray electri c fields\nand the parasitic capacitance become significant at higher f re-\nquencies: then there is a significant voltage drop across the\nsolenoid inductor coil, thus its windings are no longer equi -\npotential and an electric field emerges.\nAppendix C: Details of the susceptibility detection using a CPW\nThe coplanar waveguide or CPW is a planar RF and mi-\ncrowave transmission line whose impedance is 50 Ωat a wide\nfrequency range43,44. The CPW can be thought of as a halved\ncoaxial cable which makes the otherwise buried electric and8\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s45/s48/s46/s48/s54/s45/s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s71\n/s119/s97/s116/s101/s114/s47/s115/s97/s109/s112/s108/s101/s45/s71\n/s101/s109/s112/s116/s121\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s77/s72/s122/s41/s71\n/s115/s97/s109/s112/s108/s101/s45/s71\n/s101/s109/s112/s116/s121 \n/s32/s82/s101/s97/s108\n/s32/s73/s109\n/s71\n/s119/s97/s116/s101/s114/s45/s71\n/s101/s109/s112/s116/s121 \n/s32/s82/s101/s97/s108\n/s32/s73/s109\nFIG. 8: Reflection coefficients, Γ, with respect to the empty circuit.\nΓwater denotes the reflection coefficient when the solenoid is filled\nwith water in a quartz tube. Note that the sample gives rise to a signif-\nicant change in the reflection below 100 MHz, whereas the pres ence\nof water (dashed lines) only slightly changes it above this f requency.\nmagnetic fields available to study material parameters, ess en-\ntially as a small piece of an irradiating antenna43,44. However\nas we show below, the inevitable simultaneous presence of th e\nelectric and magnetic field hinders a meaningful analysis.\nS11 S21R* RS R* L* LS L* \nC* CS C* G* GS G* CPW section CPW section Sample drop\nPort 2 Port 1 \nFIG. 9: Upper panel: The equivalent circuit of a CPW section w ith\na sample on the top. LS,RS,CS, andGSare effective inductance,\nseries resistance, capacitance, and shunt conductance of t he small\nCPW section which contains the sample, respectively. L∗,R∗,C∗,\nandG∗are corresponding distributed circuit parameters which ar e\nnormalized to unit length. In an ideal waveguide R∗= 0 andG∗=\n0, and also Z0=/radicalbig\nL∗/C∗. Lower panel: Photo of CPW with a\ndroplet of the sample. Port 1 is labeled with green and Port 2 i s with\nred tape.\nWe show the U shaped CPW section use in our experiments\nalong with the equivalent circuit of the CPW in Fig. 9. As\nthis device has two ports, one can measure the frequency de-\npendent complex reflection ( S11) and transmission ( S21) co-\nefficients simultaneously with a VNA. We placed the sample\ndroplet on the top of the gap section of the CPW between the\ncentral conductor and the grounding side plate, where the RF\nmagnetic field component is the strongest. The presence ofthe sample influences all parameters for the waveguide sec-\ntion where it is placed: the inductance Ls, capacitance Cs, the\nseries resistance Rs, and the shunt inductance Gs. All 4 pa-\nrameters are extensive, i.e. these depend on the quantity of the\nsample and one can express the inductivity as Ls=L0(1+η/tildewideχ)\nwhereL0is the inductivity of the CPW section which is af-\nfected by the sample, ηis the relevant filing factor that is di-\nmensionless and /tildewideχis the complex magnetic susceptibility.\nTheSparameters for such a device read47:\nS11,sample=Rs+iωL s+Z0\n1+Z0(Gs+iωC s)−Z0\nRs+iωL s+Z0\n1+Z0(Gs+iωC s)+Z0(C1)\nand\nS21,sample=2Z0\n1+Z0(Gs+iωC)\nRs+iωL s+Z0\n1+Z0(Gs+iωC s)+Z0(C2)\nWe calibrated the system that without sample (empty case)\nso that the VNA shows 0 for S11and real 1 to S21on the\nentire frequency range. During the calibration, port 1 of th e\nVNA was connected to the CPW and we assembled and disas-\nsembled the necessary calibrating elements (OPEN, SHORT,\nMATCH) onto port 2 and the second end of the CPW. There-\nfore the VNA reference plane was this end of the CPW. The\ncalibration could be achieved down to Γ<5·10−4(not\nshown).\nMagnetic field \nElectric fieldε0εr ceramic\nFIG. 10: Cross section of a coplanar waveguide showing the el ectric\nand magnetic field. Note that magnetic field is present around the\ncentral conductor and that there is a significant electric fie ld in the\ntwo gaps between the central conductor and the neighboring g round\nplates.\nWe first measured the reflection transmission coefficient\nchange under the influence of a small distilled water droplet\nwith approximately the same size as that of the sample. We\nobserve a Γchange up to about 5 % (maximum value at 150\nMHz, data not shown) for both coefficients, which is a size-\nable value. We note that the solenoid investigation, which w e\ndiscussed in the main paper, gave a change in Γfor the influ-\nence of water of about 0.2 %. Clearly, this larger sensitivit y of\nthe measurement for water is due to the electric field which is\nsignificant for the CPW and is much smaller for the solenoid.\nIt is even more intriguing that the effect of the sample is\nprimarily to shift the real parts of both S11andS21by the same\namount even at DC, while leaving the imaginary components\nunchanged (data not shown). For our typical droplet size, su ch9\nas this shown in Fig. 9, this amount is ∆Γ≈0.04. Rewriting\nEqs. (C1) and (C2) in the zero frequency limit, yields:\nS11,sample, DC=Rs+Z0\n1+Z0Gs−Z0\nRs+Z0\n1+Z0Gs+Z0, (C3)\nS21,sample, DC=2Z0\n1+Z0Gs\nR+Z0\n1+Z0Gs+Z0. (C4)\nWe find that in the reasonable limit of Rsample/lessorsimilarZ0, the\ninfluence of Gsdominates and that the experimental finding\nimplies the presence of a significant shunt conductance due t o\nthe sample. We speculate that this may be due to the presence\nof excess OH−ions in the ferrofluid (the Ferrotec EMG 705\nhas a pH of 8-9), which conduct the electric current. Again,\nthis effect is the result of the finite electric field across th e gap\nof the CPW, where we place the sample.\nThe two effects, the presence of a significant capacitance\ndue to water and a shunt inductance due to the conductivity\nof the ferrofluid, occur simultaneously when using a CPW for\nthe measurement. In fact, the effect of these factors domina te\nthe reflection/transmission. This means that determining t he\nmagnetic susceptibility for a case when a finite electric fiel d is\npresent, proves to be impractical.\nAppendix D: Additional details on the theory of resonators\nCoil+Sample Trimmer \ncapacitors \nFIG. 11: Photograph of the radiofrequency resonator. The sa mple,\nmeasurement coil and the two trimmer capacitors are indicat ed.\nFig. 11. shows a photograph of the radiofrequency res-\nonator circuit which was used in the studies. Note the presen ce\nof the two trimmer capacitors, which act as frequency tuning\nand impedance matching elements.\nThe following equation was used in the main text to deter-\nmine the relation between resonator parameters and the mate -\nrial properties:\n∆ω0\nω0+i∆/parenleftbigg1\n2Q/parenrightbigg\n=−η/tildewideχ (D1)We note that the - sign before the imaginary term on the left\nhand side varies depending on the definition of the sign in the\ncomplex response function /tildewideχ. We use the convention of Ref.\n43 where /tildewideχ=χ′−iχ′′which results in the + sign in Eq. (D1).\nThe factor 2 in Eq. (9) may seem disturbing but it is the di-\nrect consequence of the Qfactor definition: Q=FWHM/ω0,\nwhere FWHM is the full width at half maximum of the res-\nonance curve (in angular frequency units). Thus 1/2Q=\nHWHM/ω0, where HWHM is the half width at half maximum\nof the resonance curve. We also recognize that a Lorentzian\nshaped resonator profile can be expressed as1\n(ω−ω0)2+1/τ2,\nwhereτis the time constant of the resonator and τ= 2Q/ω0.\nThis also means that HWHM = 1/τ.\nThis allows to express the above equation in a more com-\npact way by introducing the complex angular frequency of the\nresonator:\n/tildewideω=ω0+iω0\n2Q=ω0+i1\nτ. (D2)\nIt is interesting to note that the complex Lorentzian linesh ape\nprofile is proportional to 1/i/tildewideω. It then follows from Eq. (D2)\nthat Eq. (D1) can be expressed as:\n∆/tildewideω\nω0=−η/tildewideχ (D3)\nwhere∆/tildewideωis the shift (or change) of (the complex) /tildewideω.\nFig. 12 shows the changes in the reflection curves at the\nresonant method.\nFrequency (MHz)60 60.5 61 61.5 62|Γ|2\n00.20.40.60.81\nEmpty\nSample\nFIG. 12: The reflection curves at the resonant method. The shi ft of\nthe resonance frequency due to the sample (red) is clearly vi sible."    },    {        "title": "2001.10935v1.Metamaterials_and_Cesàro_convergence.pdf",        "content": "arXiv:2001.10935v1  [physics.class-ph]  6 Jan 2020Metamaterials and Ces` aro convergence\nYuganand Nellambakam and K. V. S. Shiv Chaitanya∗\nDepartment of Physics, BITS Pilani, Hyderabad Campus,\nJawahar Nagar, Shamirpet Mandal, Hyderabad, India 500 078.\nIn this paper, we show that the linear dielectrics and magnet ic materials in matter obey a special\nkind of mathematical property known as Ces` aro convergence . Then, we also show that the analyti-\ncal continuation of the linear permittivity & permeability to a complex plane in terms of Riemann\nzeta function. The metamaterials are fabricated materials with a negative refractive index. These\nmaterials, in turn, depend on permittivity & permeability o f the linear dielectrics and magnetic ma-\nterials. Therefore, the Ces` aro convergence property of th e linear dielectrics and magnetic materials\nmay be used to fabricate the metamaterials.\nI. INTRODUCTION\nThe discovery of artificial media, that is, metamaterials in the last de cade, has given rise to different\nkinds of phenomena such as perfect lens [1], Metamaterial Antenna s [2], Clocking Devices [3], Acoustic\nMetamaterials [4], Seismic Metamaterials [5] which are not exhibited by e xisting natural materials. The\nmaterials which are characterizedby negative refractive index met amaterials. The seminal paper on negative\nrefractiveindexmaterialsforthenon-dissipativemediumswasprop osedVeselago[6], wherebothpermittivity\nand permeability are simultaneously negative. Further, he has show n electromagnetic wave propagation\nin negative refractive index materials exhibits a unique property suc h as phase velocity is antiparallel to\nthe direction of energy flow, the reversal of the Doppler effect an d Cerenkov radiation. In an isotropic\ndielectricmagnetic medium with dissipation, a general condition for ph ase velocity directed oppositely to the\npower flow was derived by Lakhtakia [7–9]. In the same paper, it has b een shown that the real parts of both\nthe permittivity and the permeability need not be both negative. Vala nju et al. [10] have shown that the\ngroup fronts refract positively even when phase fronts refract negatively for a negative index material. In the\nreferences [11–13], the relationship between the Kramers-Kronig relations and negative index of refraction is\ninvestigated. From the above discussion, it is clear that for materia l in a dispersive medium, the refractive is\ncomplex. Therefore, the refractive index being complex constrain ts permittivity and permeability also to be\ncomplex. The refractive index is defined as nr=√ǫrµr, wherenris refractive index, ǫ=ǫ0ǫris permittivity\nhereǫ0= 8.85×10−12C2\nNM2is permittivity of free space, ǫris relative permittivity and µ=µ0µrhere\nµ0= 4π×10−7N/A2permeability of free space and µris relative permeability. For the sake of brevity, we\ntakeǫ0=µ0= 1 throughout this paper.\nThe electricfield inside asphereofhomogeneouslineardielectric mate rialis placedin anotherwiseuniform\nelectric field E0can be calculated by two approaches, first as a boundary value pro blem, and the second\n∗chaitanya@hyderabad.bits-pilani.ac.in2\nmethod is by using successive approximations [16]. But, we encounte r an issue when using the method of\nsuccessive approximations is that the result is not valid for all values of permittivity. A similar situation\narises when a linear spherical magnetic material placed in a uniform ma gnetic field B0, which can also be\ncalculatedby twoapproaches,first asaboundaryvalue probleman d the secondmethod is byusingsuccessive\napproximations [16]. Therefore, in this paper, we address this prob lem by showing that linear dielectrics and\nmagnetic materials in matter obey a special kind of mathematical pro perty is known as Ces` aro convergence.\nThis Ces` aro convergence enables to extend the results to all valu es of permittivity by analytical continuation\nto a complex plane in terms of Riemann zeta function. The metamater ials are fabricated materials with\na negative refractive index. These materials, in turn, depend on pe rmittivity & permeability of the linear\ndielectrics and magnetic materials. Therefore, the Ces` aro conve rgence property of the linear dielectrics and\nmagnetic materials may be used to fabricate the metamaterials\nThe paper is organized as follows: in section II, we give a brief introdu ction to the origin of the problem\nthat is the electric field inside a sphere of homogeneous linear dielectr ic and spherical magnetic materials\nplaced in a uniform magnetic field and two approaches. In section III , we discuss our solution in terms of\nCes` aro convergence. In section IV, we show that the Ces` aro c onvergence leads to the analytical continuation\nto a complex plane in terms of Riemann zeta function. Finally, we conclu de the paper in section V.\nII. LINEAR PERMITTIVITY & PERMEABILITY\nThe electricfield inside asphereofhomogeneouslineardielectric mate rialis placedin anotherwiseuniform\nelectric field E0is given by [16]\nE=3\nǫr+2E0 (1)\nwhereE0is the inside field. One can also arrive at the same result by the following method of successive\napproximations that is by considering the initial field inside the sphere isE0, which give rise to the polar-\nizationP0=ǫ0χeE0. This polarization P0generates a field of its own, say, E1=−1\n3ǫ0P0=−χe\n3E0, which\nin turn modifies the polarization by an amount P1=ǫ0χeE1=−ǫ0χ2\ne\n3E0, which further generates the field\nE2=−1\n3ǫ0P1=χ2\ne\n9E0, and so on. Therefore, the resulting field is given by [16]\nE=E0+E1+E2+ =∞/summationdisplay\n0/parenleftBig\n−χe\n3/parenrightBign\nE0 (2)\nIt is clear that the equation (2) is a geometric series and summed exp licitly:\nE=1\n1+χe\n3E0=3\nǫr+2E0 (3)\nwhereǫr= 1 +χewhich agrees with equation (1). Readers should note that geometr ic series of the form\n/summationtext∞\n0xn=1\n1−xis valid only for values of −1< x <1, here−1 and 1 are also excluded. Therefore, in the\ncase of equation (3) requires that χe<3 else the infinite series diverges. But from equation (1) the result is\nsubject to no such restriction hence equations (1) and (2) are inc onsistent. For example χe= 3 the series\nE=E0+E1+E2+ =∞/summationdisplay\n0/parenleftBig\n−χe\n3/parenrightBign\nE0 (4)3\nreduces to\nE= (1−1+1−1+1.......)E0 (5)\nIn literature, the series 1 −1+1−1+1.......is known as Grandi’s series and is divergent.\nSimilarly, the Magnetic field of a linear spherical magnetic material plac ed in an uniform magnetic field\nB0is given by [16]\nB=µH=3B0\n(2µ0+µ)=/parenleftbigg1+χm\n1+χm/3/parenrightbigg\nB0. (6)\nOne can also arrive at the same result by the following method of succ essive approximations that is we\nconsider the initial field inside the sphere is B0magnetizes the sphere: M0=χmH0=χm\nµ(1+χm)B0. This\nmagnetization sets up a field within the sphere B1=2\n3µ0M0=2\n3χm\n1+χmB0=2\n3κB0, whereκ=χm\n1+χm, which\nin turn modifies the magnetizes of sphere by an additional amount M1=κ\nµ0B1. This sets up an additional\nfield in the sphere B2=2\n3µ0M1=2\n3κB1=/parenleftbig2κ\n3/parenrightbig2B0,and so on. Therefore the resulting field is given by\n[16]\nB=B0+B1+B2+... (7)\n=B0+(2κ/3)B0+(2κ/3)2B0+.... (8)\n= [1+(2 κ/3)+(2κ/3)2+....]B0 (9)\n=B0\n(1−2κ/3)=3B0\n3−2χm/(1+χm)(10)\n=(3+3χm)B0\n3+3χm−2χm=/parenleftbigg1+χm\n1+χm/3/parenrightbigg\nB0. (11)\nFor the value of χm=−3\n5givesκ=−3\n2the equation (1) reduces to\nB= (1−1+1−1+1.......)B0 (12)\nAgain we arrive at Grandi’s series. It should be noted that equations (6) and (11) are inconsistent.\nIII. CES `ARO CONVERGENCE\nIt should be noted that equations (1) and (3)and equations (6) an d (11) are inconsistent as equation (1)\nand equation (6) are valid for the all values of χeandχmrespectively. Hence, the equations (3) and (11)\nalso should be valid for all values of χeandχmrespectively. One of the cases, where equations (3) and (11)\ndiverges for χe=−3 andχm=−3\n5and the series generated are given by equations (5) and (12) resp ectively\nare known as Grandi’s series in literature. The Grandi’s series Q is in gen eral is a divergent series is given\nby\nQ= 1−1+1−1+1.......=+∞/summationdisplay\nn=0Qj=+∞/summationdisplay\nn=0(−1)n(13)\nBut, this series converges to a value of1\n2for a special kind of convergence known as the Ces` aro converge nce.\nIn a geometric series the sequence of partial sums convergesto a real number, for a series which obeys Ces` aro4\nsum the average of partial sums converges to a real number. The Ces` aro sum is defined as\nσn=n−1/summationdisplay\nj=0(1−j\nn)aj=s0+s1+..sn−1\nn=1\nnn−1/summationdisplay\nj=0sj (14)\nA series/summationtextn\nj=0ajis called Ces` aro summable satisfies the following theorem:\nTheorem 1 Suppose that/summationtextn\nj=0ajis a convergent series with sum, say L. Then/summationtextn\nj=0ajis Ces` aro\nsummable to L.\nlim\nn→∞sn=L∈R⇒lim\nn→∞σn=L∈R. (15)\nThe proof is given in [14]. Following are the properties of Ces` aro sums : If/summationtext\nnan= A and/summationtext\nnbn= B are\nconvergent series, then\ni. Sum-Difference Rule:/summationtext\nn(an±bn) =/summationtext\nnan±/summationtext\nnbn= A±B\nii. Constant Multiple Rule :/summationtext\nnc an= c/summationtext\nnan= cA for any real number c.\niii. The product of AB=/summationtext\nnan/summationtext\nnbnis also as Ces` aro sums.\nThe sequence in equation (13) has two possibilities; one case is seque nce ends with an even number of\nterms, and the other argument is series ends with an odd number of terms. For the even of terms, the partial\nsums add to s2n= 0, and for the odd number of terms, the partial sums add to s2n+1= 1, then the average\nof the even and odd is 1 /2. Then, by followings the definition (14)\nσ2n+1=1\n2n+1(1+0+1+0+ ...+1) =n+1\n2n+1(16)\nσ2n=1\n2n(1+0+1+0+ ...+0) =1\n2(17)\nThe readersshould not that the terms of sequence in the equation s (16) and (17) aregeneratedfrom equation\n13 as\nQ0= 1,;Q0+Q1= 0,;Q0+Q1+Q2= 1,;Q0+Q1+Q2+Q3= 0,...... (18)\nThen by applying the theorem1 we get\nlim\nn→∞σ2n= lim\nn→∞σ2n+1=1\n2. (19)\nThe refractive index nis defined as nr=√ǫrµr, whereǫris relative permittivity and µrrelative perme-\nability. Since the ǫris relative permittivity and µrrelative permeability both satisfy the sequence given in\nequation (13) we get refractive index to be\nn2\nr=µν= (1−1+1−1+....)2(20)\nThe sequence (20) satisfies the following identity\nnr= (1−1+1−1+....)2= (1−1+1−1+...)x(1−1+1−1+...)\n= 1−2+3−4+5+.....∞/summationdisplay\nj=0(−1)jj. (21)5\nBy applying the third property of Ces` aro sums that is the product ofAB=/summationtext\nnan/summationtext\nnbnis also as Ces` aro\nsums one gets\nnr= (1−1+1−1+....)2= (1/2)2= 1/4. (22)\nThus, we have shown that the equations (1) and (3) and the equat ions (6) and (11) are valid for the all values\nofχelinear permittivity and χmlinear permeability as the equations (3) and (11) obeys Ces` aro con vergence,\nin the processes linear permittivity & permeability becomes complex an d we will discuss in next section.\nIV. ANALYTICAL CONTINUATION TO RIEMANN ZETA( ζ) FUNCTION\nAs pointed out in the previous section linear permittivity & permeability in equations (1) and (6) will\nbecome complex as , as Ces` aro sum is defined for complex number [1 5]. For example consider the geometric\nseries in equation (2) by treating χeto be complex then the value of χe=−3 is falls inside the domain of\nconvergence [15]. This process is known as the analytical continuat ion. One of the well-known examples of\nanalytical continuation is the Riemann Zeta ( ζ) function and is given by\nζ(s) =+∞/summationdisplay\nk=11\nks= 1+1\n2s+1\n3s+1\n4s+1\n5s+.... (23)\nThe Riemann Zeta is defined for the value s >1, and for all other values below less than or equal to one on\nthe real line, it diverges. Riemann has extended the domain of conve rgence of ζfunction to the entire real\nline by allowing s=σ+iωto be complex and relating it to the Dirichlet eta function\nη(s) =+∞/summationdisplay\nk=1(−1)k\nks= 1−1\n2s+1\n3s−1\n4s+1\n5s+...... (24)\nand this series converges for the value of s >1 and diverges for the value of s≤1. By allowing sto be\ncomplex the domain of Dirichlet eta function (24) is extended to entir e complex plane where the divergent\nseries obeys the Ces` aro convergence. For example when s= 0 we recover Grandi’s series (13) and s= 1 we\nrecover the series in equation (21). The analytic continuation of ζ(s) in terms of Dirichlet Eta Function η(s)\nis given by\nη(s) = 1−1\n2s+1\n3s−1\n4s+1\n5s+.... (25)\n=/parenleftbigg\n1+1\n2s+1\n3s+1\n4s+1\n5s+../parenrightbigg\n−/parenleftbigg2\n2s+2\n4s+2\n6s+2\n8s+../parenrightbigg\n=ζ(s)−1\n2s−1ζ(s) = (1−21−s)ζ(s) (26)\nFor the value of s= 0 we recover the (5) and (12) respectively whose Riemann Zeta fu nction values are\nζ(0) =−1/2 and refractive index in equation (22) to be ζ(0)2= 1/4.\nV. CONCLUSION\nIn this paper, we have shown that the linear dielectrics and magnetic materials in matter obey a special\nkind of mathematical property known as Ces` aro convergence. T hen, we have also shown that the analytical6\ncontinuation of the linear permittivity & permeability to complex plane in terms of Riemann zeta function.\nThe metamaterials are fabricated materials with a negative refract ive index. These materials, in turn,\ndepend on permittivity & permeability of the linear dielectrics and magn etic materials. Therefore, the\nCes` aro convergence property of the linear dielectrics and magne tic materials may be used to fabricate the\nmetamaterials.\nAcknowledgments\nKVSSC acknowledges the Department of Science and Technology, G ovt of India (fast-track scheme (D. O.\nNo: MTR/2018/001046)), for financial support.\n[1] J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).\n[2] Kamil, Boratay Alici, Ekmel ¨ ozbay Physica Status Solid i B. 244 (4): 11921196\n[3] J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 178 0 (2006).\n[4] Shu Zhang, Leilei Yin, and Nicholas Fang, Phys. Rev. Lett . 102, 194301, (2009)\n[5] M. Brun, S. Guenneau, and A. B. Movchan, Appl. Phys. Lett. 94, 061903 (2009)\n[6] V. G. Veselago, Sov Phys Usp, 10, 509, (1968).\n[7] R. A. Depine, A. Lakhtakia, Microw. Opt. Technol. Lett., Vol. 41, pp. 315-316 (2004)\n[8] MartinWMcCall, AkhleshLakhtakiaandWernerSWeiglhof er,European JournalofPhysics, Volume23, Number\n3 (2002)\n[9] T. G. Mackay and A. Lakhtakia, Phys. Rev. B 79, 235121 (200 9)\n[10] P. M. Valanju, R. M. Walser, and A. P. Valanju, Phys. Rev. Lett. 88, 187401 (2002)\n[11] Alkim Akyurtlu, Adil-Gerai Kussow,Phy Rev A 82, 055802 (2010)\n[12] K.-E. Peiponen, V. Lucarini, E. M. Vartiainen, and J. J. Saarinen, Eur. Phys. J. B 41, 61 (2004).\n[13] P. Kinsler, Phys. Rev. A 79, 023839 (2009).\n[14] E. Ces` aro, Bull. Sci. Math., 14 : 1 (1890) pp. 114120\n[15] Stephen Semmes, arXiv:1008.2467v1 [math.CA] (2010)\n[16] David J. Griffiths, ”Introduction to Electrodynamics”, Pearson, Cambridge University Press (1981)"    },    {        "title": "2001.11389v2.Computational_Search_for_Magnetic_and_Non_magnetic_2D_Topological_Materials_using_Unified_Spin_orbit_Spillage_Screening.pdf",        "content": "1 \n Computational Search for Magnetic and Non -magnetic 2D Topological \nMaterials using Unified Spin -orbit Spillage  Screening  \nKamal Choudhary1, Kevin F. Garrity1, Jie Jiang2, Ruth Pachter2, Francesca Tavazza1 \n1 Materials Science and Engineering Division, Nationa l Institute of Standards and Technology, Gaithersburg, \nMaryland 20899, USA.  \n2 Materials Directorate, Air Force Research Laboratory, Wright –Patterson Air Force Base, Ohio45433, USA.  \n \nAbstract  \nTwo-dimensional topological  materials  (TMs)  have a variety of properties that make them \nattractive for applications including spintronics an d quantum comput ation . However, t here are \nonly a few such experimentally  known  materials . To help discover new 2D TMs, w e develop a  \nunified and  computationally  inexpensive  approach to identify  magnetic and non -magnetic  2D \nTMs, including  gapped and semi -metallic topological classifications, in a high -throughput way  \nusing density functional theory -based spin -orbit spillage,  Wannier -interpolation, and related \ntechniques . We first compute the spin-orbit spillage  for the ~1000 2D materials  in the JARVIS -\nDFT data set (https://www.ctcms.nist.gov/~knc6/JVASP.html  ), resulti ng in 122 materials with high -\nspillage values . Then, we use Wannier -interpolation to carry -out Z 2, Chern -number, anomalous \nHall conductiv ity, Curie temperature, and edge state calculations  to further support the prediction s.  \nWe identify various topological ly non -trivial classes  such as quantum spin -hall insulators (QSHI), \nquantum anomalous -hall insulators (QAHI), and semimetals. For a few predicted materials, we \nrun G0W0+SOC and DFT +U calculations. We find that as we introduce many -body effects, only \na few materials retain non -trivial band -topology, suggesting the importance of high -level DFT \nmethods in predicting 2D to pological materials. However, as an initial step, the automated spillage \nscreening and Wannier -approach provide useful predictions  for finding new topological materials  \nand to  narrow down candidates for experimental synthesis and characterization . \n \n \nCorresponding author:  Kamal  Choudhary (E -mail: kamal.choudhary@nist.gov ) 2 \n 1 Introduction  \nIn recent  years, there has been a huge upsurge in  topological  materials  research , following the \npredictions and observations of Dirac, Weyl , and Ma jorana fermions in condensed matter \nsystems1,2. Several classes of topological materials have been propose d for applications in error -\nreduced quantum computing3-6, or as high mobility and dissip ationless conductors . While there \nhave been several recent detailed  screening  efforts  for 3D  non-magnetic topological materials7-17, \nsuch systematic search es for 2D materials are still developing18-20, especially for magnetic systems . \nNevertheless , 2D materials could be more important than 3D ones because of the ir unique potential \nas miniaturized device s and their tunabil ity via layering or functionalization21.  \n2D topo logical insulating phases can be classified into two primary types: quantum spin Hall \ninsulators (QSHI22), which have time -reversal  symmetry (TRS),  and quantum anomalous Hall \ninsulators (QAHI23,24), which lack TRS . QSHI s, characterized by  a Z2 invariant , have an insulating \nbulk and Dirac cone edge features. E xamples include  graphene, silicene25, germanene26, stanine27, \nand 1T’ metal dichalcogenides28,29. Quantum anomalous Hall insulators  (QAH I), characterized by \na Z invariant known as the Chern number, are mag netic materials with a bulk gap and quantized \nconducting edge channels, even in the absence of an external m agnetic field . Thus far, QAHI -like \nbehavior has been  observed experimentally in very few systems : Cr,V doped (Bi,Sb ) 2Te323  and \nMnBi 2Te430-33, under hi ghly controlled conditions . While there have been many theoretical works \npredicting QAHIs in 2D materials, su rfaces, or  interfaces, a systematic investigation of monolayer \n2D materials is lacking . Some previously explored examples include transition metal halide s such \nas CoBr 2, FeCl 3, NiRuCl 6, V2O3, FeCl 3, RuCl 334-39.   3 \n Compared to gapped topological systems, semimetals in two -dimensions with spin -orbit coupling \nare relatively under -explored. For example, g raphene i s a well -known example of a two -\ndimensional semimetal; however, the Dirac point in graphene  is not robust to t he addition of a \nfinite amount of spin -orbit coupling22. This splitting of s ymmetry -protected band crossings  by \nspin-orbit coupling occurs for most crossings protected by two-dimensional point group \nsymmetries . Also, u nlike in three dimensions, where Weyl points can occur at generic points in \nthe Brillion  zone,  Weyl points will no t occur  generically in two dimensions  without the tuning of \nsome external parameter40. However, it is possible for non -symmorphic symmetries to protect \nDirac or Weyl points in two -dimensions41, and we find several examples of this in our work.  While \n2D insulting topolog ical materials are relatively easy to classify  using topological  indices such as \nZ2 and the Chern number , find ing topological semimetals is non-trivial. Hence, having a universal \nstrategy for screening both topological insulator s and semimetals is highly d esirable.  \nIn our previous work8, we d iscovered several classes of non -magnetic topo logical 3D materials \nusing spin -orbit spillage42 to pe rform the initial screening step . The spillage technique is based on \ncomparing density functio nal theory (DFT) wave -functions obtained with - and without spin -orbit \ncoupling.  Materials with h igh spillage value  (discussed  later)  are considered non -trivial.   In this \npaper, we extend this approach to screen the JARVIS -DFT 2D database to search for topological \ninsulators and  semimetals, with and without TRS . The JARVIS -DFT database i s a part of Materials \nGenome Initiative  (MGI) at National Institute of Standards  and Technology  (NIST ) and contains \nabout 40000 3D and 1000 2D materials with their DFT -computed str uctural, energetics43, elastic44, \noptoe lectronic45, thermoelectric46, piezoelectric, dielectric, infrared47, solar -efficiency48,49, and \ntopological8 properties.  Most of the 2D  crystal structures  we consider  are derived from  structures \nin the experimental Inorganic Crystal Structure Database  (ICSD) database22, implying  that most  4 \n of them  should be experimentally synthesizable. In this work, we screen materials from the \nJARVIS -DFT 2D database , searching for low bandgap  materials with  high-spillage8 values , \nidentifying candidate  2D TMs .  Then , for insulati ng compounds, we systematically carry out Z 2 \ncalculations for  non-magn etic materials and Chern number calculations for magnetic materials.  \nFor metals, we search for band crossin gs near the Fermi level.  We also predict surface  (2D)  and \nedge (1D)  band -structure s, Curie temperature s, and anomalous Hall conductivity.  For a subset of \npredicted materials, we run G 0W0+SOC and DFT+U calculations. We find that as we introduce  \nimprove d treatments of  many -body effects, only a few materials retain a non-trivial band -topol ogy, \nsuggesting the importance of high er-level electronic structure  methods in predicting 2D \ntopological materials. How ever, as an initial step, the automated spillage s creening and Wannier  \nanalysis provide useful predictions of potential topological mater ials, narrow ing down candidate  \nmaterials  for experimental synthesis and characterization.  \n \n \n2 Results and discussion  \nAs mentioned above, the spin -orbit spillage criter ion is applicable for both magnetic/non -magnetic \nand metallic/insulating class es of mater ials. A flow -chart describing our computational search \nusing the spillage as well as the traditional Wanni er-based method is shown in Fig. 1 . Starting \nfrom 963 2D materi als in the JARVIS -DFT database,  we first screen for materials with \nOptB88 vdW bandgap s < 1.5 eV , because SOC -induced band -invers ion is limited to the magnitude \nof SOC . This leads to 506 materials. Then we carry out spin-polarized calculation s with and \nwithou t SOC, and compute spillage using Eq.1. Setting a threshold of 0.5, we find 1 22 materials 5 \n with high -spillage. Note that for ma gnetic materials , we also screen in terms of the magnetic \nmoment, selecting only cases with magnetic moment  > 0.5 B. As a computat ional note, for the \nspin-polarized calculations , we set the initial magnetic moment to a high initial value of 6  B per \natom, to search for high spin configurations . After this initial screening, we find 19 magnetic -\ninsulating, 40 magnetic -metallic, 10 non -magnetic insulating, and 53 non -magnetic metallic \nmaterials.  In Table. 1, we present selected examples of each class of topol ogical materials that we \nconsider . The f ull list is given in the supplementary information.  Clearly, metallic topological \ncandidat es outnumber insulators. Note that previous 2D topological material search es were mostly  \nlimited to insulators, but our approa ch can be extended to semimetals  as well. Some of the \npredicted materials have already been experimentally synthesized , including  Bi2TeI50, RuCl 351, \nFeTe52 etc., but the experimental confirmation of their topological properties is  still ongoing .   \nIn the remaining part of the discussion , we analyze the overall distribution of topological  material s, \nand then we focus on individual topological classes , exploring a few examples. Information for \nother similar materials is provided  in the JAVRIS -DFT d atabase . Finally , we discuss a few cases \nof DFT+U and G0W0+SOC calculations  as a way to unders tand the limits of semi -local DFT \ncalculations as screening tools for topol ogical properties.  \n \n \n \n 6 \n  \n \nFig. 1 Flow -chart showing the screening and analysis methodol ogy. \n \n \n \n \n \n \n \n \n7 \n 2.1 Spin -orbit spillage analysis  \n \n \nFig. 2 Spillage assisted screening of 2D top ological materials  in JARVIS -DFT database . a) \nspillage distribution of materials . Materials with spillage ≥ 0.5 are likely to be topological , b) \nmagnetic moment distribution for high -spillage materials , c) bandgap -distribution , d) chemical \nprototype distribution . Zero bandgap materials should be topological -semimetals., e) space -group  \ndistribution for hi gh spillage materials, f) spillage of bulk  (3D)  vs monolayers  (2D)  for non -\nmagnetic systems.   \n \nIn Figure 2 we show the spillage -based distribution for two-dimensional materials.  We find that \n122 materials have high spillage  (with a threshold 0.5) . The spil lage criteri on does, therefore , \neliminate  many  materials , as shown in Fig. 2a.  The 122 selected  materials include both magnetic  \nand non-magnetic  materials, as well as both metals and ins ulators,  and contains examples with a \n8 \n variety of chemical prototypes a nd crystal structure s. The magnetic moments of topological \nmaterials range  from 0 to 6 B, as shown in Fig. 2b. The most common topological 2D material  \ntypes  are AB 2, ABC, AB 3, and AB. Note that in experimental synthesis it might be easier to \nsynthesize simple  chemical prototype s such as AB or 1:1 structure s. Hence, having a variety of \nchemical prototypes provides a vast amount of opportunity for synthesis. We find that most of the \nhigh-spillage materials belong to highly symmetric space  groups,  as shown in  Fig. 2e . This is \nconsistent with our previous 3D top ological materials search. A comparison of 3D and 2D spillage \nin 2f indicates that the 2D -monolayer spillage values are generally lower than their 3D-bulk \ncounterpart s, but the trend is rather weak . This may be explained in part by quantum confinement \nin 2D -monolayers tending to increase band gaps.   \nTable 1. Examples of various classes of 2D topological materials using spillage (η) and Wannier \ncalculations.  JID represents the JARVIS -ID, Spg. the spac egroup, E g the bandgap  (eV), Ef the \nexfoliation energy  (meV/atom) , η the spillage, TopoClass the topological class of the material, T c \nthe Curie temperature. Here, QSHI, QAHI, NM -Semi, M -Semi represents quantum spin Hall \ninsulators, quantum anomalous Hall insu lators, non -magnetic semimetals and magnetic semi -\nmetals.  \n \nFormula  JID Spg. Eg (eV) η Ef (meV/atom)  TopoClass  Tc (K) \nBi2TeI 6901  𝑃3̅𝑚1 0.053  3.1 43.9 QSHI  - \nBiI 6955    C2/m  0.145  3.1 77.7 QSHI  - \nHfTe 5 19987    Pmmn  0.074  0.61 88.3 QSHI  - \nTaIrTe 2 6238    P21/m 0.034  1.02 100.4  QSHI  - \nZrFeCl 6 13600    P312  0.011  1.01 72.0 QAHI  5.3 \nCoBr 2 6034  𝑃3̅𝑚1 0.019  1.00 75.5 QAHI  17.7 \nVAg(PSe 3)2 60525    C2 0.018  0.50 74.7 QAHI  - 9 \n Ti2Te2P 27864  𝑃3̅𝑚1 0.0 1.06 58.9 NM-Semi  - \nZrTiSe 4 27780    P2/m  0.0 2.00 96.1 NM-Semi  - \nMoS 2 730 𝑅3̅𝑚 0.0 0.98 89.4 NM-Semi  - \nAuTe 2 27775  𝑃3̅𝑚1 0.0 0.50 317.6  NM-Semi  - \nMnSe  14431    P4/nmm  0.0 0.82 81.8 M-Semi  -- \nFePSe 3 27940  𝑃3̅𝑚1 0 1.07 74.9 M-Semi  148.3  \nFeTe  6667    P4/nmm  0.0 1.2 90.2 M-Semi  224.7  \nCoO 2 31379    C2/m 0.0 2.01 - M-Semi  - \nTiCl 3 13632  𝑃3̅𝑚1 0.0 0.62 72.4 M-Semi  -- \nVBr 2O 6832    Pmmm  0.0 0.45 67.7 M-Semi  - \nCo (OH) 2 28106    C2/m  0.0 2.1 - M-Semi  -- \n \n \n2.2 Individual case studies  \n2.2.1 Quantum Spin  Hall Insulator ( QSHI ) \nSome of the semiconducting non -magnetic materials with high -spillage  that we identified are: BiI  \n(JVASP -6955) , TiI3 (JVASP -6118 ), TaIrTe 2 (JVASP -6238), Bi2TeI (JVASP -6901) and Bi \n(JVASP -20002).  Materials such as 2D -Bi53 are experimentally known as QSHI . The fact that we \nfind them  using the spillage criteria further  supp orts the applicability of this  method to 2D \nmaterials.  In order to computationally confirm that they are QSHI, we calculate the Z 2 index for a \nfew candidates  (BiI, TiI 3, Bi, Bi 2TeI). We find these materials indeed have non -zero indices . We \nshow example s of high spillage QSHI in Fig.  3. We observe high spillage peaks for all three 10 \n materials (Fig. 3a,d,g ), which occur  near the Z and Gamma point s. It is difficult to observe band -\ninver sion in Fig. c because all the states near E f are derived from Bi orbitals, s o the orbital -\nprojected bandstructure doesn’t show any obvious band -inversion. However, the spillage method \nsuggests such inversion as in Fig. 3e.  In all of these cases , we find that the bulk ( Fig. 2b,e,h ) is \ninsulating while the edge states  (Fig. 2c,f,i)  are conducting , with Dirac dispersions at appropriate \nhigh symmetry points , as expected for QSHI .  \n \n \nFig. 3 Examples of quantum  spin Hall insulators (Q SHI) with spillage  (a,d,g) , surface  (b,e,h) , and \nedge bandstructure  (c,f,i)  plots.  (a,b,c) for BiI, (d ,e,f) for Bi 2TeI, (g,h,i) for HfTe 5. In (c,f,i) the \ncolor scale represent the electron occupation.  The Fermi energy is set to zero in all band structure \nplots.  \n  \n \n \n \n11 \n 2.2.2 Quantum Anomalous  Hall Insulator ( QAHI ) \nNext, we investigate the candidates for QAHI materials . We have 19 such candid ates after \nscreening for spillage  > 0.5, bandgap  > 0.0, magnetic moment  > 0.5. We carry out Chern number \ncalculations , find ing several QAHI  material s. Note that these were obtained using DFT \ncalculations only, i.e. without DFT +U or G 0W0 correction s; such corrections  will be discussed \nlater. Some of the materials with non -zero Chern numbers are: CoCl 2 (JVASP -8915) , VAg(PSe 3)2 \n(JVASP -60525), ZrFeCl 6 (JVASP -13600). We show the surface and edge band -structure s as well \nas anomalo us Hall conductivity  (AHC)  in Fig  4 for VAg(PSe 3)2 and ZrFeCl 6. They both have \nChern number equal to 1.  Correspondingly , there is a single edge channel that connects the valence \nband ( VB) to the conduction band (CB) (Fig 4 c, d) , which carries the quantize d AHC  (Fig 4 d, h) . \nLike QSHI, QAHI materials have gapped surface s (Fig 4 b, f) but conducting edges.   \n \n \nFig. 4 Examples of quantum anomalous  Hall insulators (Q AHI) with spillage  (a,e), surface  (b,f), \nand edge bandstructure  (c,g), and anomalous Hall cond uctivi ty (AHC)  (d,h)  plots.  (a,b,c,d) for \nVAg(PSe 3)2, (e,f,g,h) for ZrFeCl 6. The Fermi energy is set to zero in all band structure plots.  \n12 \n  \n2.2.3 Non-magnetic semimetals  \n \nFig. 5 Examples of  non-magnetic  (a,b,c,d)  and magnetic  (e,f,g,h)  topological semimet als with \nspillage (a,c,e,g) and surface bandstructure (b,d,f,g) plots . a-b) Ti 2TeP, c -d) AuTe 2 e-f) TiCl 3 g-h) \nMnSe  \n \nNext, we identify the non-magnetic topological semimetals. This is more challenging than \nanalyzi ng insulator s because there are no specific  Z2/Chern-like indices for such materials. Rather \nwe look for Weyl/Dirac nodes between the nominal valence and conduction bands . Examples of \nsemimetals we found and confirmed with Wannier based methods are MoS 2 (JVASP -730), AuTe 2 \n(JVASP -27775),  Ti2Te2P (JVASP-27864) and ZrTiSe 4 (JVASP -27780) . The complete list of \nfound topological semimetals is given in the supplementary information section , together with \ntheir spillage values.  As is evident from Fig. 5 a and c , the spillage  plots  have spiky peaks that \nrepresent band -inversion.  For Ti 2Te2P the band -crossing is slightly  below the Fermi level while \nfor ZrTiSe 4 it is very close to the Fermi level.  It is often possible to shift the Fermi -level of 2D \n13 \n materials via electrostatic or chemical doping , so band -crossin gs slightly away from  Fermi -level  \ncan still have observable effects . \n2.2.4 Magnetic semimetals  \nFinally, we look  at ferromagnetic phases with band crossing s, which are a ll Weyl crossings. We \nfind band crossings near the Fermi level  in MnSe  (JVASP -14431 ), FeTe (JVASP-6667 ), CoO 2 \n(JVASP -31379) , TaFeTe 3 (JVASP -60603) , TiCl 3 (JVASP -13632) , Co(HO) 2(JVASP -28106)  all of \nwhich have non -symmorphic symmetries  that can protect band crossings41. Also , we find crossings \nin VBr 2O (JVASP -6832 ), which has a double point group ( mmm ) that allows two -dimensional \nirreducible representations.  In Fig 5e, we discuss TiCl 3 (JVASP -13632), a representative magnetic \nsemimetal. The Weyl point occurs just above the Fermi level along the high symme try line between \nГ and M , as shown in Fig. 5f . Unfortunately, the valence band dips below the Fermi level at K, \nadding extra bands to the Fermi level.  Similarly,  in the case of MnSe the crossings are above the \nFermi -level  as shown in Fig. 5g and h . Note that the bandgaps a nd magnetic moments of the above \nsystems are heavily dependent on the calculation method . This limitation of the methodology  is \ndiscussed later. \n2.3 Magnetic ordering  \nIn addition to the electronic structure, we report the magnetic ordering of the structure s and Curie \ntemperature obtain ed using the method described in Ref.54 model. Many of the examples we \nconsider have either lower energy anti-ferromagnetic ( AFM ) phases or are ferromagnetic ( FM) \nbut with spins in -plane, but for the  remaining  materials , which we predict to be FM at finite \ntemperature, we present their  Curie temp eratures and Chern number s in Table 1. We find that Tc \nof FeTe is unusually high because of the high J parameter resulting from the AFM configuration. 14 \n The FePS 3 and FeTe  have Tc higher than CrI 3, which suggests they can be used for above liquid  \nnitrogen  temperature Tc topological  applications . On the other hand, the Tc of ZrFeCl 6 is very low , \nwhich is consistent with the large separation between Fe atoms in that crystal structure . As a next \nstep, it would be interesting to study the layer dependen ce of Tc, which is beyond the scope of \npresent work . \n2.4 Limitations of  GGA -based  Calculations  \nNext, we investigate the reliability of our calculation s for a few systems using GGA +U55,56 and \nG0W0+SOC57-59 methods.  We note that fully ab initio  beyond -DFT methods like G0W0 are very \ntime-consuming , and hence unfeasible for a high-throughput search , while GGA +U is \ncomputationally fast, but has an adjustab le parameter that limits predictive power.  In Fig.  6a we \nshow the GGA +U dependence of bandgaps and Chern numbers of ZrFeCl 6. We find that as we \nincrease the U parameter, ZrFeCl 6 has a non-zero C hern number only for  U < 0.2. This indicates \nthat Chern behavi or can be quite sensitive to U. Similar behavior was also observed for FeX 3 \n(X=Cl,Br,I) by Li35, where the critical value of U varies from 0.43 to 0.80 eV. In Fig. 6b we show \nhow G0W0+SOC can change the band gaps for VAgP(Se 3)2, an example QAHI . In this cas e, \nG0W0+SOC  increase s the bandgap  (0.018 eV vs 0.103 eV) , but the band -shapes remain very \nsimilar , and the topology is non -trivial even in the  G0W0+SOC  calculation . However, for the other \ncases , we found that the band  gap increase was substantial  (Table. 2 ), and caused the bands to \nbecome un -inverted, resulting in trivial materials . We note that  G0W0+SOC  can also predict  \nincorrect ba nd-structures , especially for correlated transition metal compounds60. Here we are \nusing only single -shot GW (G 0W0), and more accurate results could be obtained by using fully \nself-consistent (sc)-GW63, which is not carried out here du e to the very high computational cost. \nAlso, note that the bandgaps, magnetic moments can also depend on the selection of 15 \n pseudopotentials. Fully accurate ab initio  calculations of 2D magnetic materials remain very \nchallenging, even for a single  material, with techniques like dynamical  mean field theory ( DMFT ) \nand quantum Monte-Carlo ( QMC ) as possible approaches for future work.  \n \nTable. 2 Comparison of PBE+SOC and G 0W0+SOC bandgaps.  \nEg (eV) PBE+SOC  G0W0+SOC  \nZrFeCl 6 0.011  1.302  \nVAg(PSe 3)2 0.018  0.103 \nRuCl 3 0.0 0.50 \n \n \nFig. 6 Effect of beyond DFT methods on the electronic structure using DFT+U and G 0W0 methods.  \na) DFT+U effect on Chern number for ZrFeCl 6, b) bandstructure of VAg(PSe 3)2 without (red) and \nwith G0W0+SOC (blue) method .  \n \n16 \n  \n3 Conclusions  \nWe have presented a comprehensive search of 2D topological materi als, including  both magnetic  \nand n on-magnetic  materials, and considering insulating and metallic phases. Using the JARVIS -\nDFT 2D material dataset , we first identify materials with high spin -orbit spillage , resulting in 1 22 \nmaterials. Then we use Wannier -interpolation to carry -out Z 2, Chern -number, anomalous Hall \nconductivity, Curie temperature , and  surface and edge state calculations to identify top ological \ninsulators and semimetals. For a subset of predicted QAHI materials, we run G 0W0+SOC and \nGGA+U calcul ations. We find that as we introduce many -body effects , only a few materials retain \na non-trivial band -topology , suggesting that higher-level electronic structure methods will be \nnecessary to fully analyze the topological properties of  2D materials . Howeve r, we believe that as \nan initial step, the automated spillage screening and Wannier -approach provide s useful predictions \nfor finding new topological materials.  \n \n4 Methods  \nDFT calculations were carried out using  the Vienna Ab -initio simulation package (VASP )61,62 \nsoftware using the workflow given on our github page ( https://github.com/usnistgov/jarvis  ). \nPlease note commercial software is identified to specify procedures. Such identification does not \nimply recommendation by National Institute of Standards and Technology (NIST ). We use  the \nOptB88vdW  functional63, which gives accurate lattic e parameters for both vdW and non -vdW \n(3D-bulk) solids43. We optimize the cryst al-structures of  the bulk and monolayer phases  using  \nVASP with OptB88vdW . The initial screening step for <1.5 eV bandgap materials is done with  17 \n OptB88vdW bandgaps  from the JARVIS -DFT database . Because SOC is not currently \nimplemented for OptB88vdW  in VASP , we carry out spin-polarized PBE and spin-orbit PBE \ncalculations  in orde r to calculate  the spillage for each material.   Such an approach has been  \nvalidated  by Refs. 8,64. The crystal structure was optim ized until the forces on the ions were less \nthan 0.01 eV/ Å and energy less than 10-6 eV. We also use G0W0+SOC and DFT+U on selected \nstructures. We use Wannier9065 and Wannier -tools66 to perform  the Wannier -based evaluation of \ntopological invariants .  \nFirst, out of 963 2D monolayer materials , we identify materials w ith bandgap  < 1.5 eV, and heavy \nelements  (atomic weight≥65 ). We calculate the exfoliation energy of a 2D material a s the \ndifference in energy per atom for bulk and monolayer counterparts43. Then we us e the spillage \ntechnique  to quickly narrow down the number of possible materials.  As introduced in Ref.42, we \ncalculate the spin -orbit spillage , 𝜂(𝐤), given by the following equation:  \n𝜂(𝐤)=𝑛𝑜𝑐𝑐(𝐤)−Tr(𝑃𝑃̃)                                                                                                            (1) \nwhere, \n𝑃(𝐤)=∑ |𝜓𝑛𝐤⟩⟨𝜓𝑛𝐤|𝑛𝑜𝑐𝑐(𝐤)\n𝑛=1  is the projector onto the occupied wavefunctions without SOC, and \n𝑃̃ is the same project or with SOC  for band n and k -point k. We use a k-dependent occupan cy \n𝑛𝑜𝑐𝑐(𝐤) of the non -spin-orbit calculation so that we can treat metals, which have varying number \nof occupied elect rons at each k -point8. Here, ‘Tr’ deno tes trace over the occupied bands. We can \nwrite the spillage equivalently as:  \n𝜂(𝐤)=𝑛𝑜𝑐𝑐(𝐤)−∑ |𝑀𝑚𝑛(𝐤)|2 𝑛𝑜𝑐𝑐(𝐤)\n𝑚,𝑛=1               (2) 18 \n where 𝑀𝑚𝑛(𝐤)= 〈𝜓𝑚𝐤|𝜓̃𝑛𝐤〉 is the overlap between occupied Bloch functions  with and without \nSOC at the same wave vector k. If the SOC does not change th e character of the occupied \nwavefunctions, the spillage will be near zero, while  band inversion will result in a large spillage.  \nIn the case of i nsulating topological materia ls driven by spin -orbit based band inversion , the \nspillage will be at least 1.0 for Chern insulators or 2.0 for  TRS-invariant topological insulators at \nthe k -point (s) where band in version occurs42.  In other wo rds, t he spillage can be vi ewed as the \nnumber of ba nd-inverted electro ns at each k -point . For topological metal s and semimetals,  or cases \nwhere the inclu sion of SOC opens a gap, th e spi llage is not require d to b e an integer, but we find \nempirically  that a high spill age value is an indication of a ch ange in the band structure d ue to SOC \nthat can indicate topological behavior . We choose a threshold value of  𝜂 = 0.5 at any k -point for \nour screening  procedure, based o n compari son with known  topolog ical semim etals8. Our scr eening \nmethod can also detect  topological materials with small SOC , like the small band  gap that is \nopened at the  Dirac point in graph ene22 (JVASP -667). However, the screening method is not  \nexpected to  work for semimetals with topological features that are not caused  or modified by spin-\norbit coupling.   \nAfter spillage calculations , we run Wannier based Chern and Z 2-index calculations for these \nmaterials.  \nThe Chern number , C is calculated  over the Brillouin  zone, BZ,  as: \n𝐶=1\n2𝜋∑∫𝑑2𝒌𝛺𝑛 𝑛         (3) \n𝛺𝑛(𝒌)=−Im〈∇𝒌𝑢𝑛𝒌|×|∇𝒌𝑢𝑛𝒌〉=∑2Im〈𝜓𝑛𝑘|𝑣̂𝑥|𝜓𝑚𝑘〉〈𝜓𝑚𝑘|𝑣̂𝑦|𝜓𝑛𝑘〉\n(ꞷ𝑚−ꞷ𝑛)2 𝑚≠𝑛    (4) 19 \n Here,  𝛺𝑛is the Berry curvature , unk being the periodic part of the Bloch wave in the  nth band , 𝐸𝑛=\nћꞷ𝑛, vx and vy are velocity operators . The Ber ry curvature  as a function  of k is given by:  \n𝛺(𝒌)=∑∫𝑓𝑛𝑘𝛺𝑛(𝒌)𝑛          (5) \nThen, t he intrinsic anomalous Hall conductivity  (AHC)  𝜎𝑥𝑦 is given by:  \n𝜎𝑥𝑦=−𝑒2\nћ∫𝑑3𝒌\n(2𝜋)3𝛺(𝒌)         (6) \nIn addition to se arching for gapped phases, we also search for Dirac and Weyl semimetals by \nnumerically searching for band crossings between the highest occupied and lowest unoccupied \nband, using the algorithm from WannierTools66. This search for crossings can be performed \nefficiently because it takes a dvantage of Wannier -based band interpolation. In an ideal case, the \nband crossings will be the only points at the Fe rmi level; however, in most cases , we find additional \ntrivial metallic states at the Fermi level.  The surface spectrum was calculated by usi ng the Wannier \nfunctions and the iterative Green’s function method67,68 .  \nFor magnetic systems, we primarily screen the ferromagnetic phase, with spins oriented out of the \nplane, which we expect is possible to achieve experimentally in most cases , using an external field \nif necessary. We initialize the mag netic moment with 6 𝜇𝐵 for spin -polarized calculations. For a \nsubset of interesting compounds, we perform a set of three additional calculations: ferromagnetic \nwith spins in -plane, and antiferromagnetic with spins in - and out -of-plane. With th ese energ ies, \nwe can fit a minimal magnetic model and calculate an estimated Curie temperature for \nferromagnetic materials with out -of-plane spins, using the method of Ref.54, which takes into \naccount exchange constants and magnetic anisotropy. Magnetic anisotropy that favors out -of-\nplane spin is cruc ial to enable long -range magnetic ordering in two -dimensions  following the 20 \n Mermin -Wagner theorem69. We consider a simple  Heisenberg model Hamiltonian with nearest -\nneighbor exchange interactions J, single -ion anisotropy A, and nearest neighbor anisotropic \nexchange B.: \n𝐻=−1\n2∑𝐽𝑖𝑗𝑺𝑖𝑺𝑗 𝑖𝑗 −𝐴∑(𝑆𝑖𝑧)2\n𝑗 −1\n2∑𝐵𝑖𝑗𝑆���𝑧𝑆𝑗𝑧\n𝑖𝑗   (7) \nwith Jij, A, Bij > 0. The sums run over  all magnetic sites and Jij = J, Bij = B if i and j are nearest \nneighbors and  zero otherwise. The m aximum value of 𝑆𝑖𝑧 denoted by S. For an Ising model, which \ncorresponds to the limit of 𝐴→∞, the critical transition temperature is given by:  \n𝑇𝑐𝐼𝑠𝑖𝑛𝑔=𝑆2𝐽𝑇𝑐̃\n𝑘𝐵           (8) \nwhere 𝑇𝑐̃ is a dimensionless critical t emperature with values of 1.52, 2.27, 2.27, and 3.64 for the \nhoneycomb, quadratic, Kagomé, and hexagonal lattices respective ly. However, systems with finite \nA, we instead use: 54 \n𝑇𝑐=𝑇𝑐𝐼𝑠𝑖𝑛𝑔𝑓(𝛥\n𝐽(2𝑆−1))         (9) \nwith 𝛥=𝐴(2𝑆−1)+𝐵𝑆𝑁𝑛𝑛  and 𝑓(𝑥)=tanh1\n4[6\n𝑁𝑛𝑛𝑙𝑜𝑔(1+𝛾𝑥)]. \nTo better  account for correlation effects  and self -interaction error in  describing transition metal \nelements, we apply the DFT+U method  to a subset of ma terials55,56. For a few sys tems, we also \ncarry out  a systematic U -scan by varying the U parameter from 0 to 3 eV and monitor changes in \nthe band structure70. We also  perform  G0W0+SOC57-59 calculations with an ENCUTGW parameter \n(energy cutoff for response functio n) of 333.3 eV  for a few materials  to analyze the impact  of \nmany -body effects.  \n5 Data availability  21 \n The electronic structure data is made available at the JARVIS -DFT website: \nhttps://www.ctcms.nist.gov/~knc6/JVASP .html  and http://jarvis.nist.gov  .  \n6 Contributions  \nKC and KG  developed the workflow . KC  carried out the DFT calculations  and analyzed  the DFT \ndata. 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B.,   85, 115104 (2012).  \n \n "    },    {        "title": "2002.05225v2.Accelerated_design_of_Fe_based_soft_magnetic_materials_using_machine_learning_and_stochastic_optimization.pdf",        "content": "Accelerated design of Fe-based soft magnetic materials using\nmachine learning and stochastic optimization\nYuhao Wanga,1, Yefan Tianb,1, Tanner Kirka, Omar Larisc, Joseph H. Ross, Jr.b,d, Ronald\nD. Noebee, Vladimir Keyline, Raymundo Arr\u0013 oyavea,d,\u0003\naDepartment of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA\nbDepartment of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA\ncDepartment of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge,\nMA 02139, USA\ndDepartment of Materials Science and Engineering, Texas A&M University, College Station, TX 77843,\nUSA\neMaterials and Structures Division, NASA Glenn Research Center, Cleveland, OH 44135, USA\nAbstract\nMachine learning was utilized to e\u000eciently boost the development of soft magnetic materials.\nThe design process includes building a database composed of published experimental results,\napplying machine learning methods on the database, identifying the trends of magnetic\nproperties in soft magnetic materials, and accelerating the design of next-generation soft\nmagnetic nanocrystalline materials through the use of numerical optimization. Machine\nlearning regression models were trained to predict magnetic saturation ( BS), coercivity ( HC)\nand magnetostriction ( \u0015), with a stochastic optimization framework being used to further\noptimize the corresponding magnetic properties. To verify the feasibility of the machine\nlearning model, several optimized soft magnetic materials { speci\fed in terms of compositions\nand thermomechanical treatments { have been predicted and then prepared and tested,\nshowing good agreement between predictions and experiments, proving the reliability of the\ndesigned model. Two rounds of optimization-testing iterations were conducted to search for\nbetter properties.\nKeywords: machine learning, soft magnetic properties, nanocrystalline, materials design\n1. Introduction\n1.1. Motivation\nThe pursuit of increased e\u000eciency in energy conversion and transformation requires a\nnew generation of energy materials. Soft magnetic materials are capable of rapidly switching\ntheir magnetic polarization under relatively small magnetic \felds. They typically have small\n\u0003Corresponding author.\n1Authors contributed equally to this work.\nPreprint submitted to Acta Materialia May 6, 2020arXiv:2002.05225v2  [cond-mat.mtrl-sci]  4 May 2020intrinsic coercivity and are used primarily to enhance or channel the \rux produced by an\nelectric current. These alloys are used in a large number of electromagnetic distribution,\nconversion, and generation devices, such as transformers, converters, inductors, motors,\ngenerators, and even sensors.\nIn the current materials science community, the accelerated discovery and design of\nnew energy materials has gained considerable attention in light of the many societal and\nenvironmental challenges we currently face. Soft magnetic materials are crucial as they\nare essential elements of electro-magnetic energy transformation technologies. For example,\nthe power transformer is a critical component of the solar energy conversion system, whose\nperformance is ultimately limited by the magnetic properties of the materials used to build\nthe cores. In 1988, Yoshizawa et al. presented a new nanocrystalline soft magnetic material\nreferred to as FINEMET, which exhibits extraordinary soft magnetic performance [1]. This\nalloy was prepared by partially crystallizing an amorphous Fe-Si-B alloy with minor addition\nof Cu and Nb. This unusual combination of chemistry and processing conditions led to\nan ultra\fne grain structure in an amorphous matrix resulting in excellent soft magnetic\nproperties. The resulting soft magnetic properties of FINEMET type alloys, relevant to\nelectromagnetic energy conversion devices, are a unique combination of low energy losses,\nlow magnetostriction, and high magnetic saturation, up to 1.3 T. This was achieved through\nan ultra\fne composite microstructure of cubic-DO 3structured Fe-Si grains with grain sizes\nof 10-15 nm in a continuous amorphous matrix, providing a new path for designing next-\ngeneration soft magnetic materials.\n1.2. FINEMET-type soft magnetic materials\nThe target material system in this work is FINEMET-type soft magnetic nanocrystalline\nalloys whose properties are categorized into two groups, intrinsic and extrinsic properties.\nIntrinsic properties include magnetic saturation ( BS), magnetocrystalline anisotropy ( K1),\nmagnetostriction ( \u0015), and Curie temperature ( TC).K1and\u0015indirectly in\ruence the hys-\nteretic behavior ( B-Hloop) for each type of core material by in\ruencing coercivity and\ncore losses of the material. Extrinsic properties include permeability ( \u0016), susceptibility ( \u001f),\ncoercivity ( HC), remanence ( Mr), and core losses ( Pcv). These are in\ruenced not only by\nthe microstructure, but also the geometry of materials, the di\u000berent forms of anisotropy,\nand the e\u000bect of switching frequency of the applied \felds[2].\nAmong these soft magnetic properties, most can be obtained from its unique hysteresis\nloop - known as the B-Hcurve, shown in Fig. 1(a), where B is the \rux density generated by\nan electromagnetic coil of the given material as a function of applied magnetic \feld strength,\nH. From this curve the following terms can be de\fned: (a) Coercivity ( HC) is the intensity\nof the applied magnetic \feld required to reduce the residual \rux density to zero after the\nmagnetization of the sample has been driven to saturation. Thus, coercivity measures the\nresistance of a ferromagnetic material to be demagnetized. (b) Magnetic saturation ( BS) is\nthe limit to which the \rux density can be generated by the core as the domains in the material\nbecome fully aligned. It can be determined directly from the hysteresis loop at high \felds.\nLarge values of \rux density are desirable since most applications need a device that is light\nin mass and/or small in volume. (c) Permeability ( \u0016):\u0016=B=H = 1 +\u001f, is the parameter\n2Figure 1: (a) B-Hloop. (b) Coercivity HCvs. grain size Dfor various soft magnetic metallic alloys.\nReprinted from Ref. [3]. (c) Schematic representation of the random anisotropy model for grains embedded\nin an ideally soft ferromagnetic matrix. The double arrows indicate the random \ructuating anisotropy axis,\nthe hatched area represents the ferromagnetic correlation volume determined by the exchange length Lex.\nReprinted from Ref. [3].\nthat describes the \rux density, B, produced by a given applied \feld, H. Permeability can\nvary over many orders of magnitude and should be optimized for a given application. For\nexample, EMI \flters usually require large values to produce substantial changes in magnetic\n\rux density in small \felds. For other applications, such as \flter inductors, permeability\ndoes not necessarily need to be high but needs to be constant so that the core does not\nsaturate readily. (d) Core loss is one of the most essential properties of the material as it\nis a direct measure of the heat generated by the magnetic material in A/C applications. It\nis the area swept out by the hysteresis loop, which should be minimized to provide a high\nenergy e\u000eciency for the core. Contributions to the core loss include hysteretic sources from\nlocal and uniform anisotropies and eddy currents at high frequencies.\nMaximizing BSand minimizing HCare the most important design objectives for most\napplications requiring soft magnetic materials and therefore were the design goals in this\nstudy. Since \u0016heavily depends on the application and can range over several orders of\nmagnitude, even for a \fxed composition, depending on secondary processing conditions, it\nwas therefore not a parameter that was optimized or considered in this study. Based on the\nnature of FINEMET-type nanocrystalline alloys, several constraints have been incorporated\nin this study. The magnetic transition metal element has been set to Fe in this work and\nother elements, such as Co and Ni are excluded, because at relatively small additions they\nwill tend to decrease B S. The composition of Fe ranged from 60%-90%. The percentages of\nthe remaining elements in total varied from 10%-40%. Although the early transition metal\nelement is Nb in current commercial FINEMET alloys, other elements, such as Zr, Hf, Ta,\nMo or even combinations of di\u000berent early transition metal elements were considered. In\ncommercial alloys, metalloids B and Si are added to promote glass formation in the precursor\nand we also allowed for P. The noble metal elements are selected from Cu, Ag, or Au serving\nas nucleating agents for the ferromagnetic nanocrystalline phase.\nThe random anisotropy model [4] provides a concise and explicit picture for under-\nstanding the soft magnetic properties of nanocrystalline ferromagnetic materials, such as\n3FINEMET. As illustrated in Fig. 1, the microstructure is characterized by a random dis-\ntribution of structural units or grains in a ferromagnetic matrix with an e\u000bective magnetic\nanisotropy with a scale D. For a \fnite number ( N) of grains within the ferromagnetic cor-\nrelation volume ( V=L3\nex), the corresponding average anisotropy constant hK1iis given\nby\nhK1i\u0019K1p\nN=K1(D\nLex)3=2; (1)\nwhich is determined by the statistical \ructuations from averaging over the grains. If there\nare no other anisotropies, the coercivity HCand the magnetic saturation BSare directly\nrelated to the average anisotropy constant hKiby\nHC=pchKi\nBS; (2)\nwherepcis a dimensionless pre-factor. These relations were initially derived for coherent\nmagnetization rotation in conventional \fne particle systems. In the regime D <Lex, how-\never, they also apply for domain wall displacements. Accordingly, coercivity was shown\nto vary with grain size as HC/D6for very \fne grained materials (Fig. 1(b)). For large\ngrain sizes, it shows the typical 1 =D-dependence and thus great soft magnetic properties\nrequire large grain sizes [5]. However, for small grain sizes, HCshows an extraordinary\nD6-dependence behavior, which provides another path to realize excellent soft magnetic\nperformance, through the generation of nanocrystalline microstructures [6, 7].\n1.3. FINEMET materials design\nMachine learning as one of the most popular and e\u000ecient statistical techniques has\nenormous potential to bring the discovery and design of soft magnetic alloys to the next\nlevel. Recently, Rajesh et al. [8] presented a combined CALPHAD and machine learning\napproach to predict nanocrystalline size and volume fraction from inputs such as composition\nand heat treatment conditions, demonstrating that machine learning is a promising tool for\nmodeling soft magnetic alloys. Although our work is in a similar material space, we have\nbeen focusing on predicting magnetic properties from experimental data mined from the\nliterature. In this work, machine learning was used as the primary technique to expand\nthe boundaries of Fe-based FINEMET-type material space and design the next-generation\nof soft magnetic materials. As a \frst step, we built an experimental database compiled\nfrom 76 journal articles published starting in 1988. The database was carefully curated to\nonly include data entries in the nanocrystalline regime. Feature engineering was applied\nso that the original inputs have been transformed into more general atomic properties.\nWe built separate models to predict each magnetic property and the number of predictive\nfeatures were reduced to between 5 and 20 for each property. Various machine learning\nalgorithms were applied to the selected data sets, and the prediction evaluated using 20-\nfold cross-validation based on the coe\u000ecient of determination ( R2). Stochastic optimization\nframework was used to guide the design of new and better soft magnetic nanocrystalline\nmaterials. The overall framework is shown in Fig. 2.\n4For Each Property\nCollect  \nLiterature \nData\nClean and \nCurate \nDatabase\nIdentify Data \nSubsets for Each \nProperty\nFeature Selection \n(Dimensionality \nReduction)\nTrain and Compare \nMachine Learning \nModels\nTrained Random \nForest Models\nDetermine Design \nConstraints and \nObjectives\nApply Differential \nEvolution \nOptimization\nOptimized \nCompositions and \nProcessing Conditions\nValidated \nData\nConduct \nExperiments\nSoft Magnet \nDatabaseData Collection\nMachine Learning\nOptimization\nExperimental Validation\nUpdate Models with Experimental Data and Re-optimizeFigure 2: Diagram of machine learning and experimental design framework.\n2. Data set overview\nTo start the soft magnetic materials design process, the \frst step was to build a database\nby collecting available properties from the relevant literature. There are hundreds of acces-\nsible publications in the literature base, which describe di\u000berent types of soft magnetic\nnanocrystalline alloys in single or a small range of alloy compositions given one or several\ntypes of thermal treatments or other types of secondary processing, with some relevant\nmeasure of various properties. Due to the limitation of not being able to access the original\ndata directly, it was challenging to collect and organize all the data from the literature to\ninterpret the relations between processing, compositions, structures, and properties. Never-\ntheless, it is a valuable task to create, maintain and preserve such a data set to share in the\nmaterials science community for collaborative research purposes. The sources of our data\nset are academic literature and patent information, from published experimental work. All\npublications assembled in the database are listed in Table 1.\nThe essential components of the database are chemical compositions, thermal processing\nconditions applied to the amorphous precursor ribbon, and soft magnetic properties. Data\ncontained in plots were extracted using WebPlotDigitizer [84]. The soft magnetic prop-\nerties of nanocrystalline materials are heavily dependent on the chemical composition and\nthe di\u000berent elements in the composition can be split into four di\u000berent categories: magnetic\ntransition metal (MTM), early transition metal (ETM), post-transition metal (PTM) and\nlate transition metal (LTM) as shown in Fig. 3. MTM includes Fe, Co, and Ni, but in\nthis \frst attempt to model behavior, we focused on Fe-based materials, therefore, Co- and\n5References Year References Year\nYoshizawa et al. [1] 1988 Saad et al. [9] 2002\nKataoka et al. [10] 1989 Skorvanek et al. [11] 2002\nHerzer et al. [12] 1990 Mitrovic et al. [13] 2002\nSuzuki et al. [14] 1990 Marin et al. [15] 2002\nYoshizawa et al. [16] 1991 Zorkovska et al. [17] 2002\nSuzuki et al. [18] 1991 Sulictanu et al. [19] 2002\nFujii et al. [20] 1991 Cremaschi et al. [21] 2002\nMakino et al. [22] 1991 Chau et al. [23] 2003\nLim et al. [24] 1993 Ponpandian et al. [25] 2003\nTomida et al. [26] 1994 Kwapulinski et al. [27] 2003\nMakino et al. [28] 1994 Crisan et al. [29] 2003\nKim et al. [30] 1995 Sovak et al. [31] 2004\nInoue et al. [32] 1995 Cremaschi et al. [33] 2004\nVlasak et al. [34] 1997 Ohnuma et al. [35] 2005\nLovas et al. [36] 1998 Chau et al. [37] 2006\nGrossinger et al. [38] 1999 Ohta et al. [39] 2007\nYoshizawa et al. [40] 1999 Lu et al. [41] 2008\nKopcewicz et al. [42] 1999 Pavlik et al. [43] 2008\nFrost et al. [44] 1999 Makino et al. [45] 2009\nFranco et al. [46] 1999 Ohnuma et al. [47] 2010\nTurtelli et al. [48] 2000 Butvin et al. [49] 2010\nXuet al. [50] 2000 Lu et al. [51] 2010\nTodd et al. [52] 2000 Makino et al. [53] 2011\nBorrego et al. [54] 2000 Kong et al. [55] 2011\nKemeny et al. [56] 2000 Urata et al. [57] 2011\nIlinsky et al. [58] 2000 Makino et al. [59] 2012\nVarga et al. [60] 2000 Sharma et al. [61] 2014\nVlasak et al. [62] 2000 Liu et al. [63] 2015\nZorkovska et al. [64] 2000 Wen et al. [65] 2015\nSolyom et al. [66] 2000 Xiang et al. [67] 2015\nLovas et al. [68] 2000 Sinha et al. [69] 2015\nKwapulinski et al. [70] 2001 Wan et al. [71] 2016\nBorrego et al. [72] 2001 Dan et al. [73] 2016\nFranco et al. [74] 2001 Li et al. [75] 2017\nMazaleyrat et al. [76] 2001 Jiang et al. [77] 2017\nWuet al. [78] 2001 Li et al. [79] 2017\nBorrego et al. [80] 2001 Jia et al. [81] 2018\nGorria et al. [82] 2001 Cao et al. [83] 2018\nTable 1: List of soft magnetic papers from which the experimental data were mined.\n6Figure 3: Elemental components of FINEMET-like-soft magnetic alloys including magnetic transition metal\n(MTM), early transition metal (ETM), post-transition metal (PTM), late transition metal (LTM) elements.\nNi-containing alloys were not considered in the database. ETM elements normally act as\ngrain re\fners, which help slow down the di\u000busion process during the thermal crystallization\nstep and help maintain a \fne grain size. PTM elements form the glassy phase. In addition,\nSi serves the critical role of producing the Fe 3Si crystalline phase in FINEMET, which has a\nnegative magnetostriction and reduces the overall magnetostriction of the alloy by balancing\nout the positive magnetostriction in the glassy phase. LTM elements like Cu, Ag, and Au\ncan act as nucleants, which form clusters in the glassy phase, responsible for nucleating a\nhigh density of grains.\nThe distribution of Fe in the FINEMET data set is shown in Fig. 4(a), the dominating\ncomposition is 73.5, which is the original atomic percentage of Fe in FINEMET. The distri-\nbution of nucleant elements is shown in Figs. 4(b) and 4(c). It is shown that only 33 data\nentries contain Au which is around 2% of overall entry count. Cu occupies the majority of\nour studied compositions, and most of the entries contain 1% of Cu. For ETM elements,\nthe distribution of our study is shown in Figs. 4(h), 4(j), 4(k), and 4(l). The majority of\nour studied compositions contain Nb as grain re\fners and most of them have a Nb fraction\naround 3%. The distribution of PTM elements are shown in Figs. 4(d), 4(e), 4(f), 4(g),\nand 4(i). B occurs in most of the entries, and sometimes appears along with P or Ge. The\ndominant composition for B is 9%. Si also occurs most of the time because of the need\nto generate the Fe 3Si crystalline phase with a negative magnetostriction and to serve as a\npotential glass former.\nWe also visualize the distribution of selective magnetic properties of interest in Fig. 5.\nBecause the distribution of coercivity is heavily skewed with a large skewness \r1= 5:1, we\nthus applied a log transformation to transform skewed data to approximately conform to\nnormality. The log-transformed data give a much smaller skewness \r1= 0:6, which can\ngreatly help with the originally observed bias. It is also evident that people tend to report\ngood results, such as high magnetic saturation (larger than 1.5 T) and low magnetostriction\n(close to 0), rather than \\unattractive\" results. In our case, high magnetic saturation data\nlarger than 1.5 T is about 49% and low magnetostriction data between \u00003 and 3 is 47%.\n7Figure 4: Distribution of various element atomic percentage (at.%) in the database.\n8Figure 5: Distribution of selective magnetic properties in the database. Solid lines are kernel density\nestimations to estimate the probability density function of data entries.\nHowever, in terms of the machine learning process, the unattractive results can be just as\nbene\fcial to the overall process. Due to this reason, we decided not to leave out any \\bad\"\ndata. The rest of data (51% for magnetic saturation and 53% for magnetostriction) can help\nwith reducing the corresponding biases to a great extent.\nIn the course of constructing the database, several di\u000eculties, which a\u000fict most materials\ndesign problems, need to be clearly stated. First, we chose FINEMET-type nanocrystalline\nalloys as the material design space, and there is a vast composition parameter space where\nfew researchers have done measurements. Several clusters of data in the material database\nrepresent trendy materials, which are only a small percentage of all candidate material\ncompositions. Second, given the di\u000berent research habits of various groups and diversity of\nequipment, some researchers did not clarify everything, such as manner in which magnetic\nsaturation was de\fned. Finally, not all publications include every soft magnetic property\nof interest. For example, compared to coercivity, magnetic saturation is more di\u000ecult to\nmeasure, which leads to the fact that we have many data points for the former and less\nfor magnetic saturation. As a result, considering the primary target is to optimize both\ncoercivity and magnetic saturation, it is helpful to split the database into separate sets,\neach for a di\u000berent magnetic property as the objective. All features are listed in Table 2.\nThe features mainly contain two di\u000berent categories: chemical components and experimental\nmeasurements.\nThe purpose of the study was to evaluate nanocrystalline FINEMET type alloys. These\nalloys are generally produced by partial crystallization of an amorphous precursor by an\nannealing treatment, which also signi\fcantly increases the complexity of the problem. To\nensure data consistency, some data points in the data sets were modi\fed by the following\nprocedures:\n1. Data which were missing an annealing temperature or annealing time, all as-quenched\ndata, and all data processed below room temperature were removed.\n2. Annealing Temperatures were rounded to every 5th degree Celsius.\n3. Annealing Times were rounded to the nearest hour or half hour depending on the\n9Chemical ElementsFe, Si, C, Al, B, P, Ga, Ge, Cu, Ag, Au, Zn, Ti, V, Cr,\nZr, Nb, Mo, Hf, Ta, W, Ce, Pr, Gd, U.\nExperimental measurementsAnnealing temperature, Annealing Time, Primary Crys-\ntallization Onset, Primary Crystallization Peak, Sec-\nondary Crystallization Peak, Longitudinal Annealing\n\feld, Transverse Annealing \feld, Ribbon Thickness,\nMagnetic Saturation ( BS), Coercivity ( HC), Permeabil-\nity (\u0016), Magnetostriction ( \u0015), Core Loss, Electrical Re-\nsistivity (\u001a), Curie Temp ( TC), Grain Size ( D).\nTable 2: Table of all features from the soft magnetic data set.\nmagnitude of the value.\n4. Data points out of the nanocrystalline regime. -i.e., grain diameters over 60 nm, were\nremoved.\n5. Any features, which are unused after data reduction were removed.\n3. Machine learning model\nThe next stage in our analysis was to build a series of predictive models using machine\nlearning techniques to relate the magnetic properties with chemical components and pro-\ncessing conditions. The approach consisted of two main steps: feature selection and model\nselection.\n3.1. Feature selection\nWe employed a \fve step process to identify features that could be removed in order\nto simplify the model, without incurring a signi\fcant loss of information. The procedure\nincludes [85]: (a) Remove the features that have over 50% missing values. (b) Remove the\nfeatures containing only one unique value. (c) Identify collinear features and remove them.\n(d) Remove zero importance features using gradient boosting decision tree algorithm. (e)\nRemove cumulative low importance features using gradient boosting decision tree algorithm.\nSpeci\fcally, step c) utilizes the Pearson correlation coe\u000ecient to identify pairs of collinear\nfeatures. For each pair above the speci\fed threshold (in absolute value), it \fnds one of the\nvariables to be removed. Steps d) and e), required a supervised learning problem with\nlabels to estimate the importance of features and in this work, a gradient boosting machine\nimplemented in the LightGBM library [86] was utilized. A gradient boosting machine works\nby utilizing multiple \"weak learners\" (in our case, decision trees) and combining them into\na strong learner. Trees are constructed in a greedy manner sequentially and the subsequent\npredictors can learn from the mistakes of the previous predictors. In a single tree, a decision\nnode splits data into two parts each time based on one of the features, and the relative\nrank (i.e. depth) of a feature can be used to assess the relative importance of that feature\nwith respect to the target value. Features used at the top level of the tree have a larger\ncontribution to the \fnal prediction based on the input samples. The expected fraction of\n10the samples they contribute to can thus be used as an estimate of the relative importance of\nthe features. In a gradient boosting machine, one can average the importance of the features\nover a sequence of decision trees to reduce the variance of the estimation in order to use it\nfor feature selection [87]. For step d), features with zero importance were removed. Step e)\nbuilds o\u000b the feature importance from step d) and removes the lowest important features\nnot needed to reach a speci\fed cumulative total feature importance speci\fed by users. For\nexample, if we input a cumulative total feature importance value 0.99, it can \fnd the lowest\nimportant features that are not needed to reach 99% of the total feature importance and\nremove them.\nAfter the feature selection procedure, the remaining features are shown in Table 3. There\nCoercivity Curie Temperature Grain Size\nFe Si Fe\nSi Nb Si\nB Annealing temperature (K) B\nP Tc;0(K) Nb\nGe Ribbon Thickness ( \u0016m) Annealing temperature (K)\nCu \u0001 T0 Ribbon Thickness ( \u0016m)\nAu\nNb\nMo\nAnnealing temperature (K)\nTc;1(K)\nRibbon Thickness ( \u0016m)\n\u0001T1\nMagnetic Saturation Magnetostriction Permeability\nFe Fe Fe\nSi Si Cu\nB B Nb\nP Zr Annealing temperature (K)\nZr Nb Annealing Time (s)\nNb Ta Ribbon Thickness ( \u0016m)\nAnnealing temperature (K) Annealing temperature (K)\nAnnealing Time (s)\nTc;2(K)\nRibbon Thickness ( \u0016m)\nTable 3: Remaining features after feature selection for di\u000berent materials properties. Tc;0,Tc;1, andTc;2\nrepresent Primary Crystallization Onset, Primary Crystallization Peak, and Secondary Crystallization peak,\nrespectively. \u0001 T0and \u0001T1are the di\u000berence of Tc;0andTc;1with annealing temperature, respectively.\nare 13 features for coercivity; 6 features for Curie temperature; 6 features for grain size; 10\nfeatures for magnetic saturation; 7 features for magnetostriction and 6 features for Perme-\n11Figure 6:R2value of di\u000berent machine learning algorithms.\nability. Note that initially we tried to add atomic properties to our feature space but later\ndecided to remove them during the feature selection process. It is due to the relatively\nlimited range of element selection we have and the strong correlation between the atomic\nproperties and the chemical compositions. Furthermore, using elemental fraction is more\nstraightforward in the inverse modeling process during optimization, avoiding additional\nsteps in mapping atomic properties back to composition. The following machine learning\nmodels were built based on the selected feature sets for each property.\n3.2. Machine learning algorithms and results\nFive di\u000berent machine learning algorithms including linear regression, support vector\nmachines, decision trees, k-nearest neighbors and random forest, were utilized and compared\nwith each other in the process of building predictive models [87]. The comparison of the\ncoe\u000ecient of determination ( R2) score from 20-fold cross-validation is shown in Fig. 6. Note\nthat we performed a natural log transformation on coercivity to \fx its skewness. Based\non theR2score, it is evident that the random forest model is the best. Random forest\nregression(RFR) [88] is an ensemble of regression trees, induced from bootstrap samples of\nthe training data, using random feature selection in the tree induction process. Prediction\nis then made by averaging the outputs of the ensemble of trees. Random forest generally\nexhibits a substantial performance improvement over the single tree classi\fer, such as CART\nand C4.5. It yields a generalization error rate that compares favorably to AdaBoost, yet is\nmore robust to noise [89].\nPredicted values of magnetic saturation, coercivity and magnetostriction from random\n12Figure 7: Comparison between predicted values from the machine learning model and experimental values\nfor (a) magnetic saturation, (b) coercivity, and (c) magnetostriction.\nforest models are compared with actual experimental values in Fig. 7. The resulting RFR\nachieves a good agreement with the experimentally measured properties. Investigating the\nregression data closer, it is apparent that the RFR model underestimates coercivity within\nthe low coercivity region. This is potentially due to the high sensitivity of coercivity to the\nprocessing conditions. Furthermore, for all three properties, the data is not perfectly uni-\nformly distributed across the entire range of interest. The discrepancy between predictions\nand measurements likely arises from a lack of data within the corresponding regions.\n4. Material design and optimization\nBased on the predictive models built above, we attempted to \fnd optimized material\ncompositions and heat treatment conditions that can achieve large magnetic saturation and\nminimum loss in the presence of trade-o\u000bs between these two con\ricting objectives. In the\nmaterials informatics \feld, machine learning models often have limitations when predicting\nbeyond the limits of training data [90, 91], and furthermore the random forest model is\nmathematically unable to extrapolate. Therefore, our main goal here is to achieve better\nbalance between incompatible properties near the Pareto front and discover novel materials\nthat can achieve similar or better performance to FINEMET. To achieve this objective,\nwe performed two rounds of optimizations incrementally using the di\u000berential evolution\nalgorithm. Experimental measurements conducted after the \frst round of optimization was\nadded to the data set to try to improve the results in the second round. For the \frst round,\nthe input space of the optimization process was the combination of features in the coercivity\nmodel, magnetostriction model, and magnetic saturation model, which are shown in Table\n3. For the second round, the input space of the optimization process was the combination\nof features in the coercivity model and magnetic saturation model.\n134.1. Di\u000berential Evolution\nDi\u000berential Evolution (DE) [92] is a stochastic population based method that is useful\nfor global optimization problems. It utilizes NP D-dimensional parameter vectors\nxi;G;i= 1;2;:::;NP (3)\nas a population for each generation G. The initial vector population is chosen randomly,\nassuming a uniform probability distribution. DE generates new parameter vectors by adding\nthe weighted di\u000berence between two population vectors to a third vector, which is called\nmutation. The mutated solution is mixed with other candidate solutions to create a trial\ncandidate. In this study, the \\best1bin\" strategy was utilized. In this strategy, two members\nof the population are randomly chosen. Their di\u000berence is used to mutate the trial candidate\n\u0016i;G+1=xr1;G+F\u0001(xr2;G\u0000xr3;G); (4)\nwhiler1;r2;r32f1;2;:::;NPgare random di\u000berent indexes and Fis a constant parameter\n2[0;2] that controls the amplitude of the di\u000berential variation ( xr2;G\u0000xr3;G). In order to\nincrease the diversity of the perturbed parameter vectors, a crossover is introduced. To this\nend, the trial candidate\n\u0017i;G+1= (\u00171i;G+1;\u00172i;G+1;:::;\u0017Di;G +1) (5)\nis formed, where \u0017ji;G+1(j= 1;2;:::;D ) is determined by a binomial distribution. A random\nnumber2[0;1) is generated, if this number is less than the recombination constant that is\ndetermined by the user, then \u0017ji;G+1is loaded from \u0016ji;G+1, otherwise it is loaded from the\noriginal candidate xji;G+1. It is ensured that \u0017i;G+1gets at least one parameter from \u0016i;G+1.\nThe choice of whether to use trial candidate \u0017i;G+1or the original candidate xi;G+1is made\nusing the greedy criterion. Once the trial candidate is built, its \ftness is assessed. If the\ntrial is better than the original candidate, then it takes its place. If it is also better than\nthe best overall candidate, it replaces that value too.\n4.2. Optimization Results\nOur primary goal in designing improved Fe-based soft magnetic alloys is to minimize the\ncore loss to help reduce the energy waste during operation. A secondary goal is to maximize\nthe magnetic saturation. To ensure the existence of Fe-Si phase, a constraint was employed\nto ensure the atomic percentage of Si was no less than 3%. Both design targets can contribute\nto the design of high energy e\u000eciency. Properties like coercivity and magnetostriction, which\ndirectly a\u000bect the core loss could serve as our targets for loss minimization.\nThe \frst choice is to formulate the problem as a single-objective optimization with the\nobjective function being the magnetic saturation. To satisfy the prerequisite of low core loss,\nwe reformulated two objectives as constraints to restrict the design space while maximizing\nthe objective function. The constraints were described as ln(coercivity) not exceeding C0\n(A/m) and magnetostriction not exceeding M0(\u000210\u00006).\n14Feature space Fe, Si, Cu, Ta, Mo, Nb, Zr, B, P, Ta,ts, ln(HC),\u0015,BS\nObjective function V=\u0000\u000b1(BS) +\u000b2ln(HC)\nConstraints Si>3%;\u0015 <3 (\u000210\u00006);\nTa, Mo, Nb, Zr could be constrained to zero in certain cases.\nTable 4: First round multi-objective strategy problem formulation. Where Tais annealing temperature (K)\nandtsis annealing time (s).\nFeature space Fe, Si, Cu, Ta, Mo, Nb, Zr, B, P, Ta,ts, ln(HC),\u0015,BS\nObjective function V=\u0000BS\nConstraints Si>3%;\u0015 <3 (\u000210\u00006); ln(Hc)<-1.5 or 0.5;\nTa, Mo, Nb, Zr could be constrained to zero in certain cases.\nTable 5: First round single-objective strategy problem formulation.\nThe second choice is to formulate the problem as a multi-objective optimization. We only\nreformulate magnetostriction as the constraint and de\fne a composite objective function to\nbe minimized as:\nV=\u0000\u000b1(BS) +\u000b2ln(HC) (6)\nwhere\u000b1and\u000b2values the importance of each of the properties to achieve a balance between\ncon\ricting objectives. The \frst round of optimization emphasized achieving a low coercivity\nand the second round emphasized achieving a high magnetic saturation. For the \frst round,\nthe optimization ran on the composition space of Fe, Si, Cu Ta, Mo, Nb, Zr, B, and P.\nWe constrained our composition space further so that Si was larger than 3% to ensure the\nexistence of the Fe 3Si phase. We tried a total of four di\u000berent strategies of single-objective\nand multi-objective approaches. In single-objective methods, the value of C0was chosen to\nbe\u00001:5 or\u00000:5 andM0was chosen to be 3. In multi-objective methods, M0was also set to\n3 and two di\u000berent weight combinations have been explored in our calculations: \u000b1=\u000b2= 1\nand\u000b1= 4,\u000b2= 1. For the second round, we added the additional experimental results,\nwhich we generated following the \frst round of optimization to our database and re-trained\nthe machine learning model. The second round of optimization ran on a smaller composition\nspace of Fe, Si, Cu, Mo, Nb, B, Ge, and P where Si needed to be larger than 3%. In\nthis round, our model only ran on coercivity and magnetic saturation because of the high\ncorrelation between coercivity and magnetostriction. For the second round, we tried the\nsingle-objective approach and focused on maximizing magnetic saturation with constraints\nofC0to be two separate values of 0.5 or 0. The problem formulations of both \frst and\nsecond round optimizations are shown in Tables 4 to 6.\nFig. 8 shows the di\u000berent combination of trade-o\u000b surface plots for the results of all the\noptimization methods we tried, including both single-objective and multi-objective methods.\nFig. 8(a)-(c) are the plots of experimental measurements from the database and Fig. 8(d)-\n(g) represent the optimization results achieved by applying DE optimization to the RFR\nmodels. When there are two or more objectives, solutions rarely exist that optimize all at\n15Feature space Fe, Si, Cu, Mo, Nb, B, Ge, P, Ta,ts, ln(HC),BS\nObjective function V=\u0000BS\nConstraints Si>3%; Fe>75%; ln(Hc) <0 or 0.5;\nMo, Ge, P could be constrained to zero in certain cases.\nTable 6: Second round problem formulation.\nFigure 8: Trade-o\u000b surface of di\u000berent combinations of magnetic properties. (a)-(c) are from the experi-\nmental data set and (d)-(g) are from the machine learning optimizations. (d)-(f) are for \frst iteration and\n(g) is for second iteration. nis the number of points contained in the plots. The selected results marked by\nindex are shown in Table 7. (h) and (i) are experimental measurements of B-Hloops for di\u000berent samples\nand compared with a commercial FINEMET-like alloy composition processed similar to the experimental\nalloys investigated. (h) is for \frst iteration and (i) is for second iteration.\n16Index Fe Si Cu Ta Mo Nb Zr B P Tats ln(HC)\u0015 B S\u000b1\u000b2C0\n1 74.09 13.67 0.56 0 0 3.05 0.36 8.14 0.14 817 3959 \u00002.53 2.52 1.25 4 1 Inf\n2 74.12 13.38 0.71 0 0 2.69 0.76 8.25 0.09 816 4660 \u00002.53 2.52 1.25 4 1 Inf\n3 73.86 14.22 0.66 0 0 2.69 0 8.19 0.37 816 1206 \u00002.48 2.40 1.25 4 1 Inf\n4 73.93 14.22 0.62 0 0 2.79 0 8.14 0.30 816 121 \u00002.48 2.40 1.25 4 1 Inf\n5 73.74 14.18 0.52 0 0 2.95 0 8.14 0.48 817 4285 \u00002.54 2.41 1.25 1 1 Inf\n6 73.96 14.16 0.61 0 0 2.61 0 8.31 0.35 817 3609 \u00002.54 2.41 1.25 1 1 Inf\n7 73.53 13.35 0.68 0 0.17 3.43 0.24 8.51 0.09 816 5052 \u00002.54 2.50 1.25 1 1 Inf\n8 73.68 13.30 0.70 1.09 0.05 2.78 0.02 8.23 0.15 816 6086 \u00002.53 2.45 1.25 1 1 Inf\n9 74.01 13.47 0.65 0.50 0 2.75 0.20 8.30 0.11 817 3490 \u00002.54 2.50 1.25 1 1 Inf\n10 73.40 14.21 0.69 0.35 0.36 2.72 0 8.02 0.26 816 5091 \u00002.53 2.44 1.25 1 1 Inf\n11 77.13 9.04 0.29 0.97 0.01 3.29 1.17 7.92 0.19 783 907 \u00000:93 1.85 1.46 1 0 -0.5\n12 78.04 9.19 0.31 0.59 0 3.45 0 8.15 0.28 784 4290 \u00000:82 1.85 1.47 1 0 -0.5\n13 79.08 9.26 0.29 0 0.21 3.33 0 7.80 0.04 784 3196 \u00000:80 1.85 1.47 1 0 -0.5\n14 76.67 8.45 0.37 0.58 0 3.12 2.81 7.59 0.42 786 3489 \u00000:84 2.02 1.45 1 0 -0.5\n15 76.62 8.35 0.59 0 1.44 2.55 2.41 7.92 0.12 784 4356 \u00000:72 2.02 1.45 1 0 -0.5\n16 78.49 8.53 0.47 1.56 0.28 2.57 0 8.04 0.06 783 991 \u00000:63 1.73 1.46 1 0 -0.5\n17 79.90 8.96 0.61 0 0 2.54 0 7.98 0.02 785 1145 \u00000.71 1.84 1.47 1 0 -0.5\n18 79.79 8.92 0.57 0 0 2.47 0 7.99 0.26 784 1066 \u00000.71 1.84 1.47 1 0 -0.5\n19 78.69 8.80 0.54 0 0 2.50 1.32 7.92 0.23 784 6038 \u00000.75 2.21 1.47 1 0 -0.5\n20 79.16 8.87 0.63 0 0 2.57 0.52 7.92 0.34 783 5436 \u00000.74 2.19 1.47 1 0 -0.5\n21 83.08 4.23 1.11 0.91 0 0 0 7.09 3.58 723 1159 1.07 2.96 1.84 4 1 Inf\nTable 7: Selected \frst round Optimization results obtained by DE using Random Forest model. Where Tais\nannealing temperature (K) and tsis annealing time (s). \u000b1,\u000b2,C0are optimization parameters, Inf means\nthere's no constraint on coercivity. Constraint on magnetostriction( M0) is always set to be 3.\nIndex Fe Si Cu Mo Nb B Ge P Tats ln(HC)BSC0\n1 76.23 11.95 0.30 0.00 2.25 8.83 0.41 0.04 785 2862 -0.37 1.42 0\n2 76.31 11.96 0.33 0.21 2.21 8.99 0.00 0.00 783 2585 -0.36 1.42 0\n3 76.66 11.88 0.46 0.00 2.25 8.70 0.00 0.04 783 2938 -0.35 1.42 0\n4 76.97 11.51 0.45 0.00 2.29 8.78 0.00 0.00 663 3150 0.44 1.47 0.5\n5 77.04 11.29 0.35 0.00 2.35 8.86 0.12 0.00 661 3078 0.44 1.47 0.5\n6 76.62 11.43 0.49 0.00 2.43 8.58 0.44 0.01 662 3021 0.47 1.47 0.5\n7 76.67 11.67 0.38 0.18 2.26 8.56 0.28 0.00 663 2766 0.47 1.47 0.5\n8 76.88 11.43 0.42 0.14 2.34 8.80 0.00 0.00 662 3132 0.47 1.47 0.5\n9 76.74 11.87 0.48 0.02 2.29 8.61 0.00 0.00 663 3201 0.47 1.47 0.5\nTable 8: Selected second round optimization results obtained by DE using Random Forest model.\nonce. The objectives are normally measured in di\u000berent units, and any improvement in\none is at the loss of another [93]. It can be seen in Fig. 8(d)-(g) that there is a systematic\ntrade-o\u000b between coercivity and magnetostriction, versus magnetic saturation. Increasing\ncoercivity and magnetostriction generally led to an increase of magnetic saturation.\nSolutions marked by numbers and shown in Table 7 and Table 8 are identi\fed as optimum\nsolutions based on the trade-o\u000b surface described by the ln(coercivity)-magnetic saturation\nplot, as these two are de\fned as our main objectives. Further selection in the optimum set\ncould be based on di\u000berent application scenarios and di\u000berent weighting strategies of the\ntwo competing aspects.\nTwo-dimensional t-distributed stochastic neighbor embedding (t-SNE) was used to vi-\nsualize how the optimized alloys compare to the ones presented in the published literature.\nFig. 9 displays this t-SNE mapping, which was conducted on the processing space of the\nliterature database as well as the alloys found in the \frst and second rounds of optimization,\nshown in Tables 7 and 8 respectively. In this plot, the processing space was de\fned as the\n17Figure 9: A t-SNE mapping of the compositions and processing conditions of the alloys in the literature\ndatabase as well as the alloys found from the \frst and second rounds of optimization. Note that the precise\npositions of alloys in this plot are not quantitatively meaningful.\ncomposition of every element in each alloy as well as the annealing temperature and time,\nscaled by their respective maximum and minimum values in the database. For the purpose\nof visualization, \\FINEMET-like\" alloys were de\fned as alloys containing greater than 3%\nSi while \\Non-FINEMET-like\" alloys contain less Si. These groups seem rather separable\nin Fig. 9, indicating that they represent two distinct classes of materials in the literature.\nThe alloys found in both rounds of optimization are shown among the FINEMET-like alloys,\nwhich is to be expected given the constraints on Si and Fe shown in Tables 4, 5, and 6. While\nthe optimized alloys are not shown to exist entirely separate from literature data, they do\nnot match any alloy from literature exactly. Optimized alloys also appear to lie near the\nbounds of their respective groups, indicating that further iterations of optimal experimental\ndesign could expand the boundaries of current knowledge in this space.\n5. Experimental validation\nTo experimentally validate our machine learning model, several predicted compositions\nnear the Pareto front of the ln(coercivity) - magnetic saturation plots have been synthesized.\nFor the \frst round, we chose three points No.14, No.17 and No.19 in the points region of\nintermediate value of both coercivity and magnetic saturation as shown in Fig. 8(d). While\nfor the second round, we chose one point from each of the three point regions, namely\n18No.2, No.4 and No.5, to try to test the behavior of di\u000berent regions of point segregation in\nFig. 8(g).\nThe alloys produced for this study were melted from elemental constituents and \ripped\nand remelted several times to ensure homogeneity and then cast into cigar-shaped ingots.\nThe ingots weighed approximately 60 g. The ingots were then used as melt stock for melt\nspinning using an Edward Buehler HV melt spinner using a planar \row casting process\nand wheel speed of 25.9 m/s. The cast ribbons were approximately 19.5 microns thick and\n\u001816.5 mm wide. Composition of all melt-spun ribbons were con\frmed by inductively cou-\npled plasma atomic emission spectroscopy (ICP-AES). The ribbon was wound into small\ncores, wrapped with a piece of copper wire to hold the core together and then heat-treated\nin argon after \frst pulling a vacuum to approximately 1 \u000210\u00007torr and heat-treated at the\ntimes/temperatures speci\fed. The heating rate was 3\u000eC/min and samples were cooled at\na rate of 8\u000eC/min after the speci\fed treatment. Properties of the wound cores were deter-\nmined by IEEE Standard 393-1991: Standard for Test Procedures for Magnetic Materials.\nTesting was performed at 1000 Hz.\nFrom the resulting B-Hloops as shown in Figs. 8(h) and (i), magnetic properties were\ndetermined, including BSandHC. The comparison of predicted and experimental properties\nof selected samples from both the \frst iteration and second iteration are shown in Table 9.\nThe compositions are recorded by experimental measurement of the prepared samples and\nthe heat treatment times and temperatures are the same as the model predictions and listed\nin Tables 7 and 8. It should be noted from Figs. 8(h) and (i) that sample No.19 in \frst round\nand No.2 in second round have similar performance compared to commercial FINEMET-like\nalloys, which shows that our approach is e\u000bective in identifying other compositions with very\ngood properties.\nWe collected a substantial portion of the reported experimental data, which is shown in\nFig. 8(a)-(c) as Ashby plots. Although there are a large number of data points describing\nln(coercivity)-magnetic saturation property space in Fig. 8(a), it can still be observed that\nthe high-BS, low-HCarea (top left corner) is completely empty. The ultimate goal of the\nmaterial design is to breach the boundary and reach the target area. To improve the machine\nlearning model, we need more data, and an e\u000ecient way is to explore the property space\nthat is missing, if at all possible. Another potential strategy is to continue to iterate the\ndesign process. By using experimental results of proposed compositions associated with\nbetter properties it may be possible to continue to iterate until the target properties are\nobtained.\nAs a complex material system, soft magnetic nanocrystalline alloys usually contain sev-\neral elements, which makes it di\u000ecult to decide what elements should be included. Several\nprinciples could be considered: (a) For applications of soft magnetic materials, the prices\nof elements are non-negligible factors. In our design process, the selection of late transition\nmetal is a typical issue due to the hefty price di\u000berence between Au and Cu. The price of\nCu is about 6 USD/kg, Ag about 500 USD/kg and Au over 40,000 USD/kg. (b) If possible,\nwe try not to include too many di\u000berent elements that serve the same function in the alloy.\n(c) In this work, the only magnetic element included is Fe, which can provide a pure \u000b-Fe3Si\nnanocrystalline phase. Based on these principles, we proposed the optimized compositions\n19Index Composition ln( HC) (A/m) BS(T)\nRound 1\nPredicted Measured Predicted Measured\n14 Fe 77:1Si8:5Cu0:4Nb3:1Ta0:6B7:3P0:3Zr2:8 -0.82 2.19 1.455 0.915\n17 Fe 80Si9Cu0:7Nb2:5B7:9 -0.69 3.39 1.471 0.913\n19 Fe 78:9Si8:9Cu0:6Nb2:5B7:7P0:2Zr1:2 -0.76 1.79 1.466 1.146\nRound 2\nPredicted Measured Predicted Measured\n2 Fe 76:3Si12Cu0:3Nb3:1Mo0:2Nb2:2B9 -0.36 2.58 1.42 1.195\n4 Fe 77Si11:5Cu0:4Nb2:3B8:8 0.44 3.25 1.47 1.088\n5 Fe 77Si11:3Cu0:4Nb2:4B8:8Ge0:1 0.44 3.64 1.47 1.014\nTable 9: Predicted and measured properties of soft magnetic alloys from the \frst and second round of\noptimization. The index corresponds to the value in Tables 7 and 8. Compositions are the experimental\nvalues from samples. Heat treatment time and temperature are referenced from Tables 7 and 8.\nlisted in Table 7 and Table 8 for potential experimental validations, in addition to the se-\nlected alloys con\frmed in Table 9. Note that, as shown in Table 9, there are discrepancies\nbetween predicted and measured values, which probably arises due to several facts: (a) We\nhave been predicting properties in a high-dimensional space without adequate data; (b)\nRandom forest models have limitations for predicting extreme values. Modern development\nin machine learning algorithms such as attention-based network [94] could be employed to\nimprove the performance. By running more iterations of our optimized experimental de-\nsign framework, the models will become more robust in predicting outstanding magnetic\nproperties.\n6. Summary\nIn this work, we built a general database for Fe-based FINEMET-type soft magnetic\nnanocrystalline alloys using experimental data from all available literature. Based on this,\nmachine learning techniques were applied to analyze the statistical inference of di\u000berent\nfeatures and then build predictive models to establish the relation between materials prop-\nerties and material compositions along with processing conditions. We chose the random\nforest model as our modeling tool due to its better performance compared with several other\nmachine learning methods. Optimization process has been performed to establish and then\nsolve the inverse problem that is to \fnd a suitable combination of element components and\nprocessing conditions to achieve minimum loss and maximum magnetic saturation. Experi-\nmental validations have been applied on several predicted materials, which showed that the\npredicted novel material could have similar performance as the commercial FINEMET-like\nalloys. Furthermore, the collected data set and analysis procedure can create more insight on\nhow to design the next-generation optimized Fe-based soft magnetic nanocrystalline alloys\nmotivated by various applications. The data set and analysis code are available on Github\n[95].\n20Acknowledgments\nThe authors would like to acknowledge the Data-Enabled Discovery and Design of Energy\nMaterials (D3EM) program funded through the National Science Foundation (NSF) Award\nNo. 1545403. Y.W. also acknowledges the \fnancial support of the NSF through Award No.\n1508634. and 1905325. Y.T. also acknowledges the \fnancial support of the College of Science\nStrategic Transformative Research Program (COS-STRP) at Texas A&M University and the\nRobert A. Welch Foundation, Grant No. A-1526. 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Sparks, Compositionally-restricted attention-based network for\nmaterials property prediction doi:10.26434/chemrxiv.11869026.v1 .\nURL https://chemrxiv.org/articles/Compositionally-Restricted_Attention-Based_\nNetwork_for_Materials_Property_Prediction/11869026\n[95] Machine learning approach to fe-based soft magnetic nanocrystalline materials design (2020).\nURL https://github.com/yuhaowang2018/Machine_learning_approach_to_Iron_based_soft_\nmagnetic_nanocrystalline_materials_design\n25"    },    {        "title": "2002.11595v1.Chern_number_and_orbital_magnetization_in_ribbons__polymers__and_layered_materials.pdf",        "content": "Chern number and orbital magnetization in ribbons, polymers, and layered materials\nEnrico Drigo1and Ra\u000baele Resta2;3\u0003\n1Dipartimento di Fisica, Universit\u0012 a di Trieste, Strada Costiera 11, 34151 Trieste, Italy\n2Istituto O\u000ecina dei Materiali IOM-CNR, Strada Costiera 11, 34151 Trieste, Italy and\n3Donostia International Physics Center, 20018 San Sebasti\u0013 an, Spain\n(Dated: 13{Aug{21)\nThe modern theory of orbital magnetization addresses crystalline materials at the noninteracting\nlevel: therein the observable is the k-space integral of a geometrical integrand. Alternatively, magne-\ntization admits a local representation in rspace, i.e. a \\density\", which may address noncrystalline\nand/or inhomogeneous materials as well; the Chern number admits an analogous density. Here we\nprovide the formulation for ribbons, polymers, nanowires, and layered materials, where both k-space\nandr-space integrations enter the de\fnition of the two observables.\nI. INTRODUCTION\nBy de\fnition, orbital magnetization Mis the deriva-\ntive of the macroscopic free-energy density with respect\nto the magnetic \feld (orbital term thereof, and with a mi-\nnus sign). Customarily, the \feld adopted is H; instead|\nbecause of the reasons well explained in Ref. 1|\frst-\nprinciple theory adopts the more fundamental \feld B.\nThe modern theory of orbital magnetization dates since\n2006; therein, the observable Mis cast as the k-space\nintegral of a geometrical integrand.2The expression ad-\ndresses on the same ground trivial insulators, Chern in-\nsulators, and metals; it is also clear since then that M\nand the Chern invariant are closely related quantities on\nthe theory side. In insulators the band spectrum is al-\nways gapped, but in the Chern case Mdepends on the\nvalue of the Fermi level \u0016in the gap: this behavior is\nconsistent with the fact that, in a bounded sample, the\nspectrum is not gapped and the topologically protected\nedge states contribute to M.\nIn more recent years it has been shown that both\nthe Chern invariant and Mcan also be de\fned for a\nbounded sample, where there is no k-vector to speak of:\nboth observables can be computed by means of an r-\nspace integration, where the integrand is to be regarded\nas the dual version of its k-space counterpart.3,4While\nthek-space theory requires crystalline periodicity, the r-\nspace approach is capable of dealing with strongly disor-\ndered cases and/or macroscopically inhomogeneous ma-\nterials as well. In this paper we are going to address the\n\\hermaphrodite\" cases,5i.e. those which require an inte-\ngration over both r-space and k-space. The paradigmatic\nsystem in this class is a ribbon: a 2 dmaterials bounded\nin one Cartesian direction and lattice periodical in the\nother. The ribbon is also the simplest at the level of for-\nmulation and notations; the other hermaphrodite cases\nbasically require to adopt the same logics within di\u000berent\nnotations. In this work we provide explicit formul\u001a, thus\nextending the \frst-principle theory of both observables\nbeyond their known formulation. In the ribbon case, we\nvalidate our expressions by means of model-Hamiltonian\nsimulations.\nThe paper is organized as follows. In Sec. II we showin detail the derivation of the ribbon magnetization for-\nmula; this also sets the logics to be adopted in the other\nhermaphrodite cases. The ribbon formula for the Chern\nnumber is derived as well. In Sec. III we provide a few\ntest-case simulations based on the (by now famous) Hal-\ndane Hamiltonian.6In the following Sec. IV we provide\nexplicit formul\u001a for the case of either a stereoeregular\npolymer or a nanowire (where only the normal compo-\nnent of Mneeds a nontrivial approach) and for the case\nof a layered material (where only the in-plane component\nofMwas not accessible to the existing theory). Finally,\nin Sec. V we draw some conclusions.\nII. CHERN NUMBER AND ORBITAL\nMAGNETIZATION IN A RIBBON\nThe 2dorbital magnetization Mis a pseudoscalar with\nthe dimensions of an orbital moment per unit area, while\nthe Chern invariant is the (dimensionless) Chern number\nC12Z. In the topological case Mas a function of \u0016in\nthe gap is\nM(\u0016) =M(0)\u0000\u0016e\nhcC1; (1)\nwhere the zero of \u0016is set at the top of the valence states;\nnotice that the ribbon as a whole is gapless, but its bulk\nis insulating. We address a ribbon of width win the\nx-direction, and lattice-periodical (\\crystalline\") along y\nwith periodicity a(\"lattice constant\"). The elementary\nde\fnition of Mis given as the circulation of the micro-\nscopic orbital current per unit area:\nM=1\n2cwaZ1\n\u00001dxZa\n0dyxj(micro)\ny (x;y); (2)\nan expression dominated by boundary currents (we as-\nsume that the macroscopic current vanishes). As \frst\nproved in Ref. 4, Mis a local observable and admits a\nmicroscopic \\density\" called M(r); in the case of a ribbon\nwe have\nM=1\nwaZ1\n\u00001dxZa\n0dyM(x;y): (3)arXiv:2002.11595v1  [cond-mat.mes-hall]  26 Feb 20202\nThe macroscopic average of M(x;y) can be identi\fed\nwith (minus) the B-derivative of the free-energy density.\nEqs. (2) and (3) provide an identical Mat any width\nw, but their integrands are very di\u000berent. The trans-\nformation is similar in spirit to an integration by parts,\nandM(x;y) is nota function of the microscopic orbital\ncurrent.\nWhat remains to be done is to express Min terms of\n1dBloch orbitals, thus transforming the y-integral into\nak-integral over the 1 dBrillouin zone (BZ). We expect\nthat Eqs. (2) and (3) converge to the bulk Mvalue like\n1=w, but it will be shown that the present approach also\nallows to approach the large- wlimit in a more e\u000ecient\nway.\nIn all the hermaphrodite cases, the occupied orbitals\nobey the so-called open boundary conditions (OBCs) in\nsome Cartesian direction(s), and Born-von-K\u0012 arm\u0012 an pe-\nriodic boundary conditions (PBCs) in some other(s). In\norder to address both cases, one needs a \\bridge\" provid-\ning a seamless path between the two frameworks. The\nkey ingredient of this bridge must be the ground-state\nprojectorP(a.k.a. the one-body density matrix), whose\nvirtue is rooted in the \\nearsightedness\" principle, and\nwhich applies toP, but not to the individual eigenstates.7\nNot surprisingly, Pis one essential ingredient of our for-\nmalism.\nThe microscopic magnetization density M(r) may be\ncast in several equivalent forms; here|inspired by Ref.\n8|we adopt (in either 2 dor 3d)\nM(r) =\u0000ie\n2~chrjjH\u0000\u0016j[r;P]\u0002[r;P]jri\n=e\n~cImhrjjH\u0000\u0016j[r;P] [r;P]jri; (4)\nwherejH\u0000\u0016j= (H\u0000\u0016)(I\u00002P), i.e. it is the operator\nwhich acts as ( \u0016\u0000H) over the occupied states, and as\n(H\u0000\u0016) over the unoccupied ones. From Eq. (4) the\n\u0016-derivative of Min 2dis\nd\nd\u0016M(r) =e\nhc4\u0019ImhrjjP[x;P] [y;P]jri=\u0000e\nhcC(r);\n(5)\nwhere C(r) is a \\topological marker\" (a.k.a. Chern den-\nsity), equivalent to the one de\fned in Ref. 4; Eq. (5) is\npespicuously consistent with Eq. (1).\nEqs. (4) and (5) are obviously well de\fned within\nOBCs, but it is not so obvious that they are well de-\n\fned even in the crystalline case, where the Hamiltonian\nHis lattice periodical and Pprojects over a set of oc-\ncupied Bloch orbitals. The multiplicative operator ris a\ntrivial one within OBCs, but is \\forbidden\" within PBCs:\nit maps a vector in the PBCs Hilbert space into some-\nthing not belonging to the space.9The commutators in\nEqs. (4) and (5) e\u000bectively \\tame\" the nasty multiplica-\ntiver: this can be seen as follows. In a crystalline ma-\nterial the projector P(as well as any other legitimate\noperator) is lattice-periodical:\nhrjPjr0i=hr+RjPjr0+Ri; (6)where Ris a lattice vector. It is immediate to verify that\ni[r;P] is indeed a legitimate, lattice periodical, Hermitian\noperator.\nThe Bloch orbitals of a ribbon are j jki= eikyjujki,\nwithhx;yjujkisquare-integrable along xand periodical\nalongy; we normalize them as\nZ1\n\u00001dxZa\n0dyhx;yjujkihuj0kjx;yi=\u000ejj0: (7)\nThejujkiare eigenstates of Hk= e\u0000ikyHeikywith eigen-\nvalues\u000fjk. Within these notations, the ground-state pro-\njector per spin channel is, in the Schr odinger representa-\ntion:\nhrjPjr0i=a\n2\u0019Z\nBZdkeikyhrjPkjr0ie\u0000iky0; (8)\nPk=X\n\u000fjk\u0014\u0016jujkihujkj: (9)\nThe integrand in Eq. (8) is a periodic function of kwith\nperiod 2\u0019=a, ergo the BZ integral of its k-derivative van-\nishes:\n0 =a\n2\u0019Z\nBZdkd\ndk(eik(y\u0000y0)hrjPkjr0i) (10)\n=i(y\u0000y0)hrjPjr0i+a\n2\u0019Z\nBZdkeikyhrjP0\nkjr0ie\u0000iky0;\nwhereP0\nkis thek-derivative ofPk. In compact operator\nnotations this reads\ni[y;P] =\u0000a\n2\u0019Z\nBZdkeikyP0\nke\u0000iky; (11)\nwhile the other commutator is identically written as\n[x;P] =a\n2\u0019Z\nBZdkeiky[x;Pk]e\u0000iky0; (12)\nwe also cast the Hamiltonian in a similar form:\nH=a\n2\u0019Z\nBZdkeikyHke\u0000iky; (13)\nwhere the integrand is actually k-independent. We have\nby now all the ingredients needed to evaluate the triple\nproduct in Eq. (4), ergo in Eq. (3). The triple k-\nintegration contracts to one (details are given in the Ap-\npendix) and we get\nM(r) =e\n~cIm\u0012ia\n2\u0019Z\nBZdk\n2\u0019hrjjHk\u0000\u0016j[x;Pk]P0\nkjri\u0013\n=ea\nhcReZ\nBZdk\n2\u0019hrjjHk\u0000\u0016j[x;Pk]P0\nkjri: (14)\nThis is a periodic function of y; we take the trace over the\n1dcell to obtain the (microscopic) linear magnetization\ndensity (as a function of the bounded coordinate)\nM(x) =e\nhcReZ\nBZdk\n2\u0019TryfjHk\u0000\u0016j[x;Pk]P0\nkg:(15)3\nFIG. 1. (color online). A typical armchair Haldanium ribbon;\nthe linear cell (46 sites in the \fgure) and the central cell (4\nsites) are shown. We performed simulation up to '100 sites\nin the linear cell.\nM=1\nwZ1\n\u00001dxM(x): (16)\nEq. (15) is one of the major results of this work: it obvi-\nously yields Mvia Eq. (16), but|as shown below|can\nbe used in a more e\u000ecient way by averaging it in the\ncentral region of the ribbon (not on the whole ribbon).\nThe analogous formulae for the topological marker and\nfor the Chern number are\nC(x) =\u00004\u0019ReZ\nBZdk\n2\u0019TryfPk[x;Pk]P0\nkg: (17)\nC1=1\nwZ1\n\u00001dxC(x): (18)\nOwing to gauge invariance, the k-derivative ofPcan\nbe safely evaluated by numerical di\u000berentiation. In a\ntight-binding implementation (as the one shown below)\nthe evaluation of the trace amounts to perform multi-\nplications of small matrices. In a \frst-principle imple-\nmentation it would perhaps be more convenient to write\nthe expression in terms of the jujkiin the \\Hamiltonian\ngauge\",2i.e. using Eq. (14) as it is.\nIII. SIMULATIONS FOR AN \\HALDANIUM\"\nRIBBON\nThe paradigmatic model for validating results of the\npresent kind is the one proposed by Haldane in 1988.6It\nis a tight-binding 2 dHamiltonian on a honeycomb lat-\ntice with onsite energies \u0006\u0001, \frst neighbor hopping t1,\nand second neighbor hopping t2=jt2jei\u001e, which pro-\nvides time-reversal symmetry breaking. The model is\ninsulating at half \flling and metallic at any other \fll-\ning; according to the parameter choice the insulator can\nbe either trivial or topological (Chern number \u00061). The\nmodel has been previously used to demonstrate the local-\nity of M, and implemented for bounded samples within\nOBCs in order to address insulators (both trivial and\ntopological)4,10and metals.11We are going to benchmark\nthe present hermaphrodite results by implementing the\nmodel Hamiltonian on a periodic \\Haldanium\" ribbon,\nas the one shown in Fig. 1.\n 8.5 9 9.5 10 10.5 11\n 30  40  50  60  70  80  90  100  110Magnetization [ 10-3 e/−hc units]\nNM\nMbulkMcellFIG. 2. (color online). Orbital magnetization in the nontopo-\nlogical case as a function of the ribbon width. The symbols\nM,Mbulk, andMcellare de\fned in the text. Units of e=(~c)\nWe evaluate C1andMat \fnite widths, and we study\ntheir convergence as a function of the number of sites, us-\ning three di\u000berent expressions. In the \fgures the symbol\nMresults from Eq. (16), Mbulkresults from averaging\nover 1/2 of the sites in the central region, Mcellresults\nfrom averaging over the central cell (four sites); similar\nsymbols are adopted for C1. In the topological case the\nribbon is gapless; we therefore adopt a \\smearing\" tech-\nnique, common to many metallic simulations. In order\nto present homogeneous results, we adopt the smearing\neven in the topologically trivial case.\nAs a prototype of nontopological Haldanium we adopt\nthe parameters t1= 1,t2= 1=3, \u0001 = 1:5,\u001e=\u0019=4 at\nhalf \flling; this choice also allows benchmarking to Ref.\n-3-1.5 0 1.5 3\n-1.5 -1 -0.5  0  0.5  1  1.5  2  2.5  3  3.5Magnetization [10-1 e/−hc units]\nµMcell\nFIG. 3. Central-cell orbital magnetization Mcellin the topo-\nlogical case as a function of the Fermi level \u0016in units ofe=(~c).\nAfter Eq. (1) the \u0016-derivative of Mis\u00001=(2\u0019)'\u00000:159 in\nthe plot units.4\n 0.85 0.9 0.95 1\n 30  40  50  60  70  80  90  100  110C \nNC\nCbulkCcellfit 1/N\nFIG. 4. Convergence of the Chern number with the ribbon\nwidth.C,Cbulk, andCcellare de\fned in the text, in analogy\nto Fig. 2\n11, where some simulations adopt the same Haldanium\nparameters. As a prototypical topological ( C1= 1) case\nwe adopt a nonpolar case: t1= 1,t2= 1=3, \u0001 = 0,\n\u001e=\u0019=2; the simulations have been performed at various\n\u0016values.\nWe start displaying the nontopological case in Fig. 2.\nAs expected|and consistently with Eq. (2)|the inte-\ngral over the whole ribbon converges only like 1 =w; the\nother curves converge instead much faster, owing to the\nquasi-exponential decay of the projector Pin the present\ninsulating case. Next we switch to our topological case\nstudy; it has been proved that even in this case the pro-\njectorPhas a quasi-exponential decay,12and in fact the\nplot (not reproduced here) is qualitatively quite similar\nto Fig. 2 for any \u0016value in the bulk gap. Next we\nshow, for the same topological case, the value of Mcell,\nas de\fned above, when the Fermi level \u0016is varied across\nthe gap in Eq. (15). The perspicuous linear behavior is\ndue to the \flling of the topologically protected boundary\nstates.\nFinally in Fig. 4 we show the convergence of our topo-\nlogical markerC(x), Eq. (17), to the Chern number C1,\nwhereC,Cbulk, andCcellare de\fned analogously as for\nthe magnetization plots. Here again the convergence is\n1=wwhenC(x) is averaged over the whole ribbon, as in\nEq. (18), but it is exponential when the average is per-\nformed over an inner sample region.\nIV. POLYMERS AND LAYERED MATERIALS\nAs said above, there is a family of hermaphrodite\ncases: (i) 2 dmaterials bounded in one Cartesian direc-\ntion and lattice periodical in the other (ribbons, dealt\nwith above); (ii) 3 dmaterials bounded in 2 directions\nand lattice-periodical in the third (stereoregular poly-\nmers and nanowires, where only the normal Mcompo-nent is problematic); (iii) 3 dmaterials bounded in one di-\nrection and lattice-periodical in the remaining two (where\nthe in-plane component of Mis problematic). Above\nwe have discussed and demonstrated|via tight-binding\nsimulations|the test case of a ribbon. The correspond-\ning formul\u001a for cases (ii) and (iii) above are reported in\nthe following.\nMagnetization of polymers and nanowires\nWe deal here with a T-breaking quasi-1 dsystem, pe-\nriodic along zwith period a. The Bloch orbitals are\nj jki= eikzjujki, normalized as\nZ1\n\u00001dxZ1\n\u00001dyZa\n0dzhrjujkihuj0kjri=\u000ejj0;(19)\nand the ground-state projector is (per spin channel):\nhrjPjr0i=a\n2\u0019Z\nBZdkeikzhrjPkjr0ie\u0000ikz0;(20)\nPk=X\n\u000fjk\u0014\u0016jujkihujkj: (21)\nSince the system is microscopic in the ( x;y) plane, the\nintensive quantity of interest Mis de\fned as (minus) the\nB-derivative of the free-energy per unit length (although\nthe system is actually 3-dimensional).\nThez-component af Mis simply proportional to the\norbital moment per unit length:\nMz=\u0000e\n2acZ1\n\u00001dxZ1\n\u00001dyZa\n0dz\n\u0002[xj(micro)\ny (r)\u0000yj(micro)\nx (r)]; (22)\nthis is well de\fned since the system is bounded in the\n(x;y) directions. The normal components requires in-\nstead to be addressed via the modern theory. From the\nmain text it follows that\nMx=1\naZ1\n\u00001dxZ1\n\u00001dyZa\n0dzMx(r);\nMx(r) =e\n~cImhrjjH\u0000\u0016j[y;P] [z;P]jri:(23)\nThe commutator [ z;P] is then transformed as in the\nmain text. After contracting the k-integrals (see the Ap-\npendix) we get, similarly to the ribbon formula in Sec.\nII:\nMx(r) =ea\nhcReZ\nBZdk\n2\u0019hrjjHk\u0000\u0016j[x;Pk]P0\nkjri:(24)\nMagnetization of lattice-periodical slabs\nWe consider a 3 dsystem which is bounded in the z\ndirection and lattice periodical in the ( x;y) coordinates.5\nThe Bloch orbitals are j jki= ei(kxx+kyy)jujki, normal-\nized as\nZ1\n\u00001dzZ\nBZdkhrjujkihuj0kjri=\u000ejj0; (25)\nwhere kis the 2dBloch vector and BZ the relative Bril-\nlouin zone. The ground-state projector is (per spin chan-\nnel):\nhrjPjr0i=Ac\n(2\u0019)2Z\nBZdkeik\u0001rhrjPkjr0ie\u0000ik\u0001r0;(26)\nPk=X\n\u000fjk\u0014\u0016jujkihujkj; (27)\nwhere only the ( x;y) components of renter the products\nk\u0001r. Thejujkiare eigenstates of Hk= e\u0000ik\u0001rHeik\u0001r;\nnotice thatHkis a 3dHamiltonian, and kis a 2dBloch\nvector.\nThe intensive quantity Mof interest is the magnetiza-\ntion per unit area. The zcomponent of Mcan be derived\nfrom the standard modern theory of orbital magnetiza-\ntion, as shown in the original literature. Here we address\nthe in-plane component of M; ifAcis the area of the 2 d\nunit cell, then\nMx=1\nAcZ1\n\u00001dzZ\nAcdxdyMx(r);\nMx(r) =e\n~cImhrjjH\u0000\u0016j[y;P] [z;P]jri:(28)\nHere again we transform only the commutator [ y;P]:\ni[y;P] =\u0000AcZ\nBZdk\n(2\u0019)2eik\u0001r(@kyPk)e\u0000ik\u0001r; (29)\nwhile the other two entries in the matrix element are\nH=AcZ\nBZdk\n(2\u0019)2eik\u0001rHke\u0000ik\u0001r\n[z;P] =AcZ\nBZdk\n(2\u0019)2eik\u0001r[z;Pk]e\u0000ik\u0001r: (30)\nAfter contracting the three k-integrals (see the Ap-\npendix) we get, in analogy to the ribbon case,\nMx(r) =eAc\n~cRehrjjHk\u0000\u0016j(@kyPk) [z;Pk]jri:(31)\nV. CONCLUSIONS\nWe have shown how to extend the theory of orbital\nmagnetization beyond the two cases dealt so far in the\nliterature: periodic crystalline materials, where Mis the\nreciprocal space integral of a geometrical integrand;2and\nbounded samples (possibly noncrystalline), where the\nmagnetization density has a well de\fned expression in r-\nspace.4,11Similarly, the Chern number enjoyed a knowndual picture.3Here we have completed the theory of or-\nbital magnetization, providing explicit formul\u001a for all\nthe cases which require integration over both reciprocal\nspace and coordinate space.\nWe have also provided a formulation for the Chern\nnumberC1in a ribbon geometry; the study of its con-\nvergence as a function of the ribbon width wyields some\nimportant comments. Our formula converges like 1 =w\nwhen integrated over the whole ribbon, and instead ex-\nponentially when integrated in a more e\u000ecient way (see\ntext). When an unbounded crystalline sample is con-\nsidered,C1is computed as a k-integral on a 2 dBZ: in\nthis case even a coarse k-mesh provides the converged\nresult.13If instead we address a \rake (a sample bounded\nin both Cartesian directions), the integral over the whole\n\rake is zero: the boundary therefore yields an extraordi-\nnary negative contribution,3and the topological marker\nC(r)does not average to one over a line. The boundary\nacts as a \\reservoir\": the marker may equal one in the\nbulk only if the boundary provides a negative compen-\nsating contribution.\nThe fundamental reason for the di\u000berence between un-\nbounded samples and bounded samples is that the trace\nof the commutator of two \fnite-size matrices is zero,\nwhile the commutator of two unbounded operators may\nhave a nonzero diagonal. In our ribbon case the opera-\ntor is bounded in the xdirection and unbounded in the\nyone: the nontrivial message from Fig. 4 is that|at\nvariance with the \rake case|there are no extraordinary\nboundary contributions. A \\reservoir\" is not needed: the\naverage ofC(x) over the whole ribbon converges indeed\ntoC1, although slowly.\nLast but not least, the case of a \fnite B\feld is worth\na comment. For bulk materials a macroscopic \feld is in-\ncompatible with PBCs (except in commensurate cases);\nthe modern theory only addresses spontaneous magneti-\nzation. Instead, in all the hermaphodite cases the adop-\ntion of the appropriate Landau gauge yields a periodical\nHamiltonian. The present formulation can therefore by\napplied in principle even to cases where a \fnite B\feld\nis present. Care has to be taken, though, because of the\nubiquitous presence of Landau levels. The problem is\nhighly nonanalytical at B= 0, and the density of states\nchanges qualitatively in an abrupt way as soon as B6= 0\nis set.11\nACNOWLEDGMENTS\nUseful discussions with A. Marrazzo and R. Bianco are\ngratefully acknowledged. Work supported by the ONR\nGrant No. No. N00014-17-1-2803.6\nAPPENDIX: PRODUCTS OF\nLATTICE-PERIODICAL OPERATORS\nWe are going to make use of a simple lemma, about\nthe integral of a plane wave eikytimes a periodic function\nf(y):\nZ1\n\u00001dyeikyf(y) =2\u0019\na\u000e(k)Za\n0dy0f(y0): (32)Any ribbon-periodic operators AandBin the\nSchr odinger representation can be written as:\nhrjAjr0i=a\n2\u0019Z\nBZdkeikyhrjAkjr0ie\u0000iky;\nhrjBjr0i=a\n2\u0019Z\nBZdkeikyhrjBkjr0ie\u0000iky0;(33)\nwhereAkandBkare periodic in randr0separately . The\ndiagonal element of the product is then\nhrjABjri=a2\n(2\u0019)2Z\nBZdkZ\nBZdk0ei(k\u0000k0)yZ1\n\u00001dx\"Z1\n\u00001dy\"hrjAkjr\"iei(k0\u0000k)y\"hr\"jBk0jri\n=a\n2\u0019Z\nBZdkZ1\n\u00001dx\"Za\n0dy\"hrjAkjr\"ihr\"jBkjri\n=a\n2\u0019Z\nBZdkhrjAkBkjri: (34)\nThis contraction is associative and can be repeated for\nthree operators. It can also be generalized to systemperiodic in 2 or 3 dimensions, with an obvious change of\nnotations.\n\u0003resta@iom.cnr.it\n1D. J. Gri\u000eths, Introduction to Electrodynamics, 3rd Ed.\n(Prentice-Hall, 1999).\n2D. Vanderbilt, Berry Phases in Electronic Structure The-\nory(Cambridge University Press, Cambridge, 2018).\n3R. Bianco and R. Resta, Phys. Rev. B 84, 241106(R)\n(2011).\n4R. Bianco and R. Resta, Phys. Rev. Lett. 110, 087202\n(2013).\n5We borrow the term \"hermaphrodite\" from the nautical\nnomenclature: \\hermaphrodite brig\" is a two-masted ship\nrigged with a di\u000berent kind of sails on each mast.6F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).\n7W. Kohn, Phys. Rev. Lett. 76, 3168 (1996).\n8H. Schulz-Baldes and S. Teufel, Commun. Math. Phys.\n319, 649 (2013).\n9R. Resta, Phys. Rev. Lett. 80, 1800 (1998).\n10R. Bianco and R. Resta, Phys. Rev. B 93, 174417 (2016).\n11A. Marrazzo and R. Resta, Phys. Rev. Lett. 116, 137201\n(2016).\n12T. Thonhauser and D. Vanderbilt, Phys. Rev. B 74, 235111\n(2006).\n13T. Fukui, Y. Hatsugai, and H. Suzuki, J. Phys. Soc. Japan\n74, 1674 (2005)."    },    {        "title": "2003.00012v2.High_throughput_calculations_of_magnetic_topological_materials.pdf",        "content": "High-Throughput Calculations of Magnetic Topological Materials\nYuanfeng Xu,1Luis Elcoro,2Zhida Song,3Benjamin J. Wieder,4, 5, 3M. G. Vergniory,6, 7\nNicolas Regnault,8, 3Yulin Chen,9, 10, 11, 12Claudia Felser,13, 14and B. Andrei Bernevig3, 1, 15,\u0003\n1Max Planck Institute of Microstructure Physics, 06120 Halle, Germany\n2Department of Condensed Matter Physics, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain\n3Department of Physics, Princeton University, Princeton, New Jersey 08544, USA\n4Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA\n5Department of Physics, Northeastern University, Boston, MA 02115, USA\n6Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain\n7IKERBASQUE, Basque Foundation for Science, Bilbao, Spain\n8Laboratoire de Physique de l’Ecole normale supérieure,\nENS, Université PSL, CNRS, Sorbonne Université,\nUniversité Paris-Diderot, Sorbonne Paris Cité, Paris, France\n9School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China\n10ShanghaiTech Laboratory for Topological Physics, Shanghai 200031, China\n11Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, UK\n12State Key Laboratory of Low Dimensional Quantum Physics,\nDepartment of Physics and Collaborative Innovation Center of Quantum Matter, Tsinghua University, Beijing 100084, China\n13Max Planck Institute for Chemical Physics of Solids, Dresden D-01187, Germany\n14Center for Nanoscale Systems, Faculty of Arts and Science,\nHarvard University, 11 Oxford Street, LISE 308Cambridge, MA 021138, USA\n15Physics Department, Freie Universitat Berlin, Arnimallee 14, 14195 Berlin, Germany\n(Dated: February 11, 2023)\nThe discoveries of intrinsically magnetic topological materials, including semimetals with a large\nanomalous Hall effect and axion insulators [1–3], have directed fundamental research in solid-state\nmaterials. Topological Quantum Chemistry [4] has enabled the understanding of and the search\nfor paramagnetic topological materials [5, 6]. Using magnetic topological indices obtained from\nmagnetic topological quantum chemistry (MTQC) [7], here we perform the first high-throughput\nsearch for magnetic topological materials. here we perform a high-throughput search for magnetic\ntopologicalmaterialsbasedonfirst-principlescalculations. WeuseasourstartingpointtheMagnetic\nMaterials Database on the Bilbao Crystallographic Server, which contains more than 549 magnetic\ncompounds with magnetic structures deduced from neutron-scattering experiments, and identify\n130 enforced semimetals (for which the band crossings are implied by symmetry eigenvalues), and\ntopological insulators. For each compound, we perform complete electronic structure calculations,\nwhich include complete topological phase diagrams using different values of the Hubbard potential.\nUsing a custom code to find the magnetic co-representations of all bands in all magnetic space\ngroups, we generate data to be fed into the algorithm of MTQC to determine the topology of\neach magnetic material. Several of these materials display previously unknown topological phases,\nincluding symmetry-indicated magnetic semimetals, three-dimensional anomalous Hall insulators\nand higher-order magnetic semimetals. We analyse topological trends in the materials under varying\ninteractions: 60 per cent of the 130 topological materials have topologies sensitive to interactions,\nand the others have stable topologies under varying interactions. We provide a materials database\nforfutureexperimentalstudiesandopen-sourcecodefordiagnosingtopologiesofmagneticmaterials.\nCONTENTS\nI. Introduction 2\nII. Workflow 3\nIII. Topological phase diagrams 3\nIV. High-quality topological materials 4\nV. Consistency with previous works 6\n\u0003bernevig@princeton.eduVI. Chemical Categories 7\nVII. Discussion 7\nVIII. Conclusion 8\nIX. Methods 9\nA. A brief introduction to Magnetic Topological\nQuantum Chemistry (MTQC) 12\nB. Material statistics in the BCSMD 14\nC. Computational methods 17\n1. Convention setting of the magnetic unit\ncell 17arXiv:2003.00012v2  [cond-mat.mtrl-sci]  9 Nov 20202\n2. Parameters setting in ab initio\ncalculations 17\n3. Magnetic VASP2trace package 18\n4. Construction of Wannier tight-binding\nHamiltonian and surface states\ncalculation 18\nD. Comparison of the ground state energy\nbetween different magnetic configurations of\nseveral compounds 18\nE. Comparisons between different\nexchange-correlation potentials 19\n1. Band structure calculations with GGA\nfunctional 19\n2. Band structure calculations with\nmeta-GGA functional 19\nF. Comparisons between LDA+U and\nLDA+Gutzwiller methods 28\nG. Topological phase diagrams of the topological\nmaterials that predicted by MTQC 28\nH. Physical interpretations for the TI classified\nby MTQC 34\n1. Definitions for the stable indices of MSG\n2.4 34\n2. Definitions for the stable indices of MSG\n47.249 35\n3. Definitions for the stable indices of MSG\n81.33 35\n4. Definitions for the stable indices of MSG\n83.43 36\n5. Definitions for the stable indices of MSG\n143.1 36\n6. Stable indices of the magnetic TIs 36\nI. Compatibility-relations along high-symmetry\npaths of the symmetry enforced semimetals 45\nJ. Detailed discussion of the ideal magnetic TI\nand SMs 94\n1. Higher-order topology of the ideal Axion\ninsulator NpBi 94\n2. Topological phase diagram of the ideal\nantiferromagnetic nodal-line semimetal\nCeCo 2P2 95\n3. Topological phase diagram of the\nantiferromagnetic Dirac semimetal\nMnGeO 3 95\n4. Weyl nodes, Nodal-lines and Anomalous\nHall effect in Mn 3ZnC 96\nK. Fragile bands in the magnetic materials 98\nL. Magnetic moments for each materials with\ndifferent Coulomb interactions 103\n1. Ferro(Ferri)magnetic materials 123M. Band structures and detailed information 123\nReferences 225\nI. INTRODUCTION\nNon-magnetic topological materials have dominated\nthe landscape of topological physics for the past two\ndecades. Research in this field has led to a rapid\nsuccession of theoretical and experimental discoveries;\nnotable examples include the theoretical prediction of\nthe first topological insulators (TIs) in two [8, 9] and\nthree spatial dimensions [10], topological crystalline\ninsulators [11], Dirac and Weyl semimetals [12–\n18], and non-symmorphic topological insulators and\nsemimetals [19–23]. Though topological materials were\nonce believed to be rare and esoteric, recent advances\nin nonmagnetic topological materials have found that\nTIs and enforced semimetals (ESs) are much more\nprevalent than initially thought. In 2017 Topological\nQuantum Chemistry (TQC) and the equivalent method\nof symmetry-based indicators provided a description of\nthe universal global properties of all possible atomic limit\nband structures in all non-magnetic symmetry groups,\nin real and momentum space [4, 24–27]. This allowed\nfor a classification of the non-magnetic, non-trivial\n(topological) band structures through high-throughput\nmethods that have changed our understanding of the\nnumber of topological materials existent in nature.\nAbout 40%\u000050% of all non-magnetic materials can be\nclassified as topological at the Fermi level [5, 6, 28],\nleading to a “periodic table” of topological materials.\nThese breakthroughs in non-magnetic materials have\nnot yet been matched by similar advances in magnetic\ncompounds, due to a multitude of challenges. First,\nalthough a method for classifying band topology in the\n1651magneticandnonmagneticspacegroups(MSGsand\nSGs, respectively)wasrecentlyintroduced[29], therestill\ndoes not exist a theory similar to TQC or equivalent\nmethods [4, 24–27] by which the indicator groups in\nRef. [29] can be linked to topological (anomalous) surface\n(and hinge) states. Second, a full classification of the\nmagnetic co-representations and compatibility relations\nhas not yet been tabulated. Third, code to compute\nthe magnetic co-representations from ab initio band\nstructures does not exist. Fourth, and finally, even if\nall the above were available, the ab initio calculation\nof magnetic compounds is notoriously inaccurate for\ncomplicated magnetic structures beyond ferromagnets.\nSpecifically, unless the magnetic structure of a material\nis known a priori, then the ab initio calculation will likely\nconverge to a misleading ground state. This has rendered\nthe number of accurately predicted magnetic topological\nmaterials to be less than 10 [1–3, 30–39].\nIn the present work and in ref. [7], we present\nsubstantial advances towards solving all of the\nabove challenges—which we have made freely3\navailable to the public on internet repositories\n(https://www.cryst.ehu.es/cryst/checktopologicalmagmat)\n-covering four years of our work on the subject, and over\n70 years [40] of research on the group theory, symmetry,\nand topology of magnetic materials. We present a\nfull theory of magnetic indices, co-representations,\ncompatibility relations, code with which to compute\nthe magnetic co-representations directly from ab\ninitio calculations, and we perform full local density\napproximation (LDA) + Hubbard U calculations\non 549 magnetic structures, which have been\naccurately tabulated through the careful analysis\nof neutron-scattering data. We predict several novel\nmagnetic topological phases in real materials, including\nhigher-order magnetic Dirac semimetals with hinge\narcs [41], magnetic chiral crystals with long Fermi arcs,\nDirac semimetals with nodes not related by time-reversal\nsymmetry, Weyl points and nodal lines in non-collinear\nantiferrimagnets, and ideal axion insulators with gapped\nsurface states and chiral hinge modes [42, 43].\nII. WORKFLOW\nStarting from the material database MAGNDATA\nMAGNDATA on the Bilbao Crystallographic Server\n(BCS) (the BCSMD), which contains portable magnetic\nstructure files determined by neutron scattering\nexperiments of more than 707 magnetic structures,\nwe select 549 high-quality magnetic structures for\nthe ab initio calculations. We take the magnetic\nconfigurations provided by BCSMD as the initial inputs\nand then perform ab initio calculations incorporating\nspin–orbit coupling. LDA + U are applied for each\nmaterial with different Hubbard U parameters to\nobtain a full phase diagram. Then, we calculate\nthe character tables of the valence bands of each\nmaterial using the MagVasp2trace package. By\nfeeding the character tables into the machinery of\nMTQC, that is, the Check Topological Magnetic Mat.\n(https://www.cryst.ehu.es/cryst/checktopologicalmagmat)\nfunction on BCS [7], we identify the corresponding\nmagnetic co-representations (irreps) and classify the\nmaterials into different topological categories. Here we\ndefine six topological categories:\n1. Band representation. Insulating phase consistent\nwith atomic insulators.\n2. Enforced semimetal with Fermi degeneracy\n(ESFD). Semimetal phase with a partially filled\ndegenerate level at a high symmetry point in the\nBrillouin zone.\n3. Enforced semimetal. Semimetal phase with\nun-avoidable level crossings along high symmetry\nlines or in high-symmetry planes in the Brillouin\nzone.4. Smith-index semimetal (SISM). Semimetal phase\nwith un-avoidable level crossings at generic points\n(away from high symmetry points/lines/planes) in\nthe Brillouin zone.\n5. Stable topological insulator. Insulating phase\ninconsistent with atomic insulators. The topology\n(inconsistency with atomic insulators) is stable\nagainst being coupled to atomic insulators.\n6. Fragile topological insulator. Insulating phase\ninconsistent with atomic insulators. The topology\nis unstable against being coupled to certain atomic\ninsulators.\nFurther details about BCSMD, the calculation\nmethods, the MagVasp2trace package, the identification\nof magnetic irreps, and definitions of the topological\ncategories are given in the Methods.\nIII. TOPOLOGICAL PHASE DIAGRAMS\nWith the irreps successfully identified, we classify\n403 magnetic structures with convergent ground states\ninto the six topological categories. We find that there\nare 130 materials (about 32% of the total) that exhibit\nnontrivial topology for at least one of the U values in\nthe phase diagram. We sort these materials into four\ngroups based on their U-dependence: (1) 50 materials\nbelong to the same topological categories for all values\nof U. These are the most robust topological materials.\n(2) 49 materials, on the other hand, are nontrivial at\nU = 0 but become trivial when U is larger than a\ncritical value. (3) 20 materials have non-monotonous\ndependence on U: they belong to one topologically\nnontrivial categories at U = 0 and change to a different\ntopologically nontrivial category at a larger value of\nU. (4) Six materials are trivial at U = 0 but become\nnontrivial after a critical value of U. The topology\nof these six interesting materials is thus driven by\nelectron–electron interactions. The materials in this\ncategory are: CaCo 2P2, YbCo 2Si2, Ba 5Co5ClO 13,\nU2Ni2Sn, CeCoGe 3, and CeMnAsO. The self-consistent\ncalculations of the remaining 5 materials do not\nconverge for at least one value of U, and hence the phase\ndiagrams are not complete. Complete classifications of\nthe converged materials are tabulated in Appendix G;\nthe corresponding band structures are given in Appendix\nM. In Table Table I, we summarize the total number\nof topological materials in each magnetic space group\n(MSG) at different values of U. We have also provided\nfulltopologicalclassificationsandbandstructuresofeach\nmaterialontheTopologicalMagneticMaterialsDatabase\n(https://www.topologicalquantumchemistry.fr/magnetic).\nIn the scheme of MTQC, the stable magnetic\ntopological insulators and SISMs are characterized by\nnon-zero stable indices. These indices can be understood\nas generalizations of the Fu-Kane parity criterion for4\na three-dimensional topological insulator [44]. A\ncompletetableofthestableindicesandtheindex-implied\ntopological invariants, which include (weak) Chern\nnumbers, the axion \u0012angle, and magnetic higher-order\ntopological insulator (HOTI) indices, is given in ref.\n[7]. In Appendix H, we present examples of stable\nindices relevant to the present work, as well as their\nphysical interpretations. Alhough there are many (1,651)\nmagnetic and nonmagnetic space groups, we find in\nref. [7] that the stable indices of all of the MSGs are\ndependent on minimal indices in the set of the so-called\nminimal groups. Thus, to determine the stable indices\nof a material, we first subduce the representation of the\nMSG formed by the material to a representation of the\ncorresponding minimal group—a subgroup of the MSG\non which the indices are dependent. We then calculate\nthe indices in the minimal group. Using this method, we\nfind a tremendous variety of topological phases among\nthe magnetic materials studied in this work, including\naxion insulators [45–48], mirror topological crystalline\ninsulators [11], three-dimensional quantum anomalous\nHall insulators, and SISMs. A complete table of the\ntopology of all of the magnetic materials studied in this\nwork is provided in Appendix G.\nWe additionally discover many ESFD and enforced\nsemimetal materials, in which unavoidable electronic\nband crossings respectively occur at high-symmetry k\npoints or on high-symmetry lines or planes in the\nBrillouin zone. For each of the ESFD and enforced\nsemimetal magnetic materials, we tabulate the kpoints\nwhere unavoidable crossings occur (see Appendix I).\nWe did not discover any examples of magnetic\nmaterials for which the entire valence manifold is\nfragile topological. However, as will be discussed\nbelow, we discovered many magnetic materials with well\nisolated fragile bands in their valence manifolds, thereby\nproviding examples of magnetic fragile bands in real\nmaterials.\nIV. HIGH-QUALITY TOPOLOGICAL\nMATERIALS\nWe here select several representative “high-quality”\ntopological materials with clean band structures at\nthe Fermi level: NpBi in MSG 224.113 ( Pn\u00163m0)\n(antiferromagnetic stable topological insulator),\nCaFe 2As2in MSG 64.480 ( CAmca) (antiferromagnetic\nstable topological insulator), NpSe in MSG 228.139\n(FSd\u00163c) (antiferromagnetic ESFD), CeCo 2P2\nin MSG 126.386 ( PI4=nnc) (antiferromagnetic\nenforced semimetal), MnGeO 3in MSG 148.19 ( R\u001630)\n(antiferromagnetic enforced semimetal), Mn 3ZnC in\nMSG 139.537 ( I4=mm0m0) (non-collinear ferrimagnetic\nenforced semimetal), as shown in Fig. 1.\nTo identify the stable topologies of the\nantiferrimagnets NpBi and CaFe 2As2, we calculate\nthe stable indices subduced onto MSG 2.4 ( P\u00161),a subgroup of MSG 224.113 ( Pn\u00163m0) and 64.480\n(CAmca).\nThe stable indices in MSG 2.4 ( P\u00161) are defined using\nonly parity (inversion) eigenvalues [7, 29, 42, 49–51]:\n\u00114I=X\nKn\u0000\nKmod 4; (1)\nz2I;i=X\nK;Ki=\u0019n\u0000\nKmod 2 (2)\nwhereKsums over the eight inversion-invariant\nmomenta, and n\u0000\nKis the number of occupied states\nwith odd parity eigenvalues at momentum K.z2I;iis\nthe parity of the Chern number of the Bloch states\nin the plane ki=\u0019. As explained in Appendix H1,\n\u00114I= 1;3correspond to Weyl semimetal (WSM) phases\nwith odd-numbered Weyl points in each half of the BZ;\n\u00114I= 2indicates an axion insulator phase provided\nthat the band structure is fully gapped and the weak\nChern numbers in all directions are zero. The inversion\neigenvalues of NpBi with U=2 eV and CaFe 2As2with\nU=2 eV are tabulated in Table II. The corresponding\nband structures are shown in Fig. 2a and Fig. 2b,\nrespectively. Both NpBi and CaFe 2As2have the indices\n(\u00114I;z2I;1;z2I;2;z2I;3)=(2,0,0,0). As shown in Appendix\nH6, the\u00114Iindex of NpBi is the same for U=0, 2, 4,\n6 eV; whereas the \u00114Iindex of CaFe 2As2is 2 forU=0,\n1, 2 eV and 0 for U=3, 4 eV. We have confirmed that\nboth NpBi and CaFe 2As2have vanishing weak Chern\nnumbers, implying that the \u00114I= 2phases must be axion\ninsulators.\nAn axion insulator is defined by a nontrivial \u0012angle,\nwhich necessitates a quantized magneto-electric response\nin the bulk and chiral hinge modes on the boundary [43,\n45–48]. We have calculated the surface states of NpBi\nand find that, as expected, the (001) surface is fully\ngapped (Fig. 2a). Owing to the C3-rotation symmetry\nof the MSG 224.113, the (100) and (010) surfaces are\nalso gapped. Therefore, a cubic sample with terminating\nsurfacesinthe(100), (010)and(001)directions, asshown\ninFig.2a, exhibitscompletelygappedsurfaces. However,\nas an axion insulator, it must exhibit chiral hinge modes\nwhen terminated in an inversion-symmetric geometry\n[42, 43]. We predict that the chiral hinge modes exist\non the edges shown in Fig. 2a. More details about NpBi\nare provided in Appendix J1.\nNext, we discuss representative examples of magnetic\ntopological semimetals. The antiferrimagnet NpSe\nwithU=2 eV, 4 eV and 6 eV is an ESFD with a\npartially-filled degenerate band at the \u0000point, where\nthe lowest conduction bands and the highest valence\nbands meet in a fourfold degeneracy (Fig. 1c). The\nantiferrimagnet CeCo 2P2is an enforced semimetal at\nall theUvalues used in our calculations. For U=\n0eV and 2 eV, we predict CeCo 2P2to be a Dirac\nsemimetal protected by C4rotation symmetry. Because\nthe Dirac points in CeCo 2P2lie along a high-symmetry\nline ( \u0000Z) whose little group contains 4mm, we see that5\nTABLE I. Topological categories vary with U. Shown are the number of magnetic topological insulators/SISMs and enforced\nsemimetals/ESFDs in each MSG for different values of the Hubbard interaction U=0 eV, 2 eV and 4 eV. For U= 0, there\nare 38 topological insulators/SISMs and 73 enforced semimetals/ESFDs in total. For U= 2eV, the numbers of topological\ninsulators/SISMs and enforced semimetals/ESFDs decrease to 27 and 58, respectively. For U= 4eV, the numbers of topological\ninsulators/SISMsandenforcedsemimetals/ESFDsdecreaseto24and57, respectively. Choosingthevalueof Uforeachmaterial\nfor which the magnetic moments calculated ab initio lie closest to their experimentally measured values, there are 29 topological\ninsulators/SISMs and 62 enforced semimetals/ESFDs.\nMSGTIs/SISMs ESs/ESFDsMSGTIs/SISMs ESs/ESFDsMSGTIs/SISMs ESs/ESFDs\nU=0 U=2 U=4 U=0 U=2 U=4 U=0 U=2 U=4 U=0 U=2 U=4 U=0 U=2 U=4 U=0 U=2 U=4\n2.71 0 1 0 1 0 62.447 0 1 0 1 0 0 129.416 0 0 1 0 1 0\n4.70 1 0 0 0 0 62.450 2 2 1 2 1 2 130.432 0 1 0 1 0 1\n11.54 1 0 1 0 0 0 63.462 0 2 0 2 0 2 132.456 0 1 0 1 0 1\n11.57 1 0 0 0 0 0 63.463 0 1 0 1 0 1 134.481 3 0 1 1 2 0\n12.62 2 0 1 0 1 0 63.464 1 1 1 1 0 1 135.492 0 2 0 2 0 2\n12.63 0 0 0 0 0 0 63.466 0 0 2 0 1 1 138.528 1 0 1 0 1 0\n13.73 2 0 2 0 1 0 63.467 0 0 0 0 1 0 139.536 0 1 0 1 0 1\n14.75 0 1 0 0 0 0 64.480 3 0 2 0 0 1 139.537 0 1 0 1 0 1\n14.80 1 0 0 0 0 0 65.486 0 0 0 1 0 1 140.550 0 2 1 2 1 2\n15.89 2 1 3 0 3 0 65.489 1 0 0 1 0 1 141.556 0 1 0 0 1 0\n15.90 4 0 0 0 0 0 67.510 0 0 1 0 0 0 141.557 1 4 0 1 0 1\n18.22 0 1 0 0 0 0 70.530 0 1 0 0 0 0 148.19 0 1 0 1 0 1\n33.154 0 1 0 0 0 0 71.536 0 1 0 1 0 1 155.48 0 0 0 0 0 1\n36.178 0 0 0 0 0 1 73.553 1 0 1 0 1 0 161.69 0 1 0 1 0 0\n38.191 0 1 0 0 0 0 74.559 0 1 0 1 0 1 161.71 0 2 0 2 0 0\n49.270 0 1 0 1 0 1 85.59 0 1 0 1 0 1 165.95 0 1 0 0 0 0\n49.273 0 1 1 0 1 0 88.81 0 1 0 0 0 0 166.101 1 3 0 3 1 2\n51.295 0 1 0 1 0 1 92.114 0 1 0 1 0 1 166.97 0 1 0 1 0 1\n51.298 0 1 0 1 0 1 107.231 0 0 0 1 0 1 167.108 0 1 0 0 0 0\n53.334 0 0 0 0 1 0 114.282 0 1 0 0 0 0 185.201 1 0 0 0 0 0\n57.391 1 0 1 0 1 0 123.345 0 1 0 1 0 1 192.252 0 2 0 2 0 2\n58.398 0 1 0 0 0 0 124.360 1 3 0 4 0 4 194.268 0 0 0 0 0 1\n58.399 0 2 0 2 0 2 125.373 0 1 0 1 0 1 205.33 1 0 0 0 0 0\n59.407 0 1 0 0 0 0 126.386 0 1 0 1 0 1 222.103 0 1 0 1 0 1\n59.416 0 0 1 0 1 0 127.394 1 1 1 1 1 1 224.113 2 1 2 0 2 0\n60.431 1 0 0 0 0 0 127.397 0 1 0 1 0 1 227.131 0 1 0 0 0 0\n61.439 0 0 1 0 0 0 128.408 0 1 0 1 0 1 228.139 2 1 0 3 0 3\n62.441 0 3 0 0 0 0 128.410 0 4 0 4 0 4 Total 38 27 24 73 58 57\nTABLE II. Parities and topological indices of two magnetic topological insulators. Shown are the numbers of occupied bands\nwith odd/even parity eigenvalues at the eight inversion-invariant points ( \u0011\u000b) for the magnetic topological insulators NpX (where\nX = Sb, Bi) and XFe 2As2(where X = Ca, Ba) with U=2 eV. The\u00114I= 2phase corresponds to an axion insulator or Weyl\nsemimetal phase with even pairs of Weyl points at generic locations in the Brillouin zone interior (given vanishing weak Chern\nnumbers), and \u00114I= 1;3corresponds to a Weyl semimetal phase with odd number of Weyl points at generic locations within\neach half of the Brillouin zone. We have confirmed that the weak Chern numbers vanish in NpX and XFe 2As2, implying that\nboth materials are axion insulators.\n\u0003\u000b(0,0,0) ( \u0019,0,0) (0, \u0019,0) (\u0019,\u0019,0) (0,0,\u0019) (\u0019,0,\u0019) (0,\u0019,\u0019) (\u0019,\u0019,\u0019)(\u00114I;z2I;1;z2I;2;z2I;3)\nNpX58/22 40/40 40/40 40/40 40/40 40/40 40/40 48/32 (2,0,0,0)\nXFe 2As250/46 48/48 48/48 52/44 48/48 52/44 52/44 48/48 (2,0,0,0)\nCeCo 2P2is a higher-order topological semimetal that\nexhibits flat-band-like higher-order Fermi-arc states on\nmirror-invariant hinges, analogous to the hinge states\nrecently predicted [41] and experimentally observed [52]\nin the non-magnetic Dirac semimetal Cd 3As2. ForU= 4\neV and 6 eV, CeCo 2P2becomes a nodal ring semimetal\nprotected by the mirror symmetry Mz. As detailed in\nAppendix J2, the transition between the two enforced\nsemimetal phases is completed by two successively band\ninversions at \u0000and Z, which removes the Dirac nodeand creates the nodal ring, respectively. The band\nstructure of the nodal ring semimetal phase at U=6 eV\nis plotted in Fig. 1d. The antiferromagnet MnGeO 3is\naC3-rotation-protected Dirac semimetal, in which the\nnumber of Dirac nodes changes with the value of U.\nForU=0 eV, 1 eV, 3 eV and 4 eV, we predict MnGeO 3\nto have two Dirac nodes along the high-symmetry line\n\u0000F; forU=2 eV, we observe four Dirac nodes along\nthe same high-symmetry line. In Fig. 1e, we plot\nthe band structure of MnGeO 3usingU=4 eV. We next6\nFIG. 1. Band structures of the ‘high-quality’ magnetic topological materials predicted by MTQC. (a, b) The antiferromagnetic\naxion topological insulators, NpBi and CaFe 2As2. Although there are Fermi pockets around SandYin CaFe 2As2, the\ninsulating compatibility relations are fully satisfied. We note that there is a small gap (about 5 meV) along the path T–Y;\nthis indicates that the valence bands are well separated from the conduction bands, and thus have a well defined topology. (c)\nThe antiferromagnetic ESFD NpSe, which has a partially filled fourfold degeneracy at \u0000. (d) The antiferromagnetic nodal-line\nsemimetal CeCo 2P2. A gapless nodal ring protected by mirror symmetry lies in the Z–R–A plane. (e) The antiferromagnetic\nDirac semimetal MnGeO 3. One of the two Dirac nodes protected by the C3-rotation symmetry lies along the high-symmetry\nline\u0000–F. Note that there is a small bandgap at the \u0000point. (f) The non-collinear ferrimagnetic Weyl semimetal Mn 3ZnC.\nTwo Weyl points are pinned to the rotation-invariant line \u0000–T byC4-rotation symmetry. Mn 3ZnC also hosts nodal lines at\nthe Fermi level EF; we specifically observe five nodal rings protected by the mirror symmetry ( Mz) in the plane kz= 0. The\nsequential number of each MSG in the BNS setting and the chemical formula of each material are provided on the top of each\npanel.\npredict the non-collinear ferrimagnet Mn 3ZnC to be an\nES with symmetry-enforced Weyl points coexisting with\nthe Weyl nodal rings (Fig. 1f). Two of the Weyl points\nin Mn 3ZnC are pinned by the C4-rotation symmetry to\nthe high-symmetry line \u0000T, and we observe five nodal\nrings protected by the mirror symmetry Mzin thekz=\n0plane. In time-reversal-breaking Weyl semimetals,\ndivergent Berry curvature near Weyl points can give\nrise to a large intrinsic anomalous Hall conductivity\n[1, 2, 33, 34, 53]. We thus expect there to be a\nlarge anomalous Hall effect in Mn 3ZnC. As detailed in\nAppendix J4, we have specifically calculated the the\nanomalous Hall conductivity of Mn 3ZnC to be about 123\n\n\u00001\u0001cm\u00001.\nThesurfacestates oftheenforcedsemimetals CeCo 2P2\nand MnGeO 3are shown in Fig. 2b,c, respectively.\nBecause the bulk states of CeCo 2P2and MnGeO 3haveclean Fermi surfaces, the surface states are well separated\nfrom the bulk states, and should be observable in\nexperiment. For the Dirac semimetal MnGeO 3, we\nobserve a discontinuous Fermi surface (Fermi-arc) on the\nsurface (Fig. 2d). In Appendix J, we provide further\ndetails of our surface-state calculations.\nV. CONSISTENCY WITH PREVIOUS WORKS\nOur magnetic materials database\n(https://www.topologicalquantumchemistry.fr/magnetic)\nincludes several topological materials that have\npreviously been reported but whose topology was\nnot known to be protected by symmetry eigenvalues.\nFor example, the non-collinear magnet Mn 3Sn in MSG\n63.463 (Cm0cm0) has been reported as a magnetic7\nFIG. 2. Topological surface states of representative magnetic topological insulator and enforced semimetal phases. (a) The\n(001) surface state of the axion insulator NpBi, which has an energy gap of 30 meV. The inset shows a schematic of the chiral\nhinge states on a cubic sample. (b) The (001) surface state of the enforced semimetal CeCo 2P2. The drumhead-like topological\nsurface states connect the projections of the bulk nodal rings. (c) The (010) surface state of the enforced semimetal MnGeO 3.\nThe bulk Dirac point along the \u0016\u0000\u0000\u0016Zline is protected by C3symmetry. However, because time-reversal symmetry is broken,\nthe projected band crossing on \u0016\u0000\u0000~Z(along -kz) is no longer protected, and is instead weakly gapped. The coordinates of \u0016Z\nand ~Zon the (010) surface are (0, kz=\u0019=c) and (0,kz=\u0000\u0019=c), respectively. (d) The surface Fermi arcs connecting the Dirac\npoints on the (010) surface of MnGeO 3.\nWeyl semimetal candidate with six pairs of Weyl points\n[31, 54]. In our LDA+U calculation, for U=0 eV, 1\neV and 2 eV, we find Mn 3Sn to be classified instead\nas a magnetic topological insulator category with the\nindex\u00114I= 2.\u00114I= 2can correspond to several\ndifferent topological phases (which we emphasize are\nnot all topological insulators): (1) an axion insulator,\n(2) a three-dimensional quantum anomalous Hall state\nwith even weak Chern number (not determinable from\nsymmetry eigenvalues) [55], or (3) a Weyl semimetal\nphase with an evennumber of Weyl points in half of\nthe BZ (not determinable from symmetry eigenvalues).\nThus our calculations on Mn 3Sn forU= 0;1;2eV\nare consistent with the results in refs.[31, 54]. We\nemphasize that if the six Weyl points in half of the\nBrillouin zone were pairwise annihilated without closing\na gap at the inversion-invariant momenta, then the\ngapped phase would either be an axion insulator or a\nthree-dimensional quantum anomalous Hall state. When\nU is further increased to 3 eV and 4 eV, a topological\nphase transition occurs, driving the \u00114I= 2phase into a\ngapless enforced semimetal phase.\nVI. CHEMICAL CATEGORIES\nIn Table III, we classify the topological magnetic\nmaterials predicted by MTQC into three main\nchemical categories, and 11 sub-categories, through\na consideration of their magnetic ions and chemical\nbonding. Detailed descriptions of each category are\ngiven in the Methods. Of the materials listed in\nTable III, most antiferromagnetic insulators, which are\nwell studied experimentally in the case of the so-called\nMott insulators, appear to be trivial. We observe thatmost of the materials in Table III are identified as\ntopological enforced semimetals or ESFDs, which are\ndefined by small densities of states at the Fermi level,\nand hence lie chemically at the border between insulators\nand metals.\nVII. DISCUSSION\nA large number of the topological materials predicted\nin this work (see Appendix G for a complete tabulation)\ncan readily be synthesized into single crystals for the\nexploration of their unusual physical properties and the\nconfirmation of their topological electronic structures\nin different phase categories. These include materials\nwith non-trivial topology over the full range of U\nvalues used in our calculations (for example, Mn 3Ge,\nMn3Sn, Mn 3Ir, LuFe 4Ge2, and YFe 4Ge2), materials\nsensitive to U(for example, NdCo 2and NdCo 2P2), and\ninteraction-driven topological materials (for example,\nU2Ni2Sn and CeCuGe 3).\nWe did not find any examples of materials whose entire\nvalence manifolds are fragile topological. However, it is\nstill possible for well isolated bands within the valence\nmanifold to be fragile topological if they can be expressed\nas a difference of band representations. We find many\nexamples of energetically well isolated fragile branches\namong the occupied bands. We tabulate all the fragile\nbranches close to the Fermi level in Appendix K.\nWe emphasize that there also exist topological\ninsulators and topological semimetals (for example,\nWeyl semimetals) that cannot be diagnosed through\nsymmetry eigenvalues, which in this work are classified\nastrivialbandrepresentations[4]. Itisworthmentioning\nthat even the topologically trivial bands may also be8\nCategories Properties Materials\nI-A Non-collinear Manganese compounds Mn3GaC, Mn 3ZnC, Mn 3CuN, Mn 3Sn, Mn 3Ge, Mn 3Ir, Mn 3Pt, Mn 5Si3\nI-B Actinide Intermetallic UNiGa 5, UPtGa 5, NpRhGa 5, NpNiGa 5\nI-C Rare earth intermetallic NdCo 2, TbCo 2, NpCo 2, PrAg DyCu, NdZn, TbMg, NdMg,\nNd5Si4, Nd 5Ge4, Ho 2RhIn 8, Er 2CoGa 8, Nd 2RhIn 8, Tm 2CoGa 8,\nHo2RhIn 8, DyCo 2Ga8, TbCo 2Ga8, Er2Ni2In, CeRu 2Al10, Nd 3Ru4Al12,\nPr3Ru4Al12, ScMn 6Ge6, YFe 4Ge4, LuFe 4Ge4, CeCoGe 3\nII-A Metallic Iron pnictides LaFeAsO, CaFe 2As2, EuFe 2As2, BaFe 2As2, Fe 2As, CaFe 4As3,\nLaCrAsO, Cr 2As, CrAs, CrN\nII-B Semiconducting manganese pnictides BaMn 2As2BaMn 2Bi2, CaMnBi 2, SrMnBi 2, CaMn 2Sb2, CuMnAs,\nCuMnSb, Mn 2As\nII-C Rare earth intermetallic compounds with\nthe composition 1:2:2PrNi 2Si2, YbCo 2Si2, DyCo 2Si2, PrCo 2P2, CeCo 2P2, NdCo 2P2,\nDyCu 2Si2, CeRh 2Si2, UAu 2Si2, U 2Pd2Sn, U 2Pd2In, U 2Ni2Sn,\nU2Ni2In, U 2Rh2Sn\nII-D Rare earth ternary compounds of the\ncomposition 1:1:1CeMgPb, PrMgPb, NdMgPb, TmMgPb\nIII-A Semiconducting Actinides/Rare earth\nPnictidesHoP, UP, UP 2, UAs, NpS, NpSe, NpTe, NpSb, NpBi, U 3As4, U3P4\nIII-B Metallic oxides Ag2NiO 2, AgNiO 2, Ca 3Ru2O7, Double perovskite Sr 3CoIrO 6\nIII-C Metal to insulator transition compounds NiS2, Sr2Mn3As2O2\nIII-D Semiconducting and insulating oxides,\nborates, hydroxides, silicates, phosphateLuFeO 3, PdNiO 3, ErVO 3, DyVO 3, MnGeO 3, Tm 2Mn2O7, Yb 2Sn2O7,\nTb2Sn2O7, Ho 2Ru2O7, Er 2Ti2O7, Tb 2Ti2O7, Cd 2Os2O7, Ho 2Ru2O7,\nCr2ReO6, NiCr 2O4, MnV 2O4, Co 2SiO4, Fe 2SiO4, PrFe 3(BO 3)4,\nKCo 4(PO 4)3, CoPS 3, SrMn(VO 4)(OH), Ba 5Co5ClO 13, FeI 2\nTABLE III. The magnetic topological materials identified in this work.\ninteresting if the occupied bands form Wannier functions\ncentred at positions away from the atoms, because a\nWanniercentreshiftinthree-dimensionalinsulatorsleads\nto the appearance of topological corner states, like those\nof quantized ‘quadrupole’ insulators [41]. Topological\nphases characterized by displaced Wannier functions\nare known as obstructed atomic limits; we leave their\nhigh-throughput calculation for future studies.\nVIII. CONCLUSION\nWe have performed LDA + U calculations on\n549 existent magnetic structures and have successfully\nclassified 403 using the machinery of MTQC [7].\nWe find that 130 materials (about 32% of the\ntotal) have topological phases as we scan the U\nparameter. Our results suggest that a large\nnumber of previously synthesized magnetic materials\nare topologically nontrivial. We highlight several\n‘high-quality’ magnetic topological materials that should\nbe experimentally examined for topological response\neffects and surface (and hinge) states.\nAcknowledgements We thank U. Schmidt, I.\nWeidl, W. Shi and Y. Zhang. We acknowledge\nthe computational resources Cobra in the Max\nPlanck Computing and Data Facility (MPCDF),\nthe HPC Platform of ShanghaiTech University\nand Atlas in the Donostia International Physics\nCenter (DIPC). Y.X. is grateful to D. Liu for help\nin plotting some diagrammatic sketches. B.A.B.,\nN.R., B.J.W. and Z.S. were primarily supportedby a Department of Energy grant (DE-SC0016239),\nand partially supported by the National Science\nFoundation (EAGER grant DMR 1643312), a Simons\nInvestigator grant (404513), the Office of Naval Research\n(ONR; grant N00014-14-1-0330), the NSF-MRSEC\n(grant DMR-142051), the Packard Foundation, the\nSchmidt Fund for Innovative Research, the BSF Israel\nUS foundation (grant 2018226), the ONR (grant\nN00014-20-1-2303) and a Guggenheim Fellowship (to\nB.A.B.). Additional support was provided by the\nGordon and Betty Moore Foundation through grant\nGBMF8685 towards the Princeton theory programme.\nL.E. was supported by the Government of the Basque\nCountry (Project IT1301-19) and the Spanish Ministry\nof Science and Innovation (PID2019-106644GB-I00).\nM.G.V. acknowledges support from the Diputacion Foral\nde Gipuzkoa (DFG; grant INCIEN2019-000356) from\nGipuzkoako Foru Aldundia and the Spanish Ministerio\ndeCienciaeInnovación(grantPID2019-109905GB-C21).\nY.C. was supported by the Shanghai Municipal Science\nand Technology Major Project (grant 2018SHZDZX02)\nand a Engineering and Physical Sciences Research\nCouncil (UK) Platform Grant (grant EP/M020517/1).\nC.F. acknowledges financial support by the DFG\nunder Germany’s Excellence Strategy through the\nWürzburg-Dresden Cluster of Excellence on Complexity\nand Topology in Quantum Matter (ct.qmat EXC 2147,\nproject-id 390858490), an ERC Advanced Grant (742068\n‘TOPMAT’). Y.X. and B.A.B. were also supported by\nthe Max Planck Society.\nAuthor contributions B.A.B. conceived this\nwork; Y.X. and M.G.V. performed the first-principles9\ncalculations. L.E. wrote the code for calculating the\nirreducible representations and checking the topologies\nof materials. Y.X., Z.S., B.J.W. and B.A.B. analysed\nthe calculated results, B.J.W. determined the physical\nmeaning of the topological indices with help from L.E.,\nZ.S. and Y.X. C.F. performed chemical analysis of\nthe magnetic topological materials. N.R. built the\ntopological material database. All authors wrote the\nmain text and Y.X. and Z.S. wrote the Methods and the\nSupplementary Information.\nCompeting interests\nThe authors declare no competing interests.\nCorrespondence and requests for materials\nshould be addressed to B.A.B.\nCorresponding authors\nCorrespondence to B. Andrei Bernevig.\nIX. METHODS\nConcepts Here we give a brief introduction to MTQC\n[4, 7, 56–58] and the definitions of six topological classes.\nA magnetic band structure below the Fermi level is\npartially described by the irreducible co-representations\n(irreps) formed by the occupied electronic states at\nthe high-symmetry kpoints, which are defined as the\nmomenta whose little groups - the groups that leave\nthe momenta unchanged - are maximal subgroups of\nthe space group. If the highest occupied (valence)\nband and the lowest unoccupied (conduction) band\nare degenerate at a high-symmetry kpoint, then\nwe refer to the material as an enforced semimetal\nwith Fermi degeneracy (ESFD) [4]. Depending on\nwhether the irreps at high-symmetry points satisfy the\nso-called compatibility relations [4, 24, 25, 58] – which\ndetermine whether the occupied bands must cross with\nunoccupied bands along high-symmetry lines or planes\n(whose little groups are non-maximal) – band structures\ncan then be further classified as insulating (along\nhigh-symmetry lines and planes) or enforced semimetals\n(ES). ES-classified materials generically feature band\ncrossings along high-symmetry lines or planes. If a\nband structure satisfies the compatibility relations, it\ncan be a trivial insulator, whose occupied bands form\na BR [4], a topological semimetal with crossing nodes\nat generic momenta [Smith-index semimetal (SISM)\nor non-symmetry-indicated topological semimetal - a\nsystem which satisfies all compatibility relations but\nexhibits Weyl-type nodes], or a TI. Some of the\ntopological semimetals and insulators can be diagnosed\nthrough their irreps: If the irreps do not match a BR,\nthen the band structure must be a topological insulator\nor a SISM. There are two types of topological insulators:\nStable TIs [26, 27, 59, 60], which include crystalline and\nhigher-orderTIs(TCIsandHOTIs, respectively)[61–65],\nand fragile TIs [60, 66–70]. Stable TIs remain topological\nwhen coupled to trivial or fragile bands, whereas fragileTIs, on the other hand, can be trivialized by being\ncoupled to certain trivial bands, or even other fragile\nbands [41, 43]. In the accompanying paper [7], we\nexplicitly identify all of the symmetry-indicated stable\nelectronic (fermionic) TIs and topological semimetals,\nspecifically detailing the bulk, surface, and hinge states\nof all symmetry-indicated stable TIs, TCIs, and HOTIs\nin all 1651 spinful (double) SGs and MSGs.\nTo summarize, using MTQC, we divide an electronic\nband structure into one of six topological classes: BRs,\nESFD, ES, SISM, stable TI, and fragile TI, among which\nonly BRs are considered to be topologically trivial. If a\nband structure satisfies the compatibility relations along\nhigh-symmetry lines and planes, and has a nontrivial\nvalue of a stable index, then, unlike in the nonmagnetic\nSGs, it is possible for the bulk to be a topological\n(semi)metal [7]. We label these cases as Smith-index\nsemimetals (SISMs). See Appendix A for a more detailed\ndescription of the six topological classes.\nMagnetic Materials Database We perform\nhigh-throughput calculations of the magnetic structures\nlisted on BCSMD [71]. BCSMD contains portable\nstructure files, including magnetic structure data and\nsymmetry information, for 707 magnetic structures.\nThe magnetic structures of all the materials are\ndetermined by neutron scattering experiments. We thus\nconsider it reasonable and experimentally motivated\nto use the crystal and magnetic structures provided\non the BCSMD as the initial inputs for ab initio\ncalculations, instead of letting our theoretical ab-initio\ncodes predict the magnetic ground-state. We emphasize\nthat predictions of topological magnetic materials based\non theoretically calculated magnetic structures, rather\nthan experimentally measured structures, are more\nlikely to predict unphysical (and possibly incorrect)\nmagnetic ground states. From the 707 magnetic\nstructures on the BCSMD, we omit 63 structures with\nlattice-incommensurate magnetism and 95 alloys, as\nthey do not have translation symmetry and hence\nare not invariant under any MSG. We apply ab initio\ncalculations for the remaining 549 structures. These\nmagnetic structures belong to 261 different MSGs,\nincluding 29 chiral MSGs and 232 achiral MSGs (chiral\nMSGs are defined as MSGs without improper rotations\nor combinations of improper rotations and time reversal;\nall other MSGs are achiral). In Appendix B, we list\nthe number of materials with experimentally obtained\nmagnetic structures in each MSG.\nCalculation Methods We performed ab initio\ncalculations incorporating spin-orbital coupling (SOC)\nusing VASP [72]. Because all of the magnetic materials\non BCSMD with translation symmetry contain at least\none correlated atom with 3d,4d,4f, or5felectrons, we\napply a series of LDA+ Ucalculations for each material\nwith different Hubbard- Uparameters to obtain a full\nphase diagram. For all of the 3dvalence orbitals and the\natom Ru with 4dvalence orbitals, we take Uas 0 eV, 1\neV, 2 eV, 3 eV and 4 eV. The other atoms with 4dvalence10\nelectrons usually do not exhibit magnetism or have weak\ncorrelation effects, and hence are not considered to be\ncorrelated in our calculations. Conversely, atoms with\n4fand 5fvalence electrons have stronger correlation\neffects, so we take Ufor atoms with 4for5fvalence\nelectrons to be 0 eV, 2 eV, 4 eV and 6 eV. If a\nmaterial has both dandfelectrons near the Fermi\nlevel, we fix the Uparameter of the delectrons as 2\neV, and take Uof thefelectrons to be 0 eV, 2 eV,\n4 eV and 6 eV sequentially. We also adopt four other\nexchange-correlation functionals in the LDA + Uscheme\nto check the consistency between different functionals.\nFurther details of our first-principles calculations are\nprovided in Appendix C,D, Eand F.\nOf the 549 magnetic structures that we examined, 403\nconverged self-consistently to a magnetic ground state\nwithin an energy threshold of 10\u00005eV per cell. For\n324 of the 403 converged materials, magnetic moments\nmatching the experimental values (up to an average error\nof 50%) were obtained for at least one of the values\nof U used to obtain the material phase diagram. We\nstress that these are good agreements for calculations\non these strongly correlated states. However, for the\nother 79 materials, the calculated magnetic momenta\nalways notably diverged from the experimental values\n(see Appendix L for a complete comparison of the\nexperimental and ab initio magnetic moments). The\ndifferences can be explained as follows. First, we\nconsider only the spin components, but not the orbital\ncomponents, of the magnetic moments in our current ab\ninitio calculations. This can result in a large average\nerror for compounds with large spin–orbital coupling.\nSecond, because the average error is defined relative to\nthe experimental moments, the ‘error’ (measured as a\npercentage) is likely to be larger when the experimental\nmoments are small. In this case, the random, slight\nchanges in the numerically calculated moments have an\noutsized effect on the reported error percentage. Last\nbut not least, mean-field theory applied in the LDA+ U\ncalculations is not a good approximation for some\nstrongly correlated materials, which should be checked\nfurther with more advanced methods. Although the\npredictionofmagneticstructurewithmean-fieldtheoryis\nsometimes not reliable for strongly correlated materials,\nit is worth comparing the energy difference between the\nmagnetic structures from neutron scattering and the\nother possible magnetic structures. In Appendix D,\nwe selected several topological materials and compared\ntheirenergieswithsomepossiblemagneticconfigurations\nand different U. We find that their experimental\nmagnetic configurations have the lowest energies, and\nhence are theoretically favoured. Finally, we have\nadditionally performed self-consistent calculations of the\ncharge density at different values of U, which we used as\ninput for our band structure calculations. In Appendix\nM, we provide a complete summary of results.\nConsidering the possible underestimation of the band\ngap by generalized gradient approximations (GGA),electronic structures of 23 topological mateirals are\nfurther confirmed by the calculations using the modified\nBecke–Johnson potential [73]. As shown in Appendix\nE2, both the features of bands near Fermi level\nand the topological classes obtained from modified\nBecke–Johnson potential are consistent with LDA+ U\ncalculations. Because of the limitations of the\nLDA+Umethod, we have also performed the more\ncostly LDA+Gutzwiller [74] calculations in two of the\ntopological materials identified in this work-CeCo 2P2\nand MnGeO 3, both classified as enforced semimetal-to\nconfirm the bulk topology. As shown in Appendix\nF, the strong correlations renormalize the quasiparticle\nspectrum by a factor of quasiparticle weight, but do not\nchange the band topology. The surface-state calculations\nhavebeenperformedusingtheWannierToolspackage ??.\nIdentification of the magnetic irreps Using\nthe self-consistent charge density and potentials, we\ncalculate the Bloch wavefunctions at the high-symmetry\nmomenta in the Brillouin zone and then identify the\ncorresponding magnetic irreps using the MagVasp2trace\npackage, which is the magnetic version of Vasp2trace\npackage [75]. (See Appendix C for details about\nMagVasp2trace). Thelittlegroup Gkofahighsymmetry\npoint kis in general isomorphic to an MSG. For little\ngroups without anti-unitary operations, we calculate\nthe traces of the symmetry representations formed by\nthe wavefunctions, and then decompose the traces into\nthe characters of the small irreps of the little group\nGk. For little groups with anti-unitary operations,\nwe calculate only the traces of the unitary operations\nand decompose the representations into the irreps of\nthe maximal unitary subgroup GU\nkofGk. Since\nanti-unitary operations in general lead to additional\ndegeneracies, specifically enforcing two irreps of GU\nk\nto become degenerate and form a co-representation,\nwe check whether the additional degeneracies hold\nin the irreps obtained. Because VASP does not\nimpose anti-unitary (magnetic) symmetries, degeneracies\nlabelled by magnetic co-representations may exhibit very\nsmall splittings in band structures generated by VASP.\nIn these cases, we reduce the convergence threshold and\nre-run the self-consistent calculation until the splitting\nis specifically small ( \u001410%) compared to the smallest\nenergy gap across all of the high-symmetry momenta.\nThe algorithm and methods designed in this work are\nalso applicable to future high-throughput searches for\nmagnetic topological materials [76].\nDetails of the chemical catagories Considering\nthe magnetic ions and chemical bonding of the magnetic\nmaterials, we classify the topological magnetic materials\npredicted in this work into the following 11 chemical\ncategories.\n(I-A)Non-collinearmanganesecompounds,whichhave\nreceived considerable recent attention owing to their\nunusual combination of a large anomalous Hall effect\nand net-zero magnetic moments. The symmetry of\nthe non-collinear antiferromagnet spin structure allows11\nfor a non-vanishing Berry curvature, the origin of the\nunusualanomalousHalleffect. Examplesofnon-collinear\nmanganese compounds include the hexagonal Weyl\nsemimetals Mn 3Sn, Mn 3Ge and the well-studied cubic\nantiferromagnetic spintronic-material Mn 3Ir, as well as\ntheinverseperovskitecompoundsMn 3Y,whichrepresent\n‘stuffed’ versions of the cubic Mn 3Y compounds.\n(I-B,C) Intermetallic materials, containing rare-earth\natoms or actinide atoms, which are typically\nantiferromagnets. The variation of the Hubbard U\nchanges the band structures slightly in these materials,\nbut not the topological character.\n(II-A) The ThCr 2Si2structure and related structures,\nwhich have received attention because of the\nhigh-temperature iron pnictide superconductors in this\ngroup. Inthesematerials, thetransition-metallayersand\nthe pnictide layers form square lattices. The square nets\nof the pnictides act as a driving force for a topological\nband structure [77]. Several of the antiferromagnetic\nundoped prototypes, such as CaFe 2As2, are topological\nantiferromagnets. This suggests the possibility of\ntopological superconductivity in these materials, like\nthat recently found in FeTe 0:55Se0:45[78].\n(II-B) Semiconducting manganese pnictines, which\noccur when iron is substituted with manganese, leading\nto materials that are trivial when insulating and gapped,\nbut which become topological antiferromagnets when\ntheir gap is closed. By increasing the Hubbard U, the\nantiferromagnetic phases of these compounds can be\nconverted into trivial insulators. The antiferromagnetic\ninsulators and semimetals in this class can also be\nconverted into ferromagnetic metals by doping.\n(II-C)Transitionmetalsincombinationwithrareearth\nor actinide atoms form compounds of the ThCr 2Si2\nstructure type. Here the antiferromagnetic ordering\ncomes from the Thorium-position in the ThCr 2Si2\nstructure type.\n(II-D) Rare earth ternary compounds of the\ncomposition 1:1:1.\n(III-A,B,C,D) The third class of magnetic materials\nare ionic compounds, of which most have been\nexperimentally determined to be insulating. Within\nthe density functional approximation, several of the\ncompounds have been identified as topological nontrivial\nmetals, such as oxides, borates, hydroxides, silicates,\nphosphates and FeI 2. By increasing the Hubbard U, a\ntopologicallytrivialgapcanbeopenedinthesematerials.\nAdditional data and discussion can be found online in\nthe Supplementary information.\nData availability All data are available\nin the Supplementary Information and at\nhttps://www.topologicalquantumchemistry.fr/magnetic.\nThe codes required to calculate the character\ntable of magnetic materials are available at\nhttps://www.cryst.ehu.es/cryst/checktopologicalmagmat.12\nAppendix A: A brief introduction to Magnetic Topological Quantum Chemistry (MTQC)\nThe symmetry group property of a band structure is fully described by the multiplicities of the irreducible\nco-representations (irreps) formed by the occupied bands at all the maximal K-points. In the present paper, we\ndefine the 1st band to the Neth band as the “occupied” bands, where Neis the number of electrons. Maximal k-points\nare defined as the high symmetry momenta whose little groups are maximal subgroups of the magnetic space group.\nThe maximal K-points of each magnetic space group are supplied in the magnetic vasp2trace package. We denote the\nlittle group at the momentum KasGK, theith irrep ofGKas\u001ai\nK, and its multiplicity of \u001ai\nKformed by the occupied\nbands asm(\u001ai\nK). For example, the Brillouin zone (BZ) of the 2D space group generated from inversion ( I) and\ntranslations has four maximal k-points: \u0000 (0;0),X (\u0019;0),Y (0;\u0019),M (\u0019;\u0019), all of which have the same little group:\nCi=fE;Ig. HereEis the identity. Cihas two types of irreps: the even(+) and the odd( \u0000). Thus a band structure\nis characterized by the eight integers m(\u001a+;\u0000\n\u0000;X;Y;M). For convenience, we introduce the symmetry-data-vector[4]\nB= (m(\u001a+\n\u0000);m(\u001a\u0000\n\u0000);m(\u001a+\nX);m(\u001a\u0000\nX);m(\u001a+\nY);m(\u001a\u0000\nY);m(\u001a+\nM);m(\u001a\u0000\nM))T: (A1)\nThe symmetry property of a band structure is fully described by the symmetry-data-vector.\nEnforced semimetal with Fermi degeneracy. In some materials, the highest occupied band and the\nlowest empty band are degenerate at some maximal K-points, and the degeneracy is protected by the MSG. We\ncall such states enforced semimetals with Fermi degeneracy (ESFD)[4, 79]. ESFD does not have a well defined\nsymmetry-data-vector B. See FIG. 1c of the main text for examples of ESFD.\nFor band structures where the occupied bands are gapped from the empty bands along all the high symmetry lines,\nthe multiplicities mnecessarily satisfy the compatibility relations [4, 25, 29, 56–58, 80]. We consider two maximal\nk-pointsK1;2and a path kbetween them. On the one hand, since Gkis a subgroup of GK1, the irreps of Gkformed\nby the occupied bands in knear toK1can be obtained by subduction of the irreps of GK1formed by occupied bands\natK1. On the other hand, the irreps of Gkformed by the occupied bands in knear toK2can also be obtained\nby subduction of the irreps at K2. If the irreps of Gkobtained at the two ends K1andK2are not the same, then\nthere must be a symmetry protected level crossing along the path k. In other words, in order to guarantee the path\nkis gapped, the irreps at K1andK2must reduce to the same set of irreps of Gk. This requirement establishes the\ncompatibility relations along k. The full compatibility relations can be obtained by applying this analysis to all the\ninequivalent paths in the BZ.[56]\nIn the example of 2D space group with inversion P\u00161, all the momenta except \u0000,X,Y,Mhave the same little group:\nthe identity group. Thus for any two maximal k-points, there is only one inequivalent path connecting them, and both\neven and odd irreps reduce to the identity irrep of the identity group. The compatibility-relation is nothing but the\nrestriction that the two maximal k-points have the same number of occupied bands. We can write the compatibility\nrelations as\nm(\u001a+\n\u0000) +m(\u001a\u0000\n\u0000) =m(\u001a+\nX) +m(\u001a\u0000\nX) =m(\u001a+\nY) +m(\u001a\u0000\nY) =m(\u001a+\nM) +m(\u001a\u0000\nM): (A2)\nSince we define the “occupied” bands as the 1st band to the Neth band, there are always Neoccupied levels at any\nmomentum and hence Eq.(A2) is automatically satisfied. However, most other magnetic space groups have more\ncompatibility relations than the band number restriction; these compatibility relations can be broken in materials.\nEnforcedSemimetals. Bandstructuresbreakingthecompatibilityrelationsarereferredtoasenforcedsemimetals\n(ESs). The inversion case is not a good example for ES because the compatibility-relation is satisfied by definition.\nPlease see J for example of ES.\nWe now classify the possible band structures allowed by compatibility relations. The strategy is that we first\nenumerate all the atomic insulators and then, for any given band structure from DFT, compare its irreps with those\nof the atomic insulators. A band structure must be topologically nontrivial if its irreps are not consistent with any\natomic insulator; otherwise can be either trivial/nontrivial. Following the terminology of Zak [81–83], we refer to\natomic insulators as band representations (BRs) and to the generators of the BRs as elementary BRs (EBRs).\nWe take the 2D space group with inversion P\u00161as an example to illustrate the concept of EBRs. There are four\nmaximal Wyckoff positions in each unit cell, a(0;0),b(1\n2;0),c(0;1\n2),d(1\n2;1\n2). Maximal Wyckoff positions are\ndefined as positions with site-symmetry-groups which are maximal subgroups of the space group. In this example, the\nsite-symmetry-groups of a;b;c;dare all isomorphic to Ci=fE;Ig. SinceCionly has two types of irreps (even and\nodd), we can add either sorbital (even)/ porbital (odd) at each position. We can then obtain eight different EBRs.\nTo see that they are EBRs, we consider an atomic insulator formed by two orbitals at two general positions, (x;y),\n(1\u0000x;1\u0000y), which transform into each other under the inversion operation at dposition. We can recombine the\ntwo orbitals to form a bonding state and an anti-bonding state at the dposition. Thus this atomic insulator can be\ngenerated from two EBRs at the dposition. The symmetry-data-vectors of the eight EBRs can be calculated by acting13\nthe symmetry operators on the corresponding Bloch wave functions. The wave functions are Fourier transformations\nof the local orbitals\nj\u001e\u0018;\u000b;ki=1p\nNX\nReik\u0001(R+t\u000b)j\u0018;R+t\u000bi (A3)\nHere\u0018=\u0006is the parity of the local orbital, \u000b=a;b;c;dis the Wyckoff position, t\u000bis the position vector of the\nWyckoff position, Rsums over all lattice vectors, and Nis the system size. Since Ij\u0018;R+t\u000bi=\u0018j\u0018;\u0000R\u0000t\u000bi, we\nobtain\nIj\u001e\u0018;\u000b;ki=\u00181p\nNX\nReik\u0001(R+t\u000b)j\u0018;\u0000R\u0000t\u000bi=\u00181p\nNX\nR0e\u0000ik\u0001(R0+t\u000b)j\u0018;R0+t\u000bi; (A4)\nwhere the lattice vector R0is\u0000R\u00002t\u000b. Ifkis one of the maximal k-points ( \u0000;X;Y;M), we can calculate the parity\nof the Bloch wave function as\nh\u001e\u0018;\u000b;kjIj\u001e\u0018;\u000b;ki=\u00181\nNX\nR0e\u0000i2k\u0001(R0+t\u000b)=\u0018e\u0000i2k\u0001t\u000b: (A5)\nWe have made use of the fact that 2kis a reciprocal lattice vector and hence 2k\u0001R= 0 mod 2 \u0019. For\u000b=a, the Bloch\nwave function has the same parity at all the four maximal k-points because e\u0000i2k\u0001t\u000b= 1. Thus the EBR induced\nfrom the orbital with parity \u0006at theaposition form the irreps \u001a\u0006\n\u0000,\u001a\u0006\nX,\u001a\u0006\nY,\u001a\u0006\nM. The symmetry-data-vectors (A1) of\nthese two EBRs are\nEBR +;a= (1;0;1;0;1;0;1;0)T; EBR\u0000;a= (0;1;0;1;0;1;0;1)T: (A6)\nFor\u000b=b;c;d, the Bloch wave function has different parities at the four maximal k-points because e\u0000i2k\u0001t\u000bcan\nbe either 1/\u00001. For example, for \u000b=b,e\u0000i2k\u0001t\u000bequals to 1 and\u00001at\u0000;YandX;M, respectively. Thus the\nEBR induced from the orbital with parity \u0006at thebposition form the irreps \u001a\u0006\n\u0000,\u001a\u0007\nX,\u001a\u0006\nY,\u001a\u0007\nM. The corresponding\nsymmetry-data-vectors are\nEBR +;b= (1;0;0;1;1;0;0;1)T; EBR\u0000;b= (0;1;1;0;1;0;1;0)T: (A7)\nSimilarly, one can derive the symmetry-data-vectors of EBRs induced from c;dpositions as\nEBR +;c= (1;0;1;0;0;1;0;1)T; EBR\u0000;c= (0;1;0;1;1;0;1;0)T; (A8)\nEBR +;d= (1;0;0;1;0;1;1;0)T; EBR\u0000;d= (0;1;1;0;1;0;0;1)T: (A9)\nStable TI. We consider an example where the occupied band form a single odd irrep at \u0000and three even irreps at\nX;Y;Mrespectively. The corresponding symmetry-data-vector can be written as\nB1= (0;1;1;0;1;0;1;0)T: (A10)\nB1is not one of the EBRs; It is also not a sum of EBRs, because all EBRs have even number of odd irreps in Eqs.\n(A6) to (A9). It is also not a sum of EBRs because B1has only one band but any sum of EBRs has at least two\nbands. Thus B1must be topological. According to the Fu-Kane-like formula for Chern insulators [84]\n(\u00001)C=Y\nK=\u0000;X;Y;MY\nn\u0015n(K); (A11)\nwhereCis the Chern number, nis the index of occupied bands, and \u0015n(K)is the parity of nth band at the momentum\nK, the band structure has an odd Chern number.\nOne notices that B1can be written as a linear combination of EBRs with fractional coefficients\nB1=\u00001\n2EBR\u0000;a+1\n2EBR\u0000;d+1\n2EBR\u0000;c+1\n2EBR\u0000;d; (A12)\nbut cannot be written as an integer combination of EBRs. It is a general principle that if a band structure cannot be\nwritten as a linear combination of EBRs unless the coefficients are fractional numbers, the band structure must have\nstable topology. Such stable topology implied by symmetry eigenvalues is characterized by the stable index (SI) (also14\nreferred to as symmetry-based indicator [60, 80]). Eq. (A11) can be thought as an example of SI. Readers can refer\nto supplementary information of Ref. [60, 70] for technical details.\nSmith-index semimetal. In magnetic space groups, some symmetry-data-vectors are not compatible with gapped\nstate and implies topological Weyl semimetal (WSM), even when all of the compatibility relations are satisfied. In\nthis work, the WSM phase implied by symmetry eigenvalue is named as Smith-index semimetal (SISM).\nFrom the MTQC theory, we have found several MSGs with SI corresponding to WSM phase. All of these MSGs have\na minimal subgroup MSG 2.4 ( P\u00161)/MSG 81.33 ( P\u00164). For the MSGs with minimal subgroup P\u00161(with only inversion\nsymmetry), the topologies are described by the stable indices group Z4\u0002Z3\n2. We found the stable index \u00114Imod2is\nthe parity of the Chern number difference between kz= 0andkz=\u0019planes. Thus \u00114I= 1;3correspond to the\nWSM phase with odd number of Weyl points in one half Brillouin zone [7]. For the MSGs with minimal subgroup\nP\u00164, they have the SI group Z4\u0002Z2\n2. We find one of the two z2indices can be interpreted as [7] \u000e2S=c\u0019\u0000c0\n2mod2,\nwherec0;\u0019are the Chern numbers in the kz= 0;\u0019planes. Thus, when this \u000e2Sindex is nonzero, kz= 0;\u0019planes\nmust have different Chern numbers and hence Weyl nodes must appear in between the two planes.\nFragile TI. If theBvector of a state cannot be written as a sum of EBRs, but can be written as a difference of\nEBRs, then the state is at least a fragile TI. [60, 66–70, 85–87] We say “at least” because the state can also have a\nstable topology which cannot be diagnosed through symmetry eigenvalues but through Berry phases. Now we give\nan example in the inversion case. We consider that the occupied bands form two odd irreps at \u0000and three pairs of\neven irreps at X;Y;Mrespectively. The corresponding symmetry-data-vector is double of B1,i.e.,\nB2= (0;2;2;0;2;0;2;0)T: (A13)\nSinceB2= 2B1, we can write the B2as a linear combination of EBRs with integer coefficients, and one of the\ncoefficients is negative\nB2=\u0000EBR\u0000;a+EBR\u0000;d+EBR\u0000;c+EBR\u0000;d: (A14)\nThis decomposition implies that, after being coupled to a trivial band forming the EBR\u0000;a,B2becomes trivial\nbecause it can be written as a sum of EBRs as EBR\u0000;d+EBR\u0000;c+EBR\u0000;d. Therefore, B2is at least a fragile TI.\nReaders can refer to Ref. [70] for more examples and complete classifications of eigenvalue implied fragile TIs.\nAppendix B: Material statistics in the BCSMD\nIgnoring the magnetic materials with incommensurate structures, there are 644 materials (including 95 alloys) with\n261differentMSGsintheBilbaoCrystallographicServerMagneticdatabase(BCSMD)[71,88]. Weprovidethenumber\nof materials in each MSG in Table IV. Detailed information about each of the magnetic materials can be obtained\non the BCSMD website (http://webbdcrista1.ehu.es/magndata). Based on the stable topological classifications of\nMSGs [7, 29], we classify the MSGs into four types.\nType A The MSGs that have stable topological indices, which are indicated by red color. There are 435 materials\nin BCSMD with Type A MSGs.\nType B In this type of MSGs, given the electron number, one can immediately identify whether a material is\nESFD. This type of MSGs are indicated by blue color. There are 34 materials with Type B MSGs in BCSMD.\nType C Among Type B MSGs, some also have stable topological indices, which are indicated by green color. There\nare 19 materials in BCSMD with Type C MSG.\nType D The other MSGs that do not belong to Type A/Type B are indicated by black color. There are 183\nmaterials with Type D MSG.\nWe also emphasize that for an ES/ESFD, if the crossing point occurs at a k-point whose little co-group is chiral, the\ncrossing point must necessarily carry a nonzero chiral charge [89–91]. The chiral MSGs have been tagged in Table IV.\nTABLE IV: The number of magnetic materials per magnetic space group in BCSMD\nMSG Count MSG Count MSG Count MSG Count\n1.3PS1\u0003433.144Pna21 363.462Cm0c0m 2138.528Pc42=ncm 1\n2.4P\u00161 433.147Pna020\n1 263.463Cmc0m01138.529PC42=ncm 1\n2.6P\u001610333.148Pn0a021 363.464Cm0cm04139.535I40=mmm01\n2.7PS\u00161 3433.149Pana21 163.466Ccmcm 2139.536I40=m0m0m4\n4.10Pa21\u0003733.150Pbna21 163.467Camcm 1139.537I4=mm0m02\n4.12PC21\u0003133.154PCna21 363.468CAmcm 1140.550Ic4=mcm 615\n4.7P21\u0003335.167Cm0m20164.476Cm0ca01141.554I40\n1=am0d 2\n4.9P20\n1\u0003336.174Cm0c20\n1 264.479Camca 1141.555I40\n1=amd03\n5.13C2\u0003136.176Cm0c021 164.480CAmca 13141.556I40\n1=a0m0d 3\n5.15C20\u0003136.178Camc21 465.483Cm0mm 1141.557I41=am0d08\n5.16Cc2\u0003338.191Am0m02 165.486Cmm0m02142.568I40\n1=a0cd01\n6.20Pm0138.192Aamm2 165.489Cammm 2146.10R3\u00032\n7.27Pac 139.201Abbm2 166.500CAccm 5146.12RI3\u00032\n7.29Pbc 141.217Abba2 167.510CAmma 1148.17R\u00163 5\n8.35Ccm 142.223FSmm2 169.523Fm0mm 1148.19R\u0016302\n8.36Cam 443.227Fd0d02 169.526FSmmm 3148.20RI\u00163 1\n9.39Cc0245.237Ib0a20170.530Fd0d0d 2152.35P31201\u00031\n9.40Ccc 346.243Im0a20171.536Im0m0m 2154.41P3221\u00031\n9.41Cac 349.270Pc0cm0172.543Ib0a0m 1154.44Pc3221\u00033\n10.49PC2=m 149.273Pcccm 173.551Ib0c0a 1155.48RI32\u00031\n11.54P20\n1=m0250.282Pb0an0173.553Icbca 2157.53P31m 1\n11.55Pa21=m 251.295Pmm0a0174.559Imm0a01157.55P31m01\n11.57PC21=m 351.298Pamma 174.562Ibmma 1159.64Pc31c 3\n12.58C2=m 152.310Pn0n0a 183.50PI4=m 2161.69R3c 2\n12.60C20=m 452.312Pn0na0184.58PI42=m 1161.71R3c02\n12.62C20=m0952.315Pbnna 185.59P4=n 1161.72RI3c 2\n12.63Cc2=m 853.334PBmna 185.64Pc4=n 1162.78Pc\u001631m 1\n12.64Ca2=m 553.335PCmna 186.67P42=n 1164.89P\u00163m01 2\n13.67P20=c 154.350PBcca 186.73PC42=n 3165.94P\u001630c01 1\n13.69P20=c0154.352PIcca 387.75I4=m 1165.95P\u00163c01 2\n13.70Pa2=c 155.355Pb0am 187.78I4=m03165.96Pc\u00163c1 1\n13.73PA2=c 355.356Pbam0188.81I41=a 1166.101R\u00163m05\n13.74PC2=c 455.361Pcbam 188.86Ic41=a 1166.102RI\u00163m 1\n14.75P21=c 956.369Pc0c0n 192.111P41212\u00031166.97R\u00163m 2\n14.77P20\n1=c 356.372Pbccn 192.114P4120\n120\u00031167.103R\u00163c 1\n14.78P21=c0556.373Pcccn 294.132Pc42212\u00031167.106R\u001630c01\n14.79P20\n1=c0856.3740PAccn0296.150PI43212\u00031167.107R\u00163c01\n14.80Pa21=c 2057.389PAbcm 1107.231I4m0m01167.108RI\u00163c 5\n14.81Pb21=c 157.391PCbcm 1111.255P\u0016420m01173.129P63 1\n14.82Pc21=c 658.395Pn0nm 5113.267P\u0016421m 1173.131P60\n3 1\n14.83PA21=c 158.398Pnn0m04114.282PI\u0016421c 1174.136Pc\u00166 1\n14.84PC21=c 1058.399Pn0n0m02117.305PC\u00164b2 1176.145P60\n3=m 1\n15.85C2=c 658.404PInnm 1119.319I\u00164m0201185.197P63cm 3\n15.87C20=c 459.407Pm0mn 2122.338Ic\u001642d 1185.199P60\n3c0m 2\n15.88C2=c0259.409Pm0m0n 1123.345P4=mm0m01185.200P60\n3cm01\n15.89C20=c01159.410Pmm0n01124.360Pc4=mcc 4185.201P63c0m03\n15.90Cc2=c 2859.416PImmn 1125.367P40=nbm01186.207P63m0c01\n15.91Ca2=c 360.419Pb0cn 2125.373PC4=nbm 1188.220Pc\u00166c2 1\n18.19P2120\n120\u0003160.422Pb0c0n 1126.384Pc4=nnc 1189.223P\u00166020m 1\n18.22PB21212\u0003160.431PCbcn 1126.386PI4=nnc 1189.224P\u0016602m01\n19.25P212121\u0003261.433Pbca 2127.394P40=m0bm02192.252Pc6=mcc 2\n19.27P20\n120\n121\u0003161.437Pb0c0a03127.395P4=m0b0m01193.259P60\n3=m0cm01\n19.28Pc212121\u0003161.439PCbca 1127.397PC4=mbm 1193.260P63=mc0m03\n19.29PC212121\u0003162.441Pnma 11128.408Pc4=mnc 1194.268P60\n3=m0m0c1\n20.34C22020\n1\u0003162.443Pn0ma 2128.410PI4=mnc 5203.26Fd\u00163 1\n20.37CA2221\u0003162.444Pnm0a 4129.416P40=n0m0m3205.33Pa\u00163 2\n26.66Pmc 21 262.445Pnma05129.419P4=n0m0m01216.77FS\u001643m 1\n26.68Pm0c210262.446Pn0m0a 9130.432Pc4=ncc 2222.103PIn\u00163n 1\n26.72Pbmc21 362.447Pnm0a03131.440P40\n2=m0m0c1224.113Pn\u00163m0416\n27.82Pccc2 162.448Pn0ma05132.456Pc42=mcm 1227.131Fd\u00163m01\n29.101Pc0a20\n1 462.449Pn0m0a04134.481PC42=nnm 3228.139FSd\u00163c 3\n29.104Paca21 562.450Panma 5135.492Pc42=mbc 2229.143Im\u00163m01\n29.105Pbca21 162.452Pcnma 1136.499P40\n2=mnm02230.148Ia\u00163d01\n29.110PIca21 162.453PAnma 1136.503P42=m0n0m01\n31.129Pbmn21 363.459Cm0cm 1136.506PI42=mnm 1\n32.137Pb0a20163.461Cmcm01138.525P42=nc0m01\n\u0003Chiral MSG\nIn the magnetic material database, all of the materials have distinct chemical formulae/different MSGs except for\nthe 15 materials tabulated in Table V. The 15 compounds are reported having the same chemical formulae and MSGs\nbut different magnetic moments in two independent neutron experiments. The differences between them have been\ndescribed in Table V. These differences consist in the experimental temperature/lattice parameters. In this work, we\nhave performed the abinitio calculations for all of them.\nTABLE V: The 15 compounds that have the same chemical formulae and MSG but with different magnetic moments are\ntabulated together.\nNo.Chemical Formula MSGBSCID (Mx;My;Mz)(\u0016B) Differences\n1CoSe2O5 60.4190.119 Co(3.1,0.0,0.8)Small canting along zaxis0.161 Co(3,0,0)\n2Er2BaNiO5 15.901.15 Er(7.89,0,0.25), Ni(-1.4,0,-0.64) Small difference between the magnetic\nmoments\n1.53 Er(7.23,0,0.32),Ni(-1.38,0,-0.18)\n3Cr2TeO6 58.3950.76 Cr(1,0,0) Experimental temperature is different;\nT= 93Kfor BCSID-58.395, T= 4:2Kfor\nBCSID-0.143\n0.143 Cr(2.45,0,0)\n4Cr2WO6 58.3950.75 Cr(1,0,0) Experimental temperature is different;\nT= 45Kfor BCSID-58.395, T= 4:2Kfor\nBCSID-0.143\n0.144 Cr(2.14,0,0)\n5Ho2Ru2O7 141.5570.49 Ru(0.56,0.56,0.9) Experimental temperature is different;\nT= 20Kfor BCSID-141.557, T= 0:1Kfor\nBCSID-0.51\n0.51Ho(-4.26,-4.26,-1.84),Ru(0.22,0.22,1.77)\n6Ni2SiO4 14.821.203 Ni(1,0,1) Small difference on the lattice parameter\nand experimental temperature\n1.204 Ni(1.82,0,-0.9)\n7ScMn6Ge6 192.2521.110 Mn(0,0,1.96) Experimental temperature is different;\nT= 309Kfor BCSID-1.110, T= 149for\nBCSID-1.225\n1.225 (0,0,2.08)\n8Sr2IrO4 54.3521.3 Ir(0.24,0,0) Small canting along yaxis\n1.77 Ir(0.202,0.048,0)\n9U2Rh2Sn 135.4921.103 U(0,0,0.53) Small poloarization on Rh\n1.207 U(0,0,0.5),Rh(0.04,0.04,0)\n10 Mn2O3 61.4330.40 Mn1(2.6,0,-1.6),Mn2(3.4,0,0.7) Experimental temperature is different;\nT= 2Kfor BCSID-0.40, T= 40Kfor\nBCSID-0.41\n0.41 Mn1(2.4,0,-1.4),Mn2(3.0,0,0.8)\n11Co4Nb2O9 15.880.196Co1(3.7,1.85,1.42),Co2(2.78,1.39,0.97) Small difference on the lattice parameter\nand experimental temperature\n0.197Co1(2.677,1.312,0),Co(2.842,1.953,0)\n12HoMnO3 185.1970.32 Mn(1.72,3.44,0) Experimental temperature is different;\nT= 32Kfor BCSID-0.32, T= 1:7Kfor\nBCSID-0.3317\n0.33 Mn(1.76,3.52,0),Ho(0,0,2.87)\n13 FeI2 12.623.14 Fe(0,0,1) Different lattice parameters\n1.0.13 Fe(0,0,1)\n14Co2SiO4 62.4410.218 Co1(0.94,3.14,0.47),Co2(0,3.64,0) Small difference on the experimental\ntemperature\n0.219 Co1(1.2,3.64,0.57),Co2(0,3.35,0)\n15CuMnSb 16.721.233 Mn(2.53,1.39,2.53) Small difference on the lattice parameter\nand experimental temperature\n1.265 Mn(2.25,2.25,2.25)\nAppendix C: Computational methods\n1. Convention setting of the magnetic unit cell\nWe read the crystalline parameters and magnetic moments from the magnetic structure files, whose datatype are\n’mcif’, provided by BCSMD. In the BCSMD website, lattice parameters of the magnetic unit cell are in the convention\ncalled working setting ( ~ a,~b,~ c) and it can be transformed to the standard convention ( ~ as,~bs,~ cs) by the transformation\nmatrixTs=fTj~ \u001cgas,\n(~ as;~bs;~ cs) =T\u0001(~ a;~b;~ c) +~ \u001c (C1)\nwhere the transformation matrix Ts=fTj~ \u001cgof each material has been supplied in the BCSMD website.\nWhile, in the ab initio calculations, we adopt the primitive magnetic unit cell. The primitive lattice vectors ( ~ p1,~ p2,\n~ p3) can be obtained by transforming the lattice vectors in standard convention ( ~ as,~bs,~ cs) with the transformation\nmatrixMX,\n(~ p1;~ p2;~ p3) = (~ as;~bs;~ cs)\u0001MX (C2)\nwhereMXhas been supplied in the VASP2trace package (www.cryst.ehu.es/cryst/checktopologicalmat) and Xis the\nlattice type of the magnetic unit cell.\n2. Parameters setting in ab initiocalculations\nWe perform all of the first-principle calculations using the Vienna ab initio simulation package(VASP); the\ngeneralized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) type exchange-correlation\npotential was adopted. For each material, we set the cutoff energy for plane wave basis as 1.2 times larger than\nthe suggested value in the pseudo-potential files. In the ab initio calculations, the initial magnetic moments are set\nto the experimental values provided by BCSMD website. The convergence accuracy of self-consistent calculations is\n10\u00005eV and spin-orbital coupling (SOC) has been included. For magnetic cells containing less than 50 atoms, the\nBrillouin zone (BZ) sampling is performed by using k grids with a 9 \u00029\u00029 mesh in self-consistent calculations. We\nreduce the grids to 5 \u00025\u00025 if there are more than 50 atoms in the magnetic primitive cell to save calculational costs.\nWe implement the ab initio calculations on the MPG supercomputer Cobra and Draco with 960 CPU cores in total\nand the supercomputer in ShanghaiTech University with 560 CPU cores. For benchmarking, we calculate the simple\ncompound CeCo 2P2(with 10 atoms per magnetic primitive cell) on the Cobra supercomputer with 80 Skylake cores\nat 2.4GHz. The time used is 15 min 34s for the self consistent calculations and 15 min 45s for the band structure\ncalculations with 240 k points. For the complex compound Sr 3CoIrO 6with 66 atoms per magnetic primitive cell, it\ncosts 7h 50min for the self consistent calculations and 6h 42 min 53s for the band structure calculation with 200 k\npoints.\nSince all of the magnetic materials contain at least one correlated element, we also perform the L(S)DA+U\ncalculations for all of the magnetic materials using the VASP. For the L(S)DA+U calculations, we adopt the simplified\n(rotationally invariant) approach and set the Hubbard U as 1, 2, 3, 4 eV for the delectrons and 2, 4, 6 eV for the f\nelectrons. For the materials which have both dandfelectron, we set U of delectron as 2 eV and the U of felectron\nas 2, 4, 6 eV.18\nSimilar with the TQC for paramagnetic materials, we also provide the maximal kvectors for each magnetic space\ngroup in the BCS website. Based on the self-consistent charge density files, we calculate the wave functions at the\nmagnetic maximal kvectors and obtain the characters using the MagVASP2trace package.\n3. Magnetic VASP2trace package\nIn TQC, the in house VASP2trace package [75] is used to calculate the character tables of paramagnetic materials.\nIt read the unitary symmetry operators from the output files of VASP and can identify the space group. While the\nanti-unitary symmetries are absent and VASP2trace cannot identify the magnetic space groups (MSGs).\nIn the MTQC, we revise the VASP2trace package to calculate the character tables of magnetic materials and supply\nthe symmetry file for each MSG. The magnetic VASP2trace (MagVASP2trace) [ ?] reads the magnetic symmetries\nfrom the symmetry files that we supply, instead of reading them from the output files of VASP. The symmetry file\ncontains both unitary operations and anti-unitary operations. Both SO(3) and SU(2) matrix in the symmetry files\nare written in the basis of primitive lattice vectors.\nMagVASP2trace adopts both SO(3) and SU(2) matrix in the convention used in the BCS website\n(https://www.cryst.ehu.es/) and generate the trace.txt file that contains all of the magnetic symmetry operators\nand the character tables of the occupied bands at the magnetic maximal kvectors.\n4. Construction of Wannier tight-binding Hamiltonian and surface states calculation\nWe construct the tight-binding Hamiltonians of NpBi, CeCo 2P2, MnGeO 3and Mn 3ZnC using the Wannier90\npackage [92]. We generate the maximally localized Wannier functions (MLWFs) for 5 porbitals on Bi, 5 fand 6d\norbitals on Np for the magnetic TI NpBi. For the magnetic NLSM CeCo 2P2, the MLWFs for 3 porbitals on P, 3 d\norbitals on Co, 4 fand 5dorbitals on Ce are constructed. For the magnetic DSM MnGeO 3, we generate the MLWFs\nfor 4sorbitals on Ge, 2 porbitals on O, and 3 dorbitals on Mn. For the ferrimagnetic ES Mn 3ZnC, we generate the\nMLWFs for 4s,4pand3dorbitals on Zn, 2porbitals on C and 3dorbitals on Mn.\nThe surface states are calculated with the Green’s function method using the WannierTools package [93, 94], and\nthe results are shown in FIG. 2 of main text and FIG. 32-33.\nAppendix D: Comparison of the ground state energy between different magnetic configurations of several\ncompounds\nWeselectthethreemagnetictopologicalmaterialsNpBiwithBCSID-3.7, CeCo 2P2withBCSID-1.253andMnGeO 3\nwith BCSID-0.125 to compare the energy difference between different magnetic structures, respectively. As shown in\nFigure 3, there are three possible magnetic structures for each material, where AFM-I and AFM-II are the assumed\nconfigurations and the AFM-III phase is the one obtained from neutron experiments. The relative ground state\nenergies at each U for the three materials are tabulated in Table VI. For NpBi and CeCo 2P2, AFM-III phase always\nhas the lowest ground state energy at different U. For MnGeO 3, there is only one exception, i.e. the AFM-I phase of\nMnGeO 3with U=0, that has lower energy than the AFM-III phase. With increasing U, the experimental AFM-III\nphase lowers its energy to become the lowest.\nThe comparisons in Table VI indicate that the magnetic configurations obtained from neutron experiments are\nfavorable with the lowest ground state energy.\nTABLE VI: The relative ground state energy of NpBi, CeCo 2P2and MnGeO 3in three possible magnetic structures with\ndifferent U added. The magnetic structures are shown in Figure 3, where the magnetic configurations AFM-III are obtained\nfrom neutron scattering experiments.\nMaterials (BCSID) U(eV)E(AFM-I) (eV) E(AFM-II) (eV) E(AFM-III) (eV)\nNpBi (BCSID: 3.7)0 0.079 0.091 0\n2 2.108 2.125 1.995\n4 3.049 3.05 2.999\n6 3.538 3.543 3.525\nCeCo 2P2(BCSID: 1.253)0 0.388 0.388 019\nFIG. 3. Three possible magnetic structures for (a-c)NpBi, (d-f)CeCo 2P2and (g-i) MnGeO 3, where the AFM-III phase in\n(c)(f)(i) are the ones from neutron scattering experiments.\n2 9.159 9.243 8.468\n4 10.666 10.729 9.973\n6 11.955 12.098 11.263\nMnGeO 3(BCSID: 0.125)0 -0.4 0.164 0\n1 6.845 6.532 6.437\n2 12.713 12.151 12.068\n3 17.703 17.132 17.068\n4 22.095 21.589 21.529\nAppendix E: Comparisons between different exchange-correlation potentials\n1. Band structure calculations with GGA functional\nWe adopt five different exchange-correlation functional methods, including Perdew-Wang 91 (91), AM05 (AM),\nrevised PBE (RE), revised PBE with Pade Approximation (RP) and Ceperley-Alder functional (CA), to check the\ntopologies and the band structures that obtained by the PBE (PE) method. The topology and band structure\ncomparisons of the topological materials BaFe 2As2, CeCo 2P2, NpBi, and MnGeO 3are shown in the FIG. 4-7. The\ncomparisons indicate that different exchange-correlation functional methods have minor effect on the band structures\nbut do not change the topologies of these materials.\n2. Band structure calculations with meta-GGA functional\nTo further check the band structures and topologies obtained from LDA+U calculations, we have also performed\nab initio calculations with the modified Becke-Johnson (mBJ) [73] potential for 23 topological compounds.\nThey are Mn 3Ir (BCSID-0.108), Mn 3Sn (BCSID-0.200), Mn 3Pt (BCSID-1.143), MnGeO 3(BCSID-0.125), Mn 2As\n(BCSID-1.132), CaFe 2As2(BCSID-1.52), Cd 2Os2O7(BCSID-0.2), NiCr 2O4(BCSID-0.4), PbNiO 3(BCSID-0.21),\nLuFeO 3(BCSID-0.117), LuFe 4Ge2(BCSID-0.140), NiS 2(BCSID-0.150), Mn 3Ge (BCSID-0.203), Co 2SiO4\n(BCSID-0.218), CrN (BCSID-1.28), ScMn 6Ge6(BCSID-1.110), CaCo 2P2(BCSID-1.252), CeCo 2P2(BCSID-1.253),\nGdIn 3(BCSID-1.81), Mn 3ZnC (BCSID-2.19), NpBi (BCSID-3.7), NpSe (BCSID-3.10) and NpSb (BCSID-3.12).20\nFIG. 4. The band structures and topology of BaFe 2As2obtained by different exchange-correlation functional methods. The\ntopology is maintained for different methods, indicating a TI with topological index c2= 1. The Hubbard U of 3delectron is\nset to 1 eV.\nFIG. 5. The band structures and topology of CeCo 2P2obtained by different exchange-correlation functional methods. The\ntopology is maintained ES (also NLSM) for the different methods. The Hubbard U of 3dand4felectron are set to 2 and 6\neV, respectively.\nApart from NpSe (BCSID-3.10), we find slightly different band structures with MBJ and LDA+U methods.\nHowever, comparing these 2 band structures and its topology, we can always find a value of U that reproduces\nthe MBJ calculations. As shown in FIG. 8 \u000029, we have found the correct value of U for each compound. Using the\ncorrect value of U, we can reproduce the the band structures and topology at the Fermi level consistent with the\nresults obtained from mBJ.21\nFIG. 6. The band structures andtopology of NpBiobtained by different exchange-correlation functional methods. Thetopology\nis maintained TI with topological index c2= 1for the different methods. The Hubbard U of 5felectron is set to 2 eV.\nFIG. 7. The band structures and topology of MnGeO 3obtained by different exchange-correlation functional methods. The\ntopology is maintained ES (also DSM) for the different methods. The Hubbard U of 3delectron is set to 4 eV.\nFIG. 8. Band structures of the ES Mn 3Ir obtained from LDA+U ( U= 2eV) and mBJ methods.22\nFIG. 9. Band structures of the ES MnGeO 3obtained from LDA+U ( U= 4eV) and mBJ methods.\nFIG. 10. Band structures of the ES Mn 3Sn obtained from LDA+U ( U= 1eV) and mBJ methods.\nFIG. 11. Band structures of the ES Mn 2As obtained from LDA+U ( U= 0eV) and mBJ methods.\nFIG. 12. Band structures of the TI CaFe 2As2obtained from LDA+U ( U= 2eV) and mBJ methods.23\nFIG. 13. Band structures of the ESFD Mn 3Pt obtained from LDA+U ( U= 2eV) and mBJ methods.\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ                           L                        W              XCd2Os2O7 (LCEBR) LDA+U (U=4 eV)\n0.00.5\n-0.51.0\n-1.0Energy (eV)Cd2Os2O7 (LCEBR) mBJ\n Γ                           L                        W              X\nFIG. 14. Band structures of Cd 2Os2O7obtained from LDA+U ( U= 2eV) with LCEBR phase and mBJ method with LCEBR\nphase.\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ         L          LA       T                 Y                ZNiCr2O4 (LCEBR) LDA+U (U=4 eV)\n0.00.5\n-0.51.0\n-1.0Energy (eV)NiCr2O4 (LCEBR) mBJ\n Γ         L          LA       T                 Y                Z\nFIG. 15. Band structures of the NiCr 2O4obtained from LDA+U ( U= 4eV) with LCEBR phase and mBJ method with\nLCEBR phase.\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ                F                 L                                  T     PbNiO3 (LCEBR) LDA+U (U=4 eV)\n0.00.5\n-0.51.0\n-1.0Energy (eV)    PbNiO3 (LCEBR) mBJ\n Γ                F                L                                  T \nFIG. 16. Band structures of PbNiO 3obtained from LDA+U ( U= 4eV) with LCEBR phase and mBJ methods with LCEBR\nphase.24\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ         A                       H        K              L      MLuFeO3 (LCEBR) LDA+U (U=4 eV)\n0.00.5\n-0.51.0\n-1.0Energy (eV)LuFeO3 (LCEBR) mBJ\n Γ         A                       H        K              L      M\nFIG. 17. Band structures of the LuFeO 3obtained from LDA+U ( U= 4eV) with LCEBR phase and mBJ methods with\nLCEBR phase.\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ           R        S          T     U        X      Y        ZLuFe4Ge2 (ESFD) LDA+U (U=4 eV)\n0.00.5\n-0.51.0\n-1.0Energy (eV)LuFe4Ge2 (ESFD)  mBJ\n Γ           R        S          T     U        X      Y        Z\nFIG. 18. Band structures of the ESFD LuFe 4Ge2obtained from LDA+U ( U= 4eV) and mBJ methods.\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ                      M                  R                         XNiS2 (LCEBR) LDA+U (U=4 eV)\n0.00.5\n-0.51.0\n-1.0Energy (eV)NiS2 (LCEBR) mBJ\n Γ                      M                  R                        X\nFIG. 19. Band structures of NiS 2obtained from LDA+U ( U= 2eV) with LCEBR phase and mBJ methods with LCEBR\nphase.\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ      A      L           LA    M         V           VA    YMn3Ge (TI) LDA+U (U=4 eV)\n0.00.5\n-0.51.0\n-1.0Energy (eV)Mn3Ge (TI) mBJ\n Γ      A      L           LA    M         V           VA    Y\nFIG. 20. Band structures of the TI Mn 3Ge obtained from LDA+U ( U= 4eV) and mBJ methods.25\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ          R       S         T       U      X       Y         ZCo2SiO4 (LCEBR) LDA+U (U=2 eV)\n0.00.5\n-0.51.0\n-1.0Energy (eV)Co2SiO4 (LCEBR) mBJ\n Γ          R       S         T       U      X       Y         Z\nFIG. 21. Band structures of Co 2SiO4obtained from LDA+U ( U= 2eV) with LCEBR phase and mBJ methods with LCEBR\nphase.\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ          R      S       T        U     X          Y          ZCrN (LCEBR) LDA+U (U=4 eV)\n0.00.5\n-0.51.0\n-1.0Energy (eV)CrN (LCEBR) mBJ\n Γ          R      S       T        U     X          Y          Z\nFIG. 22. Band structures of CrN obtained from LDA+U ( U= 2eV) with LCEBR phase and mBJ methods with LCEBR\nphase.\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ     A                              H      K              L     MScMn6Ge6 (ES) LDA+U (U=0)\n0.00.5\n-0.51.0\n-1.0Energy (eV)ScMn6Ge6 (ES) mBJ\n Γ     A                              H      K              L     M\nFIG. 23. Band structures of the ES ScMn 6Ge6obtained from LDA+U ( U= 0) and mBJ methods.\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ          R       S        T        U      X       Y          ZCaCo2P2 (TI) LDA+U (U=2 eV)\n0.00.5\n-0.51.0\n-1.0Energy (eV)CaCo2P2 (TI) mBJ\n Γ          R       S        T        U      X       Y          Z\nFIG. 24. Band structures of the TI CaCo 2P2obtained from LDA+U ( U= 2eV) and mBJ methods.26\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ                    A     M              R     X               ZCeCo2P2 (ES) LDA+U (U=2 eV)\n0.00.5\n-0.51.0\n-1.0Energy (eV)CeCo2P2 (ES) mBJ\n Γ                    A     M              R     X               Z\nFIG. 25. Band structures of the ESFD CeCo 2P2obtained from LDA+U ( U= 2eV) and mBJ methods.\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ              A          M             R          X            ZGdIn3 (ES) LDA+U (U=2 eV)\n0.00.5\n-0.51.0\n-1.0Energy (eV)GdIIn3 (TI) mBJ\n Γ              A          M             R          X            Z\nFIG. 26. Band structures of the ES GdIn 3obtained from LDA+U ( U= 2eV) and mBJ methods.\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n\u0003K                             M                     N        P     XMn3ZnC (ES) LDA+U (U=0)\n0.00.5\n-0.51.0\n-1.0Energy (eV)Mn3ZnC (ES) mBJ\n\u0003K                             M                     N        P     X\nFIG. 27. Band structures of the ES Mn 3ZnC obtained from LDA+U ( U= 2eV) and mBJ methods.\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ                       M                  R                       XNpBi (TI) LDA+U (U=2 eV)\n0.00.5\n-0.51.0\n-1.0Energy (eV)NpBi (TI) mBJ\n Γ                       M                  R                       X\nFIG. 28. Band structures of the TI NpBi obtained from LDA+U ( U= 2eV) and mBJ methods.27\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ                        M                R                        XNpSb (TI) LDA+U (U=2 eV)\n0.00.5\n-0.51.0\n-1.0Energy (eV)NpSb (TI) mBJ\n Γ                        M                R                        X\nFIG. 29. Band structures of the TI NpSb obtained from LDA+U ( U= 2eV) and mBJ methods.28\n0.00.5\n-0.51.0\n-1.0Energy (eV)\n Γ                           L                       W              XNpSe (ESFD) LDA+U (U=2eV)\n0.00.5\n-0.51.0\n-1.0Energy (eV)NpSe (ES) mBJ\n Γ                           L                       W              X\nFIG. 30. Band structures of NpSe obtained from LDA+U ( U= 2eV) with ESFD phase and mBJ methods with ES phase. In\nLDA+U calculations, the irreps of the bands at \u0000point on the Fermi level is 2-fold degenerate and half-filled. So, it’s in ESFD\nphase. While, in mBJ calculations, the valence band that crossing the Fermi level is 1-dimentional. It’s in the ES phase with\na symmetry protected crossing point on the \u0000Lpath.\nAppendix F: Comparisons between LDA+U and LDA+Gutzwiller methods\nTo further check the robustness of the results obtained from LDA+U calculations, we have performed the\nLDA+Gutzwiller calculations [74] for the two stable magnetic topological semimetals, MnGeO 3and CeCo 2P2.\nLDA+Gutzwiller is a many-body technics combined with DFT calculations, which has been successfully applied to\npredict correlated topological materials [95–97]. Similar to other post-LDA methods, i.e. LDA+U and LDA+DMFT,\nthe total Hamiltonian adopt in LDA+Gutzwiller can be written as,\nHtotal=HLDA +Hint+Hdc (F1)\nwithHLDAbeing the non-interacting Hamiltonian obtained by LDA+SOC, the atomic spin orbital coupling and Hint\nbeing the interacting Hamiltonian. The last term in Eq. (F1) is the double counting Hamiltonian, which needs to\nbe included to remove the local interaction energy treated by LDA already in the mean field manner. In the present\nstudy, the Kanamori-type interaction and the fully localized limit scheme for the double counting energy [98] are\nadopted.\nIn the LDA+Gutzwiller method, the Gutzwiller type wave function j Gi=^Pj 0ihas been proposed for the trial\nwave function to minimize the ground state energy, where j 0iis the non-interacting wave function and ^Pis the local\nprojector applied to adjust the probability of the local atomic configuration (in the many-particle Fock space). In\naddition, the Gutzwiller approximation is applied to evaluate the ground state energy and an effective Hamiltonian\nHeff\u0019^PHLDA^Pdescribing the quasi-particle dispersion can be obtained. For detailed description for the method\nplease refer to references [74, 99, 100].\nIn the LDA+Gutzwiller calculation, we adopt the same parameter U as LDA+U calculation. For MnGeO 3, the\nquasi-particle weight of the delectron on Mn is about 0.8 and the local magnetic moment on Mn is about 4.6 \u0016B/Mn\nwhich is consistent with LDA+U calculation (4.3 \u0016B/Mn). Compared with the band structure obtained from LDA+U\ncalculations in Figure 31(a), the quasi-particle band structure in LDA+Gutzwiller are renormalized by a factor of\n0.86, as shown in Figure 31(b).\nForCeCo 2P2, thequasi-particleweightofthe felectrononCeisabout0.25andtheoccupationof felectronisabout\n1.0/Ce. Compared with the band structure obtained from LDA+U calculations in Figure 31(c), the quasi-particle\nband structure in LDA+Gutzwiller are strongly renormalized by a factor of 0.25, as shown in Figure 31(d). Although\nthe large-size renormalization on forbitals changes the quasi-particle bands a lot, the symmetry enforced band\ncrossing along ZApath is stable.\nBoth comparisons for MnGeO 3and CeCo 2P2indicate that strong correlations only renormalize the band width by\na factor of quasi-particle weight but don’t change the topologies for the stable topological materials MnGeO 3and\nCeCo 2P2.\nAppendix G: Topological phase diagrams of the topological materials that predicted by MTQC\nWe tabulate the topological categories at different U’s for all the magnetic materials. In the Table VII and VIII,\neach material list is represented by different colors based on their phase transition trends with increasing UThe tables29\nFIG. 31. (a) Electronic band structures of MnGeO 3obtained from LDA+U and (b) LDA+Gutzwiller (b) with the on-site\nCoulomb interaction U= 4eVand the Hund’s coupling J= 0:8eV. From LDA+Gutzwiller calculations, the quasi-particle\nweight of the delectron on Mn is about 0.86 and the magnetic moments on Mn is about 4.8 \u0016B(c)-(d) Electronic band\nstructures of CeCo 2P2obtained from LDA+U and LDA+Gutzwiller, respectively. The on-site interaction of forbitals is taken\nasU= 6eV. From LDA+Gutzwiller calculations, the quasi-particle weight of the felectron on Ce is about 0.25. The magnetic\nmoments on Ce and Co are 0.0 and 0.9 \u0016B, respectively.\ncontain the material identification number in BCSMD (BCSID), chemical formula (Formula), magnetic space group\n(MSG), the correlated atoms (CA) that exhibit added Uin the ab initio calculations, topological phases with different\nUand the link to our plotted band structures (BS).\nIn Tables VII and VIII, the interaction parameter Uis varied in the range of 0, 1, 2, 3, 4 eV for delectron and 0,\n2, 4, 6 eV for felectron, respectively. Upon adding and increasing U, there are 49 of the 130 topological materials\nthat have stable topology (remain the same topology for all U), and 49 materials have nontrivial topology for weak\ncorrelation, while becoming topological trivial with strong correlations. There are 20 materials whose topologies are\nsensitive to the interaction and have topological phase transitions between TI and ES in a small interaction range.\nThere are only 5 materials that belong to trivial class in the weak correlated case and become topological nontrivial\nwhen the correlation is strong enough. The trend of ’Topological !Trivial’ upon large U is clear in our data.\nThe green color stands for nontrivial topology stable in the whole range of Uconsidered in the calculations. The\nblue color stands for topology nontrivial for weak correlation effect, but trivial with strong correlation. The yellow\ncolor stands for topology sensitive to correlations and for topological phase transitions between TI and ES in a small\ninteraction range. The red color stands for topology trivial in weak correlations, but nontrivial with increasing U.\nThe grey color stands for the cases where the self consistent calculations are not converged and the topological phase\ndiagrams are not completed. We have separated the material list into two parts: one for delectron with Coulomb in\nthe range of 0\u00184eV, another for felectron with Coulomb in the range of 0\u00186eV. The materials are ordered by\nthe MSG. We also tag the ESs/ESFDs with chiral MSGs, in which the crossing points carry nonzero chiral charges.\nWe emphasis that if a symmetry-data-vector cannot be written as an integer combination of EBRs (LCEBR) and\nthe compatibility relations are satisfied, it is diagnosed as a TI in Table VII and VIII. This TI can be stable TI/SISM\nwith stable topological index. In H, we interpret all the TIs by their topological indices. We also find that some of\nthe ES phases can be changed to TI/SISM phase by symmetry breaking. For example, the ES phase ( U= 3;4eV)\nof Mn 3Sn (with BCSID-0.200) can be calssified as SISM phase (with indice \u00114I= 3) if its MSG 63.464 ( Cm0cm0) is\nsubducted to the minimal subgroup MSG 2.4 ( P\u00161).\nTABLE VII: Topological phase diagram of the magnetic materials that have transition elements.. The interaction parameter\nUofdelectrons on the correlated atoms have been set to 0, 1, 2, 3 and 4 eV. For the material EuFe 2As2(BCS-ID: 2.1), since\nthe local magnetic moments on Eu are about 7.0 \u0016B, which is fully spin-polarized, we take the U of felectron on Eu as 4 eV\nand the U of delectron on Fe as 0, 1, 2, 3, 4 eV.\nBCS-ID Formula MSG CAU=0 U=1 U=2 U=3 U=4 BS\n0.165SrMn(VO4)(OH) 4.7(P21)\u0003Mn,V ESLCEBR LCEBR LCEBR LCEBR Table [LI]30\n1.264 CoPS3 11.57(PC21=m)CoTILCEBR LCEBR LCEBR LCEBR Table\n[LXVI]\n0.203 Mn3Ge 12.62(C20=m0)Mn TI TI TI TI TITable [LXX]\n1.0.13 FeI2 12.62(C20=m0)FeTILCEBR LCEBR LCEBR LCEBR Table\n[LXXI]\n1.201 Cr2ReO6 14.80(Pa21=c)CrTILCEBR LCEBR TBD TBD Table\n[XCVII]\n1.49 Ag2NiO2 15.90(Cc2=c)NiTILCEBR LCEBR LCEBR LCEBR Table\n[CXXXIV]\n1.50 AgNiO2 18.22(PB21212)\u0003NiES ESLCEBR LCEBR LCEBR Table\n[CXXXIX]\n1.263 Ca3Ru2O7 33.154(PCna21)RuES ESLCEBR LCEBR LCEBR Table [CLV]\n0.85 KCo4(PO4)3 58.398(Pnn0m0)CoESLCEBR LCEBR LCEBR LCEBR Table\n[CLXXIX]\n0.140 LuFe4Ge2 58.399(Pn0n0m0)FeESFD ESFD ESFD ESFD ESFD Table\n[CLXXX]\n0.27 YFe4Ge2 58.399(Pn0n0m0)FeESFD ESFD ESFD ESFD ESFD Table\n[CLXXXI]\n1.252 CaCo2P2 59.416(PImmn)CoLCEBR TI TI TI TI Table\n[CLXXXIV]\n1.88 Mn5Si3 60.431(PCbcn)Mn TI ESLCEBR LCEBR LCEBR Table\n[CLXXXVII]\n2.1 EuFe2As2 61.439(PCbca)Eu,FeLCEBR ES TI TILCEBR Table\n[CXCI]\n0.218 Co2SiO4 62.441(Pnma)CoESLCEBR LCEBR LCEBR LCEBR Table\n[CXCV]\n0.219 Co2SiO4 62.441(Pnma)CoES ESLCEBR LCEBR LCEBR Table\n[CXCVI]\n0.221 Fe2SiO4 62.441(Pnma)FeESLCEBR LCEBR LCEBR LCEBR Table\n[CXCVII]\n1.130 Cr2As 62.450(Panma)CrLCEBR TILCEBR LCEBR LCEBR Table\n[CCXIII]\n1.131 Fe2As 62.450(Panma)FeTI TI ES TI TI Table\n[CCXIV]\n1.132 Mn2As 62.450(Panma)MnES ES ES ES ES Table\n[CCXV]\n1.28 CrN 62.450(Panma)CrTI TILCEBR LCEBR LCEBR Table\n[CCXVI]\n0.199 Mn3Sn 63.463(Cmc0m0)MnES ES ES ES ES Table\n[CCXIX]\n0.200 Mn3Sn 63.464(Cm0cm0)Mn TI TI TI ES ES Table\n[CCXX]\n1.16 BaFe2As2 64.480(CAmca)FeTI TI TI TILCEBR Table\n[CCXXIV]\n1.52 CaFe2As2 64.480(CAmca)FeTI TI TILCEBR LCEBR Table\n[CCXXVI]\n2.15 Mn3Ni20P6 65.486(Cmm0m0)Mn,Ni TBD ES ES TI ES Table\n[CCXXVII]\n0.4 NiCr2O4 70.530(Fd0d0d)Ni,Cr ESLCEBR LCEBR LCEBR LCEBR Table\n[CCXXX]\n1.125 LaFeAsO 73.553(Icbca)FeTILCEBR LCEBR LCEBR LCEBR Table\n[CCXXXI]\n1.176 YbCo2Si2 73.553(Icbca)CoLCEBR TI TI TI TI Table\n[CCXXXII]\n2.5 Mn3CuN 85.59(P4=n)MnES ES ES ES ES Table\n[CCXXXIV]31\n0.64 MnV2O4 88.81(I41=a)Mn,V ESLCEBR LCEBR LCEBR LCEBR Table\n[CCXXXVII]\n1.85 alpha-Mn 114.282(PI\u0016421c)MnESLCEBR LCEBR LCEBR TBD Table\n[CCXLII]\n1.143 Mn3Pt 132.456(Pc42=mcm)MnESFD ESFD ESFD ESFD ESFD Table\n[CCXLV]\n1.146 LaCrAsO 138.528(Pc42=ncm)CrTI TI TI TI TI Table\n[CCXLIX]\n0.212 Sr2Mn3As2O2 139.536(I40=m0m0m)MnESFD ESFD ESFD ESFD ESFD Table\n[CCLI]\n2.19 Mn3ZnC 139.537(I4=mm0m0)MnES ES ES ES ES Table\n[CCLIV]\n0.125 MnGeO3 148.19(R\u001630)MnES ES ES ES ES Table\n[CCLXI]\n0.21 PbNiO3 161.69(R3c)NiESFD ESFD ESFDLCEBR LCEBR Table\n[CCLXVII]\n1.0.5 Sr3CoIrO6 165.95(P\u00163c01)CoESLCEBR LCEBR LCEBR LCEBR Table\n[CCLXXIII]\n0.108 Mn3Ir 166.101(R\u00163m0)MnES ES ES ES ES Table\n[CCLXXV]\n0.109 Mn3Pt 166.101(R\u00163m0)MnES ES ES ES ES Table\n[CCLXXVI]\n0.177 Mn3GaN 166.97(R\u00163m)MnES ES ES ESESFD Table\n[CCLXXVII]\n1.153 Mn3GaC 167.108(RI\u00163c)MnESLCEBR LCEBR LCEBR LCEBR Table\n[CCLXXX]\n0.117 LuFeO3 185.201(P63c0m0)FeTILCEBR LCEBR LCEBR LCEBR Table\n[CCLXXXV]\n1.110 ScMn6Ge6 192.252(Pc6=mcc)MnES ES ES ES ES Table\n[CCLXXXVIII]\n1.225 ScMn6Ge6 192.252(Pc6=mcc)MnES ES ES ES ES Table\n[CCLXXXIX]\n0.118 Ba5Co5ClO13 194.268(P60\n3=m0m0c)CoLCEBR LCEBR LCEBR LCEBR ES Table\n[CCXCI]\n0.150 NiS2 205.33(Pa\u00163)NiTILCEBR LCEBR LCEBR LCEBR Table\n[CCXCIII]\n0.2 Cd2Os2O7 227.131(Fd\u00163m0)OsESLCEBR LCEBR LCEBR LCEBR Table\n[CCXCV]\n\u0003Chiral MSG. The crossing points in the ES/ESFD phase with Chiral MSG must carry nonzero chiral charges. [90, 91]\nTABLE VIII: Topological phase diagram of the magnetic materials that have Rare-earth elements. The interaction parameter\nUoffelectrons on Rare-earth elements have been set to 0, 2, 4 and 6 eV. If the material also have transition elements, we\ntakeUofdelectron as 2 eV.\nBCS-ID Formula MSG CAU=0 U=2 U=4 U=6 BS\n1.206Dy2Fe2Si2C 2.7(PS\u00161) Dy,Fe TI TI TI TI Table. [CCXCVII]\n0.104 ErVO3 11.54(P20\n1=m0)Er,VLCEBR TILCEBR LCEBR Table. [CCCII]\n0.106 DyVO3 11.54(P20\n1=m0)Dy,V TILCEBR LCEBR LCEBR Table. [CCCIII]\n1.22 DyCu2Si2 12.63(Cc2=m)Dy,Cu TBD TBDLCEBR ES Table. [CCCV]\n1.140 PrMgPb 13.73(PA2=c)Pr TI TI TI TI Table. [CCCVII]\n1.141 NdMgPb 13.73(PA2=c)Nd TI TILCEBR LCEBR Table. [CCCVIII]\n0.105 ErVO3 14.75(P21=c)Er,V ESLCEBR LCEBR LCEBR Table. [CCCX]\n0.174Pr3Ru4Al12 15.89(C20=c0)Pr,Ru TI TI TI TI Table. [CCCXVI]\n0.226 NdCo2 15.89(C20=c0)Nd,Co ES TI TI TI Table. [CCCXVII]\n2.10 HoP 15.89(C20=c0)Ho TI TI TI TI Table. [CCCXVIII]32\n1.211 Dy2O2S 15.90(Cc2=c)Dy TILCEBR LCEBR LCEBR Table. [CCCXXI]\n1.216Nd2BaNiO5 15.90(Cc2=c)Nd,Ni TILCEBR LCEBR LCEBR Table. [CCCXXIII]\n1.217Tb2BaNiO5 15.90(Cc2=c)Tb,Ni TILCEBR LCEBR LCEBR Table. [CCCXXIV]\n1.43 PrNiO3 36.178(Camc21)Pr,NiLCEBR LCEBR ESLCEBR Table. [CCCXXIX]\n0.26 TmAgGe 38.191(Am0m02)Tm ESTBD TBD TBD Table. [CCCXXX]\n2.12 TbMg 49.270(Pc0cm0)Tb ES ES ES ES Table. [CCCXXXI]\n1.139 Ho2RhIn8 49.273(Pcccm)Ho ES TI TI ESTable. [CCCXXXII]\n2.11 TbMg 51.295(Pmm0a0)Tb ES ES ES ESTable. [CCCXXXIII]\n1.222 Er2CoGa8 51.298(Pamma)Er,Co ES ES ES ESTable. [CCCXXXIV]\n1.150 PrAg 53.334(PBmna)PrLCEBR TBD TITBD Table. [CCCXXXV]\n1.8CeRu2Al10 57.391(PCbcm)Ce,Ru TI TI TILCEBR Table. [CCCXXXVIII]\n0.187 CeMnAsO 59.407(Pm0mn)Ce,Mn ESLCEBR LCEBR LCEBR Table. [CCCXXXIX]\n0.185 Nd5Ge4 62.447(Pnm0a0)Nd ES ESTBD ES Table. [CCCXLI]\n1.179 NdCoAsO 62.450(Panma)Nd,Co ES TI ES ES Table. [CCCXLV]\n0.149Nd3Ru4Al12 63.462(Cm0c0m)Nd,Ru ES ES ES ES Table. [CCCXLVII]\n0.173Pr3Ru4Al12 63.462(Cm0c0m)Pr,Ru ES ES ES ESTable. [CCCXLVIII]\n3.3Ho2RhIn8 63.464(Cm0cm0)Ho ES ESTBD TBD Table. [CCCXLIX]\n1.200 U2Ni2Sn 63.466(Ccmcm)U,NiLCEBR TI TI TI Table. [CCCL]\n1.262 NpRhGa5 63.466(Ccmcm)NpLCEBR TI ESLCEBR Table. [CCCLI]\n1.195 Er2Ni2In 63.467(Camcm)Er,NiLCEBR LCEBR TILCEBR Table. [CCCLII]\n1.188 CeRh2Si2 64.480(CAmca)Ce TILCEBR ES TI Table. [CCCLIV]\n1.223Tm2CoGa8 65.489(Cammm)Tm,Co TI ES ES ES Table. [CCCLVI]\n1.142 CeMgPb 67.510(CAmma)CeLCEBR TITBD TBD Table. [CCCLVIII]\n1.0.12 UAu2Si2 71.536(Im0m0m)UES ES ES ES Table. [CCCLX]\n2.28 NpNiGa5 74.559(Imm0a0)Np,Ni ES ES ES ES Table. [CCCLXI]\n0.184 Nd5Si4 92.114(P4120\n120)\u0003Nd ES ES ES ES Table. [CCCLXIV]\n1.0.11 CeCoGe3 107.231(I4m0m0)Ce,CoLCEBR ES ESTBD Table. [CCCLXV]\n2.26 PrCo2P2 123.345(P4=mm0m0)Pr,Co ES ES ES ES Table. [CCCLXVI]\n1.162 NdMg 124.360(Pc4=mcc)Nd ES ES ES ES Table. [CCCLXVII]\n1.251 NdCo2P2 124.360(Pc4=mcc)Nd,Co ES ES ES TITable. [CCCLXVIII]\n1.255 UPtGa5 124.360(Pc4=mcc)UTI ES ES ES Table. [CCCLXIX]\n1.261 NpRhGa5 124.360(Pc4=mcc)Np ES ES ES ES Table. [CCCLXX]\n2.14 NdMg 125.373(PC4=nbm)Nd ES ES ES ES Table. [CCCLXXII]\n1.253 CeCo2P2 126.386(PI4=nnc)Ce,Co ES ES ES ESTable. [CCCLXXIII]\n0.80 U2Pd2In 127.394(P40=m0bm0)UESFD ESFD ESFD ESFD Table. [CCCLXXIV]\n0.81 U2Pd2Sn 127.394(P40=m0bm0)UTI TI TI TITable. [CCCLXXV]\n1.81 GdIn3 127.397(PC4=mbm)Gd ES ES ES ESTable. [CCCLXXVII]\n1.102 U2Ni2In 128.408(Pc4=mnc)U,Ni ES ES ES ESTable. [CCCLXXVIII]\n1.160 UP 128.410(PI4=mnc)UES ES ES ESTable. [CCCLXXIX]\n1.187 TbRh2Si2 128.410(PI4=mnc)Tb ES ES ES ES Table. [CCCLXXX]\n1.208 UAs 128.410(PI4=mnc)UES ES ES TITable. [CCCLXXXI]\n1.21 DyCo2Si2 128.410(PI4=mnc)Dy,Co ES ES ES ESTable. [CCCLXXXII]\n0.186 CeMnAsO 129.416(P40=n0m0m)Ce,MnLCEBR TI TI TITable. [CCCLXXXIII]\n1.215 UP2 130.432(Pc4=ncc)UES ES ES ESTable. [CCCLXXXV]\n2.13 UP 134.481(PC42=nnm)UTI TI TI TITable. [CCCLXXXVII]\n2.20 UAs 134.481(PC42=nnm)UTI ES TI TITable. [CCCLXXXVIII]\n2.6Nd2CuO4 134.481(PC42=nnm)Nd TILCEBR LCEBR LCEBR Table. [CCCLXXXIX]\n1.103 U2Rh2Sn 135.492(Pc42=mbc)UES ES ES ES Table. [CCCXC]\n1.207 U2Rh2Sn 135.492(Pc42=mbc)UES ES ES ES Table. [CCCXCI]\n1.254 UNiGa5 140.550(Ic4=mcm)U,Ni ES TI TILCEBR Table. [CCCXCII]\n1.82 Nd2RhIn8 140.550(Ic4=mcm)NdLCEBR ES ES ES Table. [CCCXCIII]33\n1.87TbCo2Ga8 140.550(Ic4=mcm)Tb,Co ES ES ES ES Table. [CCCXCIV]\n0.126 NpCo2 141.556(I40\n1=a0m0d)Np,Co ESTBD TI TITable. [CCCXCVIII]\n0.151Tm2Mn2O7 141.557(I41=am0d0)Tm,Mn ESLCEBR LCEBR LCEBR Table. [CD]\n0.227 NdCo2 141.557(I41=am0d0)Nd,Co ES ES ES ES Table. [CDII]\n0.48Tb2Sn2O7 141.557(I41=am0d0)Tb ESLCEBR LCEBR LCEBR Table. [CDIII]\n0.49Ho2Ru2O7 141.557(I41=am0d0)Ho,Ru ESLCEBR LCEBR ES Table. [CDIV]\n0.51Ho2Ru2O7 141.557(I41=am0d0)Ho,Ru TILCEBR LCEBR LCEBR Table. [CDV]\n1.161PrFe3(BO3)4 155.48(RI32)\u0003Pr,FeLCEBR TBD ESLCEBR Table. [CDVI]\n0.169 U3As4 161.71(R3c0) UES ESLCEBR LCEBR Table. [CDVII]\n0.170 U3P4 161.71(R3c0) UES ESLCEBR LCEBR Table. [CDVIII]\n0.228 TbCo2 166.101(R\u00163m0)Tb,Co TI ES TI TI Table. [CDX]\n0.77Tb2Ti2O7 166.101(R\u00163m0)Tb,Ti ESLCEBR LCEBR LCEBR Table. [CDXI]\n3.8 NdZn 222.103(PIn\u00163n)NdESFD ES ESESFD Table. [CDXV]\n3.12 NpSb 224.113(Pn\u00163m0)Np TI TI TI TI Table. [CDXVI]\n3.2 UO2 224.113(Pn\u00163m0)UESLCEBR LCEBR LCEBR Table. [CDXVII]\n3.7 NpBi 224.113(Pn\u00163m0)Np TI TI TI TI Table. [CDXVIII]\n3.10 NpSe 228.139(FSd\u00163c)Np TIESFD ESFD ESFD Table. [CDXIX]\n3.11 NpTe 228.139(FSd\u00163c)Np TIESFD ESFD ESFD Table. [CDXX]\n3.9 NpS 228.139(FSd\u00163c)Np ESESFD ESFD ESFD Table. [CDXXI]\n3.6 DyCu 229.143(Im\u00163m0)DyTBD ES ES ES Table. [CDXXII]\n\u0003Chiral MSG. The crossing points in the ES/ESFD phase with Chiral MSG must carry nonzero chiral charges. [90, 91]\nSome of the topological comounds in Table VIII contain both 3delement and 4f=5felement, where we set the U value of delectron\nas 2eV. For comparisons, we have also considered the empirical U values for the 3delectron and identified the topological phase diagram\nusing MTQC theory. In the Table IX, we chose 15 materials, each of which contains the 3delement V, Co, Ni/Mn. The empirical U values\nof V, Co, Ni and Mn are set as 3.25, 3.7, 6.2 and 3.9eV, respectively[ ?]. As tabulated in Table IX, the results indicate that interaction of\n3delectron almost doesn’t change the topological phase diagram for the compounds containing 4f=5felement.\nTABLE IX: Comparisions of the topological phase diagrams for some of the topological compounds containing both 3delement\nand4f=5felement. For each material, the U value of 3delement (U3d) is set as 2eVand an empirical value. The U value of\n4f=5felement (Uf) is set as 2, 4 and 6eV. The cases that have different topologies when U3dis taken 2eV and an empirical\nvalue are marked by red color.\nBCSID Formula MSG CAU3d(eV)Uf= 2eVUf= 4eVUf= 6eV\n2.26PrCo2P2 123.345(P4=mm0m0)Pr,Co2 ES ES ES\n3.7 ES ES ES\n1.251NdCo2P2 124.360(Pc4=mcc)Nd,Co2 ES ES TI\n3.7 ES ES ES\n1.253CeCo2P2 126.386(PI4=nnc)Ce,Co2 ES ES ES\n3.7 ES ES ES\n1.102 U2Ni2In 128.408(Pc4=mnc)U,Ni2 ES ES ES\n6.2 ES ES ES\n1.21DyCo2Si2 128.410(PI4=mnc)Dy,Co2 ES ES ES\n3.7 ES ES ES\n0.105 ErVO3 14.75(P21=c)Er,V2LCEBR LCEBR LCEBR\n3.25LCEBR LCEBR LCEBR\n1.254 UNiGa5 140.550(Ic4=mcm)U,Ni2 TI TILCEBR\n6.2 TI TI TI\n0.151Tm2Mn2O7 141.557(I41=am0d0)Tm,Mn2LCEBR LCEBR LCEBR\n3.9LCEBR LCEBR LCEBR\n0.227 NdCo2 141.557(I41=am0d0)Nd,Co2 ES ES ES\n3.7 ES ES ES\n1.222Er2CoGa8 51.298(Pamma)Er,Co2 ES ES ES34\n3.7 ES ES ES\n0.187CeMnAsO 59.407(Pm0mn)Ce,Mn2LCEBR LCEBR LCEBR\n3.9LCEBR LCEBR LCEBR\n1.200U2Ni2Sn 63.466(Ccmcm)U,Ni2 TI TI TI\n6.2LCEBR TI TI\n1.195 Er2Ni2In 63.467(Camcm)Er,Ni2LCEBR TILCEBR\n6.2LCEBR LCEBR LCEBR\n1.223Tm2CoGa8 65.489(Cammm)Tm,Co2 ES ES ES\n3.7 ES ES ES\n2.28NpNiGa5 74.559(Imm0a0)Np,Ni2 ES ES ES\n6.2 ES ES ES\nAppendix H: Physical interpretations for the TI classified by MTQC\nIn MTQC theory [7], in order to have a physical interpretation for the SI, we reduce the SI of MSG to one of its subgroup, and then\ninterpret the SI of the subgroup using the layer construction approach [26]. In this work, all the involved SI are subduced onto MSG 2.4\n(P\u00161), MSG 47.24 ( Pmmm), MSG 81.33 ( P\u00164), MSG 83.43 ( P4=m),/MSG 143.1 ( P3). In the following, we provide the definitions of SI of\nthese groups.\n1. Definitions for the stable indices of MSG 2.4\nMSG 2.4 ( P\u00161)has the SI group Z4\u0002Z3\n2. The four SI (\u00114I;z2I;1;z2I;2;z2I;3)are defined as\n\u00114I=X\nKn\u0000\nK=X\nK1\n2(n\u0000\nK\u0000n+\nK) mod 4; (H1)\nz2I;i=1;2;3=Cki=\u0019mod 2 =X\nK;Ki=\u0019n\u0000\nKmod 2; (H2)\nwhereKsums over the eight inversion-invariant momenta, and n\u0006\nKmeans the number of occupied even/odd states at the momentum K.\nz2I;iis the parity of Chern number in the plane ki=\u0019.\u00114Imod 2is the parity of the Chern number difference between kz= 0and\nkz=\u0019planes. Thus \u00114I= 1;3correspond to Weyl semimetal (WSM) phase. For \u00114I= 2corresponds to an axion insulator phase/3D\nQAHI, which can not be distinguished from SI but from Wilson loop [ ?]/surface state calculations.\nLayer constructions Now make use of the layer constructions of the gapped states to build a mapping from the SI to topological\ninvariants. Here we first give the SI of several layer constructions.\n1. A Chern layer with C=\u00061atx= 0gives the SI (2100).\n2. A Chern layer with C=\u00061atx=1\n2gives the SI (0100).\n3. A Chern layer with C=\u00061aty= 0gives the SI (2010).\n4. A Chern layer with C=\u00061aty=1\n2gives the SI (0010).\n5. A Chern layer with C=\u00061atz= 0gives the SI (2001).\n6. A Chern layer with C=\u00061atz=1\n2gives the SI (0001).\nWe take the Chern layers with C= 1atz= 0and1\n2as two examples to show how to calculate the SI of layer-constructed states.\nSince the SI indicate stable topological invariants (rather than fragile topological invariants), states having the same stable topological\ninvariants must have the same SI. For layer-constructed states, the stable topological invariants are completely determined by the positions\nand directions of the layers [26] and the Chern numbers of layers. Other details about the layer constructions are irrelevant to determine\nthe SI. We first consider a single layer with C= 1atz= 0. Thez= 0plane is inversion-invariant, and there are four momenta in the\n2D Brillouin zone, i.e.,(kx;ky) = (0;0);(0;\u0019);(\u0019;0);(\u0019;\u0019). The Bloch states at these four momenta are either even/odd under the\ninversion. According to the Fu-Kane formula\n(\u00001)C=Y\nKY\nn2occ\u00150\nn(K); (H3)\nwhereKgoes over the four inversion-invariant momenta and \u00150\nK;nis the parity of the nth occupied band at K, in aC= 1state, the total\nparity of the Bloch states at the four momenta must be odd. Here we assume the Chern layer has a single occupied band and the parities\nat(0;0);(0;\u0019);(\u0019;0);(\u0019;\u0019)are\u0000;+;+;+, respectively. Then we copy the layer to all the integer zpositions, i.e.,z= 0;\u00061;\u00062\u0001\u0001\u0001, to\nconstruct the 3D state. Supposing the 2D Bloch state of a single layer at zisj kx;ky;zi, then the 3D state Bloch state is given by\nj ki=1pNzX\nz=0;\u00061\u0001\u0001\u0001eizkzj kx;ky;zi; (H4)35\nwhereNzis the period in the z-direction. Let us calculate the inversion eigenvalues of the 3D state. For (kx;ky) =\n(0;0);(0;\u0019);(\u0019;0);(\u0019;\u0019), under the inversion operator ^P, the 2D state at zfirst gains an factor \u00150(kx;ky)and then transforms to\n\u0000z. Thus we obtain\n^Pj ki=\u00150(kx;ky)1pNzX\nz=0;\u00061\u0001\u0001\u0001e\u0000izkzj kx;ky;zi: (H5)\nForkz= 0;\u0019, we further obtain\n^Pj ki=\u00150(kx;ky)1pNzX\nz=0;\u00061\u0001\u0001\u0001eizkzj kx;ky;zi=\u00150(kx;ky)j ki: (H6)\nThus the parities of the 3D state are given by \u0015(kx;ky;kz) =\u00150(kx;ky). We obtain the parities of the 3D state at (kx;ky;kz) =\n(0;0;0);(\u0019;0;0);(0;\u0019;0);(\u0019;\u0019; 0);(0;0;\u0019);(\u0019;0;\u0019);(0;\u0019;\u0019 );(\u0019;\u0019;\u0019 )as\u0000;+;+;+;\u0000;+;+;+, respectively. Substituting the parities\ninto Eqs. (H1) and (H2), we obtain the SI as (2001).\nThen we consider the same 2D layers locating at half-integer positions, i.e.,z=\u00061\n2;\u00063\n2\u0001\u0001\u0001. The 3D Bloch state can be written as\nj ki=1pNzX\nz=\u00061\n2;\u00063\n2\u0001\u0001\u0001eizkzj kx;ky;zi: (H7)\nFor(kx;ky) = (0;0);(0;\u0019);(\u0019;0);(\u0019;\u0019), we have the inversion action as\n^Pj ki=\u00150(kx;ky)1pNzX\nz=\u00061\n2;\u00063\n2\u0001\u0001\u0001e\u0000izkzj kx;ky;zi: (H8)\nForkz= 0;\u0019, we havee\u0000izkz=e\u00002izkz\u0002eizkz=eikz\u0002eizkzand hence\n^Pj ki=\u00150(kx;ky)eikz1pNzX\nz=\u00061\n2;\u00063\n2\u0001\u0001\u0001eizkzj kx;ky;zi=\u00150(kx;ky)eikzj ki: (H9)\nThus the parities of the 3D state are given by \u0015(kx;ky;kz) =\u00150(kx;ky)eikz. We obtain the parities of the 3D state at (kx;ky;kz) =\n(0;0;0);(\u0019;0;0);(0;\u0019;0);(\u0019;\u0019; 0);(0;0;\u0019);(\u0019;0;\u0019);(0;\u0019;\u0019 );(\u0019;\u0019;\u0019 )as\u0000;+;+;+;+;\u0000;\u0000;\u0000, respectively. Substituting the parities\ninto Eqs. (H1) and (H2), we obtain the SI as (0001).\nThe SI of other layer constructions can be similarly calculated.\nThe inversion Z2invariant and axion insulator We find that \u00114I= 2iff the origin point (000)is occupied by odd Chern layers.\nHence, for gapped state, we define the inversion- Z2invariant as\n\u00110\n2I=1\n2\u00114Imod 2: (H10)\nWhen the total Chern number is zero, the axion \u0012-angle is given by \u0012=\u00112I\u0019. For example, the state consists of C= 1layer at the z= 0\nplane andC=\u00001Chern layer at the z=1\n2plane, which has the SI (2000), has zero total Chern number and \u00110\n2I= 1. This state is axion\ninsulator. One can see this from the boundary state: for a finite centrosymmetric sample centered at the origin, the Chern layer at z= 0\ncontributes to a chiral hinge mode, whereas the chiral modes from all the other layers cancel each pairwise. On the other hand, the state\nconsists of C= 1Chern layers at both the z= 0;1\n2planes, which also has the SI (2000), has total Chern number 2 and also \u00110\n2I= 1.\nThis state is a 3D QAH, for which the \u0012-angle is ill-defined. ( \u0012can still be constructed if the Chern numbers are non-zero, but will be\norigin-dependent [101] and loses the physical meaning of magnetoelectric response.)\nIn the rest, we will directly give the SI and the corresponding interpretations. Readers might refer to [7] for more details.\n2. Definitions for the stable indices of MSG 47.249\nMSG 47.249 ( Pmmm )hastheSIgroup Z4\u0002Z3\n2. Becauseoftheanti-commutingmirrorsymmetries, allstatesattheinversion-invariant\nmomenta are doubly degenerate. The SI can be chosen as same as the MSG 47.250 Pmmm 10[26] because for this group TRS does not\nchange the irreps. The Z4factor is\nz4=X\nK1\n4(n\u0000\nK\u0000n+\nK) mod 4; (H11)\nwhereKsums over all inversion-invariant momenta and n\u0006\nKis the number of occupied even/odd states at K.z4can be thought as\nEq. (H1) except that n\u0000\nKare replaced by the number of odd doublets. Odd z4corresponds to axion insulator and z4= 2corresponds to a\nhigher-order TI (HOTI) jointly protected by mirrors and C2rotations [7]. The three Z2factors are the mirror Chern number parities in\nthek1;2;3=\u0019planes\nz2w;i=1;2;3=Cm;ki=\u0019mod 2 =X\nK;Ki=\u00191\n2n\u0000\nKmod 2: (H12)\nBecause of the anti-commuting mirror symmetries, net Chern numbers in all the directions vanishes.\n3. Definitions for the stable indices of MSG 81.33\nMSG 81.33( P\u00164)has the SI group Z4\u0002Z2\n2. We choose the Z4factor as the Chern number in the kz=\u0019plane mod 4\nz4S=C\u0019mod 4 = 2Nocc\u00001\n2n1\n2\nZ+1\n2n\u00001\n2\nZ\u00003\n2n3\n2\nZ+3\n2n\u00003\n2\nZ\u00001\n2n1\n2\nA+1\n2n\u00001\n2\nA\u00003\n2n3\n2\nA+3\n2n\u00003\n2\nA\u0000n1\n2\nR+n\u00001\n2\nRmod 4;(H13)36\nwheren1\n2;\u00001\n2;3\n2;\u00003\n2\nZ;Aare the numbers of occupied states with S4eigenvalues e\u0000i\u0019\n4,ei\u0019\n4,e\u0000i3\u0019\n4,ei3\u0019\n4, andn1\n2;\u00001\n2\nRare the numbers of\noccupied states with C2eigenvalues e\u0000i\u0019\n4,ei\u0019\n4. We choose the first Z2factor as the difference of Chern numbers in the kz=\u0019plane and\nthekz= 0plane over 2 mod 2\n\u000e2S=\u0000n3\n2\nZ+n\u00003\n2\nZ\u0000n3\n2\nA+n\u00003\n2\nA+n3\n2\n\u0000\u0000n\u00003\n2\n\u0000+n3\n2\nM\u0000n\u00003\n2\nMmod 2: (H14)\nThe\u000e2S= 1phase is a WSM with 2 Weyl points between kz= 0andkz=\u0019. One may be curious why C\u0019\u0000C0is an even number.\nThe answer is that it is enforced by the compatibility relations: the kz= 0andkz=\u0019planes must have the same C2eigenvalues and\nhence the same parity of Chern numbers. The second Z2factor is\nz2=X\nK=\u0000;M;Z;An1\n2\nK\u0000n\u00003\n2\nK\n2mod 2: (H15)\nForz4S= 0and\u000e2S= 0,z2corresponds to axion insulator/3D QAHIstate [7].\n4. Definitions for the stable indices of MSG 83.43\nMSG 83.43 ( P4=m)has the SI group Z3\n4. We choose the three SI as\n\u000e4m=C+\n\u0019\u0000C\u0000\n0mod 4 =X\nK=Z;A\u0012\n\u00001\n2n1\n2;+i\nK+1\n2n\u00001\n2;+i\nK\u00003\n2n3\n2;+i\nK+3\n2n\u00003\n2;+i\nK\u0013\n\u0000n1\n2;+i\nR+n\u00001\n2;+i\nR\n\u0000X\nK=\u0000;M\u0012\n\u00001\n2n1\n2;\u0000i\nK+1\n2n\u00001\n2;\u0000i\nK\u00003\n2n3\n2;\u0000i\nK+3\n2n\u00003\n2;\u0000i\nK\u0013\n+n1\n2;\u0000i\nX+n\u00001\n2;\u0000i\nXmod 4;(H16)\nz+\n4m;\u0019=C+\n\u0019mod 4 =Nocc+X\nK=Z;A\u0012\n\u00001\n2n1\n2;+i\nK+1\n2n\u00001\n2;+i\nK\u00003\n2n3\n2;+i\nK+3\n2n\u00003\n2;+i\nK\u0013\n\u0000n1\n2;+i\nR+n\u00001\n2;+i\nRmod 4; (H17)\nz\u0000\n4m;\u0019=C\u0000\n\u0019mod 4 =Nocc+X\nK=Z;A\u0012\n\u00001\n2n1\n2;\u0000i\nK+1\n2n\u00001\n2;\u0000i\nK\u00003\n2n3\n2;\u0000i\nK+3\n2n\u00003\n2;\u0000i\nK\u0013\n\u0000n1\n2;\u0000i\nR+n\u00001\n2;\u0000i\nRmod 4: (H18)\nHereC\u0006i\n0;\u0019represents Chern number in the \u0006imirror sector in the kz= 0;\u0019plane.\n5. Definitions for the stable indices of MSG 143.1\nMSG 143.1 ( P3)has the SI group Z3. According to the Chern number formula, the Chern number Cis related to the C3eigenvalues\nei2\u0019\n3C= (\u00001)NoccY\nn2occ\u0012n(\u0000)\u0012n(K)\u0012n(KA); (H19)\nwhere\u0012n(\u0000;K;K0)is theC3eigenvalue of the nth occupied state at the corresponding momentum. (One should not confuse the C3\neigenvalues \u0012nwith the axion theta angle \u0012). We define the SI as,\nz3R=X\nK=A;H;HA\u0012\nn\u00001\n2\nK\u0000n3\n2\nK\u0013\nmod 3: (H20)\n6. Stable indices of the magnetic TIs\nUsing the SI defined above, we have explained the physical meaning for all of the TIs obtained from MTQC. We tabulate all of the TIs\ndiagnosed by MTQC in Table X. For each material, we list the identification number in BCSMD (BCSID), chemical formula (Formula),\nCoulomb interaction strength in LDA+U calculations (U), magnetic space group (MSG), SI of the MSG calculated by the machinery\nof MTQC (Indices(MTQC)), whose physical meaning is unclear, the SI and the corresponding subgroup (Min-sMSG), and the physical\ninterpretation of the Indices (Interpretations).\nTABLE X: Topological indices and the physical interpretations of the TIs predicted by MTQC.\nBCSID Formula U(eV) MSG Indices(MTQC) Min-sMSG Indices Interpretations\n1.206Dy2Fe2Si2C 0 2.7 (PS\u00161)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n1.206Dy2Fe2Si2C 2 2.7 (PS\u00161)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI37\n1.206Dy2Fe2Si2C 4 2.7 (PS\u00161)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n1.206Dy2Fe2Si2C 6 2.7 (PS\u00161)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n0.104 ErVO3 211.54 (P20\n1=m0)c21= 0,c22= 0,\nc4 = 22.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n0.106 DyVO3 011.54 (P20\n1=m0)c21= 1,c22= 1,\nc4 = 12.4 (P\u00161)z2I;1= 1,\nz2I;2= 0,\nz2I;3= 1,\n\u00114I= 1SISM\n1.264 CoPS3 011.57 (PC21=m)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n0.203 Mn3Ge 012.62 (C20=m0)c21= 0,c22= 0,\nc4 = 22.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n0.203 Mn3Ge 112.62 (C20=m0)c21= 0,c22= 0,\nc4 = 22.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n0.203 Mn3Ge 212.62 (C20=m0)c21= 0,c22= 0,\nc4 = 22.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n0.203 Mn3Ge 312.62 (C20=m0)c21= 0,c22= 0,\nc4 = 22.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n0.203 Mn3Ge 412.62 (C20=m0)c21= 0,c22= 0,\nc4 = 22.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n1.0.13 FeI2 012.62 (C20=m0)c21= 1,c22= 1,\nc4 = 32.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 1SISM\n1.140 PrMgPb 013.73 (PA2=c)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n1.140 PrMgPb 213.73 (PA2=c)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n1.140 PrMgPb 413.73 (PA2=c)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI38\n1.140 PrMgPb 613.73 (PA2=c)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n1.141 NdMgPb 013.73 (PA2=c)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n1.141 NdMgPb 213.73 (PA2=c)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n1.201 Cr2ReO6 014.80 (Pa21=c)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n0.174Pr3Ru4Al12 015.89 (C20=c0)c2 = 1,c4 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 1SISM\n0.174Pr3Ru4Al12 215.89 (C20=c0)c2 = 1,c4 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 1SISM\n0.174Pr3Ru4Al12 415.89 (C20=c0)c2 = 1,c4 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 1SISM\n0.174Pr3Ru4Al12 615.89 (C20=c0)c2 = 1,c4 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 1SISM\n0.226 NdCo2 215.89 (C20=c0)c2 = 1,c4 = 3 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 3SISM\n0.226 NdCo2 415.89 (C20=c0)c2 = 0,c4 = 2 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n0.226 NdCo2 615.89 (C20=c0)c2 = 1,c4 = 3 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 3SISM\n2.10 HoP 015.89 (C20=c0)c2 = 1,c4 = 3 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 3SISM\n2.10 HoP 215.89 (C20=c0)c2 = 1,c4 = 0 2.4 (P\u00161)z2I;1= 1,\nz2I;2= 1,\nz2I;3= 0,\n\u00114I= 03D QAHI\n2.10 HoP 415.89 (C20=c0)c2 = 1,c4 = 0 2.4 (P\u00161)z2I;1= 1,\nz2I;2= 1,\nz2I;3= 0,\n\u00114I= 03D QAHI39\n2.10 HoP 615.89 (C20=c0)c2 = 1,c4 = 0 2.4 (P\u00161)z2I;1= 1,\nz2I;2= 1,\nz2I;3= 0,\n\u00114I= 03D QAHI\n1.211 Dy2O2S 0 15.90 (Cc2=c)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n1.216Nd2BaNiO5 0 15.90 (Cc2=c)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n1.217Tb2BaNiO5 0 15.90 (Cc2=c)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n1.49Ag2NiO2 0 15.90 (Cc2=c)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n1.139Ho2RhIn8 249.273 (Pcccm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.139Ho2RhIn8 449.273 (Pcccm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.150 PrAg 453.334 (PBmna)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.8CeRu2Al10 057.391 (PCbcm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.8CeRu2Al10 257.391 (PCbcm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.8CeRu2Al10 457.391 (PCbcm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.252 CaCo2P2 159.416 (PImmn)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.252 CaCo2P2 259.416 (PImmn)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.252 CaCo2P2 359.416 (PImmn)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI40\n1.252 CaCo2P2 459.416 (PImmn)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.88 Mn5Si3 060.431 (PCbcn)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n2.1EuFe2As2 261.439 (PCbca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n2.1EuFe2As2 361.439 (PCbca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.130 Cr2As 162.450 (Panma)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.131 Fe2As 062.450 (Panma)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.131 Fe2As 162.450 (Panma)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.131 Fe2As 362.450 (Panma)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.131 Fe2As 462.450 (Panma)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.179NdCoAsO 262.450 (Panma)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.28 CrN 062.450 (Panma)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.28 CrN 162.450 (Panma)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n0.200 Mn3Sn 063.464 (Cm0cm0)c21= 1,c22= 0 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n0.200 Mn3Sn 163.464 (Cm0cm0)c21= 1,c22= 0 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI41\n0.200 Mn3Sn 263.464 (Cm0cm0)c21= 1,c22= 0 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n1.200 U2Ni2Sn 263.466 (Ccmcm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.200 U2Ni2Sn 463.466 (Ccmcm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.200 U2Ni2Sn 663.466 (Ccmcm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.262NpRhGa5 263.466 (Ccmcm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.195 Er2Ni2In 463.467 (Camcm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.16BaFe2As2 064.480 (CAmca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.16BaFe2As2 164.480 (CAmca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.16BaFe2As2 264.480 (CAmca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.16BaFe2As2 364.480 (CAmca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.188CeRh2Si2 064.480 (CAmca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.188CeRh2Si2 664.480 (CAmca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.52CaFe2As2 064.480 (CAmca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.52CaFe2As2 164.480 (CAmca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI42\n1.52CaFe2As2 264.480 (CAmca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n2.15Mn3Ni20P6 365.486 (Cmm0m0)c21= 1,c22= 0 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n1.142 CeMgPb 267.510 (CAmma)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.125 LaFeAsO 073.553 (Icbca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.176YbCo2Si2 173.553 (Icbca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.176YbCo2Si2 273.553 (Icbca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.176YbCo2Si2 373.553 (Icbca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.176YbCo2Si2 473.553 (Icbca)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n2.13 UP 0134.481 (PC42=nnm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n2.13 UP 2134.481 (PC42=nnm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n2.13 UP 4134.481 (PC42=nnm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n2.13 UP 6134.481 (PC42=nnm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n2.20 UAs 0134.481 (PC42=nnm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n2.20 UAs 4134.481 (PC42=nnm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI43\n2.20 UAs 6134.481 (PC42=nnm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n2.6Nd2CuO4 0134.481 (PC42=nnm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.146 LaCrAsO 0138.528 (Pc42=ncm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.146 LaCrAsO 1138.528 (Pc42=ncm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.146 LaCrAsO 2138.528 (Pc42=ncm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.146 LaCrAsO 3138.528 (Pc42=ncm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.146 LaCrAsO 4138.528 (Pc42=ncm)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n0.228 TbCo2 0166.101 (R\u00163m0)c2 = 1,c4 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 3SISM\n0.228 TbCo2 4166.101 (R\u00163m0)c2 = 1,c4 = 3 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 1SISM\n0.228 TbCo2 6166.101 (R\u00163m0)c2 = 1,c4 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 3SISM\n0.150 NiS2 0 205.33 (Pa\u00163)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI/3D QAHI\n3.12 NpSb 0224.113 (Pn\u00163m0)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n3.12 NpSb 2224.113 (Pn\u00163m0)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n3.12 NpSb 4224.113 (Pn\u00163m0)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI44\n3.12 NpSb 6224.113 (Pn\u00163m0)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n3.7 NpBi 0224.113 (Pn\u00163m0)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n3.7 NpBi 2224.113 (Pn\u00163m0)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n3.7 NpBi 4224.113 (Pn\u00163m0)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n3.7 NpBi 6224.113 (Pn\u00163m0)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n3.10 NpSe 0228.139 (FSd\u00163c)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n3.11 NpTe 0228.139 (FSd\u00163c)c2 = 1 2.4 (P\u00161)z2I;1= 0,\nz2I;2= 0,\nz2I;3= 0,\n\u00114I= 2AXI\n1.223Tm2CoGa8 065.489 (Cammm)c2 = 0,c4 = 1 47.249 (Pmmm)z2w;1= 0,\nz2w;2= 0,\nz2w;3= 1,\nz4= 3AXI/MTCI\n1.254 UNiGa5 2140.550 (Ic4=mcm)c4 = 3 47.249 (Pmmm)z2w;1= 0,\nz2w;2= 0,\nz2w;3= 0,\nz4= 3AXI\n1.254 UNiGa5 4140.550 (Ic4=mcm)c4 = 3 47.249 (Pmmm)z2w;1= 0,\nz2w;2= 0,\nz2w;3= 0,\nz4= 3AXI\n0.81U2Pd2Sn 0127.394 (P40=m0bm0)c2 = 1 81.33 (P\u00164)\u000e2S= 1,\nz2= 1,\nz4S= 0SISM\u0003\n0.81U2Pd2Sn 2127.394 (P40=m0bm0)c2 = 1 81.33 (P\u00164)\u000e2S= 1,\nz2= 1,\nz4S= 0SISM\u0003\n0.81U2Pd2Sn 4127.394 (P40=m0bm0)c2 = 1 81.33 (P\u00164)\u000e2S= 1,\nz2= 1,\nz4S= 0SISM\u0003\n0.81U2Pd2Sn 6127.394 (P40=m0bm0)c2 = 1 81.33 (P\u00164)\u000e2S= 1,\nz2= 1,\nz4S= 0SISM\u0003\n0.186CeMnAsO 2129.416 (P40=n0m0m)c2 = 1 81.33 (P\u00164)\u000e2S= 1,\nz2= 1,\nz4S= 0SISM\u0003\n0.186CeMnAsO 4129.416 (P40=n0m0m)c2 = 1 81.33 (P\u00164)\u000e2S= 1,\nz2= 1,\nz4S= 0SISM\u000345\n0.186CeMnAsO 6129.416 (P40=n0m0m)c2 = 1 81.33 (P\u00164)\u000e2S= 1,\nz2= 1,\nz4S= 0SISM\u0003\n0.126 NpCo2 4141.556 (I40\n1=a0m0d)c2 = 1 81.33 (P\u00164)\u000e2S= 0,\nz2= 0,\nz4S= 23D QAHI\n1.251 NdCo2P2 6124.360 (Pc4=mcc)c4 = 3 83.43 (P4=m)z+\n4m;\u0019= 0,\nz\u0000\n4m;\u0019= 0,\n\u000e4m= 3AXI/MTCI\n1.255 UPtGa5 0124.360 (Pc4=mcc)c4 = 3 83.43 (P4=m)z+\n4m;\u0019= 0,\nz\u0000\n4m;\u0019= 0,\n\u000e4m= 3AXI/MTCI\n1.208 UAs 6128.410 (PI4=mnc)c4 = 3 83.43 (P4=m)z+\n4m;\u0019= 0,\nz\u0000\n4m;\u0019= 0,\n\u000e4m= 3AXI/MTCI\n0.51Ho2Ru2O7 0141.557 (I41=am0d0)c21= 1,c22= 188.81 (I41=a)\u00110\n2I= 1,\nz2= 03D QAHI\n0.117 LuFeO3 0185.201 (P63c0m0)c3 = 1 143.1 (P3)z3R= 2 3D QAHI\n\u0003Compatible with SISM\nAppendix I: Compatibility-relations along high-symmetry paths of the symmetry enforced semimetals\nIn TABLE XI, we check all the compatibility relations between maximal kvectors for the magnetic ESs in TABLE VII-VIII. Once band\nstructures break the compatibility relations between two maximal kvectors, band crossings occur and form nodal points/nodal-lines.\nTABLE XI: Compatibility-relations of the magnetic ESs. For each material, the first line lists BCSID, chemical formula, MSG\nand the value of U in LDA+U calculations. The 1st-3rd coloumns are two maximal kvectors (k1andk2) and the intermediate\npath between them. The 4th coloumn identifies whether compatibility-relations between k1andk2are satisfied. If the answer\nis ’no’, there have symmetry enforced band crossings between k1andk2. The 5th coloumn gives the location of band crossings.\n’Line’ stands for the band crossings are protected by (screw-)rotational symmetries and there are isolated degenerate points\non the line. ’Plane’ stands for the band crossings are protected by (glide-)mirror symmetries and the crossing points form\nnodal-lines on the plane. The coordinates of kvectors are written in the reciprocal conventional lattice, which are provided in\ntheCoRepresentation subsection of BCS website.\nBCSID: 0.165; Formula: SrMn(VO4)(OH); MSG: 4.7 ( P21); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,v,0) Z:(0,1/2,0) no Line\nA:(1/2,0,1/2) U:(1/2,v,1/2) E:(1/2,1/2,1/2) yes Line\nB:(0,0,1/2) V:(0,v,1/2) D:(0,1/2,1/2) yes Line\nC:(1/2,1/2,0) W:(1/2,v,0) Y:(1/2,0,0) yes Line\nBCSID: 1.22; Formula: DyCu2Si2; MSG: 12.63 ( Cc2=m); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nA:(0,0,1/2) U:(0,v,1/2) M:(0,1,1/2) no Line\nBCSID: 0.105; Formula: ErVO3; MSG: 14.75 ( P21=c); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,v,0) Z:(0,1/2,0) yes Line\nA:(1/2,0,1/2) U:(1/2,v,1/2) E:(1/2,1/2,1/2) yes Line\nB:(0,0,1/2) V:(0,v,1/2) D:(0,1/2,1/2) yes Line\nC:(1/2,1/2,0) W:(1/2,v,0) Y:(1/2,0,0) no Line\n\u0000:(0,0,0) F:(u,0,w) A:(1/2,0,1/2) yes Plane\n\u0000:(0,0,0) F:(u,0,w) B:(0,0,1/2) yes Plane46\n\u0000:(0,0,0) F:(u,0,w) Y:(1/2,0,0) no Plane\nA:(1/2,0,1/2) F:(u,0,w) B:(0,0,1/2) yes Plane\nA:(1/2,0,1/2) F:(u,0,w) Y:(1/2,0,0) no Plane\nB:(0,0,1/2) F:(u,0,w) Y:(1/2,0,0) no Plane\nC:(1/2,1/2,0) G:(u,1/2,w) D:(0,1/2,1/2) yes Plane\nC:(1/2,1/2,0) G:(u,1/2,w) E:(1/2,1/2,1/2) yes Plane\nC:(1/2,1/2,0) G:(u,1/2,w) Z:(0,1/2,0) yes Plane\nD:(0,1/2,1/2) G:(u,1/2,w) E:(1/2,1/2,1/2) yes Plane\nD:(0,1/2,1/2) G:(u,1/2,w) Z:(0,1/2,0) yes Plane\nE:(1/2,1/2,1/2) G:(u,1/2,w) Z:(0,1/2,0) yes Plane\nBCSID: 1.263; Formula: Ca3Ru2O7; MSG: 33.154 ( PCna21); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,0,0) X:(1/2,0,0) yes Line\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1/2,0) yes Line\nR:(1/2,1/2,1/2) E:(u,1/2,1/2) T:(0,1/2,1/2) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) yes Line\nS:(1/2,1/2,0) C:(u,1/2,0) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nU:(1/2,0,1/2) G:(1/2,0,w) X:(1/2,0,0) no Line\nU:(1/2,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) K:(0,v,w) T:(0,1/2,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) U:(1/2,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) X:(1/2,0,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1/2,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) X:(1/2,0,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) Z:(0,0,1/2) yes Plane\nX:(1/2,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nY:(0,1/2,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nBCSID: 1.263; Formula: Ca3Ru2O7; MSG: 33.154 ( PCna21); U= 1eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,0,0) X:(1/2,0,0) yes Line\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1/2,0) yes Line\nR:(1/2,1/2,1/2) E:(u,1/2,1/2) T:(0,1/2,1/2) yes Line47\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) yes Line\nS:(1/2,1/2,0) C:(u,1/2,0) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nU:(1/2,0,1/2) G:(1/2,0,w) X:(1/2,0,0) no Line\nU:(1/2,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) K:(0,v,w) T:(0,1/2,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) U:(1/2,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) X:(1/2,0,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1/2,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) X:(1/2,0,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) Z:(0,0,1/2) yes Plane\nX:(1/2,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nY:(0,1/2,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nBCSID: 1.43; Formula: PrNiO3; MSG: 36.178 ( Camc21); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1,0) yes Line\n\u0000:(0,0,0) SM:(u,0,0) Y:(1,0,0) yes Line\nR:(1/2,1/2,1/2) D:(1/2,1/2,w) S:(1/2,1/2,0) no Line\nT:(0,1,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) K:(0,v,w) T:(0,1,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) T:(1,0,1/2) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1,0) yes Plane\n\u0000:(0,0,0) M:(u,0,w) Y:(1,0,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nT:(0,1,1/2) K:(0,v,w) Y:(0,1,0) yes Plane\nT:(1,0,1/2) M:(u,0,w) Y:(1,0,0) yes Plane\nT:(0,1,1/2) K:(0,v,w) Z:(0,0,1/2) yes Plane\nT:(1,0,1/2) M:(u,0,w) Z:(0,0,1/2) yes Plane\nY:(0,1,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nY:(1,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nBCSID: 0.26; Formula: TmAgGe; MSG: 38.191 ( Am0m02); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane48\n\u0000:(0,0,0) SM:(0,0,w) Y:(0,0,1) no Line\nT:(1/2,0,-1) A:(1/2,0,w) Z:(1/2,0,0) yes Line\nBCSID: 2.12; Formula: TbMg; MSG: 49.270 ( Pc0cm0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,0,0) X:(1/2,0,0) no Line\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nR:(1/2,1/2,1/2) P:(1/2,v,1/2) U:(1/2,0,1/2) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) yes Line\nS:(1/2,1/2,0) C:(u,1/2,0) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\nU:(1/2,0,1/2) G:(1/2,0,w) X:(1/2,0,0) yes Line\n\u0000:(0,0,0) M:(u,0,w) U:(1/2,0,1/2) no Plane\n\u0000:(0,0,0) M:(u,0,w) X:(1/2,0,0) no Plane\n\u0000:(0,0,0) M:(u,0,w) Z:(0,0,1/2) no Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) X:(1/2,0,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) Z:(0,0,1/2) yes Plane\nX:(1/2,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nBCSID: 2.12; Formula: TbMg; MSG: 49.270 ( Pc0cm0); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,0,0) X:(1/2,0,0) no Line\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nR:(1/2,1/2,1/2) P:(1/2,v,1/2) U:(1/2,0,1/2) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) yes Line\nS:(1/2,1/2,0) C:(u,1/2,0) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\nU:(1/2,0,1/2) G:(1/2,0,w) X:(1/2,0,0) yes Line\n\u0000:(0,0,0) M:(u,0,w) U:(1/2,0,1/2) no Plane\n\u0000:(0,0,0) M:(u,0,w) X:(1/2,0,0) no Plane\n\u0000:(0,0,0) M:(u,0,w) Z:(0,0,1/2) no Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) X:(1/2,0,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) Z:(0,0,1/2) yes Plane49\nX:(1/2,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nBCSID: 2.12; Formula: TbMg; MSG: 49.270 ( Pc0cm0); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,0,0) X:(1/2,0,0) yes Line\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) no Line\nR:(1/2,1/2,1/2) P:(1/2,v,1/2) U:(1/2,0,1/2) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) yes Line\nS:(1/2,1/2,0) C:(u,1/2,0) Y:(0,1/2,0) no Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\nU:(1/2,0,1/2) G:(1/2,0,w) X:(1/2,0,0) yes Line\n\u0000:(0,0,0) M:(u,0,w) U:(1/2,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) X:(1/2,0,0) yes Plane\n\u0000:(0,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) no Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) no Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) no Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) X:(1/2,0,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) Z:(0,0,1/2) yes Plane\nX:(1/2,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nBCSID: 2.12; Formula: TbMg; MSG: 49.270 ( Pc0cm0); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,0,0) X:(1/2,0,0) no Line\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nR:(1/2,1/2,1/2) P:(1/2,v,1/2) U:(1/2,0,1/2) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) no Line\nS:(1/2,1/2,0) C:(u,1/2,0) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\nU:(1/2,0,1/2) G:(1/2,0,w) X:(1/2,0,0) no Line\n\u0000:(0,0,0) M:(u,0,w) U:(1/2,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) X:(1/2,0,0) no Plane\n\u0000:(0,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) X:(1/2,0,0) no Plane\nU:(1/2,0,1/2) M:(u,0,w) Z:(0,0,1/2) yes Plane\nX:(1/2,0,0) M:(u,0,w) Z:(0,0,1/2) no Plane50\nBCSID: 1.139; Formula: Ho2RhIn8; MSG: 49.273 ( Pcccm); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) E:(u,1/2,1/2) T:(0,1/2,1/2) no Line\nR:(1/2,1/2,1/2) P:(1/2,v,1/2) U:(1/2,0,1/2) no Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) no Line\nU:(1/2,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) no Line\nR:(1/2,1/2,1/2) W:(u,v,1/2) T:(0,1/2,1/2) no Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) no Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) no Plane\nU:(1/2,0,1/2) W:(u,v,1/2) Z:(0,0,1/2) no Plane\nBCSID: 2.11; Formula: TbMg; MSG: 51.295 ( Pmm0a0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,0,0) X:(1/2,0,0) no Line\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) E:(u,1/2,1/2) T:(0,1/2,1/2) no Line\nR:(1/2,1/2,1/2) P:(1/2,v,1/2) U:(1/2,0,1/2) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) yes Line\nS:(1/2,1/2,0) C:(u,1/2,0) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) no Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) no Line\nU:(1/2,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) K:(0,v,w) T:(0,1/2,1/2) no Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1/2,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Y:(0,1/2,0) no Plane\nT:(0,1/2,1/2) K:(0,v,w) Z:(0,0,1/2) no Plane\nU:(1/2,0,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nY:(0,1/2,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nBCSID: 2.11; Formula: TbMg; MSG: 51.295 ( Pmm0a0); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,0,0) X:(1/2,0,0) no Line\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) E:(u,1/2,1/2) T:(0,1/2,1/2) yes Line\nR:(1/2,1/2,1/2) P:(1/2,v,1/2) U:(1/2,0,1/2) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) yes Line\nS:(1/2,1/2,0) C:(u,1/2,0) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) no Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) no Line\nU:(1/2,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) yes Line51\n\u0000:(0,0,0) K:(0,v,w) T:(0,1/2,1/2) no Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1/2,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Y:(0,1/2,0) no Plane\nT:(0,1/2,1/2) K:(0,v,w) Z:(0,0,1/2) no Plane\nU:(1/2,0,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nY:(0,1/2,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nBCSID: 2.11; Formula: TbMg; MSG: 51.295 ( Pmm0a0); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,0,0) X:(1/2,0,0) yes Line\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) E:(u,1/2,1/2) T:(0,1/2,1/2) yes Line\nR:(1/2,1/2,1/2) P:(1/2,v,1/2) U:(1/2,0,1/2) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) yes Line\nS:(1/2,1/2,0) C:(u,1/2,0) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) no Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) no Line\nU:(1/2,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) K:(0,v,w) T:(0,1/2,1/2) no Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1/2,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Y:(0,1/2,0) no Plane\nT:(0,1/2,1/2) K:(0,v,w) Z:(0,0,1/2) no Plane\nU:(1/2,0,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nY:(0,1/2,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nBCSID: 2.11; Formula: TbMg; MSG: 51.295 ( Pmm0a0); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,0,0) X:(1/2,0,0) yes Line\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) E:(u,1/2,1/2) T:(0,1/2,1/2) yes Line\nR:(1/2,1/2,1/2) P:(1/2,v,1/2) U:(1/2,0,1/2) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) yes Line\nS:(1/2,1/2,0) C:(u,1/2,0) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) no Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) no Line\nU:(1/2,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) K:(0,v,w) T:(0,1/2,1/2) no Plane52\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1/2,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Y:(0,1/2,0) no Plane\nT:(0,1/2,1/2) K:(0,v,w) Z:(0,0,1/2) no Plane\nU:(1/2,0,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nY:(0,1/2,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nBCSID: 1.222; Formula: Er2CoGa8; MSG: 51.298 ( Pamma); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) P:(1/2,v,1/2) U:(1/2,0,1/2) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) no Line\nBCSID: 1.222; Formula: Er2CoGa8; MSG: 51.298 ( Pamma); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) P:(1/2,v,1/2) U:(1/2,0,1/2) no Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) no Line\nBCSID: 1.3; Formula: Sr2IrO4; MSG: 54.352 ( PIcca); U= 1eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) E:(u,1/2,1/2) T:(0,1/2,1/2) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) no Line\nU:(1/2,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) no Line\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) no Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) no Plane\nU:(1/2,0,1/2) W:(u,v,1/2) Z:(0,0,1/2) no Plane\nBCSID: 0.85; Formula: KCo4(PO4)3; MSG: 58.398 ( Pnn0m0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,0,0) X:(1/2,0,0) yes Line\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) P:(1/2,v,1/2) U:(1/2,0,1/2) yes Line\nU:(1/2,0,1/2) G:(1/2,0,w) X:(1/2,0,0) yes Line\nU:(1/2,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) no Line\n\u0000:(0,0,0) K:(0,v,w) T:(0,1/2,1/2) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1/2,0) yes Plane53\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nY:(0,1/2,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nBCSID: 0.187; Formula: CeMnAsO; MSG: 59.407 ( Pm0mn); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nR:(1/2,1/2,1/2) E:(u,1/2,1/2) T:(0,1/2,1/2) no Line\nS:(1/2,1/2,0) C:(u,1/2,0) Y:(0,1/2,0) no Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) no Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) no Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) no Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) no Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nBCSID: 1.88; Formula: Mn5Si3; MSG: 60.431 ( PCbcn); U= 1eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) no Line\nR:(1/2,1/2,1/2) P:(1/2,v,1/2) U:(1/2,0,1/2) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) yes Line\nS:(1/2,1/2,0) C:(u,1/2,0) Y:(0,1/2,0) no Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nU:(1/2,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) no Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) no Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) no Plane\nT:(0,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 2.1; Formula: EuFe2As2; MSG: 61.439 ( PCbca); U= 1eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nR:(1/2,1/2,1/2) E:(u,1/2,1/2) T:(0,1/2,1/2) yes Line\nR:(1/2,1/2,1/2) P:(1/2,v,1/2) U:(1/2,0,1/2) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) yes Line54\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) no Line\nU:(1/2,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) W:(u,v,1/2) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 0.218; Formula: Co2SiO4; MSG: 62.441 ( Pnma); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) no Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) V:(u,v,0) S:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) T:(0,1/2,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) U:(1/2,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) X:(1/2,0,0) yes Plane\n\u0000:(0,0,0) V:(u,v,0) X:(1/2,0,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1/2,0) yes Plane\n\u0000:(0,0,0) V:(u,v,0) Y:(0,1/2,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) V:(u,v,0) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nS:(1/2,1/2,0) V:(u,v,0) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Z:(0,0,1/2) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) X:(1/2,0,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nX:(1/2,0,0) V:(u,v,0) Y:(0,1/2,0) yes Plane\nX:(1/2,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nY:(0,1/2,0) K:(0,v,w) Z:(0,0,1/2) yes Plane55\nBCSID: 0.219; Formula: Co2SiO4; MSG: 62.441 ( Pnma); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) no Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) V:(u,v,0) S:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) T:(0,1/2,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) U:(1/2,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) X:(1/2,0,0) yes Plane\n\u0000:(0,0,0) V:(u,v,0) X:(1/2,0,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1/2,0) yes Plane\n\u0000:(0,0,0) V:(u,v,0) Y:(0,1/2,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) V:(u,v,0) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nS:(1/2,1/2,0) V:(u,v,0) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Z:(0,0,1/2) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) X:(1/2,0,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nX:(1/2,0,0) V:(u,v,0) Y:(0,1/2,0) yes Plane\nX:(1/2,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nY:(0,1/2,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nBCSID: 0.219; Formula: Co2SiO4; MSG: 62.441 ( Pnma); U= 1eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) no Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) V:(u,v,0) S:(1/2,1/2,0) yes Plane56\n\u0000:(0,0,0) K:(0,v,w) T:(0,1/2,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) U:(1/2,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) X:(1/2,0,0) yes Plane\n\u0000:(0,0,0) V:(u,v,0) X:(1/2,0,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1/2,0) yes Plane\n\u0000:(0,0,0) V:(u,v,0) Y:(0,1/2,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) V:(u,v,0) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nS:(1/2,1/2,0) V:(u,v,0) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Z:(0,0,1/2) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) X:(1/2,0,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nX:(1/2,0,0) V:(u,v,0) Y:(0,1/2,0) yes Plane\nX:(1/2,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nY:(0,1/2,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nBCSID: 0.221; Formula: Fe2SiO4; MSG: 62.441 ( Pnma); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) no Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) no Line\n\u0000:(0,0,0) V:(u,v,0) S:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) T:(0,1/2,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) U:(1/2,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) X:(1/2,0,0) yes Plane\n\u0000:(0,0,0) V:(u,v,0) X:(1/2,0,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1/2,0) yes Plane\n\u0000:(0,0,0) V:(u,v,0) Y:(0,1/2,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane57\nR:(1/2,1/2,1/2) L:(1/2,v,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) V:(u,v,0) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nS:(1/2,1/2,0) V:(u,v,0) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Z:(0,0,1/2) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) X:(1/2,0,0) yes Plane\nU:(1/2,0,1/2) M:(u,0,w) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nX:(1/2,0,0) V:(u,v,0) Y:(0,1/2,0) yes Plane\nX:(1/2,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nY:(0,1/2,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nBCSID: 0.185; Formula: Nd5Ge4; MSG: 62.447 ( Pnm0a0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,0,0) X:(1/2,0,0) no Line\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nR:(1/2,1/2,1/2) E:(u,1/2,1/2) T:(0,1/2,1/2) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) yes Line\n\u0000:(0,0,0) K:(0,v,w) T:(0,1/2,1/2) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1/2,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) U:(1/2,0,1/2) yes Plane\nS:(1/2,1/2,0) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) K:(0,v,w) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) L:(1/2,v,w) X:(1/2,0,0) yes Plane\nY:(0,1/2,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nBCSID: 1.131; Formula: Fe2As; MSG: 62.450 ( Panma); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane58\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) no Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.132; Formula: Mn2As; MSG: 62.450 ( Panma); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) no Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.132; Formula: Mn2As; MSG: 62.450 ( Panma); U= 1eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) no Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane59\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.132; Formula: Mn2As; MSG: 62.450 ( Panma); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) no Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.132; Formula: Mn2As; MSG: 62.450 ( Panma); U= 3eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) no Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.179; Formula: NdCoAsO; MSG: 62.450 ( Panma); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) no Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) T:(0,1/2,1/2) yes Plane60\nR:(1/2,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.179; Formula: NdCoAsO; MSG: 62.450 ( Panma); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) Q:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nS:(1/2,1/2,0) D:(1/2,v,0) X:(1/2,0,0) no Line\nT:(0,1/2,1/2) H:(0,1/2,w) Y:(0,1/2,0) yes Line\nT:(0,1/2,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,1/2) N:(u,1/2,w) S:(1/2,1/2,0) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) T:(0,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nR:(1/2,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nR:(1/2,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) T:(0,1/2,1/2) yes Plane\nS:(1/2,1/2,0) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) U:(1/2,0,1/2) yes Plane\nT:(0,1/2,1/2) N:(u,1/2,w) Y:(0,1/2,0) yes Plane\nT:(0,1/2,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nU:(1/2,0,1/2) W:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 0.149; Formula: Nd3Ru4Al12; MSG: 63.462 ( Cm0c0m); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1,0) yes Line\n\u0000:(0,0,0) SM:(u,0,0) Y:(1,0,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nT:(1,0,1/2) H:(1,0,w) Y:(1,0,0) yes Line\nT:(1,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,-1/2) D:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nRA:(1/2,-1/2,-1/2) DA:(1/2,-1/2,w) SA:(1/2,-1/2,0) yes Line\n\u0000:(0,0,0) P:(u,v,0) Y:(0,1,0) yes Plane\n\u0000:(0,0,0) P:(u,v,0) S:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) P:(u,v,0) SA:(-1/2,1/2,0) yes Plane\nT:(1,0,1/2) Q:(u,v,1/2) Z:(0,0,1/2) yes Plane\nT:(1,0,1/2) Q:(u,v,1/2) R:(1/2,1/2,1/2) yes Plane\nT:(1,0,1/2) Q:(u,v,1/2) RA:(1/2,-1/2,1/2) yes Plane\nY:(1,0,0) P:(u,v,0) S:(1/2,1/2,0) yes Plane\nY:(1,0,0) P:(u,v,0) SA:(1/2,-1/2,0) yes Plane\nZ:(0,0,1/2) Q:(u,v,1/2) R:(1/2,1/2,1/2) yes Plane\nZ:(0,0,1/2) Q:(u,v,1/2) RA:(-1/2,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) Q:(u,v,1/2) RA:(-1/2,1/2,1/2) yes Plane\nS:(1/2,1/2,0) P:(u,v,0) SA:(-1/2,1/2,0) yes Plane61\nBCSID: 0.149; Formula: Nd3Ru4Al12; MSG: 63.462 ( Cm0c0m); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1,0) yes Line\n\u0000:(0,0,0) SM:(u,0,0) Y:(1,0,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nT:(1,0,1/2) H:(1,0,w) Y:(1,0,0) yes Line\nT:(1,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,-1/2) D:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nRA:(1/2,-1/2,-1/2) DA:(1/2,-1/2,w) SA:(1/2,-1/2,0) yes Line\n\u0000:(0,0,0) P:(u,v,0) Y:(0,1,0) yes Plane\n\u0000:(0,0,0) P:(u,v,0) S:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) P:(u,v,0) SA:(-1/2,1/2,0) yes Plane\nT:(1,0,1/2) Q:(u,v,1/2) Z:(0,0,1/2) yes Plane\nT:(1,0,1/2) Q:(u,v,1/2) R:(1/2,1/2,1/2) yes Plane\nT:(1,0,1/2) Q:(u,v,1/2) RA:(1/2,-1/2,1/2) yes Plane\nY:(1,0,0) P:(u,v,0) S:(1/2,1/2,0) yes Plane\nY:(1,0,0) P:(u,v,0) SA:(1/2,-1/2,0) yes Plane\nZ:(0,0,1/2) Q:(u,v,1/2) R:(1/2,1/2,1/2) yes Plane\nZ:(0,0,1/2) Q:(u,v,1/2) RA:(-1/2,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) Q:(u,v,1/2) RA:(-1/2,1/2,1/2) yes Plane\nS:(1/2,1/2,0) P:(u,v,0) SA:(-1/2,1/2,0) yes Plane\nBCSID: 0.173; Formula: Pr3Ru4Al12; MSG: 63.462 ( Cm0c0m); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1,0) yes Line\n\u0000:(0,0,0) SM:(u,0,0) Y:(1,0,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nT:(1,0,1/2) H:(1,0,w) Y:(1,0,0) yes Line\nT:(1,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,-1/2) D:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nRA:(1/2,-1/2,-1/2) DA:(1/2,-1/2,w) SA:(1/2,-1/2,0) yes Line\n\u0000:(0,0,0) P:(u,v,0) Y:(0,1,0) yes Plane\n\u0000:(0,0,0) P:(u,v,0) S:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) P:(u,v,0) SA:(-1/2,1/2,0) yes Plane\nT:(1,0,1/2) Q:(u,v,1/2) Z:(0,0,1/2) yes Plane\nT:(1,0,1/2) Q:(u,v,1/2) R:(1/2,1/2,1/2) yes Plane\nT:(1,0,1/2) Q:(u,v,1/2) RA:(1/2,-1/2,1/2) yes Plane\nY:(1,0,0) P:(u,v,0) S:(1/2,1/2,0) yes Plane\nY:(1,0,0) P:(u,v,0) SA:(1/2,-1/2,0) yes Plane\nZ:(0,0,1/2) Q:(u,v,1/2) R:(1/2,1/2,1/2) yes Plane\nZ:(0,0,1/2) Q:(u,v,1/2) RA:(-1/2,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) Q:(u,v,1/2) RA:(-1/2,1/2,1/2) yes Plane\nS:(1/2,1/2,0) P:(u,v,0) SA:(-1/2,1/2,0) yes Plane\nBCSID: 0.173; Formula: Pr3Ru4Al12; MSG: 63.462 ( Cm0c0m); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1,0) yes Line\n\u0000:(0,0,0) SM:(u,0,0) Y:(1,0,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nT:(1,0,1/2) H:(1,0,w) Y:(1,0,0) yes Line\nT:(1,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) yes Line62\nR:(1/2,1/2,-1/2) D:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nRA:(1/2,-1/2,-1/2) DA:(1/2,-1/2,w) SA:(1/2,-1/2,0) yes Line\n\u0000:(0,0,0) P:(u,v,0) Y:(0,1,0) yes Plane\n\u0000:(0,0,0) P:(u,v,0) S:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) P:(u,v,0) SA:(-1/2,1/2,0) yes Plane\nT:(1,0,1/2) Q:(u,v,1/2) Z:(0,0,1/2) yes Plane\nT:(1,0,1/2) Q:(u,v,1/2) R:(1/2,1/2,1/2) yes Plane\nT:(1,0,1/2) Q:(u,v,1/2) RA:(1/2,-1/2,1/2) yes Plane\nY:(1,0,0) P:(u,v,0) S:(1/2,1/2,0) yes Plane\nY:(1,0,0) P:(u,v,0) SA:(1/2,-1/2,0) yes Plane\nZ:(0,0,1/2) Q:(u,v,1/2) R:(1/2,1/2,1/2) yes Plane\nZ:(0,0,1/2) Q:(u,v,1/2) RA:(-1/2,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) Q:(u,v,1/2) RA:(-1/2,1/2,1/2) yes Plane\nS:(1/2,1/2,0) P:(u,v,0) SA:(-1/2,1/2,0) yes Plane\nBCSID: 0.173; Formula: Pr3Ru4Al12; MSG: 63.462 ( Cm0c0m); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1,0) yes Line\n\u0000:(0,0,0) SM:(u,0,0) Y:(1,0,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nT:(1,0,1/2) H:(1,0,w) Y:(1,0,0) yes Line\nT:(1,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) yes Line\nR:(1/2,1/2,-1/2) D:(1/2,1/2,w) S:(1/2,1/2,0) yes Line\nRA:(1/2,-1/2,-1/2) DA:(1/2,-1/2,w) SA:(1/2,-1/2,0) yes Line\n\u0000:(0,0,0) P:(u,v,0) Y:(0,1,0) yes Plane\n\u0000:(0,0,0) P:(u,v,0) S:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) P:(u,v,0) SA:(-1/2,1/2,0) yes Plane\nT:(1,0,1/2) Q:(u,v,1/2) Z:(0,0,1/2) yes Plane\nT:(1,0,1/2) Q:(u,v,1/2) R:(1/2,1/2,1/2) yes Plane\nT:(1,0,1/2) Q:(u,v,1/2) RA:(1/2,-1/2,1/2) yes Plane\nY:(1,0,0) P:(u,v,0) S:(1/2,1/2,0) yes Plane\nY:(1,0,0) P:(u,v,0) SA:(1/2,-1/2,0) yes Plane\nZ:(0,0,1/2) Q:(u,v,1/2) R:(1/2,1/2,1/2) yes Plane\nZ:(0,0,1/2) Q:(u,v,1/2) RA:(-1/2,1/2,1/2) yes Plane\nR:(1/2,1/2,1/2) Q:(u,v,1/2) RA:(-1/2,1/2,1/2) yes Plane\nS:(1/2,1/2,0) P:(u,v,0) SA:(-1/2,1/2,0) yes Plane\nBCSID: 0.199; Formula: Mn3Sn; MSG: 63.463 ( Cmc0m0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1,0) yes Line\n\u0000:(0,0,0) SM:(u,0,0) Y:(1,0,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nT:(1,0,1/2) H:(1,0,w) Y:(1,0,0) no Line\nT:(0,1,1/2) B:(0,v,1/2) Z:(0,0,1/2) no Line\n\u0000:(0,0,0) K:(0,v,w) T:(0,1,1/2) no Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) no Plane\nT:(0,1,1/2) K:(0,v,w) Y:(0,1,0) no Plane\nT:(0,1,1/2) K:(0,v,w) Z:(0,0,1/2) no Plane\nY:(0,1,0) K:(0,v,w) Z:(0,0,1/2) no Plane63\nBCSID: 3.3; Formula: Ho2RhIn8; MSG: 63.464 ( Cm0cm0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1,0) no Line\n\u0000:(0,0,0) SM:(u,0,0) Y:(1,0,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nT:(1,0,1/2) H:(1,0,w) Y:(1,0,0) yes Line\n\u0000:(0,0,0) M:(u,0,w) T:(1,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) Y:(1,0,0) yes Plane\n\u0000:(0,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nT:(1,0,1/2) M:(u,0,w) Y:(1,0,0) yes Plane\nT:(1,0,1/2) M:(u,0,w) Z:(0,0,1/2) yes Plane\nY:(1,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nBCSID: 3.3; Formula: Ho2RhIn8; MSG: 63.464 ( Cm0cm0); U= 1eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1,0) yes Line\n\u0000:(0,0,0) SM:(u,0,0) Y:(1,0,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nT:(1,0,1/2) H:(1,0,w) Y:(1,0,0) no Line\n\u0000:(0,0,0) M:(u,0,w) T:(1,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) Y:(1,0,0) no Plane\n\u0000:(0,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nT:(1,0,1/2) M:(u,0,w) Y:(1,0,0) no Plane\nT:(1,0,1/2) M:(u,0,w) Z:(0,0,1/2) yes Plane\nY:(1,0,0) M:(u,0,w) Z:(0,0,1/2) no Plane\nBCSID: 3.3; Formula: Ho2RhIn8; MSG: 63.464 ( Cm0cm0); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1,0) yes Line\n\u0000:(0,0,0) SM:(u,0,0) Y:(1,0,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nT:(1,0,1/2) H:(1,0,w) Y:(1,0,0) no Line\n\u0000:(0,0,0) M:(u,0,w) T:(1,0,1/2) yes Plane\n\u0000:(0,0,0) M:(u,0,w) Y:(1,0,0) no Plane\n\u0000:(0,0,0) M:(u,0,w) Z:(0,0,1/2) yes Plane\nT:(1,0,1/2) M:(u,0,w) Y:(1,0,0) no Plane\nT:(1,0,1/2) M:(u,0,w) Z:(0,0,1/2) yes Plane\nY:(1,0,0) M:(u,0,w) Z:(0,0,1/2) no Plane\nBCSID: 1.262; Formula: NpRhGa5; MSG: 63.466 ( Ccmcm); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nT:(1,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) no Line\nBCSID: 1.145; Formula: Mn3Ni20P6; MSG: 64.480 ( CAmca); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nT:(1,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) no Line\nBCSID: 1.188; Formula: CeRh2Si2; MSG: 64.480 ( CAmca); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nT:(1,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) no Line64\nBCSID: 2.15; Formula: Mn3Ni20P6; MSG: 65.486 ( Cmm0m0); U= 1eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1,0) yes Line\n\u0000:(0,0,0) SM:(u,0,0) Y:(1,0,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nT:(1,0,1/2) H:(1,0,w) Y:(1,0,0) no Line\nT:(1,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) no Line\nT:(0,1,1/2) B:(0,v,1/2) Z:(0,0,1/2) no Line\n\u0000:(0,0,0) K:(0,v,w) T:(0,1,1/2) no Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nT:(0,1,1/2) K:(0,v,w) Y:(0,1,0) no Plane\nT:(0,1,1/2) K:(0,v,w) Z:(0,0,1/2) no Plane\nY:(0,1,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nBCSID: 2.15; Formula: Mn3Ni20P6; MSG: 65.486 ( Cmm0m0); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1,0) yes Line\n\u0000:(0,0,0) SM:(u,0,0) Y:(1,0,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nT:(1,0,1/2) H:(1,0,w) Y:(1,0,0) yes Line\nT:(1,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) yes Line\nT:(0,1,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) K:(0,v,w) T:(0,1,1/2) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nT:(0,1,1/2) K:(0,v,w) Y:(0,1,0) yes Plane\nT:(0,1,1/2) K:(0,v,w) Z:(0,0,1/2) yes Plane\nY:(0,1,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nBCSID: 2.15; Formula: Mn3Ni20P6; MSG: 65.486 ( Cmm0m0); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1,0) yes Line\n\u0000:(0,0,0) SM:(u,0,0) Y:(1,0,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nT:(1,0,1/2) H:(1,0,w) Y:(1,0,0) yes Line\nT:(1,0,1/2) A:(u,0,1/2) Z:(0,0,1/2) yes Line\nT:(0,1,1/2) B:(0,v,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) K:(0,v,w) T:(0,1,1/2) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Y:(0,1,0) yes Plane\n\u0000:(0,0,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nT:(0,1,1/2) K:(0,v,w) Y:(0,1,0) yes Plane\nT:(0,1,1/2) K:(0,v,w) Z:(0,0,1/2) yes Plane\nY:(0,1,0) K:(0,v,w) Z:(0,0,1/2) yes Plane\nBCSID: 1.223; Formula: Tm2CoGa8; MSG: 65.489 ( Cammm); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\nR:(1/2,1/2,1/2) D:(1/2,1/2,w) S:(1/2,1/2,0) no Line\nBCSID: 1.223; Formula: Tm2CoGa8; MSG: 65.489 ( Cammm); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane65\nR:(1/2,1/2,1/2) D:(1/2,1/2,w) S:(1/2,1/2,0) no Line\nBCSID: 0.4; Formula: NiCr2O4; MSG: 70.530 ( Fd0d0d); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,0,0) T:(1,0,0) yes Line\n\u0000:(0,0,0) DT:(0,v,0) Y:(0,1,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1) no Line\nT:(0,1,1) H:(0,1,w) Y:(0,1,0) yes Line\n\u0000:(0,0,0) M:(u,v,0) T:(1,0,0) yes Plane\n\u0000:(0,0,0) M:(u,v,0) Y:(0,1,0) yes Plane\n\u0000:(0,0,0) M:(u,v,0) Z:(1,1,0) yes Plane\nT:(1,0,0) M:(u,v,0) Y:(0,1,0) yes Plane\nT:(1,0,0) M:(u,v,0) Z:(1,1,0) yes Plane\nY:(0,1,0) M:(u,v,0) Z:(1,1,0) yes Plane\nBCSID: 1.0.12; Formula: UAu2Si2; MSG: 71.536 ( Im0m0m); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) X:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) X:(1,0,0) no Line\nW:(1/2,1/2,1/2) P:(1/2,1/2,w) T:(1/2,1/2,0) yes Line\nW:(1/2,1/2,1/2) P:(1/2,1/2,w) TA:(1/2,1/2,1) yes Line\nT:(1/2,1/2,2) P:(1/2,1/2,w) TA:(1/2,1/2,1) yes Line\n\u0000:(0,0,0) C:(u,v,0) X:(0,1,0) no Plane\n\u0000:(0,0,0) C:(u,v,0) T:(1/2,1/2,0) no Plane\n\u0000:(0,0,0) C:(u,v,0) TA:(-1/2,1/2,0) no Plane\nX:(2,1,0) C:(u,v,0) T:(3/2,3/2,0) no Plane\nX:(2,1,0) C:(u,v,0) TA:(3/2,1/2,0) no Plane\nT:(3/2,-1/2,0) C:(u,v,0) TA:(1/2,-1/2,0) yes Plane\nBCSID: 1.0.12; Formula: UAu2Si2; MSG: 71.536 ( Im0m0m); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) X:(0,0,1) yes Line\n\u0000:(0,0,0) SM:(u,0,0) X:(1,0,0) no Line\nW:(1/2,1/2,1/2) P:(1/2,1/2,w) T:(1/2,1/2,0) yes Line\nW:(1/2,1/2,1/2) P:(1/2,1/2,w) TA:(1/2,1/2,1) yes Line\nT:(1/2,1/2,2) P:(1/2,1/2,w) TA:(1/2,1/2,1) yes Line\n\u0000:(0,0,0) C:(u,v,0) X:(0,1,0) no Plane\n\u0000:(0,0,0) C:(u,v,0) T:(1/2,1/2,0) no Plane\n\u0000:(0,0,0) C:(u,v,0) TA:(-1/2,1/2,0) no Plane\nX:(2,1,0) C:(u,v,0) T:(3/2,3/2,0) yes Plane\nX:(2,1,0) C:(u,v,0) TA:(3/2,1/2,0) yes Plane\nT:(3/2,-1/2,0) C:(u,v,0) TA:(1/2,-1/2,0) yes Plane\nBCSID: 1.0.12; Formula: UAu2Si2; MSG: 71.536 ( Im0m0m); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) X:(0,0,1) yes Line\n\u0000:(0,0,0) SM:(u,0,0) X:(1,0,0) no Line66\nW:(1/2,1/2,1/2) P:(1/2,1/2,w) T:(1/2,1/2,0) yes Line\nW:(1/2,1/2,1/2) P:(1/2,1/2,w) TA:(1/2,1/2,1) yes Line\nT:(1/2,1/2,2) P:(1/2,1/2,w) TA:(1/2,1/2,1) yes Line\n\u0000:(0,0,0) C:(u,v,0) X:(0,1,0) no Plane\n\u0000:(0,0,0) C:(u,v,0) T:(1/2,1/2,0) no Plane\n\u0000:(0,0,0) C:(u,v,0) TA:(-1/2,1/2,0) no Plane\nX:(2,1,0) C:(u,v,0) T:(3/2,3/2,0) yes Plane\nX:(2,1,0) C:(u,v,0) TA:(3/2,1/2,0) yes Plane\nT:(3/2,-1/2,0) C:(u,v,0) TA:(1/2,-1/2,0) yes Plane\nBCSID: 2.28; Formula: NpNiGa5; MSG: 74.559 ( Imm0a0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) X:(0,0,1) yes Line\n\u0000:(0,0,0) SM:(u,0,0) X:(1,0,0) no Line\nW:(1/2,1/2,1/2) D:(u,1/2,1/2) S:(0,1/2,1/2) yes Line\nW:(1/2,1/2,1/2) D:(u,1/2,1/2) SA:(1,1/2,1/2) yes Line\nS:(0,1/2,1/2) D:(u,1/2,1/2) SA:(1,1/2,1/2) yes Line\n\u0000:(0,0,0) A:(0,v,w) X:(0,0,1) yes Plane\n\u0000:(0,0,0) A:(0,v,w) S:(0,1/2,1/2) yes Plane\n\u0000:(0,0,0) A:(0,v,w) SA:(0,-1/2,1/2) yes Plane\nX:(0,2,1) A:(0,v,w) S:(0,3/2,3/2) yes Plane\nX:(0,2,1) A:(0,v,w) SA:(0,3/2,1/2) yes Plane\nS:(0,1/2,1/2) A:(0,v,w) SA:(0,-1/2,1/2) yes Plane\nBCSID: 2.28; Formula: NpNiGa5; MSG: 74.559 ( Imm0a0); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) X:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) X:(1,0,0) yes Line\nW:(1/2,1/2,1/2) D:(u,1/2,1/2) S:(0,1/2,1/2) yes Line\nW:(1/2,1/2,1/2) D:(u,1/2,1/2) SA:(1,1/2,1/2) yes Line\nS:(0,1/2,1/2) D:(u,1/2,1/2) SA:(1,1/2,1/2) yes Line\n\u0000:(0,0,0) A:(0,v,w) X:(0,0,1) no Plane\n\u0000:(0,0,0) A:(0,v,w) S:(0,1/2,1/2) no Plane\n\u0000:(0,0,0) A:(0,v,w) SA:(0,-1/2,1/2) no Plane\nX:(0,2,1) A:(0,v,w) S:(0,3/2,3/2) yes Plane\nX:(0,2,1) A:(0,v,w) SA:(0,3/2,1/2) yes Plane\nS:(0,1/2,1/2) A:(0,v,w) SA:(0,-1/2,1/2) yes Plane\nBCSID: 2.28; Formula: NpNiGa5; MSG: 74.559 ( Imm0a0); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) X:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) X:(1,0,0) yes Line\nW:(1/2,1/2,1/2) D:(u,1/2,1/2) S:(0,1/2,1/2) yes Line\nW:(1/2,1/2,1/2) D:(u,1/2,1/2) SA:(1,1/2,1/2) yes Line\nS:(0,1/2,1/2) D:(u,1/2,1/2) SA:(1,1/2,1/2) yes Line\n\u0000:(0,0,0) A:(0,v,w) X:(0,0,1) no Plane\n\u0000:(0,0,0) A:(0,v,w) S:(0,1/2,1/2) no Plane\n\u0000:(0,0,0) A:(0,v,w) SA:(0,-1/2,1/2) no Plane67\nX:(0,2,1) A:(0,v,w) S:(0,3/2,3/2) yes Plane\nX:(0,2,1) A:(0,v,w) SA:(0,3/2,1/2) yes Plane\nS:(0,1/2,1/2) A:(0,v,w) SA:(0,-1/2,1/2) yes Plane\nBCSID: 2.5; Formula: Mn3CuN; MSG: 85.59 ( P4=n); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,u,0) M:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) D:(u,v,0) M:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) D:(u,v,0) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) D:(u,v,0) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 2.5; Formula: Mn3CuN; MSG: 85.59 ( P4=n); U= 1eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,u,0) M:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) D:(u,v,0) M:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) D:(u,v,0) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) D:(u,v,0) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 2.5; Formula: Mn3CuN; MSG: 85.59 ( P4=n); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,u,0) M:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line68\n\u0000:(0,0,0) D:(u,v,0) M:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) D:(u,v,0) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) D:(u,v,0) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 2.5; Formula: Mn3CuN; MSG: 85.59 ( P4=n); U= 3eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,u,0) M:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) D:(u,v,0) M:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) D:(u,v,0) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) D:(u,v,0) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 2.5; Formula: Mn3CuN; MSG: 85.59 ( P4=n); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,u,0) M:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) D:(u,v,0) M:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) D:(u,v,0) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) D:(u,v,0) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 0.207; Formula: TlFe1.6Se2; MSG: 87.75 ( I4=m); U= 1eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) yes Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) no Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) no Line\nM:(1,0,0) Y:(u,1-u,0) X:(1/2,1/2,0) yes Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) no Plane69\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) no Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) yes Plane\nBCSID: 0.207; Formula: TlFe1.6Se2; MSG: 87.75 ( I4=m); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) yes Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) yes Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) no Line\nM:(1,0,0) Y:(u,1-u,0) X:(1/2,1/2,0) no Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) yes Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) no Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) no Plane\nBCSID: 0.207; Formula: TlFe1.6Se2; MSG: 87.75 ( I4=m); U= 3eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) yes Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) no Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) no Line\nM:(1,0,0) Y:(u,1-u,0) X:(1/2,1/2,0) yes Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) no Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) no Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) yes Plane\nBCSID: 0.64; Formula: MnV2O4; MSG: 88.81 ( I41=a); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) yes Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) yes Line\nM:(1,0,0) Y:(u,1-u,0) X:(1/2,1/2,0) yes Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) yes Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) yes Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) yes Plane\nBCSID: 1.0.11; Formula: CeCoGe3; MSG: 107.231 ( I4m0m0); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\nBCSID: 1.85; Formula: alpha-Mn; MSG: 114.282 ( PI\u0000421c); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,u,0) M:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1/2,0) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) no Line\n\u0000:(0,0,0) C:(u,u,w) A:(1/2,1/2,1/2) yes Plane\n\u0000:(0,0,0) C:(u,u,w) M:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) C:(u,u,w) Z:(0,0,1/2) yes Plane\nA:(1/2,1/2,1/2) C:(u,u,w) M:(1/2,1/2,0) yes Plane70\nA:(1/2,1/2,1/2) C:(u,u,w) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) C:(u,u,w) Z:(0,0,1/2) yes Plane\nBCSID: 2.26; Formula: PrCo2P2; MSG: 123.345 ( P4=mm0m0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,u,0) M:(1/2,1/2,0) no Line\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1/2,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) no Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\n\u0000:(0,0,0) D:(u,v,0) M:(1/2,1/2,0) no Plane\n\u0000:(0,0,0) D:(u,v,0) X:(0,1/2,0) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) D:(u,v,0) X:(0,1/2,0) no Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 2.26; Formula: PrCo2P2; MSG: 123.345 ( P4=mm0m0); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,u,0) M:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1/2,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) no Line\n\u0000:(0,0,0) D:(u,v,0) M:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) D:(u,v,0) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) D:(u,v,0) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nBCSID: 2.26; Formula: PrCo2P2; MSG: 123.345 ( P4=mm0m0); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,u,0) M:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1/2,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) no Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) no Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) no Line\n\u0000:(0,0,0) D:(u,v,0) M:(1/2,1/2,0) yes Plane71\n\u0000:(0,0,0) D:(u,v,0) X:(0,1/2,0) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nM:(1/2,1/2,0) D:(u,v,0) X:(0,1/2,0) no Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nBCSID: 2.26; Formula: PrCo2P2; MSG: 123.345 ( P4=mm0m0); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,u,0) M:(1/2,1/2,0) no Line\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1/2,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) no Line\n\u0000:(0,0,0) D:(u,v,0) M:(1/2,1/2,0) no Plane\n\u0000:(0,0,0) D:(u,v,0) X:(0,1/2,0) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) D:(u,v,0) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nBCSID: 1.162; Formula: NdMg; MSG: 124.360 ( Pc4=mcc); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.162; Formula: NdMg; MSG: 124.360 ( Pc4=mcc); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.162; Formula: NdMg; MSG: 124.360 ( Pc4=mcc); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line72\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.251; Formula: NdCo2P2; MSG: 124.360 ( Pc4=mcc); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nBCSID: 1.251; Formula: NdCo2P2; MSG: 124.360 ( Pc4=mcc); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.251; Formula: NdCo2P2; MSG: 124.360 ( Pc4=mcc); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.255; Formula: UPtGa5; MSG: 124.360 ( Pc4=mcc); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.255; Formula: UPtGa5; MSG: 124.360 ( Pc4=mcc); U= 4eV73\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.255; Formula: UPtGa5; MSG: 124.360 ( Pc4=mcc); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.261; Formula: NpRhGa5; MSG: 124.360 ( Pc4=mcc); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.261; Formula: NpRhGa5; MSG: 124.360 ( Pc4=mcc); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nBCSID: 1.261; Formula: NpRhGa5; MSG: 124.360 ( Pc4=mcc); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane74\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.261; Formula: NpRhGa5; MSG: 124.360 ( Pc4=mcc); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nBCSID: 2.14; Formula: NdMg; MSG: 125.373 ( PC4=nbm); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nBCSID: 2.14; Formula: NdMg; MSG: 125.373 ( PC4=nbm); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nBCSID: 2.14; Formula: NdMg; MSG: 125.373 ( PC4=nbm); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane75\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nBCSID: 2.14; Formula: NdMg; MSG: 125.373 ( PC4=nbm); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nBCSID: 1.253; Formula: CeCo2P2; MSG: 126.386 ( PI4=nnc); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.253; Formula: CeCo2P2; MSG: 126.386 ( PI4=nnc); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.253; Formula: CeCo2P2; MSG: 126.386 ( PI4=nnc); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane76\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nBCSID: 1.253; Formula: CeCo2P2; MSG: 126.386 ( PI4=nnc); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nBCSID: 1.81; Formula: GdIn3; MSG: 127.397 ( PC4=mbm); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) no Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nBCSID: 1.81; Formula: GdIn3; MSG: 127.397 ( PC4=mbm); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nBCSID: 1.81; Formula: GdIn3; MSG: 127.397 ( PC4=mbm); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) no Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line77\nBCSID: 1.81; Formula: GdIn3; MSG: 127.397 ( PC4=mbm); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nBCSID: 1.102; Formula: U2Ni2In; MSG: 128.408 ( Pc4=mnc); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.102; Formula: U2Ni2In; MSG: 128.408 ( Pc4=mnc); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.102; Formula: U2Ni2In; MSG: 128.408 ( Pc4=mnc); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane78\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nBCSID: 1.102; Formula: U2Ni2In; MSG: 128.408 ( Pc4=mnc); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) no Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) no Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) no Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.160; Formula: UP; MSG: 128.410 ( PI4=mnc); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) no Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.160; Formula: UP; MSG: 128.410 ( PI4=mnc); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nBCSID: 1.160; Formula: UP; MSG: 128.410 ( PI4=mnc); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) no Line79\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.160; Formula: UP; MSG: 128.410 ( PI4=mnc); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.187; Formula: TbRh2Si2; MSG: 128.410 ( PI4=mnc); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) no Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.187; Formula: TbRh2Si2; MSG: 128.410 ( PI4=mnc); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.187; Formula: TbRh2Si2; MSG: 128.410 ( PI4=mnc); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line80\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.187; Formula: TbRh2Si2; MSG: 128.410 ( PI4=mnc); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.208; Formula: UAs; MSG: 128.410 ( PI4=mnc); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) no Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) no Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.208; Formula: UAs; MSG: 128.410 ( PI4=mnc); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.208; Formula: UAs; MSG: 128.410 ( PI4=mnc); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) no Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane81\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.21; Formula: DyCo2Si2; MSG: 128.410 ( PI4=mnc); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.21; Formula: DyCo2Si2; MSG: 128.410 ( PI4=mnc); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) V:(1/2,1/2,w) M:(1/2,1/2,0) yes Line\nA:(1/2,1/2,1/2) T:(u,1/2,1/2) R:(0,1/2,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.215; Formula: UP2; MSG: 130.432 ( Pc4=ncc); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nBCSID: 1.215; Formula: UP2; MSG: 130.432 ( Pc4=ncc); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane82\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.215; Formula: UP2; MSG: 130.432 ( Pc4=ncc); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nBCSID: 1.215; Formula: UP2; MSG: 130.432 ( Pc4=ncc); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) no Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) no Plane\nBCSID: 2.20; Formula: UAs; MSG: 134.481 ( PC42=nnm); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) yes Line\nM:(1/2,1/2,0) Y:(u,1/2,0) X:(0,1/2,0) no Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) no Line\nR:(0,1/2,1/2) U:(0,v,1/2) Z:(0,0,1/2) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) no Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) no Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) no Plane83\nBCSID: 1.103; Formula: U2Rh2Sn; MSG: 135.492 ( Pc42=mbc); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.103; Formula: U2Rh2Sn; MSG: 135.492 ( Pc42=mbc); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.103; Formula: U2Rh2Sn; MSG: 135.492 ( Pc42=mbc); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.103; Formula: U2Rh2Sn; MSG: 135.492 ( Pc42=mbc); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line84\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.207; Formula: U2Rh2Sn; MSG: 135.492 ( Pc42=mbc); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.207; Formula: U2Rh2Sn; MSG: 135.492 ( Pc42=mbc); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.207; Formula: U2Rh2Sn; MSG: 135.492 ( Pc42=mbc); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane85\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 1.207; Formula: U2Rh2Sn; MSG: 135.492 ( Pc42=mbc); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) Z:(0,0,1/2) no Line\nA:(1/2,1/2,1/2) S:(u,u,1/2) Z:(0,0,1/2) yes Line\nR:(0,1/2,1/2) W:(0,1/2,w) X:(0,1/2,0) yes Line\nA:(1/2,1/2,1/2) F:(u,1/2,w) M:(1/2,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nA:(1/2,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nA:(1/2,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) R:(0,1/2,1/2) yes Plane\nM:(1/2,1/2,0) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) F:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(0,1/2,1/2) E:(u,v,1/2) Z:(0,0,1/2) yes Plane\nBCSID: 2.19; Formula: Mn3ZnC; MSG: 139.537 ( I4=mm0m0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) no Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) no Line\nM:(1,0,0) Y:(u,1-u,0) X:(1/2,1/2,0) no Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) no Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) no Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) no Plane\nBCSID: 2.19; Formula: Mn3ZnC; MSG: 139.537 ( I4=mm0m0); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) no Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) no Line\nM:(1,0,0) Y:(u,1-u,0) X:(1/2,1/2,0) yes Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) no Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) no Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) yes Plane\nBCSID: 2.19; Formula: Mn3ZnC; MSG: 139.537 ( I4=mm0m0); U= 3eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) no Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) no Line\nM:(1,0,0) Y:(u,1-u,0) X:(1/2,1/2,0) yes Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) no Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) no Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) yes Plane86\nBCSID: 2.19; Formula: Mn3ZnC; MSG: 139.537 ( I4=mm0m0); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) no Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) no Line\nM:(1,0,0) Y:(u,1-u,0) X:(1/2,1/2,0) yes Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) no Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) no Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) yes Plane\nBCSID: 1.80; Formula: DyCo2Ga8; MSG: 140.550 ( Ic4=mcm); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\nBCSID: 1.80; Formula: DyCo2Ga8; MSG: 140.550 ( Ic4=mcm); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\nBCSID: 1.80; Formula: DyCo2Ga8; MSG: 140.550 ( Ic4=mcm); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\nBCSID: 1.82; Formula: Nd2RhIn8; MSG: 140.550 ( Ic4=mcm); U= 4eV\nmaximalk1intermediate path maximalk2satis��ed? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\nBCSID: 1.82; Formula: Nd2RhIn8; MSG: 140.550 ( Ic4=mcm); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\nBCSID: 1.87; Formula: TbCo2Ga8; MSG: 140.550 ( Ic4=mcm); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\nBCSID: 1.87; Formula: TbCo2Ga8; MSG: 140.550 ( Ic4=mcm); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\nBCSID: 0.151; Formula: Tm2Mn2O7; MSG: 141.557 ( I41=am0d0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) no Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) no Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) no Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) no Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) yes Plane\nBCSID: 0.158; Formula: Yb2Ti2O7; MSG: 141.557 ( I41=am0d0); U= 1eV87\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) yes Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) yes Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) yes Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) yes Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) yes Plane\nBCSID: 0.227; Formula: NdCo2; MSG: 141.557 ( I41=am0d0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) yes Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) yes Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) yes Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) yes Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) yes Plane\nBCSID: 0.227; Formula: NdCo2; MSG: 141.557 ( I41=am0d0); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) yes Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) yes Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) yes Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) yes Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) yes Plane\nBCSID: 0.227; Formula: NdCo2; MSG: 141.557 ( I41=am0d0); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) yes Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) yes Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) yes Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) yes Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) yes Plane\nBCSID: 0.227; Formula: NdCo2; MSG: 141.557 ( I41=am0d0); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) no Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) no Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) no Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) no Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) yes Plane\nBCSID: 0.48; Formula: Tb2Sn2O7; MSG: 141.557 ( I41=am0d0); U= 088\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) yes Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) yes Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) yes Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) no Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) yes Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) yes Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) yes Plane\nBCSID: 0.49; Formula: Ho2Ru2O7; MSG: 141.557 ( I41=am0d0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) no Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) no Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) no Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) no Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) yes Plane\nBCSID: 0.51; Formula: Ho2Ru2O7; MSG: 141.557 ( I41=am0d0); U= 6eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) M:(0,0,1) no Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1,0,0) no Line\n\u0000:(0,0,0) DT:(u,u,0) X:(1/2,1/2,0) no Line\nP:(1/2,1/2,1/2) W:(1/2,1/2,w) X:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,v,0) M:(0,1,0) no Plane\n\u0000:(0,0,0) C:(u,v,0) X:(1/2,1/2,0) no Plane\nM:(2,1,0) C:(u,v,0) X:(3/2,3/2,0) yes Plane\nBCSID: 0.125; Formula: MnGeO3; MSG: 148.19 ( R\u000030); U= 3eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 0.125; Formula: MnGeO3; MSG: 148.19 ( R\u000030); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 1.161; Formula: PrFe3(BO3)4; MSG: 155.48 ( RI32); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,-2*u,0) F:(-1/2,1,0) yes Line\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nL:(1/2,1/2,3/2) Y:(u,u,3/2) T:(0,0,3/2) yes Line\nBCSID: 0.169; Formula: U3As4; MSG: 161.71 ( R3c0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 0.169; Formula: U3As4; MSG: 161.71 ( R3c0); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line89\nBCSID: 0.170; Formula: U3P4; MSG: 161.71 ( R3c0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 0.170; Formula: U3P4; MSG: 161.71 ( R3c0); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 1.0.5; Formula: Sr3CoIrO6; MSG: 165.95 ( P\u00003c01); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,0,w) A:(0,0,1/2) no Line\nH:(1/3,1/3,1/2) P:(1/3,1/3,w) K:(1/3,1/3,0) yes Line\nBCSID: 0.177; Formula: Mn3GaN; MSG: 166.97 ( R\u00003m); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,-2*u,0) F:(-1/2,1,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) yes Line\nL:(1/2,1/2,3/2) Y:(u,u,3/2) T:(0,0,3/2) yes Line\n\u0000:(0,0,0) C:(u,-u,w) F:(1/2,-1/2,1) no Plane\n\u0000:(0,0,0) C:(u,-u,w) L:(-1/2,1/2,1/2) yes Plane\n\u0000:(0,0,0) C:(u,-u,w) T:(0,0,3/2) yes Plane\nF:(0,1/2,1) C:(0,u,w) L:(0,-1/2,1/2) no Plane\nF:(0,1/2,1) C:(0,u,w) T:(0,1,1/2) no Plane\nL:(-1/2,1/2,1/2) C:(u,-u,w) T:(0,0,3/2) yes Plane\nBCSID: 0.177; Formula: Mn3GaN; MSG: 166.97 ( R\u00003m); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,-2*u,0) F:(-1/2,1,0) no Line\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nL:(1/2,1/2,3/2) Y:(u,u,3/2) T:(0,0,3/2) yes Line\n\u0000:(0,0,0) C:(u,-u,w) F:(1/2,-1/2,1) no Plane\n\u0000:(0,0,0) C:(u,-u,w) L:(-1/2,1/2,1/2) no Plane\n\u0000:(0,0,0) C:(u,-u,w) T:(0,0,3/2) no Plane\nF:(0,1/2,1) C:(0,u,w) L:(0,-1/2,1/2) yes Plane\nF:(0,1/2,1) C:(0,u,w) T:(0,1,1/2) yes Plane\nL:(-1/2,1/2,1/2) C:(u,-u,w) T:(0,0,3/2) yes Plane\nBCSID: 0.108; Formula: Mn3Ir; MSG: 166.101 ( R\u00003m0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 0.108; Formula: Mn3Ir; MSG: 166.101 ( R\u00003m0); U= 1eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 0.108; Formula: Mn3Ir; MSG: 166.101 ( R\u00003m0); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 0.108; Formula: Mn3Ir; MSG: 166.101 ( R\u00003m0); U= 3eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane90\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 0.108; Formula: Mn3Ir; MSG: 166.101 ( R\u00003m0); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 0.109; Formula: Mn3Pt; MSG: 166.101 ( R\u00003m0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 0.109; Formula: Mn3Pt; MSG: 166.101 ( R\u00003m0); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 0.109; Formula: Mn3Pt; MSG: 166.101 ( R\u00003m0); U= 3eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 0.109; Formula: Mn3Pt; MSG: 166.101 ( R\u00003m0); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 0.228; Formula: TbCo2; MSG: 166.101 ( R\u00003m0); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 0.77; Formula: Tb2Ti2O7; MSG: 166.101 ( R\u00003m0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 1.153; Formula: Mn3GaC; MSG: 167.108 ( RI\u00003c); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(0,0,w) T:(0,0,3/2) no Line\nBCSID: 0.33; Formula: HoMnO3; MSG: 185.197 ( P63cm); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,0,w) A:(0,0,1/2) no Line\n\u0000:(0,0,0) LD:(u,u,0) K:(1/3,1/3,0) yes Line\n\u0000:(0,0,0) LD:(u,u,0) M:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1/2,0,0) yes Line\nA:(0,0,1/2) Q:(u,u,1/2) H:(1/3,1/3,1/2) yes Line\nA:(0,0,1/2) Q:(u,u,1/2) L:(1/2,1/2,1/2) yes Line\nA:(0,0,1/2) R:(u,0,1/2) L:(1/2,0,1/2) yes Line\nH:(1/3,1/3,1/2) P:(1/3,1/3,w) K:(1/3,1/3,0) yes Line\nH:(1/3,1/3,1/2) Q:(u,u,1/2) L:(1/2,1/2,1/2) yes Line\nK:(1/3,1/3,0) LD:(u,u,0) M:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) C:(u,u,w) A:(0,0,1/2) yes Plane\n\u0000:(0,0,0) D:(u,0,w) A:(0,0,1/2) yes Plane\n\u0000:(0,0,0) C:(u,u,w) H:(1/3,1/3,1/2) yes Plane\n\u0000:(0,0,0) C:(u,u,w) K:(1/3,1/3,0) yes Plane\n\u0000:(0,0,0) C:(u,u,w) L:(1/2,1/2,1/2) yes Plane91\n\u0000:(0,0,0) D:(u,0,w) L:(1/2,0,1/2) yes Plane\n\u0000:(0,0,0) C:(u,u,w) M:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) D:(u,0,w) M:(1/2,0,0) yes Plane\nA:(0,0,1/2) C:(u,u,w) H:(1/3,1/3,1/2) yes Plane\nA:(0,0,1/2) C:(u,u,w) K:(1/3,1/3,0) yes Plane\nA:(0,0,1/2) C:(u,u,w) L:(1/2,1/2,1/2) yes Plane\nA:(0,0,1/2) D:(u,0,w) L:(1/2,0,1/2) yes Plane\nA:(0,0,1/2) C:(u,u,w) M:(1/2,1/2,0) yes Plane\nA:(0,0,1/2) D:(u,0,w) M:(1/2,0,0) yes Plane\nH:(1/3,1/3,1/2) C:(u,u,w) K:(1/3,1/3,0) yes Plane\nH:(1/3,1/3,1/2) C:(u,u,w) K:(2/3,2/3,0) yes Plane\nH:(1/3,1/3,1/2) C:(u,u,w) L:(1/2,1/2,1/2) yes Plane\nH:(1/3,1/3,1/2) C:(u,u,w) M:(1/2,1/2,0) yes Plane\nK:(1/3,1/3,0) C:(u,u,w) L:(1/2,1/2,1/2) yes Plane\nK:(1/3,1/3,0) C:(u,u,w) M:(1/2,1/2,0) yes Plane\nL:(1/2,-1,1/2) C:(u,-2*u,w) M:(1/2,-1,0) yes Plane\nL:(1/2,0,1/2) D:(u,0,w) M:(1/2,0,0) yes Plane\nBCSID: 1.110; Formula: ScMn6Ge6; MSG: 192.252 ( Pc6=mcc); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,0,w) A:(0,0,1/2) yes Line\nA:(0,0,1/2) Q:(u,u,1/2) H:(1/3,1/3,1/2) no Line\nA:(0,0,1/2) Q:(u,u,1/2) L:(1/2,1/2,1/2) yes Line\nA:(0,0,1/2) R:(u,0,1/2) L:(1/2,0,1/2) yes Line\nH:(1/3,1/3,1/2) P:(1/3,1/3,w) K:(1/3,1/3,0) yes Line\nH:(1/3,1/3,1/2) Q:(u,u,1/2) L:(1/2,1/2,1/2) no Line\nA:(0,0,1/2) E:(u,v,1/2) H:(1/3,1/3,1/2) no Plane\nA:(0,0,1/2) E:(u,v,1/2) L:(1/2,0,1/2) yes Plane\nH:(1/3,1/3,1/2) E:(u,v,1/2) L:(1/2,0,1/2) no Plane\nBCSID: 1.225; Formula: ScMn6Ge6; MSG: 192.252 ( Pc6=mcc); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,0,w) A:(0,0,1/2) yes Line\nA:(0,0,1/2) Q:(u,u,1/2) H:(1/3,1/3,1/2) no Line\nA:(0,0,1/2) Q:(u,u,1/2) L:(1/2,1/2,1/2) yes Line\nA:(0,0,1/2) R:(u,0,1/2) L:(1/2,0,1/2) yes Line\nH:(1/3,1/3,1/2) P:(1/3,1/3,w) K:(1/3,1/3,0) yes Line\nH:(1/3,1/3,1/2) Q:(u,u,1/2) L:(1/2,1/2,1/2) no Line\nA:(0,0,1/2) E:(u,v,1/2) H:(1/3,1/3,1/2) no Plane\nA:(0,0,1/2) E:(u,v,1/2) L:(1/2,0,1/2) yes Plane\nH:(1/3,1/3,1/2) E:(u,v,1/2) L:(1/2,0,1/2) no Plane\nBCSID: 1.225; Formula: ScMn6Ge6; MSG: 192.252 ( Pc6=mcc); U= 1eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,0,w) A:(0,0,1/2) yes Line\nA:(0,0,1/2) Q:(u,u,1/2) H:(1/3,1/3,1/2) no Line\nA:(0,0,1/2) Q:(u,u,1/2) L:(1/2,1/2,1/2) yes Line\nA:(0,0,1/2) R:(u,0,1/2) L:(1/2,0,1/2) yes Line\nH:(1/3,1/3,1/2) P:(1/3,1/3,w) K:(1/3,1/3,0) yes Line\nH:(1/3,1/3,1/2) Q:(u,u,1/2) L:(1/2,1/2,1/2) no Line\nA:(0,0,1/2) E:(u,v,1/2) H:(1/3,1/3,1/2) no Plane92\nA:(0,0,1/2) E:(u,v,1/2) L:(1/2,0,1/2) yes Plane\nH:(1/3,1/3,1/2) E:(u,v,1/2) L:(1/2,0,1/2) no Plane\nBCSID: 1.225; Formula: ScMn6Ge6; MSG: 192.252 ( Pc6=mcc); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,0,w) A:(0,0,1/2) yes Line\nA:(0,0,1/2) Q:(u,u,1/2) H:(1/3,1/3,1/2) no Line\nA:(0,0,1/2) Q:(u,u,1/2) L:(1/2,1/2,1/2) no Line\nA:(0,0,1/2) R:(u,0,1/2) L:(1/2,0,1/2) no Line\nH:(1/3,1/3,1/2) P:(1/3,1/3,w) K:(1/3,1/3,0) yes Line\nH:(1/3,1/3,1/2) Q:(u,u,1/2) L:(1/2,1/2,1/2) no Line\nA:(0,0,1/2) E:(u,v,1/2) H:(1/3,1/3,1/2) no Plane\nA:(0,0,1/2) E:(u,v,1/2) L:(1/2,0,1/2) no Plane\nH:(1/3,1/3,1/2) E:(u,v,1/2) L:(1/2,0,1/2) no Plane\nBCSID: 1.225; Formula: ScMn6Ge6; MSG: 192.252 ( Pc6=mcc); U= 3eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,0,w) A:(0,0,1/2) no Line\nA:(0,0,1/2) Q:(u,u,1/2) H:(1/3,1/3,1/2) no Line\nA:(0,0,1/2) Q:(u,u,1/2) L:(1/2,1/2,1/2) yes Line\nA:(0,0,1/2) R:(u,0,1/2) L:(1/2,0,1/2) yes Line\nH:(1/3,1/3,1/2) P:(1/3,1/3,w) K:(1/3,1/3,0) no Line\nH:(1/3,1/3,1/2) Q:(u,u,1/2) L:(1/2,1/2,1/2) no Line\nA:(0,0,1/2) E:(u,v,1/2) H:(1/3,1/3,1/2) no Plane\nA:(0,0,1/2) E:(u,v,1/2) L:(1/2,0,1/2) yes Plane\nH:(1/3,1/3,1/2) E:(u,v,1/2) L:(1/2,0,1/2) no Plane\nBCSID: 0.118; Formula: Ba5Co5ClO13; MSG: 194.268 ( P60\n3=m0m0c); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) DT:(0,0,w) A:(0,0,1/2) yes Line\n\u0000:(0,0,0) LD:(u,u,0) K:(1/3,1/3,0) yes Line\n\u0000:(0,0,0) LD:(u,u,0) M:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) SM:(u,0,0) M:(1/2,0,0) no Line\nH:(1/3,1/3,1/2) P:(1/3,1/3,w) K:(1/3,1/3,0) yes Line\nK:(1/3,1/3,0) LD:(u,u,0) M:(1/2,1/2,0) yes Line\nL:(1/2,0,1/2) U:(1/2,0,w) M:(1/2,0,0) yes Line\n\u0000:(0,0,0) C:(u,u,w) A:(0,0,1/2) yes Plane\n\u0000:(0,0,0) C:(u,u,w) H:(1/3,1/3,1/2) yes Plane\n\u0000:(0,0,0) C:(u,u,w) K:(1/3,1/3,0) yes Plane\n\u0000:(0,0,0) C:(u,u,w) L:(1/2,1/2,1/2) yes Plane\n\u0000:(0,0,0) C:(u,u,w) M:(1/2,1/2,0) yes Plane\nA:(0,0,1/2) C:(u,u,w) H:(1/3,1/3,1/2) yes Plane\nA:(0,0,1/2) C:(u,u,w) K:(1/3,1/3,0) yes Plane\nA:(0,0,1/2) C:(u,u,w) L:(1/2,1/2,1/2) yes Plane\nA:(0,0,1/2) C:(u,u,w) M:(1/2,1/2,0) yes Plane\nH:(1/3,1/3,1/2) C:(u,u,w) K:(1/3,1/3,0) yes Plane\nH:(1/3,1/3,1/2) C:(u,u,w) L:(1/2,1/2,1/2) yes Plane\nH:(1/3,1/3,1/2) C:(u,u,w) M:(1/2,1/2,0) yes Plane\nK:(1/3,1/3,0) C:(u,u,w) L:(1/2,1/2,1/2) yes Plane\nK:(1/3,1/3,0) C:(u,u,w) M:(1/2,1/2,0) yes Plane\nL:(1/2,-1,1/2) C:(u,-2*u,w) M:(1/2,-1,0) yes Plane93\nBCSID: 3.8; Formula: NdZn; MSG: 222.103 ( PIn\u00003n); U= 2eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(u,u,u) R:(1/2,1/2,1/2) yes Line\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1/2,0) no Line\nM:(1/2,1/2,0) Z:(u,1/2,0) X:(0,1/2,0) no Line\nR:(1/2,1/2,1/2) S:(u,1/2,u) X:(0,1/2,0) no Line\nM:(1/2,1/2,0) B:(u,1/2,w) R:(1/2,1/2,1/2) yes Plane\nM:(1/2,1/2,0) B:(u,1/2,w) X:(0,1/2,0) no Plane\nR:(1/2,1/2,1/2) B:(u,1/2,w) X:(0,1/2,0) no Plane\nBCSID: 3.8; Formula: NdZn; MSG: 222.103 ( PIn\u00003n); U= 4eV\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(u,u,u) R:(1/2,1/2,1/2) yes Line\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1/2,0) no Line\nM:(1/2,1/2,0) Z:(u,1/2,0) X:(0,1/2,0) yes Line\nR:(1/2,1/2,1/2) S:(u,1/2,u) X:(0,1/2,0) yes Line\nM:(1/2,1/2,0) B:(u,1/2,w) R:(1/2,1/2,1/2) yes Plane\nM:(1/2,1/2,0) B:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(1/2,1/2,1/2) B:(u,1/2,w) X:(0,1/2,0) yes Plane\nBCSID: 3.2; Formula: UO2; MSG: 224.113 ( Pn\u00003m0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) SM:(u,u,0) M:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) LD:(u,u,u) R:(1/2,1/2,1/2) no Line\nM:(1/2,1/2,0) Z:(u,1/2,0) X:(0,1/2,0) yes Line\nM:(-1/2,1/2,0) ZA:(-1/2,-u,0) X:(-1/2,0,0) yes Line\nR:(1/2,1/2,1/2) S:(u,1/2,u) X:(0,1/2,0) yes Line\n\u0000:(0,0,0) A:(u,v,0) M:(1/2,1/2,0) yes Plane\n\u0000:(0,0,0) A:(u,v,0) X:(0,1/2,0) yes Plane\nM:(1/2,1/2,0) B:(u,1/2,w) R:(1/2,1/2,1/2) yes Plane\nM:(1/2,1/2,0) A:(u,v,0) X:(0,1/2,0) yes Plane\nM:(1/2,1/2,0) B:(u,1/2,w) X:(0,1/2,0) yes Plane\nR:(1/2,1/2,1/2) B:(u,1/2,w) X:(0,1/2,0) yes Plane\nBCSID: 3.9; Formula: NpS; MSG: 228.139 ( FSd\u00003c); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(u,u,u) L:(1/2,1/2,1/2) no Line\n\u0000:(0,0,0) DT:(0,v,0) X:(0,1,0) yes Line\nW:(1/2,1,0) V:(u,1,0) X:(0,1,0) yes Line\nBCSID: 0.127; Formula: Dy3Al5O12; MSG: 230.148 ( Ia\u00003d0); U= 0\nmaximalk1intermediate path maximalk2satisfied? Line/Plane\n\u0000:(0,0,0) LD:(u,u,u) H:(1,1,1) no Line\n\u0000:(0,0,0) SM:(u,u,0) N:(1/2,1/2,0) yes Line\n\u0000:(0,0,0) LD:(u,u,u) P:(1/2,1/2,1/2) no Line\nH:(1,1,1) LD:(u,u,u) P:(1/2,1/2,1/2) no Line\nN:(1/2,1/2,0) D:(1/2,1/2,w) P:(1/2,1/2,1/2) yes Line\n\u0000:(0,0,0) A:(u,v,0) H:(0,1,0) yes Plane\n\u0000:(0,0,0) A:(u,v,0) N:(1/2,1/2,0) yes Plane\nH:(2,1,0) A:(u,v,0) N:(3/2,3/2,0) yes Plane94\nAppendix J: Detailed discussion of the ideal magnetic TI and SMs\n1. Higher-order topology of the ideal Axion insulator NpBi\nThe crystal structure of NpBi adopts a face-centered cubic lattice with space group Fm\u00163m(No. 225) in the high temperature\nparamagnetic phase. Below 192.5 K, it undergoes a phase transition to the antiferromagnetic phase and the magnetic unit cell adopts\nsimple cubic lattice with MSG 224.113( Pn\u00163m0), as shown in FIG. 32(a). The generators of this MSG include two-fold screw rotation along\nthe [100] direction ~C2x=fC2xj(0;0:5;0:5)g, three-fold rotation along the [111] direction C3xyz, inversionIand an anti-unitary symmetry\nC2\u0016xz\u0001T(C2\u0016xzis the two-fold rotation along the [ \u0016110] direction and Tis the time reversal symmetry). We use the experimental lattice\nconstanta= 6:37Å and the magnetic momentum of Np is initialized as 2:42\u0016Balong the (111) direction in our first-principle calculations.\nIn order to consider the interaction of the 5f electron on Np, we apply the LDA+U calculation with U=0,2,4,6 eV. We find that the band\nstructures at the four Us have the same topology. In the main text, we have calculated the parity-based stable topological indices and find\nZ4= 2, which corresponds to the axion insulator phase. Here we only chose U= 2eVto analyze the material’s topological surface states.\nIt has been proved in Ref. [43] that the C2\u0001Tsymmetry protects the gapless surface states of axion insulators, on the plane perpendicular\nto the two-fold rotation axis. Such an axion insulator also exhibits chiral hinge states [61–65, 102] between the two gapped surfaces related\nbyC2\u0001Tsymmetry.\nThere are three equivalent first Miller planes, h\u0016110i,h0\u001611iandh10\u00161ithat preserve C2\u0001Tsymmetry. We perform the surface states\ncalculations on the h\u0016110iplane where the C2\u0001Tsymmetry is preserved. As shown in FIG. 32(e), the surface Dirac cone emerges and\nstrictly on the ~\u0000~Zpath constrained by the glide mirror symmetry ~Mz=fMzj(0:5;0:5;0)g. For theh001isurface plane, it has no C2\u0001T\nsymmetry and the surface states are fully gaped as shown in FIG. 32(f). Considering the IandC3xyzsymmetries, the chiral flow of charge\non the hinges of a cubic NpBi sample is schematically shown in FIG. 32(d).\nFIG. 32. (a) The crystal and magnetic structures of NpBi with MSG 224.113( Pn\u00163m0). (b) 3D bulk BZ and the projected 2D\nsurface BZ on the h001i(blue) andh\u0016110i(pink) surfaces. (c) Electronic band structures of NpBi along the high symmetry paths\nin BZ with the interaction parameter U= 2eV. (d) Schematic of the chiral flow of charge on the hinge of NpBi. The 1D chiral\nmodes (red) on the six hinges are related to each other by IandC3xyzsymmetries. (e) Surface states calculation on the C2\u0016xy\u0001T\nsymmetrich\u0016110isurface plane. The Dirac cone surface states is along the path ~\u0000~Xconstrained by the glide mirror symmetry\n~Mz=fMzj(0:5;0:5;0)gand the dashed line indicates the \"curved Fermi level\", the band number below which equals that of\nelectrons. (f) The surface states of h001isurface plane have a full gap in the whole BZ.95\n\u0003\u000bMLCG IrrepsfC+\n4zj(0:5;0;0)gIfMzj(0:5;0:5;0)gfMyj(0:5;0;0:5)g\n\u00004=mmm 10\u0016\u00006(2)\u0000p\n22 0 0\n\u0016\u00009(2)p\n2-2 0 0\nZ4=mmm 10\u0016Z5(2)\u0000p\n20 -2i 0\n\u0016Z7(2)\u0000p\n20 2i 0\nTABLE XII. Character table of the irreducible co-representations \u0016\u00006,\u0016\u00009,\u0016Z5and \u0016Z7of MSG 126.386( PI4=nnc) at\u0000and Z\npoints, respectively. The first two columns are the momenta and their magnetic little co-group(MLCG); the third column is\nthe irreps of the MLCG; the 4th-7th columns are the characters of the symmetry generators of the unitary subgroup. The \u0016\u00006\nand\u0016\u00009irreps have different eigenvalues of C4zandI, and \u0016Z5and \u0016Z7irreps have different eigenvalues of Mz.\n2. Topological phase diagram of the ideal antiferromagnetic nodal-line semimetal CeCo 2P2\nThe space group(SG) of CeCo 2P2in the paramagnetic phase is I4=mmm. The antiferromagnetic phase transition occurs at 440 K\nbelow which the magnetic structure, as shown in FIG. 33(a), is characterized by the MSG 126.386( PI4=nnc). The generators of this MSG\ninclude inversion I, four-fold rotation ~C4z=fC4zj(0:5;0;0)g, two-fold rotation ~C2x=fC2xj(0;0:5;0:5)gand ~C2xy=fC2xyj(0;0;0:5)g,\nand the anti-unitary translation T\u001c(\u001c= (0:5;0:5;0:5)).\nConsidering that the correlation effect of the felectron on Ce is very strong, we take the Coulomb interaction strength of the 4felectron\nin the range 0\u00186eV. For convenience, we set U=2 eV for the 3delectron of the Co atom. As shown in the phase diagram Table VIII,\nCeCo 2P2is ES for all U values. But we find that there is a topological phase transition between DSM and NLSM around U= 3:85eV.\nThe band structures along \u0000\u0000Z\u0000Rpath have been plotted in the FIG. 33(c), with the Coulomb interaction U=0, 2, 3.5, 4 and 6 eV.\nWhenU < 3:85eV, both the band inversion between the highest occupied valence bands(HOVB) and the lowest unoccupied conduction\nbands(LUCB) occurs around \u0000point. The irreps of HOVB and LUCB are \u0016\u00006and \u0016\u00009, which have different C4zeigenvalues (See the\ncharacters in Table XII). Thus the band inversion between them creates two crossing points protected by C4zon thekx=ky= 0line.\nAs the Dirac points are symmetry protected on the kx=ky= 0line with co-little group 4=mmm 10, CeCo 2P2is also a higher-order\ntopological SM. [41] With U > 3:85eV, the band inversion around \u0000point is removed and the two Dirac points annihilate each other.\nHowever, a new band inversion between HOVB and LUCB appears around Zpoint. As the irreps of the two bands at Zpoint are \u0016Z5and\n\u0016Z7, which have different Mzeigenvalues, the band crossing between them form a nodal line on the Mzinvariantkz=\u0019plane. We also\nfind that for U equal to the critical point of 3.85 eV, band inversions occur both around \u0000andZpoints. So the Dirac points and nodal\nline coexist at the critical point U= 3:85eV.\nWhen we take the Coulomb interaction of felectron on Ce to be 6 eV and delectron interaction on Co to be 2 eV, the calculated\nmagnetic moments on Co is 0.94 \u0016B, which matches the experimental magnetic moments. Then the band structure in FIG. 33(d) has a\nclean Fermi surface and forms an ideal antiferromagnetic NLSM. The surface states on the h001isurface plane are calculated with the\nWannier tight-binding Hamiltonian. As shown in the FIG. 33(b), there are two branches of surface states connecting to the nodal points\nnear the Fermi level; These are distinguishable from the bulk states and can be observed by the ARPES/STM experiments.\n3. Topological phase diagram of the antiferromagnetic Dirac semimetal MnGeO 3\nMnGeO 3with ilmenite structure was reported to be an antiferromagnet with Néel temperature TN= 120K. The magnetic moment on\nthe divalent cation Mn2+is about 5\u0016B, which is realized by the fully polarized high-spin configuration. It has a hexagonal lattice with\nSGR\u00163and MSG 148.19( R\u001630) in the paramagnetic and antiferromagnetic phase, respectively. The MSG have two symmetry generators,\nthree-fold rotation C3zand the combination of inversion and time reversal symmetries P\u0001T. The experimental lattice constants a= 5:012\nÅ andb= 14:2986Å are used in the first principle calculations. Because of the absence of time reversal symmetry, a new situation not\npossible in the time reversal invariant space groups arises in the MSG 148.19( R\u001630). In this situation, all the k points on the C3zrotational\naxis are maximal k points. If the eigenvalues of C3zbetween any two points on the axis are changed, the system is predicted to be an ES.\nWe take the Coulomb interaction strength of 3delectron of Mn from 0 to 4 eV and plot the band structures along the C3zsymmetric\npath \u0016F\u0000\u0000\u0000F, as shown in the FIG. 34, where the green and red lines represent the bands that have different eigenvalues of C3zsymmetry.\nThe little co-group(LCG) of the momenta along \u0000F path is \u001630, which have two irreducible co-representations \u0016\u00005\u0016\u00006withC3=e\u0006i\u0019=3\ndoublet and \u0016\u00004\u0016\u00004withC3=\u00001;\u00001doublet, where the two doublets are stabilized by the P\u0001Tsymmetry. Without taking into account\nthe correlation effect, the bands H1 with irrep \u0016\u00005\u0016\u00006and L1 with irrep \u0016\u00004\u0016\u00004have an anti-crossing between the \u0016Fand\u0000points, which\ngenerate two Dirac points (DPs). Upon adding U, the second highest occupied band H2 moves up and there exist 2 pairs of DPs with\nU= 2eV, as shown in FIG. 34(c). With increasing Uto 3 and 4 eV, one out of the two pairs DPs annihilates, leaving one pair of DPs\nlocating on the two sides of \u0000point. In the FIG. 2(c)-(d) of the main text, we have calculated the surface states with U= 4eV, on the\nh010isurface plane. The surface states connecting to the Dirac cone on \u0000F path can be distinguished from the bulk states.\nIn symmorphic groups, Dirac nodes always appear in pairs. Since the irrep of one band is changed when it traverses though a symmetry\nprotected DP, there must be even number of DPs for the irrep to go back to itself after a period of the BZ. It indicates that the crystalline\nNielsen-Ninomiya theorem, which guarantees the doubling of the number of Dirac nodes in a crystal is still valid for the Dirac fermions in\nmagnetic crystals.96\nFIG. 33. (a) The crystal and magnetic structures of the antiferromagnetic CeCo 2P2with MSG 126.386( PI4=nnc). (b) The 3D\nBZ with high symmetry momenta and the surface BZ projected on the h001isurface plane. The C4zrotation axis along \u0000Z\nand theMzplanekz=\u0019are represented by the blue line and green plane, respectively. (c) The topological phase diagrams\nof CeCo 2P2with different Coulomb interaction strength applied. The irreps of the HOVB and LUCB are blue-colored and\nred-colored at both \u0000andZpoints. The topology with U\u00143eV belongs to DSM and is transformed to NLSM once U\u00154\neV. All of the nodal points along the high symmetry path are indicated by the black circles. (d) The energy dispersion of the\nnodal line on the kz=\u0019plane, which is protected by Mzsymmetry. (e) The drumhead-like surface states that connect the\nnodal line on the h001isurface. The Hubbard U of felectron on Ce is equal to 6 eV in (d) and (e).\n4. Weyl nodes, Nodal-lines and Anomalous Hall effect in Mn 3ZnC\nMn3ZnC adopts a cubic lattice with anti-perovskite structure in the paramagnetic phase and undergoes a phase transition to\nferromagnetic phase with Currie temperature Tc= 470K. At 215 K, there is a second order phase transition from ferromagnetic to\nnon-collinear ferrimagnetic phase accompanied with a structural transformation from cubic lattice to a body-centered tetragonal lattice,\nas shown in FIG. 35. In this article, we focus on the low-temperature non-collinear ferrimagnetic phase of Mn 3ZnC.\nThe MSG of ferrimagnetic Mn 3ZnC is MSG 139.537( I4=mm0m0), which is generated by the four-fold rotational symmetry along [001]\ndirectionC4z, inversion symmetry Iand the anti-unitary symmetry C2x\u0001T(C2xis the two-fold rotational symmetry along [100] direction\nandTis time reversal symmetry). There are two non-equivalent Mn atoms occupy the Wyckoff position 4 c(0,1/2,0) and 8 f(1/4,1/4,1/4),\nrespectively. The Mn atoms on 4 csite form the non-collinear antiferromagnetic structure with ferromagnetic canting along [001] direction.\nWhiletheMnatomson8 fsiteareferromagneticallypolarizedalong[001]direction. Inthe ab initio calculations,non-collinearferrimagnetic\nMn3ZnC is diagnosed as ES for all Uvalues (i.e. U= 0;1;2;3;4eV for 3delectron on Mn). Considering the strong correlations on 3 d\nelectron, we take U= 4eV and explain the band structure’s topology in detail.\nWhenU= 4eV, compatibility-relations of the band representations are not satisfied along \u0000Xand\u0000Tpaths. So there have symmetry\nenforced band crossings between the HOVB and LUCB along \u0000Xand\u0000T. In FIG. 35c, we plot the band structures along T\u0000\u0000\u0000Xand\nS0\u0000near the Fermi level. The irreps of HOVB and LUCB have been indicated by different colors. (See TABLE XIII for the characters\nof the irreps.) Along T\u0000path, one can easily find two crossing points, WP3 and WP4, between the HOVB and LUCB, which have\ndifferentC4zeigenvalues. So WP3 and WP4 are C4zprotected Weyl points (WPs). On the \u0000XandS0\u0000paths, there are many band\ncrossings, which have different Mzeigenvalues. So these crossing points on the kz= 0plane form the Nodal-lines (NLs) protected by\nmirror symmetry, Mz. Using WannierTools package, we find 10 pairs of WPs and 5 NLs in the first BZ. The positions of them are plotted\nin the BZ, as shown in FIG. 35. The 10 pairs of WPs can be classified to four non-equivalent types, among which WP3 and WP4 are97\nFIG. 34. The band structures and topologies evolution of MnGeO 3(MSGR\u001630) with the Hubbard Uparameter increasing from\n0 to 4 eV. The (Ne\u00001)th to (Ne+ 2)th bands are plotted, with Nebeing the number of electrons per unit cell. All of the\npoints on the \u0016F\u0000\u0000\u0000Fpath respect the same LCG \u001630, which have two different 2-dimensional irreps named as \u0016\u00004\u0016\u00004and\u0016\u00005\u0016\u00006.\nThe three lines H1, H2 and L2 have the same \u0016\u00005\u0016\u00006representation and are red-colored; the line L1 is the \u0016\u00004\u0016\u00004representation\nand is blue-colored. (a) For U= 0, L1 and H1 have an anti-crossing along the \u0016F\u0000path and generate two DPs. (b)-(e) With\nincreasing the Coulomb interaction, the lines L2 and H2 move to higher energy and much more crossing generate between H2\nand the branch of (L1, H1). The DPs formed by the HOVB and LUCB are indicated by the purple circles.\nsymmetry enforced WPs, WP1 and WP2 are generated by accidental band crossings. WPs in each type are related by C4zandMz\nsymmetries. Using Wilson loop method [ ?], we have identified the chiralities of the WPs, as shown in FIG. 35e. We tabulated the exact\npositions and chiralities of them in TABLE XIV. The 5 NLs are classified to two types. The first type (NL1) is symmetry enforced and\nlocalized around the \u0000point. The other four NLs (NL2) are generated by accidental band crossings near S0\u0000path and they are related\nbyC4zsymmetry.\nIn time reversal breaking (TRB) systems, nonzero Berry curvatures of occupied Bloch states contribute the intrinsic anomalous Hall\nconductivities (AHC). In non-collinear ferrimagnetic Mn 3ZnC, the Berry curvatures of bands near the WPs and NLs are very large, which\nwill contribute giant AHC. We calculate the AHC of ferrimagnetic Mn 3ZnC by the sum of Berry curvatures over all occupied bands,\n\u001bxy=\u00002\u0019e2\nhZ\nBZd3~k\n(2\u0019)3X\nnfn(~k)\nz\nn(~k) (J1)\nwherefn(~k)is the Fermi-Dirac distribution function, and nis the index of the occupied bands. The Berry curvature is arisen from\nKubo-formula derivation,\n\nz\nn(~k) =\u00002ImX\nm6=nh\tn~kjvxj\tm~kih\tm~kjvyj\tn~ki\n(Em(~k)\u0000En(~k))2(J2)\nwherevx(y)is the velocity operator. We calculate the intrinsic AHC with a 101 \u0002101\u0002101k-point grid in the first BZ based on the Wannier\ntight-binding Hamiltonian.\nIn FIG. 35f, we plot the energy dependence of intrinsic AHC near the Fermi level. On the Fermi level, the AHC is about -123 \n\u00001\u0001cm\u00001,\nwhich can be observed in transport experiments.\n\u0003\u000bMLCGIrrepsC+\n4z \u0003\u000bMLCGIrrepsMz\nLD(0,0,w) 42020LD5\u00001+ip\n2DT(u,u,0) m0m20DT3-i\nLD61\u0000ip\n2DT4i\nLD7\u00001\u0000ip\n2SM(u,0,0) m0m20SM 3-i\nLD81+ip\n2SM 4i\nTABLE XIII. Character table of the irreducible co-representations of MSG PI4=nncatLD(0;0;w),DT(u;u;0)andSM(u;0;0)\npoints. The first two(5th-6th) columns are the momenta and their magnetic little co-groups(MLCG); the third(7th) column\nare the irreps of the MLCG; the 4th(8th) columns are the characters of the symmetry generators of the unitary subgroup.98\nFIG. 35. Weyl nodes and Nodal-lines in ferrimagnetic Mn 3ZnC (MSG I4=mm0m0) withU= 4eV. (a) The magnetic crystal\nstructure and (b) the corresponding Brillouin zone (BZ) of non-collinear ferrimagnetic Mn 3ZnC. (c) Band structures along the\nhigh-symmetry paths T\u0000\u0000\u0000XandS0\u0000. The irreps of HOVB and LUCB are tagged by different colors. Band crossings on\nT\u0000path form the two WPs, WP3 and WP4. Band crossings on the kz= 0plane generate 5 NLs, which are protected by\nMzsymmetry. (d) Locations of WPs and NLs in the first BZ. The red(blue) balls stand for WPs with chirality +1(-1). (e)\nThe evolution of Wannier charge centers (WCCs) on the manifold that enclosing WP1, WP2, WP3 and WP4, respectively. (f)\nEnergy dependence of the intrinsic AHC in the non-collinear ferrimagnetic Mn 3ZnC. The Fermi energy is set to 0.\nAppendix K: Fragile bands in the magnetic materials\nIn Table XV, we tabulate all of the magnetic materials that have fragile bands below the Fermi level. For the materials having more\nthan one fragile brach, we provide the irreducible co-representations at high symmetry momenta of the fragile branches closest to the\nFermi level.\nTABLE XV: The fragile occupied bands in magnetic materials. In the first four columns the chemical formulae (Formula),the\nmaterial identification number on BCSMD (BCSID), the magnetic space group number (MSG) and the Coulomb interaction\nstrength (U) are tabulated. The fifth column gives the number of fragile branches (NF) in the band structure of the\ncorresponding material. In the sixth and seventh columns, the band indices (Bands) and the irreducible representations\n(Irreps) at high symmetry momenta of the fragile branch closest to the Fermi level are tabulated.\nFormula BCSID MSGU(eV)NFBands Irreps\nCeCoGe3 1.0.11107.231 41-83\u0018-81 \u0016\u00006+\u0016\u00008+\u0016M6+\u0016M8+\u0016N2+\u0016P4+\u0016X4\nalpha-Mn 1.85114.282 05-43\u0018-40 \u0016A6+\u0016A7+\u0016\u00006+\u0016M7+\u0016R2\u0016R4+\u0016R3\u0016R5+\u0016X2+\u0016X3+\u0016X4+\u0016X5+\u0016Z6+\u0016Z7\nPrCo2P2 2.26123.345 01-115\u0018-108 \u0016A10+\u0016A11+\u0016A12+\u0016A5+\u0016A6+\u0016A7+\u0016A8+\u0016A9+\u0016\u000010+\u0016\u000012+\u0016\u00006+\n\u0016\u00008+\u0016M11+\u0016M5+\u0016M6+\u0016M7+\u0016M8+\u0016M9+\u0016R3+\u0016R4+\u0016R5+\u0016R6+\n\u0016X3+\u0016X4+\u0016X5+\u0016X6+\u0016Z10+\u0016Z12+\u0016Z6+\u0016Z8\nNdCo2P2 1.251124.360 02-119\u0018-112 \u0016A5+\u0016A6+\u0016A7+\u0016A8+\u0016\u00007+\u0016\u00009+\u0016M6+\u0016M7+\u0016M8+\u0016R3+\u0016R4+\n\u0016X5+\u0016X6+\u0016Z6+\u0016Z8\nNdCo2P2 1.251124.360 41-119\u0018-112 \u0016A5+\u0016A6+\u0016A7+\u0016A8+\u0016\u00007+\u0016\u00009+\u0016M6+\u0016M7+\u0016M8+\u0016R3+\u0016R4+\n\u0016X5+\u0016X6+\u0016Z6+\u0016Z899\nWPs Position (kx,ky,kz)(Å\u00001) Chirality E\u0000EFermi(eV)\nWP1( 0.3101, 0.3101, 0.7154) +1 -0.0767\n( -0.3101, -0.3101, 0.7154) +1 -0.0767\n( -0.3101, 0.3101, 0.7154) +1 -0.0767\n( 0.3101, -0.3101, 0.7154) +1 -0.0767\n( 0.3101, 0.3101, -0.7154) -1 -0.0767\n( -0.3101, -0.3101, -0.7154) -1 -0.0767\n( -0.3101, 0.3101, -0.7154) -1 -0.0767\n( 0.3101, -0.3101, -0.7154) -1 -0.0767\nWP2( 0.0, -0.2798, 0.5009) -1 -0.2310\n( 0.0, 0.2798, 0.5009) -1 -0.2310\n( 0.2798, 0.0, 0.5009) -1 -0.2310\n( -0.2798, 0.0, 0.5009) -1 -0.2310\n( 0.0, 0.2798, -0.5009) +1 -0.2310\n( 0.0, -0.2798, -0.5009) +1 -0.2310\n( -0.2798, 0.0, -0.5009) +1 -0.2310\n( 0.2798, 0.0, -0.5009) +1 -0.2310\nWP3( 0.0, 0.0, 0.3343) +1 -0.14\n( 0.0, 0.0, -0.3343) -1 -0.14\nWP4( 0.0, 0.0, 0.2320) +1 -0.0901\n( 0.0, 0.0, -0.2320) -1 -0.0901\nTABLE XIV. The 10 pairs of WPs in the first BZ. The first column is the type of WPs. In each type, the WPs are related by\nC4zandMzsymmetries. The second column is the position of each WP in the cartesian coordinate system of BZ. The 4th-5th\ncolumns are the chirality and energy related to the Fermi level of each WP.\nUPtGa5 1.255124.360 61-37\u0018-30 \u0016A5+\u0016A7+\u0016\u00007+\u0016\u00008+\u0016\u00009+\u0016M6+\u0016M8+\u0016R3+\u0016R4+\u0016X5+\u0016X6+\u0016Z5+\n\u0016Z6+\u0016Z7+\u0016Z8\nNpRhGa5 1.261124.360 04-69\u0018-66 \u0016A5+\u0016A7+\u0016\u00006+\u0016M6+\u0016R3+\u0016R4+\u0016X6+\u0016Z5+\u0016Z7\nNpRhGa5 1.261124.360 41-167\u0018-164 \u0016A5+\u0016A8+\u0016\u00008+\u0016\u00009+\u0016M8+\u0016M9+\u0016R3+\u0016R4+\u0016X6+\u0016Z6+\u0016Z7\nFe4Si2Sn7O16 1.19712.63 11-163\u0018-160 \u0016A3\u0016A5+\u0016A4\u0016A6+\u0016\u00003\u0016\u00004+\u0016L2\u0016L3+\u0016M3\u0016M5+\u0016M4\u0016M6+\u0016V3\u0016V3+\u0016Y5\u0016Y6\nMnBr2 1.23912.63 21-11\u0018-8 \u0016A3\u0016A5+\u0016A4\u0016A6+\u0016\u00005\u0016\u00006+\u0016L2\u0016L3+\u0016M3\u0016M5+\u0016M4\u0016M6+\u0016V3\u0016V3+\u0016Y3\u0016Y4\nLi2MnO3 1.9712.63 02-119\u0018-116 \u0016A3\u0016A5+\u0016A4\u0016A6+\u0016\u00005\u0016\u00006+\u0016L2\u0016L3+\u0016M3\u0016M5+\u0016M4\u0016M6+\u0016V3\u0016V3+\u0016Y3\u0016Y4\nU2Ni2In 1.102128.408 04-155\u0018-140 \u0016A5\u0016A6+\u0016A7\u0016A8+\u0016\u00006+\u0016\u00007+\u0016M6\u0016M7+\u0016R2\u0016R7+\u0016R3\u0016R6+\u0016R4\u0016R9+\n\u0016R5\u0016R8+\u0016X3\u0016X4+\u0016Z5+\u0016Z6+\u0016Z7+\u0016Z8\nU2Ni2In 1.102128.408 22-155\u0018-140 \u0016A5\u0016A6+\u0016A7\u0016A8+\u0016\u00006+\u0016\u00007+\u0016M6\u0016M7+\u0016R2\u0016R7+\u0016R3\u0016R6+\u0016R4\u0016R9+\n\u0016R5\u0016R8+\u0016X3\u0016X4+\u0016Z5+\u0016Z6+\u0016Z7+\u0016Z8\nU2Ni2In 1.102128.408 42-155\u0018-140 \u0016A5\u0016A6+\u0016A7\u0016A8+\u0016\u00006+\u0016\u00007+\u0016M6\u0016M7+\u0016R2\u0016R7+\u0016R3\u0016R6+\u0016R4\u0016R9+\n\u0016R5\u0016R8+\u0016X3\u0016X4+\u0016Z5+\u0016Z6+\u0016Z7+\u0016Z8\nU2Ni2In 1.102128.408 62-171\u0018-164 \u0016A5\u0016A6+\u0016A7\u0016A8+\u0016\u00006+\u0016\u00007+\u0016M6\u0016M7+\u0016R2\u0016R7+\u0016R3\u0016R6+\u0016R4\u0016R9+\n\u0016R5\u0016R8+\u0016X3\u0016X4+\u0016Z5+\u0016Z6+\u0016Z7+\u0016Z8\nTbRh2Si2 1.187128.410 01-65\u0018-62 \u0016A5+\u0016A7+\u0016\u00007+\u0016M6+\u0016M7+\u0016R2\u0016R4+\u0016R3\u0016R5+\u0016X3+\u0016X4+\u0016Z6+\u0016Z8\nTbRh2Si2 1.187128.410 41-69\u0018-66 \u0016A5+\u0016A7+\u0016\u00009+\u0016M8+\u0016M9+\u0016R2\u0016R4+\u0016R3\u0016R5+\u0016X3+\u0016X4+\u0016Z6+\u0016Z8\nTbRh2Si2 1.187128.410 61-51\u0018-48 \u0016A5+\u0016A7+\u0016\u00007+\u0016M6+\u0016M7+\u0016R2\u0016R4+\u0016R3\u0016R5+\u0016X3+\u0016X4+\u0016Z6+\u0016Z8\nDyCo2Si2 1.21128.410 02-63\u0018-58 \u0016A6+\u0016A7+\u0016A8+\u0016\u00008+\u0016\u00009+\u0016M8+\u0016M9+\u0016R6\u0016R8+\u0016R7\u0016R9+\u0016X3+\u0016X4+\n\u0016Z5+\u0016Z7+\u0016Z8\nDyCo2Si2 1.21128.410 21-59\u0018-52 \u0016A5+\u0016A6+\u0016A7+\u0016A8+\u0016\u00007+\u0016\u00008+\u0016\u00009+\u0016M6+\u0016M7+\u0016M8+\u0016M9+\n\u0016R2\u0016R4+\u0016R3\u0016R5+\u0016R6\u0016R8+\u0016R7\u0016R9+\u0016X3+\u0016X4+\u0016Z5+\u0016Z6+\u0016Z7+\u0016Z8\nDyCo2Si2 1.21128.410 43-9\u0018-6 \u0016A6+\u0016A7+\u0016\u00006+\u0016\u00009+\u0016M7+\u0016M9+\u0016R6\u0016R8+\u0016R7\u0016R9+\u0016X3+\u0016X4+\u0016Z6+\u0016Z7\nCeMnAsO 0.186129.416 09-1\u00180 \u0016A7+\u0016\u00007+\u0016M6+\u0016R2\u0016R4+\u0016X2\u0016X4+\u0016Z6\nCeMnAsO 0.186129.416 36-9\u0018-8 \u0016A6+\u0016\u00006+\u0016M7+\u0016R2\u0016R4+\u0016X2\u0016X4+\u0016Z7\nCeMnAsO 0.186129.416 46-1\u00180 \u0016A7+\u0016\u00007+\u0016M7+\u0016R3\u0016R5+\u0016X3\u0016X5+\u0016Z7\nCaMnBi2 0.72129.416 04-7\u0018-6 \u0016A6+\u0016\u00006+\u0016M6+\u0016R3\u0016R5+\u0016X3\u0016X5+\u0016Z6100\nCaMnBi2 0.72129.416 14-7\u0018-6 \u0016A6+\u0016\u00006+\u0016M6+\u0016R3\u0016R5+\u0016X3\u0016X5+\u0016Z6\nCaMnBi2 0.72129.416 28-1\u00180 \u0016A7+\u0016\u00006+\u0016M6+\u0016R3\u0016R5+\u0016X3\u0016X5+\u0016Z7\nCaMnBi2 0.72129.416 35-1\u00180 \u0016A7+\u0016\u00006+\u0016M6+\u0016R3\u0016R5+\u0016X3\u0016X5+\u0016Z7\nCaMnBi2 0.72129.416 45-1\u00180 \u0016A7+\u0016\u00006+\u0016M6+\u0016R3\u0016R5+\u0016X3\u0016X5+\u0016Z7\nUPt2Si2 0.194129.419 06-19\u0018-18 \u0016A7+\u0016\u00006+\u0016M7+\u0016R2\u0016R4\nUPt2Si2 0.194129.419 25-39\u0018-38 \u0016A7+\u0016\u00007+\u0016M7+\u0016R3\u0016R5\nUPt2Si2 0.194129.419 49-13\u0018-12 \u0016A7+\u0016\u00007+\u0016M7+\u0016R3\u0016R5\nUPt2Si2 0.194129.419 68-31\u0018-28 \u0016A6+\u0016A7+\u0016\u00007+\u0016M6+\u0016M7+\u0016R3\u0016R5+\u0016X3\u0016X5+\u0016Z7\nNd2CuO4 2.6134.481 21-71\u0018-56 \u0016A6\u0016A7+\u0016\u00006+\u0016\u00007+\u0016\u00008+\u0016\u00009+\u0016M5+\u0016R3\u0016R4+\u0016X3+\u0016X4+\u0016Z5\nUNiGa5 1.254140.550 62-43\u0018-40 \u0016\u00008+\u0016\u00009+\u0016M6+\u0016M7+\u0016N2+\u0016P7+\u0016X5+\u0016X6\nNd2RhIn8 1.82140.550 41-97\u0018-90 \u0016\u00006+\u0016\u00007+\u0016\u00008+\u0016M6+\u0016M7+\u0016N2+\u0016P6+\u0016P7+\u0016X5+\u0016X6\nTbCo2Ga8 1.87140.550 21-63\u0018-60 \u0016\u00008+\u0016\u00009+\u0016M6+\u0016M7+\u0016N2+\u0016P6+\u0016X5+\u0016X6\nEr2Ru2O7 0.154141.554 01-115\u0018-112 \u0016Z3+\u0016Z4+\u0016L3+\u0016H2\u0016H4+\u0016H3+\u0016H5+\u0016Y3+\u0016Y4+\u0016T3+\u0016T4+\u0016\u00005+\u0016\u00006\nEr2Ru2O7 0.154141.554 21-103\u0018-100 \u0016Z3+\u0016Z4+\u0016L3+\u0016H2\u0016H4+\u0016H3+\u0016H5+\u0016Y3+\u0016Y4+\u0016T3+\u0016T4+\u0016\u00005+\u0016\u00006\nEr2Ru2O7 0.154141.554 41-87\u0018-84 \u0016Z3+\u0016Z4+\u0016L3+\u0016H2\u0016H4+\u0016H3+\u0016H5+\u0016Y3+\u0016Y4+\u0016T3+\u0016T4+\u0016\u00005+\u0016\u00006\nGd2Sn2O7 0.47141.555 23-99\u0018-96 \u0016X5\u0016X6+\u0016R3+\u0016R4+\u0016S3+\u0016S4+\u0016W2\u0016W4+\u0016W3+\u0016W5+\u0016T2+\u0016\u00005+\u0016\u00006\nGd2Sn2O7 0.47141.555 41-139\u0018-136 \u0016X5\u0016X6+\u0016R5+\u0016R6+\u0016S5+\u0016S6+\u0016W2\u0016W4+\u0016W3+\u0016W5+\u0016T2+\u0016\u00005+\u0016\u00006\nGd2Sn2O7 0.47141.555 61-139\u0018-136 \u0016X5\u0016X6+\u0016R5+\u0016R6+\u0016S5+\u0016S6+\u0016W2\u0016W4+\u0016W3+\u0016W5+\u0016T2+\u0016\u00005+\u0016\u00006\nTm2Mn2O7 0.151141.557 01-25\u0018-12 \u0016\u000010+\u0016\u000011+\u0016\u000012+\u0016\u00005+\u0016\u00006+\u0016\u00007+\u0016\u00008+\u0016\u00009+\u0016M3+\u0016M4+\u0016N2+\n\u0016N3+\u0016P5+\u0016P6+\u0016P7+\u0016P8+\u0016X2\nTb2Sn2O7 0.48141.557 21-119\u0018-116 \u0016\u000011+\u0016\u00006+\u0016\u00008+\u0016\u00009+\u0016M3+\u0016M4+\u0016N2+\u0016P5+\u0016P6+\u0016P7+\u0016X2\nTb2Sn2O7 0.48141.557 41-119\u0018-116 \u0016\u000011+\u0016\u00006+\u0016\u00008+\u0016\u00009+\u0016M3+\u0016M4+\u0016N2+\u0016P5+\u0016P6+\u0016P7+\u0016X2\nHo2Ru2O7 0.51141.557 02-19\u0018-16 \u0016\u00005+\u0016\u00006+\u0016\u00007+\u0016\u00008+\u0016M3+\u0016M4+\u0016N2+\u0016N3+\u0016P5+\u0016P7+\u0016X2\nLi2Mn(SO4)2 0.12214.75 01-121\u0018-118 \u0016A2+\u0016B2+\u0016C2+\u0016D3+\u0016D4+\u0016E3+\u0016E4+\u0016\u00005+\u0016\u00006+\u0016Y3+\u0016Y4+\u0016Z2\nLi2Mn(SO4)2 0.12214.75 12-101\u0018-98 \u0016A2+\u0016B2+\u0016C2+\u0016D5+\u0016D6+\u0016E3+\u0016E4+\u0016\u00005+\u0016\u00006+\u0016Y5+\u0016Y6+\u0016Z2\nLi2Mn(SO4)2 0.12214.75 21-121\u0018-118 \u0016A2+\u0016B2+\u0016C2+\u0016D3+\u0016D4+\u0016E3+\u0016E4+\u0016\u00005+\u0016\u00006+\u0016Y3+\u0016Y4+\u0016Z2\nLi2Mn(SO4)2 0.12214.75 31-121\u0018-118 \u0016A2+\u0016B2+\u0016C2+\u0016D3+\u0016D4+\u0016E3+\u0016E4+\u0016\u00005+\u0016\u00006+\u0016Y3+\u0016Y4+\u0016Z2\nLi2Co(SO4)2 0.12114.79 02-109\u0018-106 \u0016U2\u0016U3+\u0016Z2\u0016Z3+\u0016V2\u0016V3+\u0016T3\u0016T3+\u0016R2\u0016R2+\u0016X3+\u0016Y2\u0016Y3+\u0016\u00003\nLi2Co(SO4)2 0.12114.79 11-125\u0018-122 \u0016U2\u0016U3+\u0016Z2\u0016Z3+\u0016V2\u0016V3+\u0016T3\u0016T3+\u0016R2\u0016R2+\u0016X2+\u0016Y2\u0016Y3+\u0016\u00002\nLi2Co(SO4)2 0.12114.79 31-125\u0018-122 \u0016U2\u0016U3+\u0016Z2\u0016Z3+\u0016V2\u0016V3+\u0016T3\u0016T3+\u0016R2\u0016R2+\u0016X2+\u0016Y2\u0016Y3+\u0016\u00002\nLi2Co(SO4)2 0.12114.79 41-125\u0018-122 \u0016U2\u0016U3+\u0016Z2\u0016Z3+\u0016V2\u0016V3+\u0016T3\u0016T3+\u0016R2\u0016R2+\u0016X2+\u0016Y2\u0016Y3+\u0016\u00002\nNiN2O6 0.78148.17 01-87\u0018-86 \u0016F2+\u0016\u00005+\u0016\u00006+\u0016L3+\u0016T8+\u0016T9\nNiN2O6 0.78148.17 21-153\u0018-151 \u0016F2+\u0016F3+\u0016\u00007+\u0016\u00008+\u0016\u00009+\u0016L2+\u0016T5+\u0016T6+\u0016T7\nNiN2O6 0.78148.17 31-153\u0018-151 \u0016F2+\u0016F3+\u0016\u00007+\u0016\u00008+\u0016\u00009+\u0016L2+\u0016T5+\u0016T6+\u0016T7\nNiCO3 0.11315.85 11-23\u0018-20 \u0016A2+\u0016\u00005+\u0016\u00006+\u0016L3+\u0016M2+\u0016V2+\u0016V3+\u0016Y5+\u0016Y6\nCoCO3 0.11415.85 21-27\u0018-22 \u0016A2+\u0016\u00003+\u0016\u00004+\u0016\u00005+\u0016\u00006+\u0016L2+\u0016M2+\u0016V2+\u0016V3+\u0016Y3+\u0016Y4\nCoCO3 0.11415.85 31-11\u0018-8 \u0016A2+\u0016\u00005+\u0016\u00006+\u0016L3+\u0016M2+\u0016V2+\u0016V3+\u0016Y5+\u0016Y6\nCoCO3 0.11415.85 41-11\u0018-8 \u0016A2+\u0016\u00005+\u0016\u00006+\u0016L3+\u0016M2+\u0016V2+\u0016V3+\u0016Y5+\u0016Y6\nMnCO3 0.11515.85 01-19\u0018-16 \u0016A2+\u0016\u00003+\u0016\u00004+\u0016L2+\u0016M2+\u0016V2+\u0016V3+\u0016Y3+\u0016Y4\nMnCO3 0.11515.85 11-13\u0018-10 \u0016A2+\u0016\u00005+\u0016\u00006+\u0016L3+\u0016M2+\u0016V2+\u0016V3+\u0016Y5+\u0016Y6\nMnCO3 0.11515.85 41-9\u0018-6 \u0016A2+\u0016\u00005+\u0016\u00006+\u0016L3+\u0016M2+\u0016V2+\u0016V3+\u0016Y5+\u0016Y6\nFeBO3 0.11215.89 11-19\u0018-16 \u0016Z2\u0016Z3+\u0016U3+\u0016T2+\u0016R2\u0016R3+\u0016X2+\u0016X3+\u0016Y2+\u0016Y3+\u0016V3+\u0016\u00003\nFeBO3 0.11215.89 21-19\u0018-16 \u0016Z2\u0016Z3+\u0016U3+\u0016T2+\u0016R2\u0016R3+\u0016X2+\u0016X3+\u0016Y2+\u0016Y3+\u0016V3+\u0016\u00003\nFeBO3 0.11215.89 31-3\u00180 \u0016Z2\u0016Z3+\u0016U2+\u0016T3+\u0016R2\u0016R3+\u0016X2+\u0016X3+\u0016Y2+\u0016Y3+\u0016V2+\u0016\u00002\nFeBO3 0.11215.89 43-3\u00180 \u0016Z2\u0016Z3+\u0016U2+\u0016T3+\u0016R2\u0016R3+\u0016X2+\u0016X3+\u0016Y2+\u0016Y3+\u0016V2+\u0016\u00002\nPr3Ru4Al12 0.17415.89 01-145\u0018-140 \u0016Z2\u0016Z3+\u0016U2+\u0016U3+\u0016T2+\u0016T3+\u0016R2\u0016R3+\u0016X2+\u0016Y2+\u0016V2+\u0016\u00002+\u0016\u00003\nFe2O3-alpha 0.6515.89 01-23\u0018-20 \u0016Z2\u0016Z3+\u0016U3+\u0016T2+\u0016R2\u0016R3+\u0016X2+\u0016X3+\u0016Y2+\u0016Y3+\u0016V2+\u0016\u00002\nYb2O2Se 1.21415.90 21-103\u0018-100 \u0016A2+\u0016\u00005\u0016\u00006+\u0016L2\u0016L3+\u0016M2+\u0016V2\u0016V2+\u0016Y3\u0016Y4\nCoBr2 1.24515.90 01-11\u0018-8 \u0016A2+\u0016\u00005\u0016\u00006+\u0016L2\u0016L3+\u0016M2+\u0016V2\u0016V2+\u0016Y3\u0016Y4\nCoBr2 1.24515.90 11-7\u0018-4 \u0016A2+\u0016\u00005\u0016\u00006+\u0016L2\u0016L3+\u0016M2+\u0016V2\u0016V2+\u0016Y3\u0016Y4\nCoBr2 1.24515.90 21-7\u0018-4 \u0016A2+\u0016\u00005\u0016\u00006+\u0016L2\u0016L3+\u0016M2+\u0016V2\u0016V2+\u0016Y3\u0016Y4\nCoBr2 1.24515.90 31-27\u0018-24 \u0016A2+\u0016\u00003\u0016\u00004+\u0016L2\u0016L3+\u0016M2+\u0016V3\u0016V3+\u0016Y5\u0016Y6\nCoCl2 1.24615.90 01-33\u0018-30 \u0016A2+\u0016\u00003\u0016\u00004+\u0016L2\u0016L3+\u0016M2+\u0016V3\u0016V3+\u0016Y5\u0016Y6101\nCoCl2 1.24615.90 11-11\u0018-8 \u0016A2+\u0016\u00005\u0016\u00006+\u0016L2\u0016L3+\u0016M2+\u0016V2\u0016V2+\u0016Y3\u0016Y4\nCoCl2 1.24615.90 31-7\u0018-4 \u0016A2+\u0016\u00005\u0016\u00006+\u0016L2\u0016L3+\u0016M2+\u0016V2\u0016V2+\u0016Y3\u0016Y4\nNiCl2 1.24715.90 01-35\u0018-32 \u0016A2+\u0016\u00003\u0016\u00004+\u0016L2\u0016L3+\u0016M2+\u0016V3\u0016V3+\u0016Y5\u0016Y6\nNiCl2 1.24715.90 11-35\u0018-32 \u0016A2+\u0016\u00003\u0016\u00004+\u0016L2\u0016L3+\u0016M2+\u0016V3\u0016V3+\u0016Y5\u0016Y6\nNiCl2 1.24715.90 21-7\u0018-4 \u0016A2+\u0016\u00005\u0016\u00006+\u0016L2\u0016L3+\u0016M2+\u0016V2\u0016V2+\u0016Y3\u0016Y4\nNiBr2 1.24815.90 01-35\u0018-32 \u0016A2+\u0016\u00003\u0016\u00004+\u0016L2\u0016L3+\u0016M2+\u0016V3\u0016V3+\u0016Y5\u0016Y6\nNiBr2 1.24815.90 12-7\u0018-4 \u0016A2+\u0016\u00005\u0016\u00006+\u0016L2\u0016L3+\u0016M2+\u0016V2\u0016V2+\u0016Y3\u0016Y4\nNiBr2 1.24815.90 21-7\u0018-4 \u0016A2+\u0016\u00005\u0016\u00006+\u0016L2\u0016L3+\u0016M2+\u0016V2\u0016V2+\u0016Y3\u0016Y4\nNiBr2 1.24815.90 31-7\u0018-4 \u0016A2+\u0016\u00005\u0016\u00006+\u0016L2\u0016L3+\u0016M2+\u0016V2\u0016V2+\u0016Y3\u0016Y4\nAg2NiO2 1.4915.90 11-87\u0018-84 \u0016A2+\u0016\u00005\u0016\u00006+\u0016L2\u0016L3+\u0016M2+\u0016V3\u0016V3+\u0016Y3\u0016Y4\nAg2NiO2 1.4915.90 21-29\u0018-26 \u0016A2+\u0016\u00005\u0016\u00006+\u0016L2\u0016L3+\u0016M2+\u0016V2\u0016V2+\u0016Y3\u0016Y4\nSr2CoOsO6 1.7215.90 05-69\u0018-66 \u0016A2+\u0016\u00003\u0016\u00004+\u0016L2\u0016L3+\u0016M2+\u0016V2\u0016V2+\u0016Y5\u0016Y6\nSr2CoOsO6 1.7215.90 15-69\u0018-66 \u0016A2+\u0016\u00003\u0016\u00004+\u0016L2\u0016L3+\u0016M2+\u0016V2\u0016V2+\u0016Y5\u0016Y6\nSr2CoOsO6 1.7215.90 24-97\u0018-94 \u0016A2+\u0016\u00005\u0016\u00006+\u0016L2\u0016L3+\u0016M2+\u0016V3\u0016V3+\u0016Y3\u0016Y4\nSr2CoOsO6 1.7215.90 36-65\u0018-62 \u0016A2+\u0016\u00003\u0016\u00004+\u0016L2\u0016L3+\u0016M2+\u0016V3\u0016V3+\u0016Y5\u0016Y6\nSr2CoOsO6 1.7215.90 45-65\u0018-62 \u0016A2+\u0016\u00003\u0016\u00004+\u0016L2\u0016L3+\u0016M2+\u0016V2\u0016V2+\u0016Y5\u0016Y6\nVCl2 1.237159.64 12-115\u0018-114 \u0016A4+\u0016A5+\u0016\u00004\u0016\u00005+\u0016HA6+\u0016KA6+\u0016L3+\u0016L4+\u0016M3\u0016M4\nVCl2 1.237159.64 22-115\u0018-114 \u0016A4+\u0016A5+\u0016\u00004\u0016\u00005+\u0016HA6+\u0016KA6+\u0016L3+\u0016L4+\u0016M3\u0016M4\nVCl2 1.237159.64 32-115\u0018-114 \u0016A4+\u0016A5+\u0016\u00004\u0016\u00005+\u0016HA6+\u0016KA6+\u0016L3+\u0016L4+\u0016M3\u0016M4\nVCl2 1.237159.64 42-115\u0018-114 \u0016A4+\u0016A5+\u0016\u00004\u0016\u00005+\u0016HA6+\u0016KA6+\u0016L3+\u0016L4+\u0016M3\u0016M4\nSrRu2O6 1.186162.78 03-31\u0018-24 \u0016A4\u0016A6+\u0016A5\u0016A7+\u0016A8\u0016A9+\u0016\u00004\u0016\u00005+\u0016\u00008+\u0016H4\u0016H5+\u0016H6+\u0016K4\u0016K5+\u0016K6+\n\u0016L3\u0016L5+\u0016L4\u0016L6+\u0016M3\u0016M4+\u0016M5\u0016M6\nSrRu2O6 1.186162.78 13-31\u0018-24 \u0016A4\u0016A6+\u0016A5\u0016A7+\u0016A8\u0016A9+\u0016\u00004\u0016\u00005+\u0016\u00008+\u0016H4\u0016H5+\u0016H6+\u0016K4\u0016K5+\u0016K6+\n\u0016L3\u0016L5+\u0016L4\u0016L6+\u0016M3\u0016M4+\u0016M5\u0016M6\nSrRu2O6 1.186162.78 23-31\u0018-24 \u0016A4\u0016A6+\u0016A5\u0016A7+\u0016A8\u0016A9+\u0016\u00004\u0016\u00005+\u0016\u00008+\u0016H4\u0016H5+\u0016H6+\u0016K4\u0016K5+\u0016K6+\n\u0016L3\u0016L5+\u0016L4\u0016L6+\u0016M3\u0016M4+\u0016M5\u0016M6\nSrRu2O6 1.186162.78 33-31\u0018-24 \u0016A4\u0016A6+\u0016A5\u0016A7+\u0016A8\u0016A9+\u0016\u00004\u0016\u00005+\u0016\u00008+\u0016H4\u0016H5+\u0016H6+\u0016K4\u0016K5+\u0016K6+\n\u0016L3\u0016L5+\u0016L4\u0016L6+\u0016M3\u0016M4+\u0016M5\u0016M6\nSrRu2O6 1.186162.78 43-31\u0018-20 \u0016A4\u0016A6+\u0016A5\u0016A7+\u0016A8\u0016A9+\u0016\u00004\u0016\u00005+\u0016\u00008+\u0016\u00009+\u0016H4\u0016H5+\u0016H6+\u0016K4\u0016K5+\n\u0016K6+\u0016L3\u0016L5+\u0016L4\u0016L6+\u0016M3\u0016M4+\u0016M5\u0016M6\nFeBr2 1.242165.96 05-3\u0018-2 \u0016A4+\u0016\u00004\u0016\u00005+\u0016H6+\u0016K6+\u0016L2+\u0016M3\u0016M4\nFeBr2 1.242165.96 15-3\u0018-2 \u0016A4+\u0016\u00004\u0016\u00005+\u0016H6+\u0016K6+\u0016L2+\u0016M3\u0016M4\nFeBr2 1.242165.96 27-3\u0018-2 \u0016A4+\u0016\u00004\u0016\u00005+\u0016H6+\u0016K6+\u0016L2+\u0016M3\u0016M4\nFeBr2 1.242165.96 37-3\u0018-2 \u0016A4+\u0016\u00004\u0016\u00005+\u0016H6+\u0016K6+\u0016L2+\u0016M3\u0016M4\nFeBr2 1.242165.96 47-11\u0018-4 \u0016A4+\u0016A6+\u0016\u00004\u0016\u00005+\u0016\u00006\u0016\u00007+\u0016\u00009+\u0016H4\u0016H5+\u0016H6+\u0016K4\u0016K5+\u0016K6+\u0016L2+\n\u0016M3\u0016M4+\u0016M5\u0016M6\nMn3Ir 0.108166.101 31-15\u0018-11 \u0016F2+\u0016F3+\u0016\u00004+\u0016\u00005+\u0016\u00006+\u0016L2+\u0016L3+\u0016T4+\u0016T5+\u0016T6\nMn3Pt 0.109166.101 11-6\u0018-4 \u0016F2+\u0016F3+\u0016\u00004+\u0016\u00005+\u0016\u00006+\u0016L3+\u0016T4+\u0016T8+\u0016T9\nNd3Sb3Mg2O14 0.167166.101 05-4\u0018-3 \u0016F3+\u0016\u00008+\u0016\u00009+\u0016L3+\u0016T8+\u0016T9\nNd3Sb3Mg2O14 0.167166.101 24-1\u00180 \u0016F3+\u0016\u00007+\u0016\u00008+\u0016L3+\u0016T7+\u0016T8\nTbCo2 0.228166.101 02-57\u0018-56 \u0016F2+\u0016\u00005+\u0016\u00006+\u0016L2+\u0016T5+\u0016T6\nTbCo2 0.228166.101 21-36\u0018-35 \u0016F2+\u0016\u00005+\u0016\u00006+\u0016L2+\u0016T5+\u0016T6\nTb2Ti2O7 0.77166.101 67-14\u0018-10 \u0016F2+\u0016F3+\u0016\u00007+\u0016\u00008+\u0016\u00009+\u0016L2+\u0016L3+\u0016T4+\u0016T5+\u0016T8+\u0016T9\nGd2Ti2O7 1.56166.102 03-121\u0018-118 \u0016F3\u0016F4+\u0016\u00009+\u0016L3\u0016L5+\u0016L4\u0016L6+\u0016T8\u0016T9\nGd2Ti2O7 1.56166.102 41-197\u0018-194 \u0016F3\u0016F4+\u0016\u00009+\u0016L3\u0016L5+\u0016L4\u0016L6+\u0016T8\u0016T9\nGd2Ti2O7 1.56166.102 61-197\u0018-194 \u0016F3\u0016F4+\u0016\u00009+\u0016L3\u0016L5+\u0016L4\u0016L6+\u0016T8\u0016T9\nMn3Cu0.5Ge0.5N 0.74166.97 02-12\u0018-11 \u0016F5+\u0016F6+\u0016\u00008+\u0016L5+\u0016L6+\u0016T8\nFeCO3 0.116167.103 11-13\u0018-10 \u0016F3+\u0016F4+\u0016\u00006+\u0016\u00007+\u0016\u00009+\u0016L2+\u0016T4+\u0016T5\nFeCO3 0.116167.103 31-25\u0018-22 \u0016F5+\u0016F6+\u0016\u00004+\u0016\u00005+\u0016\u00008+\u0016L2+\u0016T4+\u0016T5\nAgFe3(SO4)2(OD)6 1.129167.108 04-55\u0018-50 \u0016F3\u0016F4+\u0016\u00006\u0016\u00007+\u0016\u00008+\u0016\u00009+\u0016L2+\u0016T4+\u0016T5+\u0016T6\nAgFe3(SO4)2(OD)6 1.129167.108 14-65\u0018-62 \u0016F5\u0016F6+\u0016\u00004\u0016\u00005+\u0016\u00008+\u0016L2+\u0016T4+\u0016T5\nAgFe3(SO4)2(OD)6 1.129167.108 22-143\u0018-140 \u0016F5\u0016F6+\u0016\u00008+\u0016L2+\u0016T5+\u0016T6\nAgFe3(SO4)2(OD)6 1.129167.108 31-241\u0018-238 \u0016F5\u0016F6+\u0016\u00008+\u0016L2+\u0016T5+\u0016T6\nAgFe3(SO4)2(OD)6 1.129167.108 41-241\u0018-238 \u0016F5\u0016F6+\u0016\u00008+\u0016L2+\u0016T5+\u0016T6102\nMn3GaC 1.153167.108 01-29\u0018-24 \u0016F5\u0016F6+\u0016\u00004\u0016\u00005+\u0016\u00008+\u0016\u00009+\u0016L2+\u0016T4+\u0016T6\nMn3GaC 1.153167.108 11-3\u00180 \u0016F5\u0016F6+\u0016\u00004\u0016\u00005+\u0016\u00008+\u0016L2+\u0016T4+\u0016T6\nMn3GaC 1.153167.108 21-47\u0018-44 \u0016F5\u0016F6+\u0016\u00004\u0016\u00005+\u0016\u00008+\u0016L2+\u0016T4+\u0016T6\nMn3GaC 1.153167.108 32-23\u0018-20 \u0016F5\u0016F6+\u0016\u00004\u0016\u00005+\u0016\u00008+\u0016L2+\u0016T4+\u0016T6\nMn3GaC 1.153167.108 41-29\u0018-26 \u0016F5\u0016F6+\u0016\u00004\u0016\u00005+\u0016\u00008+\u0016L2+\u0016T4+\u0016T6\nFeCl2 1.241167.108 02-9\u0018-6 \u0016F3\u0016F4+\u0016\u00006\u0016\u00007+\u0016\u00009+\u0016L2+\u0016T4+\u0016T6\nFeCl2 1.241167.108 11-9\u0018-6 \u0016F3\u0016F4+\u0016\u00006\u0016\u00007+\u0016\u00009+\u0016L2+\u0016T4+\u0016T6\nYMnO3 0.44173.131 41-199\u0018-196 \u0016A5\u0016A6+\u0016\u00005\u0016\u00006+\u0016H4\u0016H4+\u0016HA4\u0016HA4+\u0016K4+\u0016KA4+\u0016L2\u0016L2+\u0016M2\nHoMnO3 0.32185.197 68-15\u0018-14 \u0016A7+\u0016\u00007+\u0016H6+\u0016K6\nHoMnO3 0.33185.197 48-21\u0018-20 \u0016A7+\u0016\u00007+\u0016H6+\u0016K6\nYMnO3 0.6185.197 31-95\u0018-92 \u0016A7+\u0016\u00007+\u0016H4+\u0016H5+\u0016H6+\u0016K4+\u0016K5+\u0016K6+\u0016L5+\u0016M5\nLuFeO3 0.117185.201 21-181\u0018-178 \u0016A10\u0016A9+\u0016A11\u0016A12+\u0016\u000010+\u0016\u000011+\u0016\u000012+\u0016\u00009+\u0016H4\u0016H4+\u0016K4+\u0016L3\u0016L4+\n\u0016M3+\u0016M4\nLuFeO3 0.117185.201 31-181\u0018-178 \u0016A10\u0016A9+\u0016A11\u0016A12+\u0016\u000010+\u0016\u000011+\u0016\u000012+\u0016\u00009+\u0016H4\u0016H4+\u0016K4+\u0016L3\u0016L4+\n\u0016M3+\u0016M4\nLuFeO3 0.117185.201 41-181\u0018-178 \u0016A10\u0016A9+\u0016A11\u0016A12+\u0016\u000010+\u0016\u000011+\u0016\u000012+\u0016\u00009+\u0016H4\u0016H4+\u0016K4+\u0016L3\u0016L4+\n\u0016M3+\u0016M4\nScMnO3 0.7185.201 01-199\u0018-196 \u0016A10\u0016A9+\u0016A11\u0016A12+\u0016\u000010+\u0016\u000011+\u0016\u000012+\u0016\u00009+\u0016H4\u0016H4+\u0016K4+\u0016L3\u0016L4+\n\u0016M3+\u0016M4\nScMn6Ge6 1.110192.252 03-167\u0018-164 \u0016A11+\u0016A8+\u0016\u000011+\u0016\u00009+\u0016H4\u0016H5+\u0016H9+\u0016K7+\u0016K8+\u0016L3+\u0016L4+\u0016M5+\u0016M6\nScMn6Ge6 1.110192.252 43-189\u0018-186 \u0016A12+\u0016A8+\u0016\u000011+\u0016\u00008+\u0016H4\u0016H5+\u0016H6\u0016H7+\u0016K7+\u0016L3+\u0016L4+\u0016M5+\u0016M6\nScMn6Ge6 1.225192.252 06-31\u0018-22 \u0016A10+\u0016A11+\u0016A7+\u0016A8+\u0016\u000012+\u0016\u00007+\u0016\u00008+\u0016\u00009+\u0016H4\u0016H5+\u0016H6\u0016H7+\n\u0016H8+\u0016H9+\u0016K7+\u0016K8+\u0016K9+\u0016L3+\u0016L4+\u0016M5+\u0016M6\nScMn6Ge6 1.225192.252 13-281\u0018-278 \u0016A11+\u0016A9+\u0016\u000012+\u0016\u00009+\u0016H4\u0016H5+\u0016H6\u0016H7+\u0016K7+\u0016L3+\u0016L4+\u0016M5+\u0016M6\nScMn6Ge6 1.225192.252 24-163\u0018-152 \u0016A10+\u0016A11+\u0016A12+\u0016A7+\u0016A8+\u0016\u000010+\u0016\u000011+\u0016\u00007+\u0016\u00008+\u0016\u00009+\u0016H4\u0016H5+\n\u0016H6\u0016H7+\u0016H8+\u0016H9+\u0016K7+\u0016K8+\u0016K9+\u0016L3+\u0016L4+\u0016M5+\u0016M6\nScMn6Ge6 1.225192.252 32-281\u0018-278 \u0016A11+\u0016A9+\u0016\u000012+\u0016\u00009+\u0016H4\u0016H5+\u0016H6\u0016H7+\u0016K7+\u0016L3+\u0016L4+\u0016M5+\u0016M6\nScMn6Ge6 1.225192.252 42-281\u0018-278 \u0016A11+\u0016A9+\u0016\u000012+\u0016\u00009+\u0016H4\u0016H5+\u0016H6\u0016H7+\u0016K7+\u0016L3+\u0016L4+\u0016M5+\u0016M6\nBa5Co5ClO13 0.118194.268 128-13\u0018-10 \u0016A5+\u0016\u00008+\u0016\u00009+\u0016H4\u0016H5+\u0016H6+\u0016K4+\u0016K5+\u0016K6+\u0016L2+\u0016M3+\u0016M4+\u0016M5+\u0016M6\nBa5Co5ClO13 0.118194.268 221-11\u0018-10 \u0016A4+\u0016\u00006+\u0016\u00007+\u0016H6+\u0016K6+\u0016L2+\u0016M5+\u0016M6\nBa5Co5ClO13 0.118194.268 329-3\u0018-2 \u0016A4+\u0016\u00004+\u0016\u00005+\u0016H6+\u0016K6+\u0016L2+\u0016M3+\u0016M4\nBa5Co5ClO13 0.118194.268 427-5\u0018-4 \u0016A4+\u0016\u00006+\u0016\u00007+\u0016H6+\u0016K6+\u0016L2+\u0016M5+\u0016M6\nNa3Co(CO3)2Cl 0.70203.26 12-83\u0018-80 \u0016\u00008+\u0016L6+\u0016L8+\u0016L9+\u0016W2+\u0016W3+\u0016W4+\u0016W5+\u0016X3+\u0016X4\nNa3Co(CO3)2Cl 0.70203.26 21-15\u0018-8 \u0016\u00005+\u0016\u00007+\u0016\u00009+\u0016L4+\u0016L5+\u0016L6+\u0016L7+\u0016L8+\u0016L9+\u0016W2+\u0016W3+\u0016W4+\n\u0016W5+\u0016X3+\u0016X4\nNa3Co(CO3)2Cl 0.70203.26 34-11\u0018-8 \u0016\u00007+\u0016\u00009+\u0016L4+\u0016L5+\u0016L7+\u0016L9+\u0016W2+\u0016W3+\u0016W4+\u0016W5+\u0016X3+\u0016X4\nNiS2 0.150205.33 01-63\u0018-52 \u0016\u000010+\u0016\u00005+\u0016\u00009+\u0016M3+\u0016M4+\u0016R10+\u0016R11+\u0016R4+\u0016R8+\u0016R9+\u0016X3+\u0016X4\nNiS2 0.150205.33 11-63\u0018-52 \u0016\u000010+\u0016\u00005+\u0016\u00009+\u0016M3+\u0016M4+\u0016R10+\u0016R11+\u0016R4+\u0016R8+\u0016R9+\u0016X3+\u0016X4\nGd2Ti2O7 3.16216.77 06-115\u0018-112 \u0016\u00006+\u0016\u00007+\u0016L6\u0016L6+\u0016W5\u0016W6+\u0016W7\u0016W8+\u0016X6+\u0016X7\nNdZn 3.8222.103 21-59\u0018-28 \u0016\u000010+\u0016\u000011+\u0016\u00006+\u0016\u00008+\u0016\u00009+\u0016M5+\u0016R5+\u0016R6+\u0016R7+\u0016X5+\u0016X6+\u0016X7+\u0016X8\nUSb 3.12224.113 41-11\u00180\u0016\u000010+\u0016\u00005+\u0016\u00008+\u0016\u00009+\u0016M3+\u0016M4+\u0016R10+\u0016R7+\u0016R8+\u0016R9+\u0016X3+\u0016X4\nCd2Os2O7 0.2227.131 13-23\u0018-12 \u0016\u000010+\u0016\u00005+\u0016\u00006+\u0016\u00007+\u0016\u00008+\u0016\u00009+\u0016L4+\u0016L5+\u0016L6+\u0016W2+\u0016W3\u0016W5+\n\u0016W4+\u0016X3+\u0016X4\nCd2Os2O7 0.2227.131 21-23\u0018-12 \u0016\u000010+\u0016\u00005+\u0016\u00006+\u0016\u00007+\u0016\u00008+\u0016\u00009+\u0016L4+\u0016L5+\u0016L6+\u0016W2+\u0016W3\u0016W5+\n\u0016W4+\u0016X3+\u0016X4\nNpSe 3.10228.139 02-35\u0018-8 \u0016\u000011+\u0016\u00006+\u0016\u00008+\u0016\u00009+\u0016L4+\u0016L5+\u0016L6+\u0016W3\u0016W4+\u0016W5\u0016W6+\u0016W7+\u0016X5\nNpSe 3.10228.139 21-67\u0018-52 \u0016\u000010+\u0016\u000011+\u0016\u00008+\u0016\u00009+\u0016L4+\u0016L5+\u0016L6+\u0016W3\u0016W4+\u0016W5\u0016W6+\u0016W7+\u0016X5\nNpTe 3.11228.139 01-115\u0018-108 \u0016\u000011+\u0016L4+\u0016L6+\u0016W3\u0016W4+\u0016W5\u0016W6+\u0016W7+\u0016X5\nNpTe 3.11228.139 21-31\u0018-12 \u0016\u000011+\u0016\u00006+\u0016\u00008+\u0016\u00009+\u0016L4+\u0016L5+\u0016L6+\u0016W3\u0016W4+\u0016W5\u0016W6+\u0016W7+\u0016X5\nNpTe 3.11228.139 61-67\u0018-60 \u0016\u000011+\u0016\u00008+\u0016\u00009+\u0016L4+\u0016L5+\u0016L6+\u0016W5\u0016W6+\u0016W7+\u0016X5\nNpS 3.9228.139 01-35\u0018-8 \u0016\u000011+\u0016\u00006+\u0016\u00008+\u0016\u00009+\u0016L4+\u0016L5+\u0016L6+\u0016W3\u0016W4+\u0016W5\u0016W6+\u0016W7+\u0016X5\nFe2O3-alpha 0.66 2.4 01-53\u0018-50 \u0016\u00003+\u0016R2+\u0016T2+\u0016U2+\u0016U3+\u0016V2+\u0016V3+\u0016X2+\u0016Y2+\u0016Y3+\u0016Z2+\u0016Z3\nNiS2 1.167 2.7 31-101\u0018-98 \u0016\u00002\u0016\u00002+\u0016R2\u0016R3+\u0016T2\u0016T3+\u0016U2\u0016U3+\u0016V3\u0016V3+\u0016X2\u0016X2+\u0016Y2\u0016Y2+\u0016Z2\u0016Z3103\nNiS2 1.167 2.7 41-109\u0018-106 \u0016\u00002\u0016\u00002+\u0016R2\u0016R3+\u0016T2\u0016T3+\u0016U2\u0016U3+\u0016V3\u0016V3+\u0016X2\u0016X2+\u0016Y2\u0016Y2+\u0016Z2\u0016Z3\nFePSe3 1.210 2.7 11-75\u0018-72 \u0016\u00002\u0016\u00002+\u0016R2\u0016R3+\u0016T2\u0016T3+\u0016U2\u0016U3+\u0016V3\u0016V3+\u0016X3\u0016X3+\u0016Y3\u0016Y3+\u0016Z2\u0016Z3\nCrCl3 1.244 2.7 03-15\u0018-12 \u0016\u00002\u0016\u00002+\u0016R2\u0016R3+\u0016T2\u0016T3+\u0016U2\u0016U3+\u0016V3\u0016V3+\u0016X3\u0016X3+\u0016Y3\u0016Y3+\u0016Z2\u0016Z3\nCrCl3 1.244 2.7 12-87\u0018-84 \u0016\u00002\u0016\u00002+\u0016R2\u0016R3+\u0016T2\u0016T3+\u0016U2\u0016U3+\u0016V3\u0016V3+\u0016X3\u0016X3+\u0016Y3\u0016Y3+\u0016Z2\u0016Z3\nCrCl3 1.244 2.7 22-87\u0018-84 \u0016\u00002\u0016\u00002+\u0016R2\u0016R3+\u0016T2\u0016T3+\u0016U2\u0016U3+\u0016V3\u0016V3+\u0016X3\u0016X3+\u0016Y3\u0016Y3+\u0016Z2\u0016Z3\nCrCl3 1.244 2.7 33-87\u0018-84 \u0016\u00002\u0016\u00002+\u0016R2\u0016R3+\u0016T2\u0016T3+\u0016U2\u0016U3+\u0016V3\u0016V3+\u0016X3\u0016X3+\u0016Y3\u0016Y3+\u0016Z2\u0016Z3\nCrCl3 1.244 2.7 44-43\u0018-40 \u0016\u00003\u0016\u00003+\u0016R2\u0016R3+\u0016T2\u0016T3+\u0016U2\u0016U3+\u0016V2\u0016V2+\u0016X2\u0016X2+\u0016Y2\u0016Y2+\u0016Z2\u0016Z3\nBaNi2V2O8 1.256 2.7 02-39\u0018-36 \u0016\u00002\u0016\u00002+\u0016R2\u0016R3+\u0016T2\u0016T3+\u0016U2\u0016U3+\u0016V3\u0016V3+\u0016X3\u0016X3+\u0016Y3\u0016Y3+\u0016Z2\u0016Z3\nBaNi2V2O8 1.256 2.7 14-39\u0018-36 \u0016\u00002\u0016\u00002+\u0016R2\u0016R3+\u0016T2\u0016T3+\u0016U2\u0016U3+\u0016V3\u0016V3+\u0016X3\u0016X3+\u0016Y3\u0016Y3+\u0016Z2\u0016Z3\nBaNi2V2O8 1.256 2.7 25-63\u0018-60 \u0016\u00002\u0016\u00002+\u0016R2\u0016R3+\u0016T2\u0016T3+\u0016U2\u0016U3+\u0016V3\u0016V3+\u0016X3\u0016X3+\u0016Y3\u0016Y3+\u0016Z2\u0016Z3\nBaNi2V2O8 1.256 2.7 32-63\u0018-60 \u0016\u00002\u0016\u00002+\u0016R2\u0016R3+\u0016T2\u0016T3+\u0016U2\u0016U3+\u0016V3\u0016V3+\u0016X3\u0016X3+\u0016Y3\u0016Y3+\u0016Z2\u0016Z3\nBaNi2V2O8 1.256 2.7 42-15\u0018-12 \u0016\u00002\u0016\u00002+\u0016R2\u0016R3+\u0016T2\u0016T3+\u0016U2\u0016U3+\u0016V3\u0016V3+\u0016X3\u0016X3+\u0016Y3\u0016Y3+\u0016Z2\u0016Z3\nHo2RhIn8 1.13949.273 01-97\u0018-94 \u0016\u00005+\u0016R3+\u0016R4+\u0016S6+\u0016T3+\u0016T4+\u0016U3+\u0016U4+\u0016X5+\u0016Y5+\u0016Z3+\u0016Z4\nEr2CoGa8 1.22251.298 01-109\u0018-106 \u0016\u00006+\u0016R3+\u0016R4+\u0016S3+\u0016S4+\u0016T5+\u0016U3+\u0016U4+\u0016X3+\u0016X4+\u0016Y5+\u0016Z5\nEr2CoGa8 1.22251.298 21-109\u0018-106 \u0016\u00006+\u0016R3+\u0016R4+\u0016S3+\u0016S4+\u0016T5+\u0016U3+\u0016U4+\u0016X3+\u0016X4+\u0016Y5+\u0016Z5\nPrAg 1.15053.334 01-7\u0018-4 \u0016\u00006+\u0016R6\u0016R7+\u0016R8\u0016R9+\u0016S3+\u0016S4+\u0016T3+\u0016T4+\u0016U2\u0016U3+\u0016U4\u0016U5+\u0016X3+\n\u0016X4+\u0016Y6+\u0016Z3+\u0016Z4\nYMn3Al4O12 1.15858.404 32-151\u0018-148 \u0016\u00005+\u0016R3+\u0016R4+\u0016S6+\u0016T6\u0016T8+\u0016T7\u0016T9+\u0016U6\u0016U8+\u0016U7\u0016U9+\u0016X3+\u0016X4+\n\u0016Y3+\u0016Y4+\u0016Z3+\u0016Z4\nYMn3Al4O12 1.15858.404 42-151\u0018-148 \u0016\u00005+\u0016R3+\u0016R4+\u0016S6+\u0016T6\u0016T8+\u0016T7\u0016T9+\u0016U6\u0016U8+\u0016U7\u0016U9+\u0016X3+\u0016X4+\n\u0016Y3+\u0016Y4+\u0016Z3+\u0016Z4\nNd3Ru4Al12 0.14963.462 61-93\u0018-90 \u0016C2+\u0016D2+\u0016Y5+\u0016Y6+\u0016B5+\u0016B6+\u0016E2+\u0016A5+\u0016A6+\u0016\u00003+\u0016\u00004+\u0016Z2\nPr3Ru4Al12 0.17363.462 22-51\u0018-48 \u0016C2+\u0016D2+\u0016Y5+\u0016Y6+\u0016B5+\u0016B6+\u0016E2+\u0016A5+\u0016A6+\u0016\u00003+\u0016\u00004+\u0016Z2\nPr3Ru4Al12 0.17363.462 41-143\u0018-140 \u0016C2+\u0016D2+\u0016Y5+\u0016Y6+\u0016B5+\u0016B6+\u0016E2+\u0016A5+\u0016A6+\u0016\u00003+\u0016\u00004+\u0016Z2\nPr3Ru4Al12 0.17363.462 61-143\u0018-140 \u0016C2+\u0016D2+\u0016Y5+\u0016Y6+\u0016B5+\u0016B6+\u0016E2+\u0016A5+\u0016A6+\u0016\u00003+\u0016\u00004+\u0016Z2\nMn3Ni20P6 1.14564.480 01-361\u0018-358 \u0016\u00006+\u0016R3\u0016R4+\u0016S2+\u0016T3+\u0016T4+\u0016Y5+\u0016Z3+\u0016Z4\nMn3Ni20P6 1.14564.480 41-5\u00180 \u0016\u00006+\u0016R5\u0016R6+\u0016S2+\u0016T3+\u0016T4+\u0016Y5+\u0016Y6+\u0016Z3+\u0016Z4\nCeRh2Si2 1.18864.480 02-7\u0018-4 \u0016\u00005+\u0016R3\u0016R4+\u0016S2+\u0016T3+\u0016T4+\u0016Y6+\u0016Z3+\u0016Z4\nCeRh2Si2 1.18864.480 21-11\u0018-8 \u0016\u00005+\u0016R5\u0016R6+\u0016S2+\u0016T3+\u0016T4+\u0016Y6+\u0016Z3+\u0016Z4\nCeRh2Si2 1.18864.480 41-11\u0018-8 \u0016\u00005+\u0016R5\u0016R6+\u0016S2+\u0016T3+\u0016T4+\u0016Y6+\u0016Z3+\u0016Z4\nCaFe2As2 1.5264.480 21-43\u0018-40 \u0016\u00006+\u0016R5\u0016R6+\u0016S2+\u0016T3+\u0016T4+\u0016Y5+\u0016Z3+\u0016Z4\nCaFe2As2 1.5264.480 31-43\u0018-40 \u0016\u00006+\u0016R5\u0016R6+\u0016S2+\u0016T3+\u0016T4+\u0016Y5+\u0016Z3+\u0016Z4\nCaFe2As2 1.5264.480 41-43\u0018-40 \u0016\u00006+\u0016R5\u0016R6+\u0016S2+\u0016T3+\u0016T4+\u0016Y5+\u0016Z3+\u0016Z4\nMn3Ni20P6 2.1565.486 11-831\u0018-830 \u0016L2+\u0016V3+\u0016M3+\u0016M4+\u0016A5+\u0016A6+\u0016\u00005+\u0016\u00006+\u0016Y3+\u0016Y4\nGd2CuO4 1.10466.500 03-17\u0018-14 \u0016\u00005+\u0016R3\u0016R3+\u0016R4\u0016R4+\u0016S3\u0016S5+\u0016S4\u0016S6+\u0016T3+\u0016T4+\u0016Y5+\u0016Z3+\u0016Z4\nGd2CuO4 1.10466.500 11-5\u0018-2 \u0016\u00006+\u0016R3\u0016R3+\u0016R4\u0016R4+\u0016S3\u0016S5+\u0016S4\u0016S6+\u0016T3+\u0016T4+\u0016Y6+\u0016Z3+\u0016Z4\nNiCr2O4 0.470.530 41-59\u0018-50 \u0016V2+\u0016V3+\u0016M2+\u0016A2+\u0016Y3\u0016Y5+\u0016Y4\u0016Y6+\u0016\u00003+\u0016\u00004+\u0016\u00005+\u0016\u00006+\u0016L2+\u0016L3\nNpNiGa5 2.2874.559 21-67\u0018-64 \u0016L2+\u0016L3+\u0016A5+\u0016A6+\u0016M3+\u0016M4+\u0016V2\u0016V3+\u0016U3\u0016U4+\u0016Y5+\u0016Y6+\u0016\u00005+\u0016\u00006\nSr2FeOsO6 1.4783.50 01-87\u0018-84 \u0016A11\u0016A6+\u0016A8\u0016A9+\u0016\u00006\u0016\u00008+\u0016M5\u0016M8+\u0016M6\u0016M7+\u0016R5\u0016R5+\u0016R6\u0016R6+\u0016X3\u0016X5+\n\u0016X4\u0016X6+\u0016Z10\u0016Z8+\u0016Z12\u0016Z6\nBa2CoO2Ag2Se2 2.2486.73 01-179\u0018-172 \u0016A10\u0016A11+\u0016A12\u0016A9+\u0016\u000010\u0016\u000012+\u0016\u00005\u0016\u00007+\u0016M3\u0016M4+\u0016R2+\u0016X2+\u0016Z3\u0016Z4\nBa2CoO2Ag2Se2 2.2486.73 11-179\u0018-172 \u0016A10\u0016A11+\u0016A12\u0016A9+\u0016\u000010\u0016\u000012+\u0016\u00005\u0016\u00007+\u0016M3\u0016M4+\u0016R2+\u0016X2+\u0016Z3\u0016Z4\nBa2CoO2Ag2Se2 2.2486.73 32-95\u0018-88 \u0016A5\u0016A8+\u0016A6\u0016A7+\u0016\u000010\u0016\u000012+\u0016\u00005\u0016\u00007+\u0016M3\u0016M4+\u0016R2+\u0016X2+\u0016Z3\u0016Z4\nBa2CoO2Ag2Se2 2.2486.73 41-111\u0018-104 \u0016A5\u0016A8+\u0016A6\u0016A7+\u0016\u000011\u0016\u00009+\u0016\u00006\u0016\u00008+\u0016M3\u0016M4+\u0016R2+\u0016X2+\u0016Z3\u0016Z4\nAppendix L: Magnetic moments for each materials with different Coulomb interactions\nThe experimental and calculated magnetic moments on the nonequivalent magnetic atoms of each material have been listed below. As\nit only consider the moments that contributed by spin component in the VASP, it may have an estimate for those materials that have\nlarge orbital magnetic moments. For all of the magnetic materials, we define the average error of the calculated magnetic moments as\n1\nNPN\ni=1jMExp\ni\u0000MT\nij\nMExp\ni\u0002100%, whereNis the number of nonequivalent magnetic atoms, MExpis the experimental magnetic moments\nandMTis the calculated magnetic moments. We label the magnetic moments that are closest to the experimental value by red color and104\nAverage error 0\u001810%10%\u001830%30%\u001850%>50%Total\nCount 139 117 68 79403\nTABLE XVI. Statistics of the materials with the average error of calculated magnetic moments margin in 0\u001810%,10%\u001830%,\n30%\u001850%and > 50%.\ntabulate the number of materials with the error margin in 0 10%, 10% 30%, 30% 50% and >50% in Table XVI.\nTABLE XVII: Magnetic moments of the magnetic materials with 3d=4delectrons. The first four column give the chemical\nformulae (Formula), BCSID, MSG for each materials. The 5th to 11th column give the magnetic moments of each nonequivalent\nmagnetic atom, including the experimental (Exp.) magnetic moments and the calculated magnetic moments with Coulomb\ninteraction U= 0;1;2;3;4eV. The calculated magnetic moments closest to the experimental value are tagged by red color.\nNo.Formula BCSID MSGMagnetic moments ( \u0016B)\nMag. Ele. Exp. U=0 U=1 eV U=2 eV U=3 eV U=4 eV\n1BiCrO3 0.139 2.4Cr 2.62 2.66 2.73 2.79 2.84 2.88\nError( %) 1.527 4.198 6.489 8.397 9.924\n2Cr2S3 0.5 2.4Cr 1.2 2.65 2.81 2.9 2.99 3.07\nError( %) 120.833 134.167 141.667 149.167 155.833\n3Fe2O3-alpha 0.66 2.4Fe 4.22 3.56 3.8 3.94 4.05 4.13\nError( %) 15.64 9.953 6.635 4.028 2.133\n4CaMnGe2O6 0.155 2.6Mn 4.19 4.44 4.49 4.54 4.57 4.61\nError( %) 5.967 7.16 8.353 9.069 10.024\n5MnPSe3 0.180 2.6Mn 4.74 4.19 4.3 4.38 4.45 4.51\nError( %) 11.603 9.283 7.595 6.118 4.852\n6BaNi2P2O8 0.215 2.6Ni 2.0 1.57 1.63 1.67 1.7 1.73\nError( %) 21.5 18.5 16.5 15.0 13.5\n7NiSb2O6 1.113 2.7Ni 1.56 1.53 1.61 1.65 1.69 1.72\nError( %) 1.923 3.205 5.769 8.333 10.256\n8LiFeSO4F 1.155 2.7Fe 3.78 3.61 3.64 3.68 3.71 3.74\nError( %) 4.497 3.704 2.646 1.852 1.058\n9Li2Ni(WO4)2 1.159 2.7Ni 1.94 1.56 1.61 1.65 1.68 1.71\nError( %) 19.588 17.01 14.948 13.402 11.856\n10La2LiOsO6 1.166 2.7Os 1.8 1.71 1.84 1.96 2.1 2.24\nError( %) 5.0 2.222 8.889 16.667 24.444\n11NiS2 1.167 2.7Ni 1.27 0.02 1.1 1.23 1.32 1.41\nNi 0.66 0.02 1.11 1.24 1.33 1.41\nNi 1.27 0.02 1.11 1.23 1.33 1.41\nError( %) 97.939 31.388 31.393 36.725 45.229\n12Sr2CuWO6 1.177 2.7Cu 0.57 0.52 0.59 0.59 0.6 0.59\nError( %) 8.772 3.509 3.509 5.263 3.509\n13Na3Co2SbO6 1.180 2.7Co 1.79 2.55 2.58 2.63 2.69 2.73\nError( %) 42.458 44.134 46.927 50.279 52.514\n14TlMnO3 1.182 2.7Mn 3.71 3.52 3.62 3.7 3.77 3.83\nError( %) 5.121 2.426 0.27 1.617 3.235\n15YCr(BO3)2 1.190 2.7Cr 2.47 2.83 2.84 2.88 2.9 2.93\nError( %) 14.575 14.98 16.599 17.409 18.623\n16CrTe3 1.193 2.7Cr 1.98 3.45 - 3.13 3.28 3.42\nError( %) 74.242 - 58.081 65.657 72.727\n17FePSe3 1.210 2.7Fe 4.9 3.31 3.39 3.44 3.51 3.57\nError( %) 32.449 30.816 29.796 28.367 27.143\n18CuF2 1.219 2.7Cu 0.73 0.62 0.65 0.65 0.65 0.65\nError( %) 15.068 10.959 10.959 10.959 10.959\n19BaMoP2O8 1.229 2.7Mo 1.12 - 1.72 1.77 1.81 1.85105\nError( %) - 53.571 58.036 61.607 65.179\n20FeI2 1.240 2.7Fe 3.7 3.38 3.43 3.5 3.56 3.62\nError( %) 8.649 7.297 5.405 3.784 2.162\n21CrCl3 1.244 2.7Cr 3.0 2.88 2.9 2.95 3.0 3.04\nError( %) 4.0 3.333 1.667 0.0 1.333\n22BaNi2V2O8 1.256 2.7Ni 1.55 1.43 1.54 1.6 1.65 1.69\nError( %) 7.742 0.645 3.226 6.452 9.032\n23BaNi2As2O8 1.257 2.7Ni 2.0 1.55 1.61 1.65 1.69 1.72\nError( %) 22.5 19.5 17.5 15.5 14.0\n24NaMnGe2O6 1.260 2.7Mn 1.83 3.59 3.65 3.71 3.76 3.82\nError( %) 96.175 99.454 102.732 105.464 108.743\n25CuMnO2 1.57 2.7Mn 3.04 3.55 3.64 3.72 3.79 3.86\nError( %) 16.776 19.737 22.368 24.671 26.974\n26Sc2NiMnO6 2.18 2.7Mn 1.66 - 2.75 2.84 2.93 3.01\nNi 1.49 - 1.5 1.56 1.61 1.65\nError( %) - 33.167 37.891 42.28 46.032\n27YBaFe4O7 1.124 4.10Fe 2.63 - 3.61 3.72 3.61 3.63\nFe 3.11 - 3.61 3.63 3.92 4.03\nFe 2.58 - 3.34 3.41 3.48 3.54\nFe 2.39 - 3.59 3.7 3.7 3.7\nError( %) - 33.251 36.287 38.251 39.906\n28Lu2MnCoO6 1.32 4.10Co 2.56 2.36 2.46 2.52 2.58 2.63\nError( %) 7.813 3.906 1.563 0.781 2.734\n29Cs2CoCl4 1.51 4.10Co 1.65 2.42 2.49 2.54 2.6 2.65\nError( %) 46.667 50.909 53.939 57.576 60.606\n30BaNiF4 1.64 4.10Ni 2.0 1.56 1.65 1.7 1.73 1.76\nError( %) 22.0 17.5 15.0 13.5 12.0\n31Ca2Cr2O5 1.227 4.12Cr 1.71 2.71 2.82 2.78 2.89 2.9\nError( %) 58.48 64.912 62.573 69.006 69.591\n32SrMn(VO4)(OH) 0.165 4.7Mn 3.32 3.63 3.87 4.06 4.2 4.32\nError( %) 9.337 16.566 22.289 26.506 30.12\n33LiFeP2O7 0.83 4.7Fe 4.62 4.08 4.15 4.2 4.26 4.31\nError( %) 11.688 10.173 9.091 7.792 6.71\n34Cu2MnSnS4 1.100 5.16Mn 4.28 4.16 4.26 4.35 4.42 4.49\nError( %) 2.804 0.467 1.636 3.271 4.907\n35YFe3(BO3)4 1.90 5.16Fe 3.95 4.01 4.1 4.16 4.22 4.28\nError( %) 1.519 3.797 5.316 6.835 8.354\n36Na2MnF5 1.55 7.29Mn 3.3 3.62 3.66 3.7 3.75 3.79\nError( %) 9.697 10.909 12.121 13.636 14.848\n37BiMn2O5 1.75 8.36Mn 2.14 - 2.66 2.77 2.87 2.96\nMn 2.15 - 2.66 2.77 2.87 2.97\nMn 2.85 - 3.52 3.62 3.7 3.78\nMn 2.82 - 3.54 3.64 3.72 3.79\nMn 2.84 - 3.52 3.62 3.7 3.78\nMn 2.81 - 3.54 3.64 3.72 3.79\nError( %) - 24.497 28.562 32.0 35.244\n38MnTiO3 0.50 9.39Mn 3.9 4.35 4.43 4.49 4.54 4.59\nError( %) 11.538 13.59 15.128 16.41 17.692\n39CuMnSb 1.232 9.40Mn 3.84 3.72 3.94 4.13 4.27 4.37\nError( %) 3.125 2.604 7.552 11.198 13.802\n40La2O2Fe2OSe2 1.58 9.40Fe 2.83 3.29 3.37 3.45 3.51 3.57\nError( %) 16.254 19.081 21.908 24.028 26.148106\n41BaFe2Se3 1.120 9.41Fe 2.8 - 3.06 3.2 3.3 -\nError( %) - 9.286 14.286 17.857 -\n42Li2CoSiO4 1.79 9.41Co 2.92 2.51 2.57 2.62 2.67 2.71\nError( %) 14.041 11.986 10.274 8.562 7.192\n43RuCl3 1.228 10.49Ru 0.44 0.66 0.66 0.22 0.58 0.43\nError( %) 50.0 50.0 50.0 31.818 2.273\n44AgMnVO4 1.116 11.55Mn 4.0 4.41 4.48 4.53 4.57 4.61\nError( %) 10.25 12.0 13.25 14.25 15.25\n45Co2C10O8H2 1.134 11.57Co 2.98 2.6 2.65 2.69 2.73 2.77\nError( %) 12.752 11.074 9.732 8.389 7.047\n46NiPS3 1.230 11.57Ni 1.0 1.13 1.27 1.36 1.43 1.49\nError( %) 13.0 27.0 36.0 43.0 49.0\n47CoPS3 1.264 11.57Co 3.36 2.24 2.36 2.44 2.51 2.57\nError( %) 33.333 29.762 27.381 25.298 23.512\n48MnPS3 0.163 12.60Mn 4.43 4.25 4.34 4.41 4.47 4.25\nError( %) 4.063 2.032 0.451 0.903 4.063\n49CaMn2Sb2 0.92 12.60Mn 2.8 3.83 4.04 4.2 4.32 4.41\nError( %) 36.786 44.286 50.0 54.286 57.5\n50Ag2CrO2 1.0.1 12.60Cr 2.9 2.59 2.73 2.82 2.88 -\nError( %) 10.69 5.862 2.759 0.69 -\n51Mn3Ge 0.203 12.62Mn 1.7 2.78 3.24 3.58 3.82 4.01\nError( %) 63.529 90.588 110.588 124.706 135.882\n52FeI2 1.0.13 12.62Fe 1.0 3.39 3.45 3.51 3.57 3.62\nError( %) 239.0 245.0 251.0 257.0 262.0\n53CoV2O6 1.0.6 12.62Co 4.4 2.53 2.59 2.65 2.7 2.74\nError( %) 42.5 41.136 39.773 38.636 37.727\n54LuFe2O4 1.0.7 12.62Fe 4.5 3.65 3.7 3.77 - 3.9\nError( %) 18.889 17.778 16.222 - 13.333\n55FeI2 3.14 12.62Fe 1.0 3.4 3.41 3.5 3.56 -\nError( %) 240.0 241.0 250.0 256.0 -\n56FePS3 1.183 12.63Fe 4.52 3.22 3.32 3.4 3.47 3.53\nError( %) 28.761 26.549 24.779 23.23 21.903\n57Fe4Si2Sn7O16 1.197 12.63Fe 2.52 3.6 3.63 3.66 3.7 3.73\nError( %) 42.857 44.048 45.238 46.825 48.016\n58MnBr2 1.239 12.63Mn 1.0 4.42 4.48 4.52 4.57 4.6\nError( %) 342.0 348.0 352.0 357.0 360.0\n59Li2MnO3 1.97 12.63Mn 2.35 2.68 2.74 2.82 2.9 2.99\nError( %) 14.043 16.596 20.0 23.404 27.234\n60Sr2F2Fe2OS2 2.2 12.64Fe 3.3 0.1 0.09 0.08 0.07 0.06\nError( %) 96.97 97.273 97.576 97.879 98.182\n61NiWO4 1.194 13.70Ni 2.25 1.5 1.56 1.62 1.56 1.7\nError( %) 33.333 30.667 28.0 30.667 24.444\n62Ca4IrO6 1.114 13.74Ir 0.42 0.3 0.39 0.44 0.33 0.52\nError( %) 28.571 7.143 4.762 21.429 23.81\n63NaFeSO4F 1.121 13.74Fe 3.85 3.59 3.63 3.67 3.7 3.73\nError( %) 6.753 5.714 4.675 3.896 3.117\n64NaCoSO4F 1.126 13.74Co 4.06 2.61 2.67 2.71 2.75 2.79\nError( %) 35.714 34.236 33.251 32.266 31.281\n65Mn2GeO4 0.103 14.75Mn 3.4 2.95 2.87 2.92 2.98 4.2\nMn 2.96 2.92 2.86 2.91 2.97 3.06\nMn 4.47 3.97 4.22 4.33 4.41 4.45\nError( %) 8.591 8.186 6.313 4.678 9.118107\n66Li2Mn(SO4)2 0.122 14.75Mn 4.59 4.52 4.56 4.6 4.63 4.65\nError( %) 1.525 0.654 0.218 0.871 1.307\n67La2LiRuO6 0.148 14.75Ru 2.2 1.84 1.92 2.0 2.07 2.15\nError( %) 16.364 12.727 9.091 5.909 2.273\n68Ca2MnReO6 0.204 14.75Mn 4.34 4.16 - - 4.47 -\nRe 0.22 0.41 - - 0.7 -\nError( %) 45.256 - - 110.589 -\n69LiCrGe2O6 0.217 14.77Cr 2.33 2.77 2.8 2.84 2.87 2.9\nError( %) 18.884 20.172 21.888 23.176 24.464\n70LiFePO4 0.152 14.78Fe 4.09 3.55 3.6 3.64 3.68 3.71\nError( %) 13.203 11.98 11.002 10.024 9.291\n71LiFeSi2O6 0.28 14.78Fe 4.68 4.06 4.12 4.17 4.23 4.28\nError( %) 13.248 11.966 10.897 9.615 8.547\n72Li2Co(SO4)2 0.121 14.79Co 3.33 2.63 2.68 2.72 2.76 2.79\nError( %) 21.021 19.52 18.318 17.117 16.216\n73Y2MnCoO6 0.164 14.79Co 2.92 2.46 2.5 2.56 2.6 2.65\nError( %) 15.753 14.384 12.329 10.959 9.247\n74CuSb2O6 1.133 14.80Cu 0.51 - 0.66 0.7 0.73 0.76\nError( %) - 29.412 37.255 43.137 49.02\n75Li2Fe(SO4)2 1.147 14.80Fe 3.22 3.6 3.63 3.66 - 3.73\nError( %) 11.801 12.733 13.665 - 15.839\n76MnV2O6 1.196 14.80Mn 3.75 4.08 4.2 4.3 4.38 4.46\nError( %) 8.8 12.0 14.667 16.8 18.933\n77Sc2NiMnO6 1.199 14.80Mn 1.66 2.61 2.74 2.97 3.15 3.18\nError( %) 57.229 65.06 78.916 89.759 91.566\n78Cr2ReO6 1.201 14.80Cr 1.62 0.97 1.87 - 2.63 -\nRe 0.26 0.09 0.23 - 0.47 -\nError( %) 52.754 13.485 - 71.558 -\n79LiFeGe2O6 1.39 14.80Fe 4.27 3.99 4.07 4.13 4.19 4.25\nError( %) 6.557 4.684 3.279 1.874 0.468\n80CuO 1.62 14.80Cu 0.65 - 0.49 0.55 0.59 0.63\nError( %) - 24.615 15.385 9.231 3.077\n81MnPb4Sb6S14 1.63 14.80Mn 3.2 - - 4.39 4.45 4.51\nError( %) - - 37.187 39.063 40.937\n82NiTa2O6 1.112 14.82Ni 1.93 1.55 1.61 1.65 - -\nError( %) 19.689 16.58 14.508 - -\n83NaFePO4 1.117 14.82Fe 3.89 3.58 3.62 3.66 3.69 3.72\nError( %) 7.969 6.941 5.913 5.141 4.37\n84Ni2SiO4 1.203 14.82Ni 1.41 1.52 1.59 1.63 1.67 1.7\nError( %) 7.801 12.766 15.603 18.44 20.567\n85Ni2SiO4 1.204 14.82Ni 2.03 1.53 1.6 1.64 1.67 1.71\nNi 2.08 1.54 1.62 1.67 1.7 1.74\nError( %) 25.296 21.649 19.462 18.002 16.055\n86CsFe2Se3 1.26 14.82Fe 1.77 2.81 3.07 3.25 3.37 3.48\nError( %) 58.757 73.446 83.616 90.395 96.61\n87NaFeSi2O6 1.154 14.84Fe 1.73 4.06 4.12 4.18 4.23 4.28\nError( %) 134.682 138.15 141.618 144.509 147.399\n88CaCoGe2O6 1.169 14.84Co 2.93 2.59 2.63 2.67 2.71 2.76\nError( %) 11.604 10.239 8.874 7.509 5.802\n89CuSe2O5 1.2 14.84Cu 0.52 - 0.58 0.61 0.64 0.67\nError( %) - 11.538 17.308 23.077 28.846\n90Ca3Co2O6 1.60 14.84Co 1.0 0.13 2.73 2.86 2.93 3.01108\nError( %) 87.0 173.0 186.0 193.0 201.0\n91Ba3LaRu2O9 1.94 14.84Ru 1.43 0.98 1.72 1.8 1.86 1.92\nError( %) 31.469 20.28 25.874 30.07 34.266\n92NiCO3 0.113 15.85Ni 1.73 1.54 1.62 1.66 1.7 1.73\nError( %) 10.983 6.358 4.046 1.734 0.0\n93CoCO3 0.114 15.85Co 1.73 2.57 2.63 2.68 2.72 2.76\nError( %) 48.555 52.023 54.913 57.225 59.538\n94MnCO3 0.115 15.85Mn 1.73 4.49 4.54 4.58 4.61 4.64\nError( %) 159.538 162.428 164.74 166.474 168.208\n95BiCrO3 0.138 15.85Cr 2.04 2.66 2.73 2.8 2.84 2.89\nError( %) 30.392 33.824 37.255 39.216 41.667\n96Sr2CoOsO6 0.210 15.85Co 1.57 2.61 2.69 2.77 2.92 2.98\nOs 0.7 0.71 1.08 0.17 1.65 0.3\nError( %) 33.835 62.812 76.074 110.851 73.475\n97Cr2O3 0.110 15.87Cr 1.0 2.66 2.74 2.8 2.85 2.89\nError( %) 166.0 174.0 180.0 185.0 189.0\n98Co3TeO6 0.145 15.87Co 1.03 0.87 0.94 2.65 2.7 2.74\nCo 2.73 0.79 2.54 2.61 2.67 2.72\nCo 2.84 0.71 2.52 2.59 2.65 2.7\nCo 1.1 1.02 0.4 2.62 2.67 2.72\nCo 2.85 2.33 2.44 2.52 2.59 2.65\nError( %) 37.423 20.998 64.048 64.575 65.122\n99CaMnGe2O6 0.156 15.87Mn 4.17 4.46 4.51 4.55 4.58 4.62\nError( %) 6.954 8.153 9.113 9.832 10.791\n100Co4Nb2O9 0.196 15.88Co 3.5 2.49 2.45 2.62 2.67 2.71\nCo 2.6 2.46 0.84 2.63 2.67 2.73\nError( %) 17.121 48.846 13.148 13.203 13.786\n101Co4Nb2O9 0.197 15.88Co 2.32 2.44 2.55 2.63 2.67 2.73\nCo 2.52 2.39 2.54 2.61 2.66 2.71\nError( %) 5.165 5.354 8.467 10.321 12.606\n102FeBO3 0.112 15.89Fe 4.7 3.92 4.04 4.12 4.19 4.25\nError( %) 16.596 14.043 12.34 10.851 9.574\n103FeSO4F 0.128 15.89Fe 4.32 4.05 4.12 4.18 4.24 4.3\nError( %) 6.25 4.63 3.241 1.852 0.463\n104Mn3Ti2Te6 0.176 15.89Mn 4.2 4.2 4.3 4.39 4.46 4.52\nError( %) 0.0 2.381 4.524 6.19 7.619\n105Ca3LiOsO6 0.3 15.89Os 2.2 1.69 1.83 1.96 2.09 2.23\nError( %) 23.182 16.818 10.909 5.0 1.364\n106Fe2O3-alpha 0.65 15.89Fe 4.12 3.58 3.81 3.95 4.05 4.13\nError( %) 13.107 7.524 4.126 1.699 0.243\n107Sr2CuTeO6 1.168 15.90Cu 0.69 - 0.62 0.65 0.69 0.72\nError( %) - 10.145 5.797 0.0 4.348\n108CoV2O6-alpha 1.17 15.90Co 5.08 2.51 2.59 2.65 2.7 2.74\nError( %) 50.591 49.016 47.835 46.85 46.063\n109FeI2 1.209 15.90Fe 3.7 3.37 3.43 3.49 3.56 3.61\nError( %) 8.919 7.297 5.676 3.784 2.432\n110CoBr2 1.245 15.90Co 2.77 2.42 2.49 2.55 2.6 2.65\nError( %) 12.635 10.108 7.942 6.137 4.332\n111CoCl2 1.246 15.90Co 3.0 2.48 2.55 2.59 2.64 2.68\nError( %) 17.333 15.0 13.667 12.0 10.667\n112NiCl2 1.247 15.90Ni 2.11 1.43 1.5 1.54 1.59 1.63\nError( %) 32.227 28.91 27.014 24.645 22.749109\n113NiBr2 1.248 15.90Ni 2.0 1.35 1.42 1.47 1.53 1.58\nError( %) 32.5 29.0 26.5 23.5 21.0\n114MnO 1.31 15.90Mn 5.66 4.33 4.41 4.47 4.52 4.57\nError( %) 23.498 22.085 21.025 20.141 19.258\n115Ag2NiO2 1.49 15.90Ni 0.66 0.53 0.66 0.9 1.09 1.25\nNi 0.33 0.54 0.87 0.98 1.08 1.22\nError( %) 41.666 81.818 116.667 146.212 179.545\n116NiO 1.6 15.90Ni 2.45 1.34 1.44 1.52 1.58 1.63\nError( %) 45.306 41.224 37.959 35.51 33.469\n117CoO 1.69 15.90Co 3.98 2.4 - 2.55 2.63 2.69\nError( %) 39.698 - 35.93 33.92 32.412\n118CoV2O6 1.70 15.90Co 3.98 2.53 2.59 2.65 2.7 2.74\nError( %) 36.432 34.925 33.417 32.161 31.156\n119VOCl 1.37 15.91V 1.3 1.74 1.8 1.84 1.87 1.89\nError( %) 33.846 38.462 41.538 43.846 45.385\n120AgNiO2 1.50 18.22Ni 1.55 1.31 1.42 1.54 1.6 -\nError( %) 15.484 8.387 0.645 3.226 -\n121FePO4 0.17 19.25Fe 4.16 3.97 4.06 4.13 4.19 4.25\nError( %) 4.567 2.404 0.721 0.721 2.163\n122Cu3Mo2O9 0.129 19.27Cu 0.99 0.37 0.03 0.45 0.45 0.45\nError( %) 62.626 96.97 54.545 54.545 54.545\n123CoNb2O6 1.224 19.28Co 3.05 2.55 2.61 2.66 2.71 2.75\nError( %) 16.393 14.426 12.787 11.148 9.836\n124CsNiCl3 1.0.4 20.34Ni 0.6 1.42 1.5 1.55 1.59 1.64\nNi 0.4 1.42 1.5 1.55 1.59 1.64\nError( %) 195.833 212.5 222.917 231.25 241.667\n125MgV2O4 1.138 20.37V 0.47 1.6 1.71 1.78 1.84 1.88\nError( %) 240.426 263.83 278.723 291.489 300.0\n126FeSb2O4 0.97 26.66Fe 3.68 3.56 3.56 3.61 3.65 3.68\nError( %) 3.261 3.261 1.902 0.815 0.0\n127Cu3Mo2O9 0.130 26.68Cu 0.09 - 0.02 0.6 0.65 0.69\nCu 0.62 - 0.01 0.66 0.68 0.71\nError( %) - 88.083 286.559 315.95 340.591\n128Ni3B7O13Cl 0.133 29.101Ni 1.65 1.6 1.63 1.67 1.71 1.75\nNi 0.79 1.58 1.63 1.67 1.71 1.74\nError( %) 51.515 53.771 56.302 60.046 63.157\n129Ni3B7O13Br 0.135 29.101Ni 3.56 1.54 1.62 1.66 1.7 1.74\nNi 1.33 1.5 1.61 1.65 1.69 1.73\nNi 1.4 1.51 1.61 1.66 1.7 1.73\nError( %) 25.794 30.183 32.0 33.582 34.923\n130MnS2 1.18 29.105Mn 0.5 0.22 1.44 1.1 2.9 1.54\nError( %) 56.0 188.0 120.0 480.0 208.0\n131SrCo2V2O8 1.71 29.110Co 2.25 - - 2.63 2.68 2.73\nError( %) - - 16.889 19.111 21.333\n132LuMnO3 1.101 31.129Mn 3.37 3.44 3.55 3.63 3.7 3.76\nError( %) 2.077 5.341 7.715 9.792 11.573\n133GaFeO3 0.38 33.147Ga 4.7 0.02 0.02 0.02 0.01 0.01\nFe 3.9 3.72 3.89 4.0 4.09 4.16\nFe 4.5 3.73 3.89 4.0 4.09 4.16\nError( %) 40.433 37.795 37.75 37.923 38.003\n134CaBaCo4O7 0.46 33.147Co 2.83 - 2.7 2.81 2.9 2.98\nCo 2.04 - 2.32 2.43 2.52 2.6110\nCo 2.05 - 2.27 2.38 2.48 2.57\nCo 2.44 - 2.71 2.82 2.91 2.99\nError( %) - 10.028 12.873 16.56 20.164\n135GeV4S8 1.86 33.149V 0.7 0.75 1.11 1.52 1.82 2.0\nV 0.13 0.21 0.22 0.5 0.12 0.12\nError( %) 34.34 63.901 200.879 83.846 96.703\n136Ca3Ru2O7 1.263 33.154Ru 1.59 1.37 1.4 1.44 1.51 1.57\nError( %) 13.836 11.95 9.434 5.031 1.258\n137Ba2CoGe2O7 0.56 35.167Co 2.9 2.53 2.59 2.64 2.68 2.73\nError( %) 12.759 10.69 8.966 7.586 5.862\n138Ca3Mn2O7 0.23 36.174Mn 2.67 2.55 2.64 2.74 2.83 2.93\nError( %) 4.494 1.124 2.622 5.993 9.738\n139CsCoBr3 1.0.3 36.174Co 3.02 2.38 2.47 2.54 2.6 2.65\nError( %) 21.192 18.212 15.894 13.907 12.252\n140BaCuF4 0.191 36.176Cu 0.83 - 0.7 0.74 0.77 0.79\nError( %) - 15.663 10.843 7.229 4.819\n141BiMn2O5 1.74 36.178Mn 2.51 2.58 2.67 2.78 2.87 2.98\nMn 3.22 3.4 3.52 3.62 3.7 3.77\nError( %) 4.189 7.845 11.589 14.624 17.903\n142NiTa2O6 1.172 41.217Ni 1.6 1.53 1.6 1.64 1.68 1.72\nError( %) 4.375 0.0 2.5 5.0 7.5\n143Cu2V2O7 0.137 43.227Cu 0.93 0.54 0.55 0.55 0.55 0.55\nError( %) 41.935 40.86 40.86 40.86 40.86\n144SrMn2V2O8 0.62 45.237Mn 3.99 4.28 4.37 4.44 4.5 4.28\nError( %) 7.268 9.524 11.278 12.782 7.268\n145La2NiO4 1.42 53.335Ni 1.6 1.24 1.42 1.5 1.56 1.61\nError( %) 22.5 11.25 6.25 2.5 0.625\n146Sr2IrO4 1.3 54.352Ir 0.24 0.0 0.0 0.14 0.09 0.19\nError( %) 100.0 100.0 41.667 62.5 20.833\n147BaCo2V2O8 1.30 54.352Co 2.27 - - 2.63 - 2.73\nError( %) - - 15.859 - 20.264\n148Sr2IrO4 1.77 54.352Ir 0.21 0.0 0.03 0.14 0.16 0.18\nError( %) 100.0 85.714 33.333 23.81 14.286\n149La2NiO4 0.45 56.369Ni 1.68 1.28 1.44 1.51 1.57 1.62\nError( %) 23.81 14.286 10.119 6.548 3.571\n150Cu3Bi(SeO3)2O2Br 1.122 56.373Cu 0.92 - 0.62 0.64 0.67 0.7\nCu 0.9 - 0.62 0.64 0.66 0.69\nError( %) - 31.86 29.661 26.92 23.624\n151Cu3Y(SeO3)2O2Cl 1.123 56.373Cu 0.41 - 0.64 0.66 0.68 0.7\nCu 1.04 - 0.62 0.64 0.66 0.68\nError( %) - 48.241 49.719 51.196 52.674\n152La2CuO4 1.23 56.374Cu 0.17 - 0.42 0.5 0.56 0.6\nError( %) - 147.059 194.118 229.412 252.941\n153Li2VOSiO4 1.9 57.389V 0.63 0.95 0.97 0.99 1.01 1.03\nError( %) 50.794 53.968 57.143 60.317 63.492\n154Cr2TeO6 0.143 58.395Cr 2.45 2.67 2.74 2.8 2.84 2.88\nError( %) 8.98 11.837 14.286 15.918 17.551\n155Cr2WO6 0.144 58.395Cr 2.14 2.66 2.75 2.75 2.75 2.75\nError( %) 24.299 28.505 28.505 28.505 28.505\n156Cr2WO6 0.75 58.395Cr 1.0 2.66 2.75 2.81 2.75 2.75\nError( %) 166.0 175.0 181.0 175.0 175.0\n157Cr2TeO6 0.76 58.395Cr 1.0 2.68 2.74 2.8 2.84 2.88111\nError( %) 168.0 174.0 180.0 184.0 188.0\n158Mn(N(CN2))2 0.131 58.398Mn 5.01 4.42 4.48 4.52 4.56 4.6\nError( %) 11.776 10.579 9.78 8.982 8.184\n159NiF2 0.36 58.398Ni 2.0 1.61 1.69 1.73 1.76 1.78\nError( %) 19.5 15.5 13.5 12.0 11.0\n160KCo4(PO4)3 0.85 58.398Co 2.16 2.53 2.59 2.66 2.7 2.74\nCo 3.18 2.64 2.67 2.7 2.73 2.76\nCo 3.14 2.45 2.5 2.57 2.64 2.69\nError( %) 18.695 18.776 18.798 18.358 18.131\n161LuFe4Ge2 0.140 58.399Fe 0.45 1.83 2.27 2.54 2.73 2.79\nError( %) 306.667 404.444 464.444 506.667 520.0\n162YFe4Ge2 0.27 58.399Fe 0.64 1.94 2.31 2.56 2.75 2.83\nError( %) 203.125 260.938 300.0 329.687 342.188\n163YMn3Al4O12 1.158 58.404Mn 2.92 3.65 3.71 3.77 3.82 3.88\nError( %) 25.0 27.055 29.11 30.822 32.877\n164CuMnAs 0.222 59.407Mn 3.6 3.54 3.83 4.05 4.21 4.34\nError( %) 1.667 6.389 12.5 16.944 20.556\n165CaCo2P2 1.252 59.416Co 0.32 0.49 0.57 0.64 0.71 0.79\nError( %) 53.125 78.125 100.0 121.875 146.875\n166CoSe2O5 0.119 60.419Co 3.2 2.57 2.63 2.68 2.72 2.76\nError( %) 19.688 17.813 16.25 15.0 13.75\n167CoSe2O5 0.161 60.419Co 3.0 - - 2.68 - 2.68\nError( %) - - 10.667 - 10.667\n168Mn5Si3 1.88 60.431Mn 1.89 2.54 3.1 3.48 3.72 3.91\nError( %) 34.392 64.021 84.127 96.825 106.878\n169Mn2O3-alpha 0.40 61.433Mn 3.05 3.42 3.55 3.64 3.72 3.78\nMn 3.47 3.52 3.62 3.7 3.77 3.83\nMn 3.49 3.52 3.61 3.69 3.76 3.82\nMn 3.27 3.48 3.59 3.67 3.75 3.82\nMn 4.19 3.48 3.58 3.66 3.74 3.8\nError( %) 7.559 9.701 11.317 12.753 13.979\n170Mn2O3-alpha 0.41 61.433Mn 2.69 - 3.54 3.63 3.71 3.77\nMn 3.1 - 3.63 3.71 3.77 3.83\nMn 3.01 - 3.61 3.69 3.76 3.82\nMn 2.92 - 3.59 3.67 3.75 3.82\nMn 3.54 - 3.58 3.67 3.74 3.8\nError( %) - 18.541 21.313 23.705 25.755\n171Li2Ni(SO4)2 0.71 61.437Ni 2.15 - - 1.38 1.46 1.53\nError( %) - - 35.814 32.093 28.837\n172EuFe2As2 2.1 61.439Eu 6.8 6.73 6.77 6.9 6.9 6.9\nFe 0.98 0.0 2.38 2.69 2.88 3.01\nError( %) 50.515 71.65 87.98 97.674 104.306\n173Mn2GeO4 0.102 62.441Mn 4.09 4.44 4.5 4.54 4.58 4.61\nMn 4.9 4.45 4.5 4.55 4.59 4.62\nError( %) 8.871 9.094 9.072 9.153 9.214\n174NH4Fe2O6 0.168 62.441Fe 4.13 3.99 4.1 4.19 4.28 4.34\nFe 3.12 3.77 3.77 3.75 3.73 3.74\nError( %) 12.112 10.78 10.822 11.592 12.478\n175RbFe2F6 0.192 62.441Fe 3.99 3.79 3.79 3.77 3.73 3.75\nFe 4.29 3.99 4.1 4.19 4.28 4.34\nError( %) 6.003 4.72 3.923 3.375 3.591\n176Co2SiO4 0.218 62.441Co 3.88 2.44 2.6 2.65 2.7 2.74112\nCo 3.64 2.53 2.63 2.68 2.72 2.76\nError( %) 33.804 30.369 29.038 27.843 26.779\n177Co2SiO4 0.219 62.441Co 3.87 2.44 2.57 2.65 2.7 2.73\nCo 3.35 2.53 2.58 2.65 2.72 2.76\nError( %) 30.714 28.289 26.21 24.519 23.535\n178Fe2SiO4 0.221 62.441Fe 4.44 3.49 3.58 3.62 3.67 3.7\nFe 4.4 3.57 3.59 3.63 3.67 3.7\nError( %) 20.13 18.89 17.984 16.966 16.288\n179Rb2Fe2O(AsO4)2 0.90 62.441Fe 3.64 3.89 4.0 4.09 4.16 4.22\nFe 3.19 3.75 3.97 4.06 4.15 4.21\nError( %) 12.211 17.171 19.817 22.19 23.954\n180CoSO4 0.96 62.441Co 3.22 2.61 2.67 2.71 2.75 2.78\nError( %) 18.944 17.081 15.839 14.596 13.665\n181KCrF4 0.182 62.443Cr 2.11 2.78 - - - -\nCr 1.91 2.78 - - - -\nCr 2.19 2.77 - - - -\nError( %) 34.596 - - - -\n182LiNiPO4 0.88 62.444Ni 2.22 1.56 1.64 1.68 1.71 1.74\nError( %) 29.73 26.126 24.324 22.973 21.622\n183LiCoPO4 0.193 62.445Co 1.0 2.54 2.62 2.67 2.71 2.75\nError( %) 154.0 162.0 167.0 171.0 175.0\n184SrEr2O4 0.216 62.445Er 4.5 2.74 - 2.81 - 2.85\nError( %) 39.111 - 37.556 - 36.667\n185KMn4(PO4)3 0.86 62.445Mn 4.43 4.5 4.55 4.58 4.62 4.5\nMn 4.21 4.46 4.51 4.55 4.59 4.46\nMn 4.57 4.49 4.53 4.57 4.6 4.49\nError( %) 3.09 3.57 3.821 4.658 3.09\n186NaFePO4 0.87 62.445Fe 4.55 3.56 3.61 3.65 3.68 3.72\nError( %) 21.758 20.659 19.78 19.121 18.242\n187LiFePO4 0.95 62.445Fe 4.19 3.55 3.6 3.64 3.68 3.71\nError( %) 15.274 14.081 13.126 12.172 11.456\n188Mn2GeO4 0.101 62.446Mn 3.03 4.45 4.5 4.54 4.58 4.61\nMn 4.5 4.44 4.5 4.55 4.59 4.62\nError( %) 24.099 24.257 25.473 26.578 27.406\n189Mn2SiO4 0.220 62.446Mn 3.85 4.44 4.5 4.54 4.58 4.62\nMn 4.68 4.46 4.51 4.56 4.6 4.63\nError( %) 10.012 10.258 10.243 10.336 10.534\n190LaMnO3 0.1 62.448Mn 3.87 3.55 3.63 3.71 3.77 3.83\nError( %) 8.269 6.202 4.134 2.584 1.034\n191NaOsO3 0.25 62.448Os 1.0 0.97 1.37 1.65 1.88 2.08\nError( %) 3.0 37.0 65.0 88.0 108.0\n192Rb2Fe2O(AsO4)2 0.91 62.448Fe 3.1 3.88 4.01 4.09 4.16 4.22\nFe 3.09 3.74 3.94 4.06 4.14 4.21\nError( %) 23.099 28.431 31.664 34.087 36.188\n193LiMnPO4 0.24 62.449Mn 3.9 4.5 4.54 4.58 4.61 4.64\nError( %) 15.385 16.41 17.436 18.205 18.974\n194Cr2As 1.130 62.450Cr 0.4 0.01 1.76 0.01 0.6 0.05\nCr 1.34 2.1 2.63 0.25 0.57 1.05\nError( %) 77.108 218.134 89.422 53.731 54.571\n195Fe2As 1.131 62.450Fe 0.95 1.27 2.09 2.48 2.72 2.89\nFe 1.52 2.34 2.61 2.82 3.04 3.07\nError( %) 43.816 95.855 123.289 143.158 153.092113\n196Mn2As 1.132 62.450Mn 3.7 2.03 2.59 3.35 3.53 3.68\nMn 3.5 3.52 3.8 4.04 4.2 4.32\nError( %) 22.854 19.286 12.444 12.297 11.984\n197CrN 1.28 62.450Cr 2.4 2.46 2.65 2.76 2.83 2.89\nError( %) 2.5 10.417 15.0 17.917 20.417\n198Mn3O4 1.1 62.452Mn 3.49 3.56 3.7 3.85 3.83 3.9\nError( %) 2.006 6.017 10.315 9.742 11.748\n199Sr2Mn3Sb2O2 2.27 63.459Mn 4.2 3.96 4.26 4.37 4.44 4.5\nMn 3.5 3.7 3.97 4.15 4.28 4.37\nError( %) 5.714 7.428 11.31 14.0 16.0\n200Mn3Sn 0.199 63.463Mn 3.0 3.34 3.48 3.74 3.92 4.07\nError( %) 11.333 16.0 24.667 30.667 35.667\n201Mn3Sn 0.200 63.464Mn 3.0 3.16 3.51 3.77 3.97 4.13\nError( %) 5.333 17.0 25.667 32.333 37.667\n202CaIrO3 0.79 63.464Ir 1.0 0.23 0.43 0.52 0.55 -\nError( %) 77.0 57.0 48.0 45.0 -\n203Gd2CuO4 0.82 64.476Cu 1.02 - 0.25 0.42 0.48 0.53\nError( %) - 75.49 58.824 52.941 48.039\n204Mn3Ni20P6 1.145 64.480Mn 2.4 3.71 3.93 4.11 4.25 4.36\nError( %) 54.583 63.75 71.25 77.083 81.667\n205BaFe2As2 1.16 64.480Fe 0.87 2.02 2.55 2.8 2.97 3.09\nError( %) 132.184 193.103 221.839 241.379 255.172\n206K2NiF4 1.249 64.480Ni 1.0 1.55 1.65 1.69 1.73 1.76\nError( %) 55.0 65.0 69.0 73.0 76.0\n207CaFe2As2 1.52 64.480Fe 0.8 1.83 2.35 2.67 2.86 3.0\nError( %) 128.75 193.75 233.75 257.5 275.0\n208Mn3Ni20P6 2.15 65.486Mn 2.2 - - - - 4.36\nMn 2.7 - - - - 4.47\nError( %) - - - - 81.869\n209Gd2CuO4 1.104 66.500Cu 1.41 - 0.04 0.39 0.48 0.53\nError( %) - 97.163 72.34 65.957 62.411\n210SrFeO2 1.65 69.526Fe 3.1 3.35 3.43 3.51 3.56 3.61\nError( %) 8.065 10.645 13.226 14.839 16.452\n211NiCr2O4 0.4 70.530Ni 1.64 0.66 0.43 1.54 1.62 1.67\nCr 1.4 2.6 2.75 2.84 2.89 2.93\nError( %) 72.735 85.104 54.478 53.824 55.558\n212LaFeAsO 1.125 73.553Fe 0.63 1.91 2.46 2.74 2.93 3.07\nError( %) 203.175 290.476 334.921 365.079 387.302\n213YbCo2Si2 1.176 73.553Yb 1.41 0.0 0.0 0.0 3.65 -\nError( %) 100.0 100.0 100.0 158.865 -\n214Sr2FeOsO6 1.47 83.50Fe 1.83 3.82 3.96 4.05 4.13 4.2\nOs 0.48 1.36 1.64 1.83 1.99 2.15\nError( %) 146.039 179.03 201.281 220.134 238.712\n215Mn3CuN 2.5 85.59Mn 2.86 2.81 3.2 3.48 3.68 3.9\nMn 0.65 3.13 3.43 3.63 3.78 3.95\nError( %) 191.643 219.79 240.07 255.105 272.028\n216Sr2CoO2Ag2Se2 2.23 86.73Co 3.77 2.21 2.45 2.52 2.58 2.63\nError( %) 41.379 35.013 33.156 31.565 30.239\n217Ba2CoO2Ag2Se2 2.24 86.73Co 3.97 2.38 2.47 2.54 2.61 2.66\nError( %) 40.05 37.783 36.02 34.257 32.997\n218MnV2O4 0.64 88.81Mn 4.2 4.2 4.33 4.42 4.49 4.55\nV 1.3 1.66 1.73 1.78 1.82 1.86114\nError( %) 13.846 18.086 21.081 23.452 25.705\n219FeTa2O6 2.22 88.86Fe 3.69 3.55 3.58 3.62 3.66 3.69\nError( %) 3.794 2.981 1.897 0.813 0.0\n220Ba(TiO)Cu4(PO4)4 1.235 94.132Cu 0.8 - 0.65 0.68 0.7 0.72\nError( %) - 18.75 15.0 12.5 10.0\n221ZnV2O4 1.24 96.150V 0.65 1.57 1.66 1.81 1.86 1.89\nError( %) 141.538 155.385 178.462 186.154 190.769\n222MgCr2O4 3.4 111.255Cr 0.69 0.01 2.88 2.98 3.07 3.17\nError( %) 98.551 317.391 331.884 344.928 359.42\n223alpha-Mn 1.85 114.282Mn 2.83 0.01 3.52 3.81 4.0 -\nMn 1.83 0.05 0.34 0.11 0.22 -\nMn 0.74 0.84 0.11 0.05 0.21 -\nMn 0.48 0.26 0.13 0.07 0.08 -\nMn 0.59 0.24 1.18 0.2 0.08 -\nMn 0.66 0.18 0.25 0.17 0.11 -\nError( %) 64.718 70.997 74.604 75.675 -\n224GeCu2O4 1.185 122.338Cu 0.89 - 0.62 0.65 0.68 0.7\nError( %) - 30.337 26.966 23.596 21.348\n225CaMnBi2 0.72 129.416Mn 3.73 3.87 4.08 4.23 4.35 4.45\nError( %) 3.753 9.383 13.405 16.622 19.303\n226Mn3Pt 1.143 132.456Mn 3.4 3.04 3.56 3.98 4.18 4.31\nError( %) 10.588 4.706 17.059 22.941 26.765\n227MnF2 0.15 136.499Mn 4.6 4.54 4.58 4.61 4.64 4.67\nError( %) 1.304 0.435 0.217 0.87 1.522\n228CoF2 0.178 136.499Co 2.6 2.64 2.69 2.73 2.77 2.8\nError( %) 1.538 3.462 5.0 6.538 7.692\n229Fe2TeO6 0.142 136.503Fe 4.19 3.8 3.97 4.08 4.15 4.22\nError( %) 9.308 5.251 2.625 0.955 0.716\n230LaCrAsO 1.146 138.528Cr 1.57 2.63 3.05 3.35 3.56 3.71\nError( %) 67.516 94.268 113.376 126.752 136.306\n231BaMn2As2 0.18 139.536Mn 3.88 3.68 3.93 4.11 4.24 4.35\nError( %) 5.155 1.289 5.928 9.278 12.113\n232Sr2Mn3As2O2 0.212 139.536Mn 3.4 3.56 3.83 4.04 4.21 4.31\nError( %) 4.706 12.647 18.824 23.824 26.765\n233SrMnBi2 0.73 139.536Mn 3.75 3.98 4.16 4.29 4.4 4.48\nError( %) 6.133 10.933 14.4 17.333 19.467\n234BaMn2Bi2 0.89 139.536Mn 3.83 3.97 4.15 4.29 4.39 4.48\nError( %) 3.655 8.355 12.01 14.621 16.971\n235Mn3ZnC 2.19 139.537Mn 2.73 2.58 3.0 3.28 3.5 3.67\nMn 1.6 2.48 3.08 3.43 3.63 3.78\nError( %) 30.247 51.195 67.261 77.54 85.341\n236KNiF3 1.250 140.550Ni 2.22 1.52 1.64 1.69 1.73 1.76\nError( %) 31.532 26.126 23.874 22.072 20.721\n237CoAl2O4 0.58 141.556Co 1.9 2.53 2.59 2.64 2.69 2.73\nError( %) 33.158 36.316 38.947 41.579 43.684\n238Ca2MnO4 0.211 142.568Mn 2.4 2.54 2.63 2.72 2.82 2.91\nError( %) 5.833 9.583 13.333 17.5 21.25\n239LaMn3Cr4O12 1.156 146.12Mn 3.39 3.64 3.7 3.76 3.82 3.87\nCr 2.89 2.57 2.67 2.75 2.81 2.85\nError( %) 9.223 8.378 7.879 7.726 7.772\n240Ni3TeO6 1.165 146.12Ni 2.03 1.48 1.57 1.62 1.66 1.7\nError( %) 27.094 22.66 20.197 18.227 16.256115\n241NiN2O6 0.78 148.17Ni 1.34 1.56 1.62 1.66 1.7 1.73\nNi 1.33 1.56 1.62 1.66 1.7 1.73\nError( %) 16.856 21.35 24.346 27.343 29.59\n242MnGeO3 0.125 148.19Mn 4.6 3.55 3.78 3.98 4.13 4.27\nError( %) 22.826 17.826 13.478 10.217 7.174\n243MnTiO3 0.19 148.19Mn 4.55 4.36 4.43 4.5 4.55 4.59\nError( %) 4.176 2.637 1.099 0.0 0.879\n244Ba3MnNb2O9 1.0.8 157.53Mn 4.91 4.34 4.41 4.47 4.52 4.56\nError( %) 11.609 10.183 8.961 7.943 7.128\n245Ba3Nb2NiO9 1.13 159.64Ni 1.47 1.58 1.62 1.66 1.7 1.73\nError( %) 7.483 10.204 12.925 15.646 17.687\n246VCl2 1.237 159.64V 0.93 2.51 2.58 2.63 2.67 2.51\nError( %) 169.892 177.419 182.796 187.097 169.892\n247VBr2 1.238 159.64V 2.48 2.53 2.59 2.64 2.68 2.53\nError( %) 2.016 4.435 6.452 8.065 2.016\n248PbNiO3 0.21 161.69Ni 1.69 - 1.53 1.59 - 1.68\nError( %) - 9.467 5.917 - 0.592\n249CuMnSb 1.233 161.72Mn 2.94 3.76 3.97 4.13 4.26 4.24\nError( %) 27.891 35.034 40.476 44.898 44.218\n250CuMnSb 1.265 161.72Mn 3.9 3.75 3.96 4.13 4.26 4.36\nError( %) 3.846 1.538 5.897 9.231 11.795\n251SrRu2O6 1.186 162.78Ru 1.3 1.43 1.69 1.85 1.98 2.09\nError( %) 10.0 30.0 42.308 52.308 60.769\n252Co4Nb2O9 0.111 165.94Co 3.0 2.41 2.56 2.62 2.67 2.72\nError( %) 19.667 14.667 12.667 11.0 9.333\n253Sr3NiIrO6 1.0.10 165.95Ir 0.5 0.15 0.05 0.14 0.2 0.26\nIr 0.25 0.14 0.05 0.14 0.2 0.26\nNi 1.5 1.4 1.5 1.57 1.63 1.68\nNi 0.75 1.4 1.5 1.57 1.63 1.68\nError( %) 51.834 67.5 57.5 51.5 47.0\n254Sr3CoIrO6 1.0.5 165.95Co 3.6 2.53 2.57 2.61 2.65 2.7\nCo 1.8 2.53 2.57 2.61 2.66 2.7\nIr 0.6 0.38 0.17 0.01 0.1 0.18\nIr 0.3 0.38 0.17 0.01 0.1 0.18\nError( %) 33.404 46.597 66.875 56.041 46.25\n255FeBr2 1.242 165.96Fe 3.9 3.48 3.52 3.57 3.61 3.65\nError( %) 10.769 9.744 8.462 7.436 6.41\n256Mn3Ir 0.108 166.101Mn 2.45 2.87 3.34 3.69 3.94 4.13\nError( %) 17.143 36.327 50.612 60.816 68.571\n257Mn3Pt 0.109 166.101Mn 2.94 3.07 3.49 3.77 3.99 4.18\nError( %) 4.422 18.707 28.231 35.714 42.177\n258Mn3GaN 0.177 166.97Mn 1.17 2.57 3.01 3.3 3.51 3.72\nError( %) 119.658 157.265 182.051 200.0 217.949\n259FeCO3 0.116 167.103Fe 1.0 3.57 3.61 3.65 3.68 3.72\nError( %) 257.0 261.0 265.0 268.0 272.0\n260Cr2O3 0.59 167.106Cr 2.48 2.67 2.75 2.81 2.86 2.9\nError( %) 7.661 10.887 13.306 15.323 16.935\n261Mn3GaC 1.153 167.108Mn 1.82 1.78 2.78 3.24 3.46 3.64\nError( %) 2.198 52.747 78.022 90.11 100.0\n262FeCl2 1.241 167.108Fe 4.5 3.52 3.56 3.6 3.59 3.67\nError( %) 21.778 20.889 20.0 20.222 18.444\n263ScMnO3 0.8 173.129Mn 3.54 3.4 3.49 3.58 3.66 3.73116\nError( %) 3.955 1.412 1.13 3.39 5.367\n264YMnO3 0.44 173.131Mn 3.14 3.51 3.59 3.67 3.74 3.8\nError( %) 11.783 14.331 16.879 19.108 21.019\n265YMnO3 0.6 185.197Mn 2.91 3.47 3.56 3.64 3.72 3.78\nError( %) 19.244 22.337 25.086 27.835 29.897\n266LuFeO3 0.117 185.201Fe 2.9 3.58 3.85 3.97 4.06 4.14\nError( %) 23.448 32.759 36.897 40.0 42.759\n267ScMnO3 0.7 185.201Mn 3.03 3.4 3.49 3.58 3.66 3.73\nError( %) 12.211 15.182 18.152 20.792 23.102\n268CsFeCl3 1.0.14 189.223Fe 3.16 3.5 3.5 3.58 3.63 3.66\nError( %) 10.759 10.759 13.291 14.873 15.823\n269ScMn6Ge6 1.110 192.252Mn 1.96 2.08 2.41 2.76 3.5 3.82\nError( %) 6.122 22.959 40.816 78.571 94.898\n270ScMn6Ge6 1.225 192.252Mn 2.08 2.04 2.39 2.73 3.48 3.83\nError( %) 1.923 14.904 31.25 67.308 84.135\n271CsCoCl3 1.0.9 193.259Co 2.8 2.43 2.53 2.58 2.63 2.68\nCo 2.66 2.43 2.54 2.58 2.63 2.68\nError( %) 10.93 7.077 5.433 3.6 2.519\n272Ba5Co5ClO13 0.118 194.268Co 0.61 0.64 0.53 0.42 0.31 0.27\nCo 2.21 2.48 2.62 2.71 2.82 2.88\nCo 0.35 0.39 0.36 0.44 0.46 0.97\nError( %) 9.521 11.508 26.495 36.07 87.733\n273Na3Co(CO3)2Cl 0.70 203.26Co 1.73 2.6 2.65 2.7 2.74 2.77\nError( %) 50.289 53.179 56.069 58.382 60.116\n274NiS2 0.150 205.33Ni 0.99 0.59 1.09 1.24 1.37 1.45\nError( %) 40.404 10.101 25.253 38.384 46.465\n275MnTe2 0.20 205.33Mn 4.28 4.06 4.22 4.33 4.41 4.48\nError( %) 5.14 1.402 1.168 3.037 4.673\n276Cd2Os2O7 0.2 227.131Os 1.04 0.73 1.31 1.66 1.92 2.14\nError( %) 29.808 25.962 59.615 84.615 105.769\nTABLE XVIII: Magnetic moments of the magnetic materials with 4f=5felectron. The first four column give the chemical\nformulae (Formula), BCSID, MSG for each materials. The 5th to 10th column give the magnetic moments of each nonequivalent\nmagnetic atom, including the experimental (Exp.) magnetic moments and the calculated magnetic moments with Coulomb\ninteraction U= 0;2;4;6eV for 4f=5felectron. If the materials have transition metal elements at the same time, we take the\nCoulomb interactions of 3d=4delectron as 2 eV. The calculated magnetic moments closest to the experimental value are labeled\nby red color.\nNo.Formula BCSID MSGMagnetic moments ( \u0016B)\nMag. Ele. Exp. U=0 U=2 eV U=4 eV U=6 eV\n277HoCr(BO3)2 1.191 2.7 Cr 3.07 2.84 - 2.82 -\nError( %) 7.492 - 8.143 -\n278Dy2Fe2Si2C 1.206 2.7 Dy 12.11 4.81 4.87 4.99 5.14\nError( %) 60.281 59.785 58.794 57.556\n279Tm2BaNiO5 1.218 2.7Tm 3.33 1.45 1.73 1.8 1.83\nNi 1.15 0.01 1.37 1.41 1.71\nError( %) 77.793 33.589 34.277 46.871\n280Nd2NaOsO6 1.38 2.7Nd 1.61 - - 2.94 -\nOs 0.9 - - 1.79 -\nError( %) - - 90.748 -\n281U3Al2Si3 0.37 5.15U 0.16 0.1 1.69 0.56 1.61\nU 1.29 1.81 1.94 1.52 2.53\nError( %) 38.905 503.319 133.915 501.187117\n282GdBiPt 1.111 9.40 Gd 6.61 6.87 6.98 7.03 7.06\nError( %) 3.933 5.598 6.354 6.808\n283ErVO3 0.104 11.54 V 1.19 0.05 0.01 0.59 0.5\nError( %) 95.798 99.16 50.42 57.983\n284DyVO3 0.106 11.54Dy 7.76 - 4.79 4.91 4.99\nV 1.45 - 1.81 1.77 0.42\nError( %) - 31.551 29.397 53.365\n285Tb2Fe2Si2C 1.171 12.63 Tb 8.0 5.88 5.97 6.01 -\nError( %) 26.5 25.375 24.875 -\n286DyCu2Si2 1.22 12.63 Dy 8.3 4.31 4.56 4.85 4.94\nError( %) 48.072 45.06 41.566 40.482\n287CeMnAsO 0.188 13.67Ce 0.76 0.64 0.3 0.36 -\nMn 3.32 3.47 4.0 4.28 -\nError( %) 10.154 40.504 40.774 -\n288PrMgPb 1.140 13.73 Pr 1.8 2.15 2.0 2.26 2.28\nError( %) 19.444 11.111 25.556 26.667\n289NdMgPb 1.141 13.73 Pr 3.38 2.13 2.01 1.86 1.82\nError( %) 36.982 40.533 44.97 46.154\n290Ho2O2Se 1.213 13.73 Ho 9.3 3.68 3.79 3.87 3.93\nError( %) 60.43 59.247 58.387 57.742\n291ErVO3 0.105 14.75Er 8.2 2.47 2.78 2.82 2.87\nV 1.47 0.15 1.7 0.18 0.3\nError( %) 79.837 40.872 76.683 72.296\n292Nd2NaRuO6 0.39 14.75Nd 2.25 2.93 2.92 2.93 2.94\nRu 1.62 1.57 0.76 2.04 2.04\nError( %) 16.654 41.432 28.074 28.296\n293TbOOH 2.21 14.78 Tb 8.02 5.79 5.88 5.94 -\nError( %) 27.805 26.683 25.935 -\n294SrHo2O4 2.8 14.78Ho 6.08 3.75 - 3.9 3.97\nHo 7.74 3.54 - 3.89 3.96\nError( %) 46.293 - 42.799 41.771\n295BaNd2O4 1.95 14.80 Nd 2.64 - - - 2.94\nError( %) - - - 11.364\n296LiErF4 1.35 14.84 Er 2.2 2.8 2.83 2.9 2.92\nError( %) 27.273 28.636 31.818 32.727\n297Pr3Ru4Al12 0.174 15.89Pr 3.2 1.96 2.06 2.18 1.71\nPr 1.39 1.95 2.04 2.04 1.01\nError( %) 39.519 41.194 39.318 36.95\n298NdCo2 0.226 15.89Nd 2.8 3.5 0.44 0.97 3.17\nCo 0.74 1.3 0.4 1.3 1.32\nError( %) 50.338 65.116 70.517 45.796\n299HoP 2.10 15.89 Ho 8.8 3.5 3.61 3.77 3.88\nError( %) 60.227 58.977 57.159 55.909\n300Ho2BaNiO5 1.14 15.90Ho 9.06 3.65 3.85 3.92 3.99\nNi 1.39 0.05 1.42 1.43 1.43\nError( %) 78.058 29.832 29.805 29.419\n301Er2BaNiO5 1.15 15.90Ho 7.89 3.65 3.82 3.92 3.99\nNi 1.54 0.06 0.18 1.43 1.43\nError( %) 74.921 69.948 28.729 28.286\n302Dy2O2S 1.211 15.90 Dy 7.2 4.85 4.85 4.89 -\nError( %) 32.639 32.639 32.083 -\n303Dy2O2Se 1.212 15.90 Dy 9.0 4.84 4.85 4.89 4.96118\nError( %) 46.222 46.111 45.667 44.889\n304Nd2BaNiO5 1.216 15.90Nd 2.65 2.91 2.92 2.96 2.96\nNi 1.58 0.14 0.23 1.55 1.55\nError( %) 50.476 47.816 6.798 6.798\n305Tb2BaNiO5 1.217 15.90Ni 2.02 - 0.17 0.31 0.04\nTb 8.03 - 6.01 6.05 6.15\nError( %) - 58.37 54.656 60.716\n306Dy2BaNiO5 1.36 15.90Dy 7.7 4.63 4.84 4.91 5.01\nNi 1.35 0.15 1.55 0.04 0.04\nError( %) 64.379 25.978 66.636 65.987\n307Er2BaNiO5 1.53 15.90Er 7.24 2.66 2.82 2.84 -\nNi 1.39 1.27 1.43 0.02 -\nError( %) 35.947 31.964 79.668 -\n308HoMnO3 1.20 31.129 Mn 3.87 0.0 - - -\nError( %) 100.0 - - -\n309ErAuGe 1.33 33.154 Er 8.8 2.52 2.69 2.71 2.88\nError( %) 71.364 69.432 69.205 67.273\n310PrNiO3 1.43 36.178 Ni 0.93 - 0.89 0.79 0.9\nError( %) - 4.301 15.054 3.226\n311TmAgGe 0.26 38.191 Tm 6.44 1.49 1.6 1.55 1.51\nError( %) 76.863 75.155 75.932 76.553\n312TbMg 2.12 49.270 Tb 7.34 5.9 5.86 6.06 6.18\nError( %) 19.619 20.163 17.439 15.804\n313Ho2RhIn8 1.139 49.273 Ho 6.9 3.61 3.66 3.76 3.94\nError( %) 47.681 46.957 45.507 42.899\n314TbMg 2.11 51.295 Tb 7.34 5.92 6.06 6.16 6.18\nError( %) 19.346 17.439 16.076 15.804\n315Er2CoGa8 1.222 51.298 Er 4.71 2.46 2.69 2.81 2.91\nError( %) 47.771 42.887 40.34 38.217\n316PrAg 1.150 53.334 Pr 2.12 2.11 2.13 2.1 2.36\nError( %) 0.472 0.472 0.943 11.321\n317DyB4 0.22 55.355 Dy 9.8 4.72 4.81 4.86 4.98\nError( %) 51.837 50.918 50.408 49.184\n318Gd2CuO4 1.105 56.374 Cu 1.22 - 0.45 0.56 -\nError( %) - 63.115 54.098 -\n319CeRu2Al10 1.8 57.391 Ce 0.34 - 0.02 0.17 0.22\nError( %) - 94.118 50.0 35.294\n320CeMnAsO 0.187 59.407Ce 0.7 - 0.77 0.44 0.05\nMn 3.3 - 4.01 4.03 4.03\nError( %) - 15.758 29.632 57.49\n321EuZrO3 0.146 62.444 Eu 7.3 6.73 6.82 6.88 6.92\nError( %) 7.808 6.575 5.753 5.205\n322Nd5Ge4 0.185 62.447Nd 1.85 3.24 3.15 - 3.14\nNd 3.17 3.15 3.05 - 3.03\nNd 2.75 3.21 3.08 - 3.03\nError( %) 30.831 28.685 - 28.109\n323EuZrO3 0.147 62.449 Eu 6.4 6.75 6.82 6.88 6.92\nError( %) 5.469 6.562 7.5 8.125\n324DyCoO3 0.159 62.449 Dy 9.08 4.83 4.86 4.93 5.0\nError( %) 46.806 46.476 45.705 44.934\n325DyScO3 0.171 62.449 Dy 9.47 4.8 4.86 4.92 5.0119\nError( %) 49.314 48.68 48.046 47.202\n326NdCoAsO 1.179 62.450Nd 1.39 3.04 2.93 2.95 2.95\nCo 0.32 0.48 0.56 0.57 0.56\nError( %) 84.353 92.896 95.177 93.615\n327U3Ru4Al12 0.12 63.461 U 2.5 1.49 - 1.63 1.43\nError( %) 40.4 - 34.8 42.8\n328Nd3Ru4Al12 0.149 63.462Nd 0.95 0.07 0.23 0.07 -\nNd 2.66 3.21 3.11 3.07 -\nError( %) 56.654 46.354 54.023 -\n329Pr3Ru4Al12 0.173 63.462Pr 3.1 1.85 2.04 2.04 2.03\nPr 1.4 0.07 0.1 2.04 1.03\nError( %) 67.661 63.526 39.954 30.472\n330Ho2RhIn8 3.3 63.464 Ho 7.5 3.41 3.55 - -\nError( %) 54.533 52.667 - -\n331U2Ni2Sn 1.200 63.466 U 1.41 1.44 1.74 1.69 1.3\nError( %) 2.128 23.404 19.858 7.801\n332NpRhGa5 1.262 63.466 Np 0.89 2.86 3.39 3.59 3.73\nError( %) 221.348 280.899 303.371 319.101\n333Er2Ni2In 1.195 63.467 Er 7.71 - 2.7 2.78 2.91\nError( %) - 64.981 63.943 62.257\n334CeB6 3.13 64.479 Ce 0.28 0.57 - - -\nError( %) 103.571 - - -\n335CeRh2Si2 1.188 64.480 Ce 1.5 0.22 0.87 0.94 0.15\nError( %) 85.333 42.0 37.333 90.0\n336TbGe2 0.141 65.483Tb 9.45 5.81 5.92 5.99 -\nTb 7.55 5.69 5.73 5.98 -\nError( %) 31.577 30.73 28.704 -\n337Tm2CoGa8 1.223 65.489 Tm 2.35 1.23 1.4 1.48 1.78\nError( %) 47.66 40.426 37.021 24.255\n338Pr2CuO4 1.106 66.500Pr 0.08 - 1.85 0.16 1.95\nCu 0.4 - 0.04 0.02 0.37\nError( %) - 1151.25 97.5 1172.5\n339CeMgPb 1.142 67.510 Ce 1.39 0.66 0.89 0.02 0.03\nError( %) 52.518 35.971 98.561 97.842\n340EuTiO3 0.16 69.523 Eu 6.93 - - 6.87 -\nError( %) - - 0.866 -\n341UAu2Si2 1.0.12 71.536 U 0.9 0.36 2.32 2.51 2.65\nError( %) 60.0 157.778 178.889 194.444\n342NpNiGa5 2.28 74.559 Np 0.84 0.18 3.45 3.59 3.74\nError( %) 78.571 310.714 327.381 345.238\n343KTb3F12 1.59 84.58 Tb 6.95 6.23 6.41 6.63 -\nError( %) 10.36 7.77 4.604 -\n344Ho2Ge2O7 0.107 92.111 Ho 9.06 3.75 3.84 3.88 3.94\nError( %) 58.609 57.616 57.174 56.512\n345Nd5Si4 0.184 92.114Nd 2.39 3.32 3.31 3.25 -\nNd 2.47 3.22 3.17 3.12 -\nNd 2.79 3.29 3.21 3.16 -\nError( %) 29.066 27.296 25.187 -\n346CeCoGe3 1.0.11 107.231 Ce 0.36 0.07 0.46 0.06 -\nError( %) 80.556 27.778 83.333 -\n347PrCo2P2 2.26 123.345Pr 3.08 1.76 1.76 1.97 1.96\nCo 0.9 0.73 0.88 0.86 0.87120\nError( %) 30.873 22.54 20.241 19.849\n348NdMg 1.162 124.360 Nd 2.6 3.42 3.49 3.26 3.13\nError( %) 31.538 34.231 25.385 20.385\n349NdCo2P2 1.251 124.360Nd 0.74 0.71 2.99 2.96 0.38\nCo 0.64 0.75 0.91 0.01 0.87\nError( %) 10.621 173.121 199.219 42.293\n350UPtGa5 1.255 124.360 U 0.32 0.94 1.67 2.35 1.57\nError( %) 193.75 421.875 634.375 390.625\n351NpRhGa5 1.261 124.360 Np 0.32 0.05 3.39 0.35 2.79\nError( %) 84.375 959.375 9.375 771.875\n352CeMn2Ge4O12 0.189 125.367 Mn 4.61 4.41 4.42 4.43 4.45\nError( %) 4.338 4.121 3.905 3.471\n353NdMg 2.14 125.373 Nd 2.56 3.41 3.28 3.2 3.16\nError( %) 33.203 28.125 25.0 23.438\n354CeCo2P2 1.253 126.386 Co 0.94 0.78 0.96 0.95 0.93\nError( %) 17.021 2.128 1.064 1.064\n355U2Pd2In 0.80 127.394 U 1.4 1.53 1.55 2.39 2.54\nError( %) 9.286 10.714 70.714 81.429\n356U2Pd2Sn 0.81 127.394 U 1.9 1.59 1.5 2.44 2.57\nError( %) 16.316 21.053 28.421 35.263\n357GdB4 0.9 127.395 Gd 7.14 6.84 6.93 6.99 7.04\nError( %) 4.202 2.941 2.101 1.401\n358GdIn3 1.81 127.397 Gd 1.0 0.36 0.09 1.32 1.13\nError( %) 64.0 91.0 32.0 13.0\n359U2Ni2In 1.102 128.408U 0.59 1.43 1.5 1.67 2.56\nNi 0.37 0.01 0.01 0.01 0.0\nError( %) 119.835 125.768 140.174 216.949\n360UP 1.160 128.410 U 1.7 1.58 1.64 1.77 1.4\nError( %) 7.059 3.529 4.118 17.647\n361TbRh2Si2 1.187 128.410 Tb 8.5 6.07 5.93 6.0 6.04\nError( %) 28.588 30.235 29.412 28.941\n362UAs 1.208 128.410 U 1.9 1.79 1.85 2.09 2.29\nError( %) 5.789 2.632 10.0 20.526\n363DyCo2Si2 1.21 128.410 Dy 9.5 4.83 4.87 4.9 4.95\nError( %) 49.158 48.737 48.421 47.895\n364CeMnAsO 0.186 129.416 Mn 2.78 - 3.98 3.99 3.99\nError( %) - 43.165 43.525 43.525\n365UPt2Si2 0.194 129.419 U 1.67 1.54 1.78 1.72 1.55\nError( %) 7.784 6.587 2.994 7.186\n366UP2 1.215 130.432 U 1.0 1.05 1.26 1.4 1.46\nError( %) 5.0 26.0 40.0 46.0\n367CeSbTe 1.271 130.432 Ce 0.38 0.44 0.88 0.08 0.13\nError( %) 15.789 131.579 78.947 65.789\n368UP 2.13 134.481 U 1.91 1.62 1.91 2.19 2.34\nError( %) 15.183 0.0 14.66 22.513\n369UAs 2.20 134.481 U 2.26 1.74 2.07 2.26 2.43\nError( %) 23.009 8.407 0.0 7.522\n370Nd2CuO4 2.6 134.481 Cu 1.0 - 0.3 0.36 0.27\nError( %) - 70.0 64.0 73.0\n371U2Rh2Sn 1.103 135.492 U 0.53 1.17 1.42 1.58 1.53\nError( %) 120.755 167.925 198.113 188.679\n372U2Rh2Sn 1.207 135.492U 0.5 1.1 1.4 1.57 1.57121\nRh 0.06 0.0 0.01 0.01 0.01\nError( %) 110.0 131.667 148.667 148.667\n373UNiGa5 1.254 140.550 U 0.75 0.0 2.17 2.31 2.58\nError( %) 100.0 189.333 208.0 244.0\n374Nd2RhIn8 1.82 140.550 Nd 2.53 3.5 3.3 3.14 3.03\nError( %) 38.34 30.435 24.111 19.763\n375TbCo2Ga8 1.87 140.550 Tb 9.6 5.54 5.85 5.98 6.05\nError( %) 42.292 39.063 37.708 36.979\n376Er2Ru2O7 0.154 141.554Er 4.5 2.55 2.79 2.84 2.87\nRu 2.0 0.13 1.22 1.24 0.02\nError( %) 68.417 38.5 37.444 67.611\n377Er2Ti2O7 0.29 141.554 Er 3.25 2.74 2.8 2.84 2.91\nError( %) 15.692 13.846 12.615 10.462\n378Gd2Sn2O7 0.47 141.555 Gd 6.08 6.83 6.89 6.95 7.0\nError( %) 12.336 13.322 14.309 15.132\n379NpCo2 0.126 141.556 Np 0.5 0.0 0.08 0.03 0.02\nError( %) 100.0 84.0 94.0 96.0\n380GdVO4 0.198 141.556 Gd 7.0 6.83 6.89 6.93 6.97\nError( %) 2.429 1.571 1.0 0.429\n381Tm2Mn2O7 0.151 141.557Mn 1.85 - - - 2.93\nTm 2.31 - - - 1.8\nError( %) - - - 40.228\n382Yb2Sn2O7 0.157 141.557 Yb 1.05 - 0.61 0.68 0.75\nError( %) - 41.905 35.238 28.571\n383NdCo2 0.227 141.557Nd 2.43 0.09 0.21 3.12 2.68\nCo 0.59 0.24 1.18 1.7 1.07\nError( %) 77.809 95.679 108.266 45.822\n384Tb2Sn2O7 0.48 141.557 Tb 5.87 5.8 5.87 5.94 6.02\nError( %) 1.193 0.0 1.193 2.555\n385Ho2Ru2O7 0.49 141.557 Ru 1.2 1.27 1.57 0.97 0.08\nError( %) 5.833 30.833 19.167 93.333\n386Ho2Ru2O7 0.51 141.557Ho 6.3 3.67 3.84 3.9 3.97\nRu 1.8 0.46 1.57 1.63 1.81\nError( %) 58.095 25.913 23.77 18.77\n387PrFe3(BO3)4 1.161 155.48Pr 0.79 - - 2.58 -\nFe 4.32 - - 3.94 -\nError( %) - - 117.689 -\n388U3As4 0.169 161.71 U 1.9 1.69 1.73 1.59 1.54\nError( %) 11.053 8.947 16.316 18.947\n389U3P4 0.170 161.71 U 1.48 1.62 1.66 1.49 1.5\nError( %) 9.459 12.162 0.676 1.351\n390Nd3Sb3Mg2O14 0.167 166.101 Nd 1.79 2.95 2.95 2.95 2.95\nError( %) 64.804 64.804 64.804 64.804\n391TbCo2 0.228 166.101Tb 8.3 0.29 5.89 6.01 5.24\nCo 1.3 0.03 0.06 0.37 1.41\nCo 1.19 0.04 0.06 0.37 1.49\nError( %) 96.946 73.127 56.012 23.513\n392Tb2Ti2O7 0.77 166.101Tb 5.54 - - - 6.06\nTb 3.54 - - - 6.06\nError( %) - - - 40.286\n393TbMg3 1.189 167.108 Tb 9.6 5.89 - 6.08 -\nError( %) 38.646 - 36.667 -122\n394HoMnO3 0.32 185.197 Mn 2.98 3.48 0.03 0.55 -\nError( %) 16.779 98.993 81.544 -\n395HoMnO3 0.33 185.197Ho 2.87 0.09 2.31 3.9 -\nMn 3.05 0.05 0.07 0.05 -\nError( %) 97.612 58.608 67.124 -\n396NdZn 3.8 222.103 Nd 2.51 3.37 3.33 3.17 3.14\nError( %) 34.263 32.669 26.295 25.1\n397NpSb 3.12 224.113 Np 2.86 3.46 3.61 3.71 3.79\nError( %) 20.979 26.224 29.72 32.517\n398UO2 3.2 224.113 U 1.73 0.96 1.49 1.53 1.56\nError( %) 44.509 13.873 11.561 9.827\n399NpBi 3.7 224.113 Np 2.42 3.59 3.69 3.76 3.83\nError( %) 48.347 52.479 55.372 58.264\n400NpSe 3.10 228.139 Np 1.3 3.42 3.64 3.77 3.86\nError( %) 163.077 180.0 190.0 196.923\n401NpTe 3.11 228.139 Np 1.4 3.69 3.76 3.85 3.92\nError( %) 163.571 168.571 175.0 180.0\n402NpS 3.9 228.139 Np 0.69 3.22 3.51 3.69 3.81\nError( %) 366.667 408.696 434.783 452.174\n403DyCu 3.6 229.143 Dy 8.63 4.4 4.41 4.65 5.04\nError( %) 49.015 48.899 46.118 41.599123\n1. Ferro(Ferri)magnetic materials\nAmong the 403 materials, there have 51 magnetic materials with net moments/ferromagnetic canting. All of them are tabulated in\nTable XIX with the BCSID, chemical formula and the MSG. All of the MSGs of them are Type-I/Type-III MSG, which are compatible\nwith ferro(ferri)magnets.\nTABLE XIX: Magnetic materials with non-zero net moments or ferromagnetic canting.\nBCSID Formula MSGBCSID Formula MSGBCSID Formula MSG\n0.26 TmAgGe 38.191 0.36 NiF2 58.398 0.37 U3Al2Si3 5.15\n0.48 Tb2Sn2O7 141.557 0.49 Ho2Ru2O7 141.557 0.64 MnV2O4 88.81\n0.77 Tb2Ti2O7 166.101 0.78 NiN2O6 148.17 0.85 KCo4(PO4)3 58.398\n0.91 Rb2Fe2O(AsO4)2 62.448 0.121 Li2Co(SO4)2 14.79 0.122 Li2Mn(SO4)2 14.75\n0.149 Nd3Ru4Al12 63.462 0.151 Tm2Mn2O7 141.557 0.157 Yb2Sn2O7 141.557\n0.165 SrMn(VO4)(OH) 4.70.169 U3As4 161.71 0.170 U3P4 161.71\n0.173 Pr3Ru4Al12 63.462 0.174 Pr3Ru4Al12 15.89 0.176 Mn3Ti2Te6 15.89\n0.184 Nd5Si4 92.114 0.185 Nd5Ge4 62.447 0.191 BaCuF4 36.176\n0.203 Mn3Ge 12.62 0.220 Mn2SiO4 62.446 0.226 NdCo2 15.89\n0.227 NdCo2 141.557 0.228 TbCo2 166.101 1.0.11 CeCoGe3 107.231\n1.0.12 UAu2Si2 71.536 1.0.13 FeI2 12.62 2.5 Mn3CuN 85.59\n2.10 HoP 15.89 2.11 TbMg 51.295 2.12 TbMg 49.270\n2.19 Mn3ZnC 139.537 2.28 NpNiGa5 74.559 3.3 Ho2RhIn8 63.464\nAppendix M: Band structures and detailed information\nWe list the band structures for all of the 403 magnetic materials in the following tables, with the energy range set between \u00001:0\u00181:0eV\nrelative to the Fermi level. Apart from the crystal and band structures for each material, the tables also include the material identification\nnumber on BCSMD (BCSID), chemical formula (Formula), the ICSD number if available, the magnetic space group (MSG) and stable\ntopological classification of the MSG (T.C.). For the magnetic TIs diagnosed by MTQC, see H for the stable topological indices and the\nphysical interpretations of them. For the magnetic ESs diagnosed by MTQC, see I for the high-symmetry kpaths that break compatibility\nrelations.124\nBCSID Formula ICSD MSG T.C.\n0.139 BiCrO3 174406 2.4(P\u00161) Z2Z2Z2Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XX. Topology phase diagram of BiCrO3.\nBCSID Formula ICSD MSG T.C.\n0.5 Cr2S3 626604 2.4(P\u00161) Z2Z2Z2Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXI. Topology phase diagram of Cr2S3.\nBCSID Formula ICSD MSG T.C.\n0.66 Fe2O3-alpha * 2.4(P\u00161) Z2Z2Z2Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXII. Topology phase diagram of Fe2O3-alpha.\nBCSID Formula ICSD MSG T.C.\n0.155 CaMnGe2O6 * 2.6(P\u001610) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXIII. Topology phase diagram of CaMnGe2O6.125\nBCSID Formula ICSD MSG T.C.\n0.180 MnPSe3 54140 2.6(P\u001610) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXIV. Topology phase diagram of MnPSe3.\nBCSID Formula ICSD MSG T.C.\n0.215 BaNi2P2O8 411629 2.6(P\u001610) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXV. Topology phase diagram of BaNi2P2O8.\nBCSID Formula ICSD MSG T.C.\n1.113 NiSb2O6 80802 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXVI. Topology phase diagram of NiSb2O6.\nBCSID Formula ICSD MSG T.C.\n1.155 LiFeSO4F 182944 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXVII. Topology phase diagram of LiFeSO4F.126\nBCSID Formula ICSD MSG T.C.\n1.159 Li2Ni(WO4)2 92853 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXVIII. Topology phase diagram of Li2Ni(WO4)2.\nBCSID Formula ICSD MSG T.C.\n1.166 La2LiOsO6 * 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXIX. Topology phase diagram of La2LiOsO6.\nBCSID Formula ICSD MSG T.C.\n1.167 NiS2 76684 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXX. Topology phase diagram of NiS2.\nBCSID Formula ICSD MSG T.C.\n1.177 Sr2CuWO6 193615 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXXI. Topology phase diagram of Sr2CuWO6.127\nBCSID Formula ICSD MSG T.C.\n1.180 Na3Co2SbO6 245538 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXXII. Topology phase diagram of Na3Co2SbO6.\nBCSID Formula ICSD MSG T.C.\n1.182 TlMnO3 * 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXXIII. Topology phase diagram of TlMnO3.\nBCSID Formula ICSD MSG T.C.\n1.190 YCr(BO3)2 189501 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXXIV. Topology phase diagram of YCr(BO3)2.\nBCSID Formula ICSD MSG T.C.\n1.193 CrTe3 * 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXXV. Topology phase diagram of CrTe3.128\nBCSID Formula ICSD MSG T.C.\n1.210 FePSe3 54141 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXXVI. Topology phase diagram of FePSe3.\nBCSID Formula ICSD MSG T.C.\n1.219 CuF2 9790 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXXVII. Topology phase diagram of CuF2.\nBCSID Formula ICSD MSG T.C.\n1.229 BaMoP2O8 79507 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXXVIII. Topology phase diagram of BaMoP2O8.\nBCSID Formula ICSD MSG T.C.\n1.240 FeI2 52369 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XXXIX. Topology phase diagram of FeI2.129\nBCSID Formula ICSD MSG T.C.\n1.244 CrCl3 22081 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XL. Topology phase diagram of CrCl3.\nBCSID Formula ICSD MSG T.C.\n1.256 BaNi2V2O8 96087 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XLI. Topology phase diagram of BaNi2V2O8.\nBCSID Formula ICSD MSG T.C.\n1.257 BaNi2As2O8 27014 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XLII. Topology phase diagram of BaNi2As2O8.\nBCSID Formula ICSD MSG T.C.\n1.260 NaMnGe2O6 237747 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XLIII. Topology phase diagram of NaMnGe2O6.130\nBCSID Formula ICSD MSG T.C.\n1.57 CuMnO2 * 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XLIV. Topology phase diagram of CuMnO2.\nBCSID Formula ICSD MSG T.C.\n2.18 Sc2NiMnO6 251833 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XLV. Topology phase diagram of Sc2NiMnO6.\nBCSID Formula ICSD MSG T.C.\n1.124 YBaFe4O7 * 4.10(Pa21) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XLVI. Topology phase diagram of YBaFe4O7.\nBCSID Formula ICSD MSG T.C.\n1.32 Lu2MnCoO6 * 4.10(Pa21) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XLVII. Topology phase diagram of Lu2MnCoO6.131\nBCSID Formula ICSD MSG T.C.\n1.51 Cs2CoCl4 62548 4.10(Pa21) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XLVIII. Topology phase diagram of Cs2CoCl4.\nBCSID Formula ICSD MSG T.C.\n1.64 BaNiF4 23141 4.10(Pa21) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XLIX. Topology phase diagram of BaNiF4.\nBCSID Formula ICSD MSG T.C.\n1.227 Ca2Cr2O5 238797 4.12(PC21) w/o\nTopology\nU=0 , LCEBR U=1eV , TBD U=2eV , TBD U=3eV , TBD U=4eV , LCEBR\nTABLE L. Topology phase diagram of Ca2Cr2O5.\nBCSID Formula ICSD MSG T.C.\n0.165 SrMn(VO4)(OH) * 4.7(P21) w/o\nTopology\nU=0 , ES U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LI. Topology phase diagram of SrMn(VO4)(OH).132\nBCSID Formula ICSD MSG T.C.\n0.83 LiFeP2O7 95751 4.7(P21) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LII. Topology phase diagram of LiFeP2O7.\nBCSID Formula ICSD MSG T.C.\n1.100 Cu2MnSnS4 56601 5.16(Cc2) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LIII. Topology phase diagram of Cu2MnSnS4.\nBCSID Formula ICSD MSG T.C.\n1.90 YFe3(BO3)4 * 5.16(Cc2) w/o\nTopology\nU=0 , LCEBR U=1eV , TBD U=2eV , LCEBR U=3eV , TBD U=4eV , LCEBR\nTABLE LIV. Topology phase diagram of YFe3(BO3)4.\nBCSID Formula ICSD MSG T.C.\n1.55 Na2MnF5 * 7.29(Pbc) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LV. Topology phase diagram of Na2MnF5.133\nBCSID Formula ICSD MSG T.C.\n1.75 BiMn2O5 * 8.36(Cam) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LVI. Topology phase diagram of BiMn2O5.\nBCSID Formula ICSD MSG T.C.\n0.50 MnTiO3 * 9.39(Cc0) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LVII. Topology phase diagram of MnTiO3.\nBCSID Formula ICSD MSG T.C.\n1.232 CuMnSb 628385 9.40(Ccc) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LVIII. Topology phase diagram of CuMnSb.\nBCSID Formula ICSD MSG T.C.\n1.58 La2O2Fe2OSe2 * 9.40(Ccc) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LIX. Topology phase diagram of La2O2Fe2OSe2.134\nBCSID Formula ICSD MSG T.C.\n1.120 BaFe2Se3 424315 9.41(Cac) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LX. Topology phase diagram of BaFe2Se3.\nBCSID Formula ICSD MSG T.C.\n1.79 Li2CoSiO4 245536 9.41(Cac) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXI. Topology phase diagram of Li2CoSiO4.\nBCSID Formula ICSD MSG T.C.\n1.228 RuCl3 * 10.49(PC2=m) Z2\nTopology\nU=0 , TBD U=1eV , TBD U=2eV , TBD U=3eV , LCEBR U=4eV , TBD\nTABLE LXII. Topology phase diagram of RuCl3.\nBCSID Formula ICSD MSG T.C.\n1.116 AgMnVO4 246202 11.55(Pa21=m) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXIII. Topology phase diagram of AgMnVO4.135\nBCSID Formula ICSD MSG T.C.\n1.134 Co2C10O8H2 * 11.57(PC21=m) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXIV. Topology phase diagram of Co2C10O8H2.\nBCSID Formula ICSD MSG T.C.\n1.230 NiPS3 646133 11.57(PC21=m) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXV. Topology phase diagram of NiPS3.\nBCSID Formula ICSD MSG T.C.\n1.264 CoPS3 * 11.57(PC21=m) Z2\nTopology\nU=0 , TI U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXVI. Topology phase diagram of CoPS3.\nBCSID Formula ICSD MSG T.C.\n0.163 MnPS3 61391 12.60(C20=m) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXVII. Topology phase diagram of MnPS3.136\nBCSID Formula ICSD MSG T.C.\n0.92 CaMn2Sb2 163778 12.60(C20=m) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXVIII. Topology phase diagram of CaMn2Sb2.\nBCSID Formula ICSD MSG T.C.\n1.0.1 Ag2CrO2 * 12.60(C20=m) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , TBD\nTABLE LXIX. Topology phase diagram of Ag2CrO2.\nBCSID Formula ICSD MSG T.C.\n0.203 Mn3Ge 603343 12.62(C20=m0) Z2Z2Z4\nTopology\nU=0 , TI U=1eV , TI U=2eV , TI U=3eV , TI U=4eV , TI\nTABLE LXX. Topology phase diagram of Mn3Ge.\nBCSID Formula ICSD MSG T.C.\n1.0.13 FeI2 52369 12.62(C20=m0) Z2Z2Z4\nTopology\nU=0 , TI U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXI. Topology phase diagram of FeI2.137\nBCSID Formula ICSD MSG T.C.\n1.0.6 CoV2O6 263002 12.62(C20=m0) Z2Z2Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXII. Topology phase diagram of CoV2O6.\nBCSID Formula ICSD MSG T.C.\n1.0.7 LuFe2O4 68481 12.62(C20=m0) Z2Z2Z4\nTopology\nU=0 , LCEBR U=1eV , TBD U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXIII. Topology phase diagram of LuFe2O4.\nBCSID Formula ICSD MSG T.C.\n3.14 FeI2 52369 12.62(C20=m0) Z2Z2Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXIV. Topology phase diagram of FeI2.\nBCSID Formula ICSD MSG T.C.\n1.183 FePS3 * 12.63(Cc2=m) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXV. Topology phase diagram of FePS3.138\nBCSID Formula ICSD MSG T.C.\n1.197 Fe4Si2Sn7O16 407506 12.63(Cc2=m) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXVI. Topology phase diagram of Fe4Si2Sn7O16.\nBCSID Formula ICSD MSG T.C.\n1.239 MnBr2 60250 12.63(Cc2=m) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXVII. Topology phase diagram of MnBr2.\nBCSID Formula ICSD MSG T.C.\n1.97 Li2MnO3 187499 12.63(Cc2=m) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXVIII. Topology phase diagram of Li2MnO3.\nBCSID Formula ICSD MSG T.C.\n2.2 Sr2F2Fe2OS2 * 12.64(Ca2=m) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXIX. Topology phase diagram of Sr2F2Fe2OS2.139\nBCSID Formula ICSD MSG T.C.\n1.194 NiWO4 15852 13.70(Pa2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXX. Topology phase diagram of NiWO4.\nBCSID Formula ICSD MSG T.C.\n1.114 Ca4IrO6 280873 13.74(PC2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXXI. Topology phase diagram of Ca4IrO6.\nBCSID Formula ICSD MSG T.C.\n1.121 NaFeSO4F 290051 13.74(PC2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXXII. Topology phase diagram of NaFeSO4F.\nBCSID Formula ICSD MSG T.C.\n1.126 NaCoSO4F 290052 13.74(PC2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXXIII. Topology phase diagram of NaCoSO4F.140\nBCSID Formula ICSD MSG T.C.\n0.103 Mn2GeO4 * 14.75(P21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXXIV. Topology phase diagram of Mn2GeO4.\nBCSID Formula ICSD MSG T.C.\n0.122 Li2Mn(SO4)2 290292 14.75(P21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXXV. Topology phase diagram of Li2Mn(SO4)2.\nBCSID Formula ICSD MSG T.C.\n0.148 La2LiRuO6 97488 14.75(P21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXXVI. Topology phase diagram of La2LiRuO6.\nBCSID Formula ICSD MSG T.C.\n0.204 Ca2MnReO6 * 14.75(P21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , TBD U=2eV , TBD U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXXVII. Topology phase diagram of Ca2MnReO6.141\nBCSID Formula ICSD MSG T.C.\n0.217 LiCrGe2O6 * 14.77(P20\n1=c) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXXVIII. Topology phase diagram of LiCrGe2O6.\nBCSID Formula ICSD MSG T.C.\n0.152 LiFePO4 * 14.78(P21=c0) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE LXXXIX. Topology phase diagram of LiFePO4.\nBCSID Formula ICSD MSG T.C.\n0.28 LiFeSi2O6 158579 14.78(P21=c0) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XC. Topology phase diagram of LiFeSi2O6.\nBCSID Formula ICSD MSG T.C.\n0.121 Li2Co(SO4)2 * 14.79(P20\n1=c0) Z2Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XCI. Topology phase diagram of Li2Co(SO4)2.142\nBCSID Formula ICSD MSG T.C.\n0.164 Y2MnCoO6 * 14.79(P20\n1=c0) Z2Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XCII. Topology phase diagram of Y2MnCoO6.\nBCSID Formula ICSD MSG T.C.\n1.133 CuSb2O6 80576 14.80(Pa21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XCIII. Topology phase diagram of CuSb2O6.\nBCSID Formula ICSD MSG T.C.\n1.147 Li2Fe(SO4)2 * 14.80(Pa21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XCIV. Topology phase diagram of Li2Fe(SO4)2.\nBCSID Formula ICSD MSG T.C.\n1.196 MnV2O6 * 14.80(Pa21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XCV. Topology phase diagram of MnV2O6.143\nBCSID Formula ICSD MSG T.C.\n1.199 Sc2NiMnO6 251833 14.80(Pa21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , TBD U=2eV , LCEBR U=3eV , LCEBR U=4eV , TBD\nTABLE XCVI. Topology phase diagram of Sc2NiMnO6.\nBCSID Formula ICSD MSG T.C.\n1.201 Cr2ReO6 * 14.80(Pa21=c) Z2\nTopology\nU=0 , TI U=1eV , LCEBR U=2eV , LCEBR U=3eV , TBD U=4eV , TBD\nTABLE XCVII. Topology phase diagram of Cr2ReO6.\nBCSID Formula ICSD MSG T.C.\n1.39 LiFeGe2O6 * 14.80(Pa21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XCVIII. Topology phase diagram of LiFeGe2O6.\nBCSID Formula ICSD MSG T.C.\n1.62 CuO * 14.80(Pa21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE XCIX. Topology phase diagram of CuO.144\nBCSID Formula ICSD MSG T.C.\n1.63 MnPb4Sb6S14 98581 14.80(Pa21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE C. Topology phase diagram of MnPb4Sb6S14.\nBCSID Formula ICSD MSG T.C.\n1.112 NiTa2O6 247807 14.82(Pc21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CI. Topology phase diagram of NiTa2O6.\nBCSID Formula ICSD MSG T.C.\n1.117 NaFePO4 85671 14.82(Pc21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CII. Topology phase diagram of NaFePO4.\nBCSID Formula ICSD MSG T.C.\n1.203 Ni2SiO4 35675 14.82(Pc21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CIII. Topology phase diagram of Ni2SiO4.145\nBCSID Formula ICSD MSG T.C.\n1.204 Ni2SiO4 35675 14.82(Pc21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CIV. Topology phase diagram of Ni2SiO4.\nBCSID Formula ICSD MSG T.C.\n1.26 CsFe2Se3 * 14.82(Pc21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CV. Topology phase diagram of CsFe2Se3.\nBCSID Formula ICSD MSG T.C.\n1.154 NaFeSi2O6 * 14.84(PC21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CVI. Topology phase diagram of NaFeSi2O6.\nBCSID Formula ICSD MSG T.C.\n1.169 CaCoGe2O6 * 14.84(PC21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CVII. Topology phase diagram of CaCoGe2O6.146\nBCSID Formula ICSD MSG T.C.\n1.2 CuSe2O5 603 14.84(PC21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CVIII. Topology phase diagram of CuSe2O5.\nBCSID Formula ICSD MSG T.C.\n1.60 Ca3Co2O6 246282 14.84(PC21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CIX. Topology phase diagram of Ca3Co2O6.\nBCSID Formula ICSD MSG T.C.\n1.94 Ba3LaRu2O9 * 14.84(PC21=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CX. Topology phase diagram of Ba3LaRu2O9.\nBCSID Formula ICSD MSG T.C.\n0.113 NiCO3 173985 15.85(C2=c) Z2Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXI. Topology phase diagram of NiCO3.147\nBCSID Formula ICSD MSG T.C.\n0.114 CoCO3 61066 15.85(C2=c) Z2Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXII. Topology phase diagram of CoCO3.\nBCSID Formula ICSD MSG T.C.\n0.115 MnCO3 * 15.85(C2=c) Z2Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXIII. Topology phase diagram of MnCO3.\nBCSID Formula ICSD MSG T.C.\n0.138 BiCrO3 174405 15.85(C2=c) Z2Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXIV. Topology phase diagram of BiCrO3.\nBCSID Formula ICSD MSG T.C.\n0.210 Sr2CoOsO6 * 15.85(C2=c) Z2Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXV. Topology phase diagram of Sr2CoOsO6.148\nBCSID Formula ICSD MSG T.C.\n0.110 Cr2O3 75577 15.87(C20=c) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXVI. Topology phase diagram of Cr2O3.\nBCSID Formula ICSD MSG T.C.\n0.145 Co3TeO6 183805 15.87(C20=c) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXVII. Topology phase diagram of Co3TeO6.\nBCSID Formula ICSD MSG T.C.\n0.156 CaMnGe2O6 * 15.87(C20=c) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXVIII. Topology phase diagram of CaMnGe2O6.\nBCSID Formula ICSD MSG T.C.\n0.196 Co4Nb2O9 172186 15.88(C2=c0) w/o\nTopology\nU=0 , TBD U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXIX. Topology phase diagram of Co4Nb2O9.149\nBCSID Formula ICSD MSG T.C.\n0.197 Co4Nb2O9 * 15.88(C2=c0) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXX. Topology phase diagram of Co4Nb2O9.\nBCSID Formula ICSD MSG T.C.\n0.112 FeBO3 34474 15.89(C20=c0) Z2Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXI. Topology phase diagram of FeBO3.\nBCSID Formula ICSD MSG T.C.\n0.128 FeSO4F 182945 15.89(C20=c0) Z2Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXII. Topology phase diagram of FeSO4F.\nBCSID Formula ICSD MSG T.C.\n0.176 Mn3Ti2Te6 41021 15.89(C20=c0) Z2Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXIII. Topology phase diagram of Mn3Ti2Te6.150\nBCSID Formula ICSD MSG T.C.\n0.3 Ca3LiOsO6 * 15.89(C20=c0) Z2Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXIV. Topology phase diagram of Ca3LiOsO6.\nBCSID Formula ICSD MSG T.C.\n0.65 Fe2O3-alpha * 15.89(C20=c0) Z2Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXV. Topology phase diagram of Fe2O3-alpha.\nBCSID Formula ICSD MSG T.C.\n1.168 Sr2CuTeO6 * 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXVI. Topology phase diagram of Sr2CuTeO6.\nBCSID Formula ICSD MSG T.C.\n1.17 CoV2O6-alpha * 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXVII. Topology phase diagram of CoV2O6-alpha.151\nBCSID Formula ICSD MSG T.C.\n1.209 FeI2 52369 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXVIII. Topology phase diagram of FeI2.\nBCSID Formula ICSD MSG T.C.\n1.245 CoBr2 52364 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXIX. Topology phase diagram of CoBr2.\nBCSID Formula ICSD MSG T.C.\n1.246 CoCl2 44398 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXX. Topology phase diagram of CoCl2.\nBCSID Formula ICSD MSG T.C.\n1.247 NiCl2 14208 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXXI. Topology phase diagram of NiCl2.152\nBCSID Formula ICSD MSG T.C.\n1.248 NiBr2 22106 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXXII. Topology phase diagram of NiBr2.\nBCSID Formula ICSD MSG T.C.\n1.31 MnO 9864 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXXIII. Topology phase diagram of MnO.\nBCSID Formula ICSD MSG T.C.\n1.49 Ag2NiO2 * 15.90(Cc2=c) Z2\nTopology\nU=0 , TI U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXXIV. Topology phase diagram of Ag2NiO2.\nBCSID Formula ICSD MSG T.C.\n1.6 NiO * 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXXV. Topology phase diagram of NiO.153\nBCSID Formula ICSD MSG T.C.\n1.69 CoO * 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXXVI. Topology phase diagram of CoO.\nBCSID Formula ICSD MSG T.C.\n1.70 CoV2O6 263002 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXXVII. Topology phase diagram of CoV2O6.\nBCSID Formula ICSD MSG T.C.\n1.37 VOCl * 15.91(Ca2=c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXXVIII. Topology phase diagram of VOCl.\nBCSID Formula ICSD MSG T.C.\n1.50 AgNiO2 * 18.22(PB21212) w/o\nTopology\nU=0 , ES U=1eV , ES U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXXXIX. Topology phase diagram of AgNiO2.154\nBCSID Formula ICSD MSG T.C.\n0.17 FePO4 * 19.25(P212121) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXL. Topology phase diagram of FePO4.\nBCSID Formula ICSD MSG T.C.\n0.129 Cu3Mo2O9 173779 19.27(P20\n120\n121) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXLI. Topology phase diagram of Cu3Mo2O9.\nBCSID Formula ICSD MSG T.C.\n1.224 CoNb2O6 15854 19.28(Pc212121) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXLII. Topology phase diagram of CoNb2O6.\nBCSID Formula ICSD MSG T.C.\n1.0.4 CsNiCl3 59371 20.34(C22020\n1) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXLIII. Topology phase diagram of CsNiCl3.155\nBCSID Formula ICSD MSG T.C.\n1.138 MgV2O4 * 20.37(CA2221) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXLIV. Topology phase diagram of MgV2O4.\nBCSID Formula ICSD MSG T.C.\n0.97 FeSb2O4 182798 26.66(Pmc 21) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXLV. Topology phase diagram of FeSb2O4.\nBCSID Formula ICSD MSG T.C.\n0.130 Cu3Mo2O9 * 26.68(Pm0c210) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXLVI. Topology phase diagram of Cu3Mo2O9.\nBCSID Formula ICSD MSG T.C.\n0.133 Ni3B7O13Cl * 29.101(Pc0a20\n1) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXLVII. Topology phase diagram of Ni3B7O13Cl.156\nBCSID Formula ICSD MSG T.C.\n0.135 Ni3B7O13Br 79127 29.101(Pc0a20\n1) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXLVIII. Topology phase diagram of Ni3B7O13Br.\nBCSID Formula ICSD MSG T.C.\n1.18 MnS2 36545 29.105(Pbca21) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXLIX. Topology phase diagram of MnS2.\nBCSID Formula ICSD MSG T.C.\n1.71 SrCo2V2O8 400765 29.110(PIca21) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CL. Topology phase diagram of SrCo2V2O8.\nBCSID Formula ICSD MSG T.C.\n1.101 LuMnO3 * 31.129(Pbmn21) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLI. Topology phase diagram of LuMnO3.157\nBCSID Formula ICSD MSG T.C.\n0.38 GaFeO3 151722 33.147(Pna020\n1) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLII. Topology phase diagram of GaFeO3.\nBCSID Formula ICSD MSG T.C.\n0.46 CaBaCo4O7 * 33.147(Pna020\n1) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLIII. Topology phase diagram of CaBaCo4O7.\nBCSID Formula ICSD MSG T.C.\n1.86 GeV4S8 * 33.149(Pana21) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLIV. Topology phase diagram of GeV4S8.\nBCSID Formula ICSD MSG T.C.\n1.263 Ca3Ru2O7 153776 33.154(PCna21) w/o\nTopology\nU=0 , ES U=1eV , ES U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLV. Topology phase diagram of Ca3Ru2O7.158\nBCSID Formula ICSD MSG T.C.\n0.56 Ba2CoGe2O7 * 35.167(Cm0m20) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLVI. Topology phase diagram of Ba2CoGe2O7.\nBCSID Formula ICSD MSG T.C.\n0.23 Ca3Mn2O7 55666 36.174(Cm0c20\n1) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLVII. Topology phase diagram of Ca3Mn2O7.\nBCSID Formula ICSD MSG T.C.\n1.0.3 CsCoBr3 * 36.174(Cm0c20\n1) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLVIII. Topology phase diagram of CsCoBr3.\nBCSID Formula ICSD MSG T.C.\n0.191 BaCuF4 9930 36.176(Cm0c021) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLIX. Topology phase diagram of BaCuF4.159\nBCSID Formula ICSD MSG T.C.\n1.74 BiMn2O5 * 36.178(Camc21) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLX. Topology phase diagram of BiMn2O5.\nBCSID Formula ICSD MSG T.C.\n1.172 NiTa2O6 * 41.217(Abba2) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXI. Topology phase diagram of NiTa2O6.\nBCSID Formula ICSD MSG T.C.\n0.137 Cu2V2O7 * 43.227(Fd0d02) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXII. Topology phase diagram of Cu2V2O7.\nBCSID Formula ICSD MSG T.C.\n0.62 SrMn2V2O8 182224 45.237(Ib0a20) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXIII. Topology phase diagram of SrMn2V2O8.160\nBCSID Formula ICSD MSG T.C.\n1.42 La2NiO4 69753 53.335(PCmna) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXIV. Topology phase diagram of La2NiO4.\nBCSID Formula ICSD MSG T.C.\n1.3 Sr2IrO4 78261 54.352(PIcca) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXV. Topology phase diagram of Sr2IrO4.\nBCSID Formula ICSD MSG T.C.\n1.30 BaCo2V2O8 * 54.352(PIcca) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXVI. Topology phase diagram of BaCo2V2O8.\nBCSID Formula ICSD MSG T.C.\n1.77 Sr2IrO4 78261 54.352(PIcca) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXVII. Topology phase diagram of Sr2IrO4.161\nBCSID Formula ICSD MSG T.C.\n0.45 La2NiO4 69753 56.369(Pc0c0n) Z2Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXVIII. Topology phase diagram of La2NiO4.\nBCSID Formula ICSD MSG T.C.\n1.122Cu3Bi(SeO3)2O2Br 280759 56.373(Pcccn) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXIX. Topology phase diagram of Cu3Bi(SeO3)2O2Br.\nBCSID Formula ICSD MSG T.C.\n1.123Cu3Y(SeO3)2O2Cl * 56.373(Pcccn) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXX. Topology phase diagram of Cu3Y(SeO3)2O2Cl.\nBCSID Formula ICSD MSG T.C.\n1.23 La2CuO4 87969 56.374(0PAccn0) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXI. Topology phase diagram of La2CuO4.162\nBCSID Formula ICSD MSG T.C.\n1.9 Li2VOSiO4 59359 57.389(PAbcm) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXII. Topology phase diagram of Li2VOSiO4.\nBCSID Formula ICSD MSG T.C.\n0.143 Cr2TeO6 24794 58.395(Pn0nm) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXIII. Topology phase diagram of Cr2TeO6.\nBCSID Formula ICSD MSG T.C.\n0.144 Cr2WO6 24793 58.395(Pn0nm) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXIV. Topology phase diagram of Cr2WO6.\nBCSID Formula ICSD MSG T.C.\n0.75 Cr2WO6 24793 58.395(Pn0nm) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXV. Topology phase diagram of Cr2WO6.163\nBCSID Formula ICSD MSG T.C.\n0.76 Cr2TeO6 24794 58.395(Pn0nm) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXVI. Topology phase diagram of Cr2TeO6.\nBCSID Formula ICSD MSG T.C.\n0.131 Mn(N(CN2))2 * 58.398(Pnn0m0) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXVII. Topology phase diagram of Mn(N(CN2))2.\nBCSID Formula ICSD MSG T.C.\n0.36 NiF2 * 58.398(Pnn0m0) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXVIII. Topology phase diagram of NiF2.\nBCSID Formula ICSD MSG T.C.\n0.85 KCo4(PO4)3 * 58.398(Pnn0m0) Z2\nTopology\nU=0 , ES U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXIX. Topology phase diagram of KCo4(PO4)3.164\nBCSID Formula ICSD MSG T.C.\n0.140 LuFe4Ge2 186047 58.399(Pn0n0m0) w/o\nTopology\nU=0 , ESFD U=1eV , ESFD U=2eV , ESFD U=3eV , ESFD U=4eV , ESFD\nTABLE CLXXX. Topology phase diagram of LuFe4Ge2.\nBCSID Formula ICSD MSG T.C.\n0.27 YFe4Ge2 * 58.399(Pn0n0m0) w/o\nTopology\nU=0 , ESFD U=1eV , ESFD U=2eV , ESFD U=3eV , ESFD U=4eV , ESFD\nTABLE CLXXXI. Topology phase diagram of YFe4Ge2.\nBCSID Formula ICSD MSG T.C.\n1.158 YMn3Al4O12 * 58.404(PInnm) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXXII. Topology phase diagram of YMn3Al4O12.\nBCSID Formula ICSD MSG T.C.\n0.222 CuMnAs 423230 59.407(Pm0mn) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXXIII. Topology phase diagram of CuMnAs.165\nBCSID Formula ICSD MSG T.C.\n1.252 CaCo2P2 85892 59.416(PImmn) Z2\nTopology\nU=0 , LCEBR U=1eV , TI U=2eV , TI U=3eV , TI U=4eV , TI\nTABLE CLXXXIV. Topology phase diagram of CaCo2P2.\nBCSID Formula ICSD MSG T.C.\n0.119 CoSe2O5 169716 60.419(Pb0cn) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXXV. Topology phase diagram of CoSe2O5.\nBCSID Formula ICSD MSG T.C.\n0.161 CoSe2O5 169716 60.419(Pb0cn) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXXVI. Topology phase diagram of CoSe2O5.\nBCSID Formula ICSD MSG T.C.\n1.88 Mn5Si3 * 60.431(PCbcn) Z2\nTopology\nU=0 , TI U=1eV , ES U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXXVII. Topology phase diagram of Mn5Si3.166\nBCSID Formula ICSD MSG T.C.\n0.40 Mn2O3-alpha * 61.433(Pbca) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXXVIII. Topology phase diagram of Mn2O3-alpha.\nBCSID Formula ICSD MSG T.C.\n0.41 Mn2O3-alpha * 61.433(Pbca) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CLXXXIX. Topology phase diagram of Mn2O3-alpha.\nBCSID Formula ICSD MSG T.C.\n0.71 Li2Ni(SO4)2 * 61.437(Pb0c0a0) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXC. Topology phase diagram of Li2Ni(SO4)2.\nBCSID Formula ICSD MSG T.C.\n2.1 EuFe2As2 * 61.439(PCbca) Z2\nTopology\nU=0 , LCEBR U=1eV , ES U=2eV , TI U=3eV , TI U=4eV , LCEBR\nTABLE CXCI. Topology phase diagram of EuFe2As2.167\nBCSID Formula ICSD MSG T.C.\n0.102 Mn2GeO4 * 62.441(Pnma) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXCII. Topology phase diagram of Mn2GeO4.\nBCSID Formula ICSD MSG T.C.\n0.168 NH4Fe2O6 * 62.441(Pnma) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXCIII. Topology phase diagram of NH4Fe2O6.\nBCSID Formula ICSD MSG T.C.\n0.192 RbFe2F6 186748 62.441(Pnma) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXCIV. Topology phase diagram of RbFe2F6.\nBCSID Formula ICSD MSG T.C.\n0.218 Co2SiO4 260092 62.441(Pnma) Z2\nTopology\nU=0 , ES U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXCV. Topology phase diagram of Co2SiO4.168\nBCSID Formula ICSD MSG T.C.\n0.219 Co2SiO4 260092 62.441(Pnma) Z2\nTopology\nU=0 , ES U=1eV , ES U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXCVI. Topology phase diagram of Co2SiO4.\nBCSID Formula ICSD MSG T.C.\n0.221 Fe2SiO4 26375 62.441(Pnma) Z2\nTopology\nU=0 , ES U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXCVII. Topology phase diagram of Fe2SiO4.\nBCSID Formula ICSD MSG T.C.\n0.90Rb2Fe2O(AsO4)2 * 62.441(Pnma) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXCVIII. Topology phase diagram of Rb2Fe2O(AsO4)2.\nBCSID Formula ICSD MSG T.C.\n0.96 CoSO4 18175 62.441(Pnma) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CXCIX. Topology phase diagram of CoSO4.169\nBCSID Formula ICSD MSG T.C.\n0.182 KCrF4 60896 62.443(Pn0ma) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CC. Topology phase diagram of KCrF4.\nBCSID Formula ICSD MSG T.C.\n0.88 LiNiPO4 402760 62.444(Pnm0a) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCI. Topology phase diagram of LiNiPO4.\nBCSID Formula ICSD MSG T.C.\n0.193 LiCoPO4 400625 62.445(Pnma0) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCII. Topology phase diagram of LiCoPO4.\nBCSID Formula ICSD MSG T.C.\n0.216 SrEr2O4 239309 62.445(Pnma0) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCIII. Topology phase diagram of SrEr2O4.170\nBCSID Formula ICSD MSG T.C.\n0.86 KMn4(PO4)3 246135 62.445(Pnma0) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCIV. Topology phase diagram of KMn4(PO4)3.\nBCSID Formula ICSD MSG T.C.\n0.87 NaFePO4 169118 62.445(Pnma0) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCV. Topology phase diagram of NaFePO4.\nBCSID Formula ICSD MSG T.C.\n0.95 LiFePO4 * 62.445(Pnma0) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCVI. Topology phase diagram of LiFePO4.\nBCSID Formula ICSD MSG T.C.\n0.101 Mn2GeO4 * 62.446(Pn0m0a) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCVII. Topology phase diagram of Mn2GeO4.171\nBCSID Formula ICSD MSG T.C.\n0.220 Mn2SiO4 88026 62.446(Pn0m0a) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCVIII. Topology phase diagram of Mn2SiO4.\nBCSID Formula ICSD MSG T.C.\n0.1 LaMnO3 * 62.448(Pn0ma0) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCIX. Topology phase diagram of LaMnO3.\nBCSID Formula ICSD MSG T.C.\n0.25 NaOsO3 * 62.448(Pn0ma0) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCX. Topology phase diagram of NaOsO3.\nBCSID Formula ICSD MSG T.C.\n0.91Rb2Fe2O(AsO4)2 * 62.448(Pn0ma0) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXI. Topology phase diagram of Rb2Fe2O(AsO4)2.172\nBCSID Formula ICSD MSG T.C.\n0.24 LiMnPO4 25834 62.449(Pn0m0a0) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXII. Topology phase diagram of LiMnPO4.\nBCSID Formula ICSD MSG T.C.\n1.130 Cr2As 42335 62.450(Panma) Z2\nTopology\nU=0 , LCEBR U=1eV , TI U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXIII. Topology phase diagram of Cr2As.\nBCSID Formula ICSD MSG T.C.\n1.131 Fe2As 42335 62.450(Panma) Z2\nTopology\nU=0 , TI U=1eV , TI U=2eV , ES U=3eV , TI U=4eV , TI\nTABLE CCXIV. Topology phase diagram of Fe2As.\nBCSID Formula ICSD MSG T.C.\n1.132 Mn2As 164396 62.450(Panma) Z2\nTopology\nU=0 , ES U=1eV , ES U=2eV , ES U=3eV , ES U=4eV , ES\nTABLE CCXV. Topology phase diagram of Mn2As.173\nBCSID Formula ICSD MSG T.C.\n1.28 CrN * 62.450(Panma) Z2\nTopology\nU=0 , TI U=1eV , TI U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXVI. Topology phase diagram of CrN.\nBCSID Formula ICSD MSG T.C.\n1.1 Mn3O4 * 62.452(Pcnma) Z2\nTopology\nU=0 , LCEBR U=1eV , TBD U=2eV , TBD U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXVII. Topology phase diagram of Mn3O4.\nBCSID Formula ICSD MSG T.C.\n2.27 Sr2Mn3Sb2O2 81792 63.459(Cm0cm) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXVIII. Topology phase diagram of Sr2Mn3Sb2O2.\nBCSID Formula ICSD MSG T.C.\n0.199 Mn3Sn 643730 63.463(Cmc0m0) Z2Z2\nTopology\nU=0 , ES U=1eV , ES U=2eV , ES U=3eV , ES U=4eV , ES\nTABLE CCXIX. Topology phase diagram of Mn3Sn.174\nBCSID Formula ICSD MSG T.C.\n0.200 Mn3Sn 643730 63.464(Cm0cm0) Z2Z2\nTopology\nU=0 , TI U=1eV , TI U=2eV , TI U=3eV , ES U=4eV , ES\nTABLE CCXX. Topology phase diagram of Mn3Sn.\nBCSID Formula ICSD MSG T.C.\n0.79 CaIrO3 160816 63.464(Cm0cm0) Z2Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXXI. Topology phase diagram of CaIrO3.\nBCSID Formula ICSD MSG T.C.\n0.82 Gd2CuO4 75425 64.476(Cm0ca0) Z2Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXXII. Topology phase diagram of Gd2CuO4.\nBCSID Formula ICSD MSG T.C.\n1.145 Mn3Ni20P6 72351 64.480(CAmca) Z2\nTopology\nU=0 , LCEBR U=1eV , TBD U=2eV , TBD U=3eV , TBD U=4eV , LCEBR\nTABLE CCXXIII. Topology phase diagram of Mn3Ni20P6.175\nBCSID Formula ICSD MSG T.C.\n1.16 BaFe2As2 * 64.480(CAmca) Z2\nTopology\nU=0 , TI U=1eV , TI U=2eV , TI U=3eV , TI U=4eV , LCEBR\nTABLE CCXXIV. Topology phase diagram of BaFe2As2.\nBCSID Formula ICSD MSG T.C.\n1.249 K2NiF4 33520 64.480(CAmca) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXXV. Topology phase diagram of K2NiF4.\nBCSID Formula ICSD MSG T.C.\n1.52 CaFe2As2 * 64.480(CAmca) Z2\nTopology\nU=0 , TI U=1eV , TI U=2eV , TI U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXXVI. Topology phase diagram of CaFe2As2.\nBCSID Formula ICSD MSG T.C.\n2.15 Mn3Ni20P6 72351 65.486(Cmm0m0) Z2Z2\nTopology\nU=0 , TBD U=1eV , ES U=2eV , ES U=3eV , TI U=4eV , ES\nTABLE CCXXVII. Topology phase diagram of Mn3Ni20P6.176\nBCSID Formula ICSD MSG T.C.\n1.104 Gd2CuO4 65015 66.500(CAccm) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXXVIII. Topology phase diagram of Gd2CuO4.\nBCSID Formula ICSD MSG T.C.\n1.65 SrFeO2 * 69.526(FSmmm) Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXXIX. Topology phase diagram of SrFeO2.\nBCSID Formula ICSD MSG T.C.\n0.4 NiCr2O4 280061 70.530(Fd0d0d) Z2\nTopology\nU=0 , ES U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXXX. Topology phase diagram of NiCr2O4.\nBCSID Formula ICSD MSG T.C.\n1.125 LaFeAsO 180434 73.553(Icbca) Z2\nTopology\nU=0 , TI U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXXXI. Topology phase diagram of LaFeAsO.177\nBCSID Formula ICSD MSG T.C.\n1.176 YbCo2Si2 625132 73.553(Icbca) Z2\nTopology\nU=0 , LCEBR U=1eV , TI U=2eV , TI U=3eV , TI U=4eV , TI\nTABLE CCXXXII. Topology phase diagram of YbCo2Si2.\nBCSID Formula ICSD MSG T.C.\n1.47 Sr2FeOsO6 * 83.50(PI4=m) Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXXXIII. Topology phase diagram of Sr2FeOsO6.\nBCSID Formula ICSD MSG T.C.\n2.5 Mn3CuN 628356 85.59(P4=n) Z2Z4\nTopology\nU=0 , ES U=1eV , ES U=2eV , ES U=3eV , ES U=4eV , ES\nTABLE CCXXXIV. Topology phase diagram of Mn3CuN.\nBCSID Formula ICSD MSG T.C.\n2.23 Sr2CoO2Ag2Se2 * 86.73(PC42=n) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXXXV. Topology phase diagram of Sr2CoO2Ag2Se2.178\nBCSID Formula ICSD MSG T.C.\n2.24 Ba2CoO2Ag2Se2 * 86.73(PC42=n) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXXXVI. Topology phase diagram of Ba2CoO2Ag2Se2.\nBCSID Formula ICSD MSG T.C.\n0.64 MnV2O4 * 88.81(I41=a) Z2Z2\nTopology\nU=0 , ES U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXXXVII. Topology phase diagram of MnV2O4.\nBCSID Formula ICSD MSG T.C.\n2.22 FeTa2O6 201754 88.86(Ic41=a) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXXXVIII. Topology phase diagram of FeTa2O6.\nBCSID Formula ICSD MSG T.C.\n1.235Ba(TiO)Cu4(PO4)4 239146 94.132(Pc42212) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXXXIX. Topology phase diagram of Ba(TiO)Cu4(PO4)4.179\nBCSID Formula ICSD MSG T.C.\n1.24 ZnV2O4 55443 96.150(PI43212) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXL. Topology phase diagram of ZnV2O4.\nBCSID Formula ICSD MSG T.C.\n3.4 MgCr2O4 * 111.255(P\u0016420m0) Z2Z2Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXLI. Topology phase diagram of MgCr2O4.\nBCSID Formula ICSD MSG T.C.\n1.85 alpha-Mn * 114.282(PI\u0016421c) Z2\nTopology\nU=0 , ES U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , TBD\nTABLE CCXLII. Topology phase diagram of alpha-Mn.\nBCSID Formula ICSD MSG T.C.\n1.185 GeCu2O4 100796 122.338(Ic\u001642d) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXLIII. Topology phase diagram of GeCu2O4.180\nBCSID Formula ICSD MSG T.C.\n0.72 CaMnBi2 10454 129.416(P40=n0m0m) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXLIV. Topology phase diagram of CaMnBi2.\nBCSID Formula ICSD MSG T.C.\n1.143 Mn3Pt * 132.456(Pc42=mcm) Z4\nTopology\nU=0 , ESFD U=1eV , ESFD U=2eV , ESFD U=3eV , ESFD U=4eV , ESFD\nTABLE CCXLV. Topology phase diagram of Mn3Pt.\nBCSID Formula ICSD MSG T.C.\n0.15 MnF2 68735 136.499(P40\n2=mnm0) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXLVI. Topology phase diagram of MnF2.\nBCSID Formula ICSD MSG T.C.\n0.178 CoF2 98786 136.499(P40\n2=mnm0) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXLVII. Topology phase diagram of CoF2.181\nBCSID Formula ICSD MSG T.C.\n0.142 Fe2TeO6 24795 136.503(P42=m0n0m0) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXLVIII. Topology phase diagram of Fe2TeO6.\nBCSID Formula ICSD MSG T.C.\n1.146 LaCrAsO 185855 138.528(Pc42=ncm) Z2\nTopology\nU=0 , TI U=1eV , TI U=2eV , TI U=3eV , TI U=4eV , TI\nTABLE CCXLIX. Topology phase diagram of LaCrAsO.\nBCSID Formula ICSD MSG T.C.\n0.18 BaMn2As2 * 139.536(I40=m0m0m) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCL. Topology phase diagram of BaMn2As2.\nBCSID Formula ICSD MSG T.C.\n0.212 Sr2Mn3As2O2 81803 139.536(I40=m0m0m) Z2\nTopology\nU=0 , ESFD U=1eV , ESFD U=2eV , ESFD U=3eV , ESFD U=4eV , ESFD\nTABLE CCLI. Topology phase diagram of Sr2Mn3As2O2.182\nBCSID Formula ICSD MSG T.C.\n0.73 SrMnBi2 100025 139.536(I40=m0m0m) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLII. Topology phase diagram of SrMnBi2.\nBCSID Formula ICSD MSG T.C.\n0.89 BaMn2Bi2 * 139.536(I40=m0m0m) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLIII. Topology phase diagram of BaMn2Bi2.\nBCSID Formula ICSD MSG T.C.\n2.19 Mn3ZnC 618284 139.537(I4=mm0m0) Z4Z4\nTopology\nU=0 , ES U=1eV , ES U=2eV , ES U=3eV , ES U=4eV , ES\nTABLE CCLIV. Topology phase diagram of Mn3ZnC.\nBCSID Formula ICSD MSG T.C.\n1.250 KNiF3 15426 140.550(Ic4=mcm) Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLV. Topology phase diagram of KNiF3.183\nBCSID Formula ICSD MSG T.C.\n0.58 CoAl2O4 * 141.556(I40\n1=a0m0d) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLVI. Topology phase diagram of CoAl2O4.\nBCSID Formula ICSD MSG T.C.\n0.211 Ca2MnO4 99523 142.568(I40\n1=a0cd0) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLVII. Topology phase diagram of Ca2MnO4.\nBCSID Formula ICSD MSG T.C.\n1.156 LaMn3Cr4O12 * 146.12(RI3) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLVIII. Topology phase diagram of LaMn3Cr4O12.\nBCSID Formula ICSD MSG T.C.\n1.165 Ni3TeO6 240377 146.12(RI3) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLIX. Topology phase diagram of Ni3TeO6.184\nBCSID Formula ICSD MSG T.C.\n0.78 NiN2O6 * 148.17(R\u00163) Z2Z4\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLX. Topology phase diagram of NiN2O6.\nBCSID Formula ICSD MSG T.C.\n0.125 MnGeO3 174042 148.19(R\u001630) w/o\nTopology\nU=0 , ES U=1eV , ES U=2eV , ES U=3eV , ES U=4eV , ES\nTABLE CCLXI. Topology phase diagram of MnGeO3.\nBCSID Formula ICSD MSG T.C.\n0.19 MnTiO3 171579 148.19(R\u001630) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXII. Topology phase diagram of MnTiO3.\nBCSID Formula ICSD MSG T.C.\n1.0.8 Ba3MnNb2O9 * 157.53(P31m) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , TBD U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXIII. Topology phase diagram of Ba3MnNb2O9.185\nBCSID Formula ICSD MSG T.C.\n1.13 Ba3Nb2NiO9 240280 159.64(Pc31c) w/o\nTopology\nU=0 , LCEBR U=1eV , TBD U=2eV , LCEBR U=3eV , LCEBR U=4eV , TBD\nTABLE CCLXIV. Topology phase diagram of Ba3Nb2NiO9.\nBCSID Formula ICSD MSG T.C.\n1.237 VCl2 246905 159.64(Pc31c) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXV. Topology phase diagram of VCl2.\nBCSID Formula ICSD MSG T.C.\n1.238 VBr2 246906 159.64(Pc31c) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXVI. Topology phase diagram of VBr2.\nBCSID Formula ICSD MSG T.C.\n0.21 PbNiO3 * 161.69(R3c) w/o\nTopology\nU=0 , ESFD U=1eV , ESFD U=2eV , ESFD U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXVII. Topology phase diagram of PbNiO3.186\nBCSID Formula ICSD MSG T.C.\n1.233 CuMnSb 628385 161.72(RI3c) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXVIII. Topology phase diagram of CuMnSb.\nBCSID Formula ICSD MSG T.C.\n1.265 CuMnSb 42978 161.72(RI3c) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXIX. Topology phase diagram of CuMnSb.\nBCSID Formula ICSD MSG T.C.\n1.186 SrRu2O6 248351 162.78(Pc\u001631m) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXX. Topology phase diagram of SrRu2O6.\nBCSID Formula ICSD MSG T.C.\n0.111 Co4Nb2O9 172186 165.94(P\u001630c01) w/o\nTopology\nU=0 , TBD U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXXI. Topology phase diagram of Co4Nb2O9.187\nBCSID Formula ICSD MSG T.C.\n1.0.10 Sr3NiIrO6 80287 165.95(P\u00163c01) Z12\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXXII. Topology phase diagram of Sr3NiIrO6.\nBCSID Formula ICSD MSG T.C.\n1.0.5 Sr3CoIrO6 * 165.95(P\u00163c01) Z12\nTopology\nU=0 , ES U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXXIII. Topology phase diagram of Sr3CoIrO6.\nBCSID Formula ICSD MSG T.C.\n1.242 FeBr2 409571 165.96(Pc\u00163c1) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXXIV. Topology phase diagram of FeBr2.\nBCSID Formula ICSD MSG T.C.\n0.108 Mn3Ir * 166.101(R\u00163m0) Z2Z4\nTopology\nU=0 , ES U=1eV , ES U=2eV , ES U=3eV , ES U=4eV , ES\nTABLE CCLXXV. Topology phase diagram of Mn3Ir.188\nBCSID Formula ICSD MSG T.C.\n0.109 Mn3Pt * 166.101(R\u00163m0) Z2Z4\nTopology\nU=0 , ES U=1eV , ES U=2eV , ES U=3eV , ES U=4eV , ES\nTABLE CCLXXVI. Topology phase diagram of Mn3Pt.\nBCSID Formula ICSD MSG T.C.\n0.177 Mn3GaN 87399 166.97(R\u00163m) Z2\nTopology\nU=0 , ES U=1eV , ES U=2eV , ES U=3eV , ES U=4eV , ESFD\nTABLE CCLXXVII. Topology phase diagram of Mn3GaN.\nBCSID Formula ICSD MSG T.C.\n0.116 FeCO3 100678 167.103(R\u00163c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXXVIII. Topology phase diagram of FeCO3.\nBCSID Formula ICSD MSG T.C.\n0.59 Cr2O3 * 167.106(R\u001630c0) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXXIX. Topology phase diagram of Cr2O3.189\nBCSID Formula ICSD MSG T.C.\n1.153 Mn3GaC 23586 167.108(RI\u00163c) Z2\nTopology\nU=0 , ES U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXXX. Topology phase diagram of Mn3GaC.\nBCSID Formula ICSD MSG T.C.\n1.241 FeCl2 44397 167.108(RI\u00163c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXXXI. Topology phase diagram of FeCl2.\nBCSID Formula ICSD MSG T.C.\n0.8 ScMnO3 50694 173.129(P63) Z3\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXXXII. Topology phase diagram of ScMnO3.\nBCSID Formula ICSD MSG T.C.\n0.44 YMnO3 * 173.131(P60\n3) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXXXIII. Topology phase diagram of YMnO3.190\nBCSID Formula ICSD MSG T.C.\n0.6 YMnO3 * 185.197(P63cm) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXXXIV. Topology phase diagram of YMnO3.\nBCSID Formula ICSD MSG T.C.\n0.117 LuFeO3 183152 185.201(P63c0m0) Z3\nTopology\nU=0 , TI U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXXXV. Topology phase diagram of LuFeO3.\nBCSID Formula ICSD MSG T.C.\n0.7 ScMnO3 50694 185.201(P63c0m0) Z3\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXXXVI. Topology phase diagram of ScMnO3.\nBCSID Formula ICSD MSG T.C.\n1.0.14 CsFeCl3 300249 189.223(P\u00166020m) w/o\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCLXXXVII. Topology phase diagram of CsFeCl3.191\nBCSID Formula ICSD MSG T.C.\n1.110 ScMn6Ge6 * 192.252(Pc6=mcc) Z6\nTopology\nU=0 , ES U=1eV , ES U=2eV , ES U=3eV , ES U=4eV , ES\nTABLE CCLXXXVIII. Topology phase diagram of ScMn6Ge6.\nBCSID Formula ICSD MSG T.C.\n1.225 ScMn6Ge6 192907 192.252(Pc6=mcc) Z6\nTopology\nU=0 , ES U=1eV , ES U=2eV , ES U=3eV , ES U=4eV , ES\nTABLE CCLXXXIX. Topology phase diagram of ScMn6Ge6.\nBCSID Formula ICSD MSG T.C.\n1.0.9 CsCoCl3 27511 193.259(P60\n3=m0cm0) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXC. Topology phase diagram of CsCoCl3.\nBCSID Formula ICSD MSG T.C.\n0.118 Ba5Co5ClO13 99371 194.268(P60\n3=m0m0c) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , ES\nTABLE CCXCI. Topology phase diagram of Ba5Co5ClO13.192\nBCSID Formula ICSD MSG T.C.\n0.70 Na3Co(CO3)2Cl 262045 203.26(Fd\u00163) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXCII. Topology phase diagram of Na3Co(CO3)2Cl.\nBCSID Formula ICSD MSG T.C.\n0.150 NiS2 76684 205.33(Pa\u00163) Z2\nTopology\nU=0 , TI U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXCIII. Topology phase diagram of NiS2.\nBCSID Formula ICSD MSG T.C.\n0.20 MnTe2 * 205.33(Pa\u00163) Z2\nTopology\nU=0 , LCEBR U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXCIV. Topology phase diagram of MnTe2.\nBCSID Formula ICSD MSG T.C.\n0.2 Cd2Os2O7 155771 227.131(Fd\u00163m0) Z2\nTopology\nU=0 , ES U=1eV , LCEBR U=2eV , LCEBR U=3eV , LCEBR U=4eV , LCEBR\nTABLE CCXCV. Topology phase diagram of Cd2Os2O7.193\nBCS ID Formula ICSD MSG T.C.\n1.191 HoCr(BO3)2 189502 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=2eV , TBD U=4eV , LCEBR U=6eV , TBD\nTABLE CCXCVI. Topology phase diagram of HoCr(BO3)2.\nBCS ID Formula ICSD MSG T.C.\n1.206 Dy2Fe2Si2C * 2.7(PS\u00161) Z2\nTopology\nU=0 , TI U=2eV , TI U=4eV , TI U=6eV , TI\nTABLE CCXCVII. Topology phase diagram of Dy2Fe2Si2C.\nBCS ID Formula ICSD MSG T.C.\n1.218 Tm2BaNiO5 72631 2.7(PS\u00161) Z2\nTopology\nU=0 , TBD U=2eV , TBD U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCXCVIII. Topology phase diagram of Tm2BaNiO5.\nBCS ID Formula ICSD MSG T.C.\n1.38 Nd2NaOsO6 * 2.7(PS\u00161) Z2\nTopology\nU=0 , LCEBR U=2eV , TBD U=4eV , LCEBR U=6eV , TBD\nTABLE CCXCIX. Topology phase diagram of Nd2NaOsO6.194\nBCS ID Formula ICSD MSG T.C.\n0.37 U3Al2Si3 * 5.15(C20) w/o\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCC. Topology phase diagram of U3Al2Si3.\nBCS ID Formula ICSD MSG T.C.\n1.111 GdBiPt 58786 9.40(Ccc) w/o\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCI. Topology phase diagram of GdBiPt.\nBCS ID Formula ICSD MSG T.C.\n0.104 ErVO3 185838 11.54(P20\n1=m0) Z2Z2Z4\nTopology\nU=0 , LCEBR U=2eV , TI U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCII. Topology phase diagram of ErVO3.\nBCS ID Formula ICSD MSG T.C.\n0.106 DyVO3 40392 11.54(P20\n1=m0) Z2Z2Z4\nTopology\nU=0 , TI U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCIII. Topology phase diagram of DyVO3.195\nBCS ID Formula ICSD MSG T.C.\n1.171 Tb2Fe2Si2C * 12.63(Cc2=m) Z2\nTopology\nU=0 , LCEBR U=2eV , TBD U=4eV , TBD U=6eV , TBD\nTABLE CCCIV. Topology phase diagram of Tb2Fe2Si2C.\nBCS ID Formula ICSD MSG T.C.\n1.22 DyCu2Si2 627185 12.63(Cc2=m) Z2\nTopology\nU=0 , TBD U=2eV , TBD U=4eV , LCEBR U=6eV , ES\nTABLE CCCV. Topology phase diagram of DyCu2Si2.\nBCS ID Formula ICSD MSG T.C.\n0.188 CeMnAsO 195499 13.67(P20=c) w/o\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCVI. Topology phase diagram of CeMnAsO.\nBCS ID Formula ICSD MSG T.C.\n1.140 PrMgPb * 13.73(PA2=c) Z2\nTopology\nU=0 , TI U=2eV , TI U=4eV , TI U=6eV , TI\nTABLE CCCVII. Topology phase diagram of PrMgPb.196\nBCS ID Formula ICSD MSG T.C.\n1.141 NdMgPb * 13.73(PA2=c) Z2\nTopology\nU=0 , TI U=2eV , TI U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCVIII. Topology phase diagram of NdMgPb.\nBCS ID Formula ICSD MSG T.C.\n1.213 Ho2O2Se 25809 13.73(PA2=c) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCIX. Topology phase diagram of Ho2O2Se.\nBCS ID Formula ICSD MSG T.C.\n0.105 ErVO3 185838 14.75(P21=c) Z2\nTopology\nU=0 , ES U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCX. Topology phase diagram of ErVO3.\nBCS ID Formula ICSD MSG T.C.\n0.39 Nd2NaRuO6 * 14.75(P21=c) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXI. Topology phase diagram of Nd2NaRuO6.197\nBCS ID Formula ICSD MSG T.C.\n2.21 TbOOH 6164 14.78(P21=c0) w/o\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , TBD\nTABLE CCCXII. Topology phase diagram of TbOOH.\nBCS ID Formula ICSD MSG T.C.\n2.8 SrHo2O4 246550 14.78(P21=c0) w/o\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXIII. Topology phase diagram of SrHo2O4.\nBCS ID Formula ICSD MSG T.C.\n1.95 BaNd2O4 78486 14.80(Pa21=c) Z2\nTopology\nU=0 , LCEBR U=2eV , TBD U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXIV. Topology phase diagram of BaNd2O4.\nBCS ID Formula ICSD MSG T.C.\n1.35 LiErF4 * 14.84(PC21=c) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXV. Topology phase diagram of LiErF4.198\nBCS ID Formula ICSD MSG T.C.\n0.174 Pr3Ru4Al12 * 15.89(C20=c0) Z2Z4\nTopology\nU=0 , TI U=2eV , TI U=4eV , TI U=6eV , TI\nTABLE CCCXVI. Topology phase diagram of Pr3Ru4Al12.\nBCS ID Formula ICSD MSG T.C.\n0.226 NdCo2 246554 15.89(C20=c0) Z2Z4\nTopology\nU=0 , ES U=2eV , TI U=4eV , TI U=6eV , TI\nTABLE CCCXVII. Topology phase diagram of NdCo2.\nBCS ID Formula ICSD MSG T.C.\n2.10 HoP * 15.89(C20=c0) Z2Z4\nTopology\nU=0 , TI U=2eV , TI U=4eV , TI U=6eV , TI\nTABLE CCCXVIII. Topology phase diagram of HoP.\nBCS ID Formula ICSD MSG T.C.\n1.14 Ho2BaNiO5 67930 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXIX. Topology phase diagram of Ho2BaNiO5.199\nBCS ID Formula ICSD MSG T.C.\n1.15 Er2BaNiO5 72630 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXX. Topology phase diagram of Er2BaNiO5.\nBCS ID Formula ICSD MSG T.C.\n1.211 Dy2O2S * 15.90(Cc2=c) Z2\nTopology\nU=0 , TI U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXXI. Topology phase diagram of Dy2O2S.\nBCS ID Formula ICSD MSG T.C.\n1.212 Dy2O2Se * 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXXII. Topology phase diagram of Dy2O2Se.\nBCS ID Formula ICSD MSG T.C.\n1.216 Nd2BaNiO5 72626 15.90(Cc2=c) Z2\nTopology\nU=0 , TI U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXXIII. Topology phase diagram of Nd2BaNiO5.200\nBCS ID Formula ICSD MSG T.C.\n1.217 Tb2BaNiO5 66078 15.90(Cc2=c) Z2\nTopology\nU=0 , TI U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXXIV. Topology phase diagram of Tb2BaNiO5.\nBCS ID Formula ICSD MSG T.C.\n1.36 Dy2BaNiO5 72627 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXXV. Topology phase diagram of Dy2BaNiO5.\nBCS ID Formula ICSD MSG T.C.\n1.53 Er2BaNiO5 72630 15.90(Cc2=c) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXXVI. Topology phase diagram of Er2BaNiO5.\nBCS ID Formula ICSD MSG T.C.\n1.20 HoMnO3 * 31.129(Pbmn21) w/o\nTopology\nU=0 , LCEBR U=2eV , TBD U=4eV , TBD U=6eV , TBD\nTABLE CCCXXVII. Topology phase diagram of HoMnO3.201\nBCS ID Formula ICSD MSG T.C.\n1.33 ErAuGe * 33.154(PCna21) w/o\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXXVIII. Topology phase diagram of ErAuGe.\nBCS ID Formula ICSD MSG T.C.\n1.43 PrNiO3 67718 36.178(Camc21) w/o\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , ES U=6eV , LCEBR\nTABLE CCCXXIX. Topology phase diagram of PrNiO3.\nBCS ID Formula ICSD MSG T.C.\n0.26 TmAgGe 164587 38.191(Am0m02) w/o\nTopology\nU=0 , ES U=2eV , TBD U=4eV , TBD U=6eV , TBD\nTABLE CCCXXX. Topology phase diagram of TmAgGe.\nBCS ID Formula ICSD MSG T.C.\n2.12 TbMg 104880 49.270(Pc0cm0) Z2Z2\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCXXXI. Topology phase diagram of TbMg.202\nBCS ID Formula ICSD MSG T.C.\n1.139 Ho2RhIn8 * 49.273(Pcccm) Z2\nTopology\nU=0 , ES U=2eV , TI U=4eV , TI U=6eV , ES\nTABLE CCCXXXII. Topology phase diagram of Ho2RhIn8.\nBCS ID Formula ICSD MSG T.C.\n2.11 TbMg 104880 51.295(Pmm0a0) Z2\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCXXXIII. Topology phase diagram of TbMg.\nBCS ID Formula ICSD MSG T.C.\n1.222 Er2CoGa8 169772 51.298(Pamma) Z2Z2\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCXXXIV. Topology phase diagram of Er2CoGa8.\nBCS ID Formula ICSD MSG T.C.\n1.150 PrAg 605674 53.334(PBmna) Z2\nTopology\nU=0 , LCEBR U=2eV , TBD U=4eV , TI U=6eV , TBD\nTABLE CCCXXXV. Topology phase diagram of PrAg.203\nBCS ID Formula ICSD MSG T.C.\n0.22 DyB4 68651 55.355(Pb0am) w/o\nTopology\nU=0 , LCEBR U=2eV , TBD U=4eV , TBD U=6eV , TBD\nTABLE CCCXXXVI. Topology phase diagram of DyB4.\nBCS ID Formula ICSD MSG T.C.\n1.105 Gd2CuO4 75425 56.374(0PAccn0) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXXXVII. Topology phase diagram of Gd2CuO4.\nBCS ID Formula ICSD MSG T.C.\n1.8 CeRu2Al10 59912 57.391(PCbcm) Z2\nTopology\nU=0 , TI U=2eV , TI U=4eV , TI U=6eV , LCEBR\nTABLE CCCXXXVIII. Topology phase diagram of CeRu2Al10.\nBCS ID Formula ICSD MSG T.C.\n0.187 CeMnAsO 195499 59.407(Pm0mn) w/o\nTopology\nU=0 , ES U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXXXIX. Topology phase diagram of CeMnAsO.204\nBCS ID Formula ICSD MSG T.C.\n0.146 EuZrO3 * 62.444(Pnm0a) w/o\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXL. Topology phase diagram of EuZrO3.\nBCS ID Formula ICSD MSG T.C.\n0.185 Nd5Ge4 190405 62.447(Pnm0a0) Z2\nTopology\nU=0 , ES U=2eV , ES U=4eV , TBD U=6eV , ES\nTABLE CCCXLI. Topology phase diagram of Nd5Ge4.\nBCS ID Formula ICSD MSG T.C.\n0.147 EuZrO3 * 62.449(Pn0m0a0) w/o\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXLII. Topology phase diagram of EuZrO3.\nBCS ID Formula ICSD MSG T.C.\n0.159 DyCoO3 190949 62.449(Pn0m0a0) w/o\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXLIII. Topology phase diagram of DyCoO3.205\nBCS ID Formula ICSD MSG T.C.\n0.171 DyScO3 99545 62.449(Pn0m0a0) w/o\nTopology\nU=0 , TBD U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXLIV. Topology phase diagram of DyScO3.\nBCS ID Formula ICSD MSG T.C.\n1.179 NdCoAsO 180774 62.450(Panma) Z2\nTopology\nU=0 , ES U=2eV , TI U=4eV , ES U=6eV , ES\nTABLE CCCXLV. Topology phase diagram of NdCoAsO.\nBCS ID Formula ICSD MSG T.C.\n0.12 U3Ru4Al12 163985 63.461(Cmcm0) w/o\nTopology\nU=0 , LCEBR U=2eV , TBD U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXLVI. Topology phase diagram of U3Ru4Al12.\nBCS ID Formula ICSD MSG T.C.\n0.149 Nd3Ru4Al12 * 63.462(Cm0c0m) Z2\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCXLVII. Topology phase diagram of Nd3Ru4Al12.206\nBCS ID Formula ICSD MSG T.C.\n0.173 Pr3Ru4Al12 * 63.462(Cm0c0m) Z2\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCXLVIII. Topology phase diagram of Pr3Ru4Al12.\nBCS ID Formula ICSD MSG T.C.\n3.3 Ho2RhIn8 * 63.464(Cm0cm0) Z2Z2\nTopology\nU=0 , ES U=2eV , ES U=4eV , TBD U=6eV , TBD\nTABLE CCCXLIX. Topology phase diagram of Ho2RhIn8.\nBCS ID Formula ICSD MSG T.C.\n1.200 U2Ni2Sn * 63.466(Ccmcm) Z2\nTopology\nU=0 , LCEBR U=2eV , TI U=4eV , TI U=6eV , TI\nTABLE CCCL. Topology phase diagram of U2Ni2Sn.\nBCS ID Formula ICSD MSG T.C.\n1.262 NpRhGa5 * 63.466(Ccmcm) Z2\nTopology\nU=0 , LCEBR U=2eV , TI U=4eV , ES U=6eV , LCEBR\nTABLE CCCLI. Topology phase diagram of NpRhGa5.207\nBCS ID Formula ICSD MSG T.C.\n1.195 Er2Ni2In * 63.467(Camcm) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , TI U=6eV , LCEBR\nTABLE CCCLII. Topology phase diagram of Er2Ni2In.\nBCS ID Formula ICSD MSG T.C.\n3.13 CeB6 67404 64.479(Camca) Z2\nTopology\nU=0 , LCEBR U=2eV , TBD U=4eV , TBD U=6eV , TBD\nTABLE CCCLIII. Topology phase diagram of CeB6.\nBCS ID Formula ICSD MSG T.C.\n1.188 CeRh2Si2 621959 64.480(CAmca) Z2\nTopology\nU=0 , TI U=2eV , LCEBR U=4eV , ES U=6eV , TI\nTABLE CCCLIV. Topology phase diagram of CeRh2Si2.\nBCS ID Formula ICSD MSG T.C.\n0.141 TbGe2 56030 65.483(Cm0mm) w/o\nTopology\nU=0 , LCEBR U=2eV , TBD U=4eV , TBD U=6eV , LCEBR\nTABLE CCCLV. Topology phase diagram of TbGe2.208\nBCS ID Formula ICSD MSG T.C.\n1.223 Tm2CoGa8 169773 65.489(Cammm) Z2Z4\nTopology\nU=0 , TI U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLVI. Topology phase diagram of Tm2CoGa8.\nBCS ID Formula ICSD MSG T.C.\n1.106 Pr2CuO4 202884 66.500(CAccm) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , TBD U=6eV , TBD\nTABLE CCCLVII. Topology phase diagram of Pr2CuO4.\nBCS ID Formula ICSD MSG T.C.\n1.142 CeMgPb * 67.510(CAmma) Z2\nTopology\nU=0 , LCEBR U=2eV , TI U=4eV , TBD U=6eV , TBD\nTABLE CCCLVIII. Topology phase diagram of CeMgPb.\nBCS ID Formula ICSD MSG T.C.\n0.16 EuTiO3 * 69.523(Fm0mm) w/o\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCLIX. Topology phase diagram of EuTiO3.209\nBCS ID Formula ICSD MSG T.C.\n1.0.12 UAu2Si2 * 71.536(Im0m0m) Z2Z2\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLX. Topology phase diagram of UAu2Si2.\nBCS ID Formula ICSD MSG T.C.\n2.28 NpNiGa5 * 74.559(Imm0a0) Z2Z2\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLXI. Topology phase diagram of NpNiGa5.\nBCS ID Formula ICSD MSG T.C.\n1.59 KTb3F12 51125 84.58(PI42=m) Z4\nTopology\nU=0 , TBD U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCLXII. Topology phase diagram of KTb3F12.\nBCS ID Formula ICSD MSG T.C.\n0.107 Ho2Ge2O7 161912 92.111(P41212) w/o\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCLXIII. Topology phase diagram of Ho2Ge2O7.210\nBCS ID Formula ICSD MSG T.C.\n0.184 Nd5Si4 190404 92.114(P4120\n120) w/o\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLXIV. Topology phase diagram of Nd5Si4.\nBCS ID Formula ICSD MSG T.C.\n1.0.11 CeCoGe3 190701 107.231(I4m0m0) Z2\nTopology\nU=0 , LCEBR U=2eV , ES U=4eV , ES U=6eV , TBD\nTABLE CCCLXV. Topology phase diagram of CeCoGe3.\nBCS ID Formula ICSD MSG T.C.\n2.26 PrCo2P2 73650 123.345(P4=mm0m0) Z4Z4Z4\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLXVI. Topology phase diagram of PrCo2P2.\nBCS ID Formula ICSD MSG T.C.\n1.162 NdMg * 124.360(Pc4=mcc) Z4\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLXVII. Topology phase diagram of NdMg.211\nBCS ID Formula ICSD MSG T.C.\n1.251 NdCo2P2 73652 124.360(Pc4=mcc) Z4\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , TI\nTABLE CCCLXVIII. Topology phase diagram of NdCo2P2.\nBCS ID Formula ICSD MSG T.C.\n1.255 UPtGa5 * 124.360(Pc4=mcc) Z4\nTopology\nU=0 , TI U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLXIX. Topology phase diagram of UPtGa5.\nBCS ID Formula ICSD MSG T.C.\n1.261 NpRhGa5 * 124.360(Pc4=mcc) Z4\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLXX. Topology phase diagram of NpRhGa5.\nBCS ID Formula ICSD MSG T.C.\n0.189 CeMn2Ge4O12 * 125.367(P40=nbm0) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCLXXI. Topology phase diagram of CeMn2Ge4O12.212\nBCS ID Formula ICSD MSG T.C.\n2.14 NdMg * 125.373(PC4=nbm) Z2\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLXXII. Topology phase diagram of NdMg.\nBCS ID Formula ICSD MSG T.C.\n1.253 CeCo2P2 85895 126.386(PI4=nnc) Z2\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLXXIII. Topology phase diagram of CeCo2P2.\nBCS ID Formula ICSD MSG T.C.\n0.80 U2Pd2In 106867 127.394(P40=m0bm0) Z2\nTopology\nU=0 , ESFD U=2eV , ESFD U=4eV , ESFD U=6eV , ESFD\nTABLE CCCLXXIV. Topology phase diagram of U2Pd2In.\nBCS ID Formula ICSD MSG T.C.\n0.81 U2Pd2Sn 658413 127.394(P40=m0bm0) Z2\nTopology\nU=0 , TI U=2eV , TI U=4eV , TI U=6eV , TI\nTABLE CCCLXXV. Topology phase diagram of U2Pd2Sn.213\nBCS ID Formula ICSD MSG T.C.\n0.9 GdB4 * 127.395(P4=m0b0m0) w/o\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCLXXVI. Topology phase diagram of GdB4.\nBCS ID Formula ICSD MSG T.C.\n1.81 GdIn3 * 127.397(PC4=mbm) Z4Z4\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLXXVII. Topology phase diagram of GdIn3.\nBCS ID Formula ICSD MSG T.C.\n1.102 U2Ni2In * 128.408(Pc4=mnc) Z4\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLXXVIII. Topology phase diagram of U2Ni2In.\nBCS ID Formula ICSD MSG T.C.\n1.160 UP 648255 128.410(PI4=mnc) Z4\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLXXIX. Topology phase diagram of UP.214\nBCS ID Formula ICSD MSG T.C.\n1.187 TbRh2Si2 650329 128.410(PI4=mnc) Z4\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLXXX. Topology phase diagram of TbRh2Si2.\nBCS ID Formula ICSD MSG T.C.\n1.208 UAs * 128.410(PI4=mnc) Z4\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , TI\nTABLE CCCLXXXI. Topology phase diagram of UAs.\nBCS ID Formula ICSD MSG T.C.\n1.21 DyCo2Si2 622705 128.410(PI4=mnc) Z4\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLXXXII. Topology phase diagram of DyCo2Si2.\nBCS ID Formula ICSD MSG T.C.\n0.186 CeMnAsO 195499 129.416(P40=n0m0m) Z2\nTopology\nU=0 , LCEBR U=2eV , TI U=4eV , TI U=6eV , TI\nTABLE CCCLXXXIII. Topology phase diagram of CeMnAsO.215\nBCS ID Formula ICSD MSG T.C.\n0.194 UPt2Si2 57472 129.419(P4=n0m0m0) w/o\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCLXXXIV. Topology phase diagram of UPt2Si2.\nBCS ID Formula ICSD MSG T.C.\n1.215 UP2 77853 130.432(Pc4=ncc) Z2\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCLXXXV. Topology phase diagram of UP2.\nBCS ID Formula ICSD MSG T.C.\n1.271 CeSbTe * 130.432(Pc4=ncc) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCLXXXVI. Topology phase diagram of CeSbTe.\nBCS ID Formula ICSD MSG T.C.\n2.13 UP 648255 134.481(PC42=nnm) Z2\nTopology\nU=0 , TI U=2eV , TI U=4eV , TI U=6eV , TI\nTABLE CCCLXXXVII. Topology phase diagram of UP.216\nBCS ID Formula ICSD MSG T.C.\n2.20 UAs * 134.481(PC42=nnm) Z2\nTopology\nU=0 , TI U=2eV , ES U=4eV , TI U=6eV , TI\nTABLE CCCLXXXVIII. Topology phase diagram of UAs.\nBCS ID Formula ICSD MSG T.C.\n2.6 Nd2CuO4 202885 134.481(PC42=nnm) Z2\nTopology\nU=0 , TI U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCLXXXIX. Topology phase diagram of Nd2CuO4.\nBCS ID Formula ICSD MSG T.C.\n1.103 U2Rh2Sn 246630 135.492(Pc42=mbc) Z4\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCXC. Topology phase diagram of U2Rh2Sn.\nBCS ID Formula ICSD MSG T.C.\n1.207 U2Rh2Sn * 135.492(Pc42=mbc) Z4\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCXCI. Topology phase diagram of U2Rh2Sn.217\nBCS ID Formula ICSD MSG T.C.\n1.254 UNiGa5 * 140.550(Ic4=mcm) Z4\nTopology\nU=0 , ES U=2eV , TI U=4eV , TI U=6eV , LCEBR\nTABLE CCCXCII. Topology phase diagram of UNiGa5.\nBCS ID Formula ICSD MSG T.C.\n1.82 Nd2RhIn8 * 140.550(Ic4=mcm) Z4\nTopology\nU=0 , LCEBR U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCXCIII. Topology phase diagram of Nd2RhIn8.\nBCS ID Formula ICSD MSG T.C.\n1.87 TbCo2Ga8 623196 140.550(Ic4=mcm) Z4\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CCCXCIV. Topology phase diagram of TbCo2Ga8.\nBCS ID Formula ICSD MSG T.C.\n0.154 Er2Ru2O7 97533 141.554(I40\n1=am0d) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXCV. Topology phase diagram of Er2Ru2O7.218\nBCS ID Formula ICSD MSG T.C.\n0.29 Er2Ti2O7 24152 141.554(I40\n1=am0d) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXCVI. Topology phase diagram of Er2Ti2O7.\nBCS ID Formula ICSD MSG T.C.\n0.47 Gd2Sn2O7 84753 141.555(I40\n1=amd0) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXCVII. Topology phase diagram of Gd2Sn2O7.\nBCS ID Formula ICSD MSG T.C.\n0.126 NpCo2 102606 141.556(I40\n1=a0m0d) Z2\nTopology\nU=0 , ES U=2eV , TBD U=4eV , TI U=6eV , TI\nTABLE CCCXCVIII. Topology phase diagram of NpCo2.\nBCS ID Formula ICSD MSG T.C.\n0.198 GdVO4 238223 141.556(I40\n1=a0m0d) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CCCXCIX. Topology phase diagram of GdVO4.219\nBCS ID Formula ICSD MSG T.C.\n0.151 Tm2Mn2O7 * 141.557(I41=am0d0) Z2Z2\nTopology\nU=0 , ES U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CD. Topology phase diagram of Tm2Mn2O7.\nBCS ID Formula ICSD MSG T.C.\n0.157 Yb2Sn2O7 * 141.557(I41=am0d0) Z2Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CDI. Topology phase diagram of Yb2Sn2O7.\nBCS ID Formula ICSD MSG T.C.\n0.227 NdCo2 246555 141.557(I41=am0d0) Z2Z2\nTopology\nU=0 , ES U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CDII. Topology phase diagram of NdCo2.\nBCS ID Formula ICSD MSG T.C.\n0.48 Tb2Sn2O7 * 141.557(I41=am0d0) Z2Z2\nTopology\nU=0 , ES U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CDIII. Topology phase diagram of Tb2Sn2O7.220\nBCS ID Formula ICSD MSG T.C.\n0.49 Ho2Ru2O7 96730 141.557(I41=am0d0) Z2Z2\nTopology\nU=0 , ES U=2eV , LCEBR U=4eV , LCEBR U=6eV , ES\nTABLE CDIV. Topology phase diagram of Ho2Ru2O7.\nBCS ID Formula ICSD MSG T.C.\n0.51 Ho2Ru2O7 96730 141.557(I41=am0d0) Z2Z2\nTopology\nU=0 , TI U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CDV. Topology phase diagram of Ho2Ru2O7.\nBCS ID Formula ICSD MSG T.C.\n1.161 PrFe3(BO3)4 * 155.48(RI32) w/o\nTopology\nU=0 , LCEBR U=2eV , TBD U=4eV , ES U=6eV , LCEBR\nTABLE CDVI. Topology phase diagram of PrFe3(BO3)4.\nBCS ID Formula ICSD MSG T.C.\n0.169 U3As4 42168 161.71(R3c0) w/o\nTopology\nU=0 , ES U=2eV , ES U=4eV , LCEBR U=6eV , LCEBR\nTABLE CDVII. Topology phase diagram of U3As4.221\nBCS ID Formula ICSD MSG T.C.\n0.170 U3P4 648243 161.71(R3c0) w/o\nTopology\nU=0 , ES U=2eV , ES U=4eV , LCEBR U=6eV , LCEBR\nTABLE CDVIII. Topology phase diagram of U3P4.\nBCS ID Formula ICSD MSG T.C.\n0.167 Nd3Sb3Mg2O14 * 166.101(R\u00163m0) Z2Z4\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CDIX. Topology phase diagram of Nd3Sb3Mg2O14.\nBCS ID Formula ICSD MSG T.C.\n0.228 TbCo2 152582 166.101(R\u00163m0) Z2Z4\nTopology\nU=0 , TI U=2eV , ES U=4eV , TI U=6eV , TI\nTABLE CDX. Topology phase diagram of TbCo2.\nBCS ID Formula ICSD MSG T.C.\n0.77 Tb2Ti2O7 151747 166.101(R\u00163m0) Z2Z4\nTopology\nU=0 , ES U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CDXI. Topology phase diagram of Tb2Ti2O7.222\nBCS ID Formula ICSD MSG T.C.\n1.189 TbMg3 9750 167.108(RI\u00163c) Z2\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , TBD U=6eV , TBD\nTABLE CDXII. Topology phase diagram of TbMg3.\nBCS ID Formula ICSD MSG T.C.\n0.32 HoMnO3 92838 185.197(P63cm) w/o\nTopology\nU=0 , LCEBR U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CDXIII. Topology phase diagram of HoMnO3.\nBCS ID Formula ICSD MSG T.C.\n0.33 HoMnO3 92838 185.197(P63cm) w/o\nTopology\nU=0 , LCEBR U=2eV , TBD U=4eV , TBD U=6eV , TBD\nTABLE CDXIV. Topology phase diagram of HoMnO3.\nBCS ID Formula ICSD MSG T.C.\n3.8 NdZn 646083 222.103(PIn\u00163n) Z2\nTopology\nU=0 , ESFD U=2eV , ES U=4eV , ES U=6eV , ESFD\nTABLE CDXV. Topology phase diagram of NdZn.223\nBCS ID Formula ICSD MSG T.C.\n3.12 NpSb * 224.113(Pn\u00163m0) Z2\nTopology\nU=0 , TI U=2eV , TI U=4eV , TI U=6eV , TI\nTABLE CDXVI. Topology phase diagram of NpSb.\nBCS ID Formula ICSD MSG T.C.\n3.2 UO2 647595 224.113(Pn\u00163m0) Z2\nTopology\nU=0 , ES U=2eV , LCEBR U=4eV , LCEBR U=6eV , LCEBR\nTABLE CDXVII. Topology phase diagram of UO2.\nBCS ID Formula ICSD MSG T.C.\n3.7 NpBi 106952 224.113(Pn\u00163m0) Z2\nTopology\nU=0 , TI U=2eV , TI U=4eV , TI U=6eV , TI\nTABLE CDXVIII. Topology phase diagram of NpBi.\nBCS ID Formula ICSD MSG T.C.\n3.10 NpSe * 228.139(FSd\u00163c) Z2\nTopology\nU=0 , TI U=2eV , ESFD U=4eV , ESFD U=6eV , ESFD\nTABLE CDXIX. Topology phase diagram of NpSe.224\nBCS ID Formula ICSD MSG T.C.\n3.11 NpTe * 228.139(FSd\u00163c) Z2\nTopology\nU=0 , TI U=2eV , ESFD U=4eV , ESFD U=6eV , ESFD\nTABLE CDXX. Topology phase diagram of NpTe.\nBCS ID Formula ICSD MSG T.C.\n3.9 NpS 44382 228.139(FSd\u00163c) Z2\nTopology\nU=0 , ES U=2eV , ESFD U=4eV , ESFD U=6eV , ESFD\nTABLE CDXXI. Topology phase diagram of NpS.\nBCS ID Formula ICSD MSG T.C.\n3.6 DyCu * 229.143(Im\u00163m0) Z4\nTopology\nU=0 , TBD U=2eV , ES U=4eV , ES U=6eV , ES\nTABLE CDXXII. 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The nontrivial topology originates primarily from the CrI 3layer while the non-\nmagnetic element induces the charge transfer process and proximity enhanced spin-orbit coupling.\nDue to these unique properties, our topological magnetic vdw-HS overcomes the weak magnetiza-\ntion via proximity e\u000bect in previous designs since the magnetization and topology coexist in the\nsame magnetic layer. Speci\fcally, our systems of bilayer CrI 3/Sb and trilayer CrI 3/Sb/CrI 3exhibit\ndi\u000berent topological ground state ranging from antiferromagnetic topological crystalline insulator\n(CM= 2) to a QAHE. These nontrivial topological transition is shown to be switchable in a tri-\nlayer con\fguration due to the magnetic switching from antiferromagnetism to ferromangetism in the\npresence an external perpendicular electric \feld with value as small as 0.05 eV/ \u0017A. Thus our study\nproposes a realistic system to design switchable magnetic topological device with electric \feld.\nI. INTRODUCTION\nTwo dimensional magnetic topological materials repre-\nsent a novel platform which exhibit the quantum anoma-\nlous Hall e\u000bect (QAHE) [1,2]. To realize the quantum\nanomalous Hall (QAH) state, the material system needs\nto break time reversal symmetry while maintaining a\nnontrivial 2D bulk gap protected by a nonzero Chern\nnumber. As a result, di\u000berent pathways with an empha-\nsis on inducing magnetization on a nontrivial topological\nmaterial have been proposed. Speci\fcally adatoms [3-6],\nimpurity doping [7-9], interfacing between magnetic sub-\nstrates and topological materials [10-16] have been pro-\nposed as possible strategy to realize a room-temperature\nQAHE. However, these systems require the incorporation\nof external impurity or the interfacing with a non 2D ma-\nterial which can be experimentally challenging. Alterna-\ntively, the material discovery approach has also suggested\nvarious 2D materials with large spin orbit interaction [17-\n19] as a mean to achieve a high-temperature QAHE, but\nthis method also introduces new challenges in terms of\nmaterial fabrication, processing, and characterization.\nTwo dimensional (2D) vertical van der waals het-\nerostructures have emerged as a new functional mate-\nrial class with \rexible functional properties since the het-\nerostructure can exhibit new quantum properties which\nare absent in the individual layers [20]. These structures\nrequire relatively low cost fabrication method which al-\nlows the combination of di\u000berent materials with tailored\nelectronic property in various vertical architecture. Con-\nsequently, 2D-HS represents an ideal platform to realize\nthe QAHE by combining 2D materials with large spin-\norbit coupling (SOC) and magnetic interaction. Among\nmany 2D materials, CrI 3is a new class of magnetic\nsemiconductor which has demonstrated tunable exchange\n\u0003phamad@ornl.gov\nyganeshp@ornl.govcoupling with high Curie temperature of 45 K in the\nferromagnetic phase [21-26]. In addition, di\u000berent het-\nerostructure con\fguration of CrI 3and other material like\ngraphene [27, 28] or Bi 2Se3[29] have been proposed to\nrealize the QAHE.\nA principle design of many magnetic topological vdw-\nHS is the proximity e\u000bect in which the magnetization is\ninduced on a non-magnetic but topologically non-trivial\nmaterial [27-29]. However, the drawback of this approach\nis that the magnetic moments on the topological insula-\ntor (TI) tend to be small since most topological mate-\nrials are primarily composed of materials with delocal-\nizedp-orbital, while most magnetic materials predomi-\nnantly originate from elements with half-\flled dbands.\nCrI3monolayer is a unique magnetic material which has\nnontrivial topology with high Chern numbers inside the\nconduction and valence band. As a result, it has been\nsuggested that the nontrivial topology and strong mag-\nnetization can be exploited by extreme doping up to one\nelectron(hole)/unit cell to move the Fermi level inside\nthe conduction or valence bands [30]. This high level of\ndoping is unfeasible for experimental synthesis, but the\nFermi level can be tuned signi\fcantly in a heterostruc-\nture con\fguration by creating large band o\u000bsets.\nMotivated by this idea, we \frst search for combina-\ntion of 2D monolayer materials and CrI 3as shown in ta-\nble I to design a heterostructure with the lowest strain.\nSince Ge and Sb both have larger SOC than Si and min-\nimal lattice mismatch with CrI 3, we focus our study on\nthe di\u000berent vertical vdw-HS of these materials. In ad-\ndition, germanene is (Ge) a topological insulator [31],\nwhile antimonene is a normal insulator, which can be\nturned into a topological insulator through a large ten-\nsile strain [32]. In these hetereostructures, the individual\nlayers can induce di\u000berent charge transfer process thus\nresulting di\u000berent bands contributing the Fermi level. In\naddition to the charge transfer process, other mechanisms\nlike proximity enhanced SOC and spin transfer between\nthe magnetic and non-magnetic layer, can further compli-arXiv:2003.05840v1  [cond-mat.mtrl-sci]  12 Mar 20202\ncate the topological properties in the heterostructure. In\nour study, by characterizing the electronic, magnetic and\ntopological properties of the di\u000berent layers in the het-\nerostructure, we demonstrate that the bilayer Sb/CrI 3\nand trilayer CrI 3/Sb/CrI 3con\fgurations posses nontriv-\nial topological properties with Chern number C= \u00061.\nThese topological property originates primarily from the\nCrI3layer which is electron doped from the elemental Sb\nmonolayer thus resulting in QAHE with edge states close\nto the Fermi level. While system with weak charge trans-\nfer like Ge/CrI 3or CrI 3Ge/CrI 3remain trivial metal.\nThe Sb also induces large SOC on the CrI 3layer through\nproximity e\u000bect thus opening up a possibility for high\ntemperature QAHE. In addition, the trilayer system ex-\nhibits a tunable topological transition from an antifer-\nromagnetic topological crystalline insulator (TCI) when\nthe CrI 3layers couple antiferromagnetically to a QAHE\nwhen CrI 3layers are ferromagnetically aligned. We show\nthat this magnetic transition can be induced by a small\nperpendicular electric \feld of 0.05 eV/ \u0017A, consitant with\nrecent experiments on bilayer CrI 3. Such a value has\nbeen recently realized in a monolayer and bilayer Na 3Bi\nsystem [32] to induce the topological phase transition.\nConsequently, our results demonstrate a realistic plat-\nform of controlling a magnetic topological system which\nis suitable for experimental realization.\nTABLE I. Lattice constants and strain of di\u000berent 2D mono-\nlayer elements and CrI 3\n2D materials Lattice( \u0017A) vdw-HS Lattice mismatch(%)\nSi 3.87 CrI 3/Si/CrI 3 -3.13\nGe 4.04 CrI 3/Ge/CrI 3 1.3\nSb 4.12 CrI 3/Sb/CrI 3 3.11\nCrI3 6.91 N/A N/A\nII. COMPUTATIONAL METHODS\nThe \frst principle calculations were done using the\nVASP package with the projected augmented wave\nmethod (PAW) [33] and the PerdewBurke-Ernzerhof\nfunctional [35]. An energy cuto\u000b of 600 eV was used for\nthe structural relaxation and the electronic calculations.\nThe DFT+U [36] with U=3.0 eV and J=0.9 eV was used\nin Cr's dto account for the correlation e\u000bect. For the in-\ndividual monolayers, a dense k-point mesh of 21 \u000221\u00021\nwas used to determine the equilibrium lattice constants,\nwhile a k-mesh of 9 \u00029\u00021 was used for the structural\nrelaxation of the heterostructure. To avoid the inter-\naction between the image layers in the monolayers and\nheterostructure con\fgurations, vacuum was included on\nthe top and bottom of the con\fguration with an overall\nthickness of 20 \u0017A. In the vdw-HS structures, the in-plane\nlattice constants were set at the values of the monolayers\nof Ge/Sb so that the lattice mismatch between CrI 3and\nthe monolayers do not a\u000bect their topology. To reducethe lattice mismatch, a con\fgurationp\n2\u0002p\n2 was used\nfor the mono elemental layers. The geometrical struc-\ntures of the vdw-HS are summarized in Table I. The\nmagnetic ground state in the trilayer structure was stud-\nied by setting the interlayer coupling between the CrI 3\nlayers to be ferromagnetic (FM) and antiferromagnetic\n(AFM). The spin-orbit coupling was included in a self-\nconsistent approach to determine the nontrivial topolog-\nical property. To calculate the edge state in the magnetic\ntopological phase, a Wannier tight binding method [37]\nwas used to obtain the Green's function [38] spectrum\nas implemented in the WannierTools code. The Wannier\nbasis was calculated from the Wannier90 codes [39] with\nthe orbital characteristic of Cr's dorbitals, I's porbitals,\nGe/Sb and Te's porbitals.\nIII. RESULTS AND DISCUSSIONS\nThe heterostructuring process can signi\fcantly modify\nthe electronic properties of both the CrI 3as well as the\nGe/Sb layers. As shown in Fig. 2b and 2f, the ferromag-\nnetic monolayer CrI 3maintains the half-semiconducting\ngroundstate even in the presence of small tensile strain\nin the Sb. Consequently, a band gap is clearly visible for\nboth of the spin-channel of CrI 3. Electronically, similar\nto previous theoretical study [40-42] our monolayer\nCrI3's density of state is composed of a valence band\ndominated by the I- porbitals and the Cr's fully \flled\nt2gband slightly below the I- porbitals, while the\nconduction band is composed primarily of the empty\nCr's egorbital. For the monolayer Ge and Sb, the\ndensity of state also reveals a Dirac dispersion in Ge\n[Fig. 2a] and a large semiconducting gap in Sb [Fig. 2e]\nconsistent with previous ab-inito studies [30,31]. When\nwe create a bilayer structure between CrI3 3and elemen-\ntal Ge(Sb), this results in a signi\fcant modi\fcation of\nthe electronic-structure. For one, the bands undergo a\nrigid shift so as to pin the Fermi-level to the bottom\nof the conduction band as illustrated in Figs. 2c and\n2g. Secondly, the gap in the spin-up channel is closed,\nmaking CrI 3half-metallic. This suggests that CrI 3is\nelectron-doped. Complimentarily, the Fermi-levle is\npinned to the top of the valence bande-edge in Sb and\nGe. This is reminiscent of hole-doping, and results in\nan increase in the density-of-states at the Fermi-level\nfor both these systems [Figs. 2c and 2g]. This leads\nto a net charge-transfer from the Ge(Sb) layer to CrI 3,\nas seen in Figs. S2. Interestingly, CrI 3also induces\nspin-polarization to the Ge(Sb) layer { a proximity\ninduced magnetization-e\u000bect. Consequently, the closing\nof the band-gap due to this charge-transfer e\u000bect and the\nmixing of Cr's partially occupied conduction bands with\nthe valence bands of Ge (Sb) gives rise to a situation\nwhere new topologies can emerge. But given that we see\nproximity induced magnetization in Ge(Sb) and expect\nproximity-induced SOC in CrI 3, it is not clear which\nstates give rise to the non-trivial topology.3\nFIG. 1. Schematic illustration of di\u000berent CrI 3based vdw-heterostructuctues with Ge/Sb. a) Bilayer. b) Trilayer\nFIG. 2. a), (e) Density of states of monolayer Ge and Sb. (b), (f) Partial density of states of Cr-d and I-p in monolayer\nferromagnetic CrI 3calculated with Ge and Sb in-plane lattice respectively. (c), (g) Partial density of states of Cr-d,I-p and\nGe-p (Sb-p) in ferromagnetic CrI 3/Ge(Sb) heterostructure respectively. (d),(h) Partial density of states of Cr-d, I-p and Ge-p\nin antiferromagnetic CrI 3/Ge(Sb)/CrI 3trilayer respectively.\nThe ground-state magnetic-arrangement of a CrI 3\nbilayer is antiferromagnetic. Our magnetic calculations\nshow that even with a layer of Ge(Sb) in our trilayer\ncon\fguration, this coupling remains antiferromagnetic.\nSimilar to the result in the bilayer structure, CrI 3is\nelectron-doped while the Ge(Sb) is hole-doped, indicat-\ning a charge-transfer between them. However, there is\na key di\u000berence { while time-reversal-symmetry (TRS)\nis broken in the bilayer, it is preserved in the trilayer\ncon\fguration. This is enforced by a di\u000berent degree of\nhybdridization between the monolayer and the CrI 3layer\nin the trilayer structure. This should result in di\u000berent\ntopologies in the bilayer and the trilayer con\fgurations.\nIn a bilayer structure, due to the hybridization of the\nGe(Sb) states, opening of the gap and time-reversal-\nsymmetry breaking due to the ferromagnetic CrI 3layer,\nthe system could possess new interesting topologies. For\nexample, a large hybridization gap exists between the\nCr'sdand Ge's pbands in the spin-up channel [Figs.\nS1], while the spin-down channel is dominated by Ge'spbands, which further exhibits a massive Dirac cone\nbecause inversion symmetry is broken in the bilayer.\nSpin-orbit has negligible e\u000bect on the band structure\n[Figs. S1]. As a result, the Chern number calculation\nreveals a C=0 which indicates a trivial metallic state in\nthe Ge/CrI 3bilayer. Our result is consistent with recent\nstudy of the germanene on CrI 3substrate [43].\nDi\u000berent from the Cr/CrI 3the Sb/CrI 3bilayer\nstructure, there is a massive SOC induced splitting for\nthe Cr-derived conduction bands [Fig. 3(c)], indicating\nthat it is the Sb-layer that is giving rise to a large\nproximity induced SOC in CrI 3. In conjunction with\nferromagnetism, this gives rise to a quantum anomalous\nhall e\u000bect with C = 1 and an edge states along the (110)\ndirection as shown in Fig. 3(d). This nontrivial topology\noriginates primarily from the large SOC induced parity\nexchange between the Cr's dand I's porbitals. Given\nthis proximity induced SOC in CrI 3, and the resultant\nlarge splitting of the CrI 3bands, it is pertinent to ask\nwhat topological transitions can be realized by further4\nFIG. 3. Electronic and topological property of bilayer Sb-CrI3 heterostructure. a) Band structure of spin-up states. b) Band\nstructure of spin down states. c) Band structure with SOC. d) Edge state along (110) direction.\nFIG. 4. Electronic and topological property of trilayer CrI 3/Sb/Cr 3heterostructure without and with perpendicular electric\n\fled. a) Band structure wit SOC without electric \feld in the AFM coupling. b) Topological edge state along the (110) edge\nstate in the AFM coupling. c) Band structure with SOC in the FM coupling with E = 0.05 eV/ \u0017A. d) Topological edge state\nalong the (110) edge state in the FM coupling with E = 0.05 eV/ \u0017A.\nchanging the chemical-potential to occupy more of the\nconduction-band. For example, there is a much larger\nChern gap 25.4 meV above the Fermi level in the energy\nregion 0.14-0.2 eV around the K point. The topological\ncalculation incuding states up to 0.1 eV, corresponding\nto a Fermi-level tuned up to this value, shows a C value\nof 1, indicative of a quantum anamolous hall e\u000bect\n(QAHE) a single edge state as shown in Fig. 3d. As a\nresult, we demonstrate that proximity induced SOC via\nheterostructuring can enable us to tune the system to\nposses a multitude of non-trivial topologies, that could\neven survive high-temperatures.\nTo further understand the di\u000berent contribution of\nthe di\u000berent layers on the nontrivial/trivial topology\nin the bilayer con\fguration, we deconstruct the total\nHamiltonian in the Wannier basis as following Hmn\ntotal =\nHmn\nGe=Sb+ Hmn\nCrI 3+ Hmn\nGe=Sb\u0000CrI 3. The individual Hamil-\ntonian is obtained by setting the hopping components of\nthe other orbitals equal to zero. For instance to obtain\nHmn\nGe=Sbonly the hoppings between the Ge/Sb's porbitals\nare retained in total Hamiltonian while the hoppingcontaining the orbitals on CrI 3is set to zero. For the\nGe-based bilayer heterostructure, the individual Ge\nexhibits a QAHE with C=1 as shown in Fig. S3, while\nCrI3is topological trivial when examining the bands up\nto 0.2 eV. Interestingly, when the hamiltonian contains\nboth of the Ge's and CrI 3hopping but no interlayer\nhopping, the system becomes topological nontrivial with\nC=1. When the interlayer hopping is included, a small\ngap is opened at the \u0000 point in the band structure, but\nthis results in overall in a topological trivial metal state.\nOn the other hand, in the Sb-based bilayer, the Sb-only\nlayer is a normal metal, while the CrI 3only band is\ntopological nontrivial. When the hamiltonian contains\nboth of contribution from Sb and CrI 3but without\nand with the interlayer hopping, the topology remains\nnontrivial. As a result, in the Ge-based bilayer, the\ninterlayer interaction between Ge and CrI 3destroys the\nnontrivial topology, while in the Sb-based bilayer the\nnontrivial topology is originated primarily from the CrI 3\nlayer and the Sb layer plays the role of charge transfer\nand the proximity induced spin-orbit coupling.5\nIn the case of the trilayer heterostructure, even\nthough the magnetic ground-state between the CrI 3\nis antiferromagnetic, this magnetic interaction can be\ntuned by applying a perpendicular electric \feld as\nillustrated schematically in Fig. 1b. In the CrI 3bilayers,\nthe magnetic coupling have been shown to strongly\ndependent on the vdw bonding between the di\u000berent\nlayers [23, 24]. Speci\fcally, the interlayer ferromagnetic\ncoupling can be favourable if there is reduced distance\nbetween the interlayer Cr ions resulting in interlayer\nvdw bonding. As revealed in table S1, the applied\nexternal perpendicular electric \feld results in signi\fcant\nstructural changes in the interlayer distance between the\nCrI3layers and the buckling distance in the monolayers.\nSpeci\fcally, the buckling parameter of Ge increases with\napplied E-\feld, while the two layers of CrI 3move closer\ntogether in the case of CrI 3/Sb/CrI 3. In addition, a\nfurther examination of the charge density in the trilayer\n[Fig. S3] reveal signi\fcant changes in the bonding\ncharacteristic between the monolayer and the CrI 3\nlayers in the present of the electric \feld. In the case of\nCrI3/Ge/Cr 3, the electric \feld enhances the buckling of\nthe Ge layer results in bonding between the CrI 3layers\nand the Ge monolayer due to the charge overlapping\nbetween the Ge and I atoms [Fig. S3b]. This bonding\nfacilitates the magnetic coupling between the interlayer\nCr ions thus resulting in a ferromagnetic exchange\ninteraction. On the other hand, since the Sb layer is\nhighly buckled there is already a weak bonding between\nthe I atom and the Sb atom without the external \feld\n[Fig. S3c]. As the electric \feld is turned on, the two\nCrI3layers move closer towards the Sb layer thus further\nenhancing the bonding between the CrI 3layers and the\nSb layer [Fig. S3d]. This enhanced bonding results in\na switch in the magnetic ground state. Consequently,\nthese results suggest that the external electric \feld\nact as enhancement on the bonding between the vdw\nlayers CrI 3and the Ge/Sb layer which mediates the\nferromagnetic interaction between the Cr ions in the\ndi\u000berent Cr 3layers. This also leads to a signi\fcantly\nlower critical \feld for switching the magnetization in the\npresence of antimonene. In general, this con\fguration\nshares commonality with the bilayer con\fgureation {\nboth have broken inversion and time-reversal symmetry\nand show similar degrees of charge-transfer.\nIn the CrI 3/Ge/CrI 3, the SOC has negligible e\u000bect\non the band structure [Fig. S3] similar to the bilayer\ncon\fguration, while in CrI 3/Sb/CrI 3heterostructure\nthe nontrivial topology exists in both of AFM and FM\ncoupling. Without the external \feld, the SOC opens\na up a nontrivial topological gap around K in energy\nregion from 0.1-0.25 eV, which mainly originated from\nthe Cr's d and I'p state [Figs. 4a]. Further investigation\nof the edge state along the [110] direction reveals two\nDirac cones which implies a nontrivial mirror Chern\nnumber (MCN) [Fig. 4b]. Further analysis using the\nWannier charge center con\frm a nonzero MCN with\nC = 2 [Fig. S7]. With a small \feld of 0.05 eV/ \u0017A,the magnetic interaction is tuned to a FM state which\nalso facilitates a transtion to a Chern semimetallic\nphase. In the FM con\fguration, the CrI 3/Sb/CrI 3is a\nsemiconducting ground state [Figs. S6d and 6e] similar\nto the bilayer system. When the SOC is e\u000bective, a\ndrumhead like band structure [Fig. 4c] is observed\nand the system exhibits a QAHE with C = 1 with a\nsingle edge state as shown in Fig. 4d. As a result,\nCrI3/Sb/CrI 3HS represents a novel vdw system in\nwhich multiple nontrivial topological state is switchable\nthrough a small electric \feld.\nIV. CONCLUSIONS\nIn conclusion, we have demonstrated that the appro-\npriate combination of 2D material with 2D magnet like\nCrI3in a heterostructure architecture can give rise to\nnon-trivial magnetic topological state in di\u000berent layer\ncon\fgurations. Di\u000berent from previous study of mag-\nnetic topological heterostructure in which the topology\nis induced to the non-magnetic but topological non-\ntrivial material, our hetereostructure system exploit the\nunique topology in CrI 3by designing system in which\nthe strong bonding between the elemental element like\nSb and CrI 3occurs to induce a semimetallic transition\nin Cr I3through charge transfer process. In addition, Sb\nalso induces a large spin-orbit coupling on the magnetic\nlayer due to the large intrinsic SOC and the broken inver-\nsion symmetry thus resulting in di\u000berent nontrivial mag-\nnetic topologies. Speci\fcally, our results show that the\nSb-based trilayer systems represent a novel tunable topo-\nlogical ground state which allows a switching between an\nAFM topological crystalline insulator to a QAHE using\na very small external electric \feld. Furthermore, the Sb-\nbased bilayer also exhibits a large bulk gap of 28.6 meV\nwith nonzero Chern number which can exhibit a high-\ntemperature QAHE. As a result, this system represents\na viable strategy for further experimental exploration of\ntopological device based on vdw-HS through the manip-\nulation of the spin alignment within the ferromagnetic\nlayers.\nACKNOWLEDGEMENTS\nA.P. and P.G. were \fnancially supported by the Oak\nRidge National Laboratory's, Laboratory Directed Re-\nsearch and Development project. Part of the research\nused resources of the National Energy Research Scienti\fc\nComputing Center (NERSC), a U.S. Department of En-\nergy O\u000ece of Science User Facility operated under Con-\ntract No. DE-AC02-05CH11231. Part of this research\nused resources of the Oak Ridge Leadership Computing\nFacility, which is a DOE O\u000ece of Science User Facility\nsupported under Contract DE-AC05-00OR22725.6\nREFERENCES\n[1] C.-X. Liu, S.-C. Zhang, and X.-L. Qi, Annu. 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B 99, 165410 (2019)."    },    {        "title": "2003.13485v1.Tunable_interlayer_magnetism_and_band_topology_in_van_der_Waals_heterostructures_of_MnBi2Te4_family_materials.pdf",        "content": "Tunable interlayer m agnetism and band t opology in van der Waals \nheterostructures of MnBi 2Te4-family materials  \n \nZhe Li1,2,†, Jiaheng Li1,2,†, Ke He1,2,3*, Xiangang Wan4,5, Wenhui Duan1,2,3,6, and Yong \nXu1,2,7,* \n1State Key Laboratory of Low -Dimensional Quant um Physics, Department of Physics, Tsinghua \nUniversity, Beijing 100084 , China  \n2Frontier Science Center for Quantum Information , Beijing 100084 , China  \n3Bejing Institute of Quantum Information Science, Beijing 100193, China \n4National Laboratory of Solid Stat e Microstructures and School of Physics, Nanjing University, \nNanjing 210093, China  \n5Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China  \n6Institute for Advanced Study, Tsinghua University, Beijing 100084, China  \n7RIKEN Center f or Emergent Matter Science (CEMS), Wako, Saitama 351 -0198, Japan  \n†These authors contributed equally to this work.  \n*Correspondence to:  yongxu@mail.tsinghua.edu.cn  (Y. X. ); kehe@tsinghua.edu.cn  (K. H.) . \n \n \n \nManipulating the interlayer magnetic coupling in van der Waals magnetic \nmaterials and heterostructures is the key to tailoring their magnetic and electronic \nproperties for various electronic applications and funda mental studies in \ncondensed mat ter physics. By utilizing the MnBi 2Te4-family compounds and their \nheterostructures  as a model system , we systematically studied the  dependence of \nthe sign and strength of interlayer magnetic coupling  on constituent elements  by \nusing first-principles  calculations. It was found that  the coupling is a long -range \nsuperexchange interaction mediated by the chains of p orbitals between the \nmagnetic atoms of neighboring septuple -layers. The interlayer exchange  is always \nantiferromagne tic in the pure compound s, but can be tuned to ferromagnetic in  some combinations of  heterostructures , dictated by  d orbital occupation s. Strong \ninterlayer magnetic coupling can be realized if the medial p electrons are \ndelocalized and the d bands of magne tic atoms  are near the Fermi level.  The \nknowledge on the interlayer coupling mechanism enables us to engineer magnetic \nand topological properties of  MnBi 2Te4-family materials as well as many other \ninsulating van der Waals magnetic materials and heterostruc tures . \n \nVan der Waals (vdW) layered magnets have attracted intensive  attentions  in the \nrecent years  (1-6). Benefitting from the weak vdW  interlayer  bonding , these materials \ncan be easily exfoliated into ultrathin flakes and flexibly recombined into various  \nheterostructures with novel material properties.  Many layered magne tic materials have  \nbeen found , including  Cr2Ge2Te6, CrI 3, Fe2Ge2Te6, FePS 3, VSe 2, MnSe 2 (7-22), and the \nMnBi 2Te4-family compounds  (23-34), which provide new chances for developing \nelectron ics, spintronic s, optoelectronics,  and topological physics  (4-6, 23-39). A key \ningredient  to designing and engineering vdW magnetic materials and heterostructures  \nfor various purposes is to manipulate the interlayer magnetic coupling (IMC), i.e. , the \nmagne tic coupling across the vdW gap . The sign and strength  of IMC have been found \nto vary significantly  in different vdW magnets, and a comprehensive understanding  on \nthe underlying mechanism is still lacking.   \nThe MnBi 2Te4-family compounds as vdW layered intrinsic magnetic topological \nmaterials are particularly interesting for the rich magnetic topological phases  they host , \nincluding  the antiferromagnetic (AFM) topological insulator (TI)  (23-34, 39) and the \nmagnetic Weyl semimetal (WSM)  (23-26, 34) in the 3D b ulk, as well as the quantum \nanomalous Hall (QAH) insulator  (23-29, 33, 34) and axion insulator  (23-28) in their \n2D films . Quantized transports have been observed in MnBi 2Te4 thin flakes  at relatively \nhigh temperature s (>1K)  (28, 29, 33). Crystal lized in a rhombohedral layered structure, \nbulk tetradymite -type MnBi 2Te4 is composed of vdW -bonded septuple -layers (SLs) . \nEach SL is consisted of seven atomic layers , of triangular lattice with ABC -stacking  \n(23-25) in the sequence of Te/Bi/Te/Mn/Te/Bi/Te  (Fig.  1(a)) . The intralayer magnetic coupling is ferromagnetic (FM)  with an out-of-plane easy magnetic axis, while the IMC \nis AFM . Thus the MnBi 2Te4 bulk has an A -type AFM ground state.  \nThe MnBi 2Te4-family materials provide an excellent platform  to explore the IMC  \nmechanism of vdW magnets . First ly, the ternary -compound family includes many \ncandidate materials constituted by different combination s of nonmagnetic and magnetic \nelements (23), which enables  a systematic study on the element -dependen ce of IMC. \nSecond ly, the relatively long distance between neighboring magnetic  atomic layers  \nlocks the way of bonds forming and  forbids  many kinds of electron hopping channels  \n(see a more detailed analysis  in Supplementa l Material ), simpl ifying the discussions  on \nthe IMC mechani sm. Besides, since the topological properties of MnBi 2Te4-family \nmaterials  depend critically on the interlayer magnetic state  (23-25, 28, 33, 34), the \nknowledge on the IMC mechanism can be directly used to engineer the magnetic \ntopological quantum states a nd effects  (35-39).  \nIn this Letter , we systematically studied the constituent -element -dependence of \nthe IMC of M nBi2Te3-family materials and heterostructures by first-principles  \ncalculations . Our results reveal that the IMC is dominated by the long -distan ce \nsuperexchange mechanisms mediated by the p orbitals of non -magnetic atoms. \nRemarkably, both the sign and strength of IMC can be significantly manipulated b y \nchoosing different magnetic elements with different numbers of d electrons and \nnonmagnetic eleme nts of different electronegativities . This provides us guidelines to \nunderstand, design, and engineer the magnetic and topological properties not only for \nthis material family but also for layered magnetic  insulators  in general . \nFirst-principles calculatio ns were performed in the framework of density \nfunctional theory ( DFT) using  the Vienna  ab initio  simulation  package  (VASP)  (40). \nThe Perdew -Burke -Ernzerhof (PBE) type exchange -correlation  functional  in the \ngeneralized gradient approximation (GGA) (41) was adopted . The localized 3 d-states \nof V , Cr, Mn, Fe, Ni and 4 f-states of Eu were treated by employing the PBE+ U approach \n(U = 3.0, 4.0, 4.0, 4.0, 4.0, 5.0, respectively). The U values have been tested and used \nby previous works ( 7, 11, 19 -25, 34 ). Band struc tures for thin film s and bulks were \ncomputed by the PBE+ U method  and the modified Becke -Johnson (mBJ)  functional  (42), respectively . VdW corrections were included by the DFT-D3 method  (43). All the \ncrystal  structures  were fully relaxed. The magnetic ground  state and exchange energies \nfor each material were determined by computing energy differences between different \nmagnetic configurations. Topological s urface states or edge state s were calculated by \nusing WannierTools package based on maximally localized W annier functions  (44). \nMore calculation details are described  in the Supplementa l Material . \nAccording to the superexchange theory (45, 46), the exchange coupling between \ntwo magnetic atoms emerges when an indirect electron  hopping  between  d (or f) orbitals  \nof magnetic elements is mediated by the connecting  p orbitals. In the MnBi2Te4-family \ncompounds , magnetic atoms are bonded with six chalcogen  atoms in a slightly distorted \noctahedral geometry , where t he d orbitals of each spin state are roughly split to t riply \ndegenerate 𝑡2𝑔 states and doubly degenerate 𝑒𝑔 states.  Generally , there exist t hree \nkinds of  hopping channels  in the system , including  eg-p-eg, eg-p-t2g and t2g-p-t2g, as \nillustrated  in Fig. 1(c) . Here σ bonds are formed between eg and p orbitals , whose \nstrength is  much stronger than that of π bonds formed between t2g and p orbitals.  \nTherefore,  the eg-p-eg hopping channel s, if existing , play a dominant role in determin ing \nthe sign and strength of IMC . \nWe first analyze the IMC of MnBi 2Te4. For the Mn ions , their 3d orbitals of one \nspin (spin -up) channel are fully occupied, and the other spin channel is empty. While \nthe interlayer electron hopping between d orbitals of neighboring Mn atoms is \nprohibited in the FM configuration , it gets allowed in the AFM configu ration (Fig.  1(d)). \nTherefore , the IMC  of MnBi 2Te4 is AFM , giving an A -type AFM ground state as \nlearned from  previous theoretical and experimental works (23-28, 30-32). Further \nanalyses  on other MnBi 2Te4-family  compounds indicate that their IMC is always A FM, \nindependent of constituent elements, as confirmed by our first -principles calculations.  \nThis seems to be a  disadvantage for research and applications , since sometimes a FM \nIMC is desired , for instance, for  obtaining  a non -zero net magnetism  and for rea lizing  \nthe QAH effect.   \nIn contrast to pure compounds , vdW heterostructures of  XBi 2Te4 (X=Mn, V, Ni, \nEu) has been rarely explored, which could open new opportunities to tuning magnetic and topological properties.  In particular, we will show that the IMC ca n be driven into \nFM in the vdW heterostructures with different magnetic elements. We take  the \nMnBi 2Te4-EuBi 2Te4 (MBT-EBT) combination as an example . The schematic lattice \nstructure of the MBT -EBT  bilayer is  shown in Fig. 1(b). In EuBi 2Te4, the 4f orbitals of \nEu are far away from the Fermi level  (EF), leaving the 5d orbitals close to EF. These 5 d \norbitals are empty and p olarized by magnetic moments of the 4 f electrons, giving spin-\nup states lower than spin -down ones  (47). Although the eg-p-eg hopping is allo wed in \nboth FM and AFM IMC configurations of MnBi 2Te4-EuBi 2Te4, the energy difference \nbetween the  spin-up eg orbitals of Mn and Eu atoms is smaller in the former case, \nmaking the FM coupling energetically more favorable than the AFM one (Fig.  1(e)).  \nTheref ore, the IMC between a MnBi 2Te4 SL and an EuBi 2Te4 SL is FM (see a detailed \nanalysis in the Supplementa l Material ). Applying similar analyses to all XBi 2Te4-\nYBi 2Te4 heterostructures, we  find that generally  the FM IMC emerges when X has d \nelectrons  ≥ 5 while  Y has  d electrons < 5 . In some complicated cases , such as  \nMnBi 2Te4-VBi 2Te4, the FM and AFM IMC are comparable in strength, giving rise to \nrelatively weak IMC.  \nThe IMC mechanism is supported by our first-principles calculation s. The energy \ndiffer ences between AFM and FM configurations per unit cell ( Eex) of a series of \nXBi 2Te4-YBi 2Te4 bilayer s (X, Y = Mn, V , Ni, and Eu) are listed in TABLE 1. It is shown  \nthat MnBi 2Te4-EuBi 2Te4, M nBi 2Te4-VBi2Te4, NiBi 2Te4-EuBi 2Te4, and NiBi 2Te4-\nVBi 2Te4 favor FM IMC , and the other combinations  favor AFM IMC . The results  can \nbe understood by the superexchange theory as discussed above.  \nNoticeably, the superexchange coupling in magnetic oxides is supposed to be \nrather short -ranged (45, 46). Typically, only the cation -anion -cation coupling is \nconsidered. But the calculated IMC strength  between MnBi 2Te4-family materials is not \nnegligible though the magnetic atoms from neighboring SLs are space d by six laye rs of \nnon-magnetic atoms together with a vdW gap. Especially, | Eex| between two NiBi2Te4 \nSLs is as high as 16.5 meV . According to the traditional  superexchange theory (45, 46), \nthe IMC strength is  mainly dependent on the hopping term between magnetic atoms , which is determined by the overlapping of the relevant orbitals in real space. In \nmagnetic oxides, the electronegativity difference between oxygen (  = 3.44) and \nmagnetic metal atoms (mostly  < 2.00) is so large that the medial p electrons are rather \nlocalized  around each oxygen ion. In the MnBi 2Te4-family materials, the similar \nelectronegativities between cations (Bi) and anions (Te) lead to much more delocalized \np electrons , which may facilitate longer-range magnetic coupling.   \nTo verify t he argument , we calculate d the dependence of |Eex| on the non -magnetic \nelement s in Ni(V) 2(VI) 4 compounds, where (V)= Bi or Sb, and (VI)=Te, Se, or S. As \nshown in Fig. 2(a), replacing Te  ( = 2.10)  with Se ( = 2.55) and S ( = 2.58) \nsignificantly reduces | Eex|, from 16.5meV  to 3.5meV and 1.1meV , respectively . The \nunderlying reason is that Te 5p electrons  are more delocalized than Se 4 p and S 3 p \nelectrons in this system , and thus easier to hop to neighboring atoms along the p orbital \nchain . In contrast,  |Eex| drops to nearly zero (< 0.1meV) if artificially replacing Te with \nO, which is expected by the typical superexchange coupling for such a large interatomic \ndistance . A similar trend is al so found in the NiSb 2(VI) 4 compounds. Replacing  Bi \n( =2.02)  with Sb ( =2.05)  in NiBi 2(VI) 4 enhances | Eex| to as large as  24.7 meV in the \nNiSb 2Te4 bilayer.  The relationship of |Eex| and the electronegativity difference (χ) of \nconstituent elements are plotted for Ni(V) 2(VI) 4 compounds in Fig. 2(b) , which s hows \na clear inverse correlation.  \nTo further confirm  this argument, we analyzed element -projection band structures \nof bulk NiSb 2Te4, NiBi 2Te4, NiSb 2Se4, and NiBi 2Se4 along the Γ-Z direction (out-of-\nplane ) (Fig. 2(c)) . The wider band dispersion along the Γ-Z direction  means  the more \ndelocalized p or d electrons along out -of-plane direction. Comparing NiSb 2Te4 with \nNiBi 2Te4, the dispersion of Te bands in  the former case is obviously wider than in the \nlatter , facilitated by a closer  electronegativity between Sb and Te. More hybridized and \ncoupled with Te 5 p bands, Sb 5p bands are pushed upward by Te 5 p bands almost  above  \nNi 3d bands, with wider band dispersion than that of Bi 6 p bands.  These  band features \nwill definitely lead to larger IMC in NiSb 2Te4 than NiBi2Te4. Comparing NiSb 2Te4 with \nNiSb 2Se4, however,  two factors are needed to be considered. Firstly, Te share s closer \nelectronegativity to Sb than Se, and Sb 5p bands are located nearer to Te 5p bands than Se 4p bands. The dispersion of Sb 5 p bands in NiSb 2Te4 is thus larger than NiSb 2Se4, \nindicating more delocaliz ed p electrons . Secondly, Te 5 p orbitals are more extended  \nthan Se 4 p orbitals, leading to stronger overlapping with Ni 3 d orbitals and with Te 5p \norbitals of neighboring SLs  in real space . Figure 2(d) shows the zoom -in band structures \naround Ni 3 d bands. The 3 d band dispersion of magnetic atoms is directly related to the \nIMC strength. Clearly, the dispersions of the four compounds give band widths \nNiSb 2Te4>NiBi 2Te4> NiSb 2Se4>NiBi 2Se4, exhibiting th e same trend of IMC strength . \nThe dependence of | Eex| on magnetic elements listed in TABLE I can also be understood \nwith similar analyses. The much stronger IMC of NiBi 2Te4 bilayer than other XBi 2Te4-\nYBi 2Te4 combinations is because the 3 d bands of Ni is cl oser to and thus more strongly \nhybridized with p bands than the d bands of other magnetic elements.  \nThe understanding on the IMC mechanism of MnBi 2Te4-family enables us \ndesign ing and engineer ing topological states  by controlling the IMC . For example, bulk  \nMBT is an AFM TI in its ground state and becomes a magnetic Weyl semimetal when \nit is changed into the FM configuration by a large external magnetic field  (23-25, 27). \nThe magnetic Weyl semimetal possesses only one pair of Weyl points and is expected \nto show high -Chern -number QAH  phases  in thin films  (33, 48-52). The FM IMC \nbetween MBT and EBT suggests that one may obtain an intrinsic ferromagnetic WSM  \n(in no need of applying  external magnetic field s) in MBE -EBT heterostructures .  \nFigure 3(a) displays the calculated bulk band structure of a MBT -EBT superlattice . \nA band crossing is observed at EF near the  point along the Γ−Z direction. Except \nfor the two crossing points (the other point is along opposite  direction  symmetrically ) \nand a weakly dispersed band  connecting them, no electron or hole pocket exist at EF. \nFigure 3(b) gives the Fermi surface of the ( 𝑘𝑥−𝑘𝑧) surface states , showing a Fermi arc \nconnecting the two crossing points  (i.e. Weyl points) . The band and Fermi surface \nstructures indicate  that bulk MBT-EBT is a type-I ferromagnetic WSM possessing only \none pair of Weyl points (WPs)  (53). By checking the motions of the sum of Wannier \ncharge centers (WCCs), the two WPs show opposite chiralities, as theoretically \nexpected (Fig.  3(c)). We check ed the evolution of its band structure with the SOC \nartificially varied from 75% to 110% of the real strength.  The dependence of band gap at the Γ point on the SOC strength is depicted in Fig.  3(d). The band gap c loses twice \nat 92% and 102%, and the band closing points divide  the phase diagram into three parts, \nincluding a trivial FM insulator phase , a WSM phase, and a FM insulator phase with \nband inversion.  The last  one is a phase  named as “FMTI ”, which is topological ly \nequivalent to Cr - or V-doped (Bi,Sb) 2Te3 TIs (54, 55). When the time reversal symmetry \nis recovered above the Curie temperature , it is a typical TI . Below the Curie temperature,  \nits bulk p hase becomes topologically trivial with br oken  time-reversal symmetry,  but \nits thin films are  expect ed to be QAH insulators  with C=1 (51).   \nA 2D thin film of a magnetic WSM can show the QAH effect with the Chern \nnumber increasing with the film thickness  (33, 51, 52). Figure 4(a) shows the evolution \nof the Chern number and band gap of MBT -EBT thin films with varying film thickness. \nThe film including one MBT -EBT bilayer is a normal insulator with a zero Chern \nnumber  due to quantum confinement effects. Thicker films containing 2 to 10 MBT -\nEBT bilayers are QAH insulators  with C=1, whose  band gap reaches a maximum (a bout \ntens of meV ) in the 3-bilayer  film. In a magnetic WSM film, the increment of Chern \nnumber ( C) with film thickness satisfies  (33): ∆𝐶≈∆𝑁|𝑘𝑊′|, where |𝑘𝑊′| denotes \nthe ratio of Γ−W  to Γ−Z  along Γ−Z  direction. For bulk MBT -EBT, |𝑘𝑊′|=\n0.0717\n0.5≈1\n7, which is consistent with the calculation result  that C=2 from 11 to 17 bila yers \nand C=3 above 18 bilayers . The edge states of a 3 -bilayer film and a 12 -bilayer film are \ndisplayed in Fig. 4(b) and Fig.  4(c), respectively. Clearly, they are QAH insulators with \nC=1 and C=2, respectively, showing  one and two gapless chiral edge modes  crossing \nthe bulk gap.   \nIn this material  system , a moderate external hydrostatic pressure can lead to a band \ninversion at  the Γ point , resulting in a topological phase transition (51). Fig ures. 3(e) \nand 3(f) present the p rojection  band ( p-band) results projected by Bi (blue) and Te (red) \nunder ambient pressure and 0.2 GPa hydrostatic pressure, respectively. Under ambient \npressur e, there is only one band inversion between Bi pz and Te pz bands, which leads \nto a WSM phase. Under 0.2 GPa pressure, two band inversions occur,  opening an \ninsulating band gap. The so -called “FMTI ” thus emerges  (51). Thin films of this phase above 2 bilayers are all QAH insulator with C=1, as confirmed by edge -state results \nshown for 3 bilayers (Fig.  4(e)) and 12 bilayers (Fig.  4(f)). \nThe IMC mechanism of MBT -family materials  revealed in this work provides us \na universal framework for understanding the magnetism of insulating layered magnetic \ncompounds and their heterostructures. Below we summarize several guiding principles \nfor estimati ng the IMC of insulating layered materials qualitatively or even quantitively.   \nFirst, the rules for analyzing superexchange interactions in ferrites can basically \nalso be applied to estimate the sign and relative magnitude of IMC. Especially, eg-eg \nhoppi ng through p bands usually dominates the magnetic coupling if  present . The IMC \nof many vdW heterostructures such as XB i2Te4-CrI 3, XB i2Te4-Cr2Ge2Te6, XB i2Te4-\nFePS 3 can be estimated by this way  (see S upplementa l Material). For systems where \neg-eg is not th e dominated exchange coupling channel, more careful analyses are needed \nbecause different hopping channels  favoring  different  kinds of  magnetic states  may \ncompete (7-10, 12-18).  \nSecond, a stronger IMC requires more delocalized medial p electrons  which can \nbe realized  by similar electronegativities  between  cations and  anions . It’s worth \nmentioning  that heavy nonmetallic elements share the similar electronegativities with \nheavy metallic elements. One can choose  materials including  heavier elements near the \nmetal -insulator boundary to obtain a strong er IMC . \nThird, the IMC can be further enhanced if the d bands of magnetic ions are near \nEF, strongly hybridiz ing with the p band s. So, Ni2+-based FM or AFM insulators, for \nexample, could contribute to  a strong IMC with a neighboring layer ed magnetic \nmaterial . \nThese guiding principles enable us design ing and engineering the magnetic \nproperties of magnetic vdW materials for different purposes. The various van der Waals \nheterostructures composed of different layered magnetic materials will ma ke the \nprinciples easily  tested and widely applied  in experiment.  \nIn conclusion, we systematically studied the IMC  mechanism of the MnBi2Te4-\nfamily compounds and heterostructures . The superexchange coupling is found to \ndominate the IMC, and the delocalize d p electrons contribute to the relatively  strong  coupling strength across a long atomic distance. 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Zhan g, Topological insulators \nin Bi2Se3, Bi 2Te3 and Sb 2Te3 with a single Dirac cone on the surface. Nat. Phys. 5, \n438 (2009).  \n \n  \nFigure. 1. (a, b) Schematic lattice structures (side view)  of (a) a MnBi2Te4-MnBi 2Te4 \nbilayer and ( b) a MnBi 2Te4-EuBi 2Te4 bilayer . The red arrows indicate the spin \ndirections.  (c) Schmatic drawings of three hopping channels supporting the \nsuperexchange interlayer magnetic coupling : eg-p-eg (left), eg-p-t2g (middle)  and t2g-p-\nt2g (right).  X and Y denote two magnetic atoms in XB i2Te3-YBi2Te3. (d, e) Schemtic \ndrawings of the  energy levels of magnetic atoms  and the hopping channels between \nthem in MnBi 2Te3-MnBi 2Te3 (d) and in MnBi 2Te3-EuBi 2Te3.  \n \n  \nTABLE 1:  The magnetic ground states and the interlayer exchange energies  (Eex) which \nis the  energy difference between the A -type AFM state and FM state  of XBi 2Te3-\nYBi 2Te3 bilayers  (X, Y represents Ni, Mn, V , or Eu) . \n \n \n \n \n \n \n  \n \nFigure. 2. (a) Table of the interlayer magnetic coupling (IMC) strength (| Eex|) of six \ncompounds of Ni(Ⅴ) 2(Ⅵ) 4, in which (V) = Bi, Sb; (Ⅵ) = Te, Se, S. Electronegativity \n(χ) of each element is noted below as red digits. (b) Dependence of | Eex| on the  \nelectronegativity differen ce (Δχ) between the anions (VI) and nonmagentic cations (V) \nof six Ni(Ⅴ) 2(Ⅵ) 4. (c) Dispersions of element -projected band s along Γ−Z directions \nof NiSb 2Te4, NiBi 2Te4, NiSb 2Se4, and NiBi 2Se4. Red  curves  denote  the Ni 3d-eg bands. \nGreen and purple curves denote  the p bands  of (V) and (Ⅵ) , repectviely . (d) Zoom -in \nfigures near the Ni 3d-eg bands in the four materials.  \n \n \n \n \n \n \n \n \n \n \nFigure. 3. Band structres and  topological features of bulk M nBi2Te4-EuBi2Te4. (a) Band \nstructures of bulk MnBi 2Te4-EuBi 2Te4 superlattice . The inset  shows a weakly dispersed \nband along the -Z direction , implying the type-Ⅰ WSM nature. (b) Surface  states  on \nthe (110) plane at the energy of the two Weyl points (WP 1 and WP 2). (c) The motions \nof the sum of Wannier charge centers on a sphere surrounding  each of  two Weyl points \nin the mome ntum space . (d) Band gap at Γ as a function of the strength of spin-orbit \ncoupling . The three  areas of different colors  denote different topological phases. (e, f) \nElement -projected bandstructures of Bi pz and Te pz orbitals along Z−Γ  direction \nunder (e) zero pressure and (f) external 0. 2GPa hydrostatic pressure , respectively . “+” \nand “–” signs denote even and odd parities, respectively.  \n \n \n \n \n \n \n \nFigure. 4. Topological features of M nBi2Te4-EuBi2Te4 superlattice thin films. (a, d)  The \nband gap size at  point  (red)  and the Chern number  (blue) of MnBi 2Te4-EuBi 2Te4 thin \nfilm with increasing  film thickness  under hydrostatic pressue of (a) zero and ( d) 0.2  \nGPa. The areas  of different colors  represent  the phases with different Chern numbers . \n(b, c) Edge states of (b) a film of 3 MnBi 2Te4-EuBi 2Te4 bilayer s and (c) a film of 12 \nMnBi 2Te4-EuBi 2Te4 bilayers without external pressure, which show  one and two chiral \nedge states , respectively . (c, e) Edge states of (e) a film of 3 MnBi 2Te4-EuBi 2Te4 \nbilayers and (f) a film of 12 MnBi 2Te4-EuBi 2Te4 bilayers under 0.2GPa , both of which \nexhibit one chiral edge state .  \n \n \n \n \n \n \n \n \n"    },    {        "title": "2004.00468v1.Theoretical_investigation_on_magnetic_property_of_monolayer_CrI3_from_microscale_to_macroscale.pdf",        "content": "Theoretical investigation on magnetic property of monolayer CrI3 from microscale to macroscale\nSongrui Wei,1,\u0003Dingchen Wang,2,\u0003Yadong Wei,1,yand Han Zhang1,z\n1Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province,\nCollege of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China\n2MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, School of Science,\nState Key Laboratory for Mechanical Behavior of Materials, Xian Jiaotong University, Xian 710049, China\nI. INTRODUCTION\nMagnetic two-dimensional (2D) materials have received\ntremendous attention recently due to its potential applica-\ntion in spintronics and other magnetism related fields. To\nour knowledge, five kinds of 2D materials with intrinsic\nmagnetism have been synthesized in experiment. They are\nCrI31–4, Cr2Ge2Te6, FePS3, Fe3GeTe25,6and VSe27. Apart\nfrom the above intrinsic magnetic 2D materials, many strate-\ngies have also been proposed to induce magnetism in nor-\nmal 2D materials such as atomic modification, spin valve and\nproximity effect8–10. Various devices have also been designed\nto fulfill the basic functions of spintronics: inducing spin, ma-\nnipulating spin and detecting spin.\nAmong the above 2D materials with intrinsic and induced\nmagnetism, CrI3 is a remarkable milestone because it is the\nfirst ferromagnetic 2D material down to one atomic layer\nthickness with an acceptable and tunable Curie temperature11.\nIt is synthesized in experiment with the mechanic exfoliation\nmethod and the magnetism is verified by many measurements\nsuch as magneto-optical Kerr effect, X-ray spectroscopy and\nmagnetic circular dichroism1,12. Due to its typical ferro-\nmagnetism in in-plane direction and anti-ferromagnetism be-\ntween neighboring layers, it has promising application in giant\nmagnetoresistance effect, multiferroic phenomenon, magnetic\nstorage and spin valves.\nAccompanying with the rapid experimental progress,\nmany simulation and theoretical works have also been\nperformed13–16. On one hand, for the application purpose,\nmany methods have been proposed to increase the Curie tem-\nperature, strengthen the magnetism and stabilize the structure.\nSome devices for the application of magnetic memory stor-\nage and magnetic diode have also been designed based on the\ngiant magnetoresistance effect and the concept of magnetic\nskyrmions. On the other hand, to understand the underly-\ning physics, the origin of the magnetism in layered CrI3 have\nalso been widely investigated17–22. Some problems such as\nwhether the spin orientation is Ising type or Heisenberg type;\nis the XXZ model suitable for describing the anisotropy of\nCrI3; what is the relationship between Kitaev exchange in-\nteraction and single-ion anisotropy in CrI3 have been widely\ninvestigated8,23.\nIt has been imagined that around 2025 the magnetic semi-\nconductors may be applied to commercial use. Although\nmuch progress has been made, most theoretical researches\nare still limited to the microscopic properties which ban be\nobtained directly based on density functional theory (DFT)\nor tight binding model. Only a few works try to obtain the\nmacroscopic magnetic properties based on the microscopic re-sults while the investigated macroscopic properties are limited\nto Curie temperature and critical exponent24. For applications,\nthe macroscopic magnetic properties such as domain structure\nand hysteresis loop are important. So, a theoretical description\nof macroscopic properties of magnetic semiconductors with a\nvalid basis will help much in the application of magnetic semi-\nconductors.\nThe micromagnetic theory is a mature theory which de-\nscribes the macroscopic properties of magnetic materials such\nas domain structure and hysteresis loop11,25,26. Many software\nbased on the micromagnetic theory have also been developed\nsuch as OOMMF, mumax3 and spirit. One disadvantage of\nmicromagnetic theory is that it needs the empirical param-\neters as the input and the empirical parameters usually can\nonly be obtained from experiment. This strongly impedes\nthe application of micromagnetic theory. In this work, we\ntried to obtain the empirical parameters from first principle\nsimulations based on DFT. We constructed the quantitative\nrelationship between the empirical parameters in micromag-\nnetic theory and the energy terms in DFT. In this way, the\nmacroscopic properties can be simulated with a valid basis\nand the disadvantage of micromagnetic theory is overcome\nat the same time. In this work, the microscopic properties\ninclude the band structure, density of state, atomic magnetic\nmoment, charge density distribution and exchange coefficient\nwhile the macroscopic properties include the Curie tempera-\nture, hysteresis loop, domain structure and easy magnetization\ndirection. Most of the results are consistent with the experi-\nmental or former theoretical work. To our knowledge, there is\nno such an overall investigation of magnetic properties from\nmicroscopic to macroscopic on 2D materials. Our work will\nnot only facilitate the application of CrI3 in spintronics but\nalso provide a new research method in this field.\nII. METHOD\n(1) DFT parameters Our density functional theory\n(DFT) calculations are carried out within Perdew-Burke-\nErnzerhof (PBE) exchange-correlation functional and projec-\ntor augmented-wave (PAW) pseudopotentials as implemented\nin Vienna ab initio simulation package (V ASP). The cutoff en-\nergy for the plane-wave basis is 450 eVand a Monkhorst-Pack\nk-point mesh of 15\u000215\u00021is used in the hexagonal Brillouin\nzone. In each configuration, atoms are fully relaxed by em-\nploying conjugate-gradient (CG) method. The total energy\nand atomic forces are converged to 10-5 eVand 0.01eV=˚A\nand a large vacuum spacing of at least 20 along out-of-plane\ndirection is used in all the calculations.arXiv:2004.00468v1  [physics.comp-ph]  1 Apr 20202\n(2) Transition from output of DFT to input of micromag-\nnetic model There are at least three parameters that need to be\ncalculated from the results of DFT to perform the subsequent\nmicromagnetic simulation. They are: (a) Saturated magne-\ntizationMsat; (b) Anisotropy coefficient Ku1; (c) Exchange\ncoefficientAex. (a) Saturated magnetization MsatFirstly, the\nsize of atomic magnetic moment can be obtained directly from\nthe OUTCAR of V ASP. Secondly, the saturated magnetization\nMsatcan be calculated as:\nMsat=n\u0001M\nV(1)\nwhere n is the number of magnetic atoms in a unit cell, M is\nthe size of the atomic magnetic moment, and V is the volume\nof the unit cell. The unit is transformed properly at the same\ntime.\n(b) Anisotropy coefficient Ku1The anisotropy energy term\nin micromagnetic theory describes the total anisotropy of the\nsystem. The anisotropy coefficient describes the density of\nsuch an anisotropy. Neglecting the higher order anisotropy,\nthe anisotropy coefficient Ku1can be calculated as:\nKu1=\u0001E\nV(2)\nwhere E is the energy difference between the easy magnetiza-\ntion direction and hard magnetization direction of the whole\nsystem, and V is the volume of the whole system. In the CrI3\nsystem, the easy magnetization direction is the out-of-plane\ndirection and the hard magnetization direction is the in-plane\ndirection. In mumax3, the easy magnetization direction must\nalso be chosen.\n(c) Exchange coefficient AexThe exchange energy density\nin micromagnetic theory can be expressed as:\n\"exch=Aex(rm)2(3)\nwhereAexis the exchange coefficient and m is the normal-\nized magnetic moment. Considering the relationship between\nexchange energy density and the difference between ferro-\nmagnetic and antiferromagnetic state, the exchange coeffi-\ncientAexcan be expressed as:\nAex=EFM\u0000EAFM\n4Vr2(4)\nwhere EFM and EAFM is the energy of the system at ferro-\nmagnetic and antiferromagnetic state respectively. V is the\nvolume of the whole system and r is the distance between Cr\natoms.\n(3) The transition from hexagonal structure to rectangular\nstructure In mumax3 and many other micromagnetic software,\nthe basic magnetic moments can only be arranged in rectangu-\nlar structure. But the structure of many magnetic 2D materials\nare hexagonal structure. Here we provide a method to trans-\nform the hexagonal structure to rectangular structure as shown\nin Figure 1b. We set a 10*10 square of unit cell of CrI3 as the\nbasic magnetic moment of the micromagnetic model and the\nbasic magnetic moments are arranged in rectangular structure.Magnetic\nmoment\nof Cr\natom (\u0016B)V olume of\nprimitive\ncell ( ˚A3)ferromagnetic\nstate\nvertical\nto plane (eV)Ferromagnetic\nstate parallel\nto plane (eV)Antiferromagnetic\nstate vertical\nto plane(eV)\nCrI3 2.671 7335.87 -291.31322 -291.29861 -291.17477\nIt is worth noting that after the transformation, the above cal-\nculated empirical parameters should not be scaled accordingly\nbecause all the energy terms in mumax3 are energy densities.\n(4) The micromagnetic simulation\nIn this section, we will simulate two different sizes of mag-\nnetic structure: 2 \u0016m\u00022\u0016mand 240nm\u0002240nm. For\neach size, we will firstly simulate the domain structure and\nspontaneous magnetization changing with temperature. Then,\nfor certain typical temperature, we will simulate the hystere-\nsis loop and the evolution of domain structure with external\nmagnetic field.\nIII. RESULT & DISCUSSION\n(1) The first principle calculation part\nThe magnetic moment, volume of primitive cell and en-\nergy terms of different magnetic states of monolayer CrI3 are\nshown in section . In this work, the primitive cell includes 18\nCr atoms. The calculated magnetic moment of Cr atom is in\ngood agreement with its spin quantum number S=3. The vol-\nume used here is after the structure relaxation. The energies of\ndifferent magnetic states are about the primitive cell which in-\ncludes 18 Cr atoms. These results are consistent with precious\ntheoretical work. Figure 1B is the spin density in real space.\nIt is clear that the magnetism is originated from the Cr atoms\nand the direction of the magnetic moment is in out-of-plane\ndirection. (2) The micromagnetic theory part\nThe parameters of micromagnetic theory are shown in Fig-\nure . They are calculated from the results of first principle\ncalculation. The DM interaction is zero because the structure\nis symmetrical in vertical direction. Putting the parameters in\nFigure to mumax3 and the domain structures and spontaneous\nmagnetization in different directions can be calculated. In this\nwork, two different sizes of magnetic systems are calculated.\nThe smaller one is 240 nm\u0002240nmwhich is in a single\ndomain. The larger one is 2 \u0016m\u00022\u0016mwhich includes more\nthan one domain. On the other hand, the change of domain\nstructure and spontaneous magnetization with temperature is\nsimulated by two steps. In first step, we simulated the domain\nstructures and spontaneous magnetization at the temperature\nof 0 K, 20 K, 40 K and 60 K, as shown in Figure 2. The\nwhite color and black color represent the up and down direc-\ntions vertical to the plane respectively. The other colors mean\nthat there is a component in in-plane direction. At each tem-\nperature, the domain structures are fully relaxed. Under such\nlarge temperature intervals, the change of domain structure\nand spontaneous magnetization with temperature or namely\nthe magnetic phase is very obvious. The relaxation process\nof spontaneous magnetization is also recorded, as shown in3\nFIG. 1.\nFIG. 2.\nFigure 3.\nThe results about domain structure and spontaneous mag-\nnetization are consistent with each other. In the second step,\nwe simulated the evolution of domain structure and sponta-\nneous magnetization in the temperature range of 40 K-50 K at\nthe temperature interval of 1 K, as shown in Figure 4.\nBy carefully observing the change of domain structure and\nspontaneous magnetization, the Curie temperature ca be deter-\nmined as around 43 K. This value is very close to the experi-\nmentally measured Curie temperature of monolayer CrI3: 45\nK. It is also worth noting that while it is clearer to determine\nthe Curie temperature from the domain structure with a larger\nsample size (2 \u0016m\u00022\u0016m), it is clearer to determine the\nCurie temperature from the spontaneous magnetization with a\nsmaller sample size (240 nm\u0002240nm). That is because the\ndomain wall is important for the human eyes judgement about\nthe transition process while the spontaneous magnetization al-\nmost vanishes in a multidomain sample. So, in Figure 4 the\ndomain structure is about the sample of 2 \u0016m\u00022\u0016mwhile\nthe spontaneous magnetization is about the sample of 240 nm\n\u0002240nm. This is the first reason for why we simulate the\nsamples of these two sizes.\nFor the sample at fixed temperature, the hysteresis loop can\nbe calculated. Figure 5 shows the hysteresis loop of a 2 \u0016m\n\u00022\u0016m sample at 10 K which is a typical ferromagnetic\nstate. The evolution of domain structures at some magnetic\nfields are also recorded. When the direction of magnetic fieldSaturated\nmagnetization\n(A/m)Anisotropy\nenergy\ncoefficient\n(J/m3)Exchange\ncoefficient\n(J/m)DM interaction\ncoefficient\n(J/m)Easy\ndirection\nCrI3 6.0775E4 3.186536E51.06585\nE-120Out of\nplane\ndirection\nis along the z axis which is vertical to the 2D plane of CrI3\n(black line), there is an obvious lag of the response to the ex-\nternal magnetic field and the hysteresis loop is nearly a rect-\nangular. It means that in this case the magnetism is harder.\nIn experiment, the hysteresis loop under a vertical external\nfield is measured with the optical method by many research\ngroups. For the coercivity, the results from experiments and\nour simulation are consistent with each other. For the mag-\nnetization, as the y axis of hysteresis loop measured by an\noptical method is a percentage which represents the change of\nmagnetization qualitatively, the results from simulation and\nexperiment can not be compared quantitively. When the ex-\nternal field is along the x direction which is parallel to the\n2D plane of CrI3 (red line), the area of the hysteresis loop is\nnearly zero which means that the magnetism is very soft. To\nour knowledge, the hysteresis loop under a parallel magnetic\nfield has not been measured in experiment yet. The evolution4\nFIG. 3.\nFIG. 4.\nof domain structure with external magnetic field is consistent\nwith the hysteresis loop. There is an obvious lag of evolution\nof domain structure to external field under a vertical magnetic\nfield. ??compares the hysteresis loop of 2 \u0016m\u00022\u0016msam-\nple and 240 nm\u0002240nmsample. We did this simulationbecause in experiment, it is found that the coercivity of mono-\nlayer CrI3 measured by optical method may range from 50 to\n200 mT. This difference in coercivity is ascribed to the differ-\nent sample size which may contain one single domain or multi\ndomains. Our result supports this conjecture.\n\u0003Songrui Wei& Dingchen Wang Contributed equally to this work\nyywei@szu.edu.cn\nzhzhang@szu.edu.cn\n1L. Chen, J.-H. Chung, B. Gao, T. Chen, M. B. Stone, A. I.\nKolesnikov, Q. Huang, and P. Dai, Phys. Rev. X 8, 041028 (2018).\n2Y . Hou, J. Kim, and R. Wu, Science advances 5, eaaw1874\n(2019).\n3M. U. Farooq and J. Hong, npj 2D Materials and Applications 3,\n1 (2019).\n4C. Xu, J. Feng, H. Xiang, and L. Bellaiche, npj Computational\nMaterials 4, 1 (2018).\n5Y . Deng, Y . Yu, Y . Song, J. Zhang, N. Z. Wang, Z. Sun, Y . Yi,\nY . Z. Wu, S. Wu, J. Zhu, et al. , Nature 563, 94 (2018).\n6Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song, J. Sanchez,W. Yao, D. Xiao, X. Zhu, A. F. May, et al. , Nature materials 17,\n778 (2018).\n7M. Bonilla, S. Kolekar, Y . Ma, H. C. Diaz, V . Kalappattil, R. Das,\nT. Eggers, H. R. Gutierrez, M.-H. Phan, and M. Batzill, Nature\nnanotechnology 13, 289 (2018).\n8K. Kim, J.-U. Lee, and H. Cheong, Nanotechnology 30, 452001\n(2019).\n9M. Gibertini, M. Koperski, A. Morpurgo, and K. Novoselov, Na-\nture nanotechnology 14, 408 (2019).\n10Y . Zeng, L. Wang, S. Li, C. He, D. Zhong, and D.-X. Yao, Journal\nof Physics: Condensed Matter 31, 395502 (2019).\n11A. K. Behera, S. Chowdhury, and S. R. Das, Applied Physics\nLetters 114, 232402 (2019), https://doi.org/10.1063/1.5096782.\n12L. Thiel, Z. Wang, M. A. Tschudin, D. Rohner, I. Guti ´errez-5\nFIG. 5.\nLezama, N. Ubrig, M. Gibertini, E. Giannini, A. F. Mor-\npurgo, and P. Maletinsky, Science 364, 973 (2019),\nhttps://science.sciencemag.org/content/364/6444/973.full.pdf.\n13J. Kanamori, Journal of Physics and Chemistry of Solids 10, 87\n(1959).\n14H. Ehrenberg, M. Wiesmann, J. Garcia-Jaca, H. Weitzel, and\nH. Fuess, Journal of magnetism and magnetic materials 182, 152\n(1998).\n15J. B. Goodenough, Journal of Physics and chemistry of Solids 6,\n287 (1958).\n16P. Anderson, Physical Review 79, 350 (1950).\n17J. B. Goodenough, Physical Review 100, 564 (1955).\n18G. Guo, G. Bi, C. Cai, and H. Wu, Journal of Physics: Condensed\nMatter 30, 285303 (2018).\n19S. W. Jang, M. Y . Jeong, H. Yoon, S. Ryee, and M. J. Han, Phys-\nical Review Materials 3, 031001 (2019).\n20V . K. Gudelli and G.-Y . Guo, New Journal of Physics 21, 053012\n(2019).\n21N. Sivadas, S. Okamoto, X. Xu, C. J. Fennie, and D. Xiao, Nano\nletters 18, 7658 (2018).\n22M. Wu, Z. Li, T. Cao, and S. G. Louie, Nature communications\n10, 1 (2019).\n23D.-H. Kim, K. Kim, K.-T. Ko, J. Seo, J. S. Kim, T.-H. Jang,\nY . Kim, J.-Y . Kim, S.-W. Cheong, and J.-H. Park, Physical re-\nview letters 122, 207201 (2019).\n24T. Liu, N. Zhou, X. Li, G. Zhu, X. Wei, and J. Cao, Journal ofPhysics: Condensed Matter 31, 295801 (2019).\n25O. Besbes, S. Nikolaev, N. Meskini, and I. Solovyev, Phys. Rev.\nB99, 104432 (2019).\n26M. Deb and A. K. Ghosh, Journal of Physics: Condensed Matter\n31, 345601 (2019)."    },    {        "title": "2004.11589v1.Observation_of_magnetic_domains_in_uniaxial_magnets_via_small_angle_electron_diffraction_and_Foucault_imaging.pdf",        "content": "1 \n Observation of magnetic domains in uniaxial magnets via small -angle \nelectron diffraction  and Foucault imaging  \nHiroshi Nakajima1, Atsuhiro Kotani1, Ken Harada1,2, and Shigeo Mori1* \n1Department of Materials Science, Osaka Prefecture University, Sakai, Osaka 599 -8531, Japan.  \n2Center for Emergent Matter Science, the Institute of Physical and Chemical Research (RIKEN), \nHatoyama, Saitama 350 -0395, Japan  \n*E-mail: mori@mtr.osakafu -u.ac.jp  \n \nObservation of  magnetic domains is important in understanding  the magnetic properties of magnetic \nmaterials and devices.  In this study , we report that the magnetic domains  of M-type hexaferrite s with \nuniaxial anisotropy  can be visualized via small -angle electron diffraction  and Foucault imaging . The \nposition of the diffraction pattern spots has the same period  as that of magnetic domains in a Sc -\nsubstituted hexaferrite ( BaFe 12-x-δScxMg δO19). Conversely,  the spots were observed four times longer \nthan t he period  of magnetic  domain s in hexaferrite without substitution (BaFe 12O19), demonstrating  the \nlong-range order of the Bloch walls . When the specimen was tilted , the magnetic deflection  effect , as \nwell as the periodic spots of magnetic domains,  occurred . Thus , we were able to visualize the magnetic \ndomains with different magnetization directions and domain orientations  by selecting deflection spots . \nThe results indicate that the technique utilized in this s tudy is useful in observ ing the magnetic materials \nwith uniaxial  anisotropy.  \n \nKeywords: Small -angle electron diffraction, Foucault imaging, magnetic domains, hexagonal ferrites, \nBloch wall  \n  2 \n 1. Introduction  \nFerromagnetic materials have been extensively used in industrial appliances , such as motors,  \ninductances, and  memory devices.  Recently , hexagonal ferrites (hexaferrites), Sm –Co, and Nd–Fe–B \nmagnets have been extensively investigated to improve the magnetic properties of electric appliances  \nbecause the se materials exhibit high magnetizations , uniaxial anisotropy , and transition temperatures \nabove room temperature .1,2) The m agnetic microstructures should be significantly examined to evaluate \nthe magnetic properties of these  magnets . \nTo observe  the magnetic domains in the magnets, several methods have been used , such as x-ray \nmagnetic circular dichroism, magnetic force microscopy, spin -polarized scanning electron microscopy, \nand Lorentz remission electron microscopy (Lorentz microscopy  hereinafter ). Among them, the Lorentz \nmicroscopy has high spatial resolution and availability for controllable experimental conditions , such as \ntemperature, external magnetic field, current induction, and stress.3–6) It mainly involves five techniques , \nnamely,  Fresnel  imaging,  Foucault imaging , electron holography  (EH) , differential phase contrast \nimaging  (DPC) , and s mall-angle electron diffraction (SmAED) .7,8) The first two method s provide \nqualitative  descriptions on the spatial variation of magnetization , whereas the rest supply quantitative \ninformation  on magnetic domain structures.  Fresnel imaging  is a method  used to depict  domain walls by \ndefocusing the imaging lens , whereas Foucault imaging  is utilized to visualize  magnetic  domains by \nselecting magnetically deflected spots  with an aperture . Unlike Fresnel imaging , Foucault imaging  has \nbeen used less frequently because performing  Fresnel imaging by defocusing the imaging lens  is easy . \nRecently, optical system s employed in per forming  Foucault imaging combined with  SmAED  were  \nconstructed  using a conventional transmission electron microscope.9–12) These  optical system s allow the \nobservation  of magnetically deflected spots and the visualization  of domains that cause the \ncorresponding spots. Foucault imaging is advantageous for magnetic dom ain observations because \nmagnetic domains with particular magnetization orientations can be visualized. The SmAED is useful in \nquantitatively analyzing  magnetization and domain periods  and investigating  the symmetry of magnetic \nstructure s.13–16) Unlike other quantitative methods ( e.g., EH and DPC ), it does not require special \nequipment , such as electron biprism and segmented detector. In addition,  information obtained using the \nSmAED is colle cted from a low -angle range ; thus, the measured values are not affected by diffraction  \neffect s, such as bend contour s. However, the SmAED cannot pro vide real-space images of  magnetic \ndomains . Therefore, this electron diffraction is useful when a combination of Fresnel imaging, SmAED, \nand Foucault imaging  is utilized in Lorentz microscopy observations.  3 \n In recent observations that employ SmA ED, its applications were  only restricte d to 180 ° domains in \nferromagnetic materials9,17,18 ) and helical domains in chiral magnets19,20). As mentioned  previously , \nmagnetic materials with uniaxial anisotropy , such as hexaferrites and Nd –Fe–B magnets , are important \nfor industrial  appliances . However, observing magnetic domains with uniaxial anisotropy  via Foucault \nimaging  is difficult  because the Lorenz force is not c aused when the incident beam is parallel to the \nmagnetization . \n In this paper, we report the results  of observ ing magnetic domains with uniaxial anisotropy  in Sc-\nsubstituted M-type hexaferrite s (BaFe 12–x–δScxMg δO19)21,22 ) via the SmAED and Foucault imaging . We \nobtained the SmAED patterns  demonstrating the periodic spots in  hexaferrite s. The periodic spots in the \npattern s were generated from the Bloch wall  when the incident beam was parallel to the easy axis of the \nmagnetization . In BaFe 12O19 without substitution,  the width estimated from t he periodic position was \nfour times longer than that of magnetic domains because the Bloch walls had the ↑↑↓↓ -type orde r. By \ntilting the specime n, the magnetic deflection  from magnetic domains  was observed apart from the \nperiodic spots  in SmAED pattern s. Thus , we were able to visualize  the magnetic  domains with different \nmagnetization directions and domain orientations  using the Foucault images . This paper discloses the \nuseful technique for observing magnetic domains magnetized parallel to the observation  direction \nthrough the SmAED and Foucault imaging.  \n \n2. Experimental methods  \nFigure 1 shows  the sche matic  of the optical system  by (a) the SmAED  and (b) Foucault imaging .10,11 ) In \nthe optical system,  the o bjective lens was switched off to ensure that an external magnetic field was not \napplied to the specimen . External magnetic fields can be applied up to 200 mT by adjusting the currents \nof the objective lens and objective mini  lens in this optical system.  Condenser lenses were  strongly \nexcited also to ensure that the crossover size became small  and that the divergence angle of the \nirradiation beam was small . Then, the objective  mini lens and condenser lenses  were  adjusted  to locate \nthe crossover  at a selected -area (SA)  aperture  plane.  The SA aperture functions as an angle -limiting \naperture.  By switching from diffraction to Foucault modes, the intermediate lens I was less excited after \nselectin g the diffraction spots. By selecting magnetically deflected spots, the magnetic domains that \ncause  the corresponding spots can be  visualized in bright contrast  (Foucault imaging) . Moreover, t he \nobjective aperture can be used to limit the specimen area for SmAED . The Fresnel imagin g can also be \nperformed by rendering the intermediate lens I  out of focus.  We utilized transmission electron 4 \n microscopes (JEM -2010 and JEM -2100F, J EOL  Co. Ltd.) to observe magnetic material s with uniaxial \nanisotropy.  The accelerating voltage was 200 kV. The specimens were tilted by using a goniometer \nequipped with the microscopes. The optical system was constructed by using f ree lens control.  The \nresolution of SmAED was better than 0.5 rad in the optical system  becau se the diffraction spots of \n0.5 rad were resolved.  \nThe observation  method was demonstrated in  magnetic domain s of Sc-substituted  M-type barium \nhexaferrites  BaFe 12–x–δScxMg δO19 (𝑥=1.6,δ=0.05) (BFSMO  hereinafter ) and BaFe 12O19 without \nsubstitution .21,22 ) BFSMO and BaFe 12O19 have the ferrimagnetic state whose net magnetic moments \npoint parallel to the c axis at room temperature. The specimens for the observation were prepared as \nfollows.  The polycrystals were synthesized on the basis of the solid -state reaction from high -purity (4 N \nup) powders of BaCO 3, Fe 2O3, MgO, and Sc 2O3. The powders were weighted on the basis of  the \nrequired chemical compositions and were mixed in a mortar. Then, the mixed powders were calcined  at \n980 °C for 20 h in air. Thereafter , single crystals of the M-type barium hexaferrites were grown through  \na floating -zone method in the O2 flow.23) The single crystal s were  cleaved in  the c plane.  Thin specimens  \nwhose surface was perpendicular to  the c axis were  prepared via Ar ion milling  technique  for Lorentz \nmicroscopy observations . Moreover, t he specimen thickness was less than 100 nm. \n \n3. Results and discussion  \nFigure 2(a) shows the Fresnel image of BFSMO at the c plane  without tilting the specimen . Owing to \nthe Lorentz deflection, Bloch walls were visualized as  pairs of  bright and dark lines  [Fig. 2(b)]. Figure \n2(c) illustrates a SmAED  pattern of BFSMO obtained with the incident beam parallel to the c axis. This \npattern comprises  spots and streaks. These spot s are aligned in two directions that correspond to the V-\nshaped  magnetic domains . The spots were due to the period  of the magne tic domain walls  because the \ndistance  of approximately d = 180 nm corresponds to the spot  interval of = /d = 13.4 μrad, where \n = 2.51 pm is the wavelength of the electron. The streaks were  caused  by the Bloch walls wherein  \nmagnetization  continuously  rotate s.  \nSubsequently , we tilted the specimen by 30° to investigate the change in the diffraction pattern. \nFigure 3(a) shows the Fresnel image of BFSMO at the tilt angle of 30°. Unlik e in Fig. 2( b), the domain \nwall in Fig. 3(b) is represented with a bright or dark line. As shown in the SmAED pattern of Fig. 3(c), 5 \n the intensity of the direct beam spot  weakens  when the specimen was tilted . Before tilting the specimen,  \nthe magnetizations  in magnetic domains  were “up” (antiparallel) or “down” (parallel to the electron \nbeam)  and did not have the in -plane component , as illustrated in Fig. 2(d) . Thus, the electron that passes  \nthese domains were not affected by the Lorentz force , resulting in causing  the direct be am spot  indicated \nby a white arrow  in Fig . 2(c). However, when  the specimen was tilted, these domains had in -plane \ncomponents  [see Fig. 3(d)] ; consequently , the direct beam spot disappeared  by deflecting the electrons, \nas observed in Fig. 3(c). As shown in  Figs. 2 and 3, the magnetization direction  parallel to the electron \nbeam  can be deduced from the SmAED patterns  by tilting the specimen . The Lorentz deflection  \nincreased the intensity of the spots  indicated by A, B, C, and D  in Fig. 3(c); the periodic pat terns were \nsuperimposed on  the Lorentz deflection . The interval of the periodic spots at 6.8 μrad corresponds to \ntwice longer than the Bloch wall  period  (180 nm). The reason  for the double period  is that  a pair of  up- \nand down -magnetized domains  is the periodic structure  for the incident electrons  when the specimen \nwas tilted  [denoted by the blue mark s in Figs. 2(d) and 3(d)]. A similar phenomenon was also observed \nin another ferromagnet.24)  \nWe observed BaFe 12O19 to compare the results in BFSMO . Figure 4(a) shows the Fresnel image of \nBaFe 12O19. The domain width was approximately  250 nm. In the SmAED pattern shown in Fig. 4(b), the \nperiodic spots were observed at 2. 6 rad, which corresponds to (980 nm)−1. Accordingly, the period \nevaluated from the spots is four times longer than that of the domain width.  When observing the contrast \nat the Bloch walls in the Fresnel image, we identified that the magnetization s with right –right –left–left \n(↑↑↓↓) directions were repeated at the Bloch walls (denoted by the arrows ). These results suggest  that \nthe periodic spots were caused by the Bloch walls . This discussion  is also applicable to  the periodic \nspots in B FSMO  because the period of Bloch walls is the same as the domain width . When the \nBaFe 12O19 specimen was tilted, the Fresnel image [Fig. 4( c)] had bright or dark contrast at the Bloch \nwalls . This observation is similar to that of BFSMO. The spots in the SmAED of Fig. 4(d) are 5. 3 rad, \nwhich corresponds to  approximately twice of  the periodicity  (250 nm). There fore, the periodic spots \nwere due to  the magnetic domains , similar to the case of BFSMO , when the specimen was tilted. In \ncontrast , the periodic spots were generated by the Bloch walls before  tilting the  specimen s.  \nThe long -range  (↑↑↓↓)  order of the Bloch walls in BaFe 12O19 is difficult to be noticed  from the \nFresnel image [ Fig. 4(a)]. As demonstrated in Fig. 4, SmAED has the advantage to display unnoticeable  \nperiodic magnetic structures and to determine the magnetization directions.  The one possible origin of 6 \n the order of Bloch walls is assumed to be the balance between the dipole –dipole interaction  and stray \nfield energy.25,26 ) The dipole –dipole interaction  is long range  and expressed by the following equation:  \n𝐸dipole = 𝜇0𝑀𝑠2\n4𝜋∫𝑑𝒓∫𝑑𝒓′{𝒎(𝒓)∙𝒎(𝒓′)\n|𝒓−𝒓′|3+ 𝒎(𝒓)∙(𝒓−𝒓′) 𝒎(𝒓′)∙(𝒓−𝒓′)\n|𝒓−𝒓′|5}.      (1) \nHere , 𝑀𝑠 is saturation magnetization, 𝜇0 is vacuum permeability,  and 𝒎 is magnetization vector. The \nstray field energy represents the magnetostatic energy generated from the specimen to vacuum: \n𝐸s= 1\n2𝜇0∫𝐻𝑑 2 𝑑𝒓,     (2) \nwhere  𝐻𝑑  is stray field . The dipole –dipole interaction  favors the parallel magnetization . In contrast , the \nstray field energy  forces the magnetic domains to  form  flux-closed domain structures  (180 ° or 90 ° \ndomains) to avoid the increase in the stray field . When  only the dipole –dipole interaction  is considered, \nthe magnetizations at the adjacent Bloch walls point the same direction. However, the parallel \nmagnetizations increase the stray field energy owing to the magnetic field in vacuum. Thus, the \nantiparallel  magnetizations wer e also formed  to reduce the stray field energy , resulting  in the formation \nof the long-range (↑↑↓↓) order of Bloch walls. For BFSMO  that showed the parallel Bloch walls,  \npossible origins of the Bloch wall structures are the in -plane demagnetization field due to the wedge -\nshaped specimen and residual magnetic field from the electron microscope. However, these effects can \nbe ignored when the magnetocrystalline anisotropy  in BaFe 12O19 is large.  It is revealed that  the \nmagnetocrystalline anisotropy can be reduced by the Sc substitution  and the helicity (clockwise or \ncounterclockwise)  of the Bloch walls w as randomly distributed in BFSMO22), supporting the discussions \nthat in-plane  magnetic field s align ed with the Bloch walls . \nFurthermore , we performed  the Foucault imaging in BFSMO  to demonstrate that the present optical \nsystem is useful for the magnetic domain observation . By selecting the deflected spots (A and B), we \nwere able to obtain a Foucault image that visualizes  the up-magnetized domains as bright contrast in a \nlarge area (Fig. 5). Although the Fresnel image [Fig. 2(a)] illustrates  the Bloch wall positions, the \nFoucault image shows the magnetic domains with the  same magnetization directions.  \nIn addition,  we selectively visualized the magnetic domains with various magnetization directions \nand domain orientations . Figures  6(a)–6(f) show  the contrast change in the Foucault imaging using a  \nsingle or a pair of deflected  spots.  As presented in  Fig. 6(a), where spot s A and B were used for 7 \n visualization,  bright and dark lines were  alternately  depicted, corresponding  to up - and down -\nmagnetized  domains , respectively.  Unlike in  Fig. 6(a), the contrast of the domains in Fig. 6(b) is \ninterchanged , and the down -magnetized do mains have bright contrast  if spots  C and D were selected.  As \npresented in Fig. 6(c), we selected spots  A and C  from the up -magnetized domains in the x directions \nand the down -magnetized domains in the y directions.  Consequently , the bright  and dark contrast  is \ninterchanged where the domains change the directions . Meanwhile, t heir contrast was reversed when \nspots B and D were selected , as demonstrated in Fig. 6(d). The on ly up-magnetized domains in the x \ndirections  were bright , and the other regions were  dark when spot A was used in Fig. 6(e). Similarly , the \nup-magnetized domains in the y directions were bright , as shown in Fig. 6(f) in which spot B was \nselected . As shown in Fig. 6, the Foucault images can depict  the domains with  the selected \nmagnetization direction and domain orienta tions using the present method.  \n  \n4. Conclusion s \nWe demonstrated that the magnetic domains magnetized parallel  to the incident beam direction  can be \nvisualized via Foucault imaging and SmAED . SmAED pattern s showed periodic spots that correspond \nto the Bloch wall  periods . The ↑↑↓↓ -type Bloch walls were revealed on the basis of the SmAED pattern. \nBy tilting the specimen,  the magnetization  direction was deduced from the SmAED patterns. \nFurthermore,  the magnetic deflection  from magnetic domains  appeared , as well as  the periodic spots  \nafter tilting the specimens . The result s showed  that the magnetic domains with various magnetization \ndirections and domain orientations can be visualized by selecting the cor responding spots  as Foucault \nimages . These results will help  in observing and analyzing  the magnetic domains via Foucault imaging \nand SmAED.  \n \nAcknowledgments  \nThis work was partially supported by JSPS KAKENHI (Nos. 16H03833 and 15K13306) and a grant \nfrom the Murata Foundation.   8 \n References  \n1) R.C. Pullar, Prog. Mater. Sci. 57, 1191 (2012).  \n2) M. Sagawa, S. Hirosawa, H. Yamamoto, S. Fujimura, and Y. Matsuura, Jpn. J. Appl. Phys. 26, 785 \n(1987).  \n3) D.B. Williams an d C.B. Carter, in Transm. Electron Microsc. A Textb. Mater. Sci.  (Springer, New \nYork, 2009).  \n4) Y. Togawa, T. Kimura, K. Harada, T. Akashi, T. Matsuda, A. Tonomura, and Y. Otani, Jpn. J. Appl. \nPhys. 45, L683 (2006).  \n5) H. Nakajima, H. Kawase, K. Kurushima,  A. Kotani, T. Kimura, and S. Mori, Phys. Rev. B 96, 24431 \n(2017).  \n6) K. Shibata, J. Iwasaki, N. Kanazawa, S. Aizawa, T. Tanigaki, M. Shirai, T. Nakajima, M. Kubota, M. \nKawasaki, H.S. Park, and others, Nat. Nanotechnol. 10, 589 (2015).  \n7) M. De Graef, Introduction to Conventional Transmission Electron Microscopy  (Cambridge \nUniversity Press, 2003).  \n8) J.N. Chapman, J. Phys. D. Appl. Phys. 17, 623 (1984).  \n9) Y. Taniguchi, H. Matsumoto, and K. Harada, Appl. Phys. Lett. 101, 93101 (2012).  \n10) H. Nakajima, A. Kotani, K. Harada, Y. Ishii, and S. Mori, Surf. Interface Anal. 48, 1166 (2016).  \n11) H. Nakajima, A. Kotani, K. Harada, Y. Ishii, and S. Mori, Microscopy 65, 473 (2016).  \n12) H. Nakajima, A. Kotani, K. Harada, and S. Mori, Microscopy 67, 207 (2018).  \n13) R.H. Wade, Phys. Status Solidi 19, 847 (1967).  \n14) M.J. Goringe and J.P. Jakubovics, Philos. Mag. 15, 393 (1967).  \n15) O. Bostanjoglo and W. Vieweger, Phys. Status Solidi 32, 311 (1969).  \n16) A.G. Fitzgerald, Phys. Status Solidi 20, 351 (1973).  \n17) T. Koyama, Y. Togawa, K. Takenaka, and S. Mori, J. Appl. Phys. 111, 07B104 (2012).  \n18) A. Kotani, H. Nakajima, K. Harada, Y. Ishii, and S. Mori, Phys. Rev. B 94, 24407 (2016).  9 \n 19) Y. Togawa, T. Koyama, K. Takayanagi, S. Mori, Y. Kousaka, J. Akimitsu, S. Nishihara, K. Inoue, \nA.S. Ovchinnikov, and J. Kishine, Phys. Rev. Lett. 108, 107202 (2012).  \n20) H. Nakajima, A. Kotani, M. Mochizuki, K. Harada, and S. Mori, Appl. Phys. Lett. 111, 192401 \n(2017).  \n21) X.Z. Yu, K. Shibata, W. Koshibae, Y. Tokunaga, Y. Kan eko, T. Nagai, K. Kimoto, Y. Taguchi, N. \nNagaosa, and Y. Tokura, Phys. Rev. B 93, 134417 (2016).  \n22) H. Nakajima, A. Kotani, K. Harada, Y. Ishii, and S. Mori, Phys. Rev. B 94, 224427 (2016).  \n23) Y. Tokunaga, Y. Kaneko, D. Okuyama, S. Ishiwata, T. Arima, S.  Wakimoto, K. Kakurai, Y. \nTaguchi, and Y. Tokura, Phys. Rev. Lett. 105, 257201 (2010).  \n24) T. Koyama, S. Yano, Y. Togawa, Y. Kousaka, S. Mori, K. Inoue, J. Kishine, and J. Akimitsu, J. \nPhys. Soc. Japan 81, 43701 (2012).  \n25) A. Hubert and R. Schäfer, Magnet ic Domains: The Analysis of Magnetic Microstructures  (Springer \nScience & Business Media, 2008).  \n26) S. Chikazumi and C.D. Graham, Physics of Ferromagnetism  (Oxford University Press on Demand, \n2009).  \n \n  10 \n Figure Captions  \nFIG. 1.  Schematics of the optical system for (a) small -angle electron diffraction (SmAED) and (b) \nFoucault imaging. The intermediate lens I is adjusted when the SmAED mode is switched to the \nFoucault mode. In the Foucault mode, the magnetically  deflected spots ar e selected by using a selected -\narea aperture to visualize  the magnetic domains.  \n \nFIG. 2.  (a) Fresnel image of BaFe 12–x–δScxMg δO19 (𝑥=1.6,δ=0.05) (BFSMO)  at the c plane.  The \ndefocus is ∆f = −1.5 m (underfocus). (b) Intensity  profile along X1 –Y1 in (a). Bloch walls are \nvisualized as pairs of bright and dark lines. (c) SmAED pattern performed at a camera length of 380  m. \nThe arrows show the positions of the direct beam. (d) Schematic of the magnetic domains in (a). The red \narrows show the magnet ization , whereas the blue marks represent the periods of magnetic domains for \nthe electron.  \n \nFIG. 3.  (a) Fresnel image of BFSMO when the specimen was tilted to 30°. (b) Intensity profile along \nX2–Y2 in ( a). (c) SmAED pattern at the tilt angle of 30 °. The letters (A –D) denote the magnetic \ndeflection spots used in Foucault imaging in Fig. 6.  A Bloch wall is depicted as bright or dark line. ( d) \nSchematic of the magnetic domains in ( a). \n \nFIG. 4. (a) Fresnel image of BaFe 12O19 without tilting the specimen . The arrows show the directions of \nthe magnetization at the Bloch walls. ( b) SmAED pattern without the specimen tilt. (c) Fresnel image at \nthe tilt angle of 30 °. (d) SmAED pattern at the tilt angle of 30 °. In the Fresnel images, the defocus was \n∆f = −2.0 m (underfocus). The arrows represent the magnetization directions at (a) domain walls and \n(b) magnetic domains.  The SmAED patterns were obtained at a camera length of 300 m. \n 11 \n FIG. 5. Foucault image of BFSMO. The image was obtained by selecting the A and B spots of Fig. 3(c). \n \nFIG. 6. Foucault images when selecting the magnetic  deflection spots A –D in Fig. 3(c). The used spots \nare A and B in (a), C and D in (b), A and C in (c), B and D in (d), A in (e), and B in (f). In (a), the up-\nmagnetized domains are bright , whereas the down -magnetized ones are dark. In (b), the down -\nmagnetized domains ha ve bright contrast a nd its contrast is reversed compared with (a). (c) The u p-\nmagnetized domains along the y direction and down -magnetized domains along the x direction are bright. \nTherefore,  the bright and dark contrast is reversed where the domains bend. Meanwhile , the down -\nmagnetized domains along the y direction and up -magnetized domains along the x direction  in (d)  are \nbright. In (e), only the up-magnetized domains along the x direction are bright. In (f), only the up-\nmagnetized domains along the y direction are bright.  \n \n  12 \n  \nFig. 1.  \n13 \n  \nFIG. 2.  \n  \n14 \n  \nFIG. 3.  \n  \n15 \n  \nFIG. 4.  \n \n  \n16 \n  \n \nFIG. 5.  \n  \n17 \n  \n \n \nFIG. 6.  \n \n"    },    {        "title": "2005.00871v1.Optically_driven_ultrafast_magnetic_order_transitions_in_two_dimensional_ferrimagnets.pdf",        "content": " \n \n 1 Optically driven  ultrafast magnetic order  transitions  \nin two-dimensional  ferrimagnet s \nJunjie He,1,* Thomas Frauenheim1,2,* \n1Bremen Center for Computational Materials Science, University of Bremen, Am Fallturm 1, 2835 , \nBremen, Germany.  \n2Beijing Computational Science Research Center (CSRC) , Beijing  100193  and Shenzhen  Computational \nScience  and Applied Research (CSAR) Institute , Shenzhen  5181 10, China . \n \nE-mail: junjie.he.phy @gmail.com ; thomas.frauenheim@bccms.uni -bremen.de  \n \n                                           ABSTRACT  \nLaser -induced switching and manipulation of the spins  in magnetic materials are of great interest  to \nrevolut ionize future magnetic storage technology  and spintronics with the fastest speed  and least power  \ndissipa tive. Inspired by the recent discovery of  intrinsic two-dimensional  (2D) magnets, which  provide a \nunique plat form to explore the new phenomenon for light-control magnetism in the 2D limit , we \npropose to realize light can  efficiently tune magnetic states of 2D ferrimagnets  in early  time. Here, using \nthe 2D ferrimagnetic MXenes  (M2M’X2F2, M/M ’=Cr, V, Mo; X=C/N)  as prototype system s, our real-\ntime density functional  theory  (TDDFT) simulation show that laser pulses can directly induce ultrafast \nspin-selective charge transfer between two magnetic sublattices  (M and M ’) on a few  femtoseconds,  and \nfurther generate dramatic changes in the magne tic structure of these MXenes , including a magnetic \norder  transition  from ferrimagnetism  (FiM) to transient ferromagneti sm (FM).  The microscopic \nmechanism underpinning this ultrafast switching of magnetic order  in MXenes  is governed  by optically \ninduced inter-site spin transfer (OISTR)  effect, which theoretically enables  the ultrafast direct  optical  \n \n 2 manipulation of the magnetic state in MXenes -based 2D materials . Our results open new opportunities \nfor exploring the manipulat ion of  the spin  in 2D magnets  by optical approaches .  \nKEYWORDS  : TDDFT , 2D magnets,  MXenes , spin transfer, photo -induced  spin dynamics  \nTOC  \n \n \nIntroduction  \nIn 2017, two experimental groups have independently observed the intrinsic two -dimensional (2D) \nferromagnetism (FM) in exfoliated van der Waals (vdW) CrI 3 and CrGeTe 3 crystal by magneto -optical \ntechnique. 1,2 Subsequently, a large variety of 2D magnetic vdW crystals, including FePS 3,3 NiPS 3,4  MnPS 3, \n5 Fe3GeTe 2,6 VSe 2,7 and MnSe 28 etc. have been realized by using only adhesive tape, chemical vapour \ndeposition or molecular beam epitaxy. Interestingly, both VSe 2 and MnSe 2 were claimed to be a 2D \nferromagnet with Curie temperature (T C) above room temperature.7,8 Until now, the family of 2D magnetic \ncrystals is still rapidly growing, and many more new 2D magnets have been predicted and await discovery. \n \n \n 3 9,10,11,12,13,14 These 2D magnets with excellent properties in optical, magnetic, magneto -electric, and magneto -\noptic areas , offer new opportunities for both explor ing the manipulation of  spin or magnetism in the 2D \nlimit  and for developing the next  generation spintronic devices . Following these exciting  discoveries, \nconsiderable studies have bee n devoted to  manipulat ing spin interaction  in 2D magnets , e.g., Dzyaloshinskii -\nMoriya interaction,  interlayer magnetic order , and magnetic anisotropy  by mechanical strain,  gate voltage , \nand magnetic  field.15,16,17,18 For example, it demonstrated  that the interlayer magnetic order  of bilayer CrI 3 \ncan be  reversibly tuned from the antiferromagnetic (AFM) to the  FM phase by  applying  electrostatic doping \nor vertical electric field  gate voltage .16,17,18 \n       The light represents the fast est means to manipulate the spin structures  of matter at ultrashort time \nscales  (from femtosecond to attosecond ) that promises to revolutionize future  recording and information \nprocessing by achieving the fastest possible and least power dissipative.19 Laser induced s pin injection in \nFM/NM interface ,20 all-optical magnetization switching ,21 optically tunable magneti c anisotropy  in CrI 3 \nmonolayer, 22 photo -induced topological  states 2D materials,23 and light can manipulate the interfacial \nmagnetic proximity coupling  and valley polarization  WSe 2/CrI 3 heterostructures24 are p rominent examples of  \nlight-controlled properties in materials.  Using time-dependent density -functional theory  (TDDFT) \nsimulation, Dewhurst et al. have shown  ultra-short laser  can directly  induce a spin transfer  between sub -\nlattices  causing significant magnetic order  transition , i.e., the switching  from Ferrimagnetic (FiM)  to FM \norder  in a multi -component  magnetic system, which is governed by the  previously unknown  optically \ninduced inter -site spin transfer (OISTR)  effect in  sub-exchange and sub -spin-orbit timescales .25,26 Indeed, \nrecently, several  experiment al group s have demonstrated  the ultrafast  laser  can  direct ly and coherent ly \nmanipuate  spin of magnetic multilayer  and H eusler compounds  on sub-femtosecond time scales , and \ninduced spin transfer between two magnetic  subsystems , which  confirmed  the OISTR effect .19,27,28,29  These   \n \n 4 breakthroughs suggest  a search for ultrafast optically tunable magnetism in 2D  systems . However , to realize  \nthe OISTR effect, it is highly  desir able to develop  multi -component  (two or more ) magnetic  system s but it  is \nrare in 2D magnets .  \n        Recently, 2D transition metal carbides  or nitrides called  by MXenes have emerged as a new type \nof 2D materials with the general formula  Mn+1XnTz, where M represents an early transition metal  (such \nas Ti, V, Cr, or Mo), X is carbon  and/or nitrogen, and T stands for the surface terminations  (e.g., -OH, -\nO, or -F).30,31,32 The MXenes attract great attention due  to their potential applications in sensors, \ncatalysis, energy  storage and nanoelectronics.32 The chemical and physical properties of MXenes can be \ntuned through the choice of transition metals  and surface chemical group s.12,14,33,34,35 For example, the \nferromagnetic behavior  of bare Cr 2C and Cr 2TiC 2 has been shown to become semiconduct ing \nantiferromagnetic upon -F, -OH and -Cl functionalization .12,14 Particularly , the Mo 3N2F2 MXene  has \nbeen  found to be a ferrimagnetic half -metallicity with a  high Curie temperature .36 Therefore, MXene s \nrepresent unusual  a class of  multi -component  magnetic  material s in 2D system, which provid e an \nexcellent  platform to study the optically tunable  magnetism  and spin transfer  in 2D system.  \nIn this work, we propose to realize the optically manipulate magnetic order transition in MXenes . \nFirstly, t he ground states  DFT  calculations reveal that 2D MXenes , including Cr2VC 2F2, Mo 2VC 2F2, \nMo 2VN 2F2, Mo 3C2F2 and Mo 3N2F2, have unusual  ferrim agnetic order . Then, o ur real time time-\ndependent DFT (TDDFT) simulations on FiM MXenes have shown  that l aser pulses induce spin -\nselective charge transfer  and further generate dramatic changes in the magnetic  structure of these \nMXenes , including a magnetic order  transition  from FiM to transient FM at early time . The microscopic \nmechanism underpinning this ultrafast switching of magnetic order is governed  by optically induced  \n \n 5 inter-site spin transfer (OISTR)  effect . The results open new opportunities to manipulate  the 2D \nmagnetism by optical  approaches  \n \nFig 1: The top  (a) M2M’X 2F2 (M=Cr, Mo; M’=Cr, Mo, V; X=C/N) . (b) The ligand environment for M and M’  is \nshown .  Schematics of magnetic configurations for considered MXene s: (c) FM, (d) FiM1, (e) FiM2,  (f) AFM1, (g) \nAFM2.  The  2×1 supercells are employed for the total energy calculation.   \nResults and discussions  \nThe atomic structure MXenes -based 2D systems , including Cr 2VC 2F2, Mo 2VC 2F2, Mo 2VN 2F2, \nMo 3C2F2, and Mo 3N2F2 show in Figure 1 . The M and M’ localized in the distinctive  octahedral  \nenvironment, where M is bonded to six C/N atoms  and M’  atom , but M’  is bonded to three  C/N and \nthree F  atoms , respectively . The distinctive  chemical environment  render s the different local magnetic \nmoment for M and M ’ atoms.  Geometry optimization carried out at PBE +U level gives lattice constant  \nand geometry parameter s as summarized  in Table S1. For Mo 3N2F2 MXene , the calculated  constant \nlattice  is in good agreement  with previous calculations.36 The calculated local magnetic moments  for M \n \n \n 6 and M ’ atoms are show n in Table 1 . The M and M ’ atoms  of MXenes  predominantly  contribute to the \ntotal magnetic moments while the neighboring X  and F atoms have only a small contribution. To study  \nthe magnetic ground state structures of the  M2M’X2F2 MXenes , the collinear ferromagnetic  (FM)  and \nantiferromagnetic  (AFM ) and ferromagnetic  (FiM) states were considered , respectively . To elucidate \nthe spin configurations  of these phases, we schematically plotted the arrangement  of the local magnetic \nmoments of M and M ’ atoms  with colored  arrows as shown in Figure 1 c-g, which is considering  a 2×1 \nsuper cell of MXenes . The lowest energy has  been found for the F iM1 state and this has been set as \nreference  energy. For instance, for Cr2VC 2F2 MXenes , the EFM, EFiM2, EAFM1, and EAFM2 states have \nenergies 0.81, 0.25,  0.83, and 0.35 eV higher, respectively, than the F iM1 state. These results clearly \ndemonstrate that the  FiM1 state configuration for all considered MXenes  is the magnetic  ground state \nwith high magnetic stability , similar  to previously reported results for Mo 3N2F2 MXenes.36 The \nmagnetic anisotropy energy  (MAE)  of materials, which determines  the orientation of the magnetization \nat low -temperature,  is an important parameter  for their applications in high -density  storage or quantum \nspin processing.  The non -collinear magnetic calculations  were performed for magnetization along \nX[100], Y[010] , and Z[001] directions  for MXenes , which is summarized  in Table S1 . The resul ts \nindicate  that the easy axis of Cr2VC 2F2, Mo 3C2F2, and Mo 3N2F2 MXenes  is along the [001] orientation , \nwhile easy axis of Mo 2VN 2F2 and Mo 2VC 2F2 is along the [ 100] and [010] orientation.  The band \nstructure s with the orbital projection  for MXenes in FiM state are reported in the supporting  materials  \n(see Figure S1).  They all show the metallic  feature . Interestingly , both Mo 3N2F2 and Mo 2VN 2F2 are \npredicted  to be  half-metallic , which is a great potential for spintronic  application s. \nTable 1: Calculated structural and magnetic characteristics  of five different MXenes. a \n  \n \n 7 Structure  EFM  EFiM1  EFiM2  EAFM1  EAFM1        \nCr2VC 2F2 0.81 0 0.25 0.83 0.35 2.39 -0.43 \nMo 2VC 2F2 0.54 0 0.27 0.58 0.16 1.03 -0.96 \nMo 2VN 2F2 2.33 0 2.46 1.52 1.52 1.91 -1.90 \nMo 3C2F2 0.96 0 1.07 0.89 1.18 1.39 -0.25 \nMo 3N2F2 1.30 0 0.75 0.90 1.09 1.21 -0.73 \n \na Relative energies for FM ( EFM) , FiM1 ( EFiM1), FiM2 ( EFiM2), AFM1 ( EAFM1)  and AFM2 ( EAFM2) are \nshown in eV. Local m agnetic moment s of M (  ) and M ’ (   ) atoms  for M 2M’X2F2 MXenes are also \ngiven  in μB. \n \nFigure 2: Ultrafast laser-induced ultrafast change of magnetic order. ( a) Time evolution of local magnetic  moment  \nwith (full lines)  and without (dashed lines)  spin-orbital coupling  for Cr1, Cr2, V, C and F  atoms , which at t = 0 fs (i.e., \nin the ground state) is ferrimegn ets.  At ∼6 fs, the  V atom demagnetizes and then remagnetizes but with the reversed \nspins  direction . The magnetic order  thus switches from FiM to transient FM order . (b) The relative  magnetization \n \n \n 8 dynamics for Cr1, Cr2 and V atoms . The snapshots of magnetization density at (c) t = 0  fs, (d) t= 5.8 fs, and (e) t= 15.2 \nfs. The iso-surfa ce was set to be 0.012  e/Å3 \nNext, w e will explore  the real -time evolution of the element resolved magnetism of the  FiM \nMXenes  monolayer induced by the electron dynamics under the  influence of an external light with a \nselected frequency. We performed real -time TDDFT calculations, in which the linearly polarized (in -\nplane polarization) laser pulse with full width at half -maximum  (FWHM ) of 3.63 fs, photon energy of  \n1.63 eV  and fluence of 34.2 mJ/cm2 (the vector potential  A(t) of the laser pulse  are shown in Figure \nS2(c)).  We first examined t he real-time spin  dynamics  of local magnetic moment of Cr and V atoms for \nCr2VC 2F2 MXenes  are shown in Fig ure 2a  under the in fluence of a laser pulse.  We can see that  both the \nlocal moment Cr and V atoms change dramatically, while the global moment showed no change.  The top \nand bottom  Cr atoms (Cr1 and Cr2)  demagnetize strongly  from ~2.4 μB of initial magnetic  moment to \nfinal ~1.4 μB. Most interestingly, at ∼6 fs (slightly before peak intensity of the laser pulse) the V atom  \ndemagnetizes and then remagnetizes  (from ~ -0.4 μB to ~ 0.9 μB) but with the reversed spin  direction , \nthus leading to the FiM-FM magnetic order transitio n. After 10 fs, the Cr2VC 2F2 show s transient  FM \norder  for a long time (at least up to 100 fs, at which point  our simulation ended ).  Under the influence of \nan external laser field, the relative local  moments , i.e., the quantity ( M(t)/M(0)) changes dramatically; \nthe moment of Cr  atoms  show  about 50 % decrease , while the ultrafast loss of moment  for V atoms and \nthen increase  (~200%)  but with the reversed spin  direction  as shown in Figure 2b.  We also simulate the \nreal-time spin dynamics of Cr 2VC 2F2 MXenes  with spin-orbital coupling (SOC)  as shown in Figure 2a.  \nBecause of the total spin moment  will no longer  a good quantum number , the spin-flip of photo -excited  \nprocesses  in Cr 2VC 2F2 MXenes  will be allowed. It is clear that that the effect of SOC will be not notable   \n \n 9 at least below  36 fs. The dynamics of the magnetization density  as a function of time are shown in \nFigure 1 d. The s witching of the spin direction of V at t = 15.2 fs can also be seen in this plot.  \nThe magnetic states transition from FiM to FM in Cr 2VC 2F2 can be understood by optically  \ninduced inter -site spin transfer (OISTR) mechanism , which  is found  previously to dominate the early \ntime spin dynamics of magnetic  metal multilayers and  the Heusler compounds .25,26  The ground -state \ndensity of states  (DOS) for Cr and V atoms shown in  Figure 3a .   Because the Cr and V are \nferrimagnetic  coupling , the majority spin is oppositely oriented in each  atoms : spin -up is the majority in \nthe Cr1 and Cr2  atoms,  and spin -down is the majority in V atom. From DOS, t he occupied  Cr atoms are \ndominated by  majority states, whereas there is considerable  empty  minority  orbital  of V around [1, 3] \neV energy window .  Such ground states electronic structure will enable efficient photo -driven spin  \nselective  charge  transfer  from the majority Cr  to minority V atoms according to  the OISTR mechanism  \nas shown in  Figure 3b, result ing in the loss of magnetic moment for Cr atoms and the gain of mag netic  \nmom ent for V atoms, respectively. The OISTR mechanism  in magnetic order transition of  Cr2VC 2F2 \nMXenes  is in agreement with the previous ly reported magnetic metal  multilayers and  Heusler alloys .25,26 \nTo obtain  further  analysis for the chang ing magnetic moment  for Cr 2VC 2F2 MXenes , we show  time-\ndependent occupation changes  (Δn(t)) of  the V and Cr atoms , which is defined as the difference  of the \ntime-resolved occupation function n (t) at time t concerning  the unexcited Cr 2VC 2F2 MXenes , i.e.,  Δ\nn(t)= n (t)−n(0). As shown in Figure 3 , the most notable transient change below  the Fermi level occurs \nin the minority channel of Cr and minority channel s of V atoms . One can see that photo -excitation \nresults in  a significant loss in the number of  majority  spin carriers as a function of time, which \ncorresponds to  the demagnetization  for Cr atoms . These loss minority spin carriers are transferred  from \nthe Cr to the V  sub-lattice, thus, leading to the  enhanced  magnetic moment  for V atom , and further  \n \n 10 induced the direction reverse . It is also evidenced an obvious  gain in the number of  minority  spin \ncarriers of V  (See Figure 3d ).  For other possible excitation, the Δn(t) for the Cr minority and V \nmajority spin channel is relatively small as shown in Figure S3, which does not sign ificantly affect the \ntotal spin -transfer  process between Cr and V atoms. T he time -dependent change in the spin -down  and \nup electrons for V and Cr  are presented in Figure S4  relative to the ground -state for Cr2VC 2F2 MXenes . \nIt is clear from Fig ure S4 that majority  electrons of Cr are transferred  to the minority of V atoms, which \nis consistent with  the OISTR mechanism.  \n \nFigure 3: (a) Projected density of states (DOS) calculation for Cr1 and V in Cr2VC 2F2 MXenes . And, the favorable  \nspin transfer from Cr majority  to V minority is marked . (b) Schematic overview of the OISTR effect in Cr2VC 2F2. The \noptical excitation leads to an effective spin transfer from  the occupied Cr majority  into the V minority channel. TD -\nDFT calculations of the difference of the transient occupation compared with the unexcited case in  the (c) Cr1 minority \n \n \n 11 channel  and (d) V minority channel  at selected  time steps. For Δn(t), a negative signal arises corresponding to a loss \nof minority electrons, while a simultaneous positive signal correlating to spin gain.  \nWe now turn to explore the light response  on magnetism  for Cr 2VC 2F2 MXenes with respect to \nthe changing laser pulse parameter.  Figure 4c has plotted the magnetization dynamics  of Cr and V \natoms in Cr2VC 2F2 MXenes  under the influence of laser pulses of  various  frequencies  (fixed FWHM) \nand FWHM (fixed frequency) . Here, their time dependence of the ext ernal  vector field  is shown in \nFigure S2. It is clear that the demagnetization of Cr and V atoms is highly sensitive to  the frequency  of \nlaser , indicating that the frequency of the pulse can also be used to  tailor and optimize the process of \nphoto -induced magnetic transition for FiM MXenes. For the higher energy in 1.09, 1.63, 2.18 , and 2.72 \neV, the local magnetic moment of Cr and V atoms are obviously demagnetized, whereas only relatively \nsmall  loss of local  magnetic moment is observed for energy in 0.54 eV. On the other hand, t he \ndemagnetization time will not strongly dependent on the FWHM. We can see V and Cr atom s are shown \ndramatic demagnetization for various FWHM  value s. However, the demagnetization proc ess for var ious \nFWHM take s place after a time lag between the laser pulse and the maximum of the laser pulse  (See \nFigure S2). Such a time lag effect between  the laser pulse and the  demagnetization  in 2D MXene system \nalso is similar to the Fe, Co , and Ni metal.37  \n \n 12  \nFigure 4: The spin d ynamics of local  magnetic  moment of Cr1 and V atoms in Cr 2VC 2F2 MXenes under the \ninfluence of seven different laser pulses given by (a) frequency /fluence,  and (b) FWHM/ fluence . \n \n \n \n 13 Figure 5: Time evolution of local magnetic  moment of M and M’  atoms  for (a) Mo 3C2F2, (b) Mo 3N2F2, (c) Mo 2VC 2F2 \nand (d) Mo 2VN 2F2, respectively.  \nTo probe the validity of optically induced magnetic order transition  in other  2D MXenes materials, \nwe have performed real-time TDDFT calculations of the laser induced  change in magnetic structure for \nan extended set of 2D systems : Mo 3C2F2, Mo 3N2F2, Mo 2VC 2F2, and Mo 2VN 2F2, which have a FiM \nground state . For purpose of companion, the  same  linearly polarized (in -plane polarization) laser pulse \nwith FWHM  of 3.63 fs, photon energy of  1.63 eV , and fluence of 34.2 mJ/cm2 are employed as driven \nlaser field. Under the influen ce of an external laser field,  the local  moment changes  for these four \nmaterials  dramatically  change ; the middle layered metal (M ’) demagnetize and then remagnetize with \nreverse spin direction, while the upper and lower metal (M) shown  relatively small demagnetization as \nshown in Fig ure 5 and Figure S 5. The results demonstrate that OISTR also  dominates early -time spin \ndynamics in these FiM MXenes, which is  the same mechanism as found in the case of  Cr2VC 2F2 \nMXenes . It is also important to mention  that linearly polarized light in the x -direction  for present work  \nwas used . We find that changing polarization  direction (e.g., y-direction)  of this linearly polarized light  \ndoes not affect  the process of photo -induced magnetic order transition  of MXenes . Note  that our \nsimulations just  performed at early  spin dynamics (up to  100 fs ) following the laser pulse. This time -\nscale is completely dominated by direct  optical manipulation without the  rotation of atomic magnetic \nmoments .29 After  this time, the phonon -mediated spin -flip p rocesses  will efficiently affect the \nmagnetization  of materials , and which can lead to further a reduction in magnetization; however, such \nprocesses are not included in our TDDFT simulations.  \nTo explore how the optical manipulation of magnetism in 2D FM and AFM materials, we take \nMn 2CF2 and Cr 2CF2 as an example to simulate their spin dynamics by TDDFT with the same laser  \n \n 14 parameter as Cr 2VC 2F2 as shown in Figure S6.  Both my calculation and previous reports indicate that \nthe Mn 2CF2 and Cr 2CF2 have FM and s ymmetrical AFM order.  From Figure S6, the Mn 2CF2 show \nalmost identical demagnetization of Mn atoms, result in no change  in magnetic order.  The AFM Cr 2CF2 \nwill symmetrical demagnetization because of spin transfer  between  majority  and minority of Cr atoms \nwill be identical in both spin channels . Our simulation in 2D FM and AFM materials is in agreement \nwith the previous ly proposed demagnetization mechanism in bulk materials (e.g. NiO).20 Thus, \ndeveloping suitable 2D ferromagnetic  systems will be crucial  to explore ultrafast optical switch of \nmagnetic order in 2D limit, which will trigger the further materials search and  design for photo -induced \nmagnetic states transition in 2D magnets as well as their van der Waals heterostructures.  \nExperimentally , our work  can be verified  by ultrafast extreme ultraviolet (EUV)  high harmonic pulses  \nand time-resolved magnetic circular di chroism (MCD), which have been widely applied to  explore  the \nelement -specific spin dynamics in multi -component  magnetic systems .19,29 \n     In conclusion, we performed DFT and real-time TDDFT  calculations for  ground -states  properties  \nand optical  manipulation of magnetism  in MXenes -based 2D system . Our DFT calculations show  that  \nM2M’C2F2 MXenes, including Cr2VC 2F2, Mo 2VC 2F2, Mo 2VN 2F2, Mo 3C2F2, and Mo 3N2F2, have  \nunusual ferrimagnetic  order. Using TDDFT, we predicted that l aser pulses can directly induce ultrafast \nspin-selective charge transfer between two magnetic sublattices  (M and M ’) on a few  femtoseconds,  and \nfurther generate dramatic changes in the magnetic  structure of these MXenes , including a magnetic \norder  transition  from ferrimagnetic  (FiM) to transient ferromagnetic (FM).  The spin transfer of magnetic \nsublattice in MXenes can further  be tuned by laser parameter, which is achievable with current \nexperimental techniques . The microscopic mechanisms for ultrafast switching of magnetic order  are \ndiscussed based on  optically induced inter -site spin transfer (OISTR)  effect, which theoretically enable s  \n \n 15 the ultrafast direct  optical manipulation of the magnetic state in 2D magnets. Our results open  new \nopportunities to manipulate  the spin in 2D magnets as well as the potential applications  in spintronic s.  \nMethods and c omputational details  \nThe structure  optimizations , magnetic ground states , and band structures for MXenes  were \nperformed using the Vienna ab initio simulation package (VASP)38,39 within the generalized gradient \napproximation , using the  Perdew -Burke –Ernzerhof (PBE) exchange -correlation functional .40 \nInteractions  between electrons and nuclei  were  described by the projector -augmented wave (PAW) \nmethod.  The criteria of energy and atom force convergence were  set to 10−6 eV and 0.0 01 eV/Å, \nrespectively. A plane -wave  kinetic energy cutoff of 500 eV was employed.  The vacuum space of 15  Å \nalong  the MXenes normal was adopted for calculations on monolayers . The Brillouin zone (BZ) was \nsampled using 15 ×15×1 Gamma -centered Monkhorst -Pack grids for the calculations of relaxa tion and  \nelectronic structures .  To account  for the energy of localized 3 d orbitals of TM atoms properly, the  \nHubbard “U” correction is employed within the rotationally  invariant DFT + U approach.41 A correction \nof U = 3 eV  for V, Cr , and Mo  is employed based on the relevant  previous reports .11,12,14 \nTo identify the spin dynamics  in these MXenes materials under the influence of ultrafast laser pulses , \nwe have  performed  time-dependent density functional theory (TDDFT)  calculations . The time  evolving  \nstate functions (  ) are calculated by solving the time  dependent Kohn −Sham  (KS)  equation as follows:  \n         \n     \n       \n          \n          \n             \n                               (1) \nwhere         and σ represent  a vector potential  and Pauli matrices. The KS effective potential         \n=                            can be  decomposed into the external potential      , the classical \nHartree potential   , and the exchange -correlation (XC) potential     , respectively . The KS magnetic  \n \n 16 field can be written as         =                   , where      and     represent  the magnetic field \nof the applied laser pulse plus possibly an additional magnetic field and XC magnetic field , respectively .  \nThe last term in Eq. (1) stand for the SOC .  We only time propagate the electronic system while keeping \nthe nuclei fixed .  \n     With the   s in hand a time-resolved DOS, shown in Figure  2, can be calculated using the following  \nexpression : \n                                                                                              (2) \nwith \n                                                                                                    (3) \nhere, spin -resolved and time -dependent            is calculated from the projection of the time -propagated \norbitals              onto the ground -state Kohn -Sham orbitals at t = 0. The dynamical evolution of the \nelectronic struc ture for  MXenes  after photo -excitation s also can be analyzed by the time -dependent \nchanges of        .  \n     We employed  a fully non -collinear version of TDDFT  by full-potential  augmented plane -wave ELK \ncode .42 A regular mesh  in k-space of 7 × 7 × 1 is used, and a time step of Δt = 0.1 a .u. is employed for \nthe real time TDDFT simulation . A smearing width of 0.027 eV is used.  Laser pulses used in the present \nwork are linearly polarized (in -plane p olarization) with selected  frequency . All calculations were  \nperformed usi ng adiabatic local spin density  approximation (ALSDA)  with Hubbard U (ALSDA+U), \nwith U=3 eV for Cr, V , and Mo atoms.  \nReferences   \n \n 17                                                  \n1 Huang, B. ; Clark,  G.; Navarro -Moratalla,  E.;  Klein,  D.R. ;  Cheng,  R.; Seyler,  K.L.; Zhong,  D.;  \nSchmidgall, E. ; McGuire,  M. A.;  Cobden,  D. H. ;  Yao,  W.  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The ELK code, http://elk.sourceforge.net, 2017 .   1 Supp orting Information  “Optically driven  ultrafast \nmagnetic order transitions  in two-dimensional \nferrimagnet s” \nJunjie He,1,* Thomas Frauenheim1,2,*  \n1Bremen Center for Computational Materials Science, University of Bremen, Am Fallturm 1, 2835 , \nBremen, Germany.  \n2Beijing Computational Science Research Center (CSRC) , Beijing  100193  and Shenzhen  Computational \nScience  and Applied Research (CSAR) Institute , Shenzhen  5181 10, China . \n \nE-mail: junjie.he.phy @gmail.com ; thomas.frauenheim@bccms.uni -bremen.de  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  2  \nTable S1 : Calculated constant lattice and magnetic anisotropy of five different MXen es. The L and h  is \nthe lattice constant and height of MXenes in Å. The magnetic anisotropy between Y and Z ( EYZ) and X \nand Z ( EXZ) directions are in meV. The ES stand for the electronic structure. \nStructure  L  h         easy axis ES \nCr2VC 2F2 3.04 7.04 -1.04 -2.40 [001]  Metal  \nMo 2VC 2F2 3.26 7.07 -5.70 2.38 [010]  Metal  \nMo 2VN 2F2 3.24 7.27 2.58 2.90 [100]  Half-metal  \nMo 3C2F2 3.29 7.10 -42.11  -43.66  [001]  Metal  \nMo 3N2F2 3.35 6.87 -0.21 -0.26 [001]  Half-metal  \n \n \nFigure S1 : Band structure with orbital projection  for spin up of (a) Cr 2VC 2F2, (b) Mo 3C2F2, (c) \nMo 3N2F2, (d) Mo 2VC 2F2, (e) Mo 2VN 2F2; Spin down of (f)  Cr2VC 2F2, (g) Mo 3C2F2, (h) Mo 3N2F2, (i) \nMo 2VC 2F2, (g) Mo 2VN 2F2. \n \n \n 3  \nFigure S2:  The applied pump laser pulse  in Cr 2VC 2F2. The energy and FWHM of laser was \nmarked, respectively.  \n \n \n \n \n 4  \nFigure S 3: TDDFT calculations of the difference of the transient occupation compared with the \nunexcited case in  the  (c) Cr1 minority channel and (d) V minority channel at characteristic time steps. \nFor Δn(t), a  negative signal arises corresponding to a loss of minority electrons, while a simultaneous \npositive signal correlating to spin gain. The same x -axis scale in Δn(t) is selected with Figure 3cd  \n \n \nFigure S4: The time -dependent change in the (a) spin-up and ( b) spin-down electrons on V, Cr1 and \nCr2 atoms  relativ e to the ground -state for Cr2VC 2F2. \n \n 5  \nFigure S 5:  The relative local  moment (i.e., the quantity ( M(t) /M(0)) for  (a) Mo 3C2F2, (b) \nMo 3N2F2, (c) Mo 2VC 2F2 and (d) Mo 2VN 2F2. \n \n \nFigure S 6:  Photo-induced spin dynamics for (a) ferromagnetic MXenes Mn 2CF2 and (b) \nantiferromagnetic  MXenes Cr 2CF2. \n \n \n"    },    {        "title": "2005.09707v1.New_Way_of_Generating_Electromagnetic_Waves.pdf",        "content": "New Way of Generating Electromagnetic Waves\nAli Hossseini-Fahraji,1Majid Manteghi,1and Khai D. T. Ngo1\nBradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, Virginia 24061,\nUSA\n(Dated: 21 May 2020)\nThis paper presents a new method for generating low-frequency electromagnetic waves for navigation and communi-\ncation in challenging environments, such as underwater and underground. The main idea is to store magnetic energy\nin two different spaces using the interaction between a permanent magnet and a magnetic material. The magnetic re-\nluctance of the medium around the permanent magnet is modulated to change the magnetic flux path. The nonlinear\nproperties of magnetic material as a critical phenomenon are used for effective modulation. As a result, a time-variant\nfield is generated by the modulation of the permanent magnet flux. This non-resonant time-variant characterization\nmeans that the transmitter is not bound to the fundamental limits of the antennas and can transmit higher data rates. A\nprototype transmitter as a prove-of-concept is designed and tested based on the proposed idea. Compared to the rotating\nmagnet, the prototyped transmitter can modulate 50% of the stored energy of the permanent magnet with much lower\npower consumption.\nI. INTRODUCTION/BACKGROUND\nThe emphasis is primarily on increasing data rate, which\nleads to the use of higher frequencies and wider bandwidths\nin modern communication technology research and innova-\ntions. However, because of physical constraints, increasing\nfrequency and bandwidth in many areas of technology can-\nnot necessarily be beneficial. Communication under seawa-\nter or other challenging RF environment require very low-\nfrequency, VLF, or ultra-low-frequency, ULF signals to pene-\ntrate lossy media that block high-frequency signals. Also, new\ndevelopments in neuroscience have shown the potentials of\nULF and VLF electromagnetic, EM, waves to treat neurolog-\nical conditions such as Alzheimer’s disease, amyotrophic lat-\neral sclerosis, persistent vegetative diseases, epilepsy, stroke-\nrelated illness, tinnitus, multiple sclerosis, schizophrenia, and\ntraumatic brain injury. The main challenge is that most of\nVLF and ULF generators are large and power-hungry, which\nmake them impractical or hard to use in many applications.\nIn this paper, we present a new approach for generating EM\nwaves in a compact and low-power fashion.\nAt first, radio wave technology was developed within the\nVLF ranges. Because of the broad spectrum of radiated\nwaves and the problem of spark-gap oscillators (invented by\nHertz) interference, William Crookes proposed using sinu-\nsoidal sources in resonance structures (then called syntony)\nto minimize the transmitter and receiver bandwidth in 18921.\nIt started a race to develop a continuous wave, CW, sinusoidal\nwave generator to replace the spark-gap sources for RF ap-\nplications. Innovative structures were proposed by several re-\nsearchers (Elihu Thomson, Nikola Tesla, Reginald Fessenden,\nand many others). Finally, the spark-gap oscillators were re-\nplaced by the Alexanderson alternator (a mechanical structure\nbased on a rotating permanent magnet) in 1904. Surprisingly,\nmany variants of Alexanderson alternator have been suggested\nafter more than a century2–8, in response to a DARPA call\nfor ELF and VLF sources in recent years. Such mechanical\ngenerators (mechtenna), however, still have the same short-\ncomings as the original design, such as large size, massive\npower consumption, hard to modulate and transmit informa-\ntion, synchronization, noise, vibration, and durability problemof a mechanical structure. There have also been other versions\nof mechanical vibration proposed to generate EM waves in\nVLF ranges as well9–13.\nIn 1961, on the other hand, an analysis of EM radiation\nfrom the acoustically-driven ferromagnetic yttrium iron gar-\nnet sphere (YIG) introduced the concept of acoustic reso-\nnance as strain powered (SP) antenna. Recent studies have\nshown that in a device with smaller physical dimensions than\nthe EM wavelength, multiferroic antennas can take advantage\nof acoustic resonance to reduce antenna size14–16. As a con-\ntrast to rotating permanent magnets, strain-coupled piezoelec-\ntric and magnetostrictive composites are thus used in magne-\ntostrictive materials to control magnetic spin states17–19. Al-\nthough this technique removes the necessary inertial force in\nmechtenna, it faces challenges due to the matching rigidity\nbetween piezoelectric and magnetostrictive (i.e., low energy\ntransfer from piezoelectric to magnetostrictive) and sequen-\ntially inefficient power transfer to electromagnetic radiation.\nBesides, making this structure into bulk is also a challenge.\nAs an alternative technique, we intend to use magnetic mate-\nrial to manipulate the magnetic flux of a permanent magnet.\nThis idea is to alter the reluctance of the flux path to make\nthe magnetic flux time-variant by pushing it to take an alter-\nnative path. Our concept is based on ‘variable material’ rather\nthan ‘variable structure’ as in mechanical rotation. We take\nadvantage of a permanent magnet, which is equivalent to a\nlossless electromagnet with the winding of the superconduc-\ntor, which produces a static magnetic flux without dissipating\npower. Meanwhile, we alternate the direction of flux between\nfree space and a medium with high permeability. The perme-\nability of the magnetic material near the permanent magnet\nvaries by adjusting the current through a control coil, depend-\ning on the B-H curve of the magnetic material3.\nThere are many papers published in the last three years on\nULF antennas; however, most of them have not evaluated their\nwork with a concrete criterion. Therefore the performances of\nthese proposed antennas are difficult to assess and compare.\nWe consider a permanent magnet’s magnetic flux to be a suit-\nable reference to evaluate the performance of any ULF trans-\nmitter. Hence, from now on, we believe the field produced by\na rotating magnet to be a reference to assess the field generatedarXiv:2005.09707v1  [physics.app-ph]  19 May 20202\nby any technique with the same volume magnet. In this way,\nwe calibrate the receiving device (searching coil or any other\ntype of magnetometer), especially if we can rotate the magnet\nto the generator’s operating frequency. Also, we suggest cal-\nculating the leakage of the windings around the ferrite cores\nindependently of the permanent magnet to be able to distin-\nguish between the permanent magnet’s contributions and the\nentire field of windings.\nIn this research, the magnetic flux per volume of selected\npublished designs is compared in Table I to give a better es-\ntimate of the performance of our design. Note that most arti-\ncles do not provide details about the antenna’s total volume,\nand the information is limited to the size of the main radiating\nelement. The objective of this comparison is to determine the\nminimum volume needed to reach a field strength of 1 fT at 1\nkm.\nAs shown in Table I, the results for the radiator volume\nof 1cm3(DB/Vrad) show that the rotating magnet has the\nmaximum magnetic flux, as expected. Without any modu-\nlation, the rotating magnet generates a magnetic flux of about\n200\u000210\u00003f T=cm3, whereas any designs aimed at modulat-\ning magnet rotation reduced its efficiency significantly. Fur-\nthermore, our proposed design and the best multiferroic an-\ntenna design in the literature can generate 98 \u000210\u00003and\n13:3\u000210\u00003f T=cm3magnetic fluxes corresponding to 49%\nand 13% efficiency of these antennas, respectively. The re-\nsults show at the time of publication that the proposed design\nhas the best chance to compete with a rotating magnet with\nconsiderably lower power consumption and smaller size.\nII. THEORETICAL BACKGROUND\nThe traditional way of generating electromagnetic waves is\nto periodically exchange electric and magnetic energy stored\nin two distinct parts of the radiating system. Any or both types\nof stored energy may leak some power as radiation. We thus\nhave a specific amount of radiated power, Pr, for a maximum\namount of stored energy, Wmax, and we can calculate the an-\ntenna’s quality factor as Q=wWmax=Pr. Fundamental lim-\nits of antennas22,23tie an antenna’s quality factor to its elec-\ntrical size as Q=1=(ka)3, where ais the radius of smallest\nsurrounding sphere and k=2p=lis the wave number. That\nmeans the smaller the antenna, the more energy we need for a\ngiven radiated power to be stored. Moreover, the quality fac-\ntor is related to the antenna’s instantaneous bandwidth. The\nsimple conclusion shows that we need to store a large amount\nof energy in the antenna reactive-zone in cases of low fre-\nquency or small antennas ( a=l\u001c1), and the instantaneous\nbandwidth will be small.\nInstead of exchanging energy between electric and mag-\nnetic forms, a static stored energy (e.g., stored energy in a\npermanent magnet or an electret) can be moved, vibrated, or\nrotated without altering its form to generate a time-varying\nfield. This approach differs radically from the traditional ra-\ndiation systems and is therefore not constrained by resonance\nlimitations. However, it may not be desirable to apply any of\nthese approaches to magnets or electrets by using mechani-\ncal movements. We propose to modulate the magnetic energy\nstored around a magnet by manipulating reluctance to the sur-\n \n  a r FIG. 1. Magnetic flux density decays by1\nr3.\nroundings. Therefore, the direction of the flux or the position\nof the stored energy variates in time. The magnetic field thus\nvaries in time.\nLet us first look at the magnetic flux density of a uniformly\nmagnetized sphere, as shown in Fig. 1:\nB(r>a) =m0\n4p\u0014\u0000m\nr3+3(m:r)r\nr5\u0015\n; m=4\n3pa3M(1)\nwhere M(A/m) is the magnetic dipole moment per unit vol-\nume of the permanent magnet. As is evident from the closed-\nform magnetic flux density of spherical magnet, there is a 1 =r3\ndecay for r>a. One can compute the total magnetic energy\nstored around the magnet as:\nWm=m0jmj2\n12pa3=m0\n9VjMj2(2)\nwhere Vis the volume of the magnet with magnetization\nM. One can compute the total energy stored outside a sphere\nof radius r>aas:\nWr=\u0010a\nr\u00113\nWm (3)\nThe above equations indicate that the magnetic energy con-\ntained in the radius rsphere and the magnetic flux intensity in\nthe distance rdecrease by 1 =r3. Thus, in order to reduce the\nsize of the transmitter, a high-magnetic flux (requires more\nsophisticated material) must be modulated when selecting a\nsmall r. Otherwise, miniaturization must be sacrificed in or-\nder to modulate smaller magnetic fluxes at larger r.\nThe first approach is to use a material with a controllable\nreluctance to create a shielding layer at radius r. Ideally, one\ncan alter the shield’s reluctance from a small to substantial\nvalue. This process allows the stored energy to be temporarily\ndecoupled outside the shield from the magnet, and then allows\nthe magnet to store energy outside the shield again by increas-\ning its reluctance. In its low reluctance mode, the spherical\nshield closes the field lines that pass it and thus dissipates the\nWrenergy every half cycle. For analytical convenience, we\npresume that variation of the reluctance does not substantially\ndisrupt the magnetic flux within the shield. There are various3\nTABLE I. Comparison of different low-frequency antennas for use in underwate and underground communication.\nMethod Ref Modulationa Vradb\u0000\ncm3\u0001VAntenna=Dc\u0000\ncm3=cm\u0001 Freq (Hz) DBDB(f T)\nat1kmDB=Vrad \u0000\nf T=cm3\u0001\n\u000210\u00003\nat1km\n5 - 100 -/10 30 1 pT at 264.8 m 18.5 185.00\n13 - 100 - 100 1800 nT at 2.03 m 15.06 150.60Rotating\nMagnet\n4 - 3 3/1.6 500 800 fT at 100 m 0.8000 266.67d\n2 Electromechanically 8.4 58.5/15.6 22 600 nT at 1 m 0.6000 71.43\n3 EMR 3.62 353/8 150 100 nT at 0.3 m 0.0027 0.75Modulated\nMagnet\nRotation 6 Mechanical Shutter - - 960 1.3 nT at 1 m 0.0013 -\nPendulum Array 20 DAM 29.91 -/13.4 1030 79.4 fT at 20 m 0.0006 0.02\nPiezoelectric 21 DAM 18.9 -/9.4 35500 - - -\n18 DAM 6.4 -/25 28000 16 nT at 0.4 m 0.0010 0.16Multiferroic\n19 DAM 1 -/18 10 6.05 nT at 1.3 m 0.0133 13.30\nMotionless - EMR & DAM 3 280/14 430 170 nT at 1.2 m 0.2940 98\naEMR: Electrically Modulated Reluctance, DAM: Direct Antenna Modulation\nbThe volume of the central radiator\ncThis column describes the total size of the antenna and the largest antenna dimension extracted from the literature, where applicable.\ndThis value shows a higher value than the theory, which may be due to the magnetometer’s error.\nconstraints, including loss, size, the current required to con-\ntrol the shielding material’s reluctance, and saturation level,\nwhich dictate the proper values for r.\nWe consider the next approach to be an asymmetric system\nconsisting of a ferrite yoke (as the variable reluctance mag-\nnetic material) and a permanent magnet (as the magnetic flux\nsource), as shown in Fig. 2(a) and 2(b). Since the permanent\nmagnet attracts the ferrite yoke, the total energy stored in this\nsystem is a function of the distance from the yoke to the mag-\nnet. We simulated this structure using ANSYS Maxwell for\ndifferent materials and ranges and compared the energy of the\nsystem with the energy stored in the isolated magnet. The sim-\nulation results, as shown in Fig. 2(c), suggests that nearly half\nof the magnet’s energy is converted to kinetic energy when the\nferrite yoke contacts the magnet, and another half is still stored\naround the system. The system energy for D=1cmis about\n90% of its maximum value, as the simulation results show.\nOne can then move the yoke 1 cm away from the magnet back\nand forth and modulate the stored energy with a modulation\ndepth of 40%. We can use a mechanical resonance structure\n(i.e., a spring and a fixture) to conserve the kinetic energy. We\nintend to modulate the reluctance to make the stored energy\ntime-variant, rather than a mechanical movement.\nWhile the spherical shield offers a significant modulation\ndepth (close to 100%), it is large and three-dimensional. On\nthe other hand, the system with ferrite yoke has much smaller\ndimensions; however, it cannot provide a sufficiently broad\nmodulation depth. Therefore, we combine the two above\nmethods by putting the magnet on a ferromagnetic film with\na proper winding to modulate the magnetic flux by adjusting\nthe reluctance of the film. The design parameters include the\nferrite characteristics, in particular, the nonlinearity of its B-H\ncurve, the thickness of the ferromagnetic film, the topology\nof the structure, and the windings. The design objectives are\nhigh magnetic flux, a high modulation depth, small size, and\nlow dissipated power. One of the tasks to achieve these objec-\ntives is to utilize the relationship between magnetic flux, B,\nand magnetic field, H, effectively.\nSN\nFerrite YokeD\nt\nSNFerrite Yoke\nDt(a) (b)\n \n0 20 40 60 80 100 \nD (mm) 0.5 0.6 0.7 0.8 0.9 1 \n  T mode -20mm -Orthonol  \nT mode -2mm -Orthonol  \nN mode -20mm -Orthonol \nT mode -20mm -Metglas Normalized Energy  \n(c)\nFIG. 2. The system contains magnet and ferrite yoke. a) T mode\nb)N mode c)normalized system energy of the system compared to\nthe isolated magnet versus distance D; the legend shows the system‘s\nmode, the thickness of the ferrite yoke (t), and the type of magnetic\nmaterial used for ferrite yoke implementation.\nIII. TRANSMITTER DESIGN\nWe designed and prototyped different structures to exam-\nine our proposed approach. Figure 3(a) shows the ANSYS\nmodel of one of our designs. The permanent magnet used\nin this transmitter is a rare-earth Neodymium magnet (N52,4\n6\u00021\u00020:5cm), which is the strongest permanent magnet\navailable in the market. Also, we used seven layers of Met-\nglas sheets 2705M ( Bs=0:77T) with a total thickness of\n0.178 mm as the magnetic film. Besides, a 40-turn coil\naround a c-shape magnetic core made of amorphous AMBC\n(Bs=1:56T) with a 2\u00022cmcross-section generates the mag-\nnetic flux needed to modulate the magnetic film’s reluctance.\nWe select a low reluctance core with a reasonably broad cross-\nsection to ensure that the c-shape core works at its linear state.\nAs a result, the current through the control coil generates a\nmagnetic flux in the magnetic film.\nFigure 3 shows the flux density on the system for two dif-\nferent values for the control current. Figure 3(a) shows that\nsmall areas of the magnetic film are in saturation when the\ncontrol current is zero. The saturated film helps to spread\nmagnetic flux in the air and to store magnetic energy around\nthe magnet. The small saturated area shows that the mag-\nnetic film operates as a barrier and closes inside the magnetic\nflux. Next, we apply 0.5 A current to the control coil, and the\npattern of magnetic film saturation shifts to Fig.3(b), which\nmeans the saturated area is larger than the closed mode. In\nthis mode, the magnetic flux spreads more in space, and there\nis more energy stored around the magnet. We name this state\nof the system,“Open mode.” This system’s operating modes\nwill differ by adjusting the arrangement of the magnet or the\nmagnetic film. For example, the magnetic film may be satu-\nrated by a giant magnet with zero current. The saturated area\nof the magnetic film can then be reduced by a magnetic flux\ngenerated by the control current against the magnet’s mag-\nnetic flux. In this case, the system’s operating modes switch\nto open and closed mode for zero current and high current, re-\nspectively. One can apply a sinusoidal current to the control\ncoil to change the amount of energy stored around the mag-\nnet periodically. Figure 3(b) also shows that the magnetic flux\ndensity in the AMBC core is less than 0.13 T, indicating the\namorphous AMBC cross-section we have is higher than what\nwe needed to keep it out of saturation. One can use a smaller\ncore to reduce overall system size and weight.\nIV. MEASURED RESULTS\nAssessing the performance of the prototyped transmitter\n(Fig. 4(a)) is a significant challenge due to the lack of a re-\nliable and calibrated magnetometer. As a result, we use the\nmagnetic field of a rotating permanent magnet as a reference.\nWe also used a low-noise audio amplifier connected to an air-\ncore search coil as a receiver. Besides, the magnet used in\nthe transmitter and the one used as the rotating magnet are\nidentical. If the measurement setup is the same (the trans-\nmitter replaces the rotating magnet while the relative location\nto the search coil is the same), we can assess our transmitter\naccurately. Figure 4(b) shows the permanent magnet plastic\ncase, which connects to a Dremel 4000 rotary tool (35000\nrpm) through its main shaft (see the inset of Fig. 4(b)). A\nmetal shaft is in place to secure the other end of the plastic\nenclosure to a solid fixture when it rotates. We were able to\nrotate the magnet up to 25800 rpm (equivalent to 430Hz). Fig-\nure 4(c) shows the rotating magnet and the search coil with the\ndistance R=1:2m.\n \nR\nMetglas\nAMBC\nMagnetControl Coil\nSearch Coil\n (a)\n \n(b)\n \n \n(c)\nFIG. 3. a) Maxwell model of the prototyped system. Magnetic field\ndistribution of the Metglas b) control current is zero and the system\nis in closed mode c) Control current force the Metglas to saturation\n(Open mode).\nFigure 5(a) displays the measured signal at the output of\nthe low-noise audio amplifier connected to the search coil as\nthe rotary tool rotates the magnet. The distorted waveform is\ndue to the non-linearity of the detection circuitry (audio am-\nplifier). Next, we remove the rotary system and replace it with\nthe proposed transmitter. We used a signal generator to feed\nthe proposed transmitter via a buffer amplifier with a 430 Hz\nsinusoidal waveform. In addition to the voltage waveform on\nthe audio amplifier output, Fig. 5(b) displays the input current\nwaveform. Comparing the two voltage waveforms in Fig. 5\nis reasonable by maintaining the same method of receiving\nand measuring for both cases. Notice that the rotating magnet\nswitches its field polarity per half a cycle (swinging between\n+B(R)and\u0000B(R)or 2DBmax) while the proposed transmit-5\n \nAMBC  \nMetglas  \n Permanent \nMagnet  \n  Control Coil  \n  \n(a)\n \n(a) \nCollet  \nMetal lic shaft  Magnet inside \na plastic case   \n(b)\nR\nMagnetSearch coil \n(c)\nFIG. 4. a) Photograph of the prototyped transmitter b) photograph of\nthe permanent magnet plastic case. The inset shows the photograph\nof magnet and Dremel 4000 rotary tool c) Maxwell model of the\nmagnet rotation setup.\nter can open and close the entire magnetic flux of the magnet\n(swinging between 0 and +B(R)orDBt) at its peak. Therefore\nthe rotating magnet produces twice as much a time-varying\nmagnetic flux as our proposed transmitter produces at its ideal\nperformance. Besides, the magnetic flux maximum DBmaxis\nequal to its static value for a given permanent magnet, due to\nthe low frequency (quasi-static). Simply, a rotating magnet’s\ntime-variant magnetic flux is equal to Bmaxcoswt. From now\non, we compare the transmitter’s measured time-variant flux\nwith the permanent magnet’s static flux at the same point, and\nwe call it modulation depth.\nModulation depth =DBt\nDBmax\u0002100 (%) (4)\nMeasuring the total power required to generate a time-\nvarying magnetic flux at a given distance is a crucial factor in\nevaluating the transmitter’s performance. Based on the mea-\nsured signal shown in Fig. 5(b), the sinusoidal voltage applied\nto the control coil is 0.95 V , and the current is 0.6 A, which\n \n0 10 20 30 40 \nTime ( ms) -2 -1 0 1 2 Vmag (V) 2ΔBmax (a)\n \n0 5 10 15 20 25 30 35 40 \nTime ( ms) -1 -0.5 0 0.5 1 1.5 \n-1 -0.5 0 0.5 1 1.5 \nVmag \nIin i Vmag (V) Iin (A) ΔBt \n(b)\nFIG. 5. Measured magnetic flux of a) rotating permanent magnet and\nb) proposed transmitter.\n \n200 250 300 350 \nInput power (mW)  20 30 40 50 60 70 \n   \nModulation depth  (%) \nFIG. 6. The modulation depth as a function of the input power of the\ntransmitter.\nresults in an average power of 0.285 W, while the rotary de-\nvice needs 60 W to rotate the magnet. The modulation depth\nof the proposed transmitter and the rotary device can be com-\npared with the measured input power in mind. The measured\nflux, shown in Fig. 5, used to calculate the modulation depth\nof 51%. Note that the maximum modulation depth for the\ntransmitter is 100%, while the magnet’s modulation depth is\n200%. Figure 6 also shows the measured modulation depth of\nthe transmitter versus the input power.6\n \n1.5 1.6 1.7 \nR (m) 0.2 0.25 0.3 1/R3 curve  ∆𝑣(𝑉) \nFIG. 7. The measured field versus range. The data points show each\nindividual measurement, and the line is the result of curve fitting.\nOne approach to verifying the measurement method is to\nmeasure the magnitude of the magnetic flux at various dis-\ntances for a given sinusoidal drive current. Figure 7 shows the\noutput voltage of the receiver vs. R. The magnetic flux (which\nis linearly proportional to the output voltage) decays by 1 =R3\nas expected. Besides, this figure provides a guideline for esti-\nmating the magnitude of the time-variant magnetic flux at any\ndistance where measured/simulated data at least at one point\nin the same direction is available. The theoretical equation24\nwas used to find the magnetic flux for the rotating magnet and\nthen to determine the coefficient required to convert the ob-\ntained voltage to the magnetic flux.\nThe transmitter will, therefore, generate 0 :17mTat 1.2 m.\nIn the same way, the 1 =R3decay of the magnetic field of the\nantenna allows extrapolating the field at a distance of 1 km,\nalthough the magnetic flux of 1 km is too low to measure with\nour magnetometer. It is estimated that the magnetic flux will\nbe 0.294 fT at 1 km. This study is conducted to determine\nthe magnet volume needed to achieve a field strength of 1 fT\nat 1 km. The results show that 1 fT can be accomplished\nat 1 km with a permanent magnet volume of 10 cm3with a\npower consumption of less than 0.5 W. Also, the proposed\nantenna is compared with other current designs in Table I.\nThe magnetic field generated by volume ( DB=V) for differ-\nent designs shows that the rotary magnet systems produce the\nmaximum field with a range of approximately 0 :2f T=cm3.\nHowever, this technique has its limitations. The multifer-\nroic transmitter, which generates a magnetic field of approx-\nimately 0 :013f T=cm3, is also far from competing with the\nrotating magnet. The proposed transmitter in this paper can\ngenerate a 0 :1f T=cm3magnetic field, making it a feasible\ncandidate to compete with the rotating magnet.\nIn terms of bandwidth and data rate, the proposed trans-\nmitter does not comply with the fundamental antenna limits.\nThe conventional antenna design approaches depend on the\npractical and useful Linear Time-Invariant (LTI) systems. For\nexample, a lossless tuned electrically small antenna (ESA) at\nresonance can be treated as a second-order resonator, where\nthe stored electrical/magnetic energy in its reactive zone ex-\nchanges the stored magnetic/electric energy in the reactive\nlumped element of the antenna’s matching circuit. For exam-\nple, a 1-meter lossless resonant antenna at 1 kHz ( l=300m)\nhas a minimum Q of 1014(bandwidth of 10\u000011Hz). However,\nbandwidth can be increased by sacrificing the antenna effi-ciency that can be achieved only on the receive side, but not\non the transmitter. However, it has been shown that the fun-\ndamental limits of the antennas do not bound the non-linear\nand/or time-variant (non-LTI) antennas25.\nFor example, a time-variant field can be created while\navoiding resonance, if the stored energy in an antenna’s reac-\ntive near-zone does not transform into another type of energy\nevery half a cycle (first-order system), and time variation is\nrealized by changing the location where the energy is stored.\nTherefore, the time-variant basis of the proposed structure\ngives rise to a parametric or non-LTI system that allows us to\nchange the data transfer rate, independently from the antenna\nquality factor. As a consequence, this non-LTI system results\nin higher data rates being feasible. Moreover, it has shown\nthat the stored energy frequency can be quickly shifted (FSK)\nwithout breaching the fundamental limits26. Therefore, the\nfrequency of the field modulation in the proposed transmitter\ncan be changed from a few hundred hertz to tens of kilohertz\nwithout any restriction. Besides, any type of modulation, such\nas frequency or amplitude modulation, can be applied to the\nproposed transmitter.\nV. SIMULATION RESULTS\nWe conduct further analysis in the simulation domain after\nverifying the transmitter’s functionality in the measurement\ndomain. We used magnetostatic simulation in the software\npackage, ANSYS Maxwell, to achieve that objective. In this\nanalysis, four different cases have been simulated: 1- an iso-\nlated permanent magnet, 2- an open mode transmitter (current\nON), 3- a closed mode transmitter (current OFF), and 4- a\ndeep closed mode transmitter (reverse current ON). One can\nuse the case 1 magnetic flux to examine the effects of the elec-\ntric current and the magnetic film thickness on magnetic flux\nin case 2 and case 3. Also, the modulation depth is determined\nby subtracting from case 2 the magnetic flux in case 3 or 4 and\ndividing the result by case 1 magnetic flux. The simulation re-\nsults for different cases at R=0:88mare shown in Fig. 8.\nFigure 8(a) shows that case 2 (open-mode transmitter) gen-\nerates 54% of the flux from an isolated magnet (case 1). This\nvalue is essential as we determine the size of the magnet re-\nquired for a given application. Besides, the modulation depth\nfor case 3 and case 4 is 41% and 46%, respectively. Al-\nthough we used an approximate B-H curve for the Metglas\nfilm in the simulation domain, the results are in good agree-\nment with the measured results (51% modulation depth). Note\nthat the drive current is a balanced sinusoidal in our measure-\nment setup (plus and minus currents); therefore, we compare\nthe measured results with modulation depth in case 4 as 46%.\nNext, we analyze the time-domain behavior of the rotating\nmagnet and the proposed transmitter using a transient analy-\nsis by ANSYS Maxwell. Figure 8(b) shows the magnetic flux\nof the rotating magnet at R= 0.88 m. As we expected for a\nquasi-static case, the maximum value of the flux is equal to\nthe magnetic flux of the static magnet at the same distance R\n= 0.88 m. The same behavior is observed for the proposed\ntransmitter for two different drive currents.\nWe have also analyzed the effect of the magnetic film thick-\nness on the transmitter performance. The Metglas film avail-7\n \n87.5 88 88.5 \nDistance from source ( cm) 0 0.2 0.4 0.6 0.8 1 \nCurrent ON  \nCurrent OFF  \nCurrent ON (Reverse)  \nMagnet Alone  0.925 \n0.50 \n0.125 \n0.075 Bz(μT) \n(a)\n \n0 2 4 6 Time ( ms) -1 0 1 Bz(μT) \n(b)\n \n0 2 4 6 \nTime ( ms) 0.2 0.4 0.6 \n500 mA \n700 mA Bz(μT) \n(c)\nFIG. 8. Simulation results; a) Magnitude of the magnetic flux at 0.88\nm away from the magnet in the magnetostatic solver, b) time domain\nsolution of the rotating magnet, and c) time domain solution of the\ntransmitter in the transient solver, for two different control currents.\nable comes in a roll, with a thickness of 10 mil (0.0254 mm).\nThe thickness of the magnetic film can, therefore, vary from\none layer to an integer number of layers n\u000210mil. Figure 9\nshows the simulation results for a variety of Metglas layers\nused in the magnetic film for two different drive currents. The\noptimal number of layers for drive current of 500 mA and 700\nmA is 7 and 8, respectively. Therefore, to build the magnetic\nfilm, one has to know the drive current in addition to the mag-\nnetic material’s B-H curve.\nVI. CONCLUSION\nA new method for generating electromagnetic waves using\nthe permanent magnet’s static magnetic flux has been intro-\nduced. By using reluctance modulation, the direction of the\nmagnetic flux and the location of the stored magnetic energy\nhave been changed to create a time-variant field. A method for\n \n5 10 15 \nNumber of layers  0.25 0.3 0.35 0.4 0.45 0.5 \n500 mA  \n700 mA  ∆|𝐵|(𝜇𝑇) FIG. 9. Effect of the number of layers in modulation depth, when 0.5\nA and 0.7 A, are applied as input control current.\nevaluating a ULF transmitter’s performance has also been im-\nplemented and used to assess the proposed transmitter. It has\nbeen shown that the prototype transmitter produces a time-\nvariant field with a modulation depth of 50 percent. While we\nhave not tried to minimize the size and weight of the transmit-\nter, it has realistic dimensions and weight. We also analyzed\nthe power consumption of the transmitter and the calculated\nresults. The calculations show that we can generate 1 fT of\ntime-variant magnetic flux at 1 km using a magnet volume of\n10cm3.\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.\nREFERENCES\n1W. Crookes, “Some possibilities of electricity,” The Fortnightly Review 51,\n173–181 (1892).\n2O. C. Fawole and M. Tabib-Azar, “An electromechanically modulated per-\nmanent magnet antenna for wireless communication in harsh electromag-\nnetic environments,” IEEE Transactions on Antennas and Propagation 65,\n6927–6936 (2017).\n3N. Strachen, J. Booske, and N. Behdad, “A mechanically based magneto-\ninductive transmitter with electrically modulated reluctance,” PLOS ONE\n13, e0199934 (2018).\n4H. C. Burch, A. Garraud, M. F. Mitchell, R. C. Moore, and D. P. Arnold,\n“Experimental generation of elf radio signals using a rotating magnet,”\nIEEE Transactions on Antennas and Propagation 66, 6265–6272 (2018).\n5S. G. amd Y . Liu and Y . Liu, “A rotating-magnet based mechanical antenna\n(rmbma) for elf-ulf wireless communication,” Progress In Electromagnetics\nResearch M 72, 125–133 (2018).\n6M. Gołkowski, J. Park, J. Bittle, B. Babaiahgari, R. A. L. Rorrer, and\nZ. Celinski, “Novel mechanical magnetic shutter antenna for elf /vlf ra-\ndiation,” IEEE International Symposium on Antennas and Propagation &\nUSNC/URSI National Radio Science Meeting , 65–66 (2018).\n7M. N. S. Prasad, S. Selvin, R. U. Tok, Y . Huang, and Y . Wang, “Directly\nmodulated spinning magnet arrays for ulf communications,” IEEE Radio\nand Wireless Symposium (RWS) , 171–173 (2018).\n8J. A. Bickford, A. E. Duwel, M. S. Weinberg, R. S. McNabb, D. K. Free-\nman, and P. A. Ward, “Performance of electrically small conventional and\nmechanical antennas,” IEEE Transactions on Antennas and Propagation 67,\n2209–2223 (2019).\n9F. Trouton, “Radiation of electric energy,” The Electrician 28, 302 (1892).8\n10P. G. Datskos, N. V . Lavrik, J. Tobin, and L. T. Bowland, “Using micro-\nelectro-mechanical systems (mems) as small antennas,” Future of Instru-\nmentation International Workshop (FIIW) Proceedings , 1–4 (2012).\n11A. Madanayake et al. , “Energy-efficient ulf/vlf transmitters based on\nmechanically-rotating dipoles,” Moratuwa Engineering Research Confer-\nence (MERCon) , 230–235 (2017).\n12Z. Wang, Z. Cao, and F. Yang, “Radiated field of rotating charged parallel\nplates and its frequency spectrum,” AIP Advances 8, 025325 (2018).\n13W. R. Scott, “Electromagnetic induction sensor with a spinning magnet ex-\ncitation,” IEEE International Geoscience and Remote Sensing Symposium\n(IGARSS) , 3542–3545 (2019).\n14C. W. Nan, M. I. Bichurin, S. Dong, D. Viehland, and G. Srinivasan, “Mul-\ntiferroic magnetoelectric composites: Historical perspective, status, and fu-\nture directions,” Journal of Applied Physics 103, 031101 (2008).\n15T. Nan et al. , “Acoustically actuated ultra-compact nems magnetoelectric\nantennas,” Nat. Commun. 8, 1 (2017).\n16J. P. Domann and G. P. Carman, “Strain powered antennas,” Journal of Ap-\nplied Physics 121, 044905 (2017).\n17T. Ueno, J. Qiu, and J. Tani, “Magnetic force control with composite of\ngiant magnetostrictive and piezoelectric materials,” IEEE Transactions on\nMagnetics 39, 3534–3540 (2003).\n18J. Xu et al. , “A low frequency mechanical transmitter based on magneto-\nelectric heterostructures operated at their resonance frequency,” Sensors 19,\n853 (2019).19J. D. Schneider et al. , “Experimental demonstration and operating princi-\nples of a multiferroic antenna,” Journal of Applied Physics 126, 224104\n(2019).\n20M. N. S. Prasad et al. , “Magnetic pendulum arrays for efficient ulf trans-\nmission,” Scientific Reports 9, 13220 (2019).\n21M. A. Kemp et al. , “A high q piezoelectric resonator as a portable vlf trans-\nmitter,” Nature Communications 10, 1715 (2019).\n22L. J. Chu, “Physical limitations of omni-directional antennas,” Journal of\nApplied Physics 19, 1163–1175 (1948).\n23H. A. Wheeler, “Fundamental limitations of small antennas,” Proceedings\nof the IRE 35, 1479–1484 (1947).\n24M. Manteghi, “A navigation and positining system for unmanned underwa-\nter vehicles based on a mechanical antenna,” IEEE International Sympo-\nsium on Antennas and Propagation & USNC/URSI National Radio Science\nMeeting , 1997–1998 (2017).\n25M. Manteghi, “Fundamental limits, bandwidth, and information rate of\nelectrically small antennas: Increasing the throughput of an antenna with-\nout violating the thermodynamic q-factor,” IEEE Antennas and Propagation\nMagazine 61, 14–26 (2019).\n26M. Salehi, M. Manteghi, S. Y . Suh, S. Sajuyigbe, and H. G. Skinner, “A\nwideband frequency-shift keying modulation technique using transient state\nof a small antenna (invited paper),” Progress In Electromagnetics Research\n143, 421–445 (2013)."    },    {        "title": "2005.10727v1.Antiferromagnet_semiconductor_van_der_Waals_heterostructures__interlayer_interplay_of_exciton_with_magnetic_ordering.pdf",        "content": " 1 Antiferromagnet –semiconductor van der Waals \nheterostructures: interlayer interplay of exciton \nwith magnetic ordering  \nMasaru Onga1,2, Yusuke Sugita2, Toshiya Ideue1,2, Yuji Nakagawa1,2, Ryuji Suzuki1,2, Yukitoshi \nMotome2, Yoshihiro Iwasa1,2,3 *  \n1 Quantum -Phase Electronics Center (QPEC), The University of Tokyo, Tokyo 113 -8656, Japan \n2 Department of Applied Physics, The University of Tokyo, Tokyo 113 -8656, Japan \n3 RIKEN Center for Emergent Matter Science (CEMS), Wako 351 -0198, Japan \n \nABSTRUCT  \nVan der Waals (vdW) heterostructures have attracted great interest because of their rich material \ncombinations.The discovery of two -dimensional magnets has provided a new platform for \nmagnetic vdW heterointerfaces; however, research on m agnetic vdW heterointerfaces has been \nlimited to those with ferromagnetic surfaces. Here we report a magnetic vdW heterointerface using \nlayered intralayer -antiferromagnetic MPSe 3 (M=Mn, Fe) and monolayer transition metal \ndichalcogenides (TMDs). We found an anomalous upshift of the excitonic peak in monolayer \nTMDs below the antiferromagnetic transition temperature in the MPSe 3, capturing a signature of \nthe interlayer exciton -magnon co upling. This is a concept extended from single materials to  2 heterointerfaces. Moreover, this coupling strongly depends on the in -plane magnetic structure and \nstacking direction, showing its sensitivity to their magnetic interfaces. Our finding offers an \nopportunity to investigate interactions between elementary excitations in different materials across \ninterfaces and to search for new functions of magnetic vdW heterointerfaces.  \nKeywords: Van der Waals heterostructure, layered magnet, transition metal dichal cogenide, \nexciton, antiferromagnet , photoluminescence  \n \nTEXT  \nApart from conventional heterostructures with III -V compound semiconductors or complex \noxides, van der Waals (vdW) heterostructures attract much attention in multiple fields owing to \ntheir wide extensibility1. Because of the weak vdW force between their layers, we can freely \ncombine any types of cleavable materials to make heterointerfaces regardless of their crystal \nstructures. This provides a wide platform for making heterointerfaces with noteworthy physical \nproperties: ultra -high mobility and quantum Hall effect2; vertical PN junctions3; and moiré physics \nrelated to Mott insulating states4, superconductivity5, and excitonic states6–9. Research on magnetic \nvdW heterointerfaces has greatly d eveloped since the discovery of two -dimensional (2D) \nferromagnets such as CrGeTe 3 and CrI 310,11. In particular, an optical study of a monolayer \nWSe 2/CrI 3 heterostructure has clarified the interfacial magnetic coupling between a 2D \nnonmagnetic semiconductor  and a ferromagnetic surface of the 2D magnetic insulator12.  \nHowever, there has been no study focusing on antiferromagnetic properties (especially intralayer \nantiferromagnetic ordering) in magnetic vdW heterostructures, although antiferromagnets have \nrecently been found effective for realizing spintronic functions13. Because antiferromagnets have  3 a variety of spin orderings with distinct magnetic symmetry groups, we can anticipate unique \nmagnetic properties of the heterointerfaces and control their functio nalities by choosing \nappropriate magnets. The antiferromagnetic state is intrinsically free from stray fields, implying \nspintronic properties of the antiferromagnets are robust against external perturbation. More \nimportantly, the energy gaps of the spin ex citations in antiferromagnets are generally two or three \norders of magnitude larger than those in ferromagnets. This can enhance the coupling between \nlight and magnetic excitations in antiferromagnets from the viewpoint of opto -spintronics14. \nConstructing heterointerfaces with materials of high optical quality makes it possible to use the \nantiferromagnetic properties efficiently with other optical systems.  \nHere we report the optical properties of the vdW  heterointerfaces of a layered intralayer -\nantiferromagnetic insulator ( MPSe 3; M=Mn or Fe) and a nonmagnetic monolayer direct -gap \nsemiconductor (MoSe 2) (Fig. 1a). The photoluminescence (PL) from the monolayer M oSe2 is \nmeasured to probe the coupling between the antiferromagnet and the semiconductor via the vdW \ninterface. We find an additional upshift of the excitonic peaks of MoSe 2 below the Néel \ntemperature TN, even though the antiferromagnetic ordering does not occur inside the MoSe 2 but \nin the neighbouring  MPSe 3. The underlying origin of the shift cannot be explained by a \nhomogenous exchange field like the Zeeman splitting in the ferromagnetic vdW interface4,15. \nRather, our experimental and theoretical results suggest that the coupling is due to interlayer \nexciton –magnon coupling as shown in Fig. 1b: excitons in a semiconductor layer couple with \nmagnetic excitations in the neighbouring antiferromagnet layer. The interlayer coupling can be \nturned on and off by intentionally selecting the magnetic ordering and  stacking angle, which is \nenabled by the vdW nature of these heterointerfaces in marked contrast to single bulk magnets.   4 Monolayer transition metal dichalcogenides (TMDs) including MoSe 2 have been intensively \nstudied and have excellent electrical/optical p roperties as direct -gap semiconductors with large \nexciton binding energy and Zeeman -type spin -orbit coupling16,17. In particular, excitonic peaks \nfrom monolayer MoSe 2 are separated enough to resolve the multiple excitonic states18, and thus \nMoSe 2 is the mo st appropriate TMD for studying in detail the optical response at the interface for \nour purpose. For the antiferromagnet, we chose transition metal phosphorus trichalcogenides \n(MPX3; M: transition metal; X: chalcogenides), which have been studied in bulk f orm with various \nintralayer antiferromagnetic orderings at M such as the Néel -type in MnPSe 3 and the zigzag -type \nin FePSe 3 (the details are given in section §1 of the Su pporting  Information (SI))19. MPSe 3 is \ncomposed of magnetic ions ( M) forming a honeycomb lattice within each layer and a P 2Se6 ligand \nat the centre of the honeycomb (Fig. 1a). Each M ion is octahedrally coordinated with six Se atoms \nfrom the ligands. Atomically thin FePS 3 was recently repor ted to show an antiferromagnetic phase \ntransition even in the monolayer20,21, suggesting that this family has magnetic ordering even in the \nthin-flake form. Note that MPX3 possesses intralayer antiferromagnetic ordering in contrast to the \ninterlayer antife rromagnetic ordering in multilayer CrI 311, meaning that our study is distinct from \nthat on heterointerfaces with multilayer CrI 312 (which has the ferromagnetic intralayer ordering at \neach layer).  \nWe fabricated vdW interfaces of monolayer MoSe 2 and multilay er MPSe 3 with a thickness of \ntens of nanometers. The interfaces were fabricated via a mechanical exfoliation and an all -dry \ntransfer method22 inside a glovebox with an inert atmosphere to prevent sample degradation (Fig. \n2a). We controlled the stacking ang les according to the relation between the edge of the cleaved \nflakes and the crystal direction23. The zigzag -edge of MoSe 2 was aligned parallel to the zigzag -\nedge of honeycomb Mn2+ (sample A1) to realize nearly commensurate stacking between the 2 ×  5 2 super lattice of MoSe 2 (lattice constant aMoSe2  = 0.328 nm) and the unit cell of MPSe 3 (aMnPSe3  = \n0.639 nm) with a lattice mismatch of 2.6% (see the details in SI section §2).  \nFigure 2b is the Raman spectrum of a typical heterostructure, monolayer MoSe 2/thin Mn PSe 3. \nClear MnPSe 3 peaks appear at 149 cm–1, 175 cm–1, and 222 cm–1, which is consistent with a \nprevious study24. There is also a relatively weak peak of monolayer MoSe 2 at 244 cm–1. The Raman \nintensity of MnPSe 3 is known to be enhanced below TN24, and can thus be an optical probe of an \nantiferromagnetic transition in the exfoliated thin flakes. Figure 2c (and Fig. S2 in SI section §3) \nshows the temperature dependence of the main peak of thin MnPSe 3 at 222 cm–1, indicating that \nthe Raman intensity is also enhanced even in exfoliated thin flakes. The upturn starts around TN=74 \nK as determined by a previous study on neutron scattering19, meaning that the magnetic orderings \nin our samples are not affected by thinning at least down to ~20 nm.  \nFigure 2d shows the PL spectra of monolayer MoSe 2 on MnPSe 3 and on SiO 2 at 6 K. Each \nspectrum has a neutral exciton (X0) peak around 1.66 eV and a trion (XT) peak around 1.62 eV. \nThe MoSe 2 peaks on MnPSe 3 are slightly broader than  those on SiO 2 owing to photocarrier \nrelaxation from MoSe 2 to MnPSe 3, as observed in a ferromagnetic vdW interface with spin \nrelaxation12. The differences in peak positions between the two samples are due not only to strain \ninduced by the substrates, but a lso to the magnetic ordering of the bottom material as we discuss \nbelow. All the typical spectra in this study and detailed discussion of the peaks are in section §4 \nand Fig. S3 of the SI.  \nWe performed PL measurements at various temperatures to clarify the  effect of the magnetic \nordering on the spectra. These spectra are shown in Fig. 3a for MoSe 2/MnPSe 3 (sample A1) and \nMoSe 2/SiO 2 (sample S1). The MoSe 2/SiO 2 spectra in Fig. 3a are uniformly shifted by +7 meV, a \ntemperature -independent difference due to the constant strain from the substrates, because we aim  6 to uncover only the effect of the magnetic ordering of MnPSe 3. Hereafter, we focus on the X0 \npeaks because we can clearly observe them at any temperature. The X0 peak positions are indicated \nwith triangle s in Fig. 3a and plotted in Fig. 3b as a function of temperature. The temperature \ndependence of MoSe 2/SiO 2 follows the well -known behaviour of semiconductor band gaps (see \nSI section §5)18. While both peaks are upshifted together on cooling above TN for MnPSe 3, we \nfound they are separated on further cooling below TN. Figure 3c is a plot of the difference between \nthe two peak positions against temperature. The marked feature in crossing TN strongly suggests \nthat the magnetic ordering inside MnPSe 3 affects the exciton of the attached monolayer MoSe 2. \nSimilar results are also observed in other samples with MoSe 2 (samples A2 and A3 in SI section \n§6), WSe 2, and MoS 2 (shown in SI section §7); therefore the observed shift is common in \nsemiconducting group -VI TMDs on MnPSe 3. These results for antiferromagnetic heterointerfaces, \nhowever, have never been observed or predicted to our knowledge despite much work on \nheterointerfaces with ferromagnetic surfaces12,15,25. \nExciton peaks in b ulk semiconducting anti ferro magnets, on the other hand, are known to be \nmodulated  below TN, which is attributed to exciton -magnon coupling (EMC)26. This means that \nan exciton in an antiferromagnet itself can directly interact with a magnon  via EMC . The energy \nof an absorbed/emitted photon corresponds to the addition/subtraction of exciton and magnon \nenergies, meaning that the EMC can cause optical peak modulation equal to the magnon energy26. \nAbsorption spectra of bulk MPX3 crystals reveal an EMC effect in d -d transi tions27 and additional \nupshifts in p -d transitions below TN28. Similar phenomena have been also reported on other \nantiferromagnetic insulators29-31. We can therefore interpret the present results as exciton –magnon \ncoupling across the interface  in analogy : the exciton in the semiconductor layer couples with the \nmagnon in the antiferromagnetic layer through interlayer interaction (Fig 1b and inset of Fig. 3c).  7 Hereafter, we call this interlayer exciton –magnon coupling  (interlayer EMC) to distinguish it from \nconventional EMC at the same atomic sites. Note that the observed energy shift in Fig. 3c is \nroughly comparable to the magnon energy scale in bulk MnPSe 3 (several meV) measured via \ninelastic electron tunnelling spectroscopy in a previous study32. \nMeanwhile,  we considered three possible origins other than the interlayer EMC: the band \nmodulation of monolayer MoSe 2 by the antiferromagnetic order in MnPSe 3, the additional strain \ndue to magnetostriction, and the bound magnetic polaron (shown in detail in SI secti ons §8 –11). \nIn particular, we conducted density functional theory (DFT) calculations for an exactly \ncommensurate TMD/MnPSe 3 system (corresponding to the ideal configuration of sample A1) to \nfind out whether the band structure of monolayer TMDs can be direc tly modulated through \ninterlayer coupling with antiferromagnets. We concluded, however, that none of these three \npossibilities could explain our experimental results, and thus attributed them to the interlayer EMC \nin analogy to the mechanism in bulk system s.  \nNext, we demonstrate that the interlayer EMC can be turned on in a particular vdW configuration. \nWe conducted the same measurements on FePSe 3 as on MnPSe 3 to reveal the effect of magnetic \norderings. MnPSe 3 shows a Néel -type antiferromagnetic ordering ( Fig. 4a) below TN = 74 K, \nwhereas a zigzag -type ordering occurs in FePSe 3 below TN = 112 K (Fig. 4b)19, causing distinct \ncoupling with the top MoSe 2 layer from MnPSe 3. Figure 4c shows the X0 peaks of MoSe 2/FePSe 3 \n(sample F1) against temperature along with those of MoSe 2/MnPSe 3 and MoSe 2/SiO 2. The \ntemperature dependence of MoSe 2/FePSe 3 is more similar to that of MoSe 2/SiO 2 than that of \nMoSe 2/MnPSe 3, indicating that the zigzag -type antiferromagnetic ordering of Fe2+ affects the \nexciton of the neighbouring  TMDs much more weakly than the Néel -type ordering of Mn2+.   8 Additionally, we inspected the effect of the stacking angle between MoSe 2 and MnPSe 3 layers, \nwhich can give more microscopic information about the interlay er EMC. We fabricated another \nsample with a different stacking angle than that of sample A1: sample E1 with the zigzag -edge of \nMoSe 2 perpendicular to the zigzag -edge of Mn2+ (⊥). In contrast to the parallel configuration in \nsample A1, the perpendicular one  is not expected to have good lattice matching, resulting in a \nmoiré pattern and an additional PL peak possibly from a moiré exciton (see SI section §2). The \ntemperature dependence of the sample E1 peaks in Fig 4d shows that the parallel configuration ( ∥) \nis essential for the additional peak shifts below TN. This dependence on stacking direction and the \nabove comparison with FePSe 3 (summarized in Fig. 4e including MoSe 2/FePSe 3 (⊥)) strongly \nsuggest that interlayer EMC is very sensitive to the types of magne tic ions (Mn2+ or Fe2+) and/or \ninterfacial symmetry (magnetic ordering patterns or the crystal direction). This indicates that it is \npossible to turn on the EMC via vdW engineering, which is a unique nature of interlayer exciton -\nmagnon coupling and was impossible to investigate  in single bulk AFMs. We note that such \nsensitivity has not been expected for ferromagnetic vdW interfaces, in which the exchange \ninteraction is basically uniform irrespective of stacking angle.  \nThe detailed theor ical understanding  of this interlayer EMC  remains to be established  (see SI \n§12).  The original theory on exciton -magnon coupling in bulk (formula (2) of SI §12) is based on \nthe electric dipole moment generated by the combination of an orbital excitation (exciton) at one \nsublattice and a spin excit ation (magnon) at the other sublattice via exchange interaction  between \nthe sites26. In the interlayer EMC which we propose here, we suppose that interlayer exchange \ninteraction connects an exciton in the TMDs layer and a magnon in the antiferromagnetic l ayer. \nThis conceptual expansion can make it complicated and/or interesting to construct the microscopic \ntheory.  From the other points of view, tight -binding approach including interlayer hopping can  9 give us suggestions on this system similarly to the discu ssion on the interlayer exciton (interlayer \ncoupling between charged particles).  Experimentally, it would be helpful to excite  the magnons \ndirectly  by microwave or THz light  in order to prove the EMC directly . Detailed properties can be \nrevealed by investi gating stacking -angle dependence and other material combinations.  \nFrom a different perspective, interlayer coupling between elementary excitations is intriguing \nboth for basic science and applications. It would be fascinating to understand how the valley -\nexcitons (exciton with valley degrees of freedom) microscopically interact with magnetic \nexcitations.  Efficient magnon -photon (microwave -light) transduction for opto -\nspintronics/quantum -electronics could also be expected by regarding monolayer TMDs with the \ninterlayer EMC as a repeater of magneto -optical coupling of the magnetic insulators.  \nIn conclusion,  we fabricated antiferromagnet –semiconductor van der Waals heterointerfaces and \nfound an anomalous upshift of excitonic peaks, which is presumably induced  by interlayer \nexciton –magnon coupling. These features suggest that the interaction between elementary  \nexcitations such as excitons and magnons, which has been studied for single substances, can be \nconceptually expanded to artificial heterointerfaces and modulated through vdW engineering. \nThese results open a realm of magnetic vdW interfaces with versatile  functions for opto -spintronics \nand quantum electronics  and provide detailed insight into correlated excitations in condensed \nmatter physics.  \n  10 FIGURES  \n \nFigure 1. Antiferromagnet –semiconductor van der Waals heterointerfaces.  a, Crystal \nstructure of our system . The heterostructure is made of two layered materials: monolayer MoSe 2 \nand tens -of-nanometer s-thick MPSe 3. The MPSe 3 is composed of a M2+ plane with honeycomb \nstructure and a [P 2Se6]4– dimer. b, Coupling between an exciton in one layer and a magnon in the \nother layer via interlayer interaction. Our experimental data suggest that both excitations interact \nwith each other over the van der Waal s gap.   \n 11  \nFigure 2. Characterization of a monolayer MoSe 2/MnPSe 3 heterointerface.  a, Optical image \nof a heterost ructure on a SiO 2/Si substrate.  Monolayer MoSe 2 on MnPSe 3 is highlighted by the \nblack dotted lines  b, Typical Raman spectrum of the heterostructure , which includes strong peaks \n(orange arrows) from MnPSe 3 and a weak peak (a black arrow) from monolayer MoSe 2. c, \nTemperature dependence of the peak intensity at 222 cm–1 in Fig.  2b. The s udden increase of the \nintensity indicates the tens -of-nanometer s-thick MnPSe 3 has an  antiferromagnetic transition at \nalmost the same critical temperature as the bulk. The dashed line is a guide for the eyes. d, \nPhotoluminescence (PL) spectra of MoSe 2/MnPSe 3 (red) and MoSe 2/SiO 2 (blue)  at 6 K. Both \nspectra include neutral exciton (X0) and trion (XT) peaks.      \n 12  \nFigure 3. Temperature  dependence of PL peaks.  a, PL spectra at various temperatures. All \nMoSe 2/SiO 2 spectra are uniformly shifted by +7 meV for comparison. The triangles indicate the \npeak positions of X0.  b, Temperature dependence of the peak positions of MoSe 2/MnPSe 3 (red \ncircles ; sample A1) and MoSe 2/SiO 2 (blue squares ; sample S1) in Fig. 3a. The values at 140 K are \nset as the origins of the shifts. c, Peak energy shifts  of two MoSe 2/MnPSe 3 samples due  to \nantiferromagnetic ordering , where  E is the difference between the peak shifts of MPSe 3 and that \nof SiO 2 at each temperature. Temperature on the h orizontal axis is normalized by TN for MPSe 3. \nThe left and right insets show  interlayer couplings at the interface between excitons in a \nsemiconductor (red) and an antiferromagnet (blue) below and above TN, respectively.    \n 13   \nFigure 4. Magnetic -order and stacking -pattern dependence.  a, b, Antiferromagnetic ordering \nof M2+ in MPSe 3. Each spin lies within the basal plane in bulk MnPSe 3 and is parallel to the c -axis \nin bulk FePSe 3 (see SI section §1). c, Temperature dependence of the peak shift in different M2+ \nstructures . The r ed open circles  and purple squares show the peak shifts of the heterostructures \nconsist ing of Néel-type Mn2+ (sample A1) and zigzag -type Fe2+ (sample F1) , respectively, and the \nblue open diamonds indicate the shifts for MoSe 2/SiO 2 (sample S1) . d, Temperature dependence \nof peak shifts in different stackin g patterns. The r ed open circles, green circles , and blue  open  \ndiamond  represent the PL peaks of MoSe 2 on MnPSe 3 with parallel configuration (∥; sample A1) , \non MnPSe 3 with perpendicular configuration ( ⊥; sample E1 ), and on SiO 2 (sample S1 ), \n 14 respectively . e, Summary of the experimental data. The r ed circles, green circles , and purple \nsquares indicate E for MoSe 2/MnPSe 3 (∥; sample A1) , MoSe 2/MnPSe 3 (⊥; sample E1 ), \nMoSe 2/FePSe 3 (∥; sample F1),  and MoSe 2/FePSe 3 (⊥; sample G1 ) resp ectively.  \n   15 METHODS  \nSample preparation:  Single crystals of MoSe 2, MnPSe 3, and FePSe 3 were grown via a chemical \nvapour trans port technique33,34. Monolayer MoSe 2 flakes were exfoliated directly onto \npolydimethylsiloxane (PDMS), and MPSe 3 flakes were exfoliated onto SiO 2(300 nm)/Si substrates \nin a glovebox filled with N 2. We then transferred monolayer MoSe 2 onto a MPSe 3 flake on the \nsubstrate (using an all -dry transfer method22) in the glovebox, put the sample into the chamber, \nand flew Ar/H 2 (97%/3%) gas at 523 K for 2 hours to fit the flakes to each other. The thickness \nwas determined via optical contrasts and atomic force micro scopy images.  \nPhotoluminescence measurement:  We performed the optical measurements in a He -flow \ncryostat (Oxford Instruments) under a high vacuum (~5×10–6 torr) at low temperature. Raman \nspectra were recorded on a JASCO spectrometer equipped with a semicon ductor laser (532 nm). \nIn photoluminescence spectroscopy, we used a spectrometer with a liquid -nitrogen -cooled CCD \n(Horiba) and a He –Ne laser (632.8 nm; 5 mW; Melles Griot). The power was decreased to 500 \nW, and the beam was focused onto the monolayer MoS 2 flakes using a 50x objective (Olympus).  \nDFT calculation:  In DFT calculations, we used  the OpenMX code35, which is based on a linear \ncombination of pseudoatomic orbital formalism36,37. We used the Perdew -Burke -Ernzerhof  \ngeneralized gradient approximation (GGA) functional in density functional theory38, the DFT -D2 \nmethod to take van der Waals force into account39, and a 16×16×1 k -point mesh and vacuum space \ngreater than 20 Å between bilayer systems of TMD and MnPSe 3 for the calculation of the self -\nconsistent electron density and the structure relaxation. We optimised the atomic positions in the \nunit cell with the convergence criterion 10−2 eV/Å for the inter -atomic forces starting from the \nstacking structure concluded as t he most stable one in the literature40; we confirmed that all the  16 results stably converged to similar lattice structures. See the Su pporting  Information for further \ndetails of the DFT calculations.  \n \nSupporting Information . The following files are available  free of charge.  \nSupporting_information.pdf  (including supplementary discussions related to this letter)  \n \nAUTHOR INFORMATION  \nCorresponding Author  \n*E-mail: iwasa@ap.t.u -tokyo.ac.jp  \nNotes  \nThe authors declare no competing financial interest.  \n \nACKNOWLEDGMENT  \nWe thank K. Usami and A. Fujimori for helpful discussions. Some calculations were carried out \nat the Supercomputer Center of the Institute for Solid State Physics at The University of Tokyo. \nM.O., Y.S., and Y.N. were supported by the Japan S ociety for the Promotion of Science (JSPS) \nthrough the Research Fellowship for Young Scientists. T.I. was supported by JSPS KAKENHI \ngrant numbers JP19K21843,  JP19H01819, and JST PRESTO project (JPMJPR19L1) . This \nresearch was supported by a Grant -in-Aid for  Scientific Research (S) (No. 19H05602) and the A3  17 Foresight Program of the JSPS. 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Rep.  2017 , 7, 9504.  \n \n   23 Supporting  Information:  \nAntiferromagnet –semiconductor van der Waals heterostructures: interlayer \ninterplay of exciton with magnetic ordering  \n  Masaru Onga1,2, Yusuke Sugita2, Toshiya Ideue1,2, Yuji Nakagawa1,2, Ryuji Suzuki1,2, \nYukitoshi Motome2, Yoshihiro Iwasa1,2,3 * \n1 Quantum -Phase Electronics Center (QPEC), The University of Tokyo, Tokyo 113 -8656, Japan  \n2 Department of Applied Physics, The University of Tokyo, Tokyo 113 -8656, Japan  \n3 RIKEN Center  for Emergent Matter Science (CEMS), Wako 351 -0198, Japan  \n*Correspondence to: iwasa@ap.t.u -tokyo.ac.jp  \n \n \n§1. Magnetic structure s of MPX3 \nMPX3 family ( M: transition metals, X: chalcogenides ) shows various types of magnetic \nordering1. The spin orderings occur at honeycomb M2+ within the layer: for instance, MnPS 3 has \nthe Néel-type spin ordering parallel to c -axis at Mn2+ ions, while FePS 3 and FePSe 3 have the \nzigzag -type orderings also along c -axis at  Fe2+ ions.  \nThe magnetic moment of the N éel-type MnPSe 3 is supposed to lie within the basal plain \n(or slightly canted between the plain and c -axis) according to the neutron diffraction study on \npolycrystalline powder2. In our calculation shown later in § 9, however, the monolayer MnPSe 3 \nbeneath the monolayer TMDs has an easy axis along c -axis similarly to the previous first -principal \ncalculation on a bilayer MnPSe 33. We have no conclusive picture of the actual spin ordering at the \nsurface of MnPSe 3 due to t he above controversy and the experimental difficulty to detect it.  \nWe can guess that the heterostructure made of MnPS 3 (with Neel -type Mn2+) would show \nthe similar effects to that of MnPSe 3. However, the situation could be dramatically distinct because \nof the different orientations of sublattice spins in bulk MnPS 3 (along c -axis) and bulk MnPSe 3  24 (almost lied in ab -plane) as we noted above. Since the magnetoresponse of the TMDs is quite \nanisotropic owing to the Zeeman -type spin -splitting at the K -points, the excitons at the interface \nwith antiferromagnets can also be sensitive to the spin -orientation of the magnets as well as the \ntypes of spin -ordering. It would be very informative to explore the detailed experiments with other \nMPX3 in order to elu cidate the microscopic coupling at the interface. The direction of the sublattice \nmagnetic moments can also explain the M-dependency of the exciton -magnon interaction \ndiscussed in Fig.4 of the main text.  \n \n§2. Commensurability depend ing on stacking angles  \nThe 2 × 2 superlattice of MoSe 2 (lattice constant aMoSe2  = 0.328 nm) and the unit -cell of \nMPSe 3 (exactly speaking  M2P2Se6, aMn2P2Se6  = 0.639 nm and aFe2P2Se6  = 0.632 nm) present nearly \ncommensurate lattice matching with the parallel configuration: their lattice mismatches are 2.6% \nin MnPSe 3 and 3.6% in FePSe 3 as calculated from the lattice constants of bulk. Figure S1a shows \nthe image of MoSe 2/MnPSe 3 in the pa rallel case.  In contrast, we cannot expect the good lattice \nmatching with the perpendicular configuration as shown in Fig. S2b, visually presenting moiré \npatterns in this case within this scale. In the parallel case, we cannot see moiré patterns in the sc ale \nof Fig. S1a (~ 20 unit cells of MoSe 2) due to the relatively small lattice mismatch of 2.6%, meaning \nthat more than 80 unit cells are needed to see such a periodic pattern.  \n \n§3.  Detailed discussions on Raman spectra  \nWe show Raman spectra around 222cm-1 at various temperatures in Fig. S2 as raw data of \nFig. 2c in the main text. The abrupt changes around TN=74 K of the bulk MnPSe 3 support that the  25 antiferromagnetic transition of the exfoliated MnPSe 3 occurs at almost same temperature as the \nbulk,2,4. Similar behavior is also observed in the peaks around 149 and 175 cm-1. \nGenerally speaking, two -magnon scattering can cause a broad Raman peak at the two -\nmagnon frequency at Brillouin zone boundary (~9 meV  × 2 = 145 cm-1 in MnPSe 3 3). A recent \nstudy on Ram an spectroscopy of bulk/exfoliated MnPS 3 reports a small two -magnon peak above \n2.71 eV -excitation in a bulk sample5. However, no two -magnon peak has been observed within \nour measurements with 2.33 eV -excitation. Considering the previous work mentioned abov e5, it \ncould be due to the small excitation energy and/or weaker intensity of the two -magnon peak in \nexfoliated flakes than a prominent phonon peak around 149 cm-1. \n \n§4. PL spectra from MoSe 2/MPSe 3: Possible moiré excitons and strain from substrates  \nFigure  S3 display four PL spectra of sample F1 (MoSe 2/FePSe 3, ∥), A1 (MoSe 2/FePSe 3, ∥), \nS1 (MoSe 2/SiO 2), E1 (MoSe 2/MnPSe 3, ⊥), and G1 (MoSe 2/FePSe 3, ⊥) at 6 K. We also plot the \nfitting results by multi -peaks Voigt function: the spectra of sample F1, A1 and S1 can be \ndecomposed well into two peaks, while that of sample E1 and G1 includes more peaks. Here we \ndiscuss the origins and positions of the peaks in detail.  \nFollowing the previous study on MoSe 2/SiO 26, the peaks at the highest energy in each \nspectrum are a ttributed to the neutral excitons (X0), while the peaks located 30 meV lower than X0 \nare assigned as the charged excitons (trions, XT). Only in the spectrum of the perpendicular \nconfiguration of MoSe 2/(Mn, Fe)PSe 3 (sample E1, E2 and G1), we found an unknow n peak (XM) \nlocated at 10 ~ 15 meV lower than X0. This middle peak XM is assigned as neither trions nor neutral \nbiexcitons, because the binding energies from the X0 state are 30 meV both for positively and \nnegatively charged trions regardless of the doping  level6 and 20 meV for neutral biexcitons7. This  26 suggests that the XM is likely attributed to moiré excitons  affected by the moiré  potential at the \ninterface (shown in Fig. S2b)  as discovered in TMD -TMD heterostructures8-11. \nRegarding sample G1 (MoSe 2/FePS e3, perpendicular) in Fig. S3, the spectrum appears \ncomplicated and includes multiple peaks, some of which are absent in sample F1 (MoSe 2/FePSe 3, \nparallel). For peak extraction, we assumed here that the spectrum of G1 should be composed of X0 \nand XT which exist all other spectra (with the constant trion binding energy (35 meV)), and then \nfound to be decomposed into four peaks (X0, XT, and other two peaks) by fitting. One of them is \nlocated at XM of sample E1, so we also named it XM of G1 after that of E1. The last and lowest \npeak is unknown, but we assigned it as localized exciton (XL). XL has been frequently discussed \nin MoS 2 (or WSe 2) as a defect -mediated bound state.  \nThe position of each peak  can be explained by the lattice mismatch and the consequent \ninnate strain except the antiferromagnetic effect which is mainly discussed in the main text. If the \nlattice relaxation occurs at the present vdW interface, MPSe 3 induces the compressive strain t o the \nmonolayer MoSe 2 in the parallel configuration, while the lattice mismatch on the SiO 2 substrate \ncan be regarded as zero because the surface structure of SiO 2 is amorphous. Lattice reconstruction \nat the van der Waals interfaces has been directly repor ted in graphene/BN12 and \ngraphene/graphene13 systems although such reconstruction at vdW interfaces had been assumed to \nbe absent at the early stage of this field. Our fabrication procedure includes annealing at 250 ℃ \n(same condition as the report showing the reconstruction at the heterointerface12), thus we can \nexpect such lattice relaxation producing strain from the substrates.  \nSince the compressive strain enlarges the bandgap of TMDs14,15, this mechanism can \nexplain the substrate dependence of the X0 and XT positions in Fig. S3; the peak positions line up \nin order of the lattice mismatch, sample S1 < A1 < E1. According to previous studies14,15, the  27 compressive strain is deduced from the observed peaks as 0.5 ~ 1%. The value is less than the \nlattice misma tch probably because the lattice of the surface of the counterpart MPSe 3 can be \nexpanded. Sample E1 shows the peaks of X0 and XT at similar positions as sample S1 and thus \nimplies  negligibly small strain in MoSe 2, being consistent with the nearly incommens urable picture \nin the perpendicular configuration as shown in Fig. S1b. Note here that we assume the effect of \ndielectric environment can be negligible because the peak position of MoSe 2/MnPSe 3 (⊥) is not \nshifted as seen in Fig. S3 even though the dielectr ic constant of bottom MnPSe 3 is identical to \nMoSe 2/MnPSe 3 (//). \n \n§5. Temperature dependence of the X0 peaks in MoSe 2 on SiO 2 \nThe bandgaps of MoSe 2 and other TMDs are known to follow the temperature dependence \nof standard semiconductors as following16:  \n𝐸(𝑇)=𝐸0−𝑆ℏ𝜔[coth(ℏ𝜔\n2𝑘𝐵𝑇−1)]    (1) \nwhere 𝐸0 is the bandgap (exciton energy) at zero temperature, 𝑆 is dimensionless coupling \nconstant, and ℏ𝜔 is an average phonon energy. Figure S4 shows the temperature dependence of \nsample S1 (MoSe 2 on SiO 2) and the fitting curve by ( 1). We deduced  𝐸0 = 1.6514 eV, 𝑆 = 1.86, \nand ℏ𝜔 = 14 meV, which are similar to the previous report (1.657 eV, 1.96, 15 meV , respectively)6. \nIn the main text, we discuss the anomalous peak shifts in addition to this basic temperature \ndependence.  \n \n§6. Temperature dependence in other MoSe 2/MPSe 3 samples  \nTo check reproducibility of the additional shifts below TN, we fabricated and measured \nseveral samples in addition to sample A1, E1, F1, G1 and S1 in the main text. Figure S5 indicating  28 the data from sample A2, A3, E2, S2 totally agrees to the data in Fig.  4, supporting our results are \ntruly intrinsic in the system.  \n \n§7. Experimental data of WSe 2/MPSe 3 and MoS 2/MPSe 3 heterointerfaces  \nWe also conducted the experiments using TMDs other than MoSe 2 in order to check the \nuniversality of our results among group -VI TMDs. The data shown in Fig. S6 indicate that the \nphenomena observed in MoSe 2/MPSe 3 occur also in WSe 2/MPSe 3 and MoS 2/MnPSe 3 and their \nfeatures are similar quantitatively and qualitatively among TMDs.  It is not easy to discuss their \ndifferences among TMD s in detail due to the differences in their lattice constant/band \nstructure/spin -orbit interaction at present.  \nWe note that the extracted peaks from WSe 2 and MoS 2 can include the luminescence from \ncharged excitons and/or localized excitons due to their bro adness and complexity of PL peaks \nwhile PL peaks from MoSe 2 are sharp enough to separate them clearly into neutral and charged \nexcitons. Especially, further researches on the exciton -magnon interaction in charged excitons \ncould give us a tool to tune the i nteraction electrically.  \n \n§8. Electronic band structure of the heterostructure by DFT calculations  \nWe performed band structure calculations based on the density functional theory (DFT) for \nthe bilayer system composed of monolayer TMD/monolayer MnPSe 3. The unit cell used in the \ncalculations is shown in Figs. S7a and b, where we assumed the two Mn atoms are located beneath \nthe Mo and Se atoms, respectively, to model the nearly commensurate stacking in the case of the \nparallel configuration discussed in th e main text. We adopted MoS 2 as a monolayer TMD because \nit has a small lattice mismatch ( ~ 1.0%) with MnPSe 317; we fixed the in -plane lattice constant of  29 the unit cell to 6.390 Å and optimized the atomic positions in the unit cell.  In the calculations, we \nemployed the GGA+ U method taking U = 5 eV for the d orbitals of Mn, assuming the Néel -type \nantiferromagnetic order with the magnetic moments along the out -of-plane directions in the \nhoneycomb lattice formed by Mn sites.  \nThe Brillouin zone (BZ) for this sy stem is shown in Fig. S7c. The BZ of the superlattice \n(black rectangle in Fig. S7c) is one fourth of the original one of MoS 2 (orange rectangle), and \nhence, the high -symmetric points are defined in the folded positions (e.g., the K and M points \nindicated b y the green letters). Note that the new K point originates from the “ −K point” of the \noriginal BZ of MoS 2, and the new -K-M line does not correspond to “the original -K-M line” \n[e.g., the new -K line does not include the “Q  (or ) points” of the original MoS 2].  \nFigure S7d displays the band structure of the MoS 2/MnPSe 3 bilayer. For each electronic \nband, we indicate the contributions from the d orbitals of Mo and Mn separately by the colored \ncircles. Hereafter, all the origins of the energy are set  at the top of the valence bands of MoS 2. The \nbottoms/tops of the conduction/valence bands from MoS 2 are isolated from the ones of MnPSe 3. \nWe note that our results agree well with the previous results on the same system17.  \n \n§9. Electronic band structure f or different magnetic structures  \nWe conducted three kinds of calculations to clarify how the bandgap of MoS 2 is modified \nby changing the magnetism on the neighboring MnPSe 3 layer. Specifically, we controlled the \ndirection and amplitude of the magnetic mom ents, and type of the magnetic ordering of Mn2+.  \nFirst, Fig. S8 shows the band structures for different directions of the magnetic moments \nof the Néel order, m:  parallel to the c axis ( m//c) and vertical to the c axis ( m⊥c) in Figs. S8a \nand b, respectively. From the comparison of the total energies, the case with m//c is slightly more  30 stable than that with m⊥c, as in the ground state of a bilayer MnPSe 32. We find that the difference \nof the bandgaps of MoS 2 between the two cases is very small, less than 0.5 meV, as shown in Fig. \nS8c. This is one order of magnitude smaller than the PL spectral shift in the main text. We note \nthat in both cases, the conduction band edges of MoS 2 exhibit the Zeeman -type spin polarization \n(almos t aligned along the c -axis).  \nNext, in Fig. S9, we show the band structures with different amplitudes of the magnetic \nmoments, m=|m|, for the case with m//c. The results were obtained by the GGA+ U calculations \nwith the constraints on the magnetic moments18. The bandgaps of MoS 2 are hardly modified by \nthe change in the magnetic moments from 4.60 μB to 5.73 μB, as shown in Fig. S9b.  \nFinally, we compare the band structures for the Néel and zigzag -type antiferromagnetic \norder in Fig. S10. The calculations are done by assuming the double supercell to incorporate the \nzigzag -type order (Figs. S10a and b). Note that a similar zigzag order is realized in FePSe 3. Here, \nwe adopted not GGA+ U but GGA calculations because of the high computational cost for 40 atoms \nin the supercell. As shown in Fig. S10e, we find no significant difference in the bandgaps of MoS 2 \nfor the different types of antiferromagnetic orders. Note that the additional bands, whose band \nbottom is lower than 1.570 eV, predominantly originate from Mn d orbitals.  \nFrom these results, we conclude that the antiferromagnetic order in MnPSe 3 does not affect \nthe bandgap of MoS 2 significantly. The change of the bandgap for different antiferromagnetic \nstructures is too small to explain the upshift of the PL spectra in the main text. This is in stark \ncontrast to the substantial change in the bandgap for the ferromagnetic substrate suggested \ntheoretically19. The band alignment would be changed before and after the AFM transition as we \ncan see the change in MnPSe 3 bands in changing the magnetic states  (Fig. S8 -10). The band \nalignment, however, does not affect excitonic peak positions inside monolayer TMDs (which is  31 not interlayer  but intralayer  exciton) as far as the bandgaps of TMDs themselves are not altered \nas mentioned above.  \n \n§10. Strain effect due to magnetostriction in MnPSe 3 \n Magnetostriction of the antiferromagnets below TN can compress the lattice of TMDs, \nwhich may potentially explain the observed upshift through a strain effect. The observed shift (~ \n5 meV) requires roughly 0.1% compressive strain14,15. However, a previous study on bulk MnPSe 3 \nshowed that the magnetic strain  is, if any, one order of magnitude smaller (< 0.02%, below  their \nmeasurement  limit) across TN except the usual thermal expansion1. This suggests that the strain \ninduced by magnetostriction on MnPSe 3 cannot explain our experimental results.  \nTo confirm this , we performed the DFT calculations while changing the in -plane lattice \nconstant (Fig. S11). As shown in Fig. S11a, the bandgap of MoS 2 is changed linearly to the lattice \nconstant. Figure S11b shows the total energy comparison of the systems with different  lattice \nconstants for different U in the GGA+ U calculations. The result indicates that the larger electron \ninteraction leads to the larger lattice constant in the stable crystalline structure. Since U stabilizes \nthe Néel order, our results show that the a dditional strain from the Néel ordering is expected to be \ntensile, inducing the downshift opposite to the experimental data. Thus, the magnetic strain cannot \nrationalize our experimental results.  \n \n§11. Magnetic polaron effect at the magnetic van der Waals heterointerface  \nThe exciton binding energy in monolayer TMDs could be modified due to the magnetic \nordering in MnPSe 3. Magnetic -polaron picture is frequently used to describe the excitonic features  32 in magnetic semiconductors20. This picture, however, also results in the opposite peak shift since \nthe binding energy of excitons gets larger due to the magnetic polaron effect.  \nMeanwhile, two elementary excitations composing polaronic states can generally i nteract \neach other not only attractively (similarly to the bound magnetic polaron in magnetic \nsemiconductors) but also repulsively21. If there is strong repulsion between exciton ic (high energy)  \nand magnonic (low energy) states in our system, it can explai n the upshifts of the excitonic states. \nBecause this mechanism can include exciton -magnon coupling in a broad sense, we don’t mention \nit explicitly in the main text.  \nThe e ffects on the mass of exciton in TMDs from antiferromagnetic MnPSe 3 is also not \nlikely because the band edge of TMD, which determines the mass of exciton, is not so modulated \ndirectly by MnPSe 3 as discussed in § 7 and §8. \n \n§12. Interlayer exciton -magnon coupling at the heterointerfaces  \nIn the main text, we attribute the observed shift to th e effect of the interlayer exciton -\nmagnon interaction. According to previous studies in a single material, the form of Hamiltonian \ninducing the exciton -magnon interaction ( ℋEMC) under radiative electric field ( 𝐸⃗ ) can be written \nas \nℋEMC=(𝜋⃗ ∙𝐸⃗ )𝑎†𝑐†+ ℎ.𝑐.    (2) \nwhere 𝜋⃗  is coupling constant related to the exchange interaction between the ions, 𝑎† is the exciton \ncreation operator, and 𝑐† is the magnon creation operator22,23. This  original theory on exciton -\nmagnon state in d -d transition is based on the electric dipole moment generated by the combination \nof orbital excitation (exciton) at one sublattice and spin excitation (magnon) at the other sublattice. \nThe dipole moment makes the strong optical transition allowed and strong sideband peaks appear  33 below the TN 22,23. This theory is similar to the theory on two -magnon process (spin excitations at \nboth sublattices) and exciton -phonon coupling.  \nIn the interlayer EMC, naïvely thinking, we can regard 𝑎† as the exciton creation operator at \nthe TMD layer, 𝑐† as the magnon creation operator at the MnPSe 3 layer, and 𝜋⃗  as interlayer \ncoupling constant linked to the exchange coupling between TMDs and MnPSe 3. Note that the \nexchange c ouplings between TMDs and ferromagnetic surfaces were confirmed in the recent \nstudies as mentioned in the main text24,25.  \nHowever, it is challenging to construct an appropriate theory in more detailed.  First, although \nthe exciton -magnon states in d -d tran sitions (electrical -dipole forbidden/magnetic -dipole allowed) \nare quite well understood, those in charge transfer transition ( electrical -dipole allowed, same as \nexciton in TMDs) are much less investigated even in bulk as far as we know. It would be quite \nimportant to establish the detailed theory about exciton -magnon states in electrical -dipole allowed \ntransitions with originally large oscillator strength.  \nThe original formula (2) is originally made for an exciton and a magnon at localized d -orbitals \nin on e substance. In the heterointerface, however, the magnon and exciton exist in separated layers \nand are supposed to interact via interlayer exchange coupling. Moreover, the exciton in TMDs is \nWannier excitons which is not localized at the atomic sites and d istinct from Frenkel excitons in \nlocalized d -electron system. These situations can make it complicated and/or interesting to \nconstruct the microscopic theory of the interlayer exciton -magnon coupling.  \n As a result of (2), energy conservation leads to the s imple relation in the optical transition:  \n𝐸photon= 𝐸exciton±𝐸magnon .    (3) \nThe observed upshift in the luminescence can be explained by the case of the upper sign that a \nphoton is created while an exciton and a magnon are annihilated; however, the dow nshift  34 (corresponding to the lower sign, a photon and a magnon are created and an exciton is annihilated) \ncan be also observed in general. Although it is difficult to conclude the reason why we cannot \ndetect the downshift apparently, the upshift could domi nate under the sufficient population of \nthermally and/or optically excited magnons as reported in a bulk MnF 226.   \nIt would be also fascinating to discuss the magnetic field dependence to unveil detailed \nfeatures of the interlayer exciton -magnon coupling. In d-d transitions of Mn2+-based bulk materials, \nit is known that exciton -magnon peaks show almost no Zeeman splitting due to the cancellation \ncoming from the coincidence of optically and magnetically excited states inside Mn2+ orbitals27,28. \nIn our case, although there is no comprehensive theory, we can imagine that Zeeman shifts inside \nMn2+ orbitals can be observed adding to or subtracting from the usual Zeeman splitting in TMDs \nsince no exact cancellation would occur in our interlayer system now. Because  the Zeeman \nsplitting in TMDs dependents on the valley index (valley Zeeman shift), the g -factor of such a \nmagnon -coupled valley -exciton would be worth discussing from the theoretical and experimental \nviewpoints.  \nMoreover, it would be intriguing to measure such field dependence both in Faraday and Voigt \ngeometries because it would give us the exact easy -axis of the sublattice magnetic moments just \nat the interface of MnPSe 3 (which can be different from the bulk). This method can be a tool for \ndetecti ng the surface magnetic ordering of antiferromagnets at heterointerfaces.  \n   35 References:  \n(1) Flem, G. L., Brec, R., Ouvard, G., Louisy, A. & Segransan, P. Magnetic interactions in the \nlayer compounds MPX 3 (M = Mn, Fe, Ni; X = S, Se). J. Phys.: Chem. Solid  43, 455 -461 \n(1982).  \n(2) Wiedermann, A., Rossat -Mignod, J., Louisy, A., Brec, R. & Rouxel, J. Neutron diffraction \nstudy of the layered compounds MnPSe 3 and FePSe 3. 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Lett.  15, 1023 (1965).  \n(23) Imbusch, G. F., Luminescence from inorganic solids , 155 -174 (Plenum Press, 1978).  \n(24) Zhong, D. et al.  Van der Waals engineering of ferromagnetic semiconductor \nheterostructures for spin and valleytronics. Sci. Adv.  3, e1603113 (2017).   37 (25) Zhao, C. et al.  Enhanced valley splitting in monolayer WSe 2 due to magnetic exchange field. \nNat. Nanot echnol.  12, 757 -762 (2017).  \n(26) Tsuboi, T. & Ahmet, P.  Temperature dependence of the optical exciton -magnon absorption \nlines in MnF 2 crystals.  Phys. Rev. B  45, 468 -470 (1992).  \n(27) Sell, D. D. et al.  Magnetic Effects in the Optical Spectrum of MnF 2. J. Appl. Phys.  37, 1229 -\n1231 (1966).  \n(28) Gnatchenko, S. L., Kachur, I. S. & Piryatinskaya, V. G. Exciton -magnon structure of the \noptical absorption spectrum of antiferromagnetic MnPS 3. Low Temp. Phys.  37, 144–148 \n(2011).  \n(29) Cadiz, F. et al.  Excitonic Linewidth Approaching the Homogeneous Limit in MoS 2-Based \nvan der Waals Heterostructures.  Phys. Rev. X  7, 021026 (2017).   38  \nFig. S1. Stacking images.  Top views of MoSe 2/MnPSe 3 with different stacking angles: the zigzag -\nedge of MoSe 2 parallel to the zigzag -edge of honeycomb Mn2+ (same as sample A1, A2, and A3) \nand the zigzag -edge of MoSe 2 perpendicular to the zigzag -edge of Mn2+ (same as sample E1 and \nE2). Large and small square s are monolayer MnPSe 3 and MoSe 2 films, respectively. The color of \neach element is same as Fig. 1a except for the selenides in MoSe 2 (yellow here).  \n  \n 39  \nFig. \nS2. \nRaman peak around 222 cm-1 from MnPSe 3. a, b, Raman spectra around 220 cm-1 from MnPSe 3 \nat 6, 2 0, 30, 40, 50, 60, 70, 75, 80, 90, 100, 130, and 160 K. c, Temperature dependence of \nintensities and positions of the peaks in Fig S2b. Their behaviors change around TN (= 74 K) of \nthe bulk MnPSe 3. \n  \n 40  \nFig. S3. PL spectra of all MoSe 2/MPSe 3 interfaces.  PL spectra from monolayer MoSe 2 in sample \nF1 (purple), A1 (red), S1 (blue), E1 (green), and G1 (yellow) with the results of the multi -peak \nfittings by Voigt function. Grey and black lines show the fit traces and peaks for each spectrum, \nrespec tively.  \n  \n 41  \nFig. S4. Temperature dependence of PL peaks in MoSe 2 on SiO 2. The PL peaks of X0 of sample \nS1 are plotted as a function of temperature with the fitting by (1).  \n  \n 42  \nFig. S5. Temperature dependence in other MoSe 2/MnPSe 3 samples.  The PL peak shift s from \ndifferent MoSe 2 samples from Fig. 4d. Orange and Brown circles correspond to the data from \nsample A2 and A3 of MoSe 2 on MnPSe 3 with the parallel configuration. Green circles and blue \nrhombuses come from sample E2 of MoSe 2 on MnPSe 3 with the perpendi cular configuration and \nsample S2 on MoSe 2 on SiO 2, respectively.  \n \n  \n 43  \nFig. S6. Peak shifts in the heterointerfaces using WSe 2 and MoS 2. Summary of the shifts of PL \npeaks from a WSe 2 and b MoS 2. Red circles, purple squires and blue rhombuses from TMDs on \nMnPSe 3, FePSe 3 and SiO 2 respectively.  Note that the PL peaks from MoS 2 are much broader than \nothers29, which make it difficult to deduce the peak positions precisely by fitting.  \n \n  \n 44  \nFig. S7. Electronic band structure of MoS2/MnPSe 3. a, b,  Top and side views of the unit cell of \nthe MoS 2/MnPSe 3 bilayer system. c, 1st Brillouin zone (BZ) and high -symmetry points for the \nMoS 2 itself (orange rectangle and letters) and MoS 2/MnPSe 3 superlatt ice (black rectangle and \ngreen letters). The green lines indicate the symmetric lines used for the plots of the band structures \nin the following. Note that the , K, and M points are the high -symmetry points of the superlattice \nwritten in green letters in c hereafter. d, Electronic band structure of the bilayer system obtained \nby the GGA+ U calculations ( U = 5 eV). The radius of cyan and magenta circles indicates the \n 45 weight of the d orbitals of Mn and Mo, respectively (the Mn -components are multiplied by fou r \nfor clarity). We set the origin of the energy at the top of the valence band of MoS 2. \n   46  \nFig. S8. Magnetization direction dependence of electronic band structures. a, b,  Electronic \nband structures obtained by the GGA+ U calculations ( U = 5 eV) with Néel ordering of a m//c-axis \nand b m⊥c-axis. c, Enlarged figure of a and b around the conduction band bottom of MoS 2. The \ndifference of the bandgaps (< 0.5 meV) is one order of magnitude smaller than the upshift of the \nPL spectra observed in experiments (~5 meV).  \n  \n 47  \nFig. S9. Magnetic moment dependence of electronic band structures . a, Electronic band \nstructures obtained by the GGA+ U calculations  (U = 5 eV) with the constraints o n the magnetic \nmoments of Mn. The black (purple) line shows the results with the magnetic moment m = 4.60 \n(5.73) B along the c axis on each Mn site. b, Enlarged figure of a around the conduction band \nbottom of MoS 2. The deduced bandgaps of MoS 2 are almost  identical between the two cases.  \n  \n 48  \nFig. S10. Antiferromagnetic structure dependence of electronic band structures. a,  The \nrectangular double super cell including four Mn2+ sites (denoted by Mn1, 2, 3, and 4). b, 1st \nBrillouin zone (BZ) for MoS 2 (orange rectangle) and MoS 2/MnPSe 3 double supercell in a (black \n 49 rectangle). Here, we use the same , K, and M points as in Fig. S7c for comparison. Note that the \ngreen lines, which are used for the plots of the band struct ures in c and d, are not the symmetry \nlines for the BZ for this double supercell. c, d, Electronic band structures obtained by the GGA \ncalculations with c Néel order [Mn1 and 3 (Mn2 and 4) possess up (down) spin along the c axis] \nand d zigzag order [Mn1 an d 2 (Mn3 and 4) possess up (down) spin along the c axis]. The kinks \nat the K points are due to the fact that the -K-M line is not the symmetry line for the doubled \nsupercell. e, Enlarged figure around the conduction band bottom of MoS 2. The band structure s \nfrom MoS 2 are almost identical except for the additional bands from Mn d orbitals  in the case of \nzigzag order .   50  \nFig. S11. Lattice constant dependence of bandgap of TMD.  a, b, Bandgap of MoS 2 and total \nenergy as functions of the lattice constant. The purple, green, and black lines indicate the results \nby the GGA calculations for the bilayer MoS 2/MnPSe 3, the GGA+ U calculations ( U = 2 eV) for \nthe bilayer MoS 2/MnPSe 3, and the GGA calculations for  the monolayer MoS 2, respectively.  In b, \nthe total energy is measured from the lowest total energy for each case.  \n \n \n \n"    },    {        "title": "2006.01075v1.High_throughput_search_for_magnetic_and_topological_order_in_transition_metal_oxides.pdf",        "content": "High-throughput search for magnetic and topological order\nin transition metal oxides\nNathan C. Frey,1, 2Matthew K. Horton,2, 3Jason M. Munro,2Sinad\nM. Gri\u000en,4, 5, 6Kristin A. Persson,2, 3and Vivek B. Shenoy1,\u0003\n1Department of Materials Science & Engineering,\nUniversity of Pennsylvania, Philadelphia PA 19103\n2Energy Technologies Area, Lawrence Berkeley National Laboratory, Berkeley CA 94720\n3Department of Materials Science & Engineering,\nUniversity of California, Berkeley, Berkeley CA 94720\n4Department of Physics, University of California, Berkeley, California 94720, USA\n5Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA\n6Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA\n(Dated: June 2, 2020)\nThe discovery of intrinsic magnetic topological order in MnBi 2Te4has invigorated the search for\nmaterials with coexisting magnetic and topological phases. These multi-order quantum materials\nare expected to exhibit new topological phases that can be tuned with magnetic \felds, but the search\nfor such materials is stymied by di\u000eculties in predicting magnetic structure and stability. Here, we\ncompute over 27,000 unique magnetic orderings for over 3,000 transition metal oxides in the Materials\nProject database to determine their magnetic ground states and estimate their e\u000bective exchange\nparameters and critical temperatures. We perform a high-throughput band topology analysis of\ncentrosymmetric magnetic materials, calculate topological invariants, and identify 18 new candidate\nferromagnetic topological semimetals, axion insulators, and antiferromagnetic topological insulators.\nTo accelerate future e\u000borts, machine learning classi\fers are trained to predict both magnetic ground\nstates and magnetic topological order without requiring \frst-principles calculations.\nMaterials with coexisting quantum phases o\u000ber exciting\nopportunities for solid-state device applications and ex-\nploring new physics emerging from the interplay between\ne\u000bects including topology and magnetism [1]. Intrinsic\nmagnetic topological materials enable both explorations\nof fundamental condensed matter physics and next-\ngeneration technologies that rely on topological quantum\nstates. Signi\fcant progress has been made on classifying\nand discovering topological materials [2{5]. Separate ef-\nforts have recently enabled high-throughput classi\fcation\nof magnetic behavior [6, 7]. However, both the theoreti-\ncal design and experimental realization of magnetic topo-\nlogical materials are confounded by the inherent di\u000ecul-\nties in predicting and controlling magnetic order, often\narising from strong electron correlations [8{10]. At the\nintersection of these two quantum orders the focus has\nmainly been on taking prototypical topological materi-\nals like (Bi ;Sb) 2Te3and introducing magnetic dopants\n[11, 12]. This doping approach has a number of draw-\nbacks, namely that dopants are hard to control [13] and\nthe critical temperature for observing exotic physics is\nlow (below 2 K) [14]. Important challenges remain in as-\nsessing the stability (synthetic accessibility) of proposed\nmaterials [15] and coupling topological property predic-\ntion to magnetism.\nRecently, there has been a surge of interest in the\n\feld due to the experimental realization of intrinsic mag-\nnetic topological phases in the van der Waals material\nMnBi 2Te4[5, 16, 17]. The MnBi 2Te4family o\u000bers the\n\u0003Corresponding author email: vshenoy@seas.upenn.edu\frst opportunity to possibly access the quantum anoma-\nlous Hall (QAH) phase, topological axion states, and Ma-\njorana fermions in a single materials platform by control-\nling the interplay between the magnetic and topological\norders [14]. This demonstration of a true magnetic topo-\nlogical quantum material (MTQM) opens new avenues of\nresearch into the modeling, discovery, and characteriza-\ntion of magnetic topological materials. Many opportuni-\nties remain; in particular, the realization of new magnetic\ntopological phases [10] and robust order in ambient con-\nditions.\nIn this work, we develop and apply work\rows to auto-\nmate the calculation of magnetic exchange parameters,\ncritical temperatures, and topological invariants to en-\nable high-throughput discovery of MTQMs. Building\non previous work to determine magnetic ground states\nwith density functional theory (DFT) calculations [7], we\napply the work\row to a subset of over 3,000 transition\nmetal oxides (TMOs) in the Materials Project database\n[18]. We focus on TMOs because of the large range\nof tunability that has already been demonstrated, both\nthrough ab initio calculations and molecular beam epi-\ntaxial growth of single phase and heterostructured oxide\ncompounds, and the tantalizing possibility of incorporat-\ning them with oxide electronics. In fact, several previous\nworks have identi\fed potential topological oxide candi-\ndates on a case-by-case basis [19{21], though few have\nbeen successfully synthesized or measured to be topo-\nlogical [15]. The work\row is used to identify candidate\nferromagnetic topological semimetals (FMTSMs), axion\ninsulators, and antiferromagnetic topological insulators\n(AFTIs), as well as layered magnetic and topological ma-arXiv:2006.01075v1  [cond-mat.mtrl-sci]  1 Jun 20202\nterials. Moreover, the computed magnetic orderings and\na recently published data set of predicted magnetic topo-\nlogical materials [10] are used to train machine learning\n(ML) classi\fers to predict magnetic ground states and\nmagnetic topological order, which may be used to accel-\nerate exploration of the remaining 31,000+ magnetic ma-\nterials in the Materials Project. Future work will extend\nthis modular work\row to explore diverse phenomena in-\ncluding other ferroic orders, exotic topological phases,\nand new materials systems.\nResults\nCalculating magnetic and topological order. The\nwork\row presented here is graphically summarized in\nFig. 1. For any candidate magnetic material, the method\npreviously developed by some of the coauthors [7] is used\nto generate likely collinear magnetic con\fgurations based\non symmetry considerations. Exhaustive DFT calcula-\ntions are performed to compute the energies of each mag-\nnetic ordering and determine the ground state. Alter-\nnatively, the machine learning classi\fer discussed below\ncan be used to predict the ground state ordering based\nsolely on structural and elemental data to accelerate the\nground state classi\fcation. The low energy orderings are\nthen mapped to the Heisenberg model for classical spins,\n~S:\nH=\u0000X\ni;jJij~Si\u0001~Sj: (1)\nSolving the resulting system of equations yields the\nexchange parameters, Jij. The computed exchange pa-\nrameters and magnetic moments provide all the neces-\nsary inputs to obtain the critical temperature through\nMonte Carlo simulations. The crystal is represented as\na structure graph (using the NetworkX package [22])\nwhere nodes represent atoms and edges represent ex-\nchange interactions. The entire analysis has been im-\nplemented in the pymatgen [23] code and an automated\nmagnetism work\row is available in atomate [24]. Monte\nCarlo calculations are enabled in the work\row by inter-\nfacing with the VAMPIRE atomistic simulations package\n[25]. It should be noted that this method is only appli-\ncable for systems that are well described by the classical\nHeisenberg model, that is, systems with localized mag-\nnetic moments and reasonably high Curie or Nel tem-\nperatures (TC=N>30 K), such that quantum e\u000bects can\nbe neglected.\nThe second branch of the work\row diagnoses band\ntopology. Topological invariants are determined using\nthevasp2trace [4] andirvsp [26] codes to compute ir-\nreducible representations of electronic states, as well as\nthe hybrid Wannier function method in Z2Pack [27].\nAutomated work\rows to calculate topological invariants\nare implemented in the Python Topological Materials\n(pytopomat) code [28]. By coordinating work\rows, we\nare able to discover materials with coexisting quantum\nFIG. 1. Work\row diagram for high-throughput computation\nof magnetic ordering, exchange parameters, and topological\ninvariants.\norders, like magnetic topological insulators, in a high-\nthroughput context. The schematic in Fig. 1 shows one\nsuch example: a magnetic system exhibiting the quan-\ntum anomalous Hall e\u000bect.\nTransition metal oxide database. We restrict our\nsearch to the family of transition metal oxides (TMOs),\nwhich has the advantages of encompassing thousands of\ncandidate magnetic materials and having standardized\nHubbard Uvalues based on experimental enthalpies of\nformation [29]. A subset of 3,153 TMOs were consid-\nered, encompassing over 27,000 computed magnetic or-\nderings, with any combination of Ti, V, Cr, Mn, Fe, Co,\nNi, Cu, O, and any other non- f-block elements. Impor-\ntantly, only stable and metastable phases within 200 meV\nof the convex hull are included in our database. For each\nTMO, up to 16 likely magnetic orderings were generated,\nyielding a total of 923 ferromagnetic (FM) and 2,230\nantiferromagnetic (AFM) ground states. For simplicity,\nferrimagnetic (FiM) ground states were called AFM if\nthey have an anti-parallel spin con\fguration with a net\nmagnetic moment less than 0.1 \u0016B/cell, and FM if the\nnet magnetic moment in the system is greater than 0.1\n\u0016B/cell.\nA statistical summary of the data set is presented in\nFig. 2. Fig. 2(a) is a histogram of the crystal systems\ncontained in the data. All seven crystal systems are rep-\nresented, with monoclinic being the most prevalent and3\nFIG. 2. Survey of magnetic transition metal oxides in the\ndatabase. (a) Histogram of crystal systems. (b) Maximum\nmagnetic moments in each system. Clustering is observed\naround integer values. (c) Average nearest neighbor distance\nbetween magnetic ions. (d) Occurrence of 3 dblock transition\nmetal atoms across FM and AFM systems.\nhexagonal the least. Similarly, there are a variety of space\ngroups, compositions, and symmetries present in the ma-\nterials considered. There are compounds with one, two,\nthree, or more magnetic sublattices. As expected for\nTMOs, most compounds have an average coordination\nnumber of four or six. Considering the computed ground\nstates, there is a large range of maximum magnetic mo-\nments per atom, with clustering observed around the in-\nteger values of 1, 2, 3, and 4 \u0016B/atom (Fig. 2(b)). The\nhistogram of average nearest neighbor distance between\ntwo TM atoms in a compound is shown in Fig. 2(c).\nThere is again a large range of values, from 2 to 6 \u0017A,\nwith a peak around 3 \u0017A. We also show the relative oc-\ncurrence of 3 dblock transition metal species across FM\nand AFM compounds in Fig. 2(d). Mn is the most com-\nmon transition metal, occurring in 1016 compounds in\nthe database, while Cu is the least prevalent, occurring\nin fewer than 100 compounds.\nA wealth of information is available in the computed\nhigher energy orderings as well. We de\fne the energy\ngap, \u0001E=E0\u0000E1, whereE0is the ground state energy\nandE1is the energy of the \frst excited state. When the\nlow-energy orderings are successfully found, \u0001 Equanti-\n\fes the robustness of the ground state ordering. The plot\nof \u0001Ein Fig. 3(a) shows the heavy-tailed distribution of\nthe energy gaps. Over 600 compounds exhibit \u0001 E < 0:5\nmeV/unit cell and may have correspondingly small JandTC=Nvalues. An e\u000bective Jparameter can be estimated\nfrom the energy gap as jJeffj= \u0001E=(NS2), whereNis\nthe number of magnetic atoms in the unit cell and Sis\nthe magnitude of the average magnetic moment. From\nthis crude estimate, the transition temperature is given in\nthe mean \feld approximation as TMFT\nC=N= 2Jeff=(3kB),\nwherekBis Boltzmann's constant. The plot in Fig. 3(b)\nshows theJeffvalues, more or less clustered by the inte-\nger values of the maximum magnetic moment (indicated\nby the light red ovals) in each material. One represen-\ntative high Jeffmaterial is La 2NiO 4(41.7 meV), which\nhas an AFM ground state and an estimated TNof 323 K\n(measured value of 335 K [30]).\nFIG. 3. Low-energy ordering and e\u000bective exchange inter-\nactions. (a) Energy splitting between ground state and \frst\nexcited state. (b) Jeffversus maximum magnetic moment in\nthe unit cell.\nWe brie\ry highlight some promising material can-\ndidates that may be reduced into a two-dimensional\n(2D) form [31] with possible access to intriguing low-\ndimensional magnetic properties [32, 33], and materi-\nals with strong spin-orbit coupling (SOC). We apply\nthe method from Ref. [34] to identify potentially lay-\nered TMOs, which yields 105 candidates. There are also\n66 TMOs that contain either Bi or Hg and are there-\nfore expected to exhibit strong SOC. At the intersection,\nwe \fnd three Bi-containing layered magnetic materials,\nBa2Mn 2Bi2O;CoBiO 3and CrBiO 4. These three materi-\nals may be of particular interest as tunable 2D magnets\nwith strong SOC-induced magnetic anisotropy [35].\nThe primary computational burden in generating this\ndata set is calculating the relaxed geometries and energies\nof all likely magnetic orderings, as we have no a priori\nway of determining the magnetic ground states. Fur-\nther, it is useful to compute the spectrum of low-energy\nmagnetic orderings to estimate the strength of exchange\ncouplings, thereby determining the nature of magnetic\ninteractions and critical temperatures. For simple com-\npounds with small unit cells and a single type of mag-\nnetic ion, it is relatively easy to determine the ground\nstate and only a few ( <4) orderings need to be com-\nputed. However, there is a long tail to the number of\norderings required for complex structures that may have\nmany highly symmetric AFM orderings [7]. For the TMO\ndata set, nine orderings per compound are required, on4\naverage, to \fnd the ground state. Due to the computa-\ntional cost, we are limited to collinear magnetic orderings\nin this combinatorial approach, although this work is an\nimportant \frst step towards determining the noncollinear\norderings. It is highly desirable to augment these labori-\nous DFT calculations with computationally inexpensive,\nphysics-informed models that can predict magnetic be-\nhavior.\nMagnetic ordering machine learning classi\fer.\nThe size of the TMO data set and the number of easily\navailable, physically relevant descriptors suggests that a\nphysics-informed machine learning classi\fer may be able\nto predict magnetic ground states. Our goal is to use\nfeatures based purely on structural and compositional\ninformation, without any DFT calculations , to predict\nmagnetic orderings and prioritize calculations. With the\nmatminer [36] package, we have access to thousands of\ndescriptors that are potentially correlated with magnetic\nordering. Drawing on physical and chemical intuition,\nthis list was reduced to \u0018100 descriptors that are likely\nindicators of magnetic ordering, e.g. elemental dorbital\n\flling, electronegativity, and tabulated atomic magnetic\nmoments. We also generate additional features more spe-\nci\fc to magnetic compounds, including the average near-\nest neighbor distance between TM atoms, TM-O-TM\nbond angle information, TM atom coordination num-\nber, and the number of magnetic sublattices. We have\nimplemented these features in the `magnetism' module\nofpymatgen . Unsurprisingly, no features have Pearson\ncorrelation coe\u000ecients larger than 0.3 with respect to\nground state ordering. There are no features with strong\nenough linear correlation to reliably predict magnetic be-\nhavior. To further reduce the feature space, we train a\nminimal model and discard features with extremely low\nimpurity importance and then perform hierarchical clus-\ntering based on the Spearman rank correlation, removing\na feature from each cluster.\nNext, using the reduced feature set, we construct an\nensemble of machine learning classi\fers to predict the\nmagnetic behavior. For simplicity and interpretability,\na random forest classi\fer [37] was used, although other\ntechniques like Adaptive Boosting and Extra Trees per-\nform similarly. The random forest is an ensemble of\ndecision trees made up of \\leaves\" like the one shown\nschematically in Fig. 4(a). For each feature, the tree\nsplits the data set to enable classi\fcation. In the illus-\ntration in Fig. 4(a), the simpli\fed split illustrates that\nsamples with more than one magnetic sublattice are more\nlikely to be AFM than FM. To capture the complexity\nof the data, a full decision tree is more \reshed out, like\nthe one shown in Fig. 4(b). The random forest is an\nensemble of many such trees, where the predictions of\nuncorrelated trees are averaged over to reduce over\ft-\nting. 10% of the data was held as a test set, and \fve-fold\ncross-validation was used to tune the model hyperparam-\neters. Because of the class imbalance between FM and\nAFM ground states (30% of compounds are FM), the FMcompounds are synthetically oversampled using SMOTE\n[38]. This leads to good performance in \fve-fold cross-\nvalidation, as seen in the mean and median F1scores of\n0.85 for both FM and AFM classes (see the Supplemental\nMaterial (SM) for details). The trained model achieved\nanF1score of 0.85 (0.59) for AFM (FM) compounds on\nthe test set, suggesting that the synthetic oversampling\nresults in di\u000eculties generalizing to new FM compounds,\nwhile AFM systems are well characterized.\nFIG. 4. Random forest classi\fers for magnetic ground state\nprediction. (a) An example of a \\leaf\" in the decision tree.\n(b) Graphic representation of a decision tree in the random\nforest.\nThe success of the classi\fer allows us to reexamine\nthe input features and use the model feature impor-\ntances to identify nontrivial predictors of magnetic be-\nhavior. Whereas the Gini impurity-based feature impor-\ntance somewhat misleadingly shows equal contributions\nfrom many features, here we use the permutation im-\nportance, which avoids bias towards numerical and high\ncardinality features [37]. The most important features for\nclassi\fcation are shown in Fig. S1. By far the most im-\nportant descriptor is the number of magnetic sublattices.\nOther features relate to space group symmetry, delectron\ncounts, coordination number, and distances between TM\natoms; features we expect to describe magnetism. An-\nother important descriptor is the sine Coulomb matrix,\nwhich is a vectorized representation of the crystal struc-\nture that has been introduced and used in previous stud-\nies to predict atomization energies of organic molecules\n[39] and formation energies of crystals [40]. Finally, the\nstructural complexity [41] is observed to be 47% higher\non average for AFM compounds than FM. The AFM sys-\ntems exhibit an average structural complexity that is 17\nbits/unit cell higher than the FM systems. This sim-\nple metric might indicate that more structurally com-\nplex materials are more likely to favor the more complex\nAFM orderings, rather than simple FM con\fgurations.\nThis could be related to the most important descriptor,\nwhich is also a metric of magnetic lattice complexity.\nSurprisingly, these complexity metrics along with sim-\nple TM-TM atom distances and the sine Coulomb ma-\ntrix are much better predictors of magnetic ordering than\nbond angle information, as might be expected from the\nGoodenough-Kanamori rules. It is possible that more\nsophisticated features may do a better job at capturing\nthe superexchange mechanisms that govern the magnetic\nbehavior.5\nIt is clear that models like the one presented here are\nnot guaranteed to generalize beyond the material types\nthat comprise the training data [42]. However, we ex-\npect that the physical insights related to feature engi-\nneering, as well as the tested methods, will be of use in\nfuture studies. Further di\u000eculties will be encountered\nwhen constructing machine learning models for critical\ntemperature prediction, which is inherently a problem of\noutlier-detection. Fortunately, this work provides both a\nset of promising materials to consider for further study\nand the framework to automate evaluation of exchange\nparameters and critical temperatures.\nMagnetic topological material identi\fcation. Fi-\nnally, we discuss the search for nontrivial band topology\nin the magnetic TMOs. The zoo of available topologi-\ncal order is ever expanding; here, we simplify our search\nby considering classes of centrosymmetric magnetic topo-\nlogical materials that can be readily classi\fed with high-\nthroughput calculations of topological indices. We con-\nsider antiferromagnetic topological insulators (AFTIs),\nferromagnetic topological semimetals (FMTSMs), and\nferromagnetic axion insulators. In the \frst case, we\nconsider materials that exhibit an AFM ground state\nthat breaks both time-reversal (\u0002) and a primitive-lattice\ntranslational symmetry ( T1=2), but is invariant under the\ncombinationS= \u0002T1=2. The preservedSsymmetry al-\nlows for the de\fnition of a Z2topological invariant [43]\nthat lends itself to high-throughput evaluation. For fer-\nromagnets, we consider FM ground states that break \u0002\nsymmetry but preserve inversion symmetry ( I), because\ntheir band topology can be determined by the parity\neigenvalues of occupied bands at the eight time-reversal\ninvariant momenta (TRIM) in the Brillouin zone (BZ)\n(Fig. 5(a,b)). Speci\fcally, we restrict our search to fer-\nromagnets with centrosymmetric tetragonal structures,\nwhere ideal Weyl semimetal (WSM) features may ap-\npear and where the magnetization direction can tune\nthe band topology [44]. These \flters greatly simplify\nthe screening, but recent work suggests that over 30% of\nnon-magnetic [4, 45] and magnetic [10] materials exhibit\nnontrivial topology, so there are almost certainly many\nmore interesting MTQMs to uncover in the TMO data\nset than we have considered here.\nWe classify potential Z2phases in AFM systems by the\nset of indices\nZ2= (v;vxvyvz); (2a)\nv= \u0001(ki= 0) + \u0001(ki= 1=2) mod 2; (2b)\nvi= \u0001(ki= 1=2); (2c)\nwhere \u0001(ki) is the 2D topological invariant on the time-\nreversal invariant plane kiin the BZ, and kiis in reduced\ncoordinates. If v= 1, the system is a strong topological\ninsulator, while a system with v= 0 andvi= 1 for any\nviis a weak topological insulator [27, 46]. FMTSMs are\nclassi\fed by the strong topological index Z4in terms ofparity eigenvalues [8, 10, 47], de\fned as\nZ4=8X\n\u000b=1noccX\nn=11 +\u0018n(\u0003\u000b)\n2mod 4; (3)\nwhere \u0003\u000bare the eight TRIM points, nis the band in-\ndex,noccis the number of occupied bands, and \u0018n(\u0003\u000b)\nis the parity eigenvalue ( \u00061) of then-th band at \u0003 \u000b.\nZ4= 1;3 indicates a WSM phase with an odd number of\nWeyl points in half of the BZ, while Z4= 2 indicates an\naxion insulator phase with a quantized topological mag-\nnetoelectric response [48], or a WSM phase with an even\nnumber of Weyl points. Z4= 0 corresponds to a topo-\nlogically trivial phase.\nTABLE I. Candidate tetragonal ferromagnetic topological\nsemimetals and axion insulators. Theoretical materials that\nhave not yet been experimentally synthesized are labeled with\nay.\nMaterial Space\ngroupMaterials\nProject\nIDEnergy\nabove hull\n(meV/atom)Z4\nCuFe 2O4 I41=amd N/A N/A 1\nCrO 2 P42=mnm mp-19177 63 3\nSr3CaFe 4Oy\n12 P4=mmm mp-1076424 14 3\nMn3O4Fy\n2 P42=mnm mp-780777 76 2\nSr2La2Mn4O11I4=mmm mp-1218776 65 2\nMn2POy\n5 I41=amd mp-754106 27 1\nSr5Mn5O13 P4=m mp-603888 2 2\nCaV 2Oy\n4 I41=amd mvc-10887 31 2\nCdNi 2Oy\n4 I41=amd mp-756341 0 2\nCr2TeO 6 P42=mnm mp-21355 0 2\nCuCr 2O4 I41=amd mp-1103973 13 1\nLiNiOy\n2 I41=amd mp-770635 17 2\nVMg 2Oy\n4 I41=amd N/A N/A 2\nThe TMO database was screened for materials with\nFM ground states, a tetragonal crystal structure, and\ninversion symmetry, resulting in 27 candidates. By com-\nputing the Z4indices for these materials, we identify\neight materials with Z4= 2, indicating either a WSM\nphase with an even number of Weyl points in half of the\nBZ, or an axion insulator phase. Five materials have\nan odd number of Weyl points in half of the BZ, with\nZ4= 1;3. The candidate FMTSMs and axion insula-\ntors and their respective Z4indices are listed in Table\nI. We also give the unique identi\fers for the Materi-\nals Project database entries and the calculated energy\nabove the convex hull. Here, we highlight the candi-\ndate FMTSM CuCr 2O4(Fig. 5c). CuCr 2O4has an FM\nground state and Z4= 1. CuCr 2O4is a hausmannite-\nlike spinel structure with the tetragonal I41=amd space\ngroup. Cr atoms bond with O atoms to form CrO 6oc-\ntahedra that share corners with CuO 4tetrahedra. Cr3+\natoms occupy Wycko\u000b position 8 d, Cu2+occupy Wyck-\no\u000b 4a, and O2\u0000occupy Wycko\u000b 16 h. We also draw spe-\ncial attention to the spinel CdNi 2O4 (Fig. 5f), which6\nFIG. 5. Magnetic topological materials. (a) Time-reversal invariant momenta in the Brillouin zone. (b) Schematic of parity\neigenvalues of occupied bands at TRIM points. (c) The candidate ferromagnetic topological semimetal, spinel CuCr 2O4. (d)\nSchematic of a Dirac cone in an antiferromagnetic topological insulator with Ssymmetry. (e) Schematic of Weyl cones in\na ferromagnetic topological semimetal without time-reversal and with inversion symmetry. (f) The candidate ferromagnetic\naxion insulator, spinel CdNi 2O4.\nis predicted to be an FM axion insulator with Z4= 2\nand a bandgap Ebg= 0:125 eV. This material has not\nyet been successfully synthesized and represents one of\nmany promising opportunities to grow new magnetic ox-\nides and investigate their topology.\nTABLE II. Candidate antiferromagnetic topological insula-\ntors. Theoretical materials that have not yet been experi-\nmentally synthesized are labeled with a y.\nMaterial Space\ngroupMaterials\nProject\nIDEnergy\nabove hull\n(meV/atom)Z2\nFeMoClO 4P4=nmm mp-23123 6 (1;100)\nMnMoO 4 P2=c mp-19455 5 (1;001)\nCa2MnOy\n3I4=mmm mp-1227324 27 (1;000)\nSrV 3O7 Pmmn mp-510725 3 (1;010)\nLi2TiVOy\n4 P2=m N/A N/A (1;001)\nPotential AFTIs were identi\fed by screening the TMOdatabase for AFM ground states with Ssymmetry, yield-\ning 298 candidate materials. Of these, 46 are predicted to\nbe layered antiferromagnets by at least one of the meth-\nods in Refs. [34, 49, 50]. These layered systems are of\nspecial interest due to their unique and tunable topologi-\ncal and magnetic properties [16, 32]. Eight additional an-\ntiferromagnets with Ssymmetry exhibit small bandgaps\n(<0.5 eV) and are therefore likely candidates to exhibit\nband inversion. For each of these 54 materials, the Z2\ninvariant is calculated using the hybrid Wannier func-\ntion method in Z2Pack . Four layered AFTIs were iden-\nti\fed: FeMoClO 4, MnMoO 4, Ca 2MnO 3, and SrV 3O7.\nOne small bandgap AFTI was also discovered: mono-\nclinic Li 2TiVO 4in aP2=mphase. These systems and\ntheirZ2indices are listed in Table II. We highlight the\ntetragonalI4=mmm phase of Ca 2MnO 3(Fig. 6a), which\nhas a nontrivial Z2= (1; 000). It is a caswellsilverite-like\nstructure in which Ca2+ions are bonded with O atoms\nto form CaO 6octahedra and Mn2+ions bond to form\nMnO 6octahedra [51]. In the primitive cell, Ca atoms7\nFIG. 6. The candidate antiferromagnetic topological insu-\nlator, Ca 2MnO 3. (a) Crystal structure of Ca 2MnO 3in the\ntetragonal I4=mmm phase. (b) Phase diagram of bandgap\nversus the Hubbard Uvalue for Mn showing the dependence\nof the band topology on the strength of Hubbard interactions.\noccupy Wycko\u000b position 4 e, Mn occupies Wycko\u000b 2 a,\nand the O atoms occupy Wycko\u000b 2 band 4e. Because the\ntopology of the AFTI phase is sensitive to the nature of\nthe bandgap and the strength of electron correlations, we\nplot a phase diagram (Fig. 6b) for Ca 2MnO 3indicating\nthe regions where the system is a strong AFTI or a triv-\nial insulator. We \fnd that the material is a strong AFTI\nunder a wide range of Hubbard Uvalues, although it is\npredicted to be topologically trivial at U= 4 eV and for\nU > 6 eV. Future work will identify the origin of this\ncorrelation-dependent change in topological order.\nImportantly, none of the identi\fed candidate MTQMs\nwere considered in previous e\u000borts to screen the Mate-\nrials Project for topological materials [4], because the\ncorrect magnetic orderings were not available [7]. We\nhave also highlighted theoretical materials, unique to the\nMaterials Project database, that have not yet been ex-\nperimentally synthesized and do not have experimental\nstructures reported in the Inorganic Crystal Structure\nDatabase (ICSD) [52]. Theoretical materials are labeled\nwith ayin Tables I and II. Three materials (CuFe 2O4,\nVMg 2O4, and Li 2TiVO 4) relaxed into new phases not\npreviously included in the Materials Project database af-\nter determining the magnetic ground states. Notably,\nall MTQM candidates are within 100 meV per atom of\nthe convex hull, indicating that all candidate materials\nare thermodynamically stable or metastable and may be\nsynthesizable [53]. Additional details on MTQM candi-\ndates with ICSD entries and comparisons to experimental\nmeasurements of magnetic ordering are given in the SM.\nWe have extended the machine learning approach dis-\ncussed above to classify magnetic topological materials\nfrom a recently published data set [10] of 403 magnetic\nstructures containing 130 magnetic topological materi-\nals. The random forest model achieves a 0.74 F1score\non topological material classi\fcation in \fve-fold cross-\nvalidation, using primarily symmetry- and orbital-based\ndescriptors, requiring no calculations. The details are\npresented in the SM.Due to the modularity and interoperability of the\nwork\rows developed and applied here, it is straight-\nforward to extend the search to other types of quan-\ntum orders. Here, we have provided a high-throughput,\nrelatively coarse-grained method to identify promising\nMTQMs. The topological structure can be sensitive to\nthe Hubbard Uparameter value, noncollinear magnetic\norder and the resulting magnetic space group (MSG) de-\ntermination, and how the strength of SOC compares to\nthe bandgap. Future work will involve detailed stud-\nies of candidate materials with the recently introduced\nMagnetic Topological Quantum Chemistry (MTQC) [10]\nformalism, better exchange-correlation functionals (e.g.\nmeta-GGAs like SCAN [54]) to more accurately compute\nbandgaps, and careful determination of Uvalues with the\nlinear response approach [55].\nDiscussion\nWe have developed and applied a high-throughput com-\nputational work\row to determine magnetic exchange\ncouplings, critical temperatures, and topological invari-\nants of electronic band structures in magnetic materi-\nals. By studying over 3,000 transition metal oxides span-\nning all crystal systems, nearly all space groups, and a\nwide range of compositions, we have produced a data\nset of materials rich in magnetic and topological physics.\nThis enabled the training of a machine learning classi-\n\fer to predict magnetic ground states and give insight\ninto structural and chemical factors that contribute to\nmagnetic ordering. We extended this machine learn-\ning approach to classify topological order in magnetic\nmaterials from a recently published data set using only\nsymmetry- and orbital-based descriptors. We identi\fed\n\fve promising candidate antiferromagnetic topological\ninsulators ( e.gtetragonal Ca 2MnO 3), including four lay-\nered materials, as well as 13 candidate ferromagnetic\ntopological semimetals (spinel CuCr 2O4) and axion in-\nsulators (spinel CdNi 2O4).\nMethods\nDFT calculations were performed with the Vienna Ab\nInitio Simulation Package (VASP) [56, 57] and the PBE\nexchange-correlation functional [58]. Standard Materi-\nals Project input settings and Hubbard Uvalues were\nused, as described in [7]. Speci\fcally, values were set\nfor elements Co (3.32 eV), Cr (3.7 eV), Fe (5.3 eV),\nMn (3.9 eV), Ni (6.2 eV), and V (3.25 eV) with the\nrotationally invariant Hubbard correction. These val-\nues were determined by \ftting to known binary forma-\ntion enthalpies in transition metal oxides [29]. Main-\ntaining consistent Uvalues with the Materials Project\nallows for high-throughput calculations and integration\nwithin the GGA/GGA+U mixing scheme that enables\nthe construction of phase diagrams. These Uvalues\nand the magnetic ordering work\row were shown to cor-\nrectly predict non-ferromagnetic ground states in 95% of\n64 benchmark materials with experimentally determined\nnontrivial magnetic order [7]. However, it is known that8\ntopological phase diagrams for magnetic materials can be\nstrongly dependent on the strength of Hubbard interac-\ntions [10]. Machine learning models were implemented\nwithscikit\u0000learn [59].\nAcknowledgements. N.C.F. was supported by the\nDepartment of Defense through the National Defense\nScience & Engineering Graduate Fellowship program.\nV.B.S. acknowledges support from grants EFMA-542879\nand CMMI-1727717 from the U.S. National Science\nFoundation and also W911NF-16-1-0447 from the Army\nResearch O\u000ece. S.M.G. and integration with the Ma-\nterials Project infrastructure was supported by the U.S.\nDepartment of Energy, O\u000ece of Science, O\u000ece of BasicEnergy Sciences, Materials Sciences and Engineering Di-\nvision under Contract No. DE-AC02-05-CH11231 (Ma-\nterials Project program KC23MP). This research used re-\nsources of the National Energy Research Scienti\fc Com-\nputing Center (NERSC), a U.S. Department of Energy\nO\u000ece of Science User Facility operated under Contract\nNo. DE-AC02-05CH11231.\nData availability. The data used in this study\nare available at https://materialsproject.org. Ma-\nchine learning models, data used in model train-\ning, and an example Jupyter notebook are avail-\nable on GitHub at https://github.com/ncfrey/magnetic-\ntopological-materials.\n[1] Y. Tokura, K. Yasuda, and A. Tsukazaki, Nature Re-\nviews Physics 1, 126 (2019).\n[2] H. C. Po, A. Vishwanath, and H. Watanabe, Nature\nCommunications 8, 50 (2017).\n[3] B. Bradlyn, L. Elcoro, J. 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The magnetic properties of symmetrical and asymmetrical \nsized small fullerene dimers (C n for n≤50) have been investigated using spin polarized \ndensity functional theory. The interaction energies depict that small fullerene cages form \nstable dimer structures and symmetrical sized fullerene dimers are found more stable than \nasymmetrical sized dimers. The dimerization of fullerene cages in different  modes leads to \nchange in their magnetic properties. The non -magnetic fullerene cages become magnetic after \nformation of dimer (C 20-C20, C 24-C24, C 32-C32, C 40-C40, C 20-C24, C 40-C44 and C 44-C50), \nwhereas the magnetism of magnetic fullerenes is enhanced or lowered after dimerization \n(C28-C28 C36-C36, C 24-C28, C 28-C32, C 32-C36 and C 36-C40). The individual cages of dimer \nstructures show ferromagnetic interactions amongst them and resultant magnetic  moment \nstrongly depends on the type of inter -connecting bonds. The magnetism may also be \nexplained based on distortion of carbon cages and change in the density of states (DOS) in \ndimer configuration. The calculations presented show strong possibility of experimental \nsynthesis of small fullerene based magnetic dimers.  \nKeywords: Carbon nanostructures, Small fullerenes, Density functional theory, Magnetism. \n \n \n \n \n \n 3 \n 1. INTRODUCTION;  \nThe non -IPR (Isolated Pentagon Rule) fullerenes or small fullerenes (C n, n<60) are \ninteresting to study as they exhibit significant structural, electronic and magnetic properties, \nowing to their high curvature and fused adjacent pentagons [1 -5]. The fullerenes have been \nwidely studied in recent years and t hey have been explored for emerging potential applications \nin various areas of research such as nano -electronics, molecular devices, spin -electronics etc. \n[6-8]. The applications of these fullerenes in the field of chemical catalysis [9] and \npharmaceutics [10] have also become important by virtue of their particular properties like high \nchemical reactivity and small diameter.  \nIn past, carbon based systems have become increasingly interesting due to their \nsignificant magnetic properties and they can be consi dered as a possible magnetic materials \n[11]. The origin of carbon based ferromagnetism has been reported due to the dislocations, \nvacancies and impurity atoms [12]. Till date, various attempts have been made to study the \nmagnetic properties of small and la rger fullerene cages [13 -15]. Synthesis of ferromagnetic \npolymerized fullerenes has been treated by photo assisted oxidation, which show magnetization \nof order 10-3 µB per C 60 [16]. There is an introduction of strong magnetism in the fullerene cages \nwhen t hey are endohedrally doped [14, 15, 17]. When the transition metals (TM) are \nencapsulated inside small carbon cages, the magnetic behavior of small cages is altered, which \nfurnish a novel possibility to control the magnetic properties of carbon systems [17 ]. A \ntheoretical and experimental study on carbon clusters show that their magnetic moment can be \nsignificantly enhanced by appropriately choosing their size, geometry and composition [18].  \nApart from this, the fullerene cages also have ability to form dim er structures. The \nformation of C 60 dimer has been confirmed by mechanochemical synthesis experimentally using \nhigh speed vibration miling (HSVM) technique [19]. It is found that the stability of C 60 dimer is \ncomparable to that of two C 60 molecules and the  results are in agreement with theoretical 4 \n calculations [20]. The dimer structures of fullerene cages can also be obtained through \ncombination of either functionlized fullerenes or bifunctional cycloadditions to C 60 fullerene \ncages [21]. Thermal reactions of C 60/C60O/C 60O2 system lead to the formation of other dimeric \nfullerene derivatives such as C 120O2 and C 119 [22]. The sulfur containing fullerene derivative \nC120OS has been formed by thermolysis of C 120O in the presence of sulfur [23]. Scanning \ntunneling  microscopy (STM) and low -energy electron diffraction (LEED) studies on \npolymerized C 60 studies reveal that annealing of the electron -beam -modified surfaces restores \nthe fullerene lattice [24]. The heating of dimerized C 60 structures at high temperature re stores \nthe fullerenes to their pristine state [18, 24]. Some more experiments have been performed to \nproduce the carbon bridged dimers like C 121 and C 122, which could be used as the basic units of \nfullerene chain structures [25 -29]. \n The existence of C 60 dimer was also reported theoratically in metastable phases of MC 60 \n(M = K, Rb, Cs) [30] and the dimerization can occur in different phases like dumb -bell [31], \npeanut and capped nanotubes [32]. The conductance of the dimer can be tuned with doping as a \nresult of which it become more versatile in molecular electronics [33]. The magnetic properties \nof C 60 dimer indicate the presence of strong magnetic field at cage centers of the dimer and the \naddition of C -bridges change the behavior of magnetic field. The fu llerene dimers connected \nthrough BN hexagons alter the behavior of magnetic field inside the cages [34]. Ab -initio \ncalculations show that unpaired electrons of C 59N are delocalized over C 60 molecule in C 59N-\nC60 hetrodimer [33]. Nucleus Independent Chemical  Shift (NICS) and Nuclear Magnetic \nResonance (NMR) studies have reported the magnetic properties of (C 58BN) 2 dimer and the \nresults reveal that the dimerization of fullerene cages causes the major changes in magnetic \nproperties [35]. C 36 cage can also form strong inter cage bonds and may be a hexavalent \nbuilding block for fullerene compounds like dimers and polymers [36]. A computational study \n[37] on C 36 shows the dimerization of the cage, but the experimental existence of dimer is not 5 \n yet confirmed. The en dohedral derivatives of C 36 dimer are also expected to exist with their \nunique properties, making it useful for molecular devices [38]. However, no systematic study \non small fullerene dimers has been reported yet. Because of the limited study on dimer \nstructures, we are reporting for first time a systematic study on symmetrical and asymmetrical \nsized small fullerene dimers. The motivation for considering asymmetrical sized dimers comes \nfrom the experimental existence of carbon nanobuds, which are also a com bination of \nasymmetrical sized nanostructures [39].  \nIn the present work, we employed the first principle calculations on the dimers of small \nfullerene cages (C n, n≤50) based on density functional theory. Since C 60 dimer  has significant \nproperties, so it is interesting to study the interaction between small fullerene cages, which adds a \nnew dimension to dimer properties.  \n2. COMPUTATIONAL DETAILS ; \n All the calculations were performed using Spanish Initiative for electronic \nsimulation with thousands of atoms (SIESTA) computational code, which is based on density \nfunctional theory [40]. The Perdew, Burkey and Ernzerhof (PBE) functional combined with \ndouble -ζ polarized basis set were used for the geometry optimizations [41]. Klei nman – \nBylander form of non -local norm conserving pseudo potentials are used to describe core \nelectrons [42] while numerical pseudoatomic orbitals of the Sankey – Niklewski type [43] are \nused to represent the valence electrons . The energy shift parameter i s defined by range ≈ 150 \n– 350 meV to describe the size of pseudoatomic orbitals. The fineness of a finite grid is \ndefined by Mesh cut -off, whose value lie within the range 150Ry – 250Ry. In order to obtain \nground state properties, minimization of total en ergy of the system has been executed. The \nresidual forces of the system are relaxed up to 0.004 eV/Ang. The energy eigen values have \nbeen plotted in order to give density of states (DOS) spectra.  6 \n Test calculations were performed on small fullerenes (C n, n < 60) to check the \naccuracy of our results. The geometrical parameters for small fullerenes are calculated and \nshown in Table 1 and the results are in agreement with known experimental and theoretical \nresults [3, 44]. We found that our calculated values fo llow the same pattern and are in \nagreement with a recent study which employs tight binding method [45]. Since our group has \nalready study the carbon based systems so the parameters have been checked [17, 46 -47]. \nTable 1.  Average Diameter, Average Bond Dist ance and HOMO -LUMO gaps of \nsmall fullerene cages.  \nFullerene Cage  Dav (Å) Average Bond \nDistance ( Å) HOMO -LUMO \nGap (eV)  \nC20 4.14 1.48 0.75 \nC28 4.83 1.49 0.33 \nC32 5.25 1.46 1.41 \nC36 5.60 1.46 0.43 \n \n3. RESULTS AND DISCUSSION ; \n3.1  Structural Properties:  \nWe have optimized the ground state structures of symmetrical and asymmetrical sized \nsmall fullerene dimers using method described in computational details. The ground state \ngeometries of small fullerene cages are considered to form the dimer structures. In  dimeric \npattern, the small carbon cages can connect through four possible modes, i.e. (a) point -point \nmode forming a [1+1] dimer with C -C bond between two cages,  (b) side -side mode forming a \n[2+2] dimer with a 2 -fold bond, (c) face -face mode forming a [5 +5] dimer with a 5 -fold bond \nbetween two pentagonal rings, and  (d) face -face mode forming a [6+6] dimer with a 6 -fold \nbond between two hexagonal rings as shown in Fig.1. Each dimer structure with these four 7 \n \nconfigurations is optimized and the structure wh ich has minimum total energy is the most stable \nisomer.  \n \n \n \n \n(a)                                        (b)                                 (c)                                    (d)  \nFig. 1 . Four possible configurations through which fullerene cages connect  (a) [1+1] point -point \nmode, (b) [2+2] side -side mode, (c) [5+5] face -face mode between two pentagons and (d) [6+6] \nface-face mode between two hexagons.  \n3.1.1 Symmetrical Sized Dimers;  \nSymmetrical sized dimers C 20-C20, C 24-C24, C 28-C28, C 32-C32, C 36-C36, C 40-C40, C 44-\nC44 and C 50-C50 have been investigated in detail.  \nC20 with I h symmetry forms C 20-C20 dimer structure in [2+2] side -side mode as the \nmost stable geometry. We have also calculated the relative energy differences (E r) of all the \nconfigurations to check  for any isomeric structures. In C 20-C20 dimer, [5+5] and [1+1] modes \nhave E r w.r.t. [2+2] configuration of the order of 0.382 eV and 1.27 eV respectively, which \nclearly shows that [2+2] mode has the highest stability. The average diameter (D av = 4.14 Å) \nof cages remains almost same as that of single C 20 cage. The bond lengths lie in the range \n1.38 – 1.40 Å and the connecting bond length between the cages is 1.55 Å as shown in Table \n2. \nWe have calculated the interaction energies (see Table 2) of all the pos sible structures \nusing the following expression   \n                                    ΔE = E total(Cn – Cn) – 2E(C n)    8 \n where E total(Cn – Cn) and E(C n) are the total energies of the dimer and individual cages \nrespectively. The negative values of ΔE indicate  the greater stabilities of the dimer while the \npositive values show that the dimer is less stable, so the dimers with negative ΔE have a \npossibility to be formed experimentally. C 20-C20 dimer with [2+2] side -side mode require \nmore energy ( -5.43eV) to diss ociate into two cages of dimer structure as compared to other \nconfigurations (Fig. 1(a) and 1 (c)), which indicates that this configuration is the most \nfavorable one.  \nTable -2. Interaction energy (ΔE), average diameter (D av) and connecting bond length for \nsymmetrical sized dimers.  \nDimer             Interaction Energy (eV)                     D av.( Å)                     Connecting Bond Length (Å)  \n (Cn – Cn)       [1+1]    [2+2]    [5+5]    [6+6]                                    [1+1]      [2+2]         [5+5]              [6+6]  \n C20 – C20      -4.16     -5.43    -5.04         -                  4.17                 1.51        1.55          1.61                    -     \n C24 – C24      -3.27     -4.95    -4.87     -5.13               4.48                 1.53        1.56       1.54 -1.75            1.59  \n C28 – C28      -2.47     -3.42    -3.67     -2.66               4.86                 1.56        1.58       1.58 -1.60            1 .62  \n C32 – C32      -1.59     -2.54    -0.76     -0.97               5.22                 1.54        1.56       1.58 -1.62            1.59  \n C36 – C36      -2.93     -2.72    -3.00     -2.22               5.50                 1.54        1.58       1.58 -1.60            1.59  \n C40 – C40      -1.97     -2.21    -0.87     -1.96               5.69                 1.57        1.56       1.58 -1.65        1.58 -1.62 \n C44 – C44      -1.76     -2.59    -2.42     -1.33               6.00                 1.51        1.58       1.55 -1.59        1.55 -1.58 \n C50 – C50       0.02     -1.46     0.64       0.86               6.65                 1.59        1.57       1.56 -1.61        1.57 -1.60 \n  \nC24 with D 6d symmetry forms a most stable dimer with [6+6] face -face mode between \ntwo hexagons. In C 24-C24 dimer, E r for [2+2] mode w.r.t. [6+6] mode is 0.176 eV which point \ntowards the stability of [2+2] mode as well. The energy difference for other modes w.r.t. \n[6+6] mode is comparatively larger. The average diameter of C 24 cage in isolated form is 4.69  \nÅ, which decr eases to 4.48 Å when connected in [6+6] mode of dimer leading to distortion in \nthe structure. The bond lengths vary between 1.40 – 1.55 Å and the connecting bond length 9 \n between the cages is 1.59 Å. From the values of interaction energy, [6+6] configuration  has \nmaximum chances to be formed as compared to other counterparts.  \nC28 cage having T d symmetry forms a most stable structure with [5+5] face -face mode \nbetween two pentagons after their dimerization. [1+1] and [6+6] configurations have E r w.r.t. \n[5+5] of  the order of 1.20 eV and 1.01 eV respectively. [2+2] configuration has E r 0.259 eV \nwhich indicates that after [5+5] mode, it has the highest probability to be formed. The \naverage diameter (4.83 Å) of C 28-C28 dimer remains identical as compared to the indi vidual \ncage. The C -C bond lengths vary from 1.40 Å to 1.58 Å in C 28-C28 dimer and the connecting \nbond lengths lay from 1.58 – 1.60 Å. ∆E values indicate that the dimer structure with [5+5] \nmode has largest stability amongst all the other modes.  \nC32 (D3), C 40 (D2), C 44 (D2) and C 50 (D3) cages form most stable dimers when \nconnected through [2+2] side -side mode. For C 32-C32 and C 50-C50, the relative energy \ndifferences of [1+1], [5+5] and [6+6] modes are quite high w.r.t. [2+2] mode, which further \nestablishes [ 2+2] as the most stable configuration. For C 40-C40 dimer, E r value for [1+1] and \n[6+6] is almost same of the order of ~0.25eV, which shows that these two modes are \nisoenergetic. In C 44-C44 dimer, [2+2] mode is isoenergetic with [5+5] mode as the relative \nenergy difference between these two modes is 0.07eV. The D av of individual cages do not \nshow much change after dimerization except C 44-C44 dimer, where D av decreases from 6.15 Å \nto 6.0 Å. The average bond lengths for C 32-C32 and C 44-C44 dimers lie in the ra nge 1.37 Å – \n1.53 Å, whereas for C 40-C40 and C 50-C50 dimers the bond lengths vary from 1.38 Å – 1.60 Å. \nThe cages are held together through the connecting bonds of the order of 1.56 Å – 1.58 Å. \nFor C 50-C50 dimer, only [2+2] configuration is energetically favorable as all the other \nconfigurations considered show positive ∆E.  \nC36 with D 6h symmetry has a most stable dimer structure in [5+5] face -face mode \nbetween two pentagonal rings and this configuration is isoenergetic with [1+1] mode having 10 \n relative energ y difference of 0.06 eV. The connecting bond lengths for both configurations \nare shown in Table 2 and the D av of dimer structure decreases to 5.5 Å w.r.t. single cage (5.6 \nÅ). Interaction energy shows that the dimer with [5+5] mode is most favorable dimer \nstructure between other isomers of C 36-C36 dimer.  \nThe interaction energies for all symmetrical sized dimers show that these dimer \nstructures are energetically favorable and therefore, are likely to be formed. ∆E value \ndecreases from C 20-C20 to C 50-C50 dime r, which shows that the small sized dimers are more \nlikely to be formed as compared to large sized fullerene dimers.  The observed structural \nbehavior of symmetrical sized fullerene dimers can be seen from variation in average \ndiameters and connecting bond  lengths. In all the dimers except C 24-C24, C28-C28 and C 36-C36, \nthe most stable configuration with which the fullerene prefers to attach is [2+2]. C 24-C24 and \nC28-C28/C36-C36 dimer shows maximum preference to be formed in [6+6] and [5+5] mode. \nThe average  connecting bond length at which the dimers get stable is 1.58 Å. The fullerene \nhave a tendency to settle away from each other as the single C – C bond length is 1.54 Å. In \nall the dimers formed, the D av decreases w.r.t. the individual cages. However, when  the type \nof inter -cage bonding changes, there is variation in C – C bonds leading to change in the D av, \nwhich leads to redistribution of charges at localized sites.  \n3.1.2 Asymmetrical Sized Dimers;  \nWe have extended our investigation to study of asymmetrical si zed dimers C 20-C24, \nC24-C28, C 28-C32, C 32-C36, C 36-C40, C 40-C44 and C 44-C50 using the method described in \ncomputational details. C 20-C24, C 28-C32 and C 44-C50 form the most stable dimer  with [2+2] \nside-side mode. The relative energy differences are high for other configurations as compared \nto [2+2] side -side mode. The average C -C bond lengths vary from 1.37 Å – 1.57 Å for these \nthree dimer structures. There is variation of the order of 0 .10 – 0.15 Å in the D av of C 20-C24 \nand C 28-C32 in comparison to their individual counterparts, whereas for C 44-C50 Dav remains 11 \n almost same as compared to single cages. The connecting bond lengths and interaction \nenergies are tabulated in Table 3 for all th e asymmetrical sized dimers. The interaction \nenergy for these dimers is calculated using the following expression  \n                         ΔE = E total(Cn – Cm) – E(C n) – E(C m) \nwhere E total(Cn – Cm), E(C n) and E(C m) are the total energies of the dimer and i ndividual \ncages respectively and are shown in Table 3.  \nTable -3. Interaction energy (ΔE), average diameter (D av) and connecting bond length for \nasymmetrical sized dimers.  \nDimer         Interaction Energy (eV)                        D av (Å)                      Connecting Bond Length (Å)  \nCn – Cm      [1+1]     [2+2]    [5+5]    [6+6]             C n        Cm              [1+1]       [2+2]         [5+5]             [6+6]  \nC20 – C24    -3.45     -5.18     -4.93        -                4.22     4.70            1.52         1.56        1.57 -1.65           - \nC24 – C28    -2.36     -3.91     -4.29     -3.86            4.74     4.87            1.54         1.57        1.59 -1.64       1.58 -1.62 \nC28 – C32    -2.19     -2.59     -2.23     -1.84            4.73     5.39            1.55         1.57        1.58 -1.61       1.58 -1.62 \nC32 – C36    -2.33     -2.20     -2.00     -0.27            5.34     5.50            1.54         1.57        1.57 -1.62       1.55 -1.62 \nC36 – C40    -2.04     -1.99     -1.79     -0.67            5.37     5.89            1.55         1.58        1.57 -1.63       1.53 -1.69 \nC40 – C44    -1.48     -1.37     -1.13     -1.38            5.89     6.34            1.56         1.59        1.57 -1.62       1.58 -1.61 \nC44 – C50    -0.67     -1.46     -0.78     -0.06            6.13     6.61            1.57         1.57        1.58 -1.60       1.55 -1.65 \n \nTable 3 shows extra stability of [2+2] mode for C 20-C24, C28-C32 and C 44-C50 due to \nhigher interaction energy. The connecting bond lengths lie in the range 1.56 – 1.59 Å for \nthese three dimers.  \nIn case of C 24-C28 dimer, the most stable dimer forms when pentagonal ring of C 24 is \ncombined with pentagon face of C 28 cage i.e. [5+5] fac e mode. E r shows that [1+1], [2+2] \nand [6+6] modes are less stable as compared to [5+5] mode. The average diameter remains \nalmost same for both the individual cages after their dimerization, while the average C – C \nbond lengths varies between 1.39 – 1.60 e V. The connecting bond lengths vary from 1.59 – 12 \n 1.64 Å. ∆E value for [5+5] configuration is highest for C 24-C28 among all other possible \nmodes.  \nFor C 32-C36, C 36-C40 and C 40-C44 dimers, the most stable dimer structure has [1+1] \npoint -point mode. In C 32-C36 and C 40-C44 dimers, the relative energy differences are large for \n[2+2], [5+5] and [6+6] modes w.r.t. [1+1] configuration. For C 36-C40 dimer, [2+2] side -side \nmode is isoenergetic with [1+1] having energy difference of 0.06 eV. The D av in C 32-C36 \ndimer sho ws variation of 0.1 Å w.r.t. single cages. In C 36-C40 dimer, the D av for C 36 cage \ndecreases form 5.6 Å to 5.37 Å, whereas for C 40 cage it increases from 5.75 Å to 5.89 Å w.r.t. \nthe individual cages. The D av increases after dimerization for C 40-C44 dimer as  compared to \ntheir individual counterparts. The connecting bond lengths are of the order of ~1.55Å and the \naverage C – C bond lengths vary from 1.38Å to 1.60Å. Table 2 shows that these dimers have \nmaximum stability to be formed in [1+1] point -point mode.  \nThe interaction energies for asymmetrical sized dimers show similar behavior as that \nof symmetrical sized dimers. There is an increase in ∆E from C 20-C24 to C 44-C50 dimer, which \nsuggest that the dimers of small size have more chances to form as compared to  larger ones. \nFor all asymmetrical sized dimers except C 24-C28, the average connecting bond length is 1.56 \nÅ, which is comparable to single C – C bond length. In comparison to symmetrical sized \ndimers, the fullerene cages in asymmetrical sized dimers are m ore closely bound. There is \nsignificant variation in the average diameters of asymmetrical sized dimers, which point \ntowards their valu able magnetic properties.  \n3.2  Magnetic Properties;  \nTo study the magnetic properties of small fullerene dimers , spin polarized calculations \nhave been performed on all possible dimer combinations [1+1], [2+2], [5+5] and [6+6]. The \nHOMO -LUMO gaps for spin up and down electron states, density of states (DOS), total 13 \n magnetic moments (MM) and localized magnetic moments  (MM) (Fig 2) have been \ncalculated for both symmetrical and asymmetrical sized dimers as summarized below.  \n3.2.1 Symmetrical Sized Dimers;  \nThe symmetrical sized dimers with similar size were taken into consideration to \nanalyze their magnetic properties. The HOM O-LUMO energy gaps for electrons with spin up \nand spin down and total magnetic moments  for  C 20, C 24, C 28,C32, C 36, C 40, C 44,C50 with \n[1+1], [2+2], [5+5] and [6+6] modes are shown in Table 4.  \nTable -4. Total magnetic moments and HOMO -LUMO gaps for symmetrical sized \ndimers.  \nDimer  Total Magnetic Moment (µ B) Dimer                              HOMO -LUMO gaps (eV)  \nCn-Cn [1+1]  [2+2]  [5+5]  [6+6]  Cn-Cn        [1+1]  \n   ↑             ↓       [2+2]  \n   ↑            ↓       [5+5]  \n   ↑          ↓      [6+6]  \n   ↑          ↓  \nC20-C20 2.00 0.00 2.00 -   C20-C20 0.91 1.02 1.13 1.13 1.37 0.65 - - \nC24-C24 2.00 0.00 2.00 4.00   C24-C24 0.72 0.34 0.59 0.59 0.79 0.47 1.60 0.62 \nC28-C28 6.00 4.00 6.00 4.00   C28-C28 1.52 0.81 1.79 0.43 1.94 0.81 0.34 0.65 \nC32-C32 2.00 0.00 2.00 0.00   C32-C32 1.85 0.37 1.31 1.31 0.60 0.51 0.75 0.75 \nC36-C36 2.00 4.00 2.00 3.96   C36-C36 0.92 0.74 0.71 0.73 0.93 0.60 1.03 0.96 \nC40-C40 2.00 0.00 2.00 0.00   C40-C40 0.58 0.49 0.67 0.67 0.48 0.18 0.94 0.94 \nC44-C44 0.00 0.00 0.00 0.00   C44-C44 0.57 0.57 0.66 0.66 0.55 0.55 0.72 0.72 \nC50-C50 1.67 0.00 1.95 0.00   C50-C50 0.45 0.84 1.17 1.17 0.54 0.46 0.97 0.97 \n \nC20 fullerene cage with I h symmetry has zero magnetic moment in isolated form. \nHowever, the symmetry and magnetic state of C 20 has been a point of disagreement in the \nreported results, which may be due to the accuracy of the electron correlation effects in small \nfullerenes [48]. C 20 has shown magnetic to non magnetic transition and vice versa for I h and \nD3d symmetries respectively [49]. After dimerization, C 20-C20 dimer in [1+1] and [5+5] \nmodes shows magnetic behavior with total MM of 2.0 µ B, whereas the most stable 14 \n \nconfiguration w ith [2+2] mode shows non -magnetic behavior. The local MMs on all \nindividual C atoms for [5+5] mode in C 20-C20 dimer are shown in Fig 2. The major \ncontribution to total MM is contributed by second nearest neighbors (NNs) from the \nconnecting bond atoms, whic h contributes about 65% of total MM. Atoms at first NN \npositions and atoms at connecting bridges contribute about 42% and 10% respectively to total \nMM. The HOMO -LUMO gap of spin up and spin down electrons show distinct pattern in \ndimer configurations w.r.t . gap in individual building block. The non -magnetic dimer \nconfigurations show equal magnitude of HOMO -LUMO gaps for spin up and spin down \nelectron states, whereas magnetic dimers show unequal HOMO -LUMO gaps.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 15 \n \n \n \n \n \n \n \n \n \n \n \nFig.2 . Local magnetic moments of symmetrical sized small fullerene dimers in µ B. \nC24-C24 dimer exhibits high magnetism with total MM of 4.0 µ B in [6+6] mode in \ncomparison to isolated C 24 fullerene which shows 0.0 µ B magnetic moment. When connected \nwith [1+1] and [5+5] modes the dimer shows MM of 2.0 µ B, whereas in [2+2] mode the total \nMM is 0.0 µ B. To understand the origin of magnetism, local MMs on all C atoms were \ncalculated. The local MM in [6+6] mode indicate tha t first and second NNs of both cages \ncontribute ~51% and 49% respectively towards total MM, whereas there is no contribution \nfrom connecting bond atoms to total MM. However, in [5+5] mode, 80% of contribution \ncomes from second NNs. The HOMO -LUMO gaps for s pin up and spin down states are \nunequal in [1+1], [5+5] and [6+6] modes showing the magnetic behavior of dimer.  \nC28 fullerene cage has total magnetic moment of 4.0 µ B in isolated form. When it \nforms a most stable dimer in [5+5] mode, total MM increases to 6.0 µ B. The magnetic \nmoment for [2+2] and [6+6] configurations is 4.0 µ B, while for [1+1] point -point mode total \nMM is 6.0 µ B. The local MM in [5+5] mode show that there is a contribution of 23.5%, \n25.8% and 40% for first, second and third NNs respectively  towards total MM, whereas in 16 \n [2+2] mode major contribution comes from the second, third and forth NNs. The connecting \nbond atoms for both modes show a small contribution of only 1 -2% to total MM. All the \npossible configurations have significant magnetic o rder in HOMO -LUMO gaps of spin up \nand spin down states showing their magnetic behavior.  \nIn isolated form C 36 cage is magnetic with magnetic moment of 2.0 µ B. The magnetic \nmoment remains same after dimerization of the carbon cage in [5+5] mode. Total MM \nincreases to 4.0 µ B for [2+2] and [6+6] mode, while for [1+1] mode it  has value 2.0 µ B. The \nsecond, third and fifth NNs form connecting bond atoms make a contribution of 20%, 42% \nand 48.7% respectively to total MM in [5+5] mode. For [2+2] mode, the major con tribution \nof local MM is 51% for third NN and the remaining local MM are distributed evenly on first, \nsecond, fifth and sixth neighbors. The HOMO -LUMO gaps for spin up and spin down states \nhave finite energy difference, which show their magnetic behavior.  \nAs the fullerenes C 32, C40, C44 and C 50 are non -magnetic in their isolated forms and \nafter dimerization in [2+2] mode continuous to remain non -magnetic. However, [1+1] and \n[5+5] configurations acquire finite magnetic moment after the formation of dimer due  to \nchange in inter -cage bonding. In [1+1] and [5+5] modes, C 32-C32 and C 40-C40 dimers acquire \nfinite magnetic moment of 2.0 µ B, whereas [6+6] mode in both the dimers have zero \nmagnetic moment. The local MMs at C atoms of connecting bonds show no contribut ion as \nthe magnetic moment is localized away from them. The HOMO -LUMO gaps of spin up and \ndown states are equal for non -magnetic modes [2+2] and [6+6], while the magnetic dimer \nconfigurations [1+1] and [5+5] have finite energy difference in their HOMO -LUMO  gaps. \nC44-C44 dimer is found to be non -magnetic in all the four modes with zero magnetic moment. \nThe HOMO -LUMO gaps for spin up and spin down states are same showing the non -\nmagnetic behavior of the dimer. C 50-C50 dimer  is formed only in [2+2] mode and has zero 17 \n \n-1.0 -0.5 0.0 0.5 1.0-40-32-24-16-80816243240\nC20- C20DOS\nEnergy(eV) [2+2]\n [5+5]\n C20\n Ef\n-1.0 -0.5 0.0 0.5 1.0-40-32-24-16-80816243240\nC24- C24DOS\nEnergy(eV) [6+6]\n [5+5]\n C24\n Ef\n-1.0 -0.5 0.0 0.5 1.0-40-32-24-16-80816243240\nC28- C28DOS\nEnergy(eV) [5+5]\n [2+2]\n C28\n Ef\n-1.0 -0.5 0.0 0.5 1.0-40-32-24-16-80816243240\nC32- C32DOS\nEnergy(eV) [2+2]\n [1+1]\n C32\n Ef\n-1.0 -0.5 0.0 0.5 1.0-40-32-24-16-80816243240\nC36- C36DOS\nEnergy(eV) [5+5]\n [2+2]\n C36\n Ef\n-1.0 -0.5 0.0 0.5 1.0-40-32-24-16-80816243240\nC40- C40DOS\nEnergy(eV) [2+2]\n [1+1]\n C40\n Efmagnetic moment and the dimer has identical HOMO -LUMO gaps for spin up and spin down \nelectronic state.  \n   \n \n \n \n \n(a)                                                                 (b)                                                                (c) \n \n \n \n \n   \n \n \n \n      (d)                                                               (e)                                                               (f) \nFig. 3 . DOS plots for symmetrical sized small fullerene dimers.  \nTo understand the change in the electron density of dimer configuration w.r.t. \nindividual fullerene, the Density of states (DOS) are calculated and shown in Fig. 3. Fig. 3(a) \nvalidates that C 20 fullerene cage and C 20-C20 dimer  in [2+2] mode are non -magnetic and \n[5+5] mode has spin polarized states. The origin of magnetic moment in [5+5] mode comes \nfrom the 2p orbitals because of unequal spin up and down states near the Fermi level.  For \nC24-C24 dimer, [6+6] mode shows a signifi cant change in DOS w.r.t. single C 24 cage which \nfurther points to its magnetic nature. [5+5] mode also show some decrease in the \nmagnetization but there is a visible difference in up -down DOS (Fig. 3(b)). Both [6+6] and \n[5+5] modes show strong polarization  around Fermi level in 2p orbitals while C 24 fullerene \ncage is non -magnetic and has equal up and down states. Similarly, the DOS for other dimer 18 \n cages also show significant changes near Fermi level, which is due to the redistribution of \nelectrons in 2p orb itals (see Fig. 3).  \nThe symmetrical sized dimers have shown significant variation in their magnetic \nmoment w.r.t . the type of inter -cage bonding. Their magnetic moments are mainly localized \non first, second and third NNs from connecting bond atoms due to the redistribution of \ncharges in spin up and spin down electron states.  The contribution from connecting bond \natoms towards total MM is very small which may be explained due to tetra -bonding of C \natoms completing their valency by making four σ -bonds with neighboring C atoms. We have \nalso plotted projected density of states in Fig 6 for few systems, which shows that 2p-orbitals \nin each case has maximum contribution towards total MM. The Muliken charge distribution \nanalysis of the dimers suggest that the connecting bond C atoms losses charge in range 0.021 \n– 0.114 electrons, whereas the first NN gain charge of the orde r of 0.011 – 0.066 e-s. \nHowever, the gain in charge for second NN is ~0.010 – 0.038 electrons. This redistribution of \ncharges at different C sites is responsible for the variation in localized MMs.  In symmetrical \nsized dimers, some of the C atoms show ant iferromagnetic alignment w.r.t. their surrounding \nC atoms. However, the interaction between individual cages of the dimer is found to be \nferromagnetic in nature.  \n3.2.2 Asymmetrical Sized Dimers;  \nThe magnetic properties calculated for symmetrical sized dimers as described in the \nprevious section were calculated for asymmetrical sized dimers and are tabulated in Table 5. \nAsymmetrical sized dimers with small difference in size were considered to understand their \nmagnetic behavior. C 20 and C 24 in the isolated form ar e non -magnetic with zero magnetic \nmoment, when they form C 20-C24, the stable dimer structure with [2+2] mode remains non -\nmagnetic. However, when C 20-C24 connects with [1+1] and [5+5] bonding, the resultant 19 \n \nstructure becomes magnetic with magnetic moment of  1.9 µ B and 2.0 µ B respectively. The \nresults suggest dependence of magnetic behavior on type of inter -connecting mode of dimer.  \nTable -5. Total Magnetic moments and HOMO -LUMO gaps for asymmetrical sized \ndimers.  \nDimer  Total Magnetic Moment (µ B) Dimer                                  HOMO -LUMO gaps (eV)  \nCn-Cm [1+1]  [2+2]  [5+5]  [6+6]  Cn-Cm       [1+1]  \n   ↑             ↓        [2+2]  \n   ↑           ↓      [5+5]  \n ↑           ↓        [6+6]  \n    ↑           ↓  \nC20-C24 1.90 0.00 2.00 - C20-C24 0.59 0.39 0.71 0.71 0.88 0.43 - - \nC24-C28 2.00 2.00 4.00 3.95 C24-C28 0.82 0.32 0.40 0.35 0.89 0.42 0.34 0.42 \nC28-C32 4.00 2.00 4.00 2.00 C28-C32 1.66 0.39 0.66 0.43 0.90 0.60 0.34 0.70 \nC32-C36 1.95 2.00 2.00 4.00 C32-C36 0.93 0.29 0.36 0.56 0.89 0.48 0.53 0.70 \nC36-C40 2.00 3.96 2.00 4.00 C36-C40 0.66 0.73 0.64 0.47 0.52 0.24 0.89 0.67 \nC40-C44 2.00 4.00 2.00 0.00 C40-C44 0.73 0.39 0.81 0.42 0.51 0.24 0.71 0.71 \nC44-C50 1.98 2.00 1.92 0.00 C44-C50 0.88 0.55 0.85 0.61 0.68 0.64 0.90 0.90 \n \nTo understand the origin of magnetism in C 20-C24, local magnetic moments on each C \natom was calculated and are  shown in Fig 4.  The connecting bond atoms contribute very \nsmall magnetic moment towards total MM and the major contribution comes from second \nand third NNs from connecting bond which contribute 42% and 33% of total MM \nrespectively. The HOMO -LUMO gap for spin up and spin down electron states show similar \nbehavior as observed in symmetrical sized dimers.  \n \n \n \n \n \n 20 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 21 \n \n \n \n \n \n \nFig. 4 . Local magnetic moments of asymmetrical sized small fullerene dimers in µ B. \nC28 fullerene is magnetic with magnetic moment of 4 µ B in isolated form. When it \nforms a dimer with C 24, the resultant most stable dimer structure with [5+5] mode becomes \nmagnetic with total MM of 4.0 µ B. The total MM decreases in [1+1], [2+2] and [6+6] \nconfi gurations having total MM 2.0 µ B, 2.0 µ B and 3.95 µ B respectively. The local MMs \nsuggest that the maximum contribution of 70% comes from C 28 cage, while C 24 cage \ncontributes only 30% of total MM, which indicates that C 28 is inducing magnetism in C 24. \nThere  is very small contribution of ~1.5% from connecting bond atoms towards total MM, \nwhereas the maximum contribution comes from second and third NNs i.e. ~40% and 30% of \ntotal MM respectively. In [2+2] mode of C 24-C28 dimer, only C 28 cage contributes towards  \ntotal MM where local MM is distributed away from connecting bond atoms. The HOMO -\nLUMO gap for spin up and spin down states for all the modes show finite energy difference \nfor magnetic dimer.  \nIn the combination of C 28 and C 32 cages, the most stable dimer  structure [2+2] \nbecomes magnetic after dimerization with total MM of 2.0 µ B. The other modes [1+1], [5+5] \nand [6+6] of C 28-C32 show magnetic behavior having total MM of 4.0 µ B, 4.0 µ B and 2.0 µ B \nrespectively. The local MMs on each C atom were calculated t o comment on the magnetic \nbehavior of dimer structure. For most stable [2+2] mode, only C 28 cage contribute towards \ntotal MM as the cage is magnetic in isolated form and C 32 is non -magnetic. The magnetic \nmoments are evenly distributed over first, third and  fourth NNs from connecting bonds which 22 \n contribute 23%, 32% and 36% respectively. The HOMO -LUMO gaps for spin up and down \nstates have finite values.  \nIn isolated form, C 32 fullerene is non -magnetic whereas C 36 has magnetic moment of \n2.0 µ B. After dimerizati on, the resultant dimer structure in most stable [1+1] mode has total \nMM of 1.95 µ B. However, the total MM for [2+2], [5+5] and [6+6] increases to 2.0 µ B, 2.0 \nµB and 4.0 µ B respectively. In ground state [1+1] mode, both the cages contribute equally \ntowards  total MM of the dimer and the magnetic moments are evenly distributed on third, \nfifth and seventh NNs from connecting bond atoms. The local MMs for [6+6] mode of C 32-\nC36 dimer, contribution comes from C 36 cage only and contribution of ~36% comes from firs t \nand third NNs from connecting bonds. The HOMO -LUMO gap values for spin up and spin \ndown states also show the magnetic behavior of dimer in all the configurations.  \nThe combination of magnetic C 36 fullerene with non -magnetic C 40 fullerene results in \na magn etic dimer structure having total MM of 2.0 µ B in most stable [1+1] mode. In other \nconfigurations [2+2], [5+5] and [6+6], the total MM is 3.96 µ B, 2.0 µ B and 4.0 µ B \nrespectively. The local MMs show that both the cages contribute equally towards total MM. \nThe major contribution in total MM comes from first, third and fifth NNs from connecting \nbond atoms, whereas connecting bond atoms contribute ~1% towards total MM. The HOMO -\nLUMO gaps for spin up and spin down electron states show magnetic behavior of dimer and \nhave finite values in all modes.  \nC40 and C 44 cages are non magnetic in isolated form, but after dimerization the \nresultant dimer structure becomes magnetic in most stable [1+1] mode with total MM of 2.0 \nµB. [2+2] and [6+6] modes of C 40-C44 dimer are is oenergetic but they have different magnetic \nbehavior. [6+6] mode is non -magnetic with zero magnetic moment, while [2+2] has high \nmagnitude of total MM of order of 4.0 µ B. This explains the dependence of inter -cage \nbonding on the magnetic behavior of dimer formed. The local MMs suggest that each cage 23 \n \n-1.0 -0.5 0.0 0.5 1.0-40-32-24-16-80816243240\nC20- C24DOS\nEnergy(eV) [2+2]\n [5+5]\n C20\n C24\n Ef\n-1.0 -0.5 0.0 0.5 1.0-40-32-24-16-80816243240\nC24- C28DOS\nEnergy(eV) [5+5]\n [2+2]\n C24\n C28\n Ef\n-1.0 -0.5 0.0 0.5 1.0-40-32-24-16-80816243240\nC28- C32DOS\nEnergy(eV) [2+2]\n [5+5]\n C28\n C32\n Ef\n-1.0 -0.5 0.0 0.5 1.0-40-32-24-16-80816243240\nC32- C36DOS\nEnergy(eV) [1+1]\n [6+6]\n C32\n C36\n Ef\n-1.0 -0.5 0.0 0.5 1.0-40-32-24-16-80816243240\nC36- C40DOS\nEnergy(eV) [1+1]\n [2+2]\n C36\n C40\n Ef\n-1.0 -0.5 0.0 0.5 1.0-40-32-24-16-80816243240\nC40- C44DOS\nEnergy(eV) [1+1]\n [2+2]\n C40\n C44\n Ef\n-1.0 -0.5 0.0 0.5 1.0-40-32-24-16-80816243240\nC44- C50DOS\nEnergy(eV) [2+2]\n [5+5]\n C44\n C50\n Efcontributes ~50% towards total MM and the magnetic moments are not localized on \nconnecting bond atoms. The main contribution comes from first and third NNs from \nconnecting bonds with ~24% and ~20% respectively. The HOMO -LUMO gaps for spin up \nand down states are equal for [6+6] mode indicating its non -magnetic behavior.  \n   \n   \n \n \n \n(a)                                                                     (b)                                                              (c)  \n   \n \n \n \n \n        (d)                                                               (e)                                                               (f) \n \n \n \n \n \n(g) \nFig. 5 . DOS plot for asymmetrical sized small fullerene dimers.  \nThe non -magnetic C 44 and C 50 cages form a magnetic dimer structure with total MM \nof 2.0 µ B in [1+1], [2+2] and [5+5] modes. However, the dimer  in [6+6] mode remains non -\nmagnetic. The local MMs for most stable [2+2] mode shows that only C 44 cage contribute 24 \n towards total MM and it comes from first and third NNs with 48.5% and 41% respectively. \nThe HOMO -LUMO gaps for spin up and spin down states in  [1+1], [2+2] and [5+5] modes \nhave finite energy difference which shows their magnetic behavior. The [6+6] configuration \nis non -magnetic having same spin up and down HOMO -LUMO gaps.  \nThe density of state (DOS) plots have also been plotted in Fig. 5 to under stand the \nmagnetic behavior of asymmetrical sized dimers. The plots show that there is significant \nvariation in spin up and down states near the Fermi level after dimerization of cages. In C 20-\nC24 dimer, DOS plot give the magnetic behavior of [5+5] mode, a s there are spin polarized \nstates, while for [2+2] mode spin up and down states are identical (Fig. 5(a)). Similarly the \nDOS plots of other dimers show redistribution of electrons and presence of some unfilled \nstates near the Fermi level and the contributi on comes from 2p orbitals. The variation in spin \nup and down states of all these dimers points towards their magnetic nature.  \nTable -6: Contribution of 2s and 2p orbitals towards localized magnetic moment.  \nDimer  Local MM  2s-orbital  2p-orbital  \n \n \nC24-C28 \n 0.413  0.028  0.387  \n0.407  0.028  0.381  \n0.363  0.023  0.341  \n0.355  0.022  0.332  \n0.278  0.017  0.264  \n \nTherefore, the results suggest that there is a large variation in magnetic behavior of \nindividual cages when they form asymmetrical sized dimers . In some cases, both the cages \ncontribute equally towards total MM, whereas in few of them only one cage contributes \ntowards total MM. Further, the localized moments show that the connecting bond atoms are \nnot the major contributors in total MM which may be due to the change in hybridization from 25 \n \n-1.0 -0.5 0.0 0.5 1.0-101C24- C24PDOS\nEnergy(eV) 2s\n 2px\n 2py\n 2pz\n Ef\n-1.0 -0.5 0.0 0.5 1.0-101C28- C28PDOS\nEnergy(eV) 2s\n 2px\n 2py\n 2pz\n Ef\n-1.0 -0.5 0.0 0.5 1.0-101C24- C28PDOS\nEnergy(eV) 2s\n 2px\n 2py\n 2pz\n Ef\n-1.0 -0.5 0.0 0.5 1.0-101C40- C44PDOS\nEnergy(eV) 2s\n 2px\n 2py\n 2pz\n Efsp2 to sp3. There is charge imbalance in spin up and spin down states of first, second and third \nNNs from connecting bonds which makes them contribute towards total MM. The local MMs \nshow that some of the C atoms behave antiferromagnetically w.r.t. their neighboring C atoms. \nHowever, the individual cages in dimer structure interact ferromagnetically w.r.t. each other. \nTable 6 tabulates the local MM and contribution of 2s - and 2p -orbitals for five atoms of C 24-\nC28 dimer starting from maximum value of local MM. The table shows that 2p -orbitals of C \natom has major contribution to local MM, which further contributes to total MM of dimer \nstructure. The plots of projected density of states (PDOS) also show the contributio n of 2p -\norbitals, which are plotted in Fig.6. The redistribution of charges on C atoms has been studied \nusing Muliken charge analysis which shows that both the fullerene cages lose charge in the \nrange 0.014 – 0.128 e- for connecting bond atoms. The first N Ns gain charge of the order of \n0.012 – 0.076 e-s, whereas the gain in charge for second NNs is 0.002 – 0.029 e-s. This \ncharge redistribution at different sites leads to variation in magnetic moments of individual C \natoms.  \n \n \n \n \n \n \n \n \n \nFig.6;  PDOS (projected density of states) plots for one atom of few symmetrical and \nasymmetrical sized small fullerene dimers.  26 \n 4. CONCLUSIONS;  \nWe have investigated a feasibility of formation of symmetrical and asymmetrical \nsized small fullerene dimers  and their magnetic properties using spin polarized density \nfunctional theory. All possible modes ([1+1], [2+2], [5+5] and [6+6]) through which the \ncages can connect were considered. The negative values of interaction energies for all dimer \ncombinations in dicate strong possibility of their production except C 50-C50 dimer which is \nunfavorable in their [1+1], [5+5] and [6+6] modes. All the dimer configurations are bonded \nweakly with connecting bonds ranging between 1.55 Å – 1.60 Å. The interaction energy \nsugg ests higher stability of symmetrical sized dimers than asymmetrical sized dimers.  \nDimerization of the fullerenes result in significant change in electronic and magnetic \nproperties w.r.t. isolated fullerene. All fullerenes in isolated form except C 28 and C 36 are non -\nmagnetic and show interesting change in the magnetic behavior on dimerization. When two \nnon-magnetic fullerenes are combined, the resultant dimer formed is magnetic. However, the \ncombination of a magnetic and a non -magnetic fullerene leads to ind uced magnetism on non -\nmagnetic fullerene in dimer configuration. When two magnetic fullerenes are combined, \nthere is an enhancement or decrease in total magnetic moment of the resultant dimer. All \nmagnetic dimers show different HOMO -LUMO gap for spin up an d down electrons, whereas \nnon-magnetic dimers have same HOMO -LUMO gap for spin up and spin down electrons. \nMagnitude of the magnetic moment is proportional to the difference in the HOMO -LUMO \ngap of spin up and spin down electron. The origin of magnetizatio n may be understood in \nterms of structural distortion of fullerenes in dimer configuration and redistribution of charge \nfrom connecting C atoms due to formation of connecting bonds. More is the structural \ndistortion more is the magnetic moment of the dimer  structure. Further, given suitable \nexperimental conditions, these small fullerene dimer structures can be produced, which can \nfacilitate them as good magnetic materials having potential applications in spintronics.  27 \n 5. ACKNOWLEDGEMENTS;  \n Authors are grateful to Siesta group for providing their computational code. 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"    },    {        "title": "2006.14339v1.In_plane_magnetic_domains_and_Néel_like_domain_walls_in_thin_flakes_of_the_room_temperature_CrTe__2__van_der_Waals_ferromagnet.pdf",        "content": "In-plane magnetic domains and Néel-like\ndomain walls in thin flakes of the room\ntemperature CrTe 2van der Waals ferromagnet\nAnike Purbawati,yJohann Coraux,\u0003,yJan Vogel,yAbdellali Hadj-Azzem,y\nNianJheng Wu,yNedjma Bendiab,yDavid Jegouso,yJulien Renard,yLaetitia\nMarty,yVincent Bouchiat,yAndré Sulpice,yLucia Aballe,zMichael Foerster,z\nFrancesca Genuzio,{Andrea Locatelli,{Tevfik Onur Menteş,{Zheng Vitto\nHan,x,k,?Xingdan Sun,x,kManuel Núñez-Regueiro,yand Nicolas Rougemailley\nyUniv. Grenoble Alpes, CNRS, Grenoble INP, Institut NEEL, 38000 Grenoble, France\nzALBA Synchrotron Light Facility, 08290 Cerdanyola del Valles, Spain\n{Elettra-Sincrotrone Trieste S.C.p.A., S:S. 14, km 163.5 in AREA Science Park,\nBasovizza, 34149 Trieste, Italy\nxShenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese\nAcademy of Sciences, Shenyang 110016, China\nkSchool of Material Science and Engineering, University of Science and Technology of\nChina, Anhui 230026, China\n?Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006,\nP.R.China\nE-mail: johann.coraux@neel.cnrs.fr\n1arXiv:2006.14339v1  [cond-mat.mtrl-sci]  25 Jun 2020Abstract\nThe recent discovery of magnetic van der Waals materials has triggered a wealth\nof investigations in materials science, and now offers genuinely new prospects for both\nfundamental and applied research. Although the catalogue of van der Waals ferro-\nmagnets is rapidly expanding, most of them have a Curie temperature below 300 K,\na notable disadvantage for potential applications. Combining element-selective x-ray\nmagnetic imaging and magnetic force microscopy, we resolve at room temperature the\nmagnetic domains and domains walls in micron-sized flakes of the CrTe 2van der Waals\nferromagnet. Flux-closure magnetic patterns suggesting in-plane six-fold symmetry are\nobserved. Upon annealing the material above its Curie point (315 K), the magnetic\ndomains disappear. By cooling back down the sample, a different magnetic domain dis-\ntribution is obtained, indicating material stability and lack of magnetic memory upon\nthermal cycling. The domain walls presumably have Néel texture, are preferentially\noriented along directions separated by 120 degrees, and have a width of several tens\nof nanometers. Besides microscopic mapping of magnetic domains and domain walls,\nthe coercivity of the material is found to be of a few mT only, showing that the CrTe 2\ncompound is magnetically soft. The coercivity is found to increase as the volume of the\nmaterial decreases.\nKeywords\nvan der Waals ferromagnets, two-dimensional materials, layered compounds, room temper-\nature ferromagnetism, transition metal dichalcogenides, magnetic imaging\n2Introduction\nMagnetic van der Waals (vdW) materials prove to be an excellent playground in which to\nexplore magnetic phenomena in low dimensions, and are candidate building blocks for spin-\ntronic devices. On the one hand, the origin of the magnetic order in vdW ferromagnets\nis often scrutinized under the prism of canonical spin lattice models, seeking for signatures\nof two-dimensional exotic magnetic behaviors.1–7On the other, applied research is devising\nnew strategies to tune their magnetic properties through electrostatic control, opening new\navenues for applications in spintronics.3,8–11A few prototypical devices have been already\ndemonstrated, including magnetic tunnel junctions with large magneto-resistance12–17. Al-\nthough the catalogue of vdW ferromagnets is rich and extends well beyond the most studied\nCrI3and Fe 3GeTe 2compounds, the palette of available magnetic properties remains par-\ntial. In that context, computational efforts may guide the search for materials with specific\nproperties,18,19especially those with a high Curie temperature (T C).\nMeasurements of high T Cwere reported for three few- and even single-layer dichalco-\ngenides with 1T polytype,20VSe 2, VTe 2, and MnTe 2,21–24although contradicting results are\nreported in some cases.6,25A near room temperature T Cwas also reported in Fe 5GeTe 2,26\nand in Fe 3GeTe 2ultra-thin films,3the latter under the influence of ionic gating (the chemical\ncomposition of Fe 3GeTe 2being altered by the gating process itself27). Further, the Curie\ntemperature can be substantially increased (above 300 K) in 250 nm-thick Fe 3GeTe 2flakes\netched in the form of microstructures.28Interestingly, this microstructuration also induces\nan in-plane magnetization,28a rare situation in ultra-thin vdW ferromagnets. Beyond this\nparticular case, the CrCl 3compound has an intrinsic in-plane magnetization, down to few\natomic layers, but with a Curie temperature well below 300 K.29To date, there is no known\nvdW ferromagnet with thickness below 100 nm, and with an in-plane magnetization and\na TCabove room temperature. Ferromagnetism above room temperature would offer new\nopportunities to study spintronic devices. In-plane magnetization for an ultra-thin layer\nwould provide further possibilities, for instance to test remarkable phase transitions in two-\n3dimensional XY spin lattice models,30to stabilise quantum phases via proximity effects,31\nand to build synthetic antiferromagnets with tunable magnetic properties.32\nHere, we demonstrate both ferromagnetism above room temperature and in-plane mag-\nnetic anisotropy in exfoliated flakes of the 1T-CrTe 2vdW transition metal dichalcogenide,\nan established ferromagnet in its bulk form.33Information on the lateral distribution of mag-\nnetic domains is often missing in considering the properties of thin vdW ferromagnets.34,35\nTherefore, we use two magnetic imaging techniques, magnetic force microscopy (MFM) and\nx-ray magnetic circular dichroism photoemission electron microscopy (XMCD-PEEM) to\nunveil the structure of magnetic domains and their domain walls. We unveil flux-closure\nin-plane magnetic domains, uniform across the thickness of the flakes, and separated by\nNéel domain walls at the surface of the flake, typically extending across 50 nm. Comple-\nmenting our microscopy measurements by spatially-resolved Kerr magnetometry, we find a\nclose-to-full remanence that cannot be observed in bulk crystals for magnetostatic reasons.\nBesides, we observe that the coercivity is small and depends on the volume of the flake.\nFinally, based on Raman spectroscopy, x-ray absorption spectroscopy (XAS) and x-ray pho-\ntoelectron spectroscopy (XPS), we address the stability of the exfoliated CrTe 2compound in\nambient conditions, and find that an oxidation process takes place on a time scale of about\na day.\nResults and Discussion\nExfoliation of thin CrTe 2flakes\nBulk CrTe 2was synthesized in the 1T structure (Figure 1a), starting from the removal of\npotassium in a bulk KCrTe 2crystal (see Materials and Methods ), following a procedure de-\nscribed elsewhere.33To prepare thin flakes of CrTe 2, we tested several mechanical exfoliation\nprocesses, using a standard scotch tape, a viscoelastic stamp of polydimethylsiloxane, and\na thermal release tape. Whatever the process, the yield of thin ( i.e.thinner than 20 nm)\n4Figure 1: Structure, morphology and Raman reference spectra of exfoliated\nCrTe 2.(a) Crystal structure of bulk CrTe 2. (b) AFM topographs of exfoliated CrTe 2\nflakes with a thickness of the order of 10 nm (left) and 200 nm (right). The superimposed\ncurves show a height profile of the two flakes. (c) Raman spectrum of an exfoliated, 13-nm-\nthick flake prepared under inert conditions revealing two main peaks (E 2g, A1g) at 102 and\n134 cm\u00001.\nflakes proved to be much lower than in the case of other, better-known vdW materials such\nas graphene or MoS 2. The lateral extension of the flakes was found to decrease with the flake\nthickness, being of the order of 10 \u0016m for\u001810 nm-thick flakes, as assessed with atomic force\nmicroscopy (AFM, Figure 1b). The flakes were transferred onto SiO 2(285 nm)/Si and Pt(1\nnm)/Ta(10 nm)/Si host substrates. In AFM images, we often observe nanoclusters covering\na small fraction of the surface (see Figure 1b). These clusters exhibit a slight specific phase\ncontrast in AFM (few degrees difference) compared to the clean flake surface, indicative that\nthey slightly modify mechanical dissipation processes. In our local XAS and XPS experi-\nments, they exhibit no particular contrast. We tentatively ascribe them as dust particles,\nor scotch tape residues generated by the exfoliation process, that are weakly bonded to the\nsurface.\nThe Raman spectra of the freshly exfoliated flakes on SiO 2/Si feature two characteristic\nbands (Figure 1c), the E 2gand A 1gmodes expected for 1T-CrTe 2(Figure S1). In the\n5thickness range we probe in this work ( \u001810 nm to\u0018200 nm), the wavenumbers (102 and\n134 cm\u00001) of these two bands are constant, unlike their intensity, which is weaker for thicker\nflakes (see Supporting Information, Figure S1). This thickness dependence of the intensity\nis attributed to the metallic character of 1T-CrTe 2:33while optical interferences within the\ngap defined by the Si surface and the bottom CrTe 2surface are expected to enhance the\nRaman signal for a SiO 2thickness of the order of the optical wavelength,36,37the penetration\ndepth of light in metallic CrTe 2is presumably of the order of few/several 10 nm. Under such\nconditions, interference effects have a vanishing effect as the flake becomes thicker than\nseveral tens of nm.\nFigure 2: Spatially-resolved magnetometry. Focused Kerr measurements performed\nin longitudinal geometry and at room temperature on three CrTe 2flakes having different\nthicknesses, as indicated in the AFM image. On the thinnest flake (10 nm, green), no\nmagnetic signal is found, whereas a Kerr signal is clearly seen for flakes with thickness of\n30 (red), 80 (blue38), 100 (purple and red) and 125 nm (yellow). In all cases, the hysteresis\nloops show strong remanence, close to 100%, and a coercivity of a few mT only.\nSpatially-resolved magnetometry\nThe magnetic properties of exfoliated CrTe 2flakes were first characterized at room temper-\nature, using focused magneto-optical Kerr magnetometry (see Materials and Methods ). The\n6technique allows recording hysteresis loops under a focused laser spot of a size smaller than\n1\u0016m, and an external magnetic field \u00160Happlied in the plane of the film. Figure 2 shows\nexamples of hysteresis loops measured on different locations, indicated by a colored circle in\nthe AFM images. Three CrTe 2flakes with different thicknesses are probed. Except for the\nthinnest flake (10 nm), a magnetic hysteresis loop is systematically observed, providing a\ndirect evidence of room temperature ferromagnetism. The fact that no Kerr signal is mea-\nsured for flakes thinner than 20 nm could be attributed to a reduced Curie temperature, to\na limited sensitivity of our magneto-optical Kerr magnetometer, or to the degredation of the\nmaterial surface under ambient conditions (see below). The hysteresis loops are essentially\nsquare, showing a remanent magnetization close to 100%. This result indicates that the\nCrTe 2flakes are in-plane magnetized, and that magnetic domains should be larger than the\nlaser spot (\u00181\u0016m). The coercive field \u00160Hcis of the order of a few mT, a relatively low\nvalue characteristic of a magnetically soft material.\nThe coercive field varies from flake to flake, and even within a single flake (Figure 2 and\nFigure S2). To understand the origin of these variations, we first discuss the case of the\norange and purple hysteresis loops, which were both acquired on the same flake. At this\npoint it is important to remind what is occurring during the focussed Kerr measurements\nas function of the swept magnetic field. The Kerr rotation is measured within the narrow\nfootprint of the focused laser beam, but this does not mean that the characterisation is\nlocal at this scale. Indeed, the swept external magnetic field (which is applied uniformly,\nat larger scale) induces magnetisation reversal starting with the nucleation, possibly out of\nthe laser spot, of reversed domains surrounded by domain walls. Magnetisation reversal\ngoes on viathe motion of these domain walls, and hence happens with a delay away from\nthe reversed domain nucleation centers. During this delay the sweep of the magnetic field\nhas continued. This means that locally, away from the nucleation centers, magnetisation\nreversal will effectively occur at high magnetic field, i.e.the measured coercivity will be\nhigher than at the nucleation centers. This is true if domain wall propagation can occur\n7from the nucleation centers to the laser spot.\nHowever, the two regions (orange and purple) are at least partly magnetically indepen-\ndent. To maintain the smoothness of the discussion, we refrain from disclosing the analysis\nof magnetisation patterns presented in the next section, but still need to mention that this\nindependence is evident in the magnetic imaging data of the same flake (Figure 3a, top\npannel, and similar kind of data acquired with varying applied magnetic field): the top-left\nelongated triangle which corresponds to the thicker region in the top pannel of Figure 2\n(where the orange hysteresis loop was acquired) and the remaining of the flake (where the\npurple hysteresis loop was acquired) exhibit seemingly unrelated magnetisation patterns.\nWhy these regions are independent is unclear based on our experimental data. We\nsuggest that the independence has its origin in the presence of a boundary between the two\nregions, within a more or less thick wall, where the structure or composition of the material\nis altered, hence ferromagnetism is suppressed (there, the material could for instance have a\nnon magnetic 2H structure instead of the 1T one).\nThe region corresponding to the orange hysteresis loop is slightly thicker, but its area\nis much smaller than the area corresponding to the purple hysteresis loop. The former\nhence has smaller volume than the latter, and since coercivity is increased if the number of\nnucleation centres (local defects) for reversed magnetic domains is decreased,39we indeed\nexpect fewer domain nucleation centres (if we assume a constant local defect concentration\nin CrTe 2) and higher coercivity for the orange hysteresis loop than for the purple hysteresis\nloop. We transposed this analysis to 10 other hysteresis loops. In some cases, the laser spot\nilluminates two regions that are magnetically independent, in which case the hysteresis loop\nexhibits a step-like reversal ( e.g.red hysteresis loop in Figure 2). Overall, we found that\nthe increase of coercivity with decreasing volume is a robust trend. It is interesting to note\nthat lamellar materials such as CrTe 2are naturally immune to an increase in their surface\nroughness with the flake thickness. This is in contrast to more traditional thin films grown\nfrom the bottom up, where roughness, increasing with the thickness, was argued to alter\n8coercivity.40\nWe end up this section by noting the strong difference in the magnetisation measured\nwith SQUID magnetometry versusapplied magnetic field, for thin flakes (Figure 2) and for\nmacroscopic flakes. No or very little remanence and loop opening is observed in the latter\ncase, neither for in-plane nor for out-of-plane magnetic fields down to low temperatures\n(Figure S3). The saturation magnetisation MSis reached for lower in-plane than out-of-\nplane magnetic fields, which is indicative, here as well, of a preferential orientation of the\nmagnetisation in the plane of the flakes. What is the origin of this preference?\nFrom the SQUID data we deduce a relatively small MSvalue of 250 kA/m ( i.e.slightly\nless than the third of the value for permalloy and a sixth of the value in cobalt). In the case of\nan ideal flat sheet, with a perpendicular demagnetisation factor of 1, \u00160MS= 310 mT would\nbe needed to saturate the magnetisation perpendicular to the surface. A more realistic <1\nvalue of the demagnetisation factor is expected for the macroscopic CrTe 2flakes, which have\nfinite thickness and extension, hence a slightly lower field would be needed here. This field\nshould however still be larger than the \u001850 mT value that can be inferred by determining\nthe field at which a zero-field linear fit to the data reaches MS(Figure S3). This is an\nindication that shape anisotropy alone cannot account for the observed anisotropy of the\nSQUID data. Antagonist magnetocrystalline effects favouring out-of-plane magnetisation\nprovide a reasonable interpretation.\nIn-plane, flux-closure magnetization patterns and magnetic domain\nwalls\nAt first thought, the very weak or absence of remanence observed in the SQUID data cor-\nresponding to the macroscopic flakes may seem a contradiction with the 100% remanence\nobserved in the focussed Kerr data. However, we also expect globally, for each exfoliated thin\nflake, no loop opening, despite local loop openings at smaller scale (as observed in the fo-\ncussed Kerr measurements). The reason is, as we will now see, that magnetic domains in the\n9thin flakes form flux-closure patterns. To unveil these patterns, we used two complementary\nmagnetic imaging techniques (see Materials and Methods ):\n\u000fMagnetic force microscopy (MFM), to probe the magnetic stray fields emitted by the\nmagnetization texture, especially to detect magnetic domain walls.\n\u000fPhotoemission electron microscopy (PEEM) combined with x-ray magnetic circular\ndichroism (XMCD), to probe the local direction of magnetization.\nFigure 3: Magnetic texture of individual flakes - domains and domain walls. (a,b)\nXMCD-PEEM (top panels) and MFM (bottom panels) images on the two flakes ((a) 30-\n100-125 nm-thick, (b) 80 nm-thick) reported in Figure 2 and exhibiting ferromagnetism at\nroom temperature. The XMCD-PEEM images are obtained at the Cr L 3absorption edge\n(575.8 eV). (c) Analysis of the magnetic domain walls in the MFM image shown in (a). The\nthree intensity profiles are associated to the blue dashed lines un (a). (d) High-resolution\nXMCD-PEEM image of a 25 nm-thick flake. (e) Analysis of the magnetic domain wall width\nthrough the two red dashed lines shown in (d). Scale bar is 1 \u0016m in all images. The direction\nof the x-ray beam is indicated with horizontal (a,b) and vertical (d) arrows.\n10The spatial resolution of the two techniques is similar, and of the order of a few tens of\nnanometers, typically. Contrary to MFM, XMCD-PEEM is surface sensitive, and only the\ntop few nanometers below the material surface are imaged. The two techniques were applied\nto a large number of CrTe 2flakes, and in some cases to the very same flakes. Figures 3a,b\nshow the MFM and XMCD-PEEM magnetic contrasts obtained on the two magnetic flakes\nalso studied with focused Kerr magnetometry and shown in Figure 2. The two images for the\nthickest flake (Figure 3a) reveal a magnetic contrast at room temperature, and are obtained\non an exfoliated sample kept in ambient conditions for several months. As we will see below,\nalthough the material’s surface oxidizes with time, the magnetic texture of the flakes can\nbe imaged reproducibly over an extended period of time. In addition, both techniques were\nused on the same magnetic configuration (no external magnetic field was applied between the\nMFMandXMCD-PEEMexperiments), sothatthemagneticimagesreportedinFigures3a,b\ncan be compared directly.\nThe XMCD-PEEM images reveal flux-closure configurations (see also Figure 4), and\nmagnetic domains often compatible with a six-fold symmetry (see Figures 3a, 4). Consistent\nwith this observation, the walls between the domains often orient (not exclusively, though)\naccording to a six-fold symmetry (Figure S4), as shown by a statistical analysis (Figure S5) –\nthey are 120\u000emagnetic domain walls. These observations suggest the presence of a nonzero\nin-plane magnetocrystalline anisotropy, although this anisotropy might be small, since six-\nfold symmetry is not systematically observed (Figure S5). This points to a contribution\nof shape anisotropy, which promotes other preferential in-plane orientations given that the\nflakes rarely exhibit edges corresponding to specific crystallographic directions in CrTe 2\n(Figure S5; such directions would probably be distributed according to a six-fold symmetry).\nA six-fold magnetocrystalline anisotropy is consistent with the structure of 1T-CrTe 2, a\ntrigonal system with hexagonal unit cell.\nSuch six-fold anisotropy has been observed in a few thin film systems, for instance made\nof copper-cobalt superlattices,41iron-cobalt-platinum,42or cobalt-implanted zinc oxide,43\n11but not for van der Waals ferromagnets to our knowledge. In fact, for another van der\nWaals ferromagnet, K 2CuF 4, a preferred in-plane orientation was predicted with no six-fold\nin-plane anisotropy.44\nAcross domain walls the MFM phase signal, which is proportional to the gradient of the\nforce experienced by the MFM tip, first increases and then decreases (or vice versa ) above\nand below the constant value corresponding to the two domains (see the profiles measured\nperpendicular to the walls in Figure 3c). In other words, the phase signal is a function of\nthe position having a strong odd symmetry character about the center of the domain wall.\nWithin the common description relying on bond charges (giving rise to the magnetization)\nof magnetostatics, such odd symmetry character is a clearcut qualitative evidence of an\naccumulation of positive and negative charges on either sides of the domain wall.45\nCan we tell more about the texture of the magnetisation within the domain walls? In\nprinciple, several kinds of textures could account for our observations, unfortunately very few\nexperimental techniques (if any) would be able to directly resolve them. As we discussed, for\nCrTe 2flakes we observe 120\u000ein-plane magnetised domains. In such a case, the texture of the\ndomain walls is expected to be of Néel type, and not of Bloch type,46i.e.the magnetisation\nshould rotate within the surface plane all along the thickness of the domain walls. In the\nthickness range that we probed, and given the relatively small MS, the Néel walls should in\nfact be symmetric.46These expectations are not very sensitive to the anisotropy constant\nvalue,46which is unknown so far for CrTe 2. The symmetric Néel texture is associated with\nan accumulation of positive and negative charges on either sides of the wall, and is thus fully\nconsistent with our MFM observations.\nThe XMCD-PEEM images indicate that the domain wall width is of the order of several\ntens of nanometers (see intensity scans in Figures 3d,e). Finally, combining the different\ninformation provided by the two techniques, the local magnetization texture can be resolved,\nincluding that of the magnetic domain walls (see arrows in Figure 3a).\nThe thinnest flake we imaged (10 nm) does not show any MFM or dichroic magnetic\n12contrast, and no hysteresis loop is measured (see Figure 2). This observation is not a clear-\ncut evidence for T C<300 K in this case, though, and could be instead interpreted by\ninvoking the limited sensitivity of the experimental probes. The 100- and 30-nm-thick flakes\n(see Figure 3a) can be studied using both imaging techniques and with local magnetometry\nas well. However, the 80-nm-thick flake (see Figure 3b) is characterized by an unambiguous\nhysteresis loop and a faint MFM contrast, but no clear XMCD-PEEM magnetic contrast\nwithin the flake. This finding suggests that the top of this particular flake might be degraded\nover a thickness exceeding the photoelectrons’ mean-free path of the order of a nanometer,\npreventing magnetization to be probed with a surface sensitive technique such as PEEM,\neven if the flake is in fact ferromagnetic at room temperature. This surface degradation is\nalso compatible with the analysis of the AFM phase contrast which, together with Raman\nspectroscopy and x-ray absorption spectroscopy, reveals that this region of the flake might\nbe oxidized (see Figure S6). We note that the extent of the surface degradation we observe\nis flake dependent and does not seem to be related to its thickness.\nFerromagnetic-paramagnetic transition\nXMCD-PEEM imaging was also used upon sample annealing to track the temperature de-\npendence of the magnetic contrast, and to locally measure the Curie temperature T C. As\nexpected, the magnetic contrast continuously drops as the temperature increases (see Fig-\nure 4a). This behavior is observed for several CrTe 2flakes of different thicknesses (the\nthickness of each flake was determined a posteriori with AFM). For all the imaged flakes,\nthe magnetic contrast vanishes around 315 K, which is also the T Cvalue of the bulk material\n(see Figure S3).33Our results do not reveal any obvious change in T Cwhen the thickness of\nthe CrTe 2flakes is varied, at least in the range we probe in this work, for which no thickness\ndependency has been observed either for other van der Waals ferromagnets.7Our measure-\nments also show that the magnetic domain structure may vary upon cycling the temperature\nabove the Curie point, indicating that there is no significant magnetic memory of the initial\n13Figure 4: Curie temperature of individual flakes. (a) XMCD-PEEM magnetic images\n(576.6 eV) of the same flakes for different sample temperatures (the three flakes are about\n60 nm-thick). The top row shows the warming process, whereas the bottom row shows the\ncooling. The red arrow shows the position of the flake shadow, which also carries a magnetic\ncontrast. (b) Intensity of the magnetic contrast deduced from (a) and from another set of\nimages (not shown) allowing an estimate of the Curie temperature. Note that the contrast\nvaries from one data set to the other (green-blue-orange) depending, among other factors, on\nthe sample preparation conditions (time before introduction under ultrahigh vacuum) and\nsettings of the XMCD-PEEM instrument.\nstate.\nInterestingly, because of the large thickness of the flakes and the grazing incidence of\nthe x-ray beam in XMCD-PEEM measurements, a shadow is projected onto the substrate\nsurface (see red arrow in Figure 4a). The dichroic signal in the transmitted intensity seen\nwithin the shadow is identically inverse of the dichroic contrast due to the direct emission\n14from the CrTe 2surface.47–49We thus conclude that the magnetization is homogeneous within\nthe bulk of the flake, which is reliably represented by the XMCD-PEEM image obtained from\nthe surface.\nFigure 5: Sample aging and formation of an oxide layer. (a) Absorption spectra\nat the L 3and L 2Cr edges for a 20 nm-thick freshly exfoliated flake (blue curve) and after\none day under ambient conditions (red curve). (b) X-ray photoemission spectroscopy (Cr\n2p3=2;1=2core levels) of the same sample before (blue curve) and after (red curve) exposure\nto ambient conditions. (c) Raman spectra of the sample as a function of time.\nOxidation in ambient conditions\nAll magnetic vdW materials known to date are reactive materials, and often require cautious\nmanipulation.3,50We thus discuss here the stability of the CrTe 2compound. The magnetic\ncontrast observed in XMCD-PEEM shows a reduced dichroic signal for samples exposed to\nambient conditions: the contrast is typically reduced by one order of magnitude between\na freshly cleaved sample and a sample exposed to air for several days. Considering the\nmean-free path of the photoelectrons used for imaging, this observation suggests that CrTe 2\noxidizes with time upon exposure to ambient conditions, and that the oxidation is limited to\nthe outer layers (typically 1-2 nm) of the flakes. X-ray absorption spectroscopy (XAS) and\nphotoelectron spectroscopy (XPS) measurements were performed as a function of ambient\nexposure time, starting from freshly prepared CrTe 2flakes (see Figure 5). The XAS spectra\nprogressively evolve, revealing a multiplet-like Cr L 2;3absorption edges for samples exposed\n15to air, whereas they are essentially metallic on freshly cleaved flakes (see Figure 5a), with\nbinding energies close to the values obtained for clean, pure Cr.51Consistently, while Cr core\nlevel 2p 3=2;1=2binding energies (573.0, 583.4 eV) of freshly prepared flakes are close to those\nof pure Cr surfaces,52additional peaks are observed at higher binding energies (+2.4 eV)\nafter prolonged air exposure (see Figure 5b). We attribute the presence of these peaks to\nthe formation of a chromium oxide layer.52\nContrary to the x-ray measurements which require to expose the sample to air for about\n30 min, Raman spectroscopy can be performed under the ultra-clean environment of an\nAr glove box (including the exfoliation). Such freshly prepared flakes were sealed in situ\ninside a quartz tube, and Raman spectra were recorded on the sealed flakes, i.e.prior to any\nexposure to air. No noticeable difference was observed between the Raman spectra measured\nfor flakes exposed to air for about 30 min (Figure 1c,3c) and for sealed flakes.\nThe two Raman peaks (E 2g, A1g) of freshly exfoliated or sealed flakes, observed at 102\nand 134 cm\u00001, are replaced after one weak of exposure to air by two other peaks, at 126\nand 142 cm\u00001(see Figure 5c). In an intermediate degradation state, the peak at 126 cm\u00001\nessentially appears as a shoulder of the initial 134 cm\u00001peak, and the one at 142 cm\u00001as a\nmuch broader peak (see Figure 5c).\nThe\rphase of TeO 2(obtained after re-crystalisation of TeO 2glass) exhibits two Raman\npeaks at 116 and 140 cm\u00001, that are close to those we observe.53The corresponding vibration\nmodes were described with the A1 irreducible representation.53In a much related phase of\nTeO 2, the\u000bphase, the two modes are upshifted by \u00185 and\u001810 cm\u00001respectively.53\nThe glassy nature of TeO 2suggests, in our case where no special thermal treatment has\nbeen performed, the coexistence of various phases with more-or-less defined structure, and\neach characterised by Raman modes of slightly different intensity. This might account for the\nbroad peak observed around 140-150 cm\u00001at an intermediate state of the oxidation process.\nTime being a key factor in a kinetically-limited oxidation process such as here (occurring at\nroom temperature), we propose that the corresponding peak becomes narrower with time,\n16eventually leading to the (relatively) narrow peak centred at 142 cm\u00001corresponding to one\nspecific (more-or-less crystallized) TeO 2phase.\nConclusions\nCombining several spatially-resolved techniques sensitive to magnetism, room temperature\nferromagnetism is demonstrated in exfoliated CrTe 2flakes of thickness down to 20 nm. The\nCurie temperature of 315 K is similar to that of the parent bulk compound, whereas the\nweak coercivity measured experimentally shows that the material is magnetically soft. These\nresults are consistent with those reported in Ref. 54 using magneto-transport measurements.\nMagnetic imaging reveals the magnetic domain microstructure. In particular, large in-plane\nmagnetic domains are observed, forming flux-closure configurations with a tendency to six-\nfold symmetry. Such patterns explain the absence of remanence observed with ensemble-\naveraging probes used to address bulk samples, while our microscopic analysis reveals locally\nalmost-100%remanence. Themagneticdomainsareseparatedbyseveral10nmwidedomain\nwalls presumably of Néel type. These findings suggest that the magnetic patterns in the\nCrTe 2flakes results from both shape and magnetocrystalline anisotropies. This new thin\nvan der Waals ferromagnet is moderately reactive to oxidation under ambient conditions,\nwith the topmost few layers of the material being oxidized typically within a day.\nOur work introduces a new member in the growing family of van der Waals ferromagnets,\nwith in-plane easy axis and a Curie point above room temperature, which is attractive for\nboth applications and fundamental studies. In the prospect of applications, an in-plane mag-\nnetized room-temperature ferromagnet is attractive to build heterostructures hosting specific\nquantum phases, such as an anomalous quantum Hall effect,31or artificial antiferromagnets\nwith two CrTe 2layers of different thicknesses. Concerning fundamental studies, thinning the\nCrTe 2flakes down to the single layer limit is an exciting goal. This calls for an optimization\nof exfoliation processes, beyond what is already established for more traditional materials\n17like graphite and MoS 2. In addition, the use of highly sensitive magnetic probes,55should\nallow to determine whether the Curie temperature of the compound decreases with the layer\nthickness. This would tell us whether the absence of magnetic contrast that we found for\n10 nm-thin CrTe 2flakes is related to the limited sensitivity of MFM and XMCD-PEEM,\nor is rather intrinsic to the material itself, i.e.a manifestation of a truly two-dimensional\ncharacter.\nMaterials and Methods\nBulk CrTe 2synthesis. 1T-CrTe 2samples were synthesized indirectly by oxidation of\nKCrTe 2. The parent compounds were prepared by a molar mixture of the elements, Cr, K\nand Te under argon atmosphere in a glove box. This mixture was heated to the melting\npoint of both the alkali element and tellurium, and then kept at 900\u000eC for eight days in an\nevacuated quartz tube. The tubes were opened in a glove box in order to prevent oxidation.\nAlkali atom de-intercalation was then carried out by reacting the parent compounds in\nsolutions of iodine in acetonitrile. These suspensions were stirred for about 1 hour, using an\nexcess of iodine. The final product was washed with acetonitrile to remove the formed iodide,\nfiltered and dried under vacuum. A mixture of brilliant gray platelets about 3 mm-long, 0.3\nmm-thick were obtained.\nMagneto-optical focused Kerr magnetometry. A He-Ne laser (wavelength 632 nm)\nbeamof5mWpowerwaslinearlypolarized(s-polarization)anda100 \u0002objectivelensfocused\nthelaserbeamtoaspotwithadiameteroflessthan1 \u0016monthesample. Thereflectedsignal\nwas separated by a Wollaston prism into two beams with orthogonal polarizations impinging\non two identical photodiodes. The sum and difference signal of the two diodes was sent to\nan oscilloscope. The sample was placed in the gap of a small horseshoe magnet that allowed\napplying an in-plane magnetic field of up to 80 mT. The signal on the oscilloscope was\nmeasured during a field sweep at a frequency of 1 or 2 Hz with a maximum field of \u000610 mT.\n18To improve the signal to noise ratio, the hysteresis loops were averaged over about 100 field\nsweeps.\nMagnetic force microscopy. Magnetic images were obtained using a NT-MDT micro-\nscope. The magnetic layers of the scanning tips were Co 80Cr20, a semi-hard material with\ncoercivity of the order of 100 mT. This material was used to cap (50-75 nm thickness) our\nhome-made MFM tips, whose magnetisation is essentially perpendicular to the surface. Like\nusually the case in MFM experiments, we cannot provide more details about the magneti-\nsation configuration of the tip, hence we cannot extract quantitative information about the\nsample magnetisation. For the second pass image, acquired after the topographic one a tip\nlift height of about 100 nm yielded optimum MFM phase contrast and resolution, and the\ndomain structure of the flakes seemed unaffected by the scanning with the MFM magnetic\ntip (our observation with MFM and XMCD-PEEM are fully consistent).\nX-ray photoemission electron microscopy. The XAS, XMCD and XPS-PEEM mea-\nsurements were performed using the SPELEEM III microscopes (Elmitec GmbH) installed\nat the Elettra56and ALBA57facilities. The lateral resolution in XAS-PEEM nears 20 nm\nfor both microscopes. The XMCD-PEEM images shown here were obtained by subtracting\nPEEM images acquired with opposite helicities of the incoming photon beam and normalized\nto the sum of these images. Magnetic images are acquired at the Cr L 3absorption edge.\nX-ray absorption and photoelectron spectroscopy. XAS and XPS spectra were ob-\ntained from scans of either the photon energy (XAS) or the detected electron energy (E kin\nor start voltage). For each step, a PEEM image is acquired and local (XAS,XPS) spectra\nare extracted by plotting the averaged intensity in a selected area vs the scanned parameter.\nFor XPS, the binding energy was obtained as Ebind=Ephoton-Ekin-\u001eanalyzer.\nRaman spectroscopy measurements. The Raman spectra of the CrTe 2flakes were\nrecorded using a Witec Alpha 500 Raman microscope with Rayshield coupler, which is able\nto acquire low frequency spectra as low as 10 rel.cm\u00001. A laser with an excitation wavelength\nof 532 nm was focused to the sample with a 50 \u0002Mitutoyo, NA 0.42 objective. The power\n19of laser excitation measured below the microscope objective lens was set to 0.1 mW to avoid\nsample heating. A 1800 lines/mm grating was set to achieve higher resolution, \u00140.1 cm\u00001.\nThe spectra were collected with integration times of 30 s and number of accumulation of 5.\nAll measurements were performed at room temperature.\nAuthor Information\n*E-mail: johann.coraux@neel.cnr.fr\nAssociated Content\nThe authors declare no competing financial interest.\nAcknowledgement\nThis work was financially supported by the Flag-ERA JTC 2017 project ‘MORE-MXenes’.\nA.P. and N.R warmly thank Simon Le-Denmat and Olivier Fruchart for technical help during\nthe MFM measurements, and Michel Hehn for providing the Pt/Ta/Si substrates.\nSupporting Information Available\nThe following files are available free of charge.\nSupporting information comprises supplementary Raman spectroscopy data as function\nof light polarization and flake thickness, XMCD-PEEM analysis of magnetic anisotropy,\nAFM and Raman spectroscopy analysis of the flakes’ oxidation, and an analysis of the\nmagnetization versus temperature evolution in thick samples.\n20References\n1. 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Zhou2, 4\n1Department of Physics, Florida State University, Tallahassee, Florida 32306-3016, USA\n2National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310-3706, USA\n3National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, NM 87545, USA\n4Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA\n5Anhui Key Laboratory of Condensed Matter Physics at Extreme Conditions,\nHigh Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei, Anhui 230031, China\n6University of Science and Technology of China, Hefei, Anhui 230026, China\n7Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China\n8Key Laboratory of Artificial Structures and Quantum Control,\nSchool of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China\n9Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA\n(Dated: June 29, 2021)\nMagnetoelectric materials are attractive for several applications, including actuators, switches, and magnetic\nfield sensors. Typical mechanisms for achieving a strong magnetoelectric coupling are rooted in transition metal\nmagnetism. In sharp contrast, here we identify CsEr(MoO 4)2as a magnetoelectric material without magnetic\ntransition metal ions, thus ensuring that the Er ions play a key role in achieving this interesting property. Our\ndetailed study includes measurements of the structural, magnetic, and electric properties of this material. Bulk\ncharacterization and neutron powder diffraction show no evidence for structural phase transitions down to 0.3 K\nand therefore CsEr(MoO 4)2maintains the room temperature P2/c space group over a wide temperature range\nwithout external magnetic field. These same measurements also identify collinear antiferromagnetic ordering of\nthe Er3+moments below TN= 0.87 K. Complementary dielectric constant and pyroelectric current measure-\nments reveal that a ferroelectric phase ( P\u00180.5 nC/cm2) emerges when applying a modest external magnetic\nfield, which indicates that this material has a strong magnetoelectric coupling. We argue that the magnetoelectric\ncoupling in this system arises from a pseudo Jahn-Teller distortion induced by the magnetic field.\nI. INTRODUCTION\nThe interest in magnetoelectric and multiferroic materi-\nals has been growing due to possible applications in new\ndevices1,2. Many efforts have focused on developing a de-\ntailed understanding of the magnetoelectric properties and\nmechanisms in materials based on magnetic transition metal\nions. The strong magnetoelectric coupling in these sys-\ntems has been achieved based on several design princi-\nples, including magnetostriction3–5, spin current6and the in-\nverse Dzyaloshinskii-Moriya interaction7. In most cases, a\nmagnetic-field-induced modification of the ordered spin struc-\nture gives rise to changes in the ferroelectric properties8.\nWell-known multiferroics with a strong magnetoelectric cou-\npling include the rare-earth manganites with general formula\nRMnO 3, whereRis a rare-earth element9. However, it is\ngenerally argued that the role of the rare earth elements in this\nfamily is limited to a fine tuning of the exchange parameters\narising from their slightly different ionic radii10.\nIt was proposed recently that Jahn-Teller distortions (JTDs)\ncan also provide a viable mechanism for achieving multifer-\nroicity with an intrinsic magnetoelectric coupling11,12. JTDs\nare generally active for non-linear molecules with degen-\nerate electronic ground states13. JTDs occurring coopera-\ntively via an electron-phonon coupling mechanism generate\nstructural instabilities that are responsible for numerous in-\ntriguing and important physical phenomena such as high-T c\nsuperconductivity14–17and colossal magnetoresistance18–21.Although conventional JTDs are realized in systems with de-\ngenerate electronic energy levels, two non-degenerate levels\nseparated by a small energy gap can also induce a JTD-like\ninstability called a pseudo Jahn-Teller distortion (pJTD) when\nthe energy gain from the deformation is larger than the gap.\nFor a conventional JTD, group theoretical analysis shows that\nthe normal mode associated with the distortion always has\nan even representation and therefore centrosymmetry is pre-\nserved if it is already present in the higher-symmetry state. For\nthis reason, the conventional JTD can induce multiferroicity in\nnon-centrosymmetric systems only11,12, where the local elec-\ntric dipole moments and magnetic ordering typically have dif-\nferent onset temperatures (i.e. Type I multiferroics)12, which\noften results in a weak magnetoelectric coupling. On the\nother hand, this rule does not extend to the pJTD, as the two\nclosely-spaced energy levels may have opposite parity and\ntherefore can be mixed to make an odd phonon mode active as\nwell. Consequently, a unique feature of a pJTD is its ability\nto induce local electric dipole moments in centrosymmetric\nsystems22and therefore this phenomenon is responsible for\nmany common ferroelectrics23,24. Interestingly, several recent\ntheoretical studies have shown that a pJTD can also induce\nmultiferroicity with a sizable magnetoelectric coupling24–27.\nAlthough predicting and identifying JTD-induced multifer-\nroic and magnetoelectric materials has been actively pursued,\nthe focus again has mostly been limited to materials with mag-\nnetic moments arising from 3 dtransition metals. Well-known\nexamples include (Ga,Ge)(V ,Mo) 4S828–30and Ba 2TGe2O7(TarXiv:2007.04274v1  [cond-mat.mtrl-sci]  8 Jul 20202\nFIG. 1: (color online) (a) A schematic of the refined crystal and magnetic structures in the absence of an applied magnetic field. We do\nnot show the O ions for clarity. (b) A picture of a representative CsEr(MoO 4)2single crystal used in this study. Several single crystals were\ncrushed up for the HB-2A neutron powder diffraction experiment. (c) The refined neutron powder diffraction pattern collected at T= 250 K.\nThe data is plotted with black symbols, the structural model is superimposed on the data with a red curve, the ticks below the pattern show the\nexpected Bragg peaks for the CsEr(MoO 4)2crystal structure and the Al sample can, and the difference pattern is indicated by a solid black\ncurve.\n= V , Ni)12. On the other hand, it has been known for a long\ntime that rare-earth systems with local magnetic moments of-\nten exhibit structural phase transitions that originate from a\nJTD31–34, but the possibility of multiferroicity and magneto-\nelectric coupling has been rarely studied. In contrast to 3 d\ntransition metal systems, highly-localized electrons in rare-\nearth magnets generate weakly-dispersive bands, leading to\nlarge peaks in the electronic density-of-states. If these elec-\ntronic bands can be tuned by external parameters such as a\nmagnetic field so they exhibit significant overlap with peaks\nin the phonon density-of-states, and electron-phonon coupling\nis allowed by symmetry35, then strong electron-phonon inter-\nactions may result.\nIn this article, we report the discovery of a strong mag-\nnetoelectric coupling in a material where rare-earth elements\nare solely responsible for both magnetic order and ferroelec-\ntricity. We observed this magnetoelectric behavior below TN\n= 0.87 K in the alkali rare-earth double molybdate (ARDM)\nCsEr(MoO 4)2, and we argue that it arises from a field-induced\npJTD mechanism. Our comprehensive study shows that in-\ntrinsic magnetoelectric coupling may be ubiquitous in rare-\nearth compounds susceptible to JTDs and therefore calls for\ndetailed experimental and theoretical studies to identify other\nmagnetoelectric ARDMs. Specific investigations focused on\nelucidating the nature of the electronic energy levels andphonon modes involved in the pJTD for CsEr(MoO 4)2are\nalso highly desirable.\nII. EXPERIMENTAL METHODS\nSingle crystals of CsEr(MoO 4)2were grown by a flux\nmethod. Starting materials Cs 2CO3, Er2O3and MoO 3with\npurities no less than 99.9% were mixed with masses of\n0.814 g, 0.191 g and 1.152 g respectively in a covered alumina\ncrucible. The mixture was then warmed up to 1000°C with a\nramping rate of 100°C/hr and held there for 10 hours before\ncooling down to 600°C at a rate of 3°C/hour and then quench-\ning to room temperature. The CsEr(MoO 4)2single crystals\nwere isolated by washing the product with water.\nHigh-resolution neutron powder diffraction (NPD) mea-\nsurements were performed using the HB-2A powder diffrac-\ntometer at the High Flux Isotope Reactor (HFIR) of Oak\nRidge National Laboratory (ORNL). About 3 g of crushed sin-\ngle crystals were loaded in a cylindrical Al can and mounted in\nthe He-3 insert of a cryostat to achieve a base temperature of\n0.3 K. We used neutron wavelengths \u0015= 1.54 and 2.41 ˚A with\na collimation of open-21’-12’. We performed Rietveld refine-\nments of the NPD patterns with the FullProf software suite36.\nThe dc magnetic susceptibility data were obtained with a3\nQuantum Design superconducting quantum interference de-\nvice (SQUID) magnetometer. The high field dc magnetiza-\ntion was measured using a vibrating sample magnetometer\n(VSM) at the National High Magnetic Field Laboratory. The\nac susceptibility was measured with the conventional mutual\ninductance technique with a homemade setup37. The specific\nheat data were obtained using a commercial Physical Prop-\nerty Measurement System (PPMS, Quantum Design) by the\nrelaxation method.\nThe capacitance was measured on single crystal samples\nwith thin plate geometry by painting two faces with silver\nso they could be used as electrodes for an Andeen-Hagerling\nAH-2700A commercial capacitance bridge. The capacitance\nwas converted to the dielectric constant by approximating the\nsample as an infinite parallel plate capacitor. To measure the\nelectric polarization, we first measured the pyroelectric cur-\nrent under different cooling protocols and then integrated the\npyroelectric current with respect to time. The detailed proce-\ndure of the pyroelectric current measurement has been pub-\nlished elsewhere38.\nIII. RESULTS\nWe performed neutron powder diffraction (NPD) on\ncrushed single crystals to determine the crystal structure at\n250 K in the absence of an applied magnetic field. A\nschematic of the crystal structure is shown in Figure 1(a)\nand a picture of a representative single crystal is depicted in\nFigure 1(b). The crystal has a thin-plate geometry, is easily\ncleaved, and the normal to the flat face is parallel to the a-axis.\nThe Rietveld refinement of the 250 K data is presented in Fig-\nure 1(c) and reveals that the Bragg peaks can be indexed with\nthe monoclinic space group P2/c, which is consistent with\nthe previous room-temperature structure determination39. The\nlow crystal symmetry likely arises from severe distortions\nof the orthorhombic KEr(MoO 4)2structure due to the large\nionic radius of Cs35. Notably, structural chains of Er3+ions\nrun along the b-axis, which may be indicative of a low-\ndimensional magnetic system. This arrangement of magnetic\nions is in sharp contrast to the trigonal system RbFe(MoO 4)2,\nwhere the magnetic Fe ions form a frustrated triangular sub-\nlattice that generates a 120 degree spin structure41.\nWe collected additional NPD data with \u0015= 1.54 ˚A down\nto 1.5 K to look for changes in the diffraction pattern that\nwould be indicative of structural phase transitions, but none\nwere found. Therefore, CsEr(MoO 4)2retains the P2/c space\ngroup over the entire temperature range investigated. The lat-\ntice constants, fractional coordinates, and key structural pa-\nrameters at 1.5 K and 250 K are presented in Table I. No-\ntably, the Wyckoff position of the magnetic ion Er3+is 2f\nand its point symmetry is C2, ensuring a common quantiza-\ntion axiszfor all Er3+ions along the c-axis. According to\nHund’s rules, the Er3+ground-state multiplet is described by\nJ=15\n2. Due to the low local symmetry of Er3+, crystalline\nelectric fields (CEFs) from the neighboring oxygen ions are\nexpected to split the 2J+ 1degenerate electronic levels into\neight Kramers doublets. Previous work has shown that theground state doublet and first excited state have predominantly\f\fJz=\u000615\n2\u000b\nand\f\fJz=\u000613\n2\u000b\ncharacter, respectively39. In\nother words, CsEr(MoO 4)2is an Ising magnet with a strong\neasy axis anisotropy along the crystalline c-axis40.\nTABLE I: Structural parameters for CsEr(MoO 4)2extracted from the\nrefinements of the 1.54 ˚A neutron powder diffraction data. The lattice\nconstants and Er-Er intrachain distance are in ˚A and the\fangle is in\ndegrees.\nT 250 K 1.5 K\nSG P2/c P2/c\na 9.5102(9) 9.470(1)\nb 7.9196(7) 7.9062(8)\nc 5.0350(3) 5.0295(4)\n\f 91.265(7) 91.374(8)\nEr 0.5, 0.25, 0.006(3) 0.5, 0.25, 0.003(3)\nCs 0, 0.25, 0.040(3) 0.5, 0.25, 0.035(3)\nMo 0.309(1), -0.004(1), 0.519(1) 0.310(1), -0.003(2), 0.522(1)\nO1 0.437(1), 0.000(1), 0.251(1) 0.439(1), 0.004(1), 0.250(1)\nO2 0.307(1), 0.175(1), 0.746(2) 0.305(1), 0.176(1), 0.746(2)\nO3 0.140(1), 0.982(1), 0.387(1) 0.138(1), 0.982(1), 0.385(1)\nO4 0.313(1), 0.815(1), 0.732(2) 0.311(1), 0.813(1), 0.733((2)\nEr-Er 3.96 3.95\nRwp 5.61 % 6.79 %\n\u001f27.60 11.2\nAs shown in Figure 2, the temperature-dependence of the\ndc susceptibility for CsEr(MoO 4)2,\u001f(T), exhibits no signs of\nmagnetic ordering down to 2 K for both H//a-axis andH?\na-axis. A linear Curie-Weiss fit of the inverse dc susceptibility\ndata from 100-300 K yields \u0016e\u000b=10:3\u0016Band\u0012CW=\u000070K\nforH//a-axis and\u0016e\u000b= 9:2\u0016Band\u0012CW= 30 K forH\n?a-axis. These effective moment values are consistent with\nthe free ion value of \u0016e\u000b= 9:58\u0016Bfor Er3+(4I15=2). The\ndc magnetic susceptibility for H?a-axis is about 20 times\nhigher than for H//a-axis in the paramagnetic regime, in-\ndicative of a hard axis along the a-direction. The large \u0012CW\ntemperatures are not necessarily indicative of strong exchange\nbetween Er3+ions because the intrinsic values may be signif-\nicantly influenced by low-lying crystal field levels. Further-\nmore, the sign difference of \u0012CWfor the two directions likely\narises from the high degree of magnetic anisotropy in this sys-\ntem, which is a typical characteristic of rare-earth magnets.\nThe inset in Figure 2(a) shows a Curie-Weiss fit of the inverse\nsusceptibility over the low temperature range only, and the\nresulting\u00120\nCW=-2 K is likely more indicative of the true ex-\nchange interaction strength between Er3+ions. Close agree-\nment betweenj\u00120\nCWjand the ordering temperature identified\nin the low-temperature dc susceptibility and specific heat mea-\nsurements described below indicate no strong magnetic frus-\ntration in this system42, which is consistent with the arrange-\nment of the Er3+ions in the crystal structure.\nThe dc susceptibility measured below 2 K with H//a-axis\nis shown in Figure 3(a); for \u00160H= 0.01 T a broad peak is\nobserved around 1.2 K followed by a sharp peak and the on-\nset of a zero-field-cooled (ZFC) / field-cooled (FC) divergence\nat 0.87 K. Although the broad peak disappears and the sharp\npeak becomes less pronounced with increasing field, the latter4\n┴(a)\n(b)\nFIG. 2: (color online) The temperature-dependence of the dc sus-\nceptibility and its inverse for CsEr(MoO 4)2with (a)H//a-axis and\n(b)H?aaxis. The dashed lines represent fits to the Curie-Weiss\nlaw. The inset shows the inverse susceptibility vs temperature over\nthe low-temperature range only with the corresponding Curie-Weiss\nlaw fit superimposed on the data.\nis still clearly visible and essentially temperature-independent\nfor\u00160H < 3 T. A sharp lambda anomaly is also observed in\nthe specific heat data shown in Figure 3(b) at 0.87 K, which is\nconsistent with the dc susceptibility data and therefore likely\nindicative of long-range magnetic order. Notably, the entropy\nrecovered when warming through the phase transition is less\nthanRln2 expected for a rare earth magnet with a CEF ground\nstate doublet. The real part of the ac susceptibility measured\nwith zero dc field, shown in Figure 3(c), displays a broad peak\naround 1.2 K and a sharp drop around 0.87 K; these obser-\nvations are in good agreement with the dc susceptibility and\nspecific heat data. The imaginary part of the ac susceptibil-\nity measured with zero dc field, presented in Figure 3(d), ex-\nhibits a small sharp peak around 0.87 K and therefore provides\nfurther support for long-range magnetic ordering below this\ntemperature. The long-range ordering transition temperature\nidentified here agrees well with previous work39.\nThe missing entropy when warming above TNand the\nbroad peaks in the dc magnetic susceptibility and real part\nTN  = 0.87 K (a)\n(b)(c)\n(d)FIG. 3: (color online) (a) The temperature-dependence of the dc sus-\nceptibility below 2 K with H//a-axis for different applied magnetic\nfields. The data collected at different fields is vertically-offset for\nclarity. (b) The specific heat Cp, plotted asCp=T, and the entropy\nrecovered on warming as a function of temperature. (c) The real part\nof the ac susceptibility vs temperature for different dc fields H//a-\naxis. (d) The imaginary part of the ac susceptibility vs temperature\nfor different dc fields H//a-axis.\nIR BV Atom BV components\nmkamkbmkcimkaimkbimkc\n\u00001 1 1 1 0 0 0 0 0\n2 -1 0 0 0 0 0\n 2 1 0 1 0 0 0 0\n2 0 -1 0 0 0 0\n\u00002 3 1 1 0 0 0 0 0\n2 1 0 0 0 0 0\n 4 1 0 1 0 0 0 0\n2 0 1 0 0 0 0\n\u00003 5 1 0 0 1 0 0 0\n2 0 0 -1 0 0 0\n\u00004 6 1 0 0 1 0 0 0\n2 0 0 1 0 0 0\nTABLE II: Basis vectors (BV) for possible irreducible representa-\ntions of the space group P2/c withk= (0:5;0;0)and Er site (0.5,\n0.25, 0.003).\nof the ac magnetic susceptibility are hallmarks of a low-\ndimensional magnetic system. Interestingly, a simple exami-\nnation of the crystal structure reveals Er chains running along\ntheb-axis and therefore supports quasi-one-dimensional be-\nhavior, although a power law fit of the magnetic Bragg peak\nintensity for the related compound CsDy(MoO 4)2is indica-\ntive of quasi-two-dimensional Ising behavior instead43. Re-\ngardless of the specific nature of the low-dimensionality, the\nbroad peaks observed in the susceptibility measurements in-\ndicate that short-range correlations are building up with de-\ncreasing temperature before the onset of long-range magnetic\norder. The low-dimensionality of this system may also ex-\nplain the ZFC/FC divergence in the dc magnetic susceptibil-\nity belowTN, as finite correlation lengths along one or both5\ninterchain directions can produce magnetic domains that can\nbe manipulated by temperature or an applied magnetic field.\nPrevious support for this idea comes from calculating possi-\nble magnetic structures stabilized by dipolar interactions, as\nit is found that there are two antiferromagnetic chain config-\nurations with different interchain correlations that are com-\nparable in energy39. Previous work on the related compound\nCsDy(MoO 4)2is also relevant, as a finite correlation length\nalong thea-direction was identified below TNthat could be\ntuned by the cooling rate through TN43. This magnetic do-\nmain formation and rearrangement under magnetic field may\nbe responsible for the peaks observed in the the ac suscepti-\nbility data when a small dc field is applied.\nTo determine the magnetic structure below TNin the ab-\nsence of an applied magnetic field, we collected neutron pow-\nder diffraction data at 0.3 and 1.5 K using the HB-2A powder\ndiffractometer with a neutron wavelength of 2.41 ˚A. We ver-\nified that CsEr(MoO 4)2remained in the P2=cspace group at\n0.3 K and we identified magnetic Bragg peaks in the same\ndataset below TN. Interestingly, the magnetic Bragg peaks\nare not resolution-limited, which is most apparent for the ex-\ntra scattering centered at 2\u0012=19° as shown in Figure 4(a).\nThe broad nature of these peaks is likely a consequence of a\nfinite correlation length along one or more interchain direc-\ntions, as discussed above. Nonetheless, the magnetic peaks\nobserved can be indexed with the propagation vector k=(0.5\n0 0). To model the magnetic structure we first performed\na symmetry analysis using SARAh44. Ultimately, we find\nthat the best magnetic refinement of the 0.3 K diffraction pat-\ntern is achieved by using the \u00003irreducible representation\n(see Table II for details). The refined 0.3 K diffraction pat-\ntern, shown in Figure 4(b), yields an ordered Er moment of\nmc= 2:80(3)\u0016B. Although the ordered moment magni-\ntude is much smaller than the value of \u00189\u0016Bexpected for\na\f\fJz=\u000615\n2\u000b\nCEF ground state doublet, this is not surpris-\ning because the refinement assumes that the magnetic Bragg\npeaks are resolution-limited. The spin structure is schemat-\nically presented in Figure 1(a) and consists of antiferromag-\nnetic chains of Er ions running along the b-axis with the mo-\nments pointing along the c-axis. We note that this moment\ndirection is consistent with the Ising anisotropy determined\nfor this system previously39,40. The magnetic structure of\nCsEr(MoO 4)2also consists of interchain correlations that are\nferromagnetic along the c-axis and antiferromagnetic along\nthea-axis, and it is reminiscent of the ordered spin configura-\ntion for the related system KTb(WO 4)245. This spin arrange-\nment corresponds to the expected magnetic ground state for\na simple Hamiltonian with nearest-neighbor dipolar interac-\ntions only39.\nWe now turn to the magnetic properties of CsEr(MoO 4)2\nin the presence of significant applied magnetic fields. Fig-\nure 5 shows the isothermal magnetization measured with both\nH//a-axis andH?a-axis at 0.7 K and 4.4 K. The strong\ndirectional-dependence of this data provides additional sup-\nport for the large Ising anisotropy of this system. For H?\na-axis, the magnetization sharply increases below 0.5 T and\ngradually approaches a saturation value of 8.97 \u0016B, which\nis the moment size for a free Er3+ion. Below TN, similarfeatures are observed with a slightly sharper increase at low\nfields. The relatively low applied fields required for H?a-\naxis to saturate the magnetization to the Er3+free ion value\nare consistent with a CEF ground state doublet of predomi-\nnantly\f\fJz=\u000615\n2\u000b\ncharacter both above and below TN, as\nidentified in previous work39. Although we find no clear ev-\nidence of metamagnetic transitions up to 35 T for this field\norientation, previous work indicates that the sharp increase in\nlow fields arises from a metamagnetic transition below TNfor\nH//c-axis39. Presumably, this corresponds to a spin flip tran-\nsition within the Er chains. Our high-field data cannot resolve\nthis transition clearly due to the coarse point density in the\nlow-field regime and the lack of in-plane crystal orientation\ninformation. On the other hand, our magnetization curves at\n4.4 K and 0.7 K for H//a-axis show a clear upturn centered\nat 25 T, which is indicative of a distinct metamagnetic transi-\ntion. Since this transition persists well above TN, we antici-\npate that it arises from crystalline electric field effects. More\nspecifically, there is likely a crossing of crystal field levels at\n25 T due to a larger g-factor value along the a-axis for the ex-\ncited state relative to the ground state doublet. We also note\nthat since the isothermal magnetization curves above and be-\nlowTNare similar along both directions, the internal field\nfrom the magnetic ordering only generates a small splitting\nof the CEF Kramers doublets below TN, which is typical for\nrare-earth magnets46.\nFigure 6(a) shows the temperature and magnetic field de-\npendence of the dielectric constant for CsEr(MoO 4)2with E\n//a-axis, which will now be referred to as \"0(T)and\"0(H)\nrespectively. \"0(T)shows no obvious anomaly below 1.5 K in\nthe absence of an applied magnetic field. On the other hand,\nwith\u00160H= 0.5 T applied along the a-axis a peak is visible\nat 0.84 K, which is slightly lower than TN. As the magnetic\nfield increases to 1 T, the intensity of this peak increases and\nits position shifts to 0.64 K. As the field is increased further,\nthis peak continuously shifts to lower temperatures and disap-\npears by 3 T. Meanwhile, \"0(H)measured at 0.3 K with H//\na-axis displays a sharp peak at \u00160H= 1.5 T. As the tempera-\nture increases, the peak intensity reaches a maximum at 0.48\nK before rapidly decreasing as the temperature rises above\nTN. The dielectric constant is also highly anisotropic, as we\nfound that\"0(T)withH?a-axis is anomaly-free (not shown\nhere).\nFigure 7(a) shows the pyroelectric current measured at vari-\nous magnetic fields ( H) applied along the a-axis with various\npoling electric fields. The spontaneous and reversible pyro-\nelectric current confirms the ferroelectricity of this compound\nbelowTN. Although there is no obvious anomaly observed\nbelow 1 K in the absence of a magnetic field, a modest applied\nfield of 0.2 T generates a peak below 0.9 K. With increasing\nmagnetic field up to 0.8 T, the peak intensity increases and its\nposition shifts to lower temperatures. When \u00160Hincreases\nto 1 T, the pyroelectric current peak decreases slightly and is\ncentered at 0.67 K, which is consistent with the 0.64 K peak\nin the dielectric constant observed at the same magnetic field.\nTo check the switchability of the polarization, we also mea-\nsured pyroelectric current under opposite poling electric field\nof -300 V with the magnetic field held fixed at 1 T, and indeed6\nFIG. 4: (color online) (a) Neutron powder diffraction patterns collected at 1.5 K (red curve) and 0.3 K (black symbols). (b) The refined\nneutron powder diffraction pattern collected at T= 0.3 K. The data is plotted with black symbols, the refinement result is superimposed on the\ndata with a red curve, the ticks below the pattern show the expected Bragg peaks for the CsEr(MoO 4)2crystal structure, the Al sample can,\nand the CsEr(MoO 4)2magnetic structure, and the difference pattern is indicated by a solid black curve.\n1086420M (µB / Er3+)\n35302520151050µ0H (T)8.97 µB 4.4 K 0.7 KH // aH ⊥ a\nFIG. 5: (color online) The magnetization measured as a function of\napplied magnetic field at 0.7 K and 4.4 K for H//a-axis and H?\na-axis.\nthe sign of the pryoelectric current changes accordingly. Ad-\nditionally, we measured the pyroelectric current without pol-\ning electric field applied and found no measurable response\ndue to randomly-oriented ferroelectric domains. These fea-\ntures strongly suggest that ferroelectricity is induced with an\napplied magnetic field H//a-axis. We also found that the\npresence or absence of magnetic field during cooling did not\nmake any difference. Figure 7(b) shows the electric polariza-\ntion obtained by integrating the pyroelectric current with time.\nAlthough the polarization is tiny, the finite value demonstrates\nthat the magnetic field-induced structural phase transition is\nassociated with spontaneous electric dipole moments. Taken\ntogether with the evidence from dc magnetic susceptibility for\nlong-range magnetic order with H//a-axis up to 3 T (at least),this indicates that CsEr(MoO 4)2is a strong magnetoelectric\nmaterial for particular combinations of temperature and mag-\nnetic field.\nIV . DISCUSSION\nGiven the results from the various experimental techniques,\nwe describe a possible electronic energy diagram in Figure 8\nto explain how an applied magnetic field induces ferroelec-\ntricity in this compound. As discussed above, previous work\nfor CsEr(MoO 4)2has identified a CEF ground state doublet\nand a first excited state that are close to pure\f\fJz=\u000615\n2\u000b\nand\f\fJz=\u000613\n2\u000b\n, respectively39. BelowTN, the internal magnetic\nfield generated by the ordered state leads to a small splitting\nof the Kramers’ doublets but no CEF level crossings. In the\npresence of an increasing applied magnetic field with H//a-\naxis, the larger g-factor for the first excited state (i.e. g0\na) rela-\ntive to the ground state value gamake the difference in energy\nsmaller. When the necessary condition A> \u0001(H)is met for\nthe pJTD, where Ais the distortion-induced energy gain and\n\u0001(H)is the energy gap between the two states under field,\nthe structural phase transition to the ferroelectric phase oc-\ncurs. Increasing the magnetic field further generates the CEF\nlevel crossing at 25 T observed in the magnetization measure-\nments. This field-induced sequence of transitions is consistent\nwith the extremely low energy gap \u0001= 1.6 meV between the\nCEF ground state doublet and first excited state47. Therefore,\na pseudo Jahn-Teller distortion incorporating the\f\fJz=15\n2\u000b\nand\f\fJz=13\n2\u000b\nCEF levels, as illustrated in Figure 8, is a vi-\nable mechanism for inducing ferroelectricity in this material.\nIn highly anisotropic crystals such as CsEr(MoO 4)2, the\nacoustic phonon dispersions can generate a peak in the phonon\ndensity-of-states that is similar in energy to the magnetic field-\ninduced pseudo-degenerate CEF states35. If this condition is\nmet and spin-phonon coupling is allowed by symmetry, then\nstrong spin-lattice coupling may arise. Recall that the electric7\n(a)\n(b)\nFIG. 6: (color online) (a) Temperature and (b) magnetic field depen-\ndence of the dielectric constant with E//a-axis andH//a-axis. An\nanomaly is observed for applied magnetic fields between 0.5 and 2 T\nat temperatures below TN.\npolarization for CsEr(MoO 4)2appears parallel to the a-axis\nonly. It is then interesting to note that previous work on the\nsister compound CsDy(MoO 4)2identified acoustic phonons\nwith wavevectors ~kka-axis as the driving force for the co-\noperative Jahn Teller structural transition at 38 K48. There-\nfore, we infer that similar phonon modes are responsible for\nthe field-induced structural instability and hence the electric\npolarization along this direction in CsEr(MoO 4)2.\nV . CONCLUSION\nWe performed detailed experimental studies characterizing\nthe structural, magnetic, and electric properties of the alkali\nrare-earth double molybdate CsEr(MoO 4)2. We found that\nthe room-temperature P2/c space group persists down to 0.3 K\nand the zero-field magnetic structure below TN=0.87 K con-\n(a)\n(b)FIG. 7: (color online) (a) The pyroelectric current measured with H\n//a-axis andE//a-axis. (b) The electric polarization for H//a-axis\nandE//a-axis, obtained by integrating the pyroelectric current with\ntime.\nsists of antiferromagnetic chains of Er ions with Ising mo-\nments pointing along the c-axis. Most intriguingly, we identi-\nfied magnetoelectric behavior with H//a-axis belowTNin a\nfairly narrow pocket of the T\u0000Hphase diagram, consistent\nwith a theoretical prediction49. We attribute the strong mag-\nnetoelectric coupling in this material to a pseudo Jahn-Teller\ndistortion that arises due to the relatively small energy gap\nbetween the Er3+CEF ground state doublet and first excited\nstate. Therefore, we have shown that rare-earth elements can\nbe solely responsible for magnetoelectric coupling through a\nJahn-Teller mechanism. Our work calls for first principles cal-\nculations and experimental characterization of the electronic\nband structure and phonon density-of-states for CsEr(MoO 4)2\nto better understand the pJTD mechanism responsible for the\nmagnetoelectric effect in this system.\nAcknowledgments\nWe acknowledge K.M. Taddei for providing assistance\nwith the NPD refinements and G. Sala for performing the8\n|± 15/2>|± 13/2>\n∆\n|+ 15/2>| – 15/2>|+ 13/2>| – 13/2>\nHpJTDga’   >  ga\nMagnetic Field∆(H) < A \nHmetaT > TNT < TN\nMolecular\nField\nFIG. 8: (color online) Possible electronic energy level diagram for\nCsEr(MoO 4)2. The CEF ground state doublet and first excited state\nhave predominantly\f\fJz=\u000615\n2\u000b\nand\f\fJz=\u000613\n2\u000b\ncharacter, respec-\ntively, and they are separated by an energy gap \u0001of 1.6 meV47. The\ninternal magnetic field generated by the magnetic order below TN\nonly leads to a small splitting of the Kramers’ doublets, but an ap-\nplied magnetic field with H//a-axis produces a ferroelectric tran-\nsition atHpJTD, where the condition A > \u0001(H)is met. Increasing\nthe magnetic field further generates a CEF level crossing and hence a\nmetamagnetic transition at Hmeta. The small value of \u0001and the iden-\ntification of the 25 T metamagnetic transition imply that a pseudo\nJahn-Teller distortion is possible in this system.CsEr(MoO 4)2CEF point charge calculations. Q.C., Q. H.\nand H.D.Z. thank the support from NSF-DMR through Award\nDMR-1350002. This research used resources at the High\nFlux Isotope Reactor, a DOE Office of Science User Fa-\ncility operated by the Oak Ridge National Laboratory. A\nportion of this work was performed at the NHMFL, which\nis supported by National Science Foundation Cooperative\nAgreement No. DMR-1157490 and the State of Florida.\nE. S. C. and M. L. acknowledge the support from NSF-\nDMR-1309146. G. H. W. and J. M. were supported by the\nNational Natural Science Foundation of China (11774223,\nU1732154), and the Ministry of Science and Technology of\nChina (2016YFA0300501). Z.W., L.L., and Z. Q. were sup-\nported by the National Natural Science Foundation of China\nunder Grant Nos 11774352, U1832214. A portion of this\nwork was supported by the High Magnetic Field Laboratory\nof Anhui Prince.\n\u0003Electronic address: minseong.lee10k@gmail.com\nyElectronic address: aczelaa@ornl.gov\n1S.-W. Cheong and M. Mostovoy, Nature Mater. 6, 13 (2007).\n2M. Fiebig, J. Phys. D: Appl. Phys. 38, R123 (2005).\n3C. Wang, G.-C. Guo, and L. He, Phys. Rev. Lett. 99, 177202\n(2007).\n4C. Wang, G.-C. Guo, and L. He, Phys. Rev. B 77, 134113 (2008).\n5Y . J. Choi, H. T. Yi, S. Lee, Q. Huang, V . Kiryukhin, and S.-W.\nCheong, Phys. Rev. Lett. 100, 047601 (2008).\n6H. Katsura, N. Nagaosa, and A. V . Balatsky, Phys. Rev. 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Phys. 28, 755 (2002)."    },    {        "title": "2007.04984v1.Nanoscale_Magnetic_Domain_Memory.pdf",        "content": "Chapter 3\nNanoscale Magnetic Domain Memory\nKarine Chesnel\nAdditional information is available at the end of the chapter\nhttp://dx.doi.org/10.5772/intechopen.71076\nProvisional chapter© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons  \nAttribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, \nand reproduction in any medium, provided the original work is properly cited. \nDOI: 10.5772/intechopen.71076Nanoscale Magnetic Domain Memory\nKarine Chesnel\nAdditional information is available at the end of the chapter\n“Magnets are a bit like humans.  \nNot only they  attract each other, they also have the capability to remember.”\nAbstract\nMagnetic domain memory (MDM) is the ability exhibited by certain magnetic materials \nto reproduce the exact same nanoscale magnetic domain pattern, even after it has been \ncompletely erased by an external magnetic field. In this chapter, we review the various \ncircumstances under which this unusual phenomenon occurs. We explain how partial \nMDM was first observed in rough Co/Pt multilayers with perpendicular magnetization \nas a result of structural defects. We then show how 100 % MDM was achieved, even in \nsmooth ferromagnetic films, by coupling Co/Pd multilayers to an antiferromagnetic IrMn \ntemplate via exchange interactions. We describe how high MDM, extending through -\nout nearly the entirety of the magnetization process, is obtained when zero-field-cooling \nthe material below its blocking temperature where exchange couplings occur. We also \nreview the persistence of MDM through field cycling and while warming the material all \nthe way up to the blocking temperature. Additionally, we discuss the spatial dependence \nof MDM, highlighting intriguing oscillatory behaviors suggesting magnetic correlations \nand rotational symmetries at the nanoscopic scales. Finally, we review the dependence of \nMDM on cooling conditions, revealing how MDM can be fully controlled, turned on and \noff, by adjusting the magnitude of the cooling field .\nKeywords:  magnetic domains, ferromagnetic films, perpendicular magnetic anisotropy, \nmagnetic domain memory, defect-induced memory, exchange-coupling induced \nmemory\n1. Introduction to the principles of MDM\n1.1. Magnetic domain patterns in ferromagnetic films\nFerromagnetic materials are typically composed of magnetic domains. A magnetic domain \nis a region where the magnetic moments carried by individual atoms, i.e., the atomic \n© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative\nCommons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,\ndistribution, and reproduction in any medium, provided the original work is properly cited.\nspins, align in the same direction due to exchange couplings, as illustrated in Figure 1a . \nRather than forming one giant macroscopic magnetic domain, the material often breaks \ndown into a multitude of microscopic domains of different orientations, as illustrated \nin Figure 1b . At the delimitation between one magnetic domain and a neighboring one, \na coherent rotation of the atomic spins occurs. The delimitating region, called domain \nwall, may be small in respect to the domain size. Domain sizes typically range from 1 to \n100 μm in bulk ferromagnetic materials, and from 100  nm to 1 μm in thin ferromagnetic \nfilms [1–3].\nIn a ferromagnetic material, the microscopic magnetic domains of various magnetiza -\ntion directions arrange in specific ways that minimize the competing magnetic energies \npresent in the system. The three dimensional sum of the magnetic moments carried by \nthe individual domains produces, at the macroscopic scale, a net magnetization M. The \nmagnitude and the direction of the net magnetization depends on how the magnetic \nmoments of the individual domains are distributed throughout the material. While M \nmay be uniquely set by applying an external magnetic field H, following a specific mag -\nnetization procedure, the associated microscopic magnetic domain pattern, or topology, \nis usually not unique. The formation of magnetic domain topologies and their correla -\ntion with the magnetic history of the material is still today a vast field of research to be \nexplored and understood.\nThin ferromagnetic films have been extensively studied during the past decades, in particu -\nlar because of the variety of magnetic domain patterns they exhibit [ 4–8]. Depending on the \ncomposition, the crystallographic structure and the thickness of the film, the magnetization \nmay point in-plane or out-of-plane. The magnetic domains may take various shapes, from \nround bubbles to elongated, almost infinite, stripes. Some thin film materials exhibit complex \nformations such as vortices [ 9–11] and skyrmions [ 12, 13], illustrated in Figure 1c, d , which \nhave instigated a renewed interest in the recent years because of their potential applications \nin magnetic memory technologies and spintronics [ 14, 15].\nFigure 1.  (a) Schematic illustrating the formation of magnetic domains by alignment of atomic spins. The schematic \nshows a domain wall, formed by the coherent rotation of spins. (b) Schematic of a collection of magnetic domains in a \nferromagnetic material. (c) Schematic of a magnetic vortex core [ 10]. (d) Lorentz microscopy image of a skyrmion lattice \nin FeCoSi . Extracted from  Fert et al.  [14].Magnetism and Magnetic Materials 381.2. The case of thin films with perpendicular magnetic anisotropy\nThin ferromagnetic films with perpendicular magnetic anisotropy (PMA) [ 16] exhibit a partic -\nularly rich set of magnetic domain patterns. In PMA films, the domain magnetization points \nmostly out-of-plane, thus producing high demagnetization fields and leading to the formation \nof regular domain patterns and magnetic textures [ 17–21]. PMA films have attracted an accrued \nattention in the years 2000–2010s because the perpendicular magnetization has enabled signifi -\ncant domain size reductions, in comparison to the in-plane magnetic films, thus benefiting the \nmagnetic recording industry. State-of-the-art ultra-high density magnetic recording technologies \nutilizes granular PMA thin film media, where magnetic domain sizes are as small as 20–50  nm \n[22].\nIn smooth PMA films, magnetic domains take a variety of shapes and they form patterns of \nvarious topologies. Such variety of shapes and topologies is for example  observed in [Co/Pt]N \nmultilayers where the thickness of Co typically varies between 5 and 50  Å, the thickness of the \nPt layer is around 7  Å, and the number of repeat around N  = 50. In these multilayers, the PMA \nis primarily achieved by exploiting the surface anisotropy created by the layering between \nthe Co and the Pt layers, and the high number of repeats, as well as the magneto-crystalline \nanisotropy produced by the crystallographic texturing [ 23–26].\nThe magnetic domain topologies observed in these Co/Pt multilayers vary from bubble \ndomain patterns to pure maze patterns formed of long interlaced striped domains, as shown in  \nFigure 2. Both patterns in Figure 2 are observed at remanence, where the external field is H = 0. \nAt that point, the net magnetization is M ≈ 0, so the area covered by the domains of a given direc -\ntion nearly equals the area covered by the domains in the opposite direction. Both the bubble \nFigure 2.  Magnetic Force Microscopy (MFM) images of magnetic domains in a [Co(31  Å)/Pt(7 Å)]50 multilayer with \nperpendicular magnetic anisotropy. These domains patterns are measured at remanence ( H = 0). The two colors \nrepresent opposite magnetization directions, pointing out-of-plane and into the plane. Image size is 10 × 10  μm. (a) Maze \npattern forming after applying H = 2000 Oe; (b) bubble pattern forming after applying H = 9000 Oe. The saturation point \nis H = 11,400 Oe for this film. Extracted from Chesnel et  al. [27].Nanoscale Magnetic Domain Memory\nhttp://dx.doi.org/10.5772/intechopen.7107639pattern and the maze pattern, as well as a wide array of intermediate patterns, can be observed \nat remanence in the same multilayer. The topology of the remanent pattern mainly depends on \nthe magnetic history of the film. In particular, it has been shown that the remanent magnetic \ntopology drastically change when the magnitude Hm of the magnetic field applied perpendicular \nto the film changes. If the applied field is saturating, the film form a maze-like pattern at rema -\nnence. However, if Hm is set to a specific value slightly below the saturation point, a bubble pat -\ntern is achieved and the domain density is maximized [ 27]. Each time the external magnetic field \nis cycled, the magnetic domain pattern is erased. When the field is released back to H = 0, the new \nremanent domain pattern is, in this material, different from the previous one.\n1.3. Magnetic domain memory\nMagnetic domain memory (MDM) is the tendency for magnetic domains to retrieve the \nsame exact same pattern after the pattern has been erased by a saturating magnetic field. \nWhen MDM occurs, not only the magnetic domains retrieve the same type of shape and size, \nbut their distribution in space, or topology, is similar. MDM may be total, in which case the \ndomain topology is identical; or MDM may be partial, in which case the domain pattern is \nsimilar, with some topological differences.\nFerromagnetic films usually do not exhibit any MDM.  Like in the Co/Pt multilayers previ -\nously mentioned, the domain pattern at a given field point H generally does not repeat after \nthe field has been cycled [ 28]. The magnetic domains may always take the form of small bub -\nbles or of elongated stripes of specific size at that point in field, but their spatial distribution \nwill be completely different after each cycle. This is well illustrated in Figure 3, which shows \nmagnetic domain patterns in Co/Pt multilayers measured at remanence before and after \napplying a minor magnetization cycle. The initial pattern exhibits elongated striped domains \nFigure 3.  Evolution of the magnetic domain pattern in a [Co(8  Å)/Pt(7 Å)]50 multilayer while cycling the magnetic field \nH through half a minor loop. Plotted is the net magnetization M (H). The MFM images are 5 × 5  μm and are all collected \nat the same location of the film. (1) At remanence ( H = 0), before applying the field; (2) at H = 3400 Oe, domains have \nshrunk down to small bubbles; (3) back to remanence ( H = 0), after applying the field . Extracted from  Westover et al.  [28].Magnetism and Magnetic Materials 40of opposite up / down magnetization  with close to a 50% up: 50% down coverage. At the high -\nest magnetic field value, which is not quite saturating, the domains of reverse magnetization \nhave shrunk down to small sparse bubbles. Upon return to zero field, the film exhibits mag -\nnetic stripes again but these stripes are shorter and their spatial distribution, or topology, is \ncompletely different than in the initial image. In this case, the film does not exhibit any MDM.\nMDM has been so far only observed in specific thin ferromagnetic films that have spe -\ncific structural or magnetic properties. As explained in the following sections, MDM has \nfirst been discovered in rough Co/Pt films, due to the presence of defects, acting as pin -\nning sites for the domain nucleation. In that case, the observed defect-induced MDM is \npartial and only occurs in the nucleation phase of the magnetization cycle. We will see \nhow MDM can however be maximized by exploiting magnetic exchange between the \nferromagnetic (F) layer and an antiferromagnetic (AF) layer. This has been successfully \nachieved  in [Co/Pd] IrMn films where F Co/Pd multilayers are sandwiched in between \nAF IrMn layers. The film is field-cooled (FC) down below its blocking temperature, to \nallow exchange couplings to occur. Under these conditions, the magnetic domain pat -\ntern imprinted in the IrMn layer plays the role of a magnetic template. In this case, the \nobserved exchange-bias induced MDM reaches 100% throughout a large portion of the \nmagnetization process.\n2. Probing MDM\n2.1. Real space imaging\nMagnetic domain patterns may be directly measured via Magnetic Force Microscopy (MFM), \nas seen in Figures 2  and 3. The MFM technique allows the visualization of magnetic domains \nby probing the out-of-plane magnetic stray fields emanating from the surface of the film. With \na spatial resolution down to about 20–25  nm, MFM allows the detection of individual magnetic \ndomains in thin PMA ferromagnetic films, as these domains are typically 50–200  nm wide.\nThe investigation of MDM may be possible via MFM under certain experimental conditions. \nIn 2003, Kappenberger et  al. [29] showed in their study of CoO/[Co/Pt] multilayers via MFM \nthat the specific magnetic domain pattern initially observed in a particular region of the film \nwas fully recovered after magnetically saturating the material, as seen in Figure 4 . In this \nexperiment, the material was zero-field-cooled (ZFC) down to 7.5  K and then imaged at 7.5  K \nwhile applying a large magnetic field up to 7  T. The initial striped domain pattern observed in \nthe Co/Pt layer immediately after ZFC was completely erased under application of magnetic \nfield, but it was fully retrieved when the magnetic field was decreased back to near coercive \npoint Hc on the descending branch of the magnetization loop. At Hc, the net magnetization is \nnear zero, like it was during ZFC.\nSuch experiment requires very sophisticated MFM instrumentation, with cryogenic and in-\nsitu magnetic field capabilities. Only very few MFM instruments in the world allow such \nextreme temperature and field environments. Because of the difficulty to measure MFM \nimages at low temperature and under high in-situ magnetic field, the study of MDM via \nMFM is somewhat limited in practice. The magnetization of the MFM tip may reverse while Nanoscale Magnetic Domain Memory\nhttp://dx.doi.org/10.5772/intechopen.7107641the in-situ magnetic field is being varied. Also the maximum possible value for the magnetic \nfield may be smaller than the value required to saturate the material, making the collection of \nMFM images throughout the entire magnetization loop impossible. Lastly, MFM images are \ntypically micrometric, covering a few microns of the film, and a handful of magnetic domains, \nthus providing a localized view only.\n2.2. Coherent x-ray resonant magnetic scattering\nComplementary to MFM, the technique of coherent x-ray magnetic scattering [ 30, 31] is a pow -\nerful tool to study MDM.  Under coherent illumination, the material produces a speckled scatter -\ning pattern that is a unique fingerprint of the charge and magnetic configuration of the material \n[32, 33], as illustrated in Figure 5. Because x-rays are insensitive to magnetic fields, an in-situ \nmagnetic field of any value and any direction can be applied to the material while collecting \nthe x-ray scattering signal. This allows the study of MDM throughout the entire magnetization \nprocess. Also, if mounted on a cryogenic holder, the material can be cooled down under various \nFigure 5.  Layout for coherent x-ray resonant magnetic scattering (C-XRMS). The synchrotron x-ray beam is spatially \nfiltered by a pinhole to enhance the transvers coherence. The energy of the x-ray is finely tuned to a magnetic resonance, \nthus providing longitudinal coherence and magneto-optical contrast. Under coherent illumination, the magnetic \ndomains in the film produce a speckle pattern collected on a CCD camera. Extracted from Chesnel et  al. [33].\nFigure 4.  Evolution of the magnetic domain pattern in an exchange-biased CoO/ [Co/Pt] multilayer. The MFM images \nare 5 × 5 μm. (a) Magnetization loops M(H) measured perpendicular to the film at 5  K after field-cooling (FC) under 1  T \nand zero-field-cooling (ZFC) the film. (b) Initial striped domain pattern measured right after ZFC. (c) At H  = +650 mT on \nthe ascending branch of the magnetization loop, the white domains have shrunk or disappeared. (d) At H  = −400 mT on \nthe descending branch of the magnetization loop after saturating, the pattern is exactly reverse of the initial pattern in \n(a). Extracted from Kappenberger et  al. [29].Magnetism and Magnetic Materials 42field cooling conditions, and x-ray scattering can be measured at low temperature. Lastly, the \nregion illuminated by the x-rays is set by the size of the spatial filter used to obtain transverse \ncoherence. The illumination spot size, typically in the order of 20–50  μm, allows covering a \nrelatively large number of magnetic domains, thus providing useful statistical information [ 34].\nA magneto-optical contrast can be obtained by finely tuning the x-ray energy to specific res -\nonance edges associated to the magnetic elements present in the material. This technique, \ncalled x-ray resonant magnetic scattering (XRMS), offers chemical selectivity [ 35–37]. Typical \nresonant edges used in XRMS are the L2 and L3 edges which correspond to the excitation of \nelectrons from the (2p) to the (3d) electronic bands in transition metals. For iron (Fe), the L2,3 \nedges are at around 708 and 720  eV, respectively. For cobalt (Co), the L2,3 edges are at around \n778 and 791  eV, respectively. At these energies (soft x-rays), the x-ray wavelength is around \n1–2 nm, thus allowing the probing of magnetic structures in ferromagnetic films, with spatial \nresolutions down to few nanometers.\nBy collecting XRMS patterns at various magnetic field values, one can follow the evolution of \nthe magnetic configuration of the material throughout the magnetization process. An example \nof such experiment is shown in Figure 6, where XRMS patterns collected on the descending \nbranch of the magnetization loop are compared to MFM images at nearly same field values. \nAt saturation, no magnetic domain exists so the XRMS pattern has no magnetic signal, only a \npure charge background. Shortly after nucleation, sparse domains have nucleated and started \nto expand, leading to a diffuse disk around the center of the XRMS pattern. As the domain \npropagation progresses, interlaced domains of opposite directions fill the entire space, lead -\ning to a ring-like  XRMS pattern. The radius of the ring relates to the magnetic period existing \nin the material at that stage [ 38].\nCombining coherent x-rays with XRMS, the technique of coherent-XRMS (C-XRMS) turns out \nto be very useful to study MDM [ 33]. Measured in the scattering space, or the so-called recipro -\ncal space, the CXRMS pattern provides an indirect view of magnetic domains. The inversion \nof the CXRMS pattern into the real space image is a complex process due to a phase loss in \nthe intensity of the scattering signal [ 32]. However, if the x-ray beam is made coherent, the \nFigure 6.  Probing magnetic changes in a [Co/Pd]IrMn film via MFM and XRMS. (a) Magnetization loops measured at 320  K \nand at 20  K after ZFC. (b) Evolution of the magnetic domain pattern at 300  K, viewed via 10 × 10  μm images. (c) Evolution \nof the XRMS pattern at 20  K in ZFC state at similar points in magnetization. Extracted from Chesnel et  al. [38].Nanoscale Magnetic Domain Memory\nhttp://dx.doi.org/10.5772/intechopen.7107643x-ray scattering pattern produced by the material will show speckles, as illustrated in Figure 7. \nThe specific speckle pattern is a unique fingerprint of the magnetic domain pattern in the real \nspace. Comparing speckle patterns collected at different points throughout magnetization then \nallows the evaluation of MDM in the material.\n2.3. Speckle cross-correlation metrology\nTo evaluate MDM via CXRMS, speckle patterns collected at different field values are com -\npared. This comparison is typically done via cross-correlation [ 39, 40]. In the cross-correlation \nprocess, the intensity in the two images is compared pixel by pixel. In practice, this done by \nmultiplying the intensity of two image, pixel by pixel, while one image is spatially shifted in \nrespect to the other one. Mathematically, this operation may be written as follows:\n  A × B=  ∑       \ni,j   A  (  i,j )   B  (  X + i, Y + j )     (1)\nwhere A and B are the two images being correlated, A(i,j) represents the intensity of image \nA at pixel ( i,j) and B(X+i,Y+j) represents the intensity of image B at pixel ( X+i,Y+j). Note \nthat a shift of X pixels in the first direction and Y pixels in the second direction is applied \nbetween the two images. The sum takes care of covering all the pixels present in the images. \nThe resulting cross-correlation product A×B is a function of the shift ( X,Y). It forms a pattern \nFigure 7.  Extraction of speckle patterns and cross-correlation process. (a) Magnetic domain pattern in real space, \nmeasured via MFM.  The image is 10 × 10μm. (b) Associated XRMS pattern as collected on the CCD detector. (c) Speckle \npattern, or pure coherent component, extracted from the XRMS pattern (b). (d) Zoomed-in view on the speckle spots. \n(e) Correlation pattern showing a peak at its center. The area under the peak provides an estimate of the amount of \ncorrelation between two speckle patterns. Extracted from Chesnel et  al. [33].Magnetism and Magnetic Materials 44in the ( X,Y) space, which is called correlation pattern. An example of such correlation pattern \nis shown in Figure 7e . To accelerate the cross-correlation operation, which may be compu -\ntationally costly when comparing thousands of scattering images, the cross-correlation may \nbe done via fast Fourier Transform (FFT) operations. When using FFT, the correlation pat -\ntern A×B is an image of same size than A and B. It generally shows a peak around its center, \nfor which X  = 0 and Y = 0. The intensity under the peak provides the amount of correlation \nbetween the two images. The width of the peak in the ( X,Y) space corresponds to the average \nspeckle size, which is generally set by the optics.\nTo quantify MDM, the intensity in the cross-correlation pattern A×B is integrated and com -\npared to the intensity of the auto-correlated patterns A×A and B×B. A normalized correla -\ntion coefficient is thus evaluated as follows:\n  ρ=   ∑ X,Y    A × B  __________________   \n √ _____________________    ( ∑ X,Y    A × A)  ( ∑ X,Y    B × B)       (2)\nThe coefficient ρ is then comprised between 0 and 1. If the two images A and B are completely \ndifferent, ρ will be close to 0. If the two images A and B are exactly the same, ρ = 1 (or 100%). \nSince each of the correlated speckle patterns is a unique fingerprint of the magnetic domain \nconfiguration, the correlation coefficient ρ can be used to quantify MDM.  When ρ is low, mag -\nnetic domain patterns are very different, and there is no or little MDM.  When ρ is high (close \nto 1), magnetic domain patterns are very similar, and MDM is close to 100%.\n2.4. Mapping MDM\nMDM can be evaluated in a number of different ways. The most straightforward way to \nevaluate MDM is to compare magnetic speckle patterns measured at the same point H in \nmagnetic field, after a full cycle has been completed, as illustrated in Figure 8a . This compar -\nison is called returned point memory (RPM) [ 39]. An alternative way is to compare magnetic \nFigure 8.  Illustration of various cross-correlation approaches. (a) Single point correlation. RPM: cross-correlating pairs \nof speckle patterns collected at same field ( H, H). CPM: cross-correlating pairs of speckle patterns collected at opposite \nfields ( H, −H). (b) Cross-field correlations AxA: correlating any point on the ascending branch with any other point on \nthe ascending branch (c) cross-field correlations AXD: correlating any point on the ascending branch and any other point \non the descending branch. In each approach, the two points to be correlated may be separated by several loops. Extracted \nfrom Chesnel et  al. [38].Nanoscale Magnetic Domain Memory\nhttp://dx.doi.org/10.5772/intechopen.7107645speckle patterns measured at opposite field values. In that case, one point is located at field \nH on the ascending branch of the magnetization loop whereas the other point is located at \nopposite field −H is on descending branch. This comparison is called conjugate point mem -\nory (CPM). RPM and CPM can then be plotted as function of the magnetic field H, where H \ntypically varies from negative saturation to positive saturation, on the ascending branch of \nthe magnetization loop.\nA more complete exploration of MDM may be performed by cross-correlating two speckle \nimages A and B collected any field value H throughout the magnetization process [ 40]. The \nresulting correlation coefficient ρ may be then mapped in a two-dimensional field space ( H1, \nH2) where H1 is the field value of image A and H2 is the field value of image B. The resulting \nρ(H1, H2) map is called correlation map. The two field values ( H1, H2) may be both chosen on \nascending branches, either on the same branch or on two different branches separated by \none of more cycles. Such correlation map is denoted as A × A.  A similar study can be done \non the descending branch, leading to a D × D map. If the magnetization loop is symmetri -\ncal, the correlations on A × A and D × D are expected to be the same. In another approach, \none field value H1 may be chosen on the ascending branch and the other value H2 on the \ndescending branch, leading to an A × D map. Examples of such correlation maps ρ(H1, H2) \nare shown in Figure 13 .\nMore subtle measurements of MDM may exploit the spatial information contained in the \nCXRMS patterns. Instead of computing the cross-correlation on the entire image, the cor -\nrelation is computed on specific portions of the image. One can for example look at the \ndependence on the scattering vector q, which is the distance from the center of the scatter -\ning pattern to a specific point in the image. The correlation is then performed on rings of \nspecific q radii selected from the scattering patterns, as illustrated in Figure 9 . The resulting \ncorrelation coefficient ρ may be plotted as a function of q, or mapped in a ( q, H) space [40]. \nAn example of such pattern is shown in Figure 9d . Other approaches explore the angular \ndependency of ρ.\nFigure 9  . Q-selective correlation process. (a) Initial speckle pattern; (b) ring selection from the speckle pattern; the radius \nof the ring is Q; (c) correlation pattern resulting from cross-correlating ring selections of two speckle patterns; (d) example \nof ρ(Q, H) correlation map measure on a ZFC [co/Pd]/IrMn film. Extracted from Chesnel et  al. [40].Magnetism and Magnetic Materials 463. Defect-induced MDM\n3.1. First observations in Co/Pt thin films\nThe first studies of MDM via x-ray speckle correlation metrology were carried out on [Co/Pt] \nmultilayered thin films by Pierce et  al. [39] in 2003. These multilayers, made of 50 repeats of \na Co/Pt bilayers, where the Co thickness is 0.4  nm, the Pt thickness is 0.7  nm, exhibit perpen -\ndicular anisotropy, leading to the formation of microscopic striped domain patterns.\nThese magnetic correlation studies demonstrated the occurrence of microscopic magnetic \nreturn-point-memory (RPM) when these films exhibited some interfacial roughness. It was \nfound that smooth films with no roughness did not produce any RPM.  However, films with \nlarge roughness produced significant RPM in the nucleation region of the magnetization pro -\ncess. It was established that the observed microscopic magnetic memory was induced by the \npresence of defects in the film, playing the role of anchors for magnetic domains to nucleate. \nThis phenomenon is referred to as ‘defect-induced’ or ‘disorder-induced’ MDM [ 41, 42].\nIn Figure 10 , magnetic domain configurations are shown for different film roughnesses. When \nthe film is smooth (grown under 3 mT of Ar pressure), a typical interlaced stripe domain pattern \noccurs. When the film is rough (grown under 12 mT of Ar pressure), the magnetic domain pat -\ntern becomes fuzzy and the magnetic periodicity is somewhat lost. The associated XRMS scat -\ntering profile shows a well-defined peak for the 3 mT film, but a weaken peak for the 12 mT film.\nIn these studies, microscopic magnetic correlations were evaluated while cycling the mag -\nnetic field throughout minor loops and major loops. In particular RPM and CPM were \nmeasured at various field values along the ascending and the descending branches of the \nmajor magnetization loop. These measurements were carried out on films with various \nroughnesses.\nFigure 10.  Effect of roughness on the magnetic domain configuration in [Co(4  Å)/Pt(7 Å)]50 multilayers. (a) and (b) 3 x \n3 μm MFM images for films grown under different Ar pressure (a) 3 mT, (b) 12 mT. (c) XRMS profiles for films with \ndifferent roughnesses (the label indicates the Ar pressure in mT). Extracted from Pierce et  al. [41].Nanoscale Magnetic Domain Memory\nhttp://dx.doi.org/10.5772/intechopen.71076473.2. RPM and CPM dependence on magnetic field\nRPM and CPM were evaluated on the [Co(0.4  nm)/Pt(0.7  nm)]50 multilayers at different points \nthroughout the ascending and descending branches of the major loop after cycling the field \nmultiple times. Co/Pt multilayers with significant interfacial roughness exhibited non-zero \nRPM and CPM, as illustrated in Figure 11a . It was found that CPM was systematically lower \nthan RPM.  This suggested that disorder has a component that breaks spin reversal symmetry. \nIt was also found that both RPM and CPM exhibited the same trend: their value was larger at \nnucleation ( H = −1 kOe) and decreased monotonically down to near zero when the magnetic \nfield was increased toward saturation ( H = +4 kOe). The high correlation value ρ at nucleation \nsuggested that magnetic domains tend to nucleate at specific locations, which are pinned by \nthe defects. The decreasing trend down to zero correlation when field is increasing suggests \nthat once nucleated, the magnetic domains propagate rather randomly, or non-deterministi -\ncally, throughout the film, leading to decorrelation [ 43].\n3.3. Dependence on roughness\nAs expected, the observed defect-induced MDM depended  on the amount of interfacial \nroughness exhibited by the [Co(0.4  nm)/Pt(0.7  nm)]50 multilayers. If the film was relatively \nsmooth, no memory was found. If, on the contrary, the film was rough enough, some RPM \nand CPM would be detected. The dependence of MDM with interfacial roughness is illus -\ntrated in Figure 11b , where RPM and CPM are measured at the coercive point. For rough -\nness below 0.5  nm or so, no RPM and no CPM was observed. When the roughness exceeded \n0.5 nm, both RPM and CPM quickly increased and started to plateau at around 0.7  nm of \nroughness, reaching about 40% for CPM and 50% for RPM.  This behavior, measured at room \ntemperature was, to some extent, reproduced by nonzero-temperature numerical simulations \nbased on Ising models [ 41, 42].\nThe observation of microscopic magnetic memory induced by defects in [Co/Pt] mul -\ntilayered thin films opened the door to new perspectives and led to further questions, \nboth on the fundamental and applied levels. In particular, the observed CPM and RPM \nwere somewhat limited in magnitude, not exceeding 50–60% in best cases, and also they \noccurred in a limited region of the magnetization process, namely the nucleation region. \nFigure 11.  RPM and CPM correlations in rough [Co(4  Å)/Pt(7 Å)]50 multilayers. (a) RPM and CPM vs. field on the \nascending branch of the magnetization loop for a rough film grown under 8.5 mT Ar pressure. (b) RPM and CPM values \nat the coercive point, plotted as a function of interfacial roughness. Extracted from Pierce et  al. [41].Magnetism and Magnetic Materials 48Would there be ways, in some materials, to increase MDM to higher values and to extend \nit to a wider region of the magnetization loop?\n4. Exchange-bias induced MDM\nIn an attempt to increase the amount of MDM initially observed in rough ferromagnetic materials \ncame the idea of incorporating exchange couplings in the film by interlaying the ferromagnetic \n(F) layer with an antiferromagnetic (AF) layer that would play the role of a magnetic template.\nThis was successfully achieved by combining a F Co/Pd multilayer with an AF IrMn layer \n[44]. After cooling the material below its blocking temperature TB, and inducing exchange \ncouplings (EC), high MDM was observed, which extended throughout almost the entire mag -\nnetization cycle. The observed MDM reached unprecedented values as high as 100%, even \nwhen the film was smooth.\nIf the material is cooled down under a non-zero field, the EC interactions produce a net \nexchange bias (EB) in the magnetization loop. However, because the origin of MDM is micro -\nscopic, high MDM can be observed even when zero-field cooling the material (in the absence \nof field), in which case not net bias exists.\n4.1. First MDM observations in [Co/Pd]IrMn multilayers\nThe first observations of EC induced MDM were reported by Chesnel et  al. [44] in 2008, in [Co/\nPd]/IrMn multilayers. It was found that when zero-field-cooled (ZFC), the material exhibited high \nMDM. The magnetic correlations reached high values throughout a wide range of field values.\nThe material consisted of an interlay of F [Co(0.4  nm)/Pd (0.7  nm)]12 multilayers with AF \nlayers made of IrMn (2.4  nm) alloy, repeated 4 times. This film exhibited PMA, leading to \nthe formation of serpentine magnetic domains that were about 150–200  nm wide, as seen in \nFigure 12a .\nFigure 12.  MDM measurements in ZFC [Co/Pd]/IrMn multilayers. (a) 10 × 10  μm MFM image of the striped magnetic \ndomains. (b) Magnetization loops M(H) at 300 K and at 20  K in ZFC state. (c) RPM vs. field H plotted against the \nascending branch of the magnetization curve. Also plotted is the intensity of the XRMS signal. Extracted from Chesnel \net al. [44].Nanoscale Magnetic Domain Memory\nhttp://dx.doi.org/10.5772/intechopen.7107649To enhance the exchange couplings between the [Co/Pd] and the IrMn layers, the film was \nfirst demagnetized at high temperature and was then cooled down below its blocking tem -\nperature, TB ~ 300 K, down to temperatures as low as 20  K. MDM was probed at 20  K using \nx-ray speckle correlation metrology. CXRMS patterns were collected at finely spaced field \nvalues throughout the magnetization loop seen in Figure 12b  while an in-situ magnetic field \nwas cycled numerous times.\n4.2. Dependence on magnetic field throughout magnetization loop\nThe MDM observed in the ZFC [Co/Pd]/IrMn film shows an interesting behavior, which \ndrastically differs from the behavior observed in rough Co/Pt films. Figure 12c  shows the \namount RPM measured in [Co/Pd]/IrMn at 20  K after ZFC and its dependence on magnetic \nfield throughout the ascending branch magnetization loop. Unlike for rough Co/Pt films , \nRPM is low at nucleation. However, as the magnetic field increases, RPM rapidly increases \nwhen the field approaches the remanent coercive point Hcr and plateaus at values as high as \n80–90%. The plateau extends over a wide field region above the coercive point. RPM eventu -\nally decreases down to zero, when the field approaches saturation.\n4.3. Cross-field MDM maps\nIn addition to measuring RPM, cross-field magnetic correlations were further carried out \nthroughout the entire magnetization loop. Speckle patterns collected along the ascending \nbranch and along the descending branch of the magnetization loop were cross-correlated. \nResulting correlation maps ρ(H1, H2) are shown in Figure 13 , specifically the A × A map \n(correlations between two ascending branches) and the A × D map (correlations between \nascending and descending branches) measured after one field cycle. The RPM and CPM \ninformation may be sliced from these map along their diagonal. Both the A × A and A × \nD correlation maps show high MDM, with ρ reaching as high as 95  ± 5% in the central \nregion where H1 ≈ H2 ≈Hcr. The high correlation forms a plateau extending throughout \na wide region of field values from near above nucleation to near below saturation. This \nis not only true along the diagonal, where H1 = H2, but also off-diagonal where H1 ≠ H2, \nrevealing that the nanoscale magnetic domain patterns formed at these field values are all \nvery similar [ 38].\n4.4. Imprinting of a magnetic template via field cooling\nThe drastic behavioral differences between the MDM observed in ZFC [Co/Pd]/IrMn films \nand the MDM observed in rough Co/Pt films arise from the different microscopic magnetic \ninteractions. In one case, MDM is induced by defects or disorder. In the other case, MDM is \ninduced by exchange couplings. In this later case, MDM exits even in the absence of defects, \nthat is, in smooth films. In the absence of defects, MDM is low at nucleation, as observed in \nFigure 12 , because domains nucleate at random locations in the film. The reason for high \nMDM to occur at higher field values is the presence of a magnetic template imprinted in the \nAF layer during the cooling.Magnetism and Magnetic Materials 50In the specific material, the magnetic pattern gets imprinted from the F Co/Pd layer into the \nAF IrMn layer through exchange couplings via interfacial uncompensated spins. In ZFC state, \nthe imprinted pattern is typically formed of interlaced magnetic domains with opposite mag -\nnetization, pointing perpendicular to the material, either out-of-plane or into the plane, as \nillustrated in Figure 14a, b . The area covered by one magnetization direction nearly equals \nthe area covered by the other magnetization direction. When, in the ZFC state, the external \nmagnetic field is cycled, the magnetic domains in the F layer successively nucleate, propagate, \nexpand and eventually collapse at saturation. However, due to the frozen underlying magnetic  \npattern in the AF layer, the domain formation process in the F layer is not random (as it would \nbe for a single smooth F layer). The domain formation process is highly guided by exchange \ncoupling interactions with the underlying frozen template, so that when the remanent coer -\ncive point is reached, the domain pattern exactly matches the imprinted one [ 38, 44].\nGuided by the exchange interactions with the magnetic template imprinted in the AF layer, \nthe magnetic domain formation and reversal in the F layer is deterministic. The magnetic \ndomains in the F layer always form in a way to match the underlying template. Consequently, \nhigh MDM occurs from about the remanent coercive point Hcr, where the template is fully \nmatched, all the way up to nearly saturation, as domains evolve in a way to conserve the \ntemplate topology. This happens both on the ascending and the descending branches of the \nmagnetization loop, as illustrated in Figure 14c . Also the correlations between the ascending \nFigure 13.  Cross-field MDM maps measured on [Co/Pd]/IrMn multilayers at 20  K in ZFC state. (a) A × A map measured \non same cycle, (b) A × D map measured on same cycle, (c) A × A map measured after one cycle, (b) A × D map measured \nafter one cycle. Extracted from Chesnel et  al. [38].Nanoscale Magnetic Domain Memory\nhttp://dx.doi.org/10.5772/intechopen.7107651and descending branches is high. This is possible in the ZFC state because the imprinted pat -\ntern is made of long interlaced stripes with equal coverage of up and down domains.\n5. Persistence of MDM through field cycling\nWhen MDM is observed, a question that arises is if the memory persists through mul -\ntiple field cycles. This question was investigated both in disorder-induced MDM and in \nexchange-bias induced MDM.  In both cases, the magnetic correlations persisted through \nfield cycling.\n5.1. Field cycling dependence measurements\nIn the ergodic assumption, the measured correlation, which is an average over a statistical \nensemble of microscopic magnetic domains, is expected to be the same than the time averaged \nmagnetic correlation one would measure at any given location of the material. Because the \narea of the material probed by the x-rays typically includes thousands of magnetic domains \nand the system is at equilibrium, the ergodic assumption applies. It is therefore expected that \nthe magnetic correlation measured between two magnetic states separated by a given number \nof cycles should not change in time.\nFor statistical purposes, correlation coefficients measured at the same number of separat -\ning loops but at different times have been averaged. For example, if speckle patterns were \ncollected throughout five subsequent field cycles, the correlation coefficients between cycles \nFigure 14.  Illustration of the origin of MDM in exchange biased [Co/Pd]/IrMn multilayers. (a) and (b) Schematics of \nthe domain pattern forming in the F layer (top) and AF layer (bottom) in a ZFC state where a striped domain pattern is \nimprinted in the AF layer (a) at nucleation, sparse bubble nucleate randomly in the F layer, (b) at the coercive point, the \nF domain pattern matches the AF imprinted pattern. (c) Schematics showing the occurrence of MDM throughout the \nmagnetization cycle. MDM is the strongest in the central region of the magnetization loop from the remanent coercive \npoint all the way up to near saturation, both on the ascending and descending branches of the magnetization loop. \nExtracted from Refs. [ 38, 44].Magnetism and Magnetic Materials 521 & 2, cycles 2 &3, cycles 3 & 4 and cycles 4 & 5 were all averaged, producing an average \ncorrelation coefficient corresponding to one separating cycle. Similar averages were applied \nfor two separating cycles (averaging cycles 1 & 3, 2 & 4 and 3 & 5), three separating cycles, \netc. Ultimately, the average correlation coefficient ρ was studied as a function of the number \nof separating cycles [ 45].\n5.2. Dependence of MDM on number of field cycles\nRPM and CPM correlations in [Co/Pd] multilayers and in [Co/Pd]/IrMn films were measured \nthroughout many field cycles. In Figure 15a , the average RPM measured in Co/Pt films after \n1 and after 11  cycles is plotted against the field H on the ascending branch of the magnetiza -\ntion loop. Even after 11 loops, RPM is as high as after one cycle [ 39]. In Figure 15b, c , the \naverage RPM and CPM values measured in ZFC [Co/Pd]/IrMn films at specific points in \nfield are plotted as a function of number of separating cycles. Both RPM and CPM appeared \nnearly constant as the number of separating cycles was increased. In particular, the optimal \nRPM value, which occurs near the coercive region, remains within 95–100%, and the optimal \nCPM in that region remains within 90–95%. The observed persistence of MDM with field \ncycling is consistent with predictions for exchange-bias induced magnetic memory. Indeed, \nin these systems, the magnetic domain template imprinted in the AF layer is frozen, mean -\ning it does not change while the magnetic field is cycled, as long as the temperature is kept \nconstant below the blocking point. The domain pattern in the F layer will therefore tend to \nalways retrieve that same unchanged magnetic template, independently from the number of \nfield cycles [ 45].\n6. Spatial dependence of EC-induced MDM\nThe MDM results previously discussed in this chapter were obtained by cross-correlating \nentire speckle patterns altogether. The associated correlation numbers provided an ensemble \nFigure 15.  Persistence of MDM through field cycling. (a) RPM measured in Co/Pt multilayers after 1  cycle (red) and \n11 cycles (blue). (b) and (c) MDM measured in exchange-biased [Co/Pd]/IrMn films (b) RPM vs. field cycle (c) CPM vs. \nfield cycle. Extracted from Refs. [ 39, 45].Nanoscale Magnetic Domain Memory\nhttp://dx.doi.org/10.5772/intechopen.7107653average microscopic information, but no spatial dependency was probed. Spatial information \nis however included in the 2D speckle patterns which are being correlated for the estimation \nof MDM.  This spatial information can be exploited to extract information about possible spa -\ntial dependency in MDM.  Interesting oscillatory spatial dependence was found in the MDM \nexhibited by [Co/Pd]IrMn multilayers.\n6.1. Exploring spatial dependence\nEach 2D speckle pattern, such as the one in Figure 7, represents the intensity of the x-rays coher -\nently scattered by the material. Because the scattering process involves an inversion from the \nreal space to the scattering space, the spatial scale on the speckle images is an inverse of a dis -\ntance. A common quantity to locate positions in the speckle patterns is the scattering vector q. \nThe origin of the vector q is the center of the scattering pattern, as illustrated in Figure 9. The \nmagnitude of q indicates spatial scales d being probed in the real space. q and d are linked by \nan inverse relationship:  q=  2π ___ d   . The larger q is, the smaller the features in the real space. The \nspatial scales probed when collected scattering patterns such as the one in Figure 7  typically \nrange from about 50  nm up to a few microns.\nMost scattering patterns observed when probing magnetic domain patterns in PMA films \nshow a ring. The presence of the ring reveals a magnetic periodicity in the domain pattern \nwith an isotropic arrangement (no preferred direction). The radius q* of the ring represents \nthe magnetic period d*. This magnetic period is basically twice the domain width, as one \nperiod include a pair of up and down domains. In the [CoPd]/IrMn films, the observed mag -\nnetic period was typically d* ~ 300–350  nm.\nBy cross-correlating selected regions of the scattering speckle pattern rather than the entire \npattern, one obtains spatially dependent correlation numbers. This is useful to detect any spa -\ntial correlation features occurring at the nanoscale. For instance, magnetic domains patterns \nmay correlate well at the short scale (100–500  nm), but not correlate so well in the long range \n(1 μm and above) or vice versa.\n6.2. Space-field maps of MDM\nTo study the spatial behavior of MDM, in particular its dependence on q, cross-correlation is \nperformed on selected rings of the speckle patterns. The rings are concentric, centered about \nthe origin of the scattering pattern. The radius q of the ring is increased from zero to the larg -\nest size accessible in the image, thus providing a spatially dependent correlation coefficient \nρ(q) [40].\nBecause speckle patterns are collected at specific points in field H along the magnetization \nloop, the exploration of spatial dependence in MDM may be done at each field value H. \nUltimately, the correlation ρ is mapped in a two-dimensional ( q, H) space, thus probing the \nevolution of the spatial dependence in MDM with field. An example of such ρ(q, H) is shown \nin Figure 16a .Magnetism and Magnetic Materials 546.3. Oscillatory behavior of MDM in [Co/Pd] IrMn\nThe spatial and field dependence of MDM was explored in [Co/Pd] IrMn films [ 46]. The films \nwere ZFC below the blocking temperature. Speckle patterns were collected at low temperature, \nthroughout the magnetization cycle and several subsequent loops. Ring selective cross-correlations \nwere carried out. The resulting ρ(q, H) maps, averaged over subsequent cycles, showed interesting \nfeatures. In particular, slices through the maps at specific H values around the remanent coercive \npoint Hcr showed an oscillatory behavior for ρ(q), as seen in Figure 16b .\nThe oscillation observed in the ρ(q, Hc) curve in the ZFC [Co/Pd] IrMn films revealed spatially \ndependent MDM . The central peak in the ρ(q, Hc) curve seen on Figure 16b , occurred at the \nsame location q* than the ring in the scattering pattern. This suggested that MDM is strongly \ncorrelated with the magnetic period in the magnetic domain pattern, here around 400  nm. \nTwo satellite peaks were observed around the central peak, at the same distance ± Δq from \nq*. The presence of these satellite peaks suggested a spatial superstructure in MDM.  The size \nof the superstructure, set by the value Δq, was found to be about D ~ 1.5 μm. Topographical \nAFM images of the surface of the films indicated that D nearly matched the average distance \nbetween structural defects in the material.\n6.4. Azimuthal angular dependence\nSince the speckle patterns used for the evaluation of MDM are 2D images, one can explore the \nspatial dependence of MDM in at least two directions. In addition to probing the dependence \non the scattering vector q, one can investigate the angular dependence, while varying the azi -\nmuthal angle ∆ that indicates a particular location on the ring [ 47], as illustrated in Figure 17 . \nIn that case, cross-correlations are measured between two points on the ring separated by an \nangle Δ, leading to a correlation coefficient ρ(q, Δ). Such study was carried out on ZFC [Co/\nPd] IrMn films and showed some periodical variations as a function of the angle Δ as illus -\ntrated in Figure 17c . These observations suggested the existence of some hidden rotational \nsymmetries in the formation of the disordered magnetic domain patterns in these films.\nFigure 16.  Q-dependence of MDM in ZFC exchange-biased [Co/Pd]/IrMn films. (a) ρ(Q, H ) correlation map measured at \n30 K. (b) ρ(q, Hc) slices through the correlation maps at H = Hc measured at different temperatures from 30  K up to 225  K, \nshowing an oscillation at all temperatures. Extracted from Chesnel et  al. [46].Nanoscale Magnetic Domain Memory\nhttp://dx.doi.org/10.5772/intechopen.71076557. Dependence of EC-induced MDM on temperature\nThe results discussed in the previous sections show the occurrence of MDM via exchange \nbias when the film is cooled down to low temperatures, well below the blocking temperature \nTB. Most of the measurements were performed at around 20  K after ZFC cooling the material. \nA question that arises next is if the EC-induced MDM depends on temperature and how it \nbehaves at the phase transition near TB. These questions were investigated in exchange biased \n[Co/Pd]IrMn films [ 38].\n7.1. Probing temperature dependence in zero-field-cooled (ZFC) state\nTo probe the temperature dependence of MDM, the [Co/Pd]/IrMn films were first heated \nup to around 400  K, well above the blocking temperature TB ~ 300 K and demagnetized at \n400 K. The films were then ZFC down to 20  K. Speckle patterns throughout many magneti -\nzation cycles were collected in the ZFC state at 20  K, and then at higher temperatures while \nwarming the film all the way back up to above TB. Average A × A and A × D correlation maps \nwere measured at several points in temperature. For each point, the temperature was finely \ncontrolled and stabilized via a cryogenic environment. The resulting correlation maps are \nshown in Figure 18 .\n7.2. Persistence of MDM through warming up\nThe correlation maps in Figure 18a  show the occurrence of high MDM extending through -\nout a wide region of the magnetization loop at all temperatures below TB. The high MDM \noccurred for both A × A and A × D correlation maps between same branches or opposite \nbranches of the magnetization loop, respectively. At all temperatures below TB, MDM reached \na high value plateauing over a large region of field values around the coercive point.\nFigure 17.  Angular dependence of MDM in ZFC exchange-biased [Co/Pd]/IrMn films. (a) Speckle pattern collected near \nthe coercive point. (b) Selected ring from the speckle pattern (a). The angles θ and Δ are defined here. (c) Autocorrelation \nfunction plotted against Δ. Extracted from Su et  al. [47].Magnetism and Magnetic Materials 56The temperature dependence of MDM may be compared to the temperature dependence of \nthe magnetization loop. The shape of the magnetization loop in the ZFC state, as seen in \nFigure 18b  is symmetrical, centered about the origin. The loop has an hourglass-like shape; it \nis narrow at the center and opens up at the extremities, due to the presence of exchange cou -\nplings. When increasing the temperature from the ZFC state, the overall shape of the magne -\ntization loop remains the same, but the magnitude of the opening, or hysteresis, progressively \ndecreases. Despite the changes observed in the magnetization loop while warming the film up \nfrom the ZFC state, MDM remains strong and extended at all temperatures below TB. Slices \nthrough the correlation maps, shown in Figure 18c , all show the same trend: low correlation \nat nucleation, sharply increasing to reach a high correlation plateau in the central region of the \nmagnetization loop, and then sharply decreasing toward saturation.\n7.3. Disappearance of MDM above the blocking temperature\nWhen the temperature is increased above TB, MDM vanishes, as shown in Figure 18 . At that \nstage, the correlation map shows nearly zero correlation, becoming all blue.\nThe persistence of high MDM throughout warming, at all temperatures below TB, and its \nvanishing above TB confirms that MDM is here purely induced by exchange-couplings \nbetween the F and AF layers. Above TB, these exchange couplings disappear. Consequently, \nFigure 18.  Temperature dependence of MDM in ZFC exchange-biased [Co/Pd]/IrMn films. (a) A × A and A × D correlation \nmaps measured after one cycle at different temperatures while warming from 20  K up to 335  K. (b) Magnetization loop at \nvarious temperatures. (c) Slices through the A × A maps at H ≈ Hcr. Extracted from Chesnel et  al. [38].Nanoscale Magnetic Domain Memory\nhttp://dx.doi.org/10.5772/intechopen.7107657the magnetic domain template which was imprinted in the AF layer throughout the uncom -\npensated spins at the interface is lost once the magnetic field is cycled again. In the absence of \nmagnetic template, and since the film is smooth (absence of defects), the magnetic domains \nin the F layer nucleate and propagate randomly.\n8. Optimizing EC-induced MDM by adjusting field cooling conditions\nAll the studies of EC-induced MDM discussed in the previous sections were carried out on \nzero-field-cooled [Co/Pd]/IrMn films. Next question is what happens if the film is now field-\ncooled (FC), i.e., cooled in the presence of a magnetic film. Will MDM remain as high in the \nFC state as it is in the ZFC state?\nThis question was investigated by studying the magnetic correlations after cooling the [Co/\nPd]/IrMn films under various cooling field conditions. The resulting correlation maps showed \ndrastic changes as a function of the magnitude of the cooling field [ 33]. The result of this study \nis summarized in Figure 19 . Magnetic correlations were measured in [Co/Pd]/IrMn films after \nfield cooling the material under magnetic field of various magnitudes HFC, increasing from \nzero (ZFC state) all the way up to high values, near saturation value Hs.\n8.1. Dependence of MDM on cooling field magnitude\nWhen the magnitude HFC of the cooling field is near zero, high MDM, up to 100%, is observed. \nThis high MDM extends from nucleation (lower left corner of the correlation map) all the way \nup to saturation (upper right corner of the correlation map).\nFigure 19.  Cooling condition dependence of MDM in [Co/Pd]/IrMn films. (a) A × A correlation map measured at 20  K \nafter ZFC; (b) slices through map (a); (c) A × A map measured at 20  K after cooling under HFC = 2240 Oe; (d) slices through \nmap (c); (e) A × A map measured at 20  K after cooling under HFC = 2560 Oe; (f) slices through map (e); (g) A × A map \nmeasured at 20  K after cooling under HFC = 3200 Oe; (h) slices through map (g). Extracted from Chesnel et  al. [33].Magnetism and Magnetic Materials 58When HFC is increased up to moderately high values, high MDM is still observed, with a wide cor -\nrelation plateau extending throughout a large range of field values from nucleation to saturation.\nWhen HFC approaches the saturation value, which for the [Co/Pd]/IrMn film is around \nHs ≈ 3200 Oe, the measured magnetic correlations rapidly decrease. At HFC ≈ 2600 Oe, the \ncorrelation drops from about 90% at nucleation down to about 50% in the central region \nof the magnetization loop.\nWhen HFC ≈ Hs ≈ 3200 Oe, MDM has almost vanished, except at the nucleation and saturation \nextremities of the magnetization loop.\n8.2. Shaping MDM by adjusting cooling field conditions\nIn the ZFC state, the magnetic domain template imprinted in the AF layer is formed of \nlong interlaced stripes with about the same amount of domains of opposite magnetizations \n(about a 50% up: 50% down split). The imprinted domain pattern constitutes a template \nfor the domain reversal in the F layer. Because of exchange couplings occurring between \nthe Co spins in the F layer and the uncompensated interfacial Mn spins in the AF layer, \nthe domains form in a way to match the underlying template. The matching occurs rapidly \nwhen approaching the coercive point (where the up-down magnetization split is exactly \n50–50%). The matching persists at higher field values, all the way up to near saturation. This \nis possible because, even though the magnetic domains of opposite magnetization expand \nand shrink, the specific topology of the domain pattern still matches the underlying tem -\nplate, as illustrated in Figure 20 .\nFigure 20.  Illustration of the magnetic domain memory effect in the exchange-biased [Co/Pd]/IrMn film under \nvarious field cooling conditions. The imprinted pattern in the AF IrMn layer is shown in green color, the domain \npattern in the F layer is shown in orange color. Arrows indicate magnetization direction. (a, b) 5 × 5  μm MFM images \nshowing actual magnetic domains in the F Co/Pd layer at room temperature: (a) near nucleation and (b) near the \ncoercive point. (c, d) Sketches of the magnetic domain configuration for ZFC and moderate field-cooled states: (c) near \nnucleation and (d) near the coercive point. (e, f) Sketches of the magnetic domain configuration in near-saturating \nfield-cooled states: (e) near nucleation and (f) near the coercive point. Extracted from Chesnel et  al. [33].Nanoscale Magnetic Domain Memory\nhttp://dx.doi.org/10.5772/intechopen.7107659In the FC states, the magnetic domain template imprinted in the film has an unbalance of up \nand down domains, and the net magnetization is non-zero. However, if the magnitude HFC \nof the cooling field remains in a certain range above Hcr, the imprinted pattern, still formed \nof interlaced up and down domains, provides a template that entirely drives the reversal of \nthe magnetic domains in the F layer. The topology of the imprinted pattern in the FC state \nresembles that of the imprinted pattern in the ZFC state. High MDM is therefore maintained \nthroughout the entire magnetization process.\nIf the magnitude HFC of the cooling field approaches saturation value Hs, the imprinted mag -\nnetic domain pattern does not include long interlaced magnetic domains of opposite mag -\nnetization anymore but a few bubble domains sparsely scattered throughout the film. This \nimprinted template is not able to drive the magnetic domain formation throughout the entire \nreversal but only at the extremities, that are the nucleation and the saturation points, as illus -\ntrated in Figure 20 .\nThese results demonstrate the possibility to induce and control nanoscale MDM in exchange \nbiased films by adjusting the field cooling conditions. High, up to 100% MDM, extending \nthroughout the entirety of the magnetization loop can be achieved by cooled under no field or \nrelatively low field values. However, MDM can be almost eliminated, by cooling the material \nunder high magnetic field, approaching saturation and higher.\n9. Conclusion\nMagnetic domain memory (MDM) is an unusual property exhibited by certain ferromag -\nnetic films, where the microscopic magnetic domains tend to reproduce the same topologi -\ncal pattern after it has been erased by an external magnetic field. In most ferromagnetic \nmaterials, MDM does not occur. When an external magnetic field is applied and cycled, \nmicroscopic magnetic domains form and propagate throughout the material in non-deter -\nministic ways. However, it has been found that some ferromagnetic thin films with per -\npendicular magnetic anisotropy (PMA) do show significant MDM under certain structural \nand magnetic conditions. One structural condition is the presence of defects. When rough \nenough, thin Co/Pt multilayers with PMA, exhibits partial MDM occurring in the nucle -\nation phase of the magnetization process. This disorder-induced MDM is caused by the \npresence of microscopic structural defects, playing the role of pinning sites for the domain \nnucleation. Another way to induce MDM and to maximize it, even in smooth films, is incor -\nporating magnetic exchange couplings (EC) between a PMA ferromagnetic (F) film and \nan underlying antiferromagnetic (AF) film. This has been achieved by combining F [Co/\nPd] multilayers with AF IrMn layers. After demagnetizing the material and zero-field-\ncooling it below its blocking temperature, high MDM up to 100% was observed through -\nout almost the entirety of the magnetization process. This high MDM is induced by EC \ninteractions between the Co spins in the F layer and the interfacial uncompensated spin \nin the IrMn layer. When the material is cooled down below its blocking temperature, a \nspecific magnetic domain pattern gets imprinted into the AF layer. When the field is cycled \nat low temperature, the frozen imprinted AF pattern then plays the role of a template for Magnetism and Magnetic Materials 60the domain formation in the F layer. The resulting high MDM extends throughout a wide \nrange of field values, from the coercive point to nearly saturation. This EC-induced MDM \npersists through field cycling and through warming the material all the way up the block -\ning temperature, above which MDM vanishes. Additionally, it was found that the amount \nof EC-induced MDM can be varied by adjusting the magnitude of the field applied during \nthe cooling. If the material is cooled under no field or moderate field, MDM reaches high \nvalues up to 100% throughout most of the magnetization process. If, however, the mate -\nrial is cooled under a nearly saturating field, MDM vanishes, except at the nucleation and \nsaturation extremities of the magnetization cycle.\nThese observations of MDM in certain PMA ferromagnetic films opens the door to more \ninvestigations. A particular question is if there are other ways to induce and control MDM \nin a material. It has been recently found that EC-induced MDM may be affected by light, \nsuch as x-rays. If the material is illuminated by too intense x-rays, the film may lose its EC \nproperties and MDM may vanish. This finding, which resonates with the emergent all-optical \nmagnetic switching phenomena observations [ 48–50], suggest that EC-induced MDM could \nbe controlled by light illumination. Ultimately, the ability to induce and control MDM in PMA \nferromagnetic film, either by structural disorder, or by exchange couplings and light illumina -\ntion, may offer a tremendous potential for improving technological applications in the field of \nmagnetic recording and spintronics.\nAuthor details\nKarine Chesnel\nAddress all correspondence to: kchesnel@byu.edu\nDepartment of Physics & Astronomy, Brigham Young University (BYU), Provo, UT, USA\nReferences\n[1] Néel L. Les lois de l ’aimantation et de la subdivision en domaines élémentaires d ’un \nmonocristal de fer . Journal de Physique et Le Radium. 1944; 5:241\n[2] Kittel C. Theory of the Structure of Ferromagnetic Domains in  Films and Small Particles . \nPhysics Review. 1946; 70:965\n[3] Kittel C. Physical theory of ferromagnetic domains . 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Botana4  1Materials Science and Engineering, School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, Arizona 85287, USA 2 Department of Applied Physics, Aalto University, 00076 Aalto, Espoo, Finland 3Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 4 Department of Physics, Arizona State University, Tempe, Arizona, 85287, USA 5 Departamento de Fisica Aplicada, Universidade de Santiago de Compostela, E-15782 Campus Sur s/n, Santiago de Compostela, Spain  6 Instituto de Investigacions Tecnoloxicas, Universidade de Santiago de Compostela, E-15782 Campus Sur s/n, Santiago de Compostela, Spain  1. INTRODUCTION  The recent discovery of magnetism in monolayers of two-dimensional (2D) van der Waals (vdW) materials has opened new venues in Materials Science and Condensed Matter Physics. Until recently, 2D magnetism remained elusive since the existence of magnetic order in 2D is a priori not guaranteed. The story changed in 2016 when two groups provided evidence of antiferromagnetism in monolayers of FePS3 1, 2. In 2017, the presence of ferromagnetic order was proven in monolayers of CrI3 and on a bilayer of Cr2Ge2Te63, 4. The list of candidates has been growing ever since 5-7.  There are various aspects that make 2D vdW crystals with magnetic order very interesting. First, magnetic order in 2D can only happen if there is no continuous rotational symmetry, otherwise the proliferation of low-energy spin waves, that lies behind the Mermin-Wagner theorem 8, destroys magnetic order at any finite temperature. Therefore, magnetic anisotropy and spin waves control the transition temperature of 2D magnets and play a much more important role than in their 3D counterparts. Second, the electronic and mechanical properties of 2D materials can be widely tuned in various ways: by gating, proximity, and chemical functionalization, which permits to conceive devices where magnetic order is controlled at will. Third, magnetic order adds a new functionality to the set of Lego-like pieces that enriches the game of vertical integration of 2D materials in van der Waals heterostructures. The stacking of materials with magnetic order, superconducting order, spin-valley coupling, and graphene will probably result in structures with completely new and unexpected properties that we can now explore both theoretically and experimentally.  Within just three years, the discovery of 2D magnets has already opened up new opportunities in spintronics i.e. spin pumping devices, spin transfer torque, and tunneling magnetoresistance 9-11. We envision future applications that may extend into other realms, including sensing and data storage. Here, we review some of the experiments in which magnetism in strictly 2D has been confirmed. We discuss common synthesis techniques for these materials and methods for engineering their magnetic properties. Further, we analyze in detail some of the most important theoretical aspects that need to be considered to understand 2D magnets. Finally, we identify different phenomena that we anticipate will be the next steps to follow in the field.   \n 2 2. EXPERIMENTAL FINDINGS ON 2D MAGNETS TO DATE  2.1 Evidence of magnetic order in 2D  We first review some of the existing experiments providing evidence for magnetic order at the monolayer level (or close to it) and describe briefly the corresponding 2D vdW materials (see Table I).  Antiferromagnetism in FePS3. The first experimental evidence of magnetic order at finite temperature in monolayers was found in FePS3 in 2016. By monitoring the Raman peaks that arise from zone folding due to antiferromagnetic ordering (Fig. 1a), it was demonstrated that FePS3 exhibits antiferromagnetic ordering down to the monolayer limit with a TN as high at 118 K 1, 2. Ferromagnetism in CrI3. Kerr microscopy experiments have shown that ferromagnetism in this material persists down to the monolayer level (Fig. 1b) with a large critical temperature of 45 K (not far from that of the bulk ~ 61 K) 3. Magnetic order in this compound shows an out-of-plane easy axis anisotropy. The bilayer system (Fig. 1c) showed a surprising lack of Kerr signal attributed to an interlayer antiferromagnetic arrangement genuine of the bilayer. Ever since, there has been intense interest in trying to elucidate the importance of stacking order for the magnetic response of this material 12, 13.  Antiferromagnetism in CrCl3. Tunneling magnetoresistance measurements in few-layer CrCl3 provided early evidence of antiferromagnetic ordering down to bilayer samples, shown in Fig. 1d. Few-layer samples preserved the same magnetic ordering as their bulk counterparts, with in-plane easy-axis anisotropy, and antiparallel spin ordering between layers. Strikingly, ultrathin CrCl3 samples showed a tenfold increase in exchange energy, which was attributed to the different stacking order and its feedback on the out-of-plane exchange interactions at low temperatures 14.  Ferromagnetism in Cr2Ge2Te6. Magnetic order at the bilayer level was probed in Cr2Ge2Te6 by means of Kerr rotation experiments 4. The monolayer, in turn, was found to degrade rapidly. The magnetic transition temperature proved to be tunable by means of an external magnetic field. This clearly shows the potential to build devices based on 2D vdW magnets with properties that can be easily manipulated.  Ferromagnetism in Fe3GeTe2 (FGT). Itinerant ferromagnetism persists in Fe3GeTe2 down to the monolayer limit with a sizable out-of-plane magnetocrystalline anisotropy 15. Magnetism was studied by probing the Hall resistance, shown in Fig. 1e. The ferromagnetic transition temperature is suppressed relative to the bulk (205 K) but an ionic gate can raise Tc all the way up to room temperature, opening up opportunities for potential voltage-controlled magnetoelectronics16.   Ferromagnetism in transition-metal dichalcogenides (TMDs).  A strong ferromagnetic signal at room temperature has been reported at the single-layer level in VSe2. However, spontaneous ferromagnetism in this system remains a controversial issue due to the possibility of charge density wave (CDW) formation and the subsequent suppression of magnetic order 17, 18. Angle resolved photoemission spectroscopy (ARPES) and scanning tunneling microscopy (STM) have revealed an electronic reconstruction of single layer VSe2 without a detectable FM exchange splitting, casting doubts on whether magnetism originates from an induced band structure spin splitting, or if extrinsic defects come into play 19-21. Room temperature ferromagnetism has also been reported in MnSe2 films grown by MBE. From SQUID measurements in the monolayer limit, the magnetic signal is assigned to intrinsic ferromagnetism with a Tc close to room temperature 22.  \n 3  Material Magnetic order Tc Magnetic lattice Refs. FePS3 AFM zig-zag? honeycomb 1, 2 CrI3 FM 45 K honeycomb 3 CrCl3 AFM 14 K honeycomb 14 Cr2Ge2Te6 FM 45 K honeycomb 4 Fe3GeTe2 FM 300 K triangular 15 VSe2 FM 300 K triangular 17 MnSe2 FM 300 K triangular 22   Table I. vdW material systems for which long range magnetic order has been confirmed experimentally in 2D and their characteristics. AFM stands for antiferromagnetic and FM for ferromagnetic.  \nFigure 1. Characterization of magnetism in 2D. (a) Temperature dependence of the characteristic Raman peak resulting from spin order-induced folding of the Brillouin zone in FePS3 as a function of thickness2. (b,c) Magneto-optical Kerr rotation as a function of applied magnetic field in flakes of CrI3, revealing a ferromagnetic (antiferromagnetic) ground state in monolayer (bilayer) samples. The insets show optical microscope images of CrI3 (scale bars 5 µm)3 (d) Magnetoresistance versus in-plane magnetic field for bilayer, trilayer and tetralayer tunnel junctions at 4 K. The inset shows the optical microscope image of a bilayer CrCl3 tunnel junction device.14 (e) Low-temperature Hall resistance Rxy in FGT thin-flake samples and ferromagnetic hysteresis down to the monolayer limit. Inset: schematic of the Hall effect measurement on FGT flakes.15  \na \nd \ne \nb \nc  \n 4 2.2 Common techniques for the synthesis of van der Waals magnetic crystals.  After discussing the early magnetic measurements and observations done on 2D magnetic crystals, we focus on common synthesis techniques used for producing layered vdW magnetic crystals. At the time of writing this review article, there are no studies that enable researchers to produce monolayer or few-layer thick magnetic crystals at large scales using commercially compatible chemical vapor deposition (CVD) or atomic layer deposition (ALD) methods. This is mainly because of the limited environmental stability of some of these 2D magnetic crystals and/or lack of established surface chemistry routes to enable layer-by-layer deposition.  Because of these limitations, the community currently heavily relies on the production of high crystalline quality, defect free crystals that are ideally free of magnetic impurities such as Fe, Co, and Ni. Once these layered crystals are produced, a routine mechanical exfoliation 23 technique is used to isolate monolayer to few-layer thick sheets onto the desired substrates. Depending on the crystal type (halide vs. chalcogen) as well as on their phase diagrams, different crystal growth techniques are used to produce these materials. Based on the most common studies vdW magnetic crystals in the field, the popular growth techniques include chemical vapor transport (CVT), sublimation, or flux zone techniques.   Chemical vapor transport.  The first growth technique, namely chemical vapor transport (CVT), is an extremely effective and reliable method used to produce macroscale layered materials, including transition metal thiophosphates, FGT, and TMDs discussed in this article 24-26. CVT involves transporting precursors from hot to cold zone for endothermic, or cold to hot zone for exothermic reactions using transport agents 27.  To produce crystals using CVT, it is necessary to evacuate and seal stoichiometric amounts of precursors inside thick (usually 1mm-2mm) quartz ampoules before carrying out well engineered thermal processing (crystal growth). The growth temperature (cold and hot zone) is carefully selected based on the binary or ternary phase diagrams, and usually involves high temperature processes that naturally build very high vapor pressures. For these reasons, thicker wall quartz ampoules are required for the growth process.  In addition to the precursors, transport agents such as iodine and bromine are used as promoters for fast chemical reactions and larger crystal growth. For example, transition metal thiophosphate crystal synthesis (such as MnPS3, FePS3, and CoPS3) typically involves the use of halides (I2, Br2, etc.) as transport agents to produce these vdW crystals 24. Since CrI3 crystals already contain iodine in the crystal matrix, the reaction of elemental Cr and I2 in a quartz ampoule is sufficient 28.  Sublimation.  In addition to CVT, physical vapor transport (PVT or sublimation), is another method that can be used to produce high quality, single-crystals of transition metal halides 29, 30. It has proven to be a useful and low-cost technique that does not require sophisticated ampoule sealing processes necessary for chemical vapor transport reactions, although evacuated and sealed ampoules may also be used to carry out the growth 31.  For the synthesis of CrCl3 and other transition metal halides, commercially available compounds are placed in an open-ended or sealed quartz tube and positioned in a horizontal furnace with the desired temperature gradient.  No additional transport agent is required as these materials contain the necessary halide, and self-transport.  After heating the compound to their respective sublimation points, the transition metal halides transport to the cold zone of the furnace, where they nucleate and grow directly on the walls of the quartz tube.  Large, high-quality crystals can be obtained in 24-48 hrs 32.   \n 5 Flux zone growth.  While the previously discussed vapor-phase synthesis techniques can be used for transition metal halides, other classes of 2D vdW magnets, such as Cr2Ge2Te3 and Fe3GeTe2 can be obtained from solution-phase flux methods to achieve large, high-quality crystals 33-35.  For this method, stoichiometric amounts of precursors and flux (solvent) are loaded into an inert crucible, such as quartz, and vacuum sealed (~10-5 torr). In the case of Cr2Ge2Te6, excess germanium and tellurium are used to create a self-flux.  As this is a solution-based technique, where the precursors and flux are in direct contact with the crucible, careful selection of the latter must be made to ensure that no undesirable reactions occur between the crucible and the precursors during the growth.  Additionally, the inorganic flux is chosen, among other factors, to have a high solubility of the desired elements at the growth temperature, not to create any competing phases, and in many cases it can be part of the chemical composition of the resultant product.   The precursors and flux are then heated above their melting temperatures and slowly cooled over a period of several days. Growth parameters are selected based on prior knowledge of the product’s binary or ternary phase diagrams, but experimentally determining and optimizing the growth parameters is often necessary because phase diagrams have not been established for the desired material system.  After the growth is complete, the final step is to remove the flux from the crystal.  The most common method to remove the flux from tellurium-based vdW magnets is to melt the tellurium and remove it through a centrifugation process, and depending on the flux, additional steps may be required to fully remove the flux 36.  2.3 Engineering magnetism in 2D  A significant advantage of 2D materials is that their physical properties are highly tunable by means of external control parameters that include electrostatic doping, pressure, and strain. Here we highlight a few recent demonstrations of tunable magnetism in 2D materials.  Electrostatic Doping. Electrostatic doping is a powerful technique for tuning the electronic properties of 2D materials. The working principle is similar to that underlying field-effect transistors and is based on the direct transfer of electronic or ionic charges from a dielectric into the target 2D material. Electrostatic doping has a series of advantages over chemical doping of bulk materials. It is continuously controllable through a gate bias, and it is compatible with a variety of dopant species – from simple electrons/holes to specific ions/chemical functional groups – and it can be applied to most 2D materials without being hindered by phase separation issues in non-stoichiometric bulk synthesis. It has been shown that the electrostatic doping in 2D materials could unveil new physics, such as unconventional superconductivity in MoS237, twisted bilayer graphene38, or structural transitions in MoTe239.  As mentioned above, bilayer CrI3 is a layered antiferromagnet and it was found that the interlayer exchange coupling is tunable by electrostatic doping. Fig. 2a shows the schematic of the representative bilayer CrI3 device. This device is a vertical stack of a bilayer CrI3 flake and a graphite contact encapsulated between two hexagonal boron nitride (hBN) flakes and a graphite top gate. By applying the gate voltage to the insulator hBN, the electric dipoles at the hBN-CrI3 interface introduce carrier injection into the bilayer CrI3. Fig. 2b shows the reflection magnetic circular dichroism (RMCD) signal as a function of both top- and back-gate voltages near the metamagnetic transition field µ0H = 0.78 T. The red region on the right is the signal from the ↑ ↑ state, and the pink region on the left is from the layered AFM state. The dashed line boundary indicates that the metamagnetic transition could be effectively tuned by electrostatic doping40, 41.    \n 6                      Figure 2. Gate-tunable magnetism of CrI3 and FGT (a) Schematic of a dual-gated bilayer CrI3 device. (b) RMCD signal of the electrostatic doped bilayer CrI3. The dashed contour shows the tunable metamagnetic transition field through bias voltage41. (c) Schematic of the FGT device structure and measurement setup. S and D label the source and drain electrodes, respectively, and V1, V2, V3 and V4 label the voltage probes. The solid electrolyte (LiClO4 dissolved in polyethylene oxide matrix) covers both the FGT flake and the side gate. (d) Phase diagram of the trilayer FGT sample as the gate voltage and temperature are varied. The transition temperature is determined from the extrapolation of the temperature-dependent anomalous Hall resistance to zero15.  FGT is a layered ferromagnet with bulk Tc = 205 K. In monolayer FGT, however, magnetic ordering is suppressed due to thermal fluctuations of long-wavelength acoustic-like magnon modes in 2D, while the trilayer sample has a Tc ~ 100 K. Strikingly, ionic liquid gating could raise Tc to room temperature, much higher than the bulk Tc (Fig. 2d). This ionic gating method (Fig. 2c) intercalates the Li+ from LiClO4 (the transparent liquid electrolyte) onto the surface of the FGT by the gate voltage Vg, and the doping level could reach values as high as 1014cm-2, which is one order of magnitude higher than those achievable using hBN gates15.  Pressure. In a vdW material, a small change of the interlayer spacing can cause a drastic change in physical properties. In particular, if the material supports magnetism, the interlayer interactions can be modified to produce a change in the magnitude and sign of the exchange coupling. Hydrostatic pressure is a typical method for continuous control of interlayer coupling via interlayer spacing in vdW crystals42.  Fig. 3a shows a schematic of the experimental set-up of the high-pressure study of CrI3. A magnetic tunnel junction (MTJ) device was composed of bilayer CrI3 sandwiched between top and bottom multilayer graphene contacts. The entire MTJ was encapsulated by hBN to prevent sample degradation. The device was then held in a piston cylinder cell filled with oil for application of hydrostatic pressure. Magnetic states were probed by using tunneling \nV1 \nV2 \nV4 \nV3 \nS \nD \nVg \na \nb \nc \nd \nelectrolyte  \n 7 magnetoresistance measurements as shown in Fig. 3b. After removal from the cell, reflective magnetic circular dichroism (RMCD) microscopy (Fig. 3c and 3d) showed that the bilayer CrI3 irreversibly transitioned from antiferromagnetic to ferromagnetic ordering43, 44.                Figure 3. High pressure study of CrI3. (a) Schematic of a high-pressure experimental set-up. The force applied to the piston exerts pressure on the bilayer CrI3 device through oil. (b) Tunneling current vs. magnetic field H at two pressures. Insets: magnetic states and optical microscopy image of a bilayer device. (c, d) RMCD signal from the bilayer after removal from pressure cell where it was subjected to comparable pressure (2.45 GPa), and from a pristine bilayer CrI344.  Strain. 2D materials possess outstanding mechanical properties and can sustain larger strain than their bulk counterparts. Monolayer MoS2 is predicted to sustain elastic strain levels up to 11%, and monolayer FeSe up to 6%45, 46. Strain engineering has been shown to be an effective approach to tune the properties of 2D materials using various methods including substrate-lattice mismatch47, 48, mechanically actuated strain cells49, 50, and nanomechanical drumheads51.   Fig. 4a shows the experimental set-up used in the biaxial strain study of bilayer CrI3 through the nanomechanical drumhead. By applying a voltage to the silicon substrate, the electrostatic force between silicon and CrI3 can apply tensile biaxial strain to CrI3 itself. Fig. 4b shows the structure of the bilayer CrI3 device, with CrI3 encapsulated within two stable 2D materials, few-layer graphene at bottom and monolayer WSe2 on top. In addition to protecting CrI3 from degradation under ambient conditions, few-layer graphene acts as a conducting electrode, while monolayer WSe2 provides a strain gauge via measurements of the shift in the exciton energy (under the assumption that WSe2 experiences the same level of strain as CrI3. This 2D heterostructure was first assembled and then transferred on prefabricated circular microtrenches with patterned Au electrodes and Si back gate (Fig. 4c). Fig. 4d shows that the metamagnetic transition field in bilayer CrI3 could be tuned effectively by applying tensile biaxial strain51.  \na \nb \nc \nd  \n 8 Figure 4. Biaxial strain study of CrI3. (a) Schematic of the measurement system. A DC voltage Vg is imposed to apply electrostatic force to the membrane. The laser is to detect both the strain and magnetic ordering. BS: beam splitter; PD: photodetector. (d) The metamagnetic transition field as a function of gate-induced strain (symbols) and the solid line is a linear fit51.  3. THEORY OF MAGNETISM IN 2D    3.1. Background  Symmetry breaking in low dimensional systems plays a very special role in condensed matter physics. The spontaneous breaking of a continuous symmetry is not possible in two dimensions at finite temperature, unless long-range interactions come into play. Analogous propositions were posed, in the form of mathematical theorems, in the context of crystalline order by Landau and Peierls, in the context of superconductors and superfluids by Hohenberg, and in the context of magnetism by Mermin and Wagner 8, 52. The common ground of all these theorems is the existence of gapless collective excitations, the Goldstone modes, each of which is associated with the order parameter of the broken symmetry phase. In the case of magnets, these Goldstone modes are spin waves (or magnons) and in two (and one) dimensions the thermal population of these low energy excitations completely destroys long-range order. This is exemplified by computing the correction to the magnetization within spin wave theory for an isotropic ferromagnet, that yields a divergent result in two dimensions:   𝛿𝑀(𝑇)\t~∫)\t*)+,-./01\t→∞45       (1)  where δM(T) refers to refers to the correction to the magnetization due to thermal fluctuations, ρ and β are the spin wave stiffness and inverse temperature.   Spin Hamiltonian. A very wide class of magnetic materials are insulating. Therefore, charge degrees of freedom are frozen and it is possible to describe their magnetic properties in terms of spin Hamiltonians (even in the case of conducting magnetic materials, their magnetic properties can also be described fairly well with effective spin Hamiltonians). So, it is adequate to start our discussion with a brief description of a simplified spin Hamiltonian.  𝐻=\t−∑𝐽;<\t𝑺;∙𝑺<?;<@−∑𝐾;<\t𝑆;C𝑆<C?;<@−∑𝐷;\t(𝑆;C)E;+∑𝑩∙𝑺;;   (2)  The first term describes the exchange (Heisenberg) interactions, the second the anisotropic exchange, the third the single-ion anisotropy, and the fourth introduces the effect of an \nd  \n 9 external magnetic field. We note that additional anisotropic terms could be allowed such as Kitaev 53, biquadratic exchange 54, or Dzyaloshinsky-Moriya 55 interactions (DMI).  The divergent result of Eq. 1 could be avoided if (i) the spin-wave spectra has a gap or (ii) the dispersion of the spin-wave is different. Mechanism (i) corresponds to the existence of a single-ion anisotropy or anisotropic exchange, which yields ferromagnetism in CrI33 and Fe3GeTe216. Mechanism (ii) corresponds to the correction to the spin wave dispersion due to dipolar interactions. Dipolar interactions could allow stabilization of in-plane ferromagnetic order, yet such a scenario has not been confirmed in a 2D van der Waals material to date. With the previous dispersion relation, the correction to the magnetization within the linear spin wave regime becomes at low temperatures kBT « ∆ which yields a finite correction to the maximal magnetization at low enough temperatures, thus allowing for a ferromagnetic state at finite temperatures.  𝛿𝑀(𝑇)\t~∫)\t*)+,(HI-./)01\t→1J45\t𝑒0JL   (3)   3.2. Origin of magnetic anisotropies   Single ion anisotropy. The simplest anisotropic term that can be written is the so-called uniaxial single ion anisotropy, that takes the form:  𝐻MNO=\t−𝐷∑\t(𝑆;C)E;   (4)  The parameter D favors off-plane magnetism for D > 0, whereas it favors in-plane magnetism for D < 0.  For D » J, the Heisenberg model reduces to the celeb Ising model.  It must be noted the for S = 1/2, S2 = 1/4 and thus the previous term is trivial, yielding that S = 1/2 ferromagnets cannot have single ion anisotropy.  The physical origin of this term is the interplay between the local crystal field δ and the atomic spin-orbit coupling λ. Such anisotropic terms in the Hamiltonian stem from perturbation theory in the high-spin state of the ion, and crucially depend on the spin-orbit coupling of the magnetic ion. We can distinguish between two different cases, systems with orbital degeneracy and without orbital degeneracy. In systems with orbital degeneracy, the single ion anisotropy is first order in λ, yet orbital degeneracy can be easily lifted by a Jahn-Teller mechanism. In the absence of orbital degeneracy, the single ion anisotropy stems (at least) from second order perturbation in λ/δ yielding D ~ (λ/δ)2.  Single-ion anisotropy is expected to be strong for transition metals whose crystal field environment has a well-defined symmetry axis as in the 2H-transition metal dichalcogenide structure. In contrast, for approximate octahedral environments such as those in 1T-TMDs or CrI3, the magnitude of the trigonal distortion is expected to substantially impact the value of the single-ion anisotropy D.  Exchange anisotropy. The second source of a gap in the spin-wave Hamiltonian is the anisotropic exchange that takes the form:  𝐻ON=\t−𝐾∑\t𝑆;C𝑆<C?;<@   (5)   \n 10 where ij denotes sum over first neighbors. For a ferromagnet, the previous term favors a parallel off-plane alignment for K > 0, whereas for K < 0 the system favors in-plane magnetism. We note that this term yields a non-trivial contribution for a S = 1/2 system, and thus can yield a magnon gap for a S = 1/2 ferromagnet.  Physically, the origin of the anisotropic exchange K stems from the connecting atoms between two localized spins. Importantly, in this situation K is mainly controlled by the spin orbit coupling of the bonding atom, instead of the magnetic one. A particular example of this is CrI3, where the anisotropy energy is controlled by the strength of the spin-orbit coupling of iodine. Generically, two-dimensional magnets with heavy anions such as Br, I and Te are susceptible to have sizable contributions to the anisotropic exchange due to the large spin-orbit coupling of the ligand anion 56. The relative strength of the single-ion anisotropy and anisotropic exchange can be estimated from first principles methods, yet their exact values can be sensitive to the details of the method 56, 57.   Dipolar anisotropy. Dipolar interactions are an additional mechanism to stabilize magnetic ordering in two dimensions. In particular, they may allow to stabilize in-plane magnetic ordering at finite temperature. Dipolar interactions favor in-plane arrangement of spins, yielding a Hamiltonian with in-plane rotational symmetry. This leads to the so-called reorientation transition observed in ferromagnetic thin films, that stems from the thermal renormalization of the anisotropy58, Moreover, dipolar interactions modify the spin-wave spectra so that at low energies the magnon dispersion becomes Ek ∝k1/2, yielding the integral in Eq. 1 non divergent. However, van der Waals ferromagnets with in-plane anisotropy and magnetic at zero field are highly elusive, and assessing the existence of in-plane ferromagnetic ordering in some van der Waals systems remains an open question.  3.3. Heisenberg Hamiltonian: origin of magnetic exchanges  The strengths and signs of the exchange couplings between different atoms depend on microscopic details, and often arise from a complex interplay between hopping and electronic interactions. Nevertheless, for those cases in the localized limit, i.e. with the active electrons being strongly localized in the magnetic ions, the signs of the different exchange interactions can be predicted using the well-known Goodenough-Kanamori rules59.   Most of the existing two-dimensional systems that have shown magnetic long-range order in the single-layer limit present structures with some common motifs: hexagonal or triangular lattices in the plane (see Table I), cations in an octahedral environment of their neighboring anions with these octahedra being connected via edge sharing. Most of these systems such as transition metal dihalides and trihalides 60, 61, transition metal dichalcogenides crystallizing in the 1T or 2H structures 62, Cr2Ge2Te6 63, and those of the AMX3 type 64 including phosphosulphides and phosphoselenides can be interpreted in the localized electron limit since most of them are magnetic semiconductors both in their bulk and few-layer form.  In this situation, it is important to analyze the possible mechanisms for exchange in such structures. There will be an important contribution coming from direct exchange (metal-metal), and another one coming via an anion, where the cation-anion-cation path forms an angle of approximately 90 degrees. These structural details can be observed in Fig. 5 which depicts the structure of FePS365 as an example of a case of a hexagonal in-plane network and the close-up case of two neighboring octahedra sharing an edge.  \n 11  Figure 5.   a) Structure of FePS3 as seen from the top of the hexagonal plane.   b) Two metal atoms surrounded by an ionic octahedral cage. The octahedra are edge sharing. This is the typical coordination in most known 2D vdW magnets. In this situation, competition between metal-metal direct exchange and 90◦-superexchange via anions can take place as discussed in the text.  We analyze the situation for various relevant fillings of the external d shell of the cations and discuss how ferromagnetic ordering may emerge using halides as a practical example. We stress that it is important to understand the origin of magnetic exchange in these systems in order to enhance the transition temperature, which would require a large FM coupling strength, but also larger moments, and maximized in-plane coordination.  We will limit ourselves to the two-dimensional case using 3d electrons as a reference and discuss the evolution with anion size, pressure, etc. We will use mostly transition metal di- and trihalides as an example for the discussion. The important point for following our discussion would be that, as the anion increases in size (going down in their respective column in the periodic table) this gives rise to a larger cation-cation distance, which decreases the strength of the direct exchange. However, the metal-anion-metal interaction is much less affected. Thus, if both interactions have the same sign, increasing the anion size simply leads to a reduction in the magnetic transition temperature. But, if they have opposite signs, as the anion size increases, the sign of the superexchange becomes more important.  We will discuss each d-filling separately (linking them to materials in which 2D magnetism has been confirmed or theoretically proposed) with Table II compiling all cases analyzed.   Filling Cations Direct exchange 90° superexchange Competition d3 V2+, Cr3+ AFM FM Yes d5 Mn2+, Fe3+ AFM FM Yes d6 Fe2+ FM FM No d7 Co2+ FM FM No d8 Ni2+ AFM FM Yes  Table II. Summary of the magnetic couplings59 discussed in the text for edge sharing octahedra and various fillings of the 3d shell, with examples of representative cations. AFM stands for antiferromagnetic and FM for ferromagnetic. If the two couplings have opposite signs, competition is active.   d3 filling. There is a competition between an AFM direct exchange and a FM superexchange. Direct exchange decreases its strength as a larger anion is introduced and hence the cation-\n \n 12 cation distance is increased. Hence, the tendency for ferromagnetism is enlarged as the unit cell increases. In Cr-trihalides, it is experimentally observed that the FM Curie temperature increases with anion size (17 K for Cl, 33 K for Br, 68 K for I)66. In the case of V-dihalides, the AFM Neel temperature decreases as the anion size increases (36 K for Cl, 30 K for Br, 16 K for I)67, indicating a larger importance of the FM component as the cation-cation distance increases. d5 filling. There is also a competition between AFM direct exchange and FM superexchange. Additionally, there is also a competition in the superexchange between that coming from σ-bonding (mediated by the eg electrons) which is FM, and that due to π-bonding (t2g-mediated) which is AFM. In the case of high spin d5 cations, the FM component becomes important. This competition causes the appearance of a helical phase in FeCl368 and a striped phase in Mn-dihalides. d6 filling. In this case there is no competition, both direct exchange and superexchange yield a FM component. That is why in the Fe dihalides, the Curie temperature 57 is larger for FeCl2 (38 K) than for FeBr2 (14 K), a smaller Tc occurs when the cation-cation separation increases. FePS32 is another prominent example of FM in-plane ordering with this filling.  d7 filling. Again, there is no competition between direct and superexchange, both being FM. An example of this are the Co-dihalides. The Curie temperature is reduced when going from CoCl2 to CoBr261, 69 (11 K for CoI2, 19 K for CoBr2 and 25 K for CoCl2) at the same time that the anion size increases leading to a larger Co-Co distance, that decreases the magnetic interaction strength, in particular the direct component. d8 filling. In this case, there is competition between an AFM direct exchange and a FM superexchange. Evidence for this comes from the Ni-dihalides, which are FM in-plane and their Curie temperature increases as the size of the anion does, because the AFM component of the total exchange gets reduced as the cation-cation distance increases. NiI2 has a Curie temperature of 75 K and that of the smaller anion NiCl2 is 52 K61.   4. OVERALL OUTLOOK   4.1. Multiferroics Multiferroics are materials showing a coexisting magnetic and ferroelectric order. Ferroelectric order is the spontaneous development of a finite electric dipole in a material, in analogy with the magnetic ordering of ferromagnet. Ferromagnetism and ferroelectricity are known to obstruct each other. The simplest case is the one of perovskites, where displacive ferroelectricity is favored by an empty d-shell, whereas ferromagnetism requires a partially filled d-shell. As a result, realizing multiferroic orders requires non-displacive mechanism for ferroelectricity, such as charge order, spin-driven, electronic lone pairs or geometric effects. Multiferroic two-dimensional materials would have important applications, including electric reversal of magnetization70 or electrically controlling an exchange bias71. Similar effects have been obtained in two-dimensional heterostructures, as for instance in CrI3 bilayer39, 40, yet without relying on a multiferroic effect. Multiferroicity has been predicted to intrinsically appear in two-dimensional materials such as transition metal phosphorus chalcogenides72, CuBr273, 74, and VOI2,75. Interestingly, artificial multiferroics can be engineered in ferroelectric/ferromagnetic van der Waals heterostructures72.   4.2. Skyrmions  The interplay of ferromagnetic interactions, DMI and an external magnetic field can turn a skyrmion configuration energetically favorable over the spin-spiral and ferromagnetic states.  \n 13 In addition to their fundamental interest, skyrmions in 2D vdW could provide a new paradigm for low-power data storage. At this point, a few theoretical proposals of skyrmion formation in 2D vdW materials exist. These include twisting in vdW heterostructures, in particular using the example of a ferromagnetic monolayer on top of an antiferromagnetic substrate76. An exciting possibility is skyrmion formation via inversion symmetry breaking in Janus monolayers of manganese dichalcogenides that can achieve DMI values comparable to ‘traditional’ skyrmion-hosting materials 77. Another proposal shows that, in CrI3 monolayers, skyrmion spin configurations become more stable than FM ones by applying an out-of-plane electric field 78.  Recently, the first experimental observation of magnetic skyrmions in the 2D vdW ferromagnet Fe3GeTe2 was reported using high-resolution scanning transmission X-ray microscopy (STXM) and Lorentz transmission electron microscopy measurements (Fig. 6a). A skyrmion crystal state can be generated both dynamically using current pulses and statically using canted magnetic fields (Fig. 6b)79.                 Figure 6. Magnetic skyrmion lattice phase in FGT. a) Representative STXM image of skyrmion lattice stabilized over the whole FGT at Bz=0 mT and T=100 K. Scale bar, 2 μm. b) Experimental phase diagram of magnetic configurations as a function of temperature and magnetic field79.    4.3 Quantum spin liquids  Quantum spin liquids (QSLs) are a class of quantum disordered phases where reduced dimensionality, geometric frustration and quantum fluctuations completely destroy long range magnetic order down to zero temperature. QSLs are intriguing as they exhibit topological entanglement entropy as well as fractionalized excitations that obey emergent gauge fields (see Ref. 80 for a recent review). Experimental search for QLSs mostly targeted layered magnets whereas the majority of the theoretical studies are for two-dimensional models. Therefore, 2D vdW magnets provide a unique opportunity to discover new QSLs. Two promising routes are the (i) honeycomb lattices with Kitaev exchange and (ii) triangular lattices with frustrated interactions. Recently, a strongly spin–orbit-coupled vdW Mott insulator, a-RuCl3, has emerged as a prime candidate for hosting an approximate Kitaev QSL81-85. Figure 7a shows the thermal hall conductivity when applying tilted magnetic field on α-RuCl3 at different temperatures and the half-integer plateau indicates the Majorana fermion, which is a sign of Kitaev spin liquid phase. Figure 7b shows the phase diagram of a-RuCl3 in a field tilted at q = 60° (right inset). Below T ≈ JK/kB ≈ 80 K, the spin-liquid (Kitaev \na \nb  \n 14 paramagnetic) state appears and the half-integer quantized plateau of the 2D thermal Hall conductance is observed in the red area.                  Figure 7. Half-integer thermal quantum Hall Effect in 𝛼-𝑅𝑢𝐶𝑙3 (a) Half-integer thermal hall conductivity indicates the Majorana fermion, which is a sign of spin liquid phase. (b) Phase diagram of 𝛼-RuCl3 in a field tilted at θ = 60°. Below T ≈ JK/kB ≈ 80 K, the spin-liquid (Kitaev paramagnetic) state appears83.  4.4. From the synthesis perspective As mentioned above, the current technologies only enable the field to produce vdW crystals of the magnetic materials described above. While this is a logical choice for fundamental research in order to identify the most promising 2D magnetic materials, more comprehensive crystal growth studies are needed to understand how crystal growth techniques, thermal profile, and precursor types ultimately influence the fundamental behavior of vdW magnetic crystals. Current literature heavily relies on half a century-old crystal growth methods. While these techniques are well-established to produce these crystals, prior literature has given very little attention to magnetic quantum phenomena in 2D. Thus, more careful crystal growth studies are needed to pinpoint how defect density can be reduced, how crystalline quality can be improved, and magnetic impurities eliminated.  Clearly, new crystal growth techniques or recipes will be required to produce recently predicted magnetic crystals. This is a challenging task especially for ternary and quaternary systems wherein many different phases or compositions might energetically compete with each other to produce mixed-phase crystals.   In the very big picture, these crystal growth methods and isolation of mono- and few-layers of 2D magnets by exfoliation techniques present added complexities in translating these fundamental results from the laboratory setting to applications and eventually technology development. To this end, large scale growth methods will be required, in order to produce them at wafer scales. This is a big ask from the materials synthesis community when the number of theoretically predicted 2D magnetic crystals is still increasing on a daily basis. As such, fast progress is needed to quickly identify the champion magnetic materials and develop more focused synthesis techniques to produce them at large scales (centimeter to 2 inches wafer). Here, the grand challenge will likely be in retaining their structural quality and defect profiles while increasing their lateral sizes to wafer scales. Nevertheless, general 2D \nb \na  \n 15 growth techniques specific to halide, phosphosulfide, or tellurium based 2D magnetic material systems will greatly benefit the 2D magnetism community in the long run by offering the foundations of 2D growth in these material systems.   Still many questions emerge in the synthesis of atomically thin large area 2D magnetic layers; Can large area synthesis produce 2D sheets with environmental stability properties comparable to those in bulk crystals? Can we eliminate large defect densities like those observed in large area 2D transition metal dichalcogenide systems? Can we engineer defects, strain, or pressure in these 2D magnets during synthesis by using a different choice of substrates, growth cooling profiles, or introduced defects? Can these sheets be synthesized on arbitrary substrates? Can we alloy 2D magnetic materials to unleash exciting opportunities similar to those realized in traditional materials alloying? These and many other overwhelming but equally exciting questions are awaiting the materials synthesis community and only brilliant work by researchers in the field will be capable to provide solid answers.  5. ACKNOWLEDGMENTS S.T acknowledges support from DOE, NSF DMR 1552220, DMR 1955889, DMR 1904716, and NSF CMMI 1933214. D.D. acknowledges ASU for startup funds. A.B.S and O.E. acknowledge support from NSF 1904716. R.C. acknowledges support from the Alfred P. Sloan Foundation. RC and QS work was supported by the STC Center for Integrated Quantum Materials, NSF DMR 1231319. V.P. acknowledges support from the MINECO of Spain through the project PGC2018-101334-B-C21. J.L.L. acknowledges support from the Aalto Science-IT project.   6. DATA AVAILABILITY STATEMENT.  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"    },    {        "title": "2008.10770v1.Recent_advancements_in_the_study_of_intrinsic_magnetic_topological_insulators_and_magnetic_Weyl_semimetals.pdf",        "content": "1 \n Recent advancement s in the study of  intrinsic magnetic topological \ninsulators  and magnetic Weyl semimetals  \n \nWei Ning and Z hiqiang  Mao*  \n \nDepartment of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, \nUSA  \n \nAbstract  \n   The studies of  topological insulators and topological semimetals have been at frontiers of \ncondensed matter physics and material science . Both  classes of materials  are characterized by  \nrobust surface  states created  by the  topology of the bulk band structure s and exhibit exotic \ntransport properties. When magnetism is present in  topological materials and break s the time-\nreversal symmetry, more exotic quantum phenomena  can be generated , e.g. quantum anomalous \nHall effect, axion insulator, large intrinsic anomalous Hall effect, etc . In this research update, we \nbriefly summarize the recent research progress es in magnetic topological materials , including  \nintrinsic  magnetic topological insulators and magnetic Weyl semimetals . \n \n \n \n \n \n \n \n*zim1@psu.edu  \n 2 \n I. Introduction  \nMagnetic topological materials, including magnetic topological insulators  (TI) and \nmagnetic topological semimetals , have attracted broad interest s. Magnetic TIs can be achieved in \nthree different ways: magnetic doping in a TI [ 1,2], proximity of a TI to a ferromagnetic (FM) or \nan antiferromagnetic (AFM) insulator [3,4,5], or creating  intrinsic FM or AFM  order in a TI  [6]. \nThe spontaneous magnetization  induced in magnetic TIs  interacts with topological surface state s \nand open s a gap at the  surface  Dirac point , which can generate a new topological quantum state - \nquantum anomalous Hall insulator  (QAH I), when chemical potential is tuned to an appropriate \nvalue  in thin film samples . Since QAH I features spin -polarized chiral edge state, which can \nsupport dissipationless current, it carries great promise for applications in future energy saving \nelectronics.  QAH I was first  realized in thin TI films of Cr - and/or V -doped (Bi,Sb) 2Te3 [1,2]. \nThis pioneering work has generated a great  deal of interest and several review a rticles \n[7,8,9,10,11] on this topic  have been published.  In this research update, we will focus on \nreview ing recent studies on intrinsic magnetic TI  MnBi 2Te4 and its related materials .   \nIn magnetic topological semimetals, the interplay between magnetism and non -trivial \nband topology can also generate new exotic quantum states .  One remarkable example is  time \nreversal symmetry  (TRS)  breaking Weyl semimetal (WSM) state  in which  linearly dispersed, \nspin-split bands  cross  at discrete momentum points , thus resulting in  Weyl nodes.  Low energy \nexcitations near Weyl nodes behave as chiral Weyl fermions. Weyl nodes always come in pairs \nwith opposite chirality and they can be understood  as source and drain of Berry curvature in \nmomentum space. When the Weyl nodes are at or close to the Fermi level, net Berry curvature \ncan be present  due to broken TRS, which can give rise to new exotic quantum phenomena such \nas large intrinsic anomalous H all effect (AHE) [12] and anomalous Nernst effect  [13]. Like non -3 \n magnetic WSMs, magnetic WSMs are also characterized by topological surface states, i.e. \nsurface Fermi arcs  [14]. Theory has predicted  TRS breaking  WSMs can also evolve into QAHI \nwhen the dimensionality is reduced from 3D to 2D  [15]. Experimentally, several materials , \nincluding Co3Sn2S2 [16,17,18,19], Co 2MnGa  [20,21], Co2MnAl  [22], Mn 3Sn [23], GdPtBi [ 24] \nand YbMnBi 2 [25]  have been reported  to be  TRS breaking WSM states.  In this research review , \nwe will also give a brief overview on the studies of these materials . \nII. Intrinsic m agnetic topological insulator  MnBi 2Te4 and its related materials   \nAlthough quantum anomalous Hall  effect (QAHE) has been seen in thin TI films of Cr - \nand V -doped (Bi ,Sb)2Te3 [1,2], the ‘critical temperature’ required is below ~2  K, severely \nconstraining the exploration of fundamental physics and technological applications. \nInhomogeneous surface gap induced by randomly distributed magnetic dopants  is believed to be \nthe origin of low -temperature requirement for observing QAH E [26,27]. High -temperature \nQAH E has been predict ed to occur in thin films of intrinsic FM or AFM TI materials [ 15,28]. \nNevertheless, despite considerable theoretical and experimental efforts, there has been little \nprogress until the recent discovery of an intrinsic AFM TI MnBi 2Te4 [29,30,31]. MnBi 2Te4 is a \nlayered ternary tetradymite compound ; it crystallizes in a rhombohedral  structure  (space group \nR-3m), built of the stacking of  Te-Bi-Te-Mn-Te-Bi-Te septuple layers (SLs)  (Fig. 1a). SLs are \ncoupled through van der Waals bonding .  \nThe single crystals of MnBi 2Te4 can be grown either from the melt with stoichiometric \ncomposition [31, 32] or using the flux method with excessive Bi 2Te3 serving as flux  [33]. Since \nMnBi 2Te4 is metastable  [34], its single crystals can be obtained only through quenching at a \ntemperature close to 590 C. For the melt growth, the stoichiometric mixture first needs to be 4 \n heated to a high temperature (700 -1000C), then slowly cooled down to a temperature close to \n590, finally followed by annealing and quenching at this temperature [31,32]. For the flux \ngrowth, prolonged slow cooling (~2 weeks) from ~600 C to ~590 C is necessary, and the \nexcessive flux is separated through centrifuging  [33]. \nMnBi 2Te4 enables combin ation of  intrinsic  antiferromagnetism with nontrivial band \ntopology , thus giving rise to an intrinsic AFM TI  [29,30,31].  Its antiferromagnetism is produced \nby the Mn -sub-lattice, while its nontrivial band topology  is formed by inverted Bi and Te pz \nbands at the  point due to strong spin -orbital coupling  (SOC ). Its AFM state shows  an A-type \nAFM order ( TN = 25K) [31,33,35], characterized  by Mn FM layers stacked antiferromagnetically \nalong the c-axis and the ordered magnetic moments are aligned along  the c-axis [33]. A large \nspin gap as well as magnetic frustration due to large next -nearest neighbor AFM exchange have \nalso been probed in recent inelastic neutron scattering experiments  on MnBi 2Te4 [36]. On the \n(001) surface,  a large gap ( ~88 meV [31]) is opened at the surface Dirac node due to the \nbreaking of the S=T1/2 symmetry (  and T1/2 represent the time reversal and primitive translation \nsymmetry respectively)  (Fig. 1b) . Such a surface gap  was probed in ARPES measurements on \nsingle crystal samples  first by Otrokov et al. [31] (Fig. 1d)  and subsequently  by several other \ngroups  [37,38]. However, there have also been reports on  ARPES experiments which  \n[39,40,41,42] show  the surface Dirac cone state is gapless either in the paramagnetic or the AFM \nstate (Fig. 1e) .  \nMnBi 2Te4 offers an ideal platform to realize new exotic topological quantum states. \nTheory predicts it can  host not only high-temperature QAH E and axion insulator with \ntopological magnetoelectric effect in thin film samples [ 29,30,31, 43], but also an ideal Weyl \nsemimetal state with one pair of Weyl  nodes near the Fermi level in its bulk FM phase driven by 5 \n external magnetic fields or strain (Fig. 1c)  [29,30]. Moreover, chiral Majorana mode  is also \npredicted to be accessible  via interaction  between MnBi 2Te4 and a s-wave superconductor [ 44]. \nRecently, remarkable progresses have been made toward rea lizing these predicted quantum states  \n[45,46].   \nDeng et al.  [45] firstly reported the observation of quantized Hall resistance  of h/e2 (h is \nthe plank constant and e is the elemental charge) in atomically thin MnBi 2Te4 flakes with odd \nnumber of  SLs (i.e. 5 SLs) under zero magnetic fields (Fig.2a). Such quantized Hall resistance is \naccompanied by zero longitudinal resistance, which is typical behavior of QAH I. Contrasted \nwith the conventional QH E in 2D electron  gas, QAHE in MnBi 2Te4 does not originate from \nquantized Landau levels.  When  MnBi 2Te4 is exfoliated to flakes with even number of SL s (e.g. \n6 SLs) , it is found to exhibit axion insulator behavior at zero magnetic field, characterized by \nlarge longitudinal resistance and zero H all resistance [ 46] (Fig.2b); moderate magnetic fields can \ndrive the axion insulator to the Chern insulator with quantized Hall resistance of h/e2 (Fig. 2c) .   \nFurthermore, another interesting result observed in MnBi 2Te4 atomic crystals is the high \nChen number QAHE ( C=2) (Fig. 2d and Fig.2e) [47]. High Chern number QAHE is of interest in \nview of applications, since a high Chern number could enable the chiral edge states to carry \nlarger current. Quantum  confinement effect induced by dimensionality reduction should play an \nimportant role in realizing the QAHE and axion insulator in the 2D thin layers of MnBi 2Te4. In \nfew-layer thin films, the surface states should dominate its longitudinal transport properties and \nthe two surfaces (top and bottom) display half -integer Hall conductance of opposite (axion \ninsulator) or identical sign (QAHE). However, for thick flakes, interlayer coupling affects its \ntransport properties, leading to very differ ent transport behavior from few -layer thin flakes [ 47]. 6 \n This may explain why thinner MnBi 2Te4 flakes show QAHE or axion insulator, but thicker \nflake s (9 or 10 SLs) behave as a high Chern number insulator .   \n Additionally, experimental studies  on MnBi 2Te4 and Mn(Bi,Sb) 2Te4 bulk single crystals  \nhave also revealed many other interesting properties. First, Lee et al. [32] found  MnBi 2Te4 \nundergoes two magnetic transitions upon increasing magnetic field  (parallel to the c -axis) , i.e. \nthe spin -flop transition from an AFM to a canted antiferromagnetic (CAFM) state at H c1 (~3.6 T) \nand the CAFM -to-FM transition at H c2 (~7.7T) . The CAFM state show s intrinsic AHE due to the \nnon-collinear spin structure  [32]. Second , both magnetism and carrier density  in MnBi 2Te4 are \nfound to be tunable  by Sb substitution for Bi. Single crystals of the Mn(Bi 1-xSbx)2Te4 alloy series \nwith 0 x  1 can be made using a similar flux growth method used for growing MnBi 2Te4 \n[48,49,50]. Both H c1 and H c2 are suppressed by Sb substitution for Bi and merge as x approaches \n1 [49]. As x is equal or close  to 1, the system involves strong competition between FM and AFM \nphases  [49]. Both FM and AFM phases have been synthesized for MnSb 2Te4 [49, 51] and \nMnSb 1.8Bi0.2Te4 [50, 52]. FM MnSb 2Te4 is predicted to hos t either an ideal type -II Weyl \nsemimetal phase  [51], or the simplest type -I Weyl semimetal with only one pair of Weyl nodes \non the three -fold rotational axis  under strain tuning  [53], while  FM MnSb 1.8Bi0.2Te4 has been \nreported to show unusual AHE  [52]. In the AFM  Mn(Bi 1-xSbx)2Te4 series, the carrier density can \neffectively been tuned by changing Sb concentration, down  to  a minimum near x = 0.3 where \nthe carrier type also changes from electron to hole  [48,49,50]. Such a critical composition could \nfavor the observation of QAHE. The realization of QAHE state generally requires the chemical \npotential to be inside the gap to  achieve a bulk insulating state. For pristine MnBi 2Te4, a \nrelatively large gate voltage is required to tune it to such a state since as -grown MnBi 2Te4 \ncrystals are always heavily electron doped  [45] . If crystals of Mn(Bi 1-xSbx)2Te4 ( x ~0.26) was 7 \n used in the devices, QAHE can probably be seen at much smaller gate voltages.  Moreover, Sb \nsubsti tution for Bi increases the surface gap [ 54], which might increase the observation \ntemperature of QAHE.  Additionally , Lee et al .[50] recently reported  the predicted ideal Weyl \nstate can be achieved in the CAFM and FM phases of the samples  with minimal carrier density. \nThis is revealed by a magnetic -field induced electronic  phase transition at the AFM -to-FM phase \nboundary, a large intrinsic anomalous Hall effect  (see F ig. 1f) , a non -trivial  Berry phase of the \ncyclotron orbit and a large positive magnetoresistance in the FM phase  [50].  \nIn addition to MnBi 2Te4, several  van der Waals materials relevant to MnBi 2Te4, including \nMnBi 4Te7, MnBi 6Te10 and MnBi 8Te13, are also recently  reported  to be intrinsic AFM/FM \nTI/axion insulator [ 55,56,57,58,59,60,61,62,63]. These materials belong to the same family, \nwhich can be expressed as (MnBi 2Te4)(Bi 2Te3)m with m = 1, 2, 3 , …Their common structural \ncharacteristic is the alternating stacking of [MnBi 2Te4]-SLs and [Bi2Te3] quintuple layers (QL s). \nThe main difference between the m=1,2,3 members is the number of QLs ( m) sandwiched \nbetween SLs ; m=1 for MnBi 4Te7, m=2 for MnBi 6Te10, and m=3 for MnBi 8Te13. The magnetic \nproperties of these materials depend on m. With increasing m, the interlayer AFM coupling \nbecomes weak  due to increased separation distance between Mn magnetic layers. Although  \nMnBi 4Te7 and Mn Bi6Te10 remain AFM, their Neel temperature decrease to 13.0 K and 11.0 K \nrespectively  [55,57]. MnBi 8Te13, however, becomes FM with the Curie temperature of 10.5K  \n[60], indicating interlayer magnetic coupling involves competition between AFM and FM  and \nlarger separation distance favors FM coupling. We note FM MnBi 6Te10 with TC =12K as well as \na AFM -to-FM transition in MnBi 4Te7 were  also reported  [59, 64], suggesting the Gibbs energy \ndifference between AFM and FM phases for th ese composition s is very small. Band structure \ncalculations and ARPES studies  [55,58,59,60,61] have sh own all these materials host topological 8 \n phases: MnBi 4Te7 is an intrinsic AFM TI [ 55], whereas MnBi 6Te10 is either an AFM axion \ninsulator [ 61] or a FM TI [ 59]. MnBi 8Te13 is reported  to be  an intrinsic FM axion insulator [ 60]. \nAnother common property of these materials is that they all show large magnetic hysteresis and \nlow spin -flip transition fields. Therefore, they offer a new promising platform to explore novel \ntopological quantum state s, including QAH E and axion insulator  at high temperatures .     \n \nIII. Magnet ic Weyl  semimetals   \n \nThree -dimensional (3D) Dirac semimetal s, which were first theoretically predicted and \nexperimentally verified in Na 3Bi [65,66,67] and Cd 3As2 [68,69,70], can be viewed as a 3D \ngraphene.  A Dirac semimetal can transform  into a WSM  by breaking either the time -reversal  \n(TRS)  or inversion symmetry . Inversion symmetry broken WSMs were first discovered in  non-\nmagnetic  TaAs -class materials  [71,72,73,74,75]. The TRS-breaking WSM were initially \npredicted in Re 2Ir2O7 (Re=rare earth) [ 14], HgCr 2Se4 [15] and recently demonstrated  in several \nmagnetic materials systems, including  Co3Sn2S2 [16,17,18,19],  Co2MnGa  [20, 76], Co2MnAl \n[22],  Mn 3Sn/Mn 3Ge [23, 77], GdPtBi [ 24] and YbMnBi 2 [25]. In this section, we will review the \nrecent research progress in the study of  these  magnetic WSMs .  \n1）Ferromagnetic WSMs    \n1a. Kagome -lattice WSM  Co3Sn2S2: The kagome lattice  is known to host exotic quantum states  \nsuch as spin liquid [78]. Recent  studies show a layered FM compound Co3Sn2S2 with Kagome -\nlattice (space group, R -3m) hosts a TRS breaking  WSM state [16,17,18,19]. Th e magnetic \nproperties  of this material  originate  from the kagome -lattice  of cobalt, whose magnetic moments \norder ferromagnetically and are oriented along the out-of-plane  direction  in the ground state  (Fig. \n3a) [16]. Recent SR experiments showed such an out -of-plane FM order sustains up to 90K, 9 \n and then evolves into a mixed phase of the out -of-plane FM and the in -plane AFM order in the \n90-172K range, an d finally to a mixed phase of paramagnetic and FM in the 172 -175K range  \n[79]. The C 3v-rotation and inversion symmetries of this material generates a total of six nodal \nrings without considering SOC. When SOC is considered , the linear crossing points of nodal \nrings split  into three pairs of Weyl modes as shown in Fig. 3d. These Weyl  nodes are only about \n60 meV above the Fermi level  according to theoretical calculations [ 18]. \nThe experimental evidence for such a  TRS breaking WSM state of  Co3Sn2S2 was first \nrevealed in magnetotransport measurements [16,17]. This material  exhibit s not only negative \nlongitudinal magnetoresistance (LMR) [ 16], but also  large intrinsic anomalous Hall effect (AHE) \n[16,17] and large anomalous Nernst effect (ANE) [ 80,81,82]. Negative LMR is the manifestation \nof the chiral anomaly arising from the charge pumping between  paired Weyl nodes with opposite \nchirality  under parallel electric and magnetic fields . The intrinsic origin of AHE in Co 3Sn2S2 is \nevidenced by the observation s that its  anomalous Hall conductivity 𝑦𝑥𝐴 is nearly independent of \nlongitudinal conductivity 𝑥𝑥 below 90K [ 16] and linearly increases with magnetization [ 17]. \nBesides large   𝑦𝑥𝐴 (~1130 -1.cm-1 at ~90K), the anomalous Hall angle ( θH = 𝑦𝑥𝐴/𝑥𝑥) of \nCo3Sn2S2 was also found to be large, ~ 20% at 90K , about one order of magnitude larger than  \nthose of  typical magnetic systems  [83]. As shown in Fig. 3b, such large values of 𝑦𝑥𝐴 and θH \ncan be attributed to large net Berry curvature of occupied states  [16]. The steep decrease of 𝑦𝑥𝐴  \nabove 90  K is due to the fact that the out -of-plane FM phase coexists with the in -plane AFM \nphase and the volume fraction of the FM phase decreases with increasing temperature  [79]. \nFurthermore, systematic studies on the ANE of Co 3Sn2S2 by Ding et al. [80] show  that the \nanomalous Nernst response 𝑆𝑥𝑦𝐴 (the ratio of transverse electric field to the longitudinal \ntemperature gradient) is inversely proportional to the carrier mobility  , contrasted with the 10 \n ordinary Nernst response 𝑆𝑥𝑦0, which is  . This indicates that anomalous transverse \nthermoelectricity 𝑥𝑦𝐴 in Co 3Sn2S2 is determined by the Berry curvature, rather than the mean \nfree path  [80].  \nThe hallmark of the electronic structure of a WSM phase is the surface Fermi arcs  (SFAs) , \nwhich connect the projected Weyl nodes with oppo site chirality on the surface Brillouin -zone. \nSuch an expected feature for Co 3Sn2S2 has recently been demonstrated by the angle-resolved \nphotoemission spectroscopy (ARPES) [18] and scanning tunneling spectroscopy  (STS)  \nexperiments  [19]. As shown in Fig. 3c  and 3d(i) , the surface Fermi arcs are comp rised  of three \nline-segments which connect the projected Weyl  points (WP+ and WP -) near M. These three \nline-segments form a triangle -shaped surface Fermi surface, which is clearly visualized in the \nARPES [Fig. 3d(ii)] and STS experiments [18,19]. Moreover, the STS experiments also show \nthe surface Fermi -arc contour and Weyl node connectivity is termination dependent [19]. By \nmeans of in situ electron doping , the ARPES experiments also detected bulk Weyl nodes  (Fig. 3e)  \n[18].   \n1b. Heusler  alloy  FM WSM Co2MnGa and Co 2MnAl : Recent theoretical  work predicted  that \nCo-based Heusler compound Co 2XZ (X=V, Zr, Nb, Ti, Mn, Hf; Z =Si, Ge, Sn, Ga and Al)  can \nhost unique  FM WSM phase s [84,85,86]. First, its Weyl states can have the least number of \nWeyl  nodes (two), which can make the interpretation of spectroscopic and transport properties \nmuch easier. Second, the Weyl node separation in momentum space is large, giving rise to large \nanomalous Hall effect, which is of great use for applications. Third, the Weyl node location in \nmomentum space can be manipulated by controlling the magnetization direction [84,85]. These \ncharacteristics make Co 2XZ a promising platform for exploring novel magnetic Weyl physics \nand potential applications.  11 \n Although Co 2XZ allows for many different element combinations, experimental studies on \ntheir possible exotic properties induced by the expected WSM states are sparse, which is \npossibly due to the difficulty of the single crystal growth of this family of materials. Co 2MnGa is \nthe first confirmed member that have distinct properties associated with the FM WSM state  \n[20,76]. This material is a room temperature ferromagnet with the Curie temperature of 690 K \nand possesses a cubic structure with  the space group of Fm -3m (Fig.4a)  [20]. It shows a giant \nAHE , with its 𝑦𝑥𝐴 being  as large as 2000 Ω-1cm-1 at low temperature  (Fig.4b)  [20]. Furthermore, \nCo2MnGa was also found to show a giant ANE [20,76]. The Nernst signal S yx increases with \nelevating temperature, reaching a record high value of Syx ≈ 6 μV K-1 at room temperature and \napproaching  8 μV K-1 at 400 K  [20,76], which is more than one order of magnitude larger than \nthe typical values known for the ANE in other magnetic conductors. These results , together with \nthe unsaturated positive longitudinal ma gnetoconductance  (i.e. chiral anomaly ) [20], provide \ntransport evidence for Weyl fermions in Co 2MnGa.  \nRecent ARPES  studies  on Co 2MnGa  [21] unveil its characteristics of Weyl state. The \ncombination  of mirror symmetries  and FM ordering  of this material  leads to 3D Weyl nodal lines \nwith 2 -fold degeneracy, which form Hopflike links and nodal chains [ 21,87]. These nodal lines \nare protected by  mirror symmetries and give rise to drumhead surface states. Both Weyl nodal \nlines and drumhead surface states have been visualized in the ARPES experiments [ 21]. The top \npanel in Fig. 4c shows an ARPES constant energy surface  [21], from which the projection of \nWeyl nodal lines on  the ka-kb plane can be seen clearly. The distribution of calculated Berry \ncurvature on the ka-kb plane (bottom panel of Fig. 4c) matches well the shape of Weyl nodal line \nprojection  [21], indicating that the Berry curvature of Co 2MnGa predominantly stems from Weyl 12 \n nodal lines. The 𝑦𝑥𝐴 calculated from the Berry curvature is indeed consistent with the \nexperimental value [ 21].  \n Like Co 2MnGa, the L2 1 structural phase of Co 2MnAl is a lso predicted to be a FM WSM \ncandidate [ 86] and early Berry curvature calculations suggest it has the largest AHE  among the  \nCo2XZ Heusler alloys  [88]. The recent success of single crystals growth of this material has \nenable d further experimental studies  on this material . Li et al. [ 22] indeed observe d a tunable  \ngiant AHE in Co 2MnAl single crystals. Its 𝑦𝑥𝐴 is as large as 1300 Ω−1cm−1 at room temperature; \nmore noticeably, its room temperature anomalous Hall angle reaches a record value among \nmagnetic conductors, with tan θH = 0.21 , which brings the promise for practical device \napplications . Theoretical studies have further clarified the intrinsic mechanism of such a giant \nAHE [ 22]. As shown in Fig. 4e, the two lowest conduction bands and the two highest  valence \nbands cross along four Weyl nodal rings without SOC. These four nodal rings  at the kx,y,z = 0 \nplanes are  protected by  the mirror symmetries  and interconnected, forming Hopflike link s.  \nWhen SOC is considered, magnetic moments are  couple d to the lattice, thus reducing the \nsymmetry  and gapping the nodal ring s. The gapped nodal rings generate  large Berry curvature \nand thus give rise to the huge anomalous Hall conductivity . For instance, when the magnetization \nis oriented along the [001] direction, the nodal rings on the kz=0 plane do not open gaps due to \nthe preserved mirror symmetry, while the nodal rings on the kx,y = 0 planes  are gapped. Berry \ncurvature calculations showed the largest nodal ring (ring#3 in Fig. 4e) make dominant \ncontributions to the Berry curvature and the resulting  large  𝑦𝑥𝐴 (~1400 -1.cm-1) [22].  \nAnother important property of Co 2MnAl is that its band topology and resulting AHE can \nbe controlled by the rotation of magnetization axis . This  is because that  the mirror symmetries \ndepend on the magnetization orientation. For example, for M//[001] , the nodal rings on the kz=0 13 \n plane  are gapless , but  gapped on the kx,y=0 planes ; however, if M//[111], all nodal rings are \ngapped. Therefore, the rotation of magnetization leads to a cos -like angular dependence in 𝑦𝑥𝐴, \nwhich is indeed observed in experiments (Fig. 4f) [ 22]. Since Co 2MnAl is a soft ferromagnet, the \nrotation of magnetization can be driven by a weak magnetic field.  As such, this material offers \nan ideal platform to explore band topology tuning by  magnetization.  \n \n2）Antiferromagnetic WSMs   \n2a. WSM state in Chiral antiferromagne t Mn 3Sn: Besides  FM materials hosting Weyl \nfermions as summarized above , recent works also reveal ed that Weyl fermions can also exist in \nantiferromagnet ic materials . Mn 3Sn is a recently established , remarkable example [ 23]. This \nmaterial is a h exagonal antiferromagnet and exhibits  noncollinear spin ordering with TN≈ 420 K  \n[23]. Although  this material shows a very small magnetization ∼0.002 µ B/Mn, it exhibits  large \nAHE  [89] and ANE  [90]. In addition, it also exhibits positive magnetoconductance  under parallel \nelectric and magnetic fields  [23]. These distinct transport properties are associated its Weyl state \nwhich have been confirmed by both theory calculations and ARPES measurements  [23]. In this \nmaterial, t he electron –hole band crossings form a nodal ring surrounding K points  without SOC . \nWhen SOC is considered, the TRS breaking lifts the spin degeneracy and leads to band crossing \nand the formation of Weyl nodes at different energies. Since its Weyl nodes appear at points \nwhere electron and hole pockets intersect, the resulting Weyl cones  are strongly tilted , which is a \ntypical nature of a type -II Weyl semimetal  Isostructural compound Mn 3Ge is also found to host \na similar WSM state [ 77,91]. In addition, a nother layered, AFM compound YbMnBi 2 with \nsquare lattice was also reported to have a  TRS breaking,  type-II Weyl state [ 25].   \n 14 \n  2b. Magnetic -field -induced WSM in AFM Half -Heusler  alloys .  \nAnother route to generate TRS breaking WSM s is to use external magnetic  field to \ngenerate Weyl nodes in Dirac semimetals  or zero gap semiconductors. For instance, bulk Dirac \ncones in Na 3Bi have been found to evolve to Weyl cones under magnetic field [92]. The first \nexample of Weyl  nodes generated by external magnetic fields in zero gap semiconductors is \nGdPtBi [ 24, 93,94], which is a half Heusler compound  and possesses a cubic structure consisting \nof interpenetrating face centered cubic lattices and exhibits antiferromagnetic ordering with TN = \n9.2 K  [24]. The hallmark s of Weyl state in transport, including negative LMR  caused by chiral \nanomaly  [24], large intrinsic AHE  [93] and planar Hall effect (PHE)  [95], have  been \ndemonstrated in this material . Such  a magnetic field -driven Weyl state  is believed to originate \neither from the  Zeeman splitti ng by the external magnetic field [24] or from the  exchange \nsplitting of the conduction band s [94]. The finding of field -induced Weyl state  in GdPtBi  has \ninspired studies on other isostructural half Heusler compounds like NdPtBi [ 94] and TbPtBi \n[96,97]. TbPtBi was found to show an exceptionally large AHE with the anomalous Hall angle  \nof 0.68 -0.76 [ 96] ( about a few times larger than that in GdPtBi [ 93]), though its other transport \nsignatures  of Weyl state (e.g.  PHE ) are not significant. The f irst-principle electronic structure \nand the associated anomalous Hall conductivity calculations show  that the exceptionally large \nAHE in TbPtBi  does not originate from the Weyl points but that it is driven by the large net \nBerry curvature produced by the anticrossing of spin -split bands near the Fermi level  [96]. \n \n3. Outlook  15 \n       From the above overview, it can be seen clearly that the interplay between band topology and \nmagnetic state s can create unique topological quantum states which are potentially useful for \ntechnology applications. QAHI is the most promising example. However, device applications \nrequire such a state to be realized at room temperature. Although  theoretical studies show this is \npossible,  efforts are needed to discover more promising candidate materials which combine band \ntopology with room temperature magnetism. Among the current magnetic TIs, the magnetic \ntransitions all occur at low temperatures (below 50  K). TIs with room temp erature magnetism is \nhighly desired.  Materials design by theory and computations could play a key role in this regard. \nSince FM WSMs can also evolve into QAHI in the 2D limit, discovering room temperature ideal \nFM WSMs may be another route to realize high temperature QAHI.  An ideal WSM generally \nrefer s to a Weyl state with all Weyl nodes being symmetry related and at or close to the chemical \npotential, without interfered with by any other bands ; current magnetic WSMs are not ideal.  \nFurthermore , magnetic topological materials offer unique opportunities to explore the \ntopological -electronic -state’s tunability  by magnetism and new fundamental physics of \ntopological fermions.    \n \nAcknowledgement:  \nZ.Q.M. acknowledges the support from  the US National  Science Foundation  (NSF)  under grants \nDMR 1917579 , 1832031  and the Penn State 2D Crystal Consortium -Materials Innovation \nPlatform (2DCC -MIP) funded by the NSF ( cooperative agreement DMR -1539916 ). \n 16 \n Data Availability Statements ：The data that support the findings of this study are \navailable from the corresponding author upon reasonable request.  \n \nReferences:  \n \n[1] C.-Z. Chang , J. S. Zhang, X. Feng, J. Shen, Z. C. Zhang, M. H. Guo, K. Li, Y. B. Ou, P. 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Mao,  Phys . Rev. B 101 , 161105  (2020) . \n[97]R. Singha, S. Roy, A. Pariari, B. Satpati, and P. Mandal , Phys . Rev. B 99, 035110 (2019).  \n \nCaptions  \nFigure 1. (a)  Crystal and magnetic structure of MnBi 2Te4 [29]. (b)  Spin-resolved electronic \nstructure of the MnBi 2Te4 (0001) surface.[ 31] (c) The bulk band structure of MnBi 2Te4 in the \nFM phase, which shows a pair of  Weyl nodal crossing at W/W′ along the -Z directio n [30]. (d) \nBand dispersion of MnBi 2Te4 measured on the (0001) surface [31]. (e) A gapless surface Dirac 25 \n  \ncone measured on MnBi 2Te4 [39]. (f) The maximal anomalous Hall angle of the FM phase vs. \ncarrier concentration for Mn(Bi1 -xSbx)2Te4  [50]. \n \nFigure 2. (a) Magnetic field dependences of Hall resistance Ryx and longitudinal resistance Rxx  \nmeasured  on a flake with five SLs  at T = 1.4 K. Ryx reaches 0.97 h/e2 at μ 0H = 0 T and rises to \n0.998 h/e2 above μ 0H ~ 2.5 T [45]. (b)  The gate voltage ( Vg ) dependence of longitudinal \nresistivity ρxx and the derivative of Hall resistivity ρyx with respect to magnetic field measured at \nT=1.6K around zero magnetic f ield, measured on a 6 SLs device.[ 46]. (c) The gate voltage \ndependence of ρxx and ρyx at1.6K and 9 T, from which typical characteristics of a Chern  \ninsulator can be seen. The inset in (b) and (c) schematically illustrates the FM order and \nelectronic structures of the Axion and Chern insulators [ 46].  (d) Hall resistance Ryx as a function \nof magnetic field at various temperatures probed in a 10 -SL MnBi 2Te4 device. The Ryx plateau \nreaches 0.97 h/2e2 at 13 K [47].  (e) Gate voltage dependence of Rxx and Ryx at 2 K and 15 T in a \n10-SL device.  The inset schematically illustrates the FM order and electronic structure of the \nC=2 Chern insulator [ 47]. \n \nFigure 3. (a) Crystal structure of Co 3Sn2S2. The cobalt atoms form a ferromagnetic kagome \nlattice with a C3z-rotation [ 16]. (b) Temperature dependences of the anomalous Hall conductivity \n(𝐻𝐴), the longitudinal conductivity (σ) and the anomalous Hall angle ( 𝐻𝐴/σ) at zero magnetic \nfield [16]. (c) Schematic of the bulk and surface Brillouin zones along the (001) surface of \nCo3Sn2S2, which display three pairs of Weyl  nodes connected by surface Fermi arcs (SFA, \nmarked by yellow line segments) [ 18]. (d) Comparison between the calculated Fermi surface of \nboth bulk a nd surface states (i) and the experimentally measured Fermi surfaces (ii). The \nmagenta and green dots in (i) represent the Weyl points with opposite chirality [18]. (e) Linear \nband crossing at a Weyl point probed by the ARPES measurements at 10K [18].  \n 26 \n  \nFigure 4. (a) L21 ordered cubic full Heusler structure of Co 2MnGa [ 20]. (b) Temperature \ndependence of the Hall conductivity -σyx for Co 2MnGa [20]. (c) The bottom panel shows the z \ncomponent of the Berry curvature of occupied states and the top panel presents the ARPES \nconstant energy surface at the corresponding EB [21]. (d) Nodal rings and the first Brillouin zone \nof Co 2MnAl. Without SOC, there are nodal rings on mirror planes [ 22]. (e) There are four nodal \nrings centered at the Z point of the FCC Brillouin zone for Co 2MnAl [ 22]. (f) The anomalous \nHall conductivity 𝑦𝑥𝐴 of Co 2MnAl as a function of the magnetization orientation angle. The \nexperimental and theoretical results are represented by red and black circles , respectively [ 22]. \n \n \n \n "    },    {        "title": "2008.12907v1.High_throughput_Design_of_Magnetic_Materials.pdf",        "content": "High-throughput Design of Magnetic Materials\nHongbin Zhang\nInstitute of Materials Science, TU Darmstadt, 64287 Darmstadt, Germany\n(Dated: August 28, 2020)\nMaterials design based on density functional theory (DFT) calculations is an emergent \feld of\ngreat potential to accelerate the development and employment of novel materials. Magnetic mate-\nrials play an essential role in green energy applications as they provide e\u000ecient ways of harvesting,\nconverting, and utilizing energy. In this review, after a brief introduction to the major functional-\nities of magnetic materials, we demonstrated the fundamental properties which can be tackled via\nhigh-throughput DFT calculations, with a particular focus on the current challenges and feasible\nsolutions. Successful case studies are summarized on several classes of magnetic materials, followed\nby bird-view perspectives for the future.\nContents\nI. Introduction 2\nII. Main applications of magnetic materials 3\nA. Main applications 3\nIII. Criticality and sustainability 4\nIV. Main challenges and possible solutions 5\nA. New compounds and phase diagram 5\nB. Correlated nature of magnetism 7\nC. Magnetic ordering and ground states 8\nD. Magnetic \ructuations 10\nE. Magnetic anisotropy and permanent magnets 12\n1. Origin of magnetocrystalline anisotropy 12\n2. Permanent magnets: Rare-earth or not? 13\nF. Magneto-structural transitions 16\nG. Spintronics 18\nH. Magnetic topological materials 21\nI. Two-dimensional magnetic materials 23\nV. Case Studies 26\nA. High-throughput work\rows 26\nB. Heusler compounds 27\nC. Permanent magnets 30\nD. Magnetocaloric materials 30\nE. Topological materials 31\nF. 2D magnets 31\nVI. Future Perspectives 32\nA. Multi-scale modelling 32\nB. Machine learning 33\nVII. Summary 34\nReferences 36Glossary of acronyms\n2D: two-dimensional\nAFM: antiferromagnetic\nAHC: anomalous Hall conductivity\nANC: anomalous Nernst conductivity\nAMR: anisotropic magnetoresistance\nARPES: angle-resolved photoemission\nBZ: Brillouin zone\nDFT: density functional theory\nDLM: disordered local moment\nDOS: density of states\nDMFT: dynamical mean \feld theory\nDMI: Dzyaloshinskii-Moriya interaction\nDMS: dilute magnetic semiconductors\nFiM: ferrimagnetic\nFL: Fermi liquid\nFM: ferromagnetic\nFMR: ferromagnetic resonance\nFOPT: \frst-order phase transition\nGMR: giant magnetoresistance\nHM: half-metal\nHMFM: half-metallic ferromagnet\nHTP: high-throughput\nICSD: inorganic crystal structure database\niSGE: inverse spin Galvanic e\u000bect\nIFM: itinerant ferromagnet2\nIM: itinerant magnet\nMAE: magnetocrystalline anisotropy energy\nMBE: molecular beam epitaxy\nMCE: magnetocaloric e\u000bect\nMGI: materials genome initiative\nML: machine learning\nMr: remanent magnetization\nMs: saturation magnetization\nMOKE: magneto-optical Kerr e\u000bect\nMRAM: magnetic random-access memory\nMSMA: magnetic shape memory alloy\nMSMA: magnetic shape memory e\u000bect\nNFL: non-Fermi liquid\nNM: nonmagnetic\nQAHE: quantum anomalous Hall e\u000bect\nQAHI: quantum anomalous Hall insulator\nQCP: quantum critical point\nQPI: quasipartical interference\nQPT: quantum phase transition\nQSHE: quantum spin Hall e\u000bect\nRE: rare-earth\nRPA: random phase approximation\nSCR: self-consistent renormalization\nSdH: Shubnikov-de Haas\nSDW: spin density wave\nSGS: spin gapless semiconductor\nSHE: spin Hall e\u000bect\nSIC: self-interaction correction\nSOC: spin-orbit coupling\nSOT: spin-orbit torque\nSTS: scanning tunnelling spectroscopy\nSQS: special quasi-random structure\nSTT: spin-transfer torque\nTC: Curie temperature\nTI: topological insulatorTRIM: time-reversal invariant momenta\nTM: transition metal\nTMR: tunnelling magnetoresistance\nTN: N\u0013 eel temperature\nvdW: van der Waals\nXMCD: x-ray magnetic circular dichroism\nI. INTRODUCTION\nAdvanced materials play an essential role in the func-\ntioning and welfare of the society, particularly magnetic\nmaterials as one class of functional materials suscepti-\nble to external magnetic, electrical, and mechanical stim-\nuli. Such materials have a vast spectrum of applications\nthus are indispensable to resolve the current energy is-\nsue. For instance, permanent magnets can be applied for\nenergy harvesting ( e.g., wind turbine to generate elec-\ntricity) and energy conversion ( e.g., electric vehicles and\nrobotics with mechanical energies from electricity). Ac-\ncording to the BCC research report,1the global mar-\nket for soft and permanent magnets reached $32.2 billion\nin 2016 and will reach $51.7 billion by 2022. Further-\nmore, to go beyond the quantum limit of conventional\nelectronic devices, spintronics exploiting the spin degree\nof freedom of electrons provides a promising alternative\nfor energy e\u000ecient apparatus, which has attracted in-\ntensive attention in the last decades. Nonetheless, there\nare still a variety of pending fundamental problems and\nemergent phenomena to be understood. Therefore, there\nis a strong impetus to develop better understanding of\nmagnetism and magnetic materials, and to design mag-\nnetic materials with optimal performance.\nThe conventional way of discovering and employing\nmaterials is mostly based on the empirical structure-\nproperty relationships and try-and-error experiments,\nwhich are time and resource costly. Early in 2011, the\nU.S. government has launched the Materials Genome Ini-\ntiative (MGI), aiming at strategically exploring materials\ndesign.2The proposed synergistic paradigm integrating\ntheory, modelling, and experiment has proven to be suc-\ncessful, while there are still plenty of open challenges.3\nFrom the theoretical point of view, as the material prop-\nerties comprise the intrinsic (as given by the crystal struc-\nture) and extrinsic (as given by the microstructure) con-\ntributions, a multi-scale modelling framework should be\nestablished and embraced, leading to the integrated com-\nputational materials engineering approach4and the Eu-\nropean materials modelling council program.5For both\nframeworks and the counterparts, density functional the-\nory (DFT) is of vital importance, due to its capability to\nobtain accurate electronic structure and thus the intrin-\nsic properties, and essential parameters for multi-scale\nmodelling, ensuring the predictive power.3\nTill now, high-throughput (HTP) computations based\non DFT have been applied to screening for various\nfunctional materials, such as electro-catalyts,6thermo-\nelectrics,7and so on. Correspondingly, open databases\nsuch as Materials Project,8AFLOWlib,9NOMAD,10\nand OQMD11have been established, with integrated\nplatforms like AiiDA12and Atomate13available. This\nchanges the way of performing DFT calculations from\nmonitoring jobs on a few compounds to de\fning and ap-\nplying work\rows applicable on thousands of compounds,\nso that the desired properties get evaluated and opti-\nmized, e.g., the thermoelectric \fgure of merit.7\nIn contrast to the other physical properties such as\nband gaps, absorption energies for catalysts, and thermo-\nelectric properties, magnetic properties and their charac-\nterization based on DFT pose a series of unique chal-\nlenges and there has been limited exploration of design-\ning functional magnetic materials. For instance, there are\nthree key intrinsic magnetic properties, i.e., magnetiza-\ntion, magnetic anisotropy energy (MAE), and the critical\nordering temperature, which are di\u000ecult to be evaluated\nin a HTP manner. Besides the intriguing origin of mag-\nnetization ( e.g., localized or itinerant) where consistent\ntreatment requires a universal theoretical framework be-\nyond local and semilocal approximations to DFT, it is\nalready a tricky problem to identify the magnetic ground\nstates as the magnetic moments can get ordered in ferro-\nmagnetic (FM), ferrimagnetic (FiM), antiferromagnetic\n(AFM), and even incommensurate noncollinear con\fg-\nurations. Moreover, the dominant contribution to the\nMAE can be attributed to the relativistic e\u000bects, e.g.,\nspin-orbit coupling (SOC), where the accurate evaluation\ndemands good convergence with respect to the k-mesh,\nresulting in expensive computational e\u000borts. Lastly, the\ncritical ordering temperature is driven by the magnetic\nexcitations and the corresponding thermodynamic prop-\nerties cannot in principle be addressed by the standard\nDFT without extension to \fnite temperature. Due to\nsuch challenges, to the best of our knowledge, there\nare only a few limited successful stories about applying\nthe HTP method on designing magnetic materials (cf.\nSect. V for details).\nIn this review, we aim at illustrating the pending prob-\nlems and discussing possible solutions to facilitate HTP\ndesign of magnetic materials, focusing particularly on the\nintrinsic properties for crystalline compounds which can\nbe evaluated based on DFT calculations. Also, we pre-\nfer to draw mind maps with a priority on the conceptual\naspects rather than the technical and numerical details,\nwhich will be referred to the relevant literature. Corre-\nspondingly, the major applications of magnets and the\ncriticality aspects will be brie\ry summarized in Sect. II.\nIn Sect. III, we will elucidate several fundamental as-\npects which are essential for proper HTP computations\non magnetic materials, with detailed discussions on a\nfew representative classes of magnetic materials. The in-\ndepth mathematical/physical justi\fcation will not be re-\npeated but referred to necessary publications for curiousreaders, e.g., the recently published \\Handbook of Mag-\nnetism and Advanced Magnetic Materials\"14is a good\nsource for elaborated discussions on the physics of mag-\nnetic materials. In Sect. V the successful case studies\nwill be summarized, with future perspectives given in\nSect. VI.\nIt is noted that there are a big variety of magnetic ma-\nterials which are with intriguing physics or promising for\napplications, as listed in the magnetism roadmap.15{17\nWe excuse ourselves for leaving out the current dis-\ncussions on quantum magnets,18{20quantum spin liq-\nuid,21(curvilinear) nanomagnets,22,23Skyrmions,24high\nentropy alloys,25multiferroics,26and ultrafast mag-\nnetism,27which will be deferred to reviews speci\fed.\nAlso, we apologize for possible ignorance of speci\fc pub-\nlications due to the constraint on the man power.\nII. MAIN APPLICATIONS OF MAGNETIC\nMATERIALS\nA. Main applications\nThe main applications of magnetic materials are com-\npiled in Table.I, together with the representative mate-\nrials. We note that small magnetic nanoparticles (2-100\nnm in diameter) can also be used in biology and medicine\nfor imaging, diagnostics, and therapy,28which is beyond\nthe scope of this review. Most remaining applications\nare energy related. For instance, both permanent mag-\nnets ( i.e., hard magnets) and soft magnets are used to\nconvert the mechanical energy to electrical energy and\nvice versa . Whereas spintronics stands for novel devices\noperating with the spin of electrons, which are in princi-\nple more energy e\u000ecient than the established electronic\ndevices based on semiconductors.\nPermanent magnets are magnetic materials with signif-\nicant magnetization (either FM or FiM) which are mostly\napplied in generating magnetic \rux in a gap, with the\ncorresponding \fgure of merit being the maximum en-\nergy product (BH) max.29As marked by the B-H curves\nin Fig. 1, it provides an estimation of the energy stored\nin the magnets and can be enhanced by maximizing the\nhysteresis, e.g., by increasing the remanent magnetiza-\ntion Mrand coercivity H c. Such extrinsic quantities like\nMrand Hcare closely related to the microstructure of the\nmaterials, where the corresponding upper limits are given\nby the intrinsic quantities such as saturation magnetiza-\ntion Msand MAE. Nowadays, for commercially available\nNd2Fe14B, 90% of the theoretical limit of (BH) maxcan\nbe achieved,30suggesting that there is a substantial space\nto further improve the performance of other permanent\nmagnets.\nIn contrast, soft magnets are magnetic materials with\nsigni\fcant M sas well, and they are easy to be mag-\nnetized and demagnetized corresponding to high initial\nand maximal relative permeability \u0016r= B/(\u00160H), where\nB =\u00160(H+M) denotes the magnetic induction under4\nFIG. 1: (color online) Typical hysteresis curves of a FM ma-\nterial. The B-H and M-H loops are denoted by red dashed\nand blue solid lines, respectively. The black bullets marks\nthe critical values of the key quantities such as the residual\ninduction B r, remanent magnetization M r, coercivity H c, in-\ntrinsic coercivity H ciand saturation magnetization M s. The\nmaximal energy product (BH) maxis indicated by the shaded\nregion.\nmagnetic \feld H, and \u00160is the vacuum magnetic per-\nmeability. That is, the corresponding hysteresis loop as\nshown in Fig. 1 is ideally narrow for soft magnets, cor-\nresponding to vanishing hysteresis in the ideal case. In\nthis regard, most soft magnets are Fe-based alloys of cu-\nbic structures, because Fe has the largest average mo-\nment among the 3 dseries of elements in solids. The soft\nmagnets are widely applied in power generation, trans-\nmission, and distribution (Table. I). In addition to the\nMswhich is an intrinsic property, microstructures play a\nsigni\fcant role in developing soft magnets, e.g., the eddy\ncurrent losses caused by the cyclical rearrangements of\nmagnetic domains in AC magnetic \felds should be min-\nimized.31\nAnother interesting class of magnetic materials are\nthose with phase transitions (either \frst-order or second-\norder) driven by external magnetic \felds, leading to the\nmagnetocaloric e\u000bect (MCE) and magnetic shape mem-\nory e\u000bect (MSME). Compared to the 45% e\u000eciency for\nthe best gas-compressing refrigerators, the cooling e\u000e-\nciency of MCE devices based on Gd can reach 60% of the\ntheoretical limit.32Moreover, the MCE devices are highly\ncompact and less noisy, giving rise to environmental-\nfriendly solutions for ever-growing demands of cooling on\nthe global scale. Following the thermodynamic Maxwell\nrelation@S\n@B\f\f\nT=@M\n@T\f\f\nB, optimized MCE can be achieved\nupon phase transitions with signi\fcant changes in the\nmagnetization, which can be easily realized in those com-\npounds with \frst-order phase transitions (FOPTs). This\ncauses a problem about how to reduce the concomitant\nhysteresis caused by the athermal nature of FOPTs.33\nOn the other hand, the MSME is caused by the domainwall twinning induced by magnetic \felds during FOPTs\n(mostly martensitic transitions), and the corresponding\nmagnetic shape memory alloys (MSMAs) such as Ni-Mn-\nGa alloys can be applied as actuators and sensors.34\nLast but not least, magnetic materials play a pivotal\nrole in the spintronic information technologies.35As de-\ntailed in Sect. IV G, the \frst generation of spintronic de-\nvices rely on the spin-dependent transport (either di\u000bu-\nsive or tunnelling) phenomena which are best represented\nby the discovery and application of the giant magnetore-\nsistance (GMR) e\u000bect and the conjugate spin transfer\ntorque (STT).36Whereas the second generation spintron-\nics takes advantage of spin-orbit coupling (SOC) (thus\ndubbed as spin-orbitronics) and functions via the genera-\ntion, manipulation, and detection of spin current, engag-\ning both FM and AFM materials.37Many materials have\nbeen investigated including half-metals (HMs),38dilute\nmagnetic semiconductors (DMSs)39, leading to devices\nlike magnetic random-access memory (MRAM), and spin\ntransistors. Additionally, magnetic materials can also be\napplied for the storage of information in both analogue\nand digital forms, where the materials optimization is\na trade-o\u000b between competing quantities like signal to\nnoise ratio, write-ability with reasonable \felds, and long-\nterm stability against thermal \ructuations.40\nIII. CRITICALITY AND SUSTAINABILITY\nMagnetic materials are a prime example where the\nsupply risk of strategic metals, here most importantly\nrare-earth (RE) elements, might inhibit the future de-\nvelopment. In general, resource criticality and sustain-\nability is understood as a concept to assess potentials\nand risks in using raw materials for certain technologies,\nparticularly strategic metals and their functionalities in\nemerging technologies.41Such principles and evaluation\ncan be transferred to the other material classes subjected\nto HTP design and future applications. As the appli-\ncation and market for magnetic materials are expected\nto grow in future technologies, factors such as geologi-\ncal availability, geopolitical situation, economic develop-\nments, recyclability, substitutability, ecological impacts,\ncritical competing technologies and the performance of\nthe magnetic materials have to be considered at the be-\nginning in the development of new materials and their\nconstituent elements.\nTo be speci\fc, the RE elements such as Sm, Dy, and\nTb (the latter two are usually used to improve the co-\nercivity and thermal stability of the Nd-Fe-B systems42)\nare of high supply risk due to geopolitical reasons with\nan expensive price, as speci\fed in the \\Critical Materials\nStrategy\" report,43leading to a source of concern dubbed\nas the \\rare-earth crisis\".44Particularly, such RE metals\nin high demands have relatively low abundance, whereas\nthe abundant light RE elements such as La and Ce may\nalso be utilized to design volume magnets with desired\nproperties. The scenario also applies to transition metal5\nTABLE I: A summary of the main applications of magnetic materials, with representative compounds\nrequired properties materials applications\nhigh anisotropy AlNiCo\nlarge M r Ferrite power generation\npermanent magnets high coercivity H c Sm-Co electric motor\nlow permeability Nd-Fe-B robotics\nhigh Curie temperature\nlow anisotropy\nlarge M s transformer core\nsoft magnets high permeability Fe-Si inductor\nsmall coercivity low-carbon steel magnetic \feld shield\nlow hysteresis Fe-Co\nlow eddy current losses\nvibration damper\nMSMA magnetic phase transition Ni-Mn-Ga actuator\nstructural reorientation sensor\nenergy harvester\nlarge temperature change \u0001T La-Fe-Si\nmagnetocaloric minimal hysteresis Ni-Mn-X magnetic refrigeration\nmaterials mechanical stability Gd5(Si,Ge) 4\nstrong spin polarization HMs sensor\ne\u000ecient spin injection DMS MRAM\nspintronics long spin di\u000busion length spin transistor\ncontrollable interfaces\nhigh ordering temperature\nmedium coercivity\nmagnetic storage large signal-to-noise ratio Co-Cr hard disks\nshort writing time 10\u00009s FePt\nlong stability time 10 years\n(TM) and main group elements such as Co, Ga, and Ge,\nwhich are susceptible to the sustainable availability and\ncan probably be substituted with Mn, Fe, Ni, Al, etc.\nIn addition to high prices and low abundance, toxic ele-\nments like As and P are another issue, which dictate com-\nplex processing to make the resulting compounds useful.\nThat is, not all elements in the periodic table are equally\nsuitable choices for designing materials in practice.\nIV. MAIN CHALLENGES AND POSSIBLE\nSOLUTIONS\nIn this section, the fundamental aspects of designing\nmagnetic materials are discussed, with the pending prob-\nlems and possible solutions illustrated.\nA. New compounds and phase diagram\nThere have been of the order of 105inorganic com-\npounds which are experimentally known ( e.g. from the\nICSD database), which amount to a few percent of all\npossible combinatorial compositions and crystal struc-\ntures.45Therefore, a common task for materials design of\nany functionality is to screen for unreported compounds\nby evaluating the stabilities, which can be performed via\nHTP DFT calculations. Theoretically, the stabilities canbe characterized in terms of the following criteria:\n1. thermodynamical stability, which can be character-\nized by the formation energy and distance to the\nconvex hull (Fig. 2), de\fned as\n\u0001G=G(target)\u0000G(comp. phases)\u00140; (1)\nwhereG(target) and G(comp. phases) denote the\nGibbs free energy for the target and competing\nphases, respectively. We note that the Gibbs free\nenergy is a function of temperature T and pressure\nP, leading to possible metastable phases at \fnite\ntemperature/pressure. In most cases, only the for-\nmation energy with respect to the constituent ele-\nments is evaluated, but not the convex hull with re-\nspect to the other (either known or unknown) com-\npeting phases. It is observed that evaluating the\nconvex hull can reduce the number of predicted sta-\nble compounds by one order of magnitude.46Thus,\nit is recommended to carry out the evaluation of\nconvex hull routinely for reasonable predictions, us-\ning at least those relevant compounds collected in\nexisting databases as competing phases. Certainly,\nthere are always unknown phases which can jeop-\nardize the predictions, but it is believed that the\nfalse positive predictions will be signi\fcantly re-\nduced with the resulting candidates more accessible\nfor further experimental validation.6\nFIG. 2: Sketch of the convex hull for a hypothetical binary\nmetal(M) nitride.47Copyright requested.\nA few comments are in order. First of all, DFT\ncalculations are usually performed at 0 K and am-\nbient pressure, thus cannot be directly applied to\naccess the metastability at \fnite temperature or\npressure. The pressure can be easily incorporated\ninto calculations using most DFT codes, whereas\nthe temperature constraint can also be remedied\nby evaluating the Gibbs free energies as discussed\nlater in Sect. IV D. Nevertheless, given that typi-\ncal solid phase transitions occur around a few hun-\ndreds Kelvin which amounts to tens of meV, it\nis a challenging task to evaluate the phase transi-\ntion temperature accurately based on DFT calcula-\ntions. In this regard, the combination of the DFT\nand CALPHAD48methods provides a good solu-\ntion where experimental measurements can be eas-\nily incorporated, in addition to a straightforward\ngeneralization to multicomponent systems. The\nresulting phase diagram will also provide valuable\nguidance for the experimental synthesis. Further-\nmore, the stability of metastable phases can be fur-\nther enhanced by modifying the experimental pro-\ncesses. For instance, precursors can be used in or-\nder to reduce the thermodynamic barrier (Fig. 2),\ne.g., using more reactive nitride precursors like NH 3\nmay allow the synthesis of metastable phases.47\nLastly, non-equilibrium synthesis techniques such\nas molecular beam epitaxy (MBE), melt spinning,\nmechanical alloying, and speci\fc procedure to get\nnano-structured materials can also be applied to\nobtain metastable phases, as demonstrated for the\n\"-phase of MnAl.49Particularly for magnetic compounds, in most cases\nwhen evaluating the thermodynamic stability, the\nformation energies and distances to the convex hull\nare obtained assuming FM con\fgurations, as done\nin Materials Project and OQMD. This is justi-\n\fed for the majority of the systems, but we ob-\nserved that the magnetic states will change the\nenergy landscape drastically for compounds with\nstrong magneto-structural coupling, which will be\ndiscussed in detail in Sect. IV C. Another criti-\ncal problem is how to obtain reliable evaluation\nof the formation energies for the RE-based inter-\nmetallic compounds, where mixing DFT (for the\nintermetallics) and DFT+U (for the RE elements)\ncalculations are required. This is similar to the case\nof TM oxides, where additional correction terms\nare needed to get a reasonable estimation of the\nformation energies.50It is noted that it is a chal-\nlenging task to do proper DFT+U calculations in\na HTP way, where local minima occur very often\nwithout good control on the density matrix and\nadditional orbital polarization correction is needed\nto get correct orbital moments.51Therefore, a solu-\ntion to evaluate the thermodynamic stability is still\nmissing for RE-based intermetallic compounds.\n2. mechanical stability, which describes the stability\nagainst distortions with respect to small strain. It\ncan be formulated as\n\u0001E=Edistorted\u0000E0=1\n2X\ni;jCi;j\u000fi\u000fj>0;(2)\nwhereCijdenotes the elastic constant and \u000fthe\nstrain, and E0denotes the ground state energy\nfrom DFT with equilibrium lattice parameters. De-\npending on the crystalline symmetry, Eq. (2) can\nbe transformed into the generic Born stability con-\nditions,52i.e., a set of relationships for the elastic\nconstants which can be straightforwardly evaluated\nbased on DFT.\n3. dynamical stability, which describes the stability\nagainst atomic displacements due to phonons. In\nthe harmonic approximation,53it yields\nE=E0+1\n2X\nR;\u001bX\nR0;\u001b0DR;\u001b\b\u001b;\u001b0\nR;R0DR0;\u001b0\n|{z }\n>0; (3)\nwhere \b\u001b;\u001b0\nR;R0is the force constant matrix, DR;\u001b\nmarks the displacement of atom at Rin the Carte-\nsian direction \u001b. The positive de\fniteness can\nbe assured by diagonalizing the Fourier transfor-\nmation of \b\u001b;\u001b0\nR;R0, where no negative eigenvalue is\nallowed. That is, there should be no imaginary\nphonon mode in the whole Brillouin zone (BZ).\nOn the other hand, if there does exist imaginary7\nphonon modes, particularly at a few speci\fc q-\npoints, it suggests that the compounds can prob-\nably be stabilized in the correspondingly distorted\nstructure.\nTill now, the stabilities are evaluated assuming spe-\nci\fc crystal structures. Such calculations can be easily\nextended to include more structural prototypes based on\nthe recently compiled libraries of prototypes.54,55This\nhas been applied successfully to screen for stable ABO 3\nperovskite56and Heusler57compounds. One interesting\nquestion is whether there are phases with unknown crys-\ntal structures which might be stable or metastable, giving\nrise to the question of crystal structure prediction. It is\nnoted that the number of possible structures scales expo-\nnentially with respect to the number of atoms within the\nunit cell, e.g., there are 1014(1030) structures with 10\n(20) atoms per unit cell for a binary compound.58There\nhave been well established methods as implemented in\nUSPEX59and CALYPSO60to predict possible crystal\nstructures. For instance, within the NOVAMAG project,\nthe evolutionary algorithm has been applied to predict\nnovel permanent materials, leading to Fe 3Ta and Fe 5Ta\nas promising candidates.61\nTo summarize, HTP calculations can be performed to\nvalidate the stability of known compounds and to predict\npossible new compounds. The pending challenges are (a)\nhow to systematically address the stability and metasta-\nbility, ideally with phase diagrams optimized incorporat-\ning existing experimental data, (b) how to extent to the\nmulticomponent cases with chemical disorder and thus\nentropic free energy, so that the stability of high entropy\nalloys62is accessible, and (c) how to perform proper eval-\nuation of the thermodynamic stability for correlated RE\ncompounds and transition metal oxides.\nB. Correlated nature of magnetism\nFor magnetic materials, based on (a) how the magnetic\nmoments are formed and (b) the mechanism coupling the\nmagnetic moments, the corresponding theory has a bifur-\ncation into localized and itinerant pictures.63The former\ndates back to the work of Heisenberg,64which applies\nparticularly for strongly correlated TM insulators and\nRE-based materials with well-localized f-electrons.The\nlocal magnetic moments can be obtained following the\nHund's rule via\n\u0016e\u000b=gJp\nJ(J+ 1)\u0016B; (4)\nwheregJis the Land\u0013 e g-factor, Jdenotes the total an-\ngular momentum, and \u0016Bis the Bohr magneton. The\nmagnetic susceptibility above the critical ordering tem-\nperature ( i.e., in the paramagnetic (PM) states) obeys\nthe Curie-Weiss law:65\n\u001f(T) =C\nT\u0000\u0012W; (5)whereCis a material-speci\fc constant, and \u0012Wdenotes\nthe Curie-Weiss temperature where \u001fshows a singular\nbehavior. The Weiss temperature \u0012Wcan have pos-\nitive and negative values, depending on the resulting\nferromagnetic (FM) and antiferromagnetic (AFM) or-\ndering, whereas its absolute value is comparable to the\nCurie/N\u0013 eel temperature. The Weiss temperature corre-\nsponds to the molecular \feld as introduced by Weiss,66\nwhich can be obtained via a mean-\feld approximation of\nthe general Heisenberg Hamiltonian:\nH=\u00001\n2X\ni;jJijSi\u0001Sj; (6)\nwhereJijdenotes the interatomic exchange interaction.\nSpeci\fcally, Si\u0001Sjis approximated to Sz\nihSz\nji+hSz\niiSz\nj\nby neglecting the \ructuations and hence the spin \rip\ntermS+\niS\u0000\nj, leading to an e\u000bective \feld proportional to\nthe magnetization corresponding to the exchange \feld in\nspin-polarized DFT.67\nOn the other hand, for intermetallic compounds with\nmobile conduction electrons, the magnetic ordering is\ncaused by the competition between the kinetic energy\nand magnetization energy. Introducing the exchange en-\nergyI=J=N whereJis the averaged exchange integral\noriginated from the intra-atomic inter-orbital Coulomb\ninteraction and N the number of atoms in the crystal, a\nferromagnetic state can be realized if68\nI\u0017(EF)>1; (7)\nwhere\u0017(EF) is the density of states (DOS) at the Fermi\nenergyEFin the nonmagnetic state. The so-obtained\nitinerant picture has been successfully applied to under-\nstand the occurrence of ferromagnetism in Fe, Co, Ni,\nand many intermetallic compounds including such ele-\nments.69\nFIG. 3: The Rhodes-Wohlfarth ratio qc=qsfor various mag-\nnetic materials.65Copyright requested\nHowever, the real materials in general cannot be classi-\n\fed as either purely localized or itinerant and are mostly8\nat a crossovers between two limits. As shown in Fig. 3,\nwhere the ratio of Curie-Weiss constant qcand the sat-\nuration magnetization qsis plotted with respect to the\nCurie temperature, the qc=qsratio should be equal to 1\nfor localized moments as indicated by EuO, whereas for\nthe itinerant moments qc=qs\u001d1. It is interesting that\ntheqc=qsratio for Ni is close to 1 while it is a well known\nitinerant magnet. More interestingly, there exists a class\nof materials such as TiAu and ZrZn 2where there is no\npartially \flled d- orf-shells but they are still displaying\nferromagnetic behavior.65\nIt is noted that the localized moment picture based\non the Heisenberg model has limited applicability on\nthe intermetallic magnets, while the Stoner picture fails\nto quantitatively describe the \fnite temperature mag-\nnetism, e.g., it would signi\fcantly overestimate the Curie\ntemperature and leads to vanishing moment and hence\nno Curie-Weiss behavior above T C. A universal picture\ncan be developed based on the Hubbard model,70\nH=X\nij\u001btijay\ni\u001baj\u001b+X\niUni\"ni# (8)\nwheretijis the hopping parameter between di\u000berent\nsites,nis the number of electrons, and U denotes the\non-site Coulomb interaction. As a matter of fact, the\nStoner model is a mean \feld approximation of the Hub-\nbard model in the weakly correlated limit andthe Heisen-\nberg model can be derived in the strongly coupled limit\nat half-\flling.71Another uni\fed theory has been formu-\nlated by Moriya considering self-consistent renormaliza-\ntion of the spin \ructuations in the static and long-wave\nlength limits,63which works remarkably well for the weak\nitinerant magnets such as ZrZn 2.65In terms of the Hub-\nbard model, which can be further casted into the DFT\n+ dynamical mean-\feld theory (DMFT) framework,72\nwhere the hoppings are obtained at the DFT level and the\nonsite electron-electron correlations are evaluated accu-\nrately locally including allorders of Feynman diagrams.\nRecently, the DFT+DMFT methods have been applied\non ZrZn 2,73revealing a Fermi liquid (FL) behavior of the\nZr-4delectrons which are responsible for the formation of\nmagnetic moments. In this regard, it provides a univer-\nsal solution which can be applied for magnetic materials\nfrom localized to itinerant limits, which is better than\nbare DFT. Importantly, for correlated TM oxides and RE\ncompounds, the d- andf-moments are mostly localized,\nleading to narrow bands where the local quantum \ructu-\nations are signi\fcant and hence the spin \ructuations,74\nwhich are naturally included in DFT+DMFT. Neverthe-\nless, the usually performed single-site DFT+DMFT cal-\nculations cannot capture the transverse magnetic excita-\ntions particularly the long wave-length excitations criti-\ncal for the behavior around T C. Detailed discussions will\nbe presented in Sect. IV D. We noted that the classi\fca-\ntion of itinerant and localized magnetism is not absolute,\ne.g., the Ce-4fshell changes its nature depending on the\ncrystalline environment.75\nIn this sense, DFT as a mean \feld theory has beensuccessfully applied to magnets of the itinerant and lo-\ncalized nature. In the latter case, the DFT+U76method\nis usually applied to account for the strong electronic cor-\nrelations. However, there are known cases where DFT\nfails, e.g., it predicts a FM state for FeAl while exper-\nimentally the system is paramagnetic.77Also, the local\ndensity approximation (LDA) gives wrong ground state\nof Fe, i.e., LDA predicts the nonmagnetic fcc phase being\nmore stable than the FM bcc phase.78Therefore, cau-\ntions are required when performing DFT calculations on\nmagnetic materials and validation with experiments is\nalways called for. On the other hand, the DFT+DMFT\nmethod is a valuable solution which covers the whole\nrange of electronic correlations, but the key problem is\nthat the current implementations do not allow automated\nset-up and e\u000ecient computation on a huge number of\ncompounds.\nC. Magnetic ordering and ground states\nAssuming the local magnetic moments are well de-\n\fned, the Heisenberg Hamiltonian Eq. (IV B) is valid to\ndescribe the low temperature behavior, the interatomic\nexchange J ijis the key to understand the formation of\nvarious long-range ordered states.There are three types\nof mechanisms, based on the distance between the mag-\nnetic moments and how do they talk to each other.\n1.direct exchange for atoms close enough so that the\nwave functions have su\u000ecient overlap. Assuming\ntwo atoms each with one unpaired electron, the\nCoulomb potential is reduced for two electrons lo-\ncalized between the atoms when the atoms are very\nclose to each other, leading to AFM coupling based\non the Pauli's exclusion principle. On the other\nhand, when the distance between two atoms be-\ncomes larger, the electrons tend to stay separated\nfrom each other by gaining kinetic energy, result-\ning in FM coupling. Thus, the direct exchange is\nshort ranged, which is dominant for the coupling\nbetween the \frst nearest neighbors. This leads to\nthe Bethe-Slater curve, where the critical ratio be-\ntween the atomic distance and the spatial extent of\nthe 3d-orbital is about 1.5 which separate the AFM\nand FM coupling.79\n2.indirect exchange mediated by the conduction elec-\ntrons for atoms without direct overlap of the\nwave functions. It can be understood based on\nthe Friedel oscillation, where the conduction elec-\ntrons try to screen a local moment, thus forming\nlong range oscillating FM/AFM exchange coupling,\nleading to the RKKY-interaction named after Ru-\nderman, Kittel, Kasuya, and Yoshida.80It is the\ndominant exchange coupling for atoms beyond the\nnearest neighbour in TM magnets and localized\n4f-moments (mediated by the sp-electrons) in RE\nmagnets.9\n3.superexchange for insulating compounds with local-\nized moments, which is driven by kinetic energy\ngain via the virtual excitations of local spin be-\ntween magnetic ions through the bridging nonmag-\nnetic elements, such as oxygen in TM oxides. The\nsuperexchange can favor both AFM and FM cou-\nplings depending on the orbital occupation and the\nlocal geometry, which has been formulated as the\nGoodenough-Kanamori-Anderson rules.81\nWe note that in real materials, more than one type of\nexchange coupling should be considered, e.g., the direct\nexchange and RKKY exchange for intermetallic com-\npounds, and the competition of direct exchange and su-\nperexchange in TM oxides.\nBased on the discussions above, the exchange coupling\nbetween atomic moments can be either FM or AFM, de-\npending on the active orbitals, distances, and geometries.\nSuch AFM/FM exchange couplings induce various possi-\nble magnetic con\fgurations, which give rise to a compli-\ncation for HTP screening of magnetic materials. That is,\nthe total energy di\u000berence for the same compounds with\ndi\u000berent magnetic states can be signi\fcant. This can be\nroughly estimated by the Curie temperature (T C), i.e.,\nabout 0.1 eV corresponding to the T C(1043 K) of bcc\nFe. Thus, magnetic ground state is important not only\nfor the electron structure but also for the thermodynamic\nstability. However, the number of possible AFM con\fg-\nurations which should be considered to de\fne the mag-\nnetic ground state can be big, depending on the crystal\nstructures and the magnetic ions involved. Therefore,\nidentifying the magnetic ground state is the most urgent\nproblem to be solved for predictive HTP screening on\nmagnetic materials.\nThere have been several attempts trying to develop a\nsolution. The most straightforward way is to collect all\npossible magnetic con\fgurations from the literature for\na speci\fc class of materials or consider a number of most\nprobable states, such as done for the half-Heuslers.83\nHowever, it is hard to be exhaustive and more impor-\ntantly it is only applicable for one structural prototype.\nHorton et al. developed a method to enumerate possible\nmagnetic con\fgurations followed by HTP evaluation of\nthe corresponding total energies.84It is mostly applica-\nble to identify the FM ground state, as the success rate\nis about 60% for 64 selected oxides, which might due to\nthe correlated nature of such materials. Another interest-\ning method is the \fre\ry algorithm, as demonstrated for\nNiF2and Mn 3Pt,85where the magnetic con\fgurations\nare con\fned within the primitive cell. A systematic way\nto tackle the problem is to make use of maximal magnetic\nsubgroups,82where the magnetic con\fgurations are gen-\nerated in a progressive way taking the propagation vec-\ntors as a control parameter. We have implemented the\nalgorithm and applied it successfully on the binary in-\ntermetallic compounds with the Cu 3Au-type structure.\nIt is observed that the landscape of convex hull changes\nsigni\fcantly after considering the magnetic ground state,\nas shown in Fig. 4 for the binary Mn-Ir systems. That is,\nFIG. 4: The binary convex hull for Mn-Ir. Dashed (solid) lines\ndenotes the convex hull obtained assuming ferromagnetic con-\n\fgurations (with real magnetic ground states obtained via sys-\ntematic calculations by maximal magnetic subgroups). The\nmagnetic ground states for MnIr and Mn 3Ir are display on\ntop of the \fgure.82Copyright requested\nthe magnetic ground state matters not only for the elec-\ntronic structure but also for the thermodynamic stabil-\nity of magnetic materials, con\frming our speculation in\nSect. IV A. Recently, the genetic algorithm has been im-\nplemented to generate magnetic con\fgurations and com-\nbined with DFT calculations the energies for such con-\n\fgurations can be obtained in order to search for the\nmagnetic ground state.86It has been applied successfully\non FeSe, CrI 3monolayers, and UO 2, where not only the\ncollinear but also noncollinear con\fgurations can be gen-\nerated, which is also interesting for future exploration.\nFurthermore, the interatomic exchange J ijin\nEq. (IV B) can be evaluated based on DFT calculations,\nwhich can then be used to \fnd out the magnetic ground\nstates via Monte Carlo modelling. There are di\u000berent\nmethods on evaluating the J ij. The most straightfor-\nward one is the energy mapping, where the J ijfor each\npair of moments are obtained by the di\u000berence of total\nenergies for the FM and AFM con\fgurations.87Such\na method can be traced back to the broken symmetry\nmethod \frstly proposed by Noodleman88and later\ngeneralized by Yamaguchi,89where one of the spin state\nused in energy mapping is not the eigenstate of the\nHeisenberg Hamiltonian (Eq. IV B). In this regard, for\nmagnetic materials with multiple magnetic sublattices,\nthe isotropic J ijcan be obtained by performing least\nsquare \ftting of the DFT total energies of a \fnite\nnumber of imposed magnetic con\fgurations, e.g., using\nfour-state mapping method.87,90\nBased on a two-site Hubbard model, it is demonstrated\nthat the energy mapping method is only accurate in the\nstrong coupling limit, i.e., insulating states with well\nde\fned local moments.91Additional possible problems10\nfor such an approach are (a) the supercell can be large\nin order to get the long-range interatomic exchange pa-\nrameters and (b) the magnitude of local moments might\nchange for the FM and AFM calculations resulting in\nunwanted contribution from the longitudinal excitations.\nThus, this method is most applicable for systems with\nwell-de\fned local moments. In this case, the J ijcan also\nbe evaluated based on the perturbation theory, which can\nbe done with the help of Wannier functions hence there\nis no need to generate supercells.92The arti\fcial longi-\ntudinal excitations can also be suppressed by performing\nconstrained DFT calculations.93\nThere are two more systematic ways to evaluate theJij. One is based on the so-called frozen magnon\nmethod,94where the total energies for systems with im-\nposed spin-waves of various qvectors are calculated and\nafterwards a back Fourier transformation is carried out\nto parameterize the real space J ij.95This method de-\nmands the implementation of noncollinear magnetism in\nthe DFT codes. Another one is based on the magnetic\nforce theorem,96where the J ijis formulated as a linear re-\nsponse function which can be evaluated at both the DFT\nand DFT+DMFT levels.97It is noted that the e\u000bective\nspin-spin interaction is actually a second order tensor,98\nwhich yields\n1\n2X\ni;jSi\u0001Jij\u0001Sj=1\n2X\ni;j\u00141\n3Tr(Jij)Si\u0001Sj+Si(1\n2(Jij+Jt\nij)\u0000Jij)Sj+1\n2(Jij\u0000Jt\nij)Si\u0002Sj\u0015\n(9)\nwhere the Heisenberg term in Eq. (IV B) corresponds to\nthe isotropic exchange1\n3Tr(Jij), the symmetric trace-\nless part Jsym\nij=1\n2(Jij+Jt\nij)\u0000Jijis usually referred\nas anisotropic exchange, and the antisymmetric part\nJantisym\nij =1\n2(Jij\u0000Jt\nij) represents the Dzyaloshinsky-\nMoriya interaction (DMI). It is noted that DMI is the\ncrucial parameter to form Skyrmions.24Also, in order to\naddress complex magnetic orderings, e.g., in 2D magnetic\nmaterials, the full tensor should be evaluated consistently\nto construct fully-\redged spin models.\nIn short, the essential challenges are (a) how to obtain\nthe magnetic ground states in a HTP manner and (b) how\nto evaluate the exchange parameters. As we discussed,\nthere are a few possible solutions for the former question.\nRegarding the latter, as far as we are aware, there has\nbeen no reliable implementation where the exchange pa-\nrameters can be evaluated in an automated way, where\neither the DFT part of the codes needs careful adap-\ntion for materials with diverse crystal structures or only\nspeci\fc components of the exchange parameters can be\nevaluated.99\nD. Magnetic \ructuations\nThe discussions till now have been focusing on the\nmagnetic properties at zero Kelvin, however, the thermo-\ndynamic properties and the magnetic excitations of var-\nious magnetic materials are also fundamental problems\nwhich should be addressed properly based on the micro-\nscopic theory. To this goal, the magnetic \ructuations\ndriven by temperature shall be evaluated, which will de-\nstroy the long-range magnetic ordering at T C/TN, result-\ning in a paramagnetic state with \ructuating moments.\nThe working horse DFT can in principle be generalized\nto \fnite temperature,100but to the best of our knowl-edge there is no implementation into the DFT codes on\nthe market. Speci\fcally for the magnetic excitations, the\nab initio spin dynamics formalism proposed by Antropov\net al.101is accurate but it is demanding to get imple-\nmented, thus it has only been applied to simple systems\nsuch as Fe. In the following, we will focus on (a) the\natomistic spin models, (b) the disordered local moment\napproximation, and (c) the DFT+DMFT methods, with\nthe thermodynamic properties at \fnite temperature in\nmind.\nFrom the physics point of view, there are two types of\nexcitations, i.e., transversal and longitudinal excitations.\nFor itinerant magnets, the former refers to the collec-\ntive \ructuations of the magnetization directions which\ndominates at the low-temperature regime, whereas the\nlatter is about the spin-\rip excitations ( i.e., Stoner exci-\ntations) leading to the reduction of magnetic moments.\nCorrespondingly for the localized moments, the occu-\npation probability of the atomic multiplets varies with\nrespect to temperature, giving rise to the temperature\ndependent magnitude of local moments, while the inter-\natomic exchange parameters of the RKKY type will cause\ncollective transversal spin waves as well.\nOne of the most essential intrinsic magnetic proper-\nties is the magnetic ordering temperature, i.e., TC(TN)\nfor FM (AFM) materials. The theoretical evaluation of\nthe TC/TNusually starts with the Heisenberg model\n(Eq. (IV B)) with the Jijparameters obtained from DFT\ncalculations, as detailed in Sect. IV C. As the spin wave\nprecession energy is smaller than the band width and\nthe exchange splitting, it is justi\fed to neglect the pre-\ncession of magnetization due to the spin waves when\nevaluating the electronic energies, dubbed as the adia-\nbatic approximation, the critical temperature can be ob-\ntained by statistical averaging starting from the Heisen-\nberg model. Both mean-\feld approximation and ran-11\ndom phase approximation can be applied, where the for-\nmer fails to describe the low-temperature excitations.102\nIn this way, the longitudinal excitations with compara-\nble energies103can be treated based on parameterized\nmodels.95,104Classical Monte Carlo is often used, lead-\ning to \fnite speci\fc heat at zero Kelvin,105while quan-\ntum Monte Carlo su\u000bers from the sign problem when\nlong-range exchange parameters with alternative signs\nare considered. One solution to this problem is to in-troduce the quantum thermostat.106It is noted that\nmore terms such as external magnetic \felds and mag-\nnetic anisotropy can be included, leading to atomistic\nspin dynamics which is valuable to develop a multi-scale\nunderstanding of magnetic properties.107,108\nAnother fundamental quantity is the Gibbs free en-\nergy which is crucial to elucidate the magneto-structural\nphase transitions. In general, the Gibbs free energy can\nbe formulated as109\nG(T;P;em) =H(T;V;em) +PV=E(V;em) +Felectronic(V;T) +Flattice(V;T) +Fmagnetic(V;T) +PV; (10)\nwhere emdenotes the magnetization direction, T, P, and\nV refer to the temperature, pressure, and volume, re-\nspectively. The electronic, lattice, and magnetic contri-\nbutions to the free energy can be considered as indepen-\ndent of each other, because the typical time scale of elec-\ntronic, lattice, and magnetic dynamics is about 10\u000015s,\n10\u000012s, and 10\u000013s, respectively. This approximation is\nnot justi\fed anymore for the paramagnetic states where\nthe spin decoherence time is about 10\u000014s,110which will\nbe discussed in detail below.\nTo evaluate Fmagnetic, the disordered local moment\n(DLM) method111provides a valuable solution for both\nmagnetically ordered and disordered (e.g., paramagnetic)\nstates. The DLM method also adopts the adiabatic ap-\nproximation, and it includes two conceptual steps:112(1)\nthe evolution of the local moment orientations and (2)\nthe electronic structure corresponding to each speci\fc\ncon\fguration. A huge number of orientational con\fg-\nurations are required in order to accurately evaluate the\npartition function and hence the thermodynamic poten-\ntial, which can be conveniently realized using the multiple\nscattering theory formalism based on the Green's func-\ntions, namely the KKR method.113,114The recent gen-\neralization115into the relativistic limit enables accurate\nevaluation of the temperature-dependent MAE for both\nTM-based115and 4f-3dmagnets,116and also the mag-\nnetic free energies with even frustrated magnetic struc-\ntures.112Nevertheless, the codes with the KKR method\nimplemented requires sophisticated tuning for complex\ncrystalline geometries and the evaluation of forces is not\nstraightforward.\nThe DLM method can also be performed using su-\npercells by generating magnetic con\fgurations like spe-\ncial quasi-random structures (SQS), which is invented to\nmodel the chemical disorder.117,118It has been applied to\nsimulate the paramagnetic states with supercells of equal\nnumber of up and down moments, particularly to eval-\nuate the lattice dynamics as the forces can be obtained\nin a straightforward way. Moreover, con\fgurational av-\neraging should be done in order to model local defects,which can be achieved by generating a number of su-\npercells with randomly distributed moments which sum\nup to zero.118Spin-space averaging can be performed on\ntop to interpolate between the magnetically ordered and\nthe paramagnetic states, as done for bcc Fe.119In this\nway, the spin-lattice dynamics can be performed to ob-\ntain the thermodynamic properties for the paramagnetic\nstates,120which cannot be decoupled as discussed above.\nIt is noted that usually the spin-lattice dynamics is per-\nformed based on model parameters,121where the instan-\ntaneous electronic response is not considered. The DLM\nmethod with supercell has also been recently combined\nwith the atomistic spin dynamics with in general non-\ncollinear spin con\fgurations.122It is noted that the local\nmoments should be very robust in order to perform such\ncalculations, otherwise signi\fcant contribution from lon-\ngitudinal \ructuations might cause trouble.\nAnother promising method to address all the issues dis-\ncussed above is the DFT+DMFT methods. The DMFT\nmethod works by mapping the lattice Hubbard model\nonto the single-site Anderson impurity model, where\nthe crystalline environment acts on the impurity via\nthe hybridization function.72Continuous time quantum\nMonte Carlo has been considered to be the state-of-the-\nart impurity solver.123Very recently, both the total en-\nergies124and forces125can be evaluated accurately by\nperforming charge self-consistent DFT+DMFT calcula-\ntions, with a great potential addressing the paramagnetic\nstates. Nevertheless, the local Coulomb parameters are\nstill approximated, though there is a possibility to eval-\nuate such quantities based on constrained RPA calcula-\ntions.126One additional problem is that the mostly ap-\nplied single-site DFT+DMFT does not consider the in-\ntersite magnetic exchange, which will overestimate the\nmagnetic critical temperature. We suspect that maybe\natomistic modelling of the long wavelength spin waves\nwill cure the problem, instead of using expensive cluster\nDMFT.\nOverall, the key challenge to obtain the thermody-\nnamic properties of magnetic materials is the accurate12\nevaluation of the total Gibbs free energy including longi-\ntudinal and transversal \ructuations, together with cou-\npling to the lattice degree of freedom. DFT+DMFT is\nagain a decent solution despite known constraints as dis-\ncussed above, but there is no proper implementation yet\nready for HTP calculations.\nE. Magnetic anisotropy and permanent magnets\n1. Origin of magnetocrystalline anisotropy\nAs another essential intrinsic magnetic property, MAE\ncan be mostly attributed to the SOC,127which couples\nthe spontaneous magnetization direction to the under-\nlying crystal structure, i.e., breaks the continuous sym-\nmetry of magnetization by developing anisotropic energy\nsurfaces.128Like spin, SOC is originated from the rel-\nativistic e\u000bects,129which can be formulated as (in the\nPauli two-component formalism):\nHSOC=\u0000rV\n2m2c2s\u0001l=\u0018s\u0001l; (11)\nwhererVmarks the derivative of a scalar electrostatic\npotential,mthe mass of electrons, cthe speed of light,\ns(l) the spin (orbital) angular momentum operator. \u0018\nindicates the magnitude of SOC, which is de\fned for\neachl-shell of an atom, except that for the s-orbitals\nwherel= 0 leading to zero SOC. The strength of atomic\nSOC can be estimated by \u0018\u00192\n5(\u000fd5=2\u0000\u000fd3=2) for the\nd-orbitals and \u0018\u00192\n7(\u000fd7=2\u0000\u000fd5=2) for thef-orbitals, re-\nspectively.130This leads to the average magnitude of \u0018\nfor the 3d-orbitals of 3 dTM atoms is about 60 meV ( e.g,\nfrom 10 meV (Sc) to 110 meV (Cu)131), while that of the\n4f-orbitals for lanthanide elements can be as large as 0.4\neV. Therefore, the larger the atomic number is, the larger\nthe magnitude of \u0018,e.g., enhanced SOC in 4 d- and 5d-\nelements induces many fascinating phenomena.132The\nelectrostatic potential Vcan be modulated signi\fcantly\nat the surfaces or interfaces, leading to the Rashba e\u000bect\nwhich is interesting for spintronics.133\nConceptually, the MAE can be understood in such a\nway that SOC tries to recover the orbital moment which\nis quenched in solids, i.e., it is attributed to the interplay\nof exchange splitting, crystal \felds, and SOC.134Based\non the perturbation theory, the MAE driven by SOC can\nbe expressed as:135\nMAE =\u00001\n4\u0018em\u0001[\u000ehl#i\u0000\u000ehl\"i]\n|{z}\norbital moment\n+\u00182\n\u0001Eex[10:5em\u0001\u000ehTi+ 2\u000eh(l\u0010s\u0010)2i]\n| {z }\nspin \rip;(12)\nwhere emstands for the unit vector along the magnetiza-\ntion direction. T=em\u00003# \u0014r(# \u0014r\u0001em)\u0019\u00002\n7Q\u0001emdenotesthe anisotropic spin distribution, with# \u0014rindicating the\nposition unit vector, Q=l2\u00001\n3l2indicates the charge\nquadrupole moment. It is noted that the spin-\rip con-\ntribution is of higher order with respect to \u0018, which is\nsigni\fcant for compounds with signi\fcant SOC. On the\nother hand, for the orbital momentum term, it will be re-\nduced to MAE = \u00001\n4\u0018em\u0001\u000ehl#ifor strong magnets with\nthe majority channel fully occupied,136suggesting that\nthe magnetization prefers to be aligned along the direc-\ntion where the orbital moment is of larger magnitude.\nFollowing the perturbation theory, the MAE can be\nenhanced by engineering the local symmetry of the mag-\nnetic ions. It is well understood that the orbital moments\nare originated from the degeneracy of the atomic orbitals\nwhich are coupled by SOC.135Therefore, when the crys-\ntal \felds on the magnetic ions lead to the degeneracy\nwithin thefdxy, dx2+y2gorfdyz, dxzgorbitals, the MAE\ncan be signi\fcantly enhanced, depending also on the or-\nbital occupations. This suggests, local crystalline envi-\nronments of the linear, trigonal, trigonal bipyramid, pen-\ntagonal bipyramid, and tricapped trigonal prism types\nare ideal to host a possible large MAE, because the re-\nsulting degeneracy in both fdxy, dx2+y2gandfdyz, dxzg\norbital pairs.137For instance, Fe atoms in Li 2Li1\u0000xFexN\nlocated on linear chains behave like RE elements with gi-\nant anisotropy,138and Fe monolayers on InN substrates\nexhibit a giant MAE as large as 54 meV/u.c. due to the\nunderlying trigonal symmetry.139This leads to an e\u000bec-\ntive way to tailor the MAE by adjusting the local crys-\ntalline geometries, as demonstrated recently for magnetic\nmolecules.140\nTechnically, MAE is evaluated as the di\u000berence of total\nenergies between two di\u000berent magnetization directions\n(e1\nmande2\nm):\nMAE =Ee1m\u0000Ee2m\n\u0019occ.X\nk;i=1\"i(k;e1\nm)\u0000occ.X\nk;i=1\"i(k;e2\nm)(13)\nIn general, MAE is a small quantity with a typical mag-\nnitude between \u0016eV and meV, as a di\u000berence of two\nnumbers which are orders of magnitude larger. In this\nregard, it is very sensitive to the numerical details such\nas exchange-correlation functionals, k-point convergence,\nimplementation of SOC, and so on, leading to expensive\nfull relativistic calculations particularly for compounds\nwith more than 10 atoms per unit cell. To get a good\nestimation, force theorem is widely used (second line of\nEq. (IV E 1)), where MAE is approximated as the di\u000ber-\nence of the sum of DFT energies for occupied states ob-\ntained by one-step SOC calculations.141We want to em-\nphasize that the k-point integration over the irreducible\npart of the Brillouin zone de\fned by the Shubnikov group\nof the system is usually required, suggesting (a) symme-\ntry should be applied with caution, i.e., the actual sym-\nmetry depends on the magnetization direction and (b)\nthere is no hot zone with dominant contribution.142In\nthis regard, the recently developed Wannier interpola-13\ntion technique may provide a promising solution to the\nnumerical problem.143,144\n2. Permanent magnets: Rare-earth or not?\nFrom the materials point of view, there are two classes\nof widely used permanent magnets, namely, the TM-\nbased ferrite and AlNiCo, and the high performance RE-\nbased Nd 2Fe14B and SmCo 5. One crucial factor to dis-\ntinguish such two classes of materials is whether the RE\nelements are included, which causes signi\fcant MAE due\nto the competition of enhanced SOC and reduced band\nwidth for the 4 f-orbitals.131Hereafter these two classes\nof permanent magnets will be referred as RE-free and 4 f-\n3dmagnets. Given the fact that the research on these\nfour systems is relatively mature, the current main in-\nterest is to identify the so-called gap magnets ,145i.e.,\nmaterial systems with performances lying between these\ntwo classes of permanent magnets.\nSince MAE is originated from SOC, it is an inert\natomic property, i.e., its strength cannot be easily tuned\nby applying external forces which are negligible compar-\ning to the gradient of the electrostatic potential around\nthe nuclei. In addition, the exchange splitting is mostly\ndetermined by the Coulomb interaction.69Thus, the\nknob to tailor MAE is the hybridization of the atomic\nwave functions with those of the neighboring atoms,\nwhich can be wrapped up as crystal \felds. On the\none hand, for the 4 f-3dmagnets, the MAE is mainly\noriginated from the non-spherical distribution of the 4 f-\nelectrons,146which can be expressed based on the local\ncrystal \felds on the 4 f-ions.147On the other hand, for\nTM-based magnetic materials, the orbital degree of free-\ndom can be engineered to enhance the MAE by manip-\nulating the crystal \felds via \fne tuning the crystalline\nsymmetries, as discussed above following the perturba-\ntion theory. For instance, the MAE of Co atoms can be\nenhanced by three orders of magnitude, from 0.001 meV\nin fcc Co to 0.06 meV/atom in hcp Co driven by the re-\nduction of crystalline symmetry,148and further to about\n60 meV for Co adatoms on MgO149or Co dimers on Ben-\nzene attributed to the energy levels strongly a\u000bected by\nthe crystal \felds.150Such a concept can also be applied\nto improve the MAE of bulk magnetic materials, such\nas in Li 2Li1\u0000xFexN138and FeCo alloys,151where thed-\norbitals strongly coupled by SOC (such as dxyanddx2\u0000y2\norbitals) are adjusted to be degenerate and half-occupied\nby modifying the local crystalline geometry and doping.\nFor the RE-free magnets, the theoretical upper limit152\nof MAE as high as 3.6 MJ/m3. That is, there is a strong\nhope that RE-free materials can beengineered for high\nperformance permanent magnets. There have also been\na number of compounds investigated following this line,\nfocusing on the Mn-, Fe-, and Co-based compounds.153\nThe Mn-based systems are particularly interesting, as\nMn has the largest atomic moments of 5 \u0016Bwhich will\nbe reduced in solid materials due to the hybridization.However, the interatomic exchange between Mn moments\nhas a strong dependence on the interatomic distance,\ne.g., the Mn moments tend to couple antiferromagnet-\nically (ferromagnetically) if the distance is about 2.5-2.8\n\u0017A(>2.9\u0017A).29Nevertheless, there are a few systems such\nas MnAl154, MnBi155, and Mn-Ga156showing promising\nmagnetic properties to be optimized for permanent mag-\nnet applications. On the other hand, for the Fe-based\nsystems, FePt157and FeNi158have been extensively in-\nvestigated. One key problem for these two compounds is\nthat there exists strong chemical disorder with depends\non the processing. For instance, the order-disorder tran-\nsition temperature for FeNi is about 593 K, which de-\nmands extra long time of annealing to reach the ordered\nphase with strong anisotropy.159\nGiven that the M sand MAE can be obtained in a\nstraightforward way for most TM-based magnetic ma-\nterials, the main strategy for searching novel RE-free\npermanent magnets is to screen over a number of pos-\nsible chemical compositions and crystal structures. One\nparticularly interesting direction is to characterize the\nmetastable phases, such as Fe 16N2160,161and MnAl.49\nSuch phases can be obtained via non-equilibrium synthe-\nsis, such as MBE, melt spinning and mechanical alloying.\nIn this sense, the permanent magnets of thin \flms is an\noxymoron, i.e., they can provide valuable information\nabout how to engineer the intrinsic properties but it is\nchallenging to obtain bulk magnets. It is also noted that\npure DFT might not be enough to account for the mag-\nnetism in tricky cases due to the missing spin \ructuations\nas demonstrated in (FeCo) 2B.162\nFor the 4f-3dcompounds, it is fair to say that there\nis still no consistent quantitative theory at the DFT\nlevel to evaluate their intrinsic properties, though both\nSmCo 5and Nd 2Fe14B were discovered decades ago. The\nmost salient feature common to the 4 f-3dmagnetic com-\npounds is the energy hierarchy, driven by the coupling\nbetween RE-4 fand TM-3dsublattices.163First of all,\nthe TCis ensured by the strong TM-TM exchange cou-\npling. Furthermore, the MAE is originated from the in-\nterplay of RE and TM sublattices. For instance, the Y-\nbased compounds usually exhibit signi\fcant MAE, such\nas YCo5,164whereas for the RE atoms with nonzero 4 f\nmoments, the MAE is signi\fcantly enhanced due to non-\nspherical nature of the RE-4 fcharges146caused by the\ncrystal \felds.147Lastly, the exchange coupling between\nRE-4fand TM-3dmoments is activated via the RE-5 d\norbitals, which enhances the MAE by entangling the RE\nand TM sublattices. The intra-atomic 4 f-5dcoupling on\nthe RE sites is strongly ferromagnetic, while the inter-\natomic 3d-5dbetween TM and RE atoms is AFM. This\nleads to the AFM coupling between the spin moments of\nthe TM-3dand RE-4fmoments, as observed in most 4 f-\n3dmagnets.165In this regard, light RE elements are more\npreferred166as permanent magnets, because the orbital\nmoments of heavy RE elements will reduce the magneti-\nzation as they are parallel to the spin moments following\nthe third Hund's rule.14\nFIG. 5: Homologous structures of Sm-Co intermetallic compounds. (a) SmCo 5(CaCu 5-type), (b) SmCo 12(ThMn 12-type), (c)\nSm2Co17(Th 2Ni17-type, 2:17H), (d) Sm 2Co17(Th 2Zn17-type, 2:17R), (e) Sm 2Co7(Ce2Ni7-type), (f) SmCo 4B (CeCo 4B-type),\n(g) SmCo 3B2(Co3GdB 2-type), and (h) SmCo 2(MgZn 2-type, C14 Laves). (a-d) are the members of the series RE m\u0000nTM 5m+2n\n(n/m<1/2), with (m, n) = (1, 0), (2, 1), hexagonal-(3, 1), and rhombohedra-(3, 1), respectively. (a) and (e) are the members\nof the series RE n+2TM 5n+4with n = 1and 2, respectively. (a), (f), and (g) are the members of the series RE n+1TM 3n+5B2n\nwith n = 0, 1, and 1, respectively.\nThe multiple sublattice couplings have been utilized\nto construct atomistic spin models to understand the \f-\nnite temperature magnetism of 4f-3d materials, such as\nRECo 5167and recently Nd 2Fe14B.168Such models are\nideally predictive, if the parameters are obtained from\naccurate electronic structure calculations based on DFT.\nOne typical problem of such phenomenological modelling\nis the ill-description of the temperature dependency of\nthe physical properties, e.g., the magnitude of moments\nis assumed to be temperature independent.116This is-\nsue cannot be solved using the standard DFT which is\nvalid at 0 K. For instance, both the longitudinal and\ntransversal \ructuations are missing in conventional DFT\ncalculations. Additionally, for the 4 f-3dmagnets, the\nMAE and the inter-sublattice exchange energy are of the\nsame order of magnitude, thus the experimental mea-\nsurement along the hard axis does not correspond to a\nsimpli\fed (anti-)parallel con\fguration of 4f and 3d mo-\nments.163This renders the traditional way of evaluating\nthe MAE by total energy di\u000berences not applicable.169.\nAnother challenging problem is how to treat the\nstrongly correlated 4 f-electrons. Taking SmCo 5as an\nexample, there is clear experimental evidence that the\nground state multiplet J=5/2 is mixed with the excited\nmultiplets J=7/2 and J=9/2, leading to strong temper-\nature dependence of both spin and orbital moments of\nSm.170The problem arises how to perform DFT calcula-\ntions with several multiplets, which requires more thanone Slater determinants. Furthermore, Ce is an abundant\nelement which is preferred for developing novel RE-lean\npermanent magnets. The 4 felectrons in trivalent Ce\ncan be localized or delocalized, leading to the mixed va-\nlence state, as represented by the abnormal properties of\nCeCo 5.171. Such features have been con\frmed by the X-\nray magnetic circular dichroism (XMCD) measurements\nin a series of Ce-based compounds.172\nTherefore, the strongly correlated nature of 4 f-\nelectrons calls for a theoretical framework which is be-\nyond the standard DFT, where a consistent theoretical\nframework being able to treat the correlated 4f-electrons\nwith SOC and spin \ructuations consistently is required.\nMost previous DFT calculations173,174treated the 4 f-\nelectrons using the open-core approximation which does\nnot allow the hybridization of 4 f-shell with the valence\nbands. Moreover, the spin moment of RE elements are\nusually considered as parallel to that of the TM atoms.175\nIt is worthy mentioning that Patrick and Staunton pro-\nposed a relativistic DFT method based on the disor-\ndered local moment approximation to deal with \fnite\ntemperature magnetism where the self-interaction correc-\ntion (SIC) is used for the 4 f-electrons.116It is observed\nthat the resulting total moment (7.13 \u0016B) of SmCo 5is\nsigni\fcantly underestimated compared to the experimen-\ntal value of about 12 \u0016B.170In this regard, such methods\nsuch as SIC and DFT+U76based on a single Slater deter-\nminant may not be su\u000ecient to capture all the features15\nof the correlated 4 f-shells, e.g., additional non-spherical\ncorrection is needed to get the the correct orbital mo-\nments.176\nIt is believed that an approach based on multicon-\n\fgurational wave functions including SOC and accurate\nhybridization is needed, in order to treat many close-\nlying electronic states for correlated open shells ( i.e.d-\nandf-shells) driven by the interplay of SOC, Coulomb\ninteraction, and crystal \felds with comparative magni-\ntude. From the quantum chemistry point of view, the\nfull con\fgurational interaction method is exact but can\nonly be applied on cases with small basis sets. The\ncomplete active space self-consistent \feld method based\non optimally chosen e\u000bective states is promising, where\na multicon\fgurational approach can be further devel-\noped with SOC explicitly considered.177Such methods\nhave been successfully applied on the evaluation of MAE\nfor magnetic molecules,178and recently extended to the\nfour-component relativistic regime.179A similar method\nbased on exact diagonalization with optimized bath func-\ntions has been developed as a potential impurity solver\nfor DMFT calculations.180Thus, with the hybridization\naccounted for at the DFT level, such impurity solvers\nbased on the multicon\fgurational methods will make\nDFT+DMFT with full charge self-consistency a valu-\nable solution to develop a thorough understanding of the\nmagnetism in 4 f-3dintermetallic compounds. Whereas\nprevious DFT+DMFT calculations on GdCo 5181and\nYCo 5182have been performed using primitive analyti-\ncal impurity solvers. It is noted that SOC is o\u000b-diagonal\nfor the Anderson impurities with real spherical harmon-\nics as the basis, leading to possible severe sign problem\nwhen the quantum Monte Carlo methods are used as the\nimpurity solver. For such cases, either the hybridiza-\ntion function is diagonalized to obtain an e\u000bective ba-\nsis such as demonstrated for iridates in the 5d5J=1/2\nstates183,184, or the variational cluster approach can be\napplied.185,186A recently developed bonding-antibonding\nbasis o\u000bers also a possible general solution to perform\nDFT+DMFT on such materials with strong SOC.187\nFrom the materials point of view, compounds includ-\ning Fe or Co with crystal structures derived from the\nCaCu 5-type are particularly important. It is noted that\nNd2Fe14B has also a structure related to the CaCu 5-\ntype, but its maximal permanent magnet performance\nhas been reached to the theoretical limit, leaving not\nmuch space to improve further.188Interestingly, as shown\nin Fig. 5, various crystal structures can be derived from\nthe CaCu 5-type structure. Taking Sm-Co as an exam-\nple, a homologous series RE m\u0000nTM5m+2ncan be ob-\ntained, where n out of m Sm atoms are substituted by\nthe dumbbells of Co-Co pairs. For instance, the ThMn 12-\ntype structure (Fig. 5b) corresponds to (m=2, n=1),\ni.e., one Sm out of two is replaced by a Co-Co pair.\nFor Sm 2Co17(m=3, n=1), both the hexagonal Th 2Ni17-\ntype (2:17H) (Fig. 5c) and rhombohedra Th 2Zn17-type\n(2:17R) (Fig. 5d) structures can be obtained depending\non the stacking of Co dumbbells. Another homologousseries RE n+2TM5n+4can be obtained by intermixing the\nSmCo 2(MgZn 2-type) (Fig. 5h) and the SmCo 5phases,\nleading to Sm 2Co7(Fig. 5e) with n=2. Lastly, by substi-\ntuting B for preferentially Co on the 2c-sites, yet another\nhomologous series RE n+1TM3n+5B2ncan be obtained,\nleading to SmCo 4B (n=1, Fig. 5f), and SmCo 3B2(n=1,\nFig. 5g). For all the derived structures, the Kagome lay-\ners formed by the Co atoms on the 3g sites of SmCo 5are\nslightly distorted, while the changes occur mostly in the\nSmCo 2layers due to large chemical pressure.189\nSuch a plenty of phases o\u000ber an arena to understand\nthe structure-property relationship and to help us to de-\nsign new materials, particularly the mechanical under-\nstanding based on the local atomic structures. For in-\nstance, Sm 2Co17has uniaxial anisotropy while all the\nother early RE-Co 2:17 compounds show a planar be-\nhaviour.190This is caused by the competition between\nthe positive contribution from the 2 c-derived Co and the\nnegative contribution from the dumbbell Co pairs.190As\na matter of fact, the 2 c-Co atoms in the original CaCu 5-\ntype structure have both enhanced spin and orbital mo-\nments.191Thus, insight on the structure-property rela-\ntionship at the atomic level with de\fned driving forces\nfrom the electronic structure will be valuable to engineer\nnew permanent materials.\nOne particularly interesting class of compounds which\nhas drawn intensive attention recently is the Fe-\nrich metastable materials with the ThMn 12-type struc-\nture.188There have been many compounds REFe 11X or\nREFe 10X2stabilized by substituting a third element such\nas Ti and V192. Moreover, motivated by the successful\npreparation of NdFe 12193and SmFe 12194thin \flms with\noutstanding permanent magnet properties, there have\nbeen many follow-up experimental and theoretical inves-\ntigations188. The key problem though is to obtain bulk\nsamples with large coercive \feld, instead of thin \flms\nfabricated under non-equilibrium conditions. In order to\nguide experimental exploration, detailed thermodynamic\noptimization of the multicomponent phase diagram is\nneeded, where the relative formation energies in most\nprevious DFT calculations are of minor help, e.g., the\nprediction of NdFe 11Co195is disproved by detailed ex-\nperiments.196Another challenge for the metastable Fe-\nrich compounds is that the T Cis moderate. This can\nbe cured by partially substituting Co for Fe. Addition-\nally, interstitial doping with H, C, B, and N can also\nbe helpful, e.g., Sm 2Fe17N2:1has a TCas high as 743\nK, increased by 91% compared to pristine Sm 2Fe17.197\nInterestingly, the MAE for Co-rich ThMn 12-type com-\npounds shows an interesting behaviour. For instance,\nfor both RECo 11Ti and RECo 10Mo2, almost all com-\npounds have uniaxial anisotropy, except that Sm-based\ncompounds show a basal-plane MAE.198This echoes that\nonly Sm 2Co17has a uniaxial anisotropy.190This can be\neasily understood based on the crystal structure as shown\nin Fig. 5(b&d), where the uniaxial c-axis of the ThMn 12-\ntype is parallel to the Kagome plane of 3g-Co atoms, but\nperpendicular in the case of 2:17 types. That is, from the16\nMAE point of view, the 1:12 structure conjugates with\nthe 2:17 structures. Thus, we suspect that substitutional\nand interstitial doping can be utilized extensively to en-\ngineer MAE of the 2:17 and 1:12 compounds.\nOverall, although the MAE is numerically expensive\nto evaluate, it is a straightforward task to perform calcu-\nlations with SOC considered which has been reliably im-\nplemented in many DFT codes. The key issue to perform\nHTP screening of permanent magnets is to properly ac-\ncount for the magnetism such as the spin \ructuations in\nRE-free and strongly correlated 4 felectrons in RE-based\nmaterials. In addition, a systematic investigation of the\nintermetallic structure maps and the tunability of crystal\nstructures via interstitial and substitutional doping will\nbe valuable to design potential compounds. Cautions are\ndeserved particularly when trying to evaluate the MAE\nfor doped cases where the supercells might impose un-\nwanted symmetry on the MAE, and also that the FM\nground state should be con\frmed before evaluating the\nMAE.\nF. Magneto-structural transitions\nIn general, the functionalities of ferroic materials are\nmostly driven by the interplay of di\u000berent ferroic or-\nders such as ferroelasticity, ferromagnetism, and ferro-\nelectricity, which are coupled to the external mechanical,\nmagnetic, and electrical stimuli.200Correspondingly, the\ncaloric e\u000bect can be induced by the thermal response as\na function of generalized displacements such as magne-\ntization, electric polarization, strain (volume), driven by\nthe conjugate generalized forces such as magnetic \feld,\nelectric \felds, and stress (pressure), leading to magne-\ntocaloric, electrocaloric, and elastocaloric (barocaloric)\ne\u000bects.201In order to obtain optimized response, such\nmaterials are typically tuned to approach/cross a FOPT.\nTherefore, it is critical to understand the dynamics of\nFOPT in functional ferroic materials. Most FOPTs occur\nwithout long-range atomic di\u000busion, i.e., of the marten-\nsitic type, leading to intriguing and complex kinetics. For\ninstance, such solid-solid structural transitions are typi-\ncally athermal, i.e., occurring not at thermal equilibrium,\ndue to the high energy barriers and the existence of as-\nsociated metastable states. This leads to the broaden-\ning of the otherwise sharpness of FOPTs with respect to\nthe driving parameters and hence hysteresis.33Therefore,\nmechanistic understanding of such thermal hysteresis en-\ntails accurate evaluation of the thermodynamic proper-\nties, where successful control can reduce the energy loss\nduring the cycling.202\nTwo kinds of fascinating properties upon such FOPTs\nin ferroic functional materials are the shape memory203\nand caloric e\u000bects,201e.g., MSME and MCE in the con-\ntext of magnetic materials. Note that MCE can be in-\nduced in three di\u000berent ways, namely, the ordering of\nmagnetic moments (second order phase transition),204\nthe magneto-structural phase transition,205or the rota-tion of magnetization,206all driven by applying the ex-\nternal magnetic \felds. Taking Gd207with second-order\nphase transition as an example, under adiabatic condi-\ntions ( i.e., no thermal exchange between the samples and\nthe environment) where the total entropy is conserved,\napplying (removing) the external magnetic \felds leads\nto a reduction (increase) in the magnetic entropy, as\nthe magnetic moments get more ordered (disordered),\nleading to increased (decreased) temperature. That is,\nthe temperature change is mostly due to the entropy\nchange associated with the ordering/disorder of the mag-\nnetic moments. In contrast, for materials where MCE is\ndriven by the magneto-structural transitions of the \frst-\norder nature, additional enthalpy di\u000berence between two\nphases before and after the transition should be consid-\nered, although the total Gibbs free energy is continuous\nat the transition point. In this case, the resulting con-\ntribution to the entropy change comprise the lattice and\nmagnetic parts, which can be cooperative or competitive.\nA prototype system is the Heusler Ni-Mn-Sn alloys with\nthe inverse magnetocaloric e\u000bect.208\nTherefore, to get enhanced magnetocaloric perfor-\nmance which is best achieved in the vicinity of the\nphase transitions, the ideal candidates should display\n(1) large change of the magnetization \u0001 M, trigged by\nlow magnetic \feld strengths. This makes such materi-\nals also promising to harvest waste heat based on the\nthermomagnetic e\u000bect,209(2) tunable transition temper-\nature ( e.g., via composition) for the \frst-order magneto-\nstructural transitions, because the transition is sharp so\nthat a wide working temperature range can be achieved\nby successive compositions, and (3) good thermal con-\nductivity to enable e\u000ecient heat exchange.\nTo search for potential candidates with signi\fcant\nMCE, it is critical to develop mechanistic understand-\ning of the existing cases. For instance, Ni 2MnGa is the\nonly Heusler compounds with the stoichiometric compo-\nsition that undergoes a martensitic transition at 200K\nbetween the L2 1austenite and DO 22martensitic phases\n(cf. Fig. 10), which can be optimized as magnetocaloric\nmaterials. DFT calculations show that the martensitic\ntransition can be attributed to the band Jahn-Teller ef-\nfect,210which is con\frmed by neutron211and photoemis-\nsion measurements.211,212Detailed investigation to un-\nderstand the concomitant magnetic \feld induced strain\nas large as 10%211reveals that there exists a premarten-\nsitic phase between 200K and 260K211and that both\nthe martensitic and premartensitic phases have modu-\nlated structures. It is still under intensive debate about\nthe ground state structure and its origin. Kaufman et\nal.introduced the adaptive martensitic model, arguing\nthat the ground state is the L1 0type tetragonal phase\nwhich develops into modulated structures via nanotwin-\nning.213,214On the other hand, Singh et al. argued based\non the synchrotron x-ray measurements that the 7M-\nlike incommensurate phase is the ground state, which\nis caused by the phonon softening.215\nTo shed light on the nature of the magneto-structural17\nFIG. 6: Phase diagram of Ni-Mn-X.199Copyright requested.\ncoupling, the Gibbs free energy G(T;V;P) (Eq. (IV D))\nas a function of temperature and volume ( T;V) for dif-\nferent phases P. According to Eq. IV D, the electronic\ncontribution to Fele.can be easily obtained using the\nmethods as detailed in Ref. 216. To evaluate the vibra-\ntional term Flat., the quasi harmonic approximation can\nbe applied,217which has been implemented as a standard\nroutine in the Phonopy code.218It is noted that the har-\nmonic approximation is only valid to get the local min-\nima on the potential energy surface, whereas proper con-\nsideration on the transition paths and kinetics requires\naccurate evaluation of the anharmonic e\u000bects.219How-\never, when the spin-phonon interaction becomes signi\f-\ncant as in the paramagnetic states, it is still a challenging\ntask to evaluate the phonon spectra and hence the lat-\ntice free energies. The spin space average technique has\nbeen developed and applied for bcc Fe119and we be-\nlieve the recent implementation in the DMFT regime is\nvery promising.125For the magnetic free energy Fmag.,\nthe classical Monte Carlo simulations, as usually done\nbased on the Heisenberg model (Eq. (IV B)) with DFT\nderived exchange coupling Jijto obtain the Curie tem-\nperature, are not enough, as the speci\fc heat at 0K stays\n\fnite instead of the expected zero, due to the contin-\nuous symmetry of spin rotations. Additionally, quan-\ntum Monte Carlo su\u000bers from the sign problem as Jij\nis long range and changes its sign with respect to the\ndistance. One solution is to rescale the classical Monte\nCarlo results, as done for bcc Fe.105An alternative is to\nmap the free energy based on the \ftted temperature de-\npendence of the magnetization.220This has been applied\nsuccessfully to get the martensitic transition temperature\nand the stability of premartensitic phases in Ni-Mn-Ga\nsystem.221The recently developed spin dynamics with\nquantum thermostat106, the Wang-Landau technique tosample the thermodynamic density of states in the phase\nspace directly222, and the atomistic spin-lattice dynam-\nics are interesting to explore in the future.223Last but\nnot least, magnetic materials with noncollinear magnetic\nordering and thus vanishing net magnetization can also\nbe used for caloric e\u000bects such as barocaloric224and elas-\ntocaloric225e\u000bects, it is essential to evaluate the free en-\nergy properly which has been investigated recently.112\nThe magnetocaloric performance of stoichiometric\nNi2MnGa is not signi\fcant, as the martensitic transi-\ntion occurs between the FM L2 1and modulated marten-\nsitic phase. As shown in Fig. 6(a), excess Ni will bring\nthe martensitic phase transition temperature and T Cto-\ngether, and hence enhance the magnetocaloric perfor-\nmance.226Interestingly, such a phase diagram can be\napplied to interpolate the transitions in several Ni-Mn-\nX (X = Ga, In, Sn, Sb) alloys, where the dependence\nwith respect to the valence electron concentration can be\nformulated (Fig. 6(b)). The valence electron concentra-\ntion accounts the weighted number of the s/p/delectrons,\nwhich can be tuned by excess Mn atoms. For instance,\nthe martensitic transition temperature for the Ni-Mn-X\nalloys interpolates between the stoichiometric Ni 2MnX\nand the AFM NiMn alloys. In such cases, the marten-\nsitic transition occurs usually between the FM L2 1and\nthe paramagnetic L1 0phases, leading to the metamag-\nnetic transitions and hence giant magnetocaloric perfor-\nmances.205,208The magnetocaloric performance can be\nfurther enhanced by Co doping,227{229which enhances\nthe magnetization of the austenite phase. However, Co-\ndoping modi\fes the martensitic transition temperature\nsigni\fcantly, leading to a parasitic dilemma.230\nFrom the materials point of view, there are many\nother classes of materials displaying signi\fcant magne-\ntocaloric performance beyond the well-known Heusler18\nFIG. 7: Library of MCE materials231Copyright requested.\ncompounds, as summarized in Fig. 7.231Three of them,\ne.g., Gd alloys,232La-Fe-Si,233,234and MnFe(Si,P)235\nhave been integrated into devices, where La(FeSi) 13is\nconsidered as the most promising compounds for applica-\ntions, driven by the \frst-order metamagnetic phase tran-\nsition at T C.236For such materials, research has been\nfocusing on the \fne tuning of the magnetocaloric per-\nformance by chemical doping. As discussed above, the\nkey problem to overcome when optimizing and design-\ning MCE materials is to reduce the hysteresis in order to\nminimize the energy losses,237this is a challenging task\nfor HTP design where the Gibbs free energies for both the\nend phases and the intermediate phases are needed in or-\nder to have a complete thermodynamic description of the\nphase transition. In this regard, a particularly promis-\ning idea is to make use of the hysteresis rather than to\navoid it, which can be achieved in the multicaloric regime\nengaging multiple stimuli. This has been demonstrated\nrecently in Ni-Mn-In system238combining magnetic \felds\nand uniaxial strains.\nTherefore, designing novel MCE materials entails joint\ntheoretical and experimental endeavour. HTP calcu-\nlations can be applied to \frstly search for compounds\nwith polymorphs linked by group-subgroup relationships,\nand then to evaluate the Gibbs free energy for the\nmost promising candidates. Nevertheless, for such ma-\nterials with \frst-order magneto-structural transitions,\nhysteresis typically arises as a consequence of nucle-\nation, in caloric materials it occurs primarily due to\nthe domain-wall pinning, which is the net result of\nlong-range elastic strain associated with phase transi-\ntions Thus, experimental optimization on the microstruc-\ntures239(Sect. VI A) is unavoidable in order to obtain a\npractical material ready for technical applications.G. Spintronics\nTill now we have been focusing on the physical\nproperties of the equilibrium states, whereas the non-\nequilibrium transport properties of magnetic materials\nleads to many interesting properties, where spintronics is\none class of the most interesting phenomena. In contrast\nto the conventional electronic devices, the spin degree of\nfreedom of electrons has been explored to engineer more\nenergy e\u000ecient devices.35The \frst generation of spin-\ntronic applications are mostly based on the FM materi-\nals, initiated by the discovery of giant magnetoresistance\n(GMR) in magnetic multilayers in 1988,240,241e.g., with\nthe MR ratio as large as 85% between parallel and an-\ntiparallel con\fgurations of Fe layers separated by the Cr\nlayers in-between. The underlying mechanism of GMR\ncan be understood based on the two-current model,242\nwhere spin-polarized currents play a deterministic role.\nTwo important follow-up development of GMR are the\ntunnelling magnetoresistance (TMR)243and spin trans-\nfer torque (STT).244,245The TMR e\u000bect is achieved in\nmultilayers where the ferromagnetic metals are separated\nby large gap insulators such as MgO, and the MR ratio\ncan be as large as 600% in CoFeB/MgO/CoFeB by im-\nproving the surface atomic morphology.246In the case of\nSTT, by driving electric currents through the reference\nlayer, \fnite torque can be exerted on the freestanding\nlayers and hence switch their magnetization directions.\nThis enables also engineering spin transfer oscillator,247\nracetrack memory,248and nonvolatile RAM.249\nThe second generation of spintronics comprise phe-\nnomena driven by SOC, dubbed spin-orbitronics, which\ncame around 2000.36For ferromagnetic materials,\nSOC gives rise to the anisotropic magnetoresistance\n(AMR)250, where the resistivity depends on the mag-\nnetization direction, in analog to GMR but without the\nnecessity to form multilayers. The most essential advan-\ntage of spin-orbitronics is to work with spin current ( i.e.,\na \row of spin angular momentum ideally without con-\ncomitant charge current), instead of the spin polarized\ncurrent.251Correspondingly, the central subjects of spin-\norbitronics are the generation, manipulation, and detec-\ntion of spin current. Note that after considering SOC,\nspin is not any more a good quantum number, thus it\nis still an open question how to de\fne the spin current\nproperly.252\nTwo most important phenomena for spin-orbitronics\nare the spin Hall e\u000bect (SHE)291and spin-orbit torque\n(SOT).292SHE deals with the spin-charge conversion,\nwhere transversal spin current can be generated by a lon-\ngitudinal charge current. It has been observed in param-\nagnetic metals such as Ta293and Pt,294FM metals like\nFePt,295AFM metals including MnPt,296Mn80Ir20,297\nand Mn 3Sn,298semiconductors such as GaAs,299and\ntopological materials such as Bi 2Se3300and TaAs.301It\nis noted that the reciprocal e\u000bect, i.e., the inverse SHE\n(iSHE), can be applied to detect the spin current by mea-\nsuring the resulting charge current.19\nTABLE II: Selected systems implementing spin-orbitronics and AFM spintronics. Copyright requested for the \fgures.\nspin transport spin-orbit torque spin pumping\nspin current253SOT in NM jFM bilayers254spin pumping in FM jNM bilayers255\nFM metal review256PtjCo257,258NiFejPt255,259,260\nTajCoFeB258,261NiFejGaAs262(metal jsemiconductor)\nSrIrO 3jNiFe263(oxides) NiFejYIG264(metal jinsulator)\nWTe 2jNiFe265(out-of-plane)\nNiMnSb266(bulk)\nFM insulator YIG267GaMnAs jFe268GaMnAs jGaAs269\nGaMnAs270(bulk) YIGjPt253,271\nAFM metal IrMnjCoFeB272NiFejIrMn273\nIrMnjNiFe274\nMnPt jCo/Ni275,276\nMn3IrjNiFe277\nCuMnAs278(bulk)\nMn2Au279,280(bulk)\nAFM insulator Cr2O3281PtjNiO282{284YIGjNiO285,286\nFe2O3287PtjTmIG288YIGjCoO289\nBi2Se3jNiOjNiFe290\nSOT re\rects with the interaction of magnetization\ndynamics and the spin current, leading to magnetiza-\ntion switching and spin-orbit pumping. Following the\nLandau-Lifshitz-Gilbert (LLG) equation,292\ndem\ndt=\u0000\rem\u0002B|{z}\nprecession+\u000bem\u0002dem\ndt|{z}\nrelaxation+\r\nMsT\n|{z}\ntorque; (14)\nwhere\r,\u000b, andMsmark the gyromagnetic ratio, the\nGilbert damping parameter, and the saturation magne-\ntization, respectively. em,B, and Tindicate the mag-\nnetization unit vector, the e\u000bective \feld, and the total\ntorque, respectively. The torque Tis perpendicular to\nm, which can be generally expressed as292\nT=\u001cFLem\u0002\u000f|{z}\n\feld-like+\u001cDLem\u0002(em\u0002\u000f)|{z}\ndamping-like; (15)\nwhere \u000fis the unit vector of the torque. Note that the\n\feld-like and damping-like terms act on the magnetiza-\ntion like the precession and relaxation terms in the LLG\nequation (Eq. (IV G)).\nTo induce \fnite SOT, e\u000bective non-equilibrium spin\npolarization is required, which can be obtained via (a)\nSHE and (b) Edelstein e\u000bect302(i.e., inverse spin Gal-\nvanic e\u000bect(iSGE)303). From symmetry point of view,\nthe occurrence of SOT requires noncentrosymmetric sym-\nmetry, thus the heterostructures of NM/FM materials\nprovide a rich playground to investigate SOT, where\nparamagnetic materials with signi\fcant SHE have been\nthe dominant source for inducing spin injection and SOTin FM materials. For instance, SOT has been investi-\ngated in TajCoFeB,261PtjCo,257,258, SrIrO 3midNiFe,263\nand Bi 2Se3jBaFe 12O19.304On the other hand, the Edel-\nstein e\u000bect requires Rashba splitting, which can also be\napplied for spin-charge conversion and hence SOT. For\ninstance, the spin-charge conversion has been demon-\nstrated for 2DEG at the interfaces of LaAlO 3jSrTiO 3,305\nAgjBi,306, and CujBi2O3jNiFe.307Correspondingly, SOT\nis recently observed in NiFe jCuOxdriven by the Rashba\nsplitting at the interfaces.308We note that the SHE-SOT\nand iSGE-SOT (Edelstein) are entangled, e.g., can be\ncompetitive or cooperative with each other, as demon-\nstrated in (GaMn)As jFe.268\nIt is noteworthy pointing out that from the symme-\ntry perspective,258the SOT generated at the interfaces\nof FM metals and heavy metals lies in-plane for both the\nanti-damping and \feld-like contributions, thus can only\nswitch the in-plane magnetization direction e\u000eciently.\nNevertheless, it is demonstrated that in WTe 2jNiFe,265\nout-of-plane SOT can be induced by current along a\nlow-symmetry axis, which is promising for future SOT\nswitching of FM materials with perpendicular magnetic\nanisotropy. Although the SOT discussed above takes\nplace at the interfaces, it does not mean that SOT is\nprohibited in bulk materials. For instance, SOT has been\ncon\frmed in half Heusler NiMnSb.266Thus, the key is to\nbreak the inversion symmetry in order to obtain \fnite\nSOT.\nAs SOT can be summarized as torque induced by spin\ncurrent, the Onsager reciprocal e\u000bect is spin pumping,\ni.e., the generation of spin current via induced magneti-20\nzation dynamics. Usually the spin pumping is induced\nby generating non-equilibrium magnetization dynamics\nusing the ferromagnetic resonance (FMR) and hence cre-\nates a pure spin current which can be injected into the\nadjacent NM layers ( i.e., spin sink) without a charge \row\nunder zero bias voltage. For instance, it is demonstrated\nthat spin pumping can be achieved in NiFe jPt bilayers,259\nwith Pt being the spin sink, which can be conveniently\ndetected via the iSHE. In this regard, many materials\nhave been considered as spin sinks, such as heavy metals\nlike Au and Mo260and semiconductors as GaAs.262An\nintriguing aspect of spin pumping is that the polarization\nof the spin current is time-dependent, leading to both\ndc- andac-components.309It is demonstrated that the\nac-component is at least one order of magnitude larger\nthan thedc-component for NiFe jPt,255which can lead to\nfutureacspintronic devices.\nAn emergent \feld of great interest is AFM spintron-\nics, which possess several advantages over the FM coun-\nterpart.37,310,311For instance, there is no stray \feld\nfor AFM materials thus the materials are insensitive to\nthe neighboring unit, which allows denser integration of\nmemory bits. Moreover, the typical resonance frequency\nof FM materials is in the GHz range which is mostly\ndriven by MAE,312while that for AFM materials can be\nin the THz range due to the exchange interaction between\nthe moments.313This leads to ultrafast magnetization\ndynamics in AFM materials. As a matter of fact, almost\nall the spintronic phenomena existing in FM materials\nhave been observed in AFM materials, such as AMR\nin FeRh314and MnTe,315SHE in MnPt,296and tun-\nnelling AMR.316,317Nevertheless, one major challenge\nfor AFM spintronics is how to control and to detect the\nAFM ordering. It turns out that SOT can be applied to\nswitch the magnetic ordering of AFM compounds. For\ninstance, the iSGE-SOT can lead to staggered N\u0013 eel spin-\norbit torques, creating an e\u000bective \feld of opposite sign\non each magnetic sublattice. The SOT induced switch-\ning has been observed in CuMnAs278and Mn 2Au.279,280\nIt is noted that the staggered SOT has special require-\nment on the symmetry of the materials, i.e., the inversion\nsymmetry connecting the AFM magnetic sublattices in\nthe crystal structure is broken by the magnetic con\fgu-\nration.37Thus, it is an interesting question whether there\nare more materials with the demanded magnetic con\fg-\nurations for bulk AFM spintronics. To detect the AFM\nmagnetic ordering, spin Hall magnetoresistance e\u000bect can\nbe used.318\nOne particularly interesting subject is the SOT and\nspin pumping for magnetic junctions combining FM and\nAFM materials. For instance, as AFM materials dis-\nplay also signi\fcant SHE, SOT has been observed in,\ne.g., IrMnjCoFeB,272IrMnjNiFe,274MnPtjCo,275and\nMn3IrjNiFe.277It is demonstrated in MnPt jCo/Ni bilay-\ners, the resulting SOT switching behaves like an arti\f-\ncial synapses, and a programmable network of 36 SOT\ndevices can be trained to identify 3 \u00023 patterns.276This\nleads to an intriguing application of spintronic devicesfor neuromorphic computation.319Furthermore, as spin\npumping depends on the magnetic susceptibility at the\ninterfaces and thus the magnetic \ructuations, it is ob-\nserved that the spin pumping in NiFe jIrMn bilayers is\nsigni\fcantly enhanced around the AFM phase transition\ntemperature of IrMn,273consistent with the theoretical\nmodel based on enhanced interfacial spin mixing con-\nductance.320In addition to enhance the spin pumping\ne\u000eciency, such types of experiments can be applied to\ndetect the AFM ordering without being engaged with\nthe neutron scattering facilities.\nAnother emerging \feld of spintronics is magnonic spin-\ntronics321realized based on both FM and AFM insula-\ntors, as each electron carries an angular momentum of\n~=2 whereas each magnon as the quasiparticle of spin\nwave excitations carries an angular momentum of ~. The-\noretically, it is predicted that at the interfaces of NM jFM-\ninsulator bilayers, spin gets accumulation which is ac-\ncompanied by the conversion of spin current to magnon\ncurrent.322In the case of FM (ferrimagnetic) insulators,\nYIGjPt is a prototype bilayer system to demonstrate the\ntransmission of spin current at the interfaces253and spin\npumping.271,323It is also demonstrated the SOT can be\napplied to switch the magnetization directions of FM in-\nsulators, as in PtjTmIG.288The SOT and spin pumping\nhave also been observed in AFM insulators. For instance,\nSOT leads to the switching of AFM ordering of NiO in\nPtjNiO bilayers.282{284The origin of switching can be\nattributed to the non-staggered SHE-SOT which acts as\nanti-damping-like torque exerted by a spin accumulation\nat the interface, but this is a question under intensive\ndebate.284Furthermore, spin pumping can be induced in\nbilayers of FMjAFM-insulators such as YIG jNiO285. One\nintriguing aspect is the non-locality of magnetic spin cur-\nrent which can propagate 100 nm in NiO.286In general,\nmagnon number is not conserved, leading to \fnite trans-\nport length scale for magnon mediated spin current de-\ntermined by the Gilbert damping coe\u000ecient which is an\nintrinsic character of compounds. In contrast to the FM\nmaterials,256AFM materials allow long range spin trans-\nport, as observed in Cr 2O3281and Fe 2O3324which might\nbe attributed to the spin super\ruidity.325Particularly,\nthe damping of magnons can be compensated by SOT,\nwhich leads to zero damping as observed very recently in\nYIG.267Lastly, it is a fascinating idea to combine SOT\nand long di\u000busion length of spin current in AFM materi-\nals,e.g., switching magnetism using the SOT carried by\nmagnons in Bi 2Se3jNiOjNiFe.290\nIn short, spintronics is a vast \feld but there is an ob-\nvious trend that more phenomena driven by SOC are\nbeing investigated. Moreover, materials to generate, ma-\nnipulate/conduct, and detect spin current should be in-\ntegrated for devices, thus the interfacial engineering is a\ncritical issue. Following Table. IV G, both FM and AFM\nmetals/insulators can be incorporated into the spintronic\ndevices, resulting in \rexibility but also time consuming\ncombinatorial optimizations. For instance, a few FM\nmetals like Fe/Co/Ni, permalloy (NiFe), and CoFeB have21\nbeen widely applied in the spintronic heterostructures,\nas the experimental techniques to fabricate such systems\nare well established. In this regard, there is a signi\fcant\nbarrier between theoretical predictions and experimental\nimplementations, though HTP calculations can be car-\nried out in a straightforward way (cf. Sect. V).\nH. Magnetic topological materials\nPioneered by the discovery of integer326,327and\nfractional328,329quantum Hall e\u000bects, a resurgence of\nresearch on materials with nontrivial topological nature\nhas began, as exempli\fed by the 2D topological insula-\ntors (TIs) predicted in graphene330and HgTe quantum\nwells,331and shortly afterwards con\frmed experimen-\ntally in many 2D and 3D systems.332,333The key funda-\nmental concept lies on the Berry phase, which measures\nthe global geometric phase of the electronic wave func-\ntions accumulated through adiabatic evolutions.334Such\nmaterials host many fascinating properties such as quan-\ntum spin Hall e\u000bect (QSHE), which are promising for\nfuture applications such as topological spintronics.332,333\nParticularly, introducing the magnetic degree of free-\ndom not only enriches their functionalities, e.g., quantum\nanomalous Hall e\u000bect (QAHE),335axion electromagnetic\ndynamics,336,337and chiral Majorana fermions,338but\nalso enables more \rexible tunability. The relevant ma-\nterials as compiled in Tab. III, categorized based on the\ndimensionality of the resulting band touchings, namely,\ntopological insulator, nodal point semimetals (including\nWeyl, Dirac, and manyfold nodal points339), and nodal\nline semimetals.340\nAs symmetry plays an essential role in the topologi-\ncal nature of such materials,341we put forward a brief\ndiscussion before getting into particular material sys-\ntems. It is clear that the time-reversal symmetry \u0002\nis broken for all magnetically ordered compounds, and\nhence they are good candidates to search for the QAHE\nand Weyl semimetal phases, which are topologically pro-\ntected rather than symmetry protected. In addition, the\nWeyl node is a general concept which is closely related\nto the accidental degeneracy of electronic bands,342e.g.,\nthere are many Weyl points in bcc Fe.343Thus, the ul-\ntimate goal is to search for compounds where the low\nenergy electronic structure around the Fermi energy are\ndominated solely (or mostly) by the linear dispersions\naround the Weyl points.\nThe occurrence of other topological phases such as TI\nand Dirac fermions in magnetic materials requires ex-\ntra symmetry. For instance, the Kramers degeneracy is\nrequired to de\fne the Z2index for nonmagnetic TI.330\nHowever, the electronic bands in magnetic materials are\ngenerally singly degenerate except at the time-reversal\ninvariant momenta (TRIMs), therefore additional sym-\nmetry is required in order to restore the Kramers de-\ngeneracy. This leads to the prediction of AFM topologi-\ncal insulators,344which are protected by an anti-unitaryproduct symmetry S=P\u0002, wherePcan be a point\ngroup symmetry345or a nonsymmorphic symmetry op-\nerator which reverses the magnetization direction of all\nmoments. Furthermore, it is noted that the inversion\nsymmetryIrespects the magnetization direction, thus\nparity as the resulting eigenvalues of Ican still be ap-\nplied to characterize the topological phase if the com-\npound is centrosymmetric. Nevertheless, the generalized\nZ4character should be used,346e.g., 4n+2 number of odd\nparities from the occupied states at TRIMs corresponds\nto a nontrivial axion insulator. In the same way, for\ntopological semimetallic phases,339additional crystalline\nline symmetry should be present, as will be discussed for\nparticular materials below.\nTurning now to the \frst class of materials exhibit-\ning QAHE, which was predicted by Haldane.385Such\na phenomena can be realized in magnetic topologi-\ncal insulators,386with magnetism introduced by dop-\ning as observed in Cr 0:15(Bi0:1Sb0:9)1:85Te3at 30 mK.347\nThere have been many other proposals based on DFT\ncalculations to predict \fnite temperature QAHE,335\nawaiting further experimental validations. Particu-\nlarly, compounds with honeycomb lattices are a promis-\ning playground for QAHE, including functionalized\ngraphene,387,388Mn2C18H12,389etc. We want to point\nout that QAHE is de\fned for 2D systems,385an interest-\ning question is whether there exists 3D QAHI. By making\nan analogue to the 3D integer quantum Hall e\u000bect,390the\n3D QAHI can be obtained by stacking 2D QAHI on top\nof each other, or by stacking Weyl semimetals.391In the\nformer case, if the Chern number for each kz-plane (in the\nstacking direction) changes, one will end up with Weyl\nsemimetals.392It is proposed recently that for Ba 2Cr7O14\nandh-Fe3O4,3483D QAHI can be obtained due to the\npresence of inversion symmetry.\nThe 2D nature of QAHE observed in Cr-doped TIs\narises two questions: (1) whether the gap opening in the\nDirac surface states required for QAHE can be con\frmed\nexperimentally and (2) whether it is possible to engineer\naxion insulators by forming heterostructures of magnetic-\nTI/normal-TI/magnetic-TI. For the former, a band gap\nof 21 meV was observed in Mn-doped Bi 2Se3393but\nproved not originated from magnetism.394Only recently,\na comparative study of Mn-doped Bi 2Te3and Bi 2Se3re-\nveals that the perpendicular MAE for the Mn moments\nin Bi 2Te3is critical to induce a \fnite band gap of 90\nmeV opening in the Dirac surface states.395In the latter\ncase, if the magnetization directions for two magnetic-\nTIs are antiparrallel and the sandwiching normal-TI is\nthick enough, the axion insulator phase can be achieved\nwith quantized ac-response such as magneto-optical ef-\nfect and topological magnetoelectric e\u000bect;396Whereas\nQAHE with quantized dc-response corresponds to the\nparallel magnetization of two magnetic-TIs.397Several\nexperiments have been done in this direction with con-\nvincing results.398{400\nFocusing from now on only the intrinsic (bulk, non-\ndoped, non-heterostructure) magnetic topological mate-22\nTABLE III: The incomplete list of magnetic topological materials. SdH: Shubnikov-de Haas\ncompounds phase space group measurements\nCr0:15(Bi0:1Sb0:9)1:85Te3 QAHI R\u00163m transport at 30 mK347\nBa2Cr7O14 QAHI R\u00163m DFT348\nMnBi 2Te4 AFM TI R\u00163m DFT+ARPES349\nEuSn 2P2 AFM TI R\u00163m ARPES350\nEuIn 2As2 AFM axion P63/mmc DFT,351ARPES352,353\nEuSn 2As2 AFM TI R\u00163m DFT+ARPES354\nFeSe monolayers 2D AFM TI P4/nmm (bulk) DFT+ARPES+STS355\nHgCr 2Se4 FM Weyl Fd\u00163m DFT356,357\nCo2MnGa FM Weyl/nodal-line Fm\u00163m DFT,358ARPES359\nEuCd 2As2 ideal Weyl P\u00163m1 DFT+ARPES360+SdH361\nGdPtBi AFM Weyl F\u001643m DFT+transport362\nYbPtBi AFM Weyl F\u001643m DFT+ARPES+transport363\nCo3Sn2S2 FM Weyl R\u00163m DFT+transport364\nMn3Sn AFM Weyl P63/mmc DFT+APRES365\nFe3Sn2 FM Weyl R\u00163m ARPES,366STS+QPI367\n(Y/RE) 2Ir2O7 AFM Weyl/axion TI Fd\u00163m DFT368\nCeAlGe Weyl I41md transport369\nCuMnAs AFM massive Dirac Pnma DFT,370,371transport372\nGdSbTe AFM Dirac P4/nmm DFT+ARPES373\nBaFe 2As2 AFM massless Dirac I4/mmm DFT+Infrared,374ARPES375\nCaIrO 3 AFM Dirac Pnma transport+SdH376\nFeSn AFM massless Dirac P6/mmm DFT+ARPES377,378\nCaMnBi 2 AFM massive Dirac P4/nmm ARPES379\n(Sr/Ba)MnBi 2 AFM massive Dirac I4/mmm DFT+ARPES379,380\nEuMnBi 2 AFM massive Dirac I4/mmm transport+SdH381\nYbMnBi 2 AFM massive Dirac P4/nmm DFT+ARPES382\nGdAg 2 2D nodal-line { DFT+ARPES383\nGdPtTe AFM nodal-line P4/nmm DFT+ARPES373\nMnPd 2 AFM nodal-line Pnma DFT384\nrials, an inspiring system is MnBi 2Te4, which is con-\n\frmed to be an AFM TI recently.349It is an ordered\nphase with one Mn layer every septuple layer in the\nBi2Te3geometry and the Mn atoms coupled ferromagnet-\nically (antiferromagnetically) within (between) the Mn\nlayers, leading to a N\u0013 eel temperature of 24 K. Thus,\nthe nontrivial topological phase is protected by the com-\nbined symmetry S= \u0002TwithTbeing the transla-\ntional operator connecting two AFM Mn-layers.349It is\nworthy mentioning that the MnBi 2Te4phase is actually\nthe reason why the observed band gap is as large as 90\nmeV in Mn-doped Bi 2Te3.395In this regard, it is still\nan open issue about whether the Dirac surface states of\nMnBi 2Te4are gapped401{403or gapless354,404,405and the\ncorresponding magnetic origin as the gap survives above\nthe N\u0013 eel temperature.403Nevertheless, MnBi 2Te4stands\nfor a family of compounds with a general chemical for-\nmula MnBi 2n(Se/Te) 3n+1, which are predicted to host\nmany interesting topological phases.406\nThe topological materials can be designed based on\nthe chemistry in systems with comparable crystal struc-\ntures.407,408Taking three Eu-based compounds listed in\nTab. III as an example, the electronic states around the\nFermi energy are mainly derived from the In-5 s/Sn-5p\nand P-3p/As-4pstates, whereas the Eu-4 felectrons are\nlocated more than 1 eV below the Fermi energy.350,351,354Certainly the hybridization between the 4f and valence\nelectronic states around the Fermi energy is mandatory\nin order to drive the systems into the topological phases\nby the corresponding magnetic ordering. Nevertheless,\nit is suspected that given appropriate crystal structures,\nthere is still free space to choose a the chemical compo-\nsition, i.e., via substituting chemically similar magnetic\nRE elements to \fne tune the electronic structure and\nthus design novel materials. This philosophy is best man-\nifested in the class of Dirac semimetals AMnBi 2(A = Ca,\nSr, Bi, Eu, and Yb) as listed in Tab. III. For such com-\npounds, the Dirac cone can be attributed to the square\nlattice of Bi atoms, which is common to all the com-\npounds following the DFT prediction.409It is noted that\nthe Dirac cones in all the AMnBi 2compounds are mas-\nsive due to SOC, but still leads to high mobility as the\nresulting e\u000bective mass of electrons is small.410Substitut-\ning Sb for Bi leads to another class of Dirac semimetals\nAMnSb 2as predicted by DFT411and con\frmed experi-\nmentally.410,412{415\nAs discussed above, the occurrence of Dirac points\n(with four-fold degeneracy) in magnetic materials de-\nmands symmetry protection. Taking CuMnAs as an ex-\nample, it is demonstrated that the Dirac points is pro-\ntected via the combined symmetry of S=I\u0002 as a prod-\nuct of inversion Iand \u0002.370Considering additionally23\nSOC leads to lifted degeneracies depending on the mag-\nnetization directions because of the screw rotation oper-\natorS2z. The same arguments applies to FeSn377and\nMnPd 2.384Several comments are in order. Firstly, it is\nexactly due to the combined symmetry S=I\u0002 why\nthere exists staggered SOT for CuMnAs, leading to a\ndescriptor to design AFM spintronic materials.371Sec-\nondly, the magnetization direction dependent electronic\nstructure is a general feature for allmagnetic topolog-\nical materials, such as surface states of AFM TIs like\nMnBi 2Te4349and EuSn 2As2,354and also the nodal line\nand Weyl semimetals.416This o\u000bers another way to tailor\nthe application of the magnetic topological materials for\nspintronics.417Thirdly, the Dirac (Weyl) features prevail\nin AFM (FM) materials as listed in Tab. III, thus we\nsuspect that there are good chances to design AFM ma-\nterials with high mobility of topological origin.\nThe relevant orbitals responsible for the topological\nproperties for most of the materials discussed so far are\np-orbitals of the covalent elements, where the TM- dand\nRE-fstates play an auxiliary role to introduce mag-\nnetic ordering. A particularly interesting subject is to\ninvestigate the topological features in bands derived from\nmore correlated d=f-orbitals, which has recently been\nexplored extensively in ferromagnetic Kagome metallic\ncompounds in the context of Weyl semimetals. We note\nthat the Kagome lattice with AFM nearest neighbor cou-\npling has been an intensively studied \feld for the quan-\ntum spin liquid.418On the one hand, Weyl semimetals\nhost anomalous Hall conductivity419and negative mag-\nnetoresistance driven by the chiral anomaly.420,421On\nthe other hand, due to the destructive interference of the\nBloch wave functions in the Kagome lattice, \rat bands\nand Dirac bands can be formed with nontrivial Chern\nnumbers.422,423Therefore, intriguing physics is expected\nno matter whether the Fermi energy is located in the\nvicinity of the \rat band or the Dirac gap, as in the\nTmXncompounds (T = Mn, Fe, and Co; X = Sn and\nGe).378Particularly, for Fe 3Sn2, the interplay of gauge\n\rux driven by spin chirality and orbital \rux due to SOC\nleads to giant nematic energy shift depending on the\nmagnetization direction and further the spin-orbit en-\ntangled correlated electronic structure.367This o\u000bers a\nfascinating arena to investigate the emergent phenomena\nof topology and correlations in such quantum materials,\nwhich can also be generalized to superconducting mate-\nrials such as Fe-based superconductors.355,374,375\nFrom the materials point of view, there are many\npromising candidates to explore. For instance, the re-\ncently discovered FM Co 3Sn2S2shows giant anomalous\nHall conductivity364and signi\fcant anomalous Nernst\nconductivity,424, driven by the Weyl nodes around the\nFermi energy. In addition, various Heusler compounds\ndisplays also Weyl features in the electronic structure,\nincluding Co 2MnGa,358,359Co2XSn,425,426Fe2MnX,426\nand Cr 2CoAl.427The most interesting class of com-\npounds are those with noncollinear magnetic con\fgura-\ntions. It is con\frmed both theoretically and experimen-tally that Mn 3Sn is a Weyl semimetal,365,428leading to\nsigni\fcant AHC429and SHC.298It is expected that the\nWeyl points can also be found in the other noncollinear\nsystems such as Mn 3Pt430and antiperovskite,46awaiting\nfurther investigation.\nIridates are another class of materials which are pre-\ndicted to host di\u000berent kind of topological phases.431,432\nMost insulating iridates contain Ir4+ions with \fve 5 d\nelectrons, forming IrO 6octahedra with various connec-\ntivities. Due to strong atomic SOC, the t 2gshell is split\ninto low-lying J = 3/2 and high-lying half-\flled J =\n1/2 states. Therefore, moderate correlations can lead\nto insulating states, and such unique SOC-assisted insu-\nlating phase have been observed in Ruddlesden-Popper\niridates.433Pyrochlore iridates RE 2Ir2O7can also host\nsuch insulating states, which were predicted to be topo-\nlogically nontrivial.368Many theoretical studies based\non models have been carried out, supporting the non-\ntrivial topological nature,434435but no smoking-gun ev-\nidence, e.g., surface states with spin-momentum lock-\ning using ARPES, has been observed experimentally.\nOur DFT+DMFT calculations with proper evaluation\nof the topological character suggest that the insulating\nRE2Ir2O7are likely topologically trivial,184consistent\nwith the ARPES measurements.436Nevertheless, prox-\nimity to the Weyl semimetal phase is suggested, existing\nin a small phase space upon second-order phase tran-\nsition by recent experiments,437where magnetic \felds\ncan applied to manipulate the magnetic ordering of the\nRE-sublattices and thus the topological character of the\nelectronic states.438439440Last but not least, it is ob-\nserved that there exist robust Dirac points in CaIrO 3\nwith the post-perovskite structure,376leading to high mo-\nbility which is rare for correlated oxides.\nObviously, almost all known topological phases can be\nachieved in magnetic materials, which o\u000ber more degrees\nof freedom compared to the nonmagnetic cases to tai-\nlor the topological properties. Symmetry plays a crit-\nical role, e.g., semimetals with nodal line383373384and\nmany-fold nodal points,339can also be engineered be-\nyond the Dirac nodes and AFM TIs discussed above.\nUnlike nonmagnetic materials where the possible topo-\nlogical phases depend only on the crystal structure and\nthere are well compiled databases such as ICSD, there is\nunfortunately no complete database with the magnetic\nstructures collected. Also, it is fair to say that no ideal\nWeyl semimetals have been discovered till now, e.g., there\nare other bands around the Fermi energy for all the Weyl\nsemimetal materials listed in Tab. III. Therefore, there\nis a strong impetus to carry out more systematic stud-\nies on screening and designing magnetic materials with\nnontrivial topological properties.\nI. Two-dimensional magnetic materials\nPioneered by the discovery of graphene,441there has\nbeen recently a surge of interest on two-dimensional (2D)24\nmaterials, due to a vast spectrum of functionalities such\nas mechanical,442electrical,443optoelectronic,444and su-\nperconducting445properties and thus immense poten-\ntial in engineering miniaturized devices. However, the\n2D materials with intrinsic long-range magnetic order-\ning have been missing until the recently con\frmed sys-\ntems like CrI 3,446Cr2Ge2Te6,447and Fe 3GeTe 2448in the\nmonolayer limit. This enables the possibility to fabricate\nvdW heterostructures based on 2D magnets and hence\npaves the way to engineer novel spintronic devices. Nev-\nertheless, the \feld of 2D magnetism is still in its infancy,\nwith many pending problems such as the dimensional\ncrossover of magnetic ordering, tunability, and so on.\nAccording to the Mermin-Wagner theorem,449the\nlong-range ordering is strongly suppressed at \fnite tem-\nperature for systems with short-range interactions of con-\ntinuous symmetry in reduced dimensions, due to the di-\nvergent thermal \ructuations. Nevertheless, as proven\nanalytically by Onsager,450the 2D Ising model guaran-\ntees an ordered phase, protected by a gap in the spin-\nwave spectra originated from the magnetic anisotropy.451\nIn this sense, the rotational invariance of spins can\nbe broken by dipolar interaction, single-ion anisotropy,\nanisotropic exchange interactions, or external magnetic\n\felds, which will lead to magnetic ordering at \fnite tem-\nperature. The complex magnetic phase diagram is best\nrepresented by that of transition metal thiophosphates\nMPS 3(M = Mn, Fe, and Ni),452,453which are of the 2D-\nHeisenberg nature for MnPS 3, 2D-Ising for FePS 3, and\n2D-XXZ for NiPS 3, respectively. As shown in Fig. 8 for\nfew-layer systems of both FePS 3and MnPS 3, the tran-\nsition temperature remains almost the same as that of\nthe corresponding bulk phase. However, for NiPS 3, the\nmagnetic ordering is suppressed for the monolayers while\nfew-layer slabs have slightly reduced N\u0013 eel temperature\ncompared to that of the bulk.454This can be attributed\nto the 2D-XXZ nature of the e\u000bective Hamiltonian, where\nin NiPS 3monolayers it may be possible to realize the\nBKT topological phase transition.455This arises also an\ninteresting question whether there exist novel quantum\nphases when magnetic ordering is suppressed and how to\ntune the magnetic phase diagram. Also, it is suspected\nthat the interlayer exchange coupling does not play a\nsigni\fcant role for such 2D systems with intralayer AFM\nordering.\nThe interplay of interlayer exchange and magnetic\nanisotropy leads to more interesting magnetic properties\nfor CrX 3(X = Cl, Br, and I). The Cr3+ions with oc-\ntahedral environment in such compounds have the 3 d3\ncon\fguration occupying the t 2gorbitals in the major-\nity spin channel. The SOC is supposed to be quenched\nleading to negligible single-ion MAE. Nevertheless, large\nMAE can be induced by the strong atomic SOC on the\nI atoms, i.e., intersite SOC, as elaborated in Ref.456,457\nAs the atomic SOC strength is proportional to the atomic\nnumber, the MAE favors in-plane (out-of-plane) magne-\ntization for CrCl 3(CrBr 3and CrI 3), respectively. Note\nthat for the bulk phases, the intralayer exchange is FMfor all three compounds, while the interlayer exchange is\nFM for CrBr 3and CrI 3, and AFM for CrCl 3.458In this\nregard, the most surprising observation for the few-layer\nsystems is that CrI 3bilayers have AFM interlayer cou-\npling446with a reduced N\u0013 eel temperature of 46K in com-\nparison to the bulk value of 61K (Fig. 8). Additionally,\nthe FM/AFM interlayer coupling can be further tuned\nby pressure, as showed experimentally recently.459,460\nSuch a transition from FM to AFM interlayer coupling\nwas suggested to be induced by the stacking fault based\non DFT calculations, namely the rhombohedral (mono-\nclinic) stacking favors FM (AFM) interlayer exchange.461\nIt is noted that CrX 3has the structural phase transi-\ntion in the bulk phases from monoclinic to rhombohe-\ndral around 200 K, e.g., 220K for CrI 3.462Thus, a very\ninteresting question is why there is a structural phase\ntransition from the bulk rhombohedral phase to the mon-\noclinic phase in bilayers. A recent experiment suggests\nthat such a stacking fault is induced during exfoliation,\nwhere it is observed that the interlayer exchange is en-\nhanced by one order of magnitude for CrCl 3bilayers.463\nIt is noted that it is still not clear how the easy-plane XY\nmagnetic ordering in CrCl 3would behavior in the mono-\nlayer regime. Therefore, for such 2D magnets, an in-\ntriguing question to explore is the dimensional crossover,\ni.e., how the magnetic ordering changes few-layer and\nmonolayer cases in comparison to the bulk phases, as\nwe summarized in Fig. 8 for the most representative 2D\nmagnets.\nThe key to understand the magnetic behavior of such\n2D magnets is to construct a reliable spin Hamiltonian\nbased on DFT calculations. Di\u000berent ways to evaluate\nthe interatomic exchange parameters have been scru-\ntinized in Ref.464 for CrCl 3and CrI 3. It is pointed\nout that the energy mapping method, which has been\nwidely used, leads to only semi-quantitative estimation.\nWhereas the linear response theory including the ligand\nstates give more reasonable results, ideally carried out in\na self-consistently way. Particularly, exchange parame-\nters beyond the nearest neighbors should be considered,\nas indicated by \ftting the inelastic neutron scattering\non FePS 3.465Moreover, for the interlayer exchange cou-\npling, DFT calculations predict AFM coupling while the\nground state is FM for Cr 2Si2Te6,466which may be due\nto the dipole-dipole interaction which is missing in non-\nrelativistic DFT calculations.467Note that the interlayer\nexchange coupling is very sensitive to the distance, stack-\ning, gating, and probably twisting,468which can be used\nto engineer the spin Hamiltonian which should be eval-\nuated quantitatively to catch the trend. Furthermore,\nas the most of the 2D magnets have the honeycomb lat-\ntice with edge-sharing octahedra like the layered Na 2IrO3\nand\u000b-RuCl 3which are both good candidates to realize\nthe quantum spin liquid states,469the Kitaev physics be-\ncomes relevant, e.g., a comparative investigation on CrI 3\nand Cr 2Ge2Te6demonstrated that there does exist sig-\nni\fcant Kitaev interactions in both compounds.470Thus,\nfor 2D magnets, the full 3 \u00023 exchange matrix as speci-25\n\fed in Eq. (IV C) has to be considered beyond the nearest\nneighbors, in order to understand the magnetic ordering\nand emergent properties. This will enable us to engi-\nneer the parameter space to achieve the desired phases,\ne.g., to obtain 2D magnetism with critical temperature\nhigher than the room temperature. For instance, the an-\ntisymmetric DMI can stabilize skyrmions in 2D magnets,\nas observed in recent experiments on Fe 3GeTe 2nanolay-\ners.471,472\nDue to the spatial con\fnement and reduced dielec-\ntric screening, the Coulomb interaction is supposed to be\nstronger in 2D materials, which bring forth the electronic\ncorrelation physics. Taking Cr 2Ge2Te6as an example, re-\ncent ARPES measurements demonstrate that the band\ngap is about 0.38 eV at 50K,473in comparison to 0.2 eV\nat 150K,474while the band gap is only about 0.15 eV in\nthe GGA+U calculations with U = 2.0 eV which repro-\nduce the MAE.475As both the magnetic ordering and\nelectronic correlations can open up \fnite band gaps, re-\ncent DFT+DMFT calculations reveal that the band gap\nin Cr 2Si2Te6is mostly originated from the electronic cor-\nrelations.476Additionally, both the d-dtransitions and\nmolecular ligand states are crucial for the helical photo-\nluminescence of CrI 3monolayers.477Moreover, it is also\ndemonstrated that the ferromagnetic phase coexists with\nthe Kondo lattice behavior in Fe 3GeTe 2.478The corre-\nlated nature of the electronic states in 2D magnets calls\nfor further experimental investigation and detailed the-\noretical modelling. 2D magnetic materials provide also\nan interesting playground to study the interplay between\nelectronic correlations and magnetism, which is in com-\npetition with other emergent phases.\nTo tailor the 2D magnetism, voltage control of mag-\nnetism has many advantages over the mainstream meth-\nods for magnetic thin \flms such as externally magnetic\n\felds and electric currents, because they su\u000ber from poor\nenergy e\u000eciency, slow operating speed, and appalling\nsize compactness. For biased CrI 3bilayers, external elec-\ntric \felds can be applied to switch between the FM and\nAFM interlayer coupling.479,480Such a gating e\u000bect can\nbe further enhanced by sandwiching dielectric BN lay-\ners between the electrode and the CrI 3layers.481More-\nover, ionic gating on Fe 3GeTe 2monolayers can boost\ntheir Curie temperature up to the room temperature,\nprobably due to enhanced MAE but with quite irreg-\nular dependence with respect to the gating voltage.482\nInteresting questions are to understand the variation in\nthe spin Hamiltonian and thus the Curie temperature\nin such materials under \fnite electric \felds. Addition-\nally, strain has also been applied to tailor 2D magnets.\nFor instance, it is demonstrated that 2% biaxial com-\npressive strain leads to a magnetic state transition from\nAFM to FM for FePS 3monolayers,483and 1.8% tensile\nstrain changes the FM ground state into AFM for CrI 3\nmonolayers.484Moreover, strain can also induce signif-\nicant modi\fcation of the MAE, e.g., 4% tensile strain\nresults in a 73% increase of MAE for Fe 3GeTe 2with a\nmonotonous dependence in about 4% range.485Thus, it\nFIG. 8: Dependence of the magnetic ordering temperature\nwith respect to the number of layers for typical 2D mag-\nnets. The values are normalized using the corresponding\nbulk Curie/N\u0013 eel temperature, e.g., CrCl 3(TC= 17K),458\nCrBr 3(TC= 37K),458CrI3(TC= 61K),446Cr2Ge2Te6(TC\n= 68K),447MnPS 3(TN = 78K),464FePS 3(TN = 118K),465\nNiPS 3(TN= 155K),454and Fe 3GeTe 2(TC= 207K).448Lines\nare guide for the eyes. The \fgure is adapted from Ref. 486\nwith updated data. Copyright requested.\nis suspected that biaxial strain can be applied to tune\nthe magnetic properties of vdW magets e\u000bectively.\n2D magnets exhibit a wide spectrum of functionali-\nties. For instance, tunnelling magnetoresistance in het-\nerostructures of CrI3 has been observed in several set-ups\nincorporating CrI 3multilayers,487{489with the magne-\ntoresistance ratio as large as 1,000,000%.490Such a high\nratio can be attributed to perfect interfaces for the vdW\nheterostructures, which is challenging to achieve for con-\nventional heterostructures of magnetic metals and insu-\nlators such as Fe/MgO. The tunnelling can be further\ntuned via gating,491leading to implementation of spin\ntunnel \feld-e\u000bect transistors.492Moreover, metallic 2D\nmagnets displays also fascinating spintronic properties,\ne.g., spin valve has been implemented based on the sand-\nwich Fe 3GeTe 2/BN/Fe 3GeTe 2geometry, where a tun-\nnelling magnetoresistance of 160% with 66% polarization\nhas been achieved.493\nSuch transport properties are based on the charge and\nspin degrees of freedom of electrons, whereas emergent\ndegree of freedom such as valley has attracted also inten-\nsive attention, leading to valleytronics.494As most of the\ncompounds discussed above have the honeycomb lattice,\nthe valley-degeneracy in the momentum space will be\nlifted when the inversion symmetry is broken, resulting in\nvalley-dependent (opto-)electronic properties. Moreover,26\nconsidering SOC causes the spin-valley coupling,495i.e.,\nvalley-dependent optical selection rules become also spin-\ndependent, which can be activated by circularly polarized\nlights. Such phenomena can be realized in the vdW het-\nerostructures such as the CrI 3/WSe 2heterostructures496\nvia the proximity e\u000bect or in AFM 2D magnets such as\nMnPS 3.497\nFrom the materials perspective, 2D materials have\nbeen extensively investigated both experimentally and\ntheoretically. There exist several 2D material\ndatabases498{501based on HTP DFT calculations, lead-\ning to predictions of many interesting 2D magnets.502\nNevertheless, it is still a fast developing \feld, e.g.,\nthe Curie temperature of 2D FM compounds has been\npushed up to the room temperature in VTe 2503and\nCrTe 2.504It is noted that the DFT predictions should\ngo hand-in-hand with detailed experimental investiga-\ntion. For instance, VSe 2monolayers are predicted to be\na room temperature magnet505supported by a follow-\nup experiment.506However, recent experiments reveal\nthat the occurrence of the charge density wave phase will\nprohibit the magnetic ordering.507,508To go beyond the\nknown prototypes to design new 2D materials, one can ei-\nther predict more exfoliable bulk compounds with layered\nstructure or laminated compounds where etching can be\napplied. For instance, MXene is a new class of 2D materi-\nals which can be obtained from the MAX compounds,509\nwhere the HTP calculations can be helpful. It is particu-\nlarly interesting to explore such systems with 4 d/5dTM\nor RE elements, where the interplay of electronic corre-\nlations, exchange coupling, and SOC will leads to more\nintriguing properties, such as recently reported MoCl 5510\nand EuGe 2.511Last but not least, vdW materials can also\nbe obtained using the bottom-up approach such as molec-\nular beam epitaxy down to the monolayer limit, such\nas MnSe 2on GaSe substrates.512This broadens signi\f-\ncantly the materials phase space to those non-cleavable\nor metastable vdW materials and provides a straightfor-\nward way to in-situ engineer the vdW heterostructures.\nLike the 3D magnetic materials, 2D magnets displays\ncomparable properties but with a salient advantage that\nthey can form van der Waals (vdW) heterostructures\nwithout enforcing lattice matching.513This leads to a\nwide range of combinations to further tailor their prop-\nerties, e.g., the emergent twistronics514exampled by the\nsuperconductivity in magic angle bilayer graphene.515In\ncomparison to the interfaces in conventional magnetic\nheterostructures, which are impeded by dangling bonds,\nchemical di\u000busion, and all possible intrinsic/extrinsic de-\nfects, vdW heterostructures with well-de\fned interfacial\natomic structures are optimal for further engineering of\nspintronic devices.516Nevertheless, to the best of our\nknowledge, there is no 2D magnet showing magnetic or-\ndering at room temperature, which is a challenge among\nmany as discussed above for future HTP design.\nFIG. 9: A typical HTP work\row to design magnetic materials.\nBlue, yellow, green, and purple blocks denote the processes of\ncrystal structure identi\fcation, evaluation of stabilities, char-\nacterization of magnetic properties, and database curation,\nrespectively.\nV. CASE STUDIES\nA. High-throughput work\rows\nFig. 9 displays a generic work\row to perform HTP\nscreening on magnetic materials, with four processes\nhighlighted in di\u000berent colors. That is, HTP screening\ncan be performed in four essential steps:\n1. crystal structure identi\fcation. As DFT cal-\nculations need only the chemical composition\nand crystal structure as inputs, HTP calcula-\ntions mostly start with such information which\ncan be obtained via (1) databases compil-\ning known compounds, (2) substitution based\non crystal structure prototypes, and (3) crys-\ntal structure prediction using evolutional algo-\nrithms. There have been a few databases such\nas ICSD https://icsd.\fz-karlsruhe.de/index.xhtml27\nand COD http://www.crystallography.net/cod/,\nwith the crystal structures for experimentally syn-\nthesized compounds and theoretically predicted\ncrystal structures. Moreover, typical structure pro-\ntotypes have been identi\fed,54,55which o\u000ber a good\nstarting point to perform chemical substitutions.517\nThe crystal structures can also be generated based\non evolutionary algorithms using USPEX518and\nCALYPSO.60\n2. evaluation of stabilities. This is usually done in\na funnel-like way, as thermodynamic, mechanical,\nand dynamical stabilities should be systematically\naddressed, as detailed in Sect. IV A. We want to\nemphasise that the evaluation of convex hull should\nbe performed with respect to as many competing\nphases as possible, thus the corresponding DFT cal-\nculations are better done using consistent parame-\nters as cases in existing databases such as Materials\nProject,8AFLOW,9and OQMD.11\n3. characterization of magnetic properties. In Fig. 9\nwe list several fundamental magnetic properties,\nsuch as magnetization, MAE, T C/TNand trans-\nport properties (please refer to the previous sec-\ntions for detailed discussions), which can be selec-\ntively evaluated depending on the target applica-\ntions. It is worthy pointing out that a system-\natic evaluation of the thermodynamic properties\nfor magnetic materials dictates accurate Gibbs free\nenergies with signi\fcant magnetic contributions,519\nas illustrated for the Fe-N systems.520Thus, proper\nevaluation of the magnetization and magnetic or-\ndering temperature is required.\n4. database curation. Ideally, a data infrastructure is\nneeded in order to obtain, store, analyze, and share\nthe DFT calculations. Non-SQL databases ( e.g.,\nMongoDB) are mostly used because of their \rexibil-\nity with the data structure, which can be adjusted\nfor compounds with various complexity and add-\non properties evaluated in an asynchronized way.\nMoreover, the data should be \fndable, accessible,\ninteroperable, and reusable (FAIR),521so that they\ncan be shared with the community with approved\nprovenance. Last but not least, with such database\nconstructed, data mining using machine learning\ntechniques can be performed to further accelerate\nthe previous three steps and develop statistical in-\nsights on the results, as discussed in Sect. VI B.\nIt is straightforward to set up such HTP work\rows,\neither based on integrated platforms or starting from\nscratch, as many DFT codes have Python interfaces.522\nIn this regard, the atom simulation environment523is\na convenient tool, which has been interfaced to more\nthan 30 software packages performing DFT and molec-\nular simulations. Nevertheless, the following two as-\npects deserve meticulosity, which essentially distinguishHTP from conventional DFT calculations. The work-\n\row should be automated, so that the computational\ntasks can be properly distributed, monitored, and man-\naged. For instance, a typical practice is to generate\ninput \fles for thousands of compositions, optimize the\ncrystal structures, and assess the thermodynamic stabil-\nity. The corresponding work\rows can be implemented\nusing stand-alone Python packages like \freworks,524or\nusing the integrated platforms such as AiiDa12and Ato-\nmate.13Another critical point is job management and\nerror handling. Ideally, a stand-by thread shall mon-\nitor the submitted jobs, check the \fnished ones, and\n(re-)submit further jobs with proper error (due to either\nhardware or software issues) recovery. In this regard, cus-\ntodian(https://materialsproject.github.io/custodian/) is\na good option, with integrated error handling for the\nVASP, NwChem, and QChem codes. On the other hand,\nthe set-up and maintenance of such job management de-\npends on the computational environment, and also ex-\npertise with the usage of scienti\fc softwares.\nB. Heusler compounds\nHeusler compounds form an intriguing class of inter-\nmetallic systems possessing a vast variety of physical\nproperties such as HM, MSME, TI, superconductivity,\nand thermoelectricity (please refer to Ref. 525 for a com-\nprehensive review). Such versatility is originated from\nthe tunable metallic and insulating nature of the elec-\ntronic structure due to the \rexibility in the chemical\ncomposition. As shown in Fig. 10, there are two main\ngroups of Heusler compounds, namely, the half Heusler\nwith the chemical composition XYZ and the full-Heusler\nX2YZ, where X and Y are usually transition metal or\nRE elements, and Z being a main group element. For\nboth half Heusler and full Heusler systems, the crystal\nstructures can be considered to be with a skeleton of the\nZnS-type (zinc blende) formed by covalent bonding be-\ntween X and Z elements, and the Y cations will occupy\nthe octahedral sites and the vacancies. This gives rise to\nthe \rexibility to tune the electronic structure by playing\nwith the X, Y, and Z elements.525\nThe full Heuslers X 2YZ can end up with either the reg-\nular (L2 1-type) or the inverse (X \u000b-type) Heusler struc-\nture, as shown in Fig. 10. It is noted that the both the\nregular and inverse full Heusler crystal structures are de-\nrived from the closely packed fcc structure, which is one\nof the most preferred structures for ternary intermetal-\nlic systems.526Depending on the relative electronegativ-\nity of the X and Y elements, the empirical Burch's rule\nstates that the inverse Heusler structure is preferred if\nthe valence of Y is larger than that of X, e.g., Y is\nto the right of X if they are in the same row of the\nperiodic table.527Several HTP calculations have been\ndone to assess the Burch's rule, such as in Sc-528and\nPd-based529,530Heuslers. Additionally, the nonmagnetic\nL21Heusler compounds as high-strength alloys531and28\nFIG. 10: Illustration of the Heusler crystal structures for (a) (regular) full Heusler X 2YZ, (b) inverse (full) Heusler X 2YZ, (c)\nhalf Heusler XYZ, (d) quaternary Heusler XX0YZ, and (e) tetragonal (full) Heusler X 2YZ. The conventional unit cell of 16\natoms are sketched, using the Wycko\u000b positions of the F \u001643m to give a general description for the cubic (a-d) cases.\nthermoelectric532materials have been studied. For mag-\nnetic properties, Sanvito et al.57performed HTP calcula-\ntions on 236,115 X 2YZ compounds and found that 35,602\ncompounds are stable based on the formation energy,\nwhere 6,778 compounds are magnetic. Unfortunately,\nthe relative stability with respect to competing binary\nand ternary phases is expensive thus demanding to ac-\ncomplish. In a recent work,533Maet al. carried out\nDFT calculations on a small set of 405 inverse Heusler\ncompounds, but with the relative stability to the reg-\nular Heusler and the convex hull considered systemati-\ncally, including also a few possible magnetic con\fgura-\ntions.533Ten HMs have been identi\fed, which can be ex-\nplored for spintronic applications.\nFocusing on the regular Heusler systems, Ballu\u000b et\nal.534performed systematic evaluation of the convex hull\nbased on the available data in AFLOW and it is found\nthat the number of magnetic Heuslers will be reduced\nfrom 5,000 to 291. That is, the convex hull construc-\ntion will reduced the number of stable compounds by one\norder of magnitude, consistent with our observation on\nthe magnetic antiperovskites.46Therefore, although the\ndatabase of competing phases might not be complete, it\nis highly recommended to evaluate the thermodynamic\nstability including both the formation energy and convex\nhull, if not the mechanical and dynamical stabilities. Fur-\nthermore, to identify the magnetic ground states, Ballu\u000b\net al. carried out calculations on two antiferromagnetic\ncon\fgurations and obtained 70 compounds with AFM\nground states. The N\u0013 eel temperature of such compounds\nare then computed using the Monte Carlo method\nwith exchange parameters calculated explicitly using theSPRKKR (https://ebert.cup.uni-muenchen.de) code, re-\nsulting in 21 AFM Heusler compounds with N\u0013 eel tem-\nperature higher than 290 K. Such compounds are ready\nto be explored for applications on AFM spintronics.\nInterestingly, for both regular and inverse Heusler com-\npounds, tetragonal distortions can take place driven by\nthe band Jahn-Teller e\u000bect associated with high DOS\nat the Fermi energy.535HTP calculations reveal that\n62% of the 286 Heusler compounds investigated prefer\nthe tetragonal phase, due to the van Hove singulari-\nties around the Fermi energy.536Such tetragonal Heusler\ncompounds exhibit large MAE, e.g., 5.19 MJ/m3for\nFe2PtGe and 1.09 MJ/m3for Fe 2NiSn, making them\ninteresting candidates as permanent magnets and STT-\nMRAM materials.537To search for potential materials for\nSTT applications, Al-, Ga-, and Sn-based inverse Heusler\ncompounds in both cubic and tetragonal structures have\nbeen investigated, aiming at optimizing the spin polariza-\ntion and Gilbert damping for materials with perpendicu-\nlar magnetic anisotropy.538Additionally, such tetragonal\nHeusler compounds can also be used to engineer magnetic\nheterostructures with enhanced TMR e\u000bects.539\nSigni\fcant MAE can also be obtained by inducing\ntetragonal distortions on the cubic Heusler compounds.\nFor instance, the MAE of Ni 2YZ compounds can be as\nlarge as 1 MJ/m3by imposing c/a 6= 1.540To stabilize\nsuch tetragonal distortions, we examined the e\u000bect of\nlight elements (H, B, C, and N) as interstitial dopants\ninto the Heuslers and found that tetragonal distortions\ncan be universally stabilized due to the anisotropic crys-\ntalline environments for the interstitials preferentially oc-\ncupying the octahedral center.541This leads to an e\u000bec-29\ntive way to design RE-free permanent magnets. Two\npending problems though are the solubility and possi-\nble disordered distribution of the light interstitials in the\nHeusler structures. Nevertheless, it is demonstrated that\nfor the Fe-C alloys, due to interplay of anharmonicity\nand segregation, collective ordering is preferred,542en-\ntailing further studies on the behavior of interstitials in\nthe Heusler compounds. We note that the structural\nphase transition between the cubic and tetragonal phases\nin Ni 2MnGa induces the MSME and in Ni 2MnX (X =\nIn, Al, and Sn) the MCE. In this regard, there are many\nmore compounds with such martensitic transitions which\ncan host MSME and MCE, awaiting further theoretical\nand experimental investigations.\nTo go beyond the p-dcovalent bonding between the X\nand Z atoms which stabilizes the Heusler structures, Wei\net al. \frst explored the d-dhybridization in the Ni-Mn-Ti\nsystems and observed that there exist stable phases with\nsigni\fcant MSME, which o\u000bers also the possibility to im-\nprove the mechanical properties of the resulting Heusler\nsystems in comparison to the main-group-element based\nconventional cases.543Considering only the cubic phases,\nfollow-up HTP calculations revealed that 248 compounds\nare thermodynamically stable out of 36,540 prototypes\nand 22 of them have magnetic ground states compatible\nwith the primitive unit cell.57The predictions are vali-\ndated by successful synthesis of Co 2MnTi and Mn 2PtPd\nwhich adopts the tetragonal structure with space group\nI4/mmm (Fig. 10).57An interesting question is whether\nthere exist promising candidates with enhanced caloric\nperformance beyond the known Ni-Mn-Ti case.544There\nhave been several studies focusing on Zn-,545V-,546and\nCd-based547Heuslers, where possible martensitic trans-\nformation is assessed based on the Bain paths. We note\nthat in order to get enhanced MCE, the martensitic phase\ntransition temperature is better aligned within the Curie\ntemperature window.548Therefore, to predict novel all-\nd-metal Heuslers with magneto-structural functionalities,\nmore systematic HTP characterization on the thermody-\nnamic and magnetic properties for both cubic and tetrag-\nonal phases is required.\nThe \rexibility in the chemical composition for the in-\ntermetallic Heusler compounds can be further explored\nby substituting a fourth element or vacancy in X 2YZ,\nleading to quaternary XX0YZ and half XYZ Heusler\n(Fig. 10). Following the empirical rule, we performed\nHTP calculations on magnetic quaternary Heusler com-\npounds with 21, 26, and 28 valence electrons to search\nfor spin gapless semiconductors (SGSs).549Considering\nboth structural polymorphs by shu\u000fing atomic posi-\ntions and magnetic ordering compatible with the primi-\ntive cell, we identi\fed 70 unreported candidates covering\nall four types of SGSs out of 12,000 chemical composi-\ntions, with all the 22 experimentally known cases vali-\ndated. Such SGSs are promising for spintronic applica-\ntions, as they display signi\fcant anisotropic magnetore-\nsistance, and tunable AHC and tunnelling magnetoresis-\ntance.549,550We note that the semimetallic phase can alsobe found in quaternary Heuslers with the other numbers\nof electrons,551which is interesting for future studies.\nRegarding the half Heusler compounds, there have\nbeen extensive HTP calculations to screen for nonmag-\nnetic systems as thermoelectric materials,552,553where\nthe formation of defects has also been addressed in a\nHTP manner.554Maet al. carried out HTP calculations\non 378 half Heuslers and identi\fed 26 semiconductors, 45\nHMs, and 34 nearly HMs, which are thermodynamically\nstable.83Another recent work investigated the alkaline\nelement based half Heuslers and found 28 ferromagnetic\nmaterials out of 90 compositions.555\nImportantly, for half Heusler materials, it is not enough\nto consider only the binary competing phases, but also\npossible ternary compounds as there are many crys-\ntal structures which can be stabilized based on the 18-\nelectron rule for the 1:1:1 composition.556For instance,\nthe so-called hexagonal Heusler compounds are a class\nof materials which can host topological insulator557and\n(anti-)ferroelectric phases.558,559Particularly, the mag-\nnetic counterparts hexagonal Heuslers host martensitic\ntransitions, making them promising for MCE applica-\ntions.560This leads to two recent work where the compo-\nsition of the hexagonal Heusler compounds is optimized\nto further improve the MCE,561,562whereas more sys-\ntematic HTP calculations are still missing. Additionally,\nit is demonstrated that the half Heuslers with the F \u001643m\nstructure can be distorted into either P6 3mc or Pnma\nstructures, giving rise to an interesting question whether\nmagnetic materials with signi\fcant MCE can be identi-\n\fed based on HTP screening. Last but not least, like\nthe quaternary Heuslers, the \rexibility in composition\ncan also be realized in the half Heusler structure. For\ninstance, 131 quaternary double half Heusler compounds\nare predicted to be stable where Ti 2FeNiSb 2has been ex-\nperimentally synthesized showing low thermal conductiv-\nity as predicted. This paves the way to explore the tun-\nable electronic and magnetic properties of half Heusler\ncompounds.\nOne particularly interesting subject is to screen for\nHeuslers with transport properties of the topological ori-\ngin. For instance, motivated by the recent discovered\nWeyl semimetal Co 2MnGa,359an enlightening work eval-\nuated the topological transport properties ( e.g., AHC,\nANC, and MOKE which obey the same symmetry rules)\nof 255 cubic (both regular and inverse) Heusler com-\npounds in a HTP way.563By comparing the results for\nfull and inverse Heuslers systems, it is observed that the\nmirror symmetry present in the full but not the inverse\nHeusler compounds plays an essential role to induce sig-\nni\fcant linear response properties. It is noted that only\none FM (AFM) con\fguration is considered for the full\n(inverse) Heusler compounds, which might not be enough\nas the electronic structure and the derived physical prop-\nerties are subjected to the changes in the magnetic con-\n\fgurations.\nOverall, it is obvious that Heusler compounds are a\nclass of multifunctional magnetic materials, but the cur-30\nrent HTP calculations are still far away from being com-\nplete. One additional issue is how to properly treat the\nchemical and magnetic disorders in such compounds in\norder to predict the electronic and thermodynamic prop-\nerties. For instance, the MCE takes place only for Mn-\nrich Ni-Mn-Ti,543which has not yet been explained based\non DFT calculations. We believe substitutions and chem-\nically disordered systems deserve more systematic treat-\nments in the future HTP studies, particularly for Heusler\nalloys.\nC. Permanent magnets\nAs discussed above, the main tasks for HTP screen-\ning of permanent magnets are to evaluate the intrinsic\nmagnetic properties for the known and predicted phases,\nincluding both RE-free and RE-based systems. A recent\nwork564focuses on the RE-free cases, where starting from\n10,000 compounds containing Cr, Mn, Fe, Co, and Ni\navailable in the ICSD database, step-by-step screening is\ncarried out based on the chemical composition, M s, crys-\ntal structure, magnetic ground state, and MAE. Three\ncompounds, namely, Pt 2FeNi, Pt 2FeCu, and W 2FeB 2\nare \fnally recommended. It is noted that the magnetic\nground state for the compounds during the screening is\ndetermined by the literature survey, which might be la-\nbor intensive and de\fnitely cannot work for \\unreported\"\ncases.\nIn addition to HTP characterization of the known\nphases and prediction of novel compounds, tailoring the\nexisting phases on the boundary of being permanent\nmagnets is also promising, which can be done via sub-\nstitutional and interstitial doping. For instance, theoret-\nical predictions suggest that Ir/Re doping can enhanced\nthe MAE of (Fe 1\u0000xCox)2B, which is con\frmed by ex-\nperiments.565Nevertheless, cautions are deserved when\nevaluating the MAE for the doped compounds using the\nsupercell method, where the symmetry of the solution\nphases may be broken thus leading to erroneous predic-\ntions on MAE. Special techniques such as VCA and CPA\nare needed to obtain MAE as done in Ref. 565, which are\nnot generally available in the main stream codes. Fur-\nthermore, the interstitial atoms such as H, B, C, and N\ncan also be incorporated into the existing intermetallic\nphases, giving rise to structural distortions and thus sig-\nni\fcant MAE as demonstrated for the FeCo alloys.566,567\nFollowing this line, we have performed HTP calculations\nto investigate the e\u000bects of interstitials on the intermetal-\nlic compounds of both the Cu 3Au82and Heusler541types\nof crystal structures, and identi\fed quite a few candidates\ninteresting for further exploration.\nTurning now to the RE-based permanent magnets,\ndue to the challenge to treat the correlated 4 f-electrons,\napproximations are usually made. For instance, in a\nseries of work, the tight-binding linear mu\u000en-tin or-\nbital method in the atom sphere approximation has\nbeen applied to screen the 1-5/2-17/2-14-1 type,5681-12 type,569,5701-11-X type,173and the 1-13-X type571\nclasses of materials, where the 4 f-electrons are consid-\nered in the spin-polarized core approximation. One ap-\nparent problem is that the magnetization and MAE can-\nnot be obtained consistently based on such a method.\nThe Staunton group has developed a consistent theoret-\nical framework to evaluate both the magnetization and\nMAE,572which is interesting if further HTP calculations\ncan be performed. Additionally, another common prob-\nlem to access the thermodynamic stability of the RE-\nbased compounds with most DFT codes on the market.\nNevertheless, the trend with respect to chemical com-\npositions can be obtained, which can be validated with\nfurther experiments. Last but not least, the spin mo-\nments of the RE elements is usually aligned antiparallel\nto the magnetic moment of the transition metal atoms,\nwhich is usually not properly treated. For such materials,\nnot only the single ion MAE but also inter-sublattice ex-\nchange coupling should be included in order to compare\nwith experimental measurements.573\nTherefore, due to the challenges such as the correlated\nnature of 4f-electrons and its interplay with SOC, HTP\ndesign of permanent magnets is still a subject requiring\nfurther improvement on the computing methodology. As\nsuch materials are applied in big volume, it might be\nstrategic to consider \frstly the criticality of the consti-\ntute elements, e.g., the high cost and supply risk on Co,\nNd, Dy, and Tb. In this regard, compounds based on Fe\nand Mn combined with cheap elements in the periodic\ntable should be considered with high priority. In addi-\ntion, all three intrinsic properties (M s, MAE, and T C)\nshould be optimized which is numerically expensive and\nthus not usually done. In this regard, a consistent \fg-\nure of merit should be adopted, e.g., the suggestion by\nCoey to de\fne and optimize the dimensionless magnetic\nhardness parameter \u0014=\u0010\nK1\n\u00160M2s\u00111=2\n>1, which should\nbe implemented for future HTP screening.29\nD. Magnetocaloric materials\nAs discussed in Sect. IV F, MCE is driven by the in-\nterplay of various degrees of freedom leading to chal-\nlenges in performing HTP design of novel MCE mate-\nrials. For instance, rigorous evaluation of the Gibbs free\nenergies including the lattice, spin, and electronic degrees\nof freedom is a numerically expensive task, not to men-\ntion the complex nature of magnetism and phase transi-\ntion. Thus, it is of great impetus to establish a compu-\ntational \\proxy\" which correlates the MCE performance\nwith quantities easily accessible via DFT calculations.\nFollowing the concept that MCE is signi\fcant when the\nmagneto-structural phase transitions occur, Bocarsly et\nal.proposed a proxy in terms of the magnetic deforma-\ntion \u0006M=1\n3(\u00112\n1+\u00112\n2+\u00113\n3)1=2\u0002100 and \u0011=1\n2(PTP\u0000I)\nwhere P=A\u00001\nnonmag\u0001AmagwithAnonmag andAmagbe-\ning the lattice constants of the nonmagnetic and mag-31\nnetic unit cells.574Assuming the high temperature para-\nmagnetic phase can be described with the nonmagnetic\nsolution at 0K, the magnetic deformation \u0006 Mmeasures\n\\the degree to which structural and magnetic degrees of\nfreedom are coupled in a material \". In fact, there is\na universal correlation between \u0001 Sand \u0006M, no mat-\nter whether MCE is driven by FOPTs or not. Nev-\nertheless, there is no direct scaling between \u0001 Sand\n\u0006Mand it is suggested that \u0006 M>1:5% is a reason-\nable cuto\u000b to select the promising compounds. Further\nscreening on the known FM materials reveals 30 com-\npounds out of 134 systems as good candidates, where\none of them MnCoP is validated by experimental mea-\nsurements showing \u0001 S=\u00003:1J=kg=K under an applied\n\feld of 2T. Recently, this proxy has been successfully\napplied to evaluate the MCE behavior of two solid so-\nlutions Mn(Co 1\u0000xFex)Ge and (Mn 1\u0000yNiy)CoGe, where\nthe predicted optimal compositions x=0.2 and y=0.1 are\nin good agreement with the experiments.562It is inter-\nesting that such consistency is obtained by evaluating\nthe corresponding quantities via con\fgurational average\nover supercells with Boltzmann weights, instead of the\nSQS method. One point to be veri\fed is whether the\nnonmagnetic instead of the paramagnetic con\fgurations\nare good enough. Our preliminary results on a serious of\nAPVs with noncollinear magnetic ground states indicate\nthat the negative thermal expansion associated with the\nmagneto-structural transition and hence MCE is over-\nestimated using the magnetic deformation between FM\nand nonmagnetic states.\nA more systematic and computationally involved work-\n\row is the CaloriCool approach,237based on two-step\nscreening. It is suggested that the metallic alloys and\nintermetallic compounds are more promising candidates\nthan the oxides, which \\ su\u000ber from intrinsically low isen-\ntropic (a.k.a. adiabatic) temperature changes due to large\nmolar lattice speci\fc heat \". The \frst step fast screening\nis based on the phase diagrams and crystal structures\nfrom the literature and known databases, which results\nin compounds with the same chemical composition but\ndi\u000berent crystal structures. In the second step, the phys-\nical properties including mechanical, electronic, thermo-\ndynamic, kinetic properties will be evaluated, c ombin-\ning DFT with multi-scale and thermodynamic methods.\nFor instance, the thermodynamic properties can be evalu-\nated following the methods proposed in Ref. 575, which is\nforeseeably expensive and better done one after another.\nAlso, the success of the approach depends signi\fcantly\non the database used for the \frst step screening. Nev-\nertheless, it is proposed based on such design principles\nthat there should be a solution phase Zr 1\u0000xYbxMn6Sn6\nwith signi\fcant MCE due to the fact that the pristine\nZrMn 6Sn6and YbMn 6Sn6are with AFM and FM ground\nstates with critical temperatures being 580K and 300K,\nrespectively. This is very similar to the morphotropic\nphase boundary concept for ferroelectric materials, where\nthe piezoelectric response associated with \frst order fer-\nroelectric phase transition can be greatly enhanced at thecritical compositions.576\nE. Topological materials\nAs discussed in Sect. IV H, insulators and semimet-\nals of nontrivial nature are an emergent class of materi-\nals from both fundamental physics and practical appli-\ncations points of view. To design such materials, sym-\nmetry plays an essential role as explained in detail in\nSect. IV H for a few speci\fc compounds. Particularly for\nmagnetic materials, the occurrence of QAHI and Weyl\nsemimetals is notconstrained to compounds of speci\fc\nsymmetries, whereas the AFM TIs entail a product sym-\nmetryS=P\u0002 which can give rise to the demanded\nKramers degeneracy. It is noted that the symmetry ar-\ngument applies to the nonmagnetic topological materi-\nals as well, where based on topological quantum chem-\nistry577and \flling constraints578,579HTP characteriza-\ntion of nonmagnetic materials have been systematically\nperformed.580{583Such screening can also be performed\nin a more brute-force way by calculating the surface\nstates584and spin-orbit spillage.585,586\nA very fascinating work done recently is to perform\nHTP screening of AFM topological materials based on\nthe magnetic topological quantum chemistry,587as an ex-\ntension to the topological quantum chemistry.577For this\napproach, the irreducible co-representations of the occu-\npied states are evaluated at the high symmetry points\nin the BZ, and the so-called compatibility relations are\nthen taken to judge whether the compounds are topo-\nlogically trivial or not.588In this way, six categories of\nband structures can be de\fned, e.g. band representa-\ntions, enforced semimetal with Fermi degeneracy, en-\nforced semimetal, Smith-index semimetal, strong TI, and\nfragile TI, where only band representations are topolog-\nically trivial. Out of 403 well converged cases based\non DFT calculations of 707 compounds with experi-\nmentally available AFM magnetic structure collected in\nthe MAGNDATA database,589it is observed that about\n130 (\u001932%) showing nontrivial topological features,\nwhere NpBi, CaFe 2As2, NpSe, CeCo 2P2, MnGeO 3, and\nMn3ZnC are the most promising cases. We note that\none uncertainty is the U value which is essential to get\nthe correct band structure, and more sophisticated meth-\nods such as DFT+DMFT might be needed to properly\naccount for the correlations e\u000bect.184\nF. 2D magnets\nThe HTP screening of 2D magnetic materials has been\ninitiated by examining \frstly the bulk materials that are\nheld together by the van der Waals interaction, which can\nthen be possibly obtained by chemical/mechanical exfo-\nliation or deposition. Starting from the inorganic com-\npounds in the ICSD database,590two recent publications\ntried to identify 2D materials by evaluating the packing32\nratio, which is de\fned as the ratio of the covalent volume\nand the total volume of the unit cell.591,592This leads to\nabout 90 compounds which can be obtained in the 2D\nform, where about 10 compounds are found to be mag-\nnetic.592The screening criteria have then be generalized\nusing the so-called topological-scaling algorithm, which\nclassi\fes the atoms into bonded clusters and further the\ndimensionality.498Combined with the DFT evaluation\nof the exfoliation energy, 2D materials can be identi\fed\nfrom the layered solids. This algorithm has been ap-\nplied to compounds in the Materials Project database,\nwhere 826 distinct 2D materials are obtained. Interest-\ningly, 128 of them are with \fnite magnetic moments >\n1.0\u0016B/u.c., and 30 of them showing HM behavior.498\nExtended calculations starting from the compounds in\nthe combined ICSD and COD databases identi\fed 1825\npossible 2D materials, and 58 magnetic monolayers have\nbeen found out of 258 most promising cases.499Similarly,\nby evaluating the number of \\covalently connected atoms\nparticularly the ratio in supercell and primitive cells, 45\ncompounds are identi\fed to be of layered structures out\nof 3688 systems containing one of (V, Cr, Mn, Fe, Co,\nNi) in the ICSD database, leading to 15 magnetic 2D\nmaterials.593\nAs discussed in Sect. IV I, the occurrence of 2D mag-\nnetism is a tricky problem, particularly the magnetic or-\ndering temperature which is driven by the interplay of\nmagnetic anisotropy and exchange parameters. Based\non the computational 2D materials database (C2DB),500\nHTP calculations are performed to obtain the exchange\nparameters, MAE, and the critical temperature for 550\n2D materials, and it is found that there are about\n150 (50) FM (AFM) compounds being stable.502Impor-\ntantly, the critical temperatures for such compounds have\na strong dependence on the values of U, indicating that\nfurther experimental validation is indispensable. Never-\ntheless, the characterization of the other properties for\n2D materials in the HTP manner is still limited. One\nexception is the HTP screening for QAHC with in-plane\nmagnetization,594where the prototype LaCl might not\nbe a stable 2D compound. It is noted that there have\nbeen a big number of predictions on 2D magnetic and\nnonmagnetic materials hosting exotic properties but the\nfeasibility to obtain such compounds has not been sys-\ntematically addressed. Therefore, we suspect that consis-\ntent evaluations of both the stability and physical prop-\nerties are still missing, particularly in the HTP manner\nwhich can guide and get validated by future experiments.\nDespite the existing problems with stability for 2D\n(magnetic) materials, a system work\row to character-\nize their properties has been demonstrated in a recent\nwork.595Focusing the FM cases and particularly the pre-\ndiction of associated T C, it is found that only 53% of\n786 compounds predicted to be FM,500are actually sta-\nble against AFM con\fgurations generated automatically\nbased on the method developed in Ref. 84. In this regard,\nthe parameterization of the Heisenberg exchange param-\neters should be scrutinized, where explicit comparison ofthe total energies for various magnetic con\fgurations can\nbe valuable. Follow-up Monte Carlo simulations based on\n\ftted exchange parameters reveal 26 materials out of 157\nwould exhibit T Chigher than 400 K, and the results are\nmodeled using ML with an accuracy of 73%. Unfortu-\nnately, the exact AFM ground states are not addressed,\nwhich is interesting for future investigation.\nIt is well known that 2D materials o\u000ber an intrigu-\ning playground for various topological phases particularly\nfor magnetic materials including the QAHE594, AFM\nTI596, and semimetals.417HTP calculations based on\nthe spin-orbit spillage585has been carried out to screen\nfor magnetic and nonmagnetic topological materials in\n2D materials,597resulting in four insulators with QAHE\nand seven magnetic semimetals. Such calculations are\ndone on about 1000 compounds in the JARVIS database\n(https://www.ctcms.nist.gov/ \u0018knc6/JVASP.html), and\nwe suspect that more extensive calculations together with\nscreening on the magnetic ground state can be interest-\ning.\nBeyond the calculations across various crystal struc-\nture prototypes, HTP calculations on materials of the\nsame structural type substituted by di\u000berent elements\nprovide also promising candidates and insights on the\n2D magnetism.598For instance, Chittari et al. per-\nformed calculations on 54 compounds of the MAX 3type,\nwhere M = V, Cr, Mn, Fe, Co, Ni, A = Si, Ge, Sn,\nand X = S, Se, Te. It is observed that most of them\nare magnetic, hosting HMs, narrow gap semiconductors,\netc. Importantly, strain is found to be e\u000bective to tai-\nlor the competition between AFM and FM ordering in\nsuch compounds.599Similar observations are observed in\nin-plane ordered MXene (i-MXene),600which can be de-\nrived from the in-plane ordered MAX compounds with\nnano-laminated structures.601Moreover, our calculations\non the 2D materials of the AB 2(A being TM and B =\nCl, Br, I) indicate that FeI 2is a special case adopting the\n2H-type structure while all the neighboring cases have\nthe 1T-type.602This might be related to an exotic elec-\ntronic state as linear combination of the t2gorbitals in\nFeI2driven by the strong SOC of I atoms.603We found\nthat the band structure of such AB 2compounds is very\nsensitive to the value of the e\u000bective local Coulomb in-\nteraction U applied on the d-bands of TM atoms. This\nis by the way a common problem for calculations on 2D\n(magnetic) materials, where the screening of Coulomb in-\nteraction behave signi\fcantly from that in 3D bulk ma-\nterials.604\nVI. FUTURE PERSPECTIVES\nA. Multi-scale modelling\nAs discussed in previous sections, HTP calculations\nbased on DFT are capable of evaluating the intrinsic\nphysical properties, which usually set the upper limits\nfor practical performance. To make more realistic pre-33\ndictions and to directly validate with the experiments,\nmulti-scale modelling is required. The \frst issue is to\ntackle the thermodynamic properties which can be ob-\ntained by evaluating the Gibbs free energy considering\nthe contributions from the electronic, lattice, and spin de-\ngrees of freedom. Moreover, the applications of magnetic\nmaterials at elevated temperature dictate the numerical\ncalculations of the magnetic properties at \fnite temper-\nature as well, e.g., MAE at \fnite temperature.116The\nelectronic structure will also be renormalized driven by\nthe thermal \ructuations, e.g., temperature driven topo-\nlogical phase transition in Bi 2Se3.605\nThe major challenge for multi-scale modelling is how\nto make quantitative predictions for materials with struc-\ntural features of large length scales beyond the unit cells,\nincluding both topological defects ( e.g., domain wall) and\nstructural defects ( e.g., grain boundary). Taking per-\nmanent magnets as an example, as indicated in Fig. 1,\nboth the M rand Hacannot reach the corresponding the-\noretical limit of M sand MAE. MAE (represented by\nthe lowest-order uniaxial anisotropy K) sets an upper\nlimit for the intrinsic coercivity 2 K=(\u00160Ms) as for single-\ndomain particles via coherent switching,606whereas the\nreal coercivity is given by Ha=\u000b2K=(\u00160Ms)\u0000\fMs,\nwhere\u000band\fare the phenomenological parameters,\ndriven by complex magnetization switching processes\nmostly determined by the microstructures. It is noted\nthat\u000bis smaller than one (about 0.1-0.3) due to the\nextrinsic mechanisms such as inhomogeneity, misaligned\ngrains, etc., and \faccounts for the local demagnetiza-\ntion e\u000bects.607This leads to the so-called Brown's para-\ndox.608That is, the magnetization switching cannot be\nconsidered as ideally a uniform rotation of the magnetic\nmoments in a single domain, but is signi\fcantly in\ru-\nenced by the extrinsic processes such as nucleation and\ndomain wall pinning,609which are associated with struc-\ntural imperfections at nano-, micro-, and macroscopic\nscales.607Therefore, in addition to screening for novel\ncandidates for permanent magnets, the majority of the\ncurrent research are focusing on how to overcome the\nlimitations imposed by the microstructures across sev-\neral length scales.610,611\nFurthermore, the functionalities of ferroic materials are\nmostly enhanced when approaching/crossing the phase\ntransition boundary, particularly for magneto-structural\ntransitions of the \frst-order nature. Upon such struc-\ntural phase transitions, microstructures will be developed\nto accommodate the crystal structure change, e.g., for-\nmation of twin domains due to the loss of point-group\nsymmetry. Moreover, most FOPT occur without long-\nrange atomic di\u000busion, i.e., of the martensitic type, lead-\ning to intriguing and complex kinetics. As indicated\nabove, what is characteristic to the athermal FOPTs is\nthe hysteresis, leading to signi\fcant energy loss converted\nto waste heat during the cooling cycle for MCE materi-\nals.33While hysteresis typically arises as a consequence of\nnucleation, in caloric materials it occurs primarily due to\ndomain-wall pinning, which is the net result of long-rangeelastic strain associated with phase transitions of inter-\nest. When the hysteresis is too large, the reversibility of\nMCE can be hindered. Therefore, there is a great im-\npetus to decipher the microstructures developed during\nthe magneto-structural transitions, particularly the nu-\ncleation and growth of coexisting martensitic and austen-\nite phases. For instance, it is observed recently that ma-\nterials satisfying the geometric compatibility condition\ntend to exhibit lower hysteresis and thus high reversibil-\nity, leading to further stronger cofactor conditions.612\nTherefore, in order to engineer magnetic materials for\npractical applications, multi-scale modelling is indispens-\nable. To this goal, accurate DFT calculations can be per-\nformed to obtain essential parameters such as the Heisen-\nberg exchange, DMI, SOT, and exchange bias, which\nwill be fed into the atomistic108and micromagnetic mod-\nelling.22It is noted that such scale-bridging modelling is\nrequired to develop fundamental understanding of spin-\ntronic devices as well. For instance, for the SOT memris-\ntors based on the heterostructures composing AFM and\nFM materials,275the key problems are to tackle the origin\nof the memristor-like switching behavior driven possibly\nby the interplay of FM domain wall propagation/pinning\nwith the randomly distributed AFM crystalline grains,\nand to further engineer the interfacial coupling via com-\nbinatorial material combinations for optimal device per-\nformance.\nGiven the fact that the computational facilities have\nbeen signi\fcantly improved nowadays, HTP predictions\ncan be straightforwardly made, sometimes too fast in\ncomparison to the more time-consuming experimental\nvalidation. In this regard, HTP experiments are valu-\nable. Goll et al. proposed to use reaction sintering as\nan accelerating approach to develop new magnetic ma-\nterials,613which provides a promising solution to verify\nthe theoretical predictions and to further fabricate the\nmaterials. Such a HTP approach has also been imple-\nmented to screen over the compositional space, in order\nto achieve optimal MSME performance.614Nevertheless,\nwhat is really important is a mutual responsive frame-\nwork combining accurate theoretical calculations and ef-\n\fcient experimental validation.\nB. Machine learning\nAnother emergent \feld is materials informatics615\nbased on advanced analytical machine learning tech-\nniques, which can be implemented as the fourth\nparadigm616to map out the process-(micro)structure-\nproperty relationship and thus to accelerate the devel-\nopment of materials including the magnetic ones. As the\nunderlying machine learning is data hungry, it is critical\nto curate and manage databases. Although there is in-\ncreasing availability of materials database such as Mate-\nrials Project,8OQMD,617and NOMAD,10the sheer lack\nof data is currently a limiting factor. For instance, the\nexisting databases mentioned above compile mostly the34\nchemical compositions and crystal structures, whereas\nthe physical properties particularly the experimentally\nmeasured results are missing. In this regard, a recent\nwork trying to collect the experimental T Cof FM materi-\nals is very interesting, which is achieved based on natural\nlanguage processing on the collected literature.618\nFollowing this line, machine learning has been success-\nfully applied to model the T Cof FM materials, which\nis still a challenge for explicit DFT calculations as dis-\ncussed above. Dam et al. collected the T Cfor 108 bi-\nnary RE-3d magnets and the database is regressed using\nthe random forest algorithm.61927 empirical features are\ntaken as descriptors with detailed analysis on the feature\nrelevance. Two recent work620,621started with the Atom-\nWork database622and used more universal chemical and\nstructural descriptors. It is observed that the Curie tem-\nperature is mostly driven by the chemical composition,\nwhere compounds with polymorphs are to be studied in\ndetail with structural features. The other properties such\nas MAE623and magnetocaloric performance624can also\nbe \ftted, which makes it very interesting for the future.\nHTP calculations generate a lot of data which can\nbe further explored using machine learning. This has\nbeen performed on various kinds of materials and phys-\nical properties, as summarized in a recent review.625\nSuch machine learning modelling has also been applied\non magnetic materials. For instance, based on the for-\nmation energies calculated, machine learning has been\nused to model not only the thermodynamic stability of\nthe full Heusler,526,626half Heusler,627and quaternary\nHeusler628compounds, but also the spin polarization of\nsuch systems.629\nIn this sense, machine learning o\u000bers a straightfor-\nward solution to model many intrinsic properties of mag-\nnetic materials, but is still constrained by the lack of\ndatabases. For instance, the T Cand TNare collected for\nabout 10,000 compounds in the AtomWork database,622\nwith signi\fcant uncertainty for compounds with multi-\nple experimental values thus should be used with cau-\ntion. In addition, both machine learning modellings of\ntheTC620,621are done based on the random forest al-\ngorithm, whereas our test using the Gaussian kernel re-\ngression method leads to less satisfactory accuracy. This\nimplies that the data are quite heterogeneous. Although\nour modelling can distinguish the FM and AFM ground\nstate, it is still unclear about how to predict exact AFM\ncon\fguration, which requires a database of magnetic\nstructures. To the best of our knowledge, there is only\none collection of AFM structures available in the MAGN-\nDATA database, where there are 1100 compounds listed\nwith the corresponding AFM ground states.589There-\nfore, there is a strong impetus to sort out the information\nin the literature and to compile a database of magnetic\nmaterials.\nImportantly, it is suspected that the interplay of\nmachine learning and multi-scale modelling will create\neven more signi\fcant impacts on identifying the process-\nstructure-property relationships. For instance, phase\feld modelling aided with computational thermodynam-\nics can be applied to simulate the microstructure evo-\nlution at the mesoscopic scale for both bulk and inter-\nfaces,630leading to reliable process-structure mapping.\nAs demonstrated in a recent work,631the structure-\nproperty connection for permanent magnets can also\nbe established based on massive micromagnetic simula-\ntions. Thus, HTP calculations performed in a quanti-\ntative way can be applied to generate a large database\nwhere the process-structure-property relationship can be\nfurther addressed via machine learning, as there is no uni-\n\fed formalism based on math or physics to de\fne such\nlinkages explicitly. Additionally, machine learning in-\nteratomic potentials with chemistry accuracy have been\nconstructed and tested on many materials systems,632\nwhich are helpful to bridge the DFT and molecular dy-\nnamics simulations at larger length scales. We believe an\nextension of such a scheme to parameterize the Heisen-\nberg Hamiltonian (Eq. (IV B)) with on-top spin-lattice\ndynamics simulations121,223will be valuable to obtain ac-\ncurate evaluation of the thermodynamic properties for\nmagnetic materials.\nFrom the experimental perspective, adaptive design\n(i.e., active learning) based on surrogate models has been\nsuccessfully applied to guide the optimization of NiTi\nshape memory alloys633and BaTiO 3-based piezoelectric\nmaterials.634Such methods based on Bayesian optimiza-\ntion or Gaussian process work on small sample sizes but\nwith a large space of features, resulting in guidance on\nexperimental prioritization with greatly reduced number\nof experiments. This method can hopefully be applied to\noptimizing the performance of the 2-14-1 type magnets,\nas the phase diagram and underlying magnetic switch-\ning mechanism based on nucleation is well understood.\nFurthermore, the machine learning techniques are valu-\nable to automatize advanced characterization. Small an-\ngle neutron scattering is a powerful tool to determine\nthe microstructure of magnetic materials, where machine\nlearning can be applied to accelerate the experiments.635\nMeasurements on the spectral properties such as x-ray\nmagnetic circular dichroism (XMCD) can also be ana-\nlyzed on-the-\ry in an automated way based on Gaussian\nprocess modelling.636In this regard, the hyperspectral\nimages obtained in scanning transmission electronic mi-\ncroscopy can be exploited by mapping the local atomic\npositions and phase decomposition driven by thermody-\nnamics. That is, not only the local structures but also the\npossible grain boundary phases assisted with di\u000braction\nand EELS spectra can be obtained, as recently achieved\nin x-ray spectro-microscopy.637,638\nVII. SUMMARY\nIn conclusion, despite the magnetism and magnetic\nmaterials have been investigated based on quantum me-\nchanics for almost a hundred years (if properly) marked\nby the discovery of electron spin in 1928, it is fair to say35\nFIG. 11: Theoretical framework for future materials design\nthat they are not under full control of us due to the lack\nof thorough understanding, as we are facing fundamental\ndevelopments marked by progresses on AFM spintronics,\nmagnetic topological materials, and 2D magnets, and we\nare exposed to complex multi-scale problems in magneti-\nzation reversal and magneto-structural phase transitions.\nThus, we believe systematic calculations based on the\nstate-of-the-art \frst-principles methods on an extensive\nlist of compounds will at least get the pending issues bet-\nter de\fned.\nAmong them, we would emphasize a few urgent and\nimportant aspects as follows,\n•integrating the magnetic ground state searching\ninto the available HTP platforms so that auto-\nmated work\row can be de\fned. There are sev-\neral solutions developed recently as discussed in\nSect. IV C;\n•implementation of a feasible DFT+DMFT frame-\nwork for medium- (if not high-) throughput calcu-\nlations. Two essential aspects are the interplay of\nCoulomb interaction, SOC, and hybridization and\nthe treatment of magnetic \ructuations and param-\nagnetic states. This is also important for 2D mag-\nnets with enhanced quantum and thermal \ructua-\ntions;\n•consistent framework to evaluate the transport\nproperties which can be achieved based on accu-\nrate tight-binding-like models using Wannier func-\ntions;639\n•quantitative multi-scale modelling of the magneti-\nzation reversal processes and the thermodynamic\nproperties upon magneto-structural transitions inorder to master hysteresis for permanent magnets\nand magnetocaloric materials;\n•curation and management of a \rexible database of\nmagnetic materials, ready for machine learning.\nAdditionally, a common problem not only applica-\nble for magnetic materials but also generally true for\nthe other classes of materials subjected to property-\noptimization is a consistent way to consider substitution\nin both the dilute and concentrated limits. There are\nwidely used methods such as virtual crystal approxima-\ntion, coherent potential approximation,640special quasir-\nandom structures,117cluster expansion,641etc., but we\nhave not yet seen HTP calculations done systematically\nusing such methods to screen for substitutional optimiza-\ntion of magnetic materials.\nBeyond the bare HTP design, we envision a strong\ninterplay between HTP, machine learning, and multi-\nscale modelling, as sketched in Fig. 11. Machine learn-\ning is known to be data hungry, where HTP calculations\nbased on DFT can provide enough data. In addition\nto the current forward predictions combining HTP and\nML,625we foresee that inverse design can be realized,642\nas highlighted by a recent work on prediction new crystal\nstructures.643As mentioned above, the marriage between\nHTP and multi-scale modeling helps to construct accu-\nrate atomistic models and thus to make the multi-scale\nmodelling more quantitative and predictive. Last but\nnot least, it is suspected that the interaction between\nmachine learning and multi-scale modelling will be valu-\nable to understand the microstructures, dubbed as mi-\ncrostructure informatics.644Such a theoretical framework\nis generic, i.e., applicable to not only magnetic materials\nbut also the other functional and structural materials.\nAll in all, HTP calculations based on DFT are valuable\nto provide e\u000bective screening on interesting achievable\ncompounds with promising properties, as demonstrated\nfor magnetic cases in this review. As the HTP method-\nology has been mostly developed in the last decade, we\nhave still quite a few problems to be solved in order to get\ncalculations done properly for magnetic materials, in or-\nder to be predictive. 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A variety of quantum magnet s show novel magnetic correlations\nsuch as quantum spin liquids. These novel magnetic correlat ions are beyond the direct detection\nof INS. In this paper we propose a coincidence technique, coi ncidence inelastic neutron scattering\n(cINS), which can detect the two-spin magnetic correlation s of the magnetic materials. In cINS\nthere are two neutron sources and two neutron detectors with an additional coincidence detector.\nTwo neutrons from the two neutron sources are incident on the target magnetic material, and\nthey are scattered by the electron spins of the magnetic mate rial. The two scattered neutrons are\ndetected by the two neutron detectors in coincidence with th e coincidence probability described by\na two-spin Bethe-Salpeter wave function. Since the two-spi n Bethe-Salpeter wave function defines\nthe momentum-resolved dynamical wave function with two spi ns excited, cINS can explicitly detect\nthe two-spin magnetic correlations of the magnetic materia l. Thus, it can be introduced to study\nthe various spin valence bond states of the quantum magnets.\nI. INTRODUCTION\nThe novel magnetic correlations in various quantum\nmagnetshaveattractedmuchattentionin thecondensed-\nmatter field. Quantum spin liquids with strong frustra-\ntion and quantum fluctuations are one special type of\nexample1–4. One experimental technique in the study\nof these novel magnetic correlations is inelastic neutron\nscattering (INS), which can provide the single-spin dy-\nnamical responses of magnetic materials and thus can\nshow the relevant physics of single-spin excitations5–9.\nHowever, as most novel magnetic correlations in the\nquantum magnets are beyond that of the single-spin\nmagnons, the spectrum of INS cannot provide explicit\ninformation on these novel magnetic correlations. It is\nimperative to develop experimental techniques which can\nexplicitly detect these novel magnetic correlations.\nRecently, coincidence angle-resolved photoemission\nspectroscopy (cARPES) was proposed for detection of\ntwo-particle correlations of material electrons10. In this\npaper we will follow the idea of cARPES to propose\nanother coincidence technique, coincidence INS (cINS),\nwhich can explicitly detect the two-spin magnetic cor-\nrelations of magnetic materials. There are two neutron\nsources and two neutron detectors in the experimental\ninstrument of cINS, with an additional coincidence de-\ntector. The twoneutron sourcesemit twoneutronswhich\nareincidentonthetargetmagneticmaterialandarescat-\ntered by the material electron spins. These two scattered\nneutrons are then detected by the two neutron detectors\nin coincidence with the coincidence probability relevant\nto a two-spin Bethe-Salpeter wave function.\nThe two-spin Bethe-Salpeter wave function is defined\nas\nφ(ij)\nαβ(q1t1,q2t2) =/an}b∇acketle{tΨβ|TtS(i)\n⊥(q1,t1)S(j)\n⊥(q2,t2)|Ψα/an}b∇acket∇i}ht,\n(1)\nwhere|Ψα/an}b∇acket∇i}htand|Ψβ/an}b∇acket∇i}htare the eigenstates of the electron\nspins of the target magnetic material, S(i)\n⊥(q,t) is theith component of the spin operator within a perpendic-\nular plane normal to the momentum q, andTtis a time-\nordering operator. This Bethe-Salpeter wave function\ndescribes the time dynamical evolution of the magnetic\nmaterial with two spins excited at times t1andt2in\ntime ordering. The coincidence probability of cINS can\nprovide the Fourier transformation of the time dynami-\ncalBethe-Salpeterwavefunction, withthecenter-of-mass\nfrequency defined by the sum of the two transfer energies\nin the two-neutron scattering and the relative frequency\ndefined by the difference of the two transfer energies.\nTherefore, the coincidence detection of cINS can pro-\nvide the momentum-resolved dynamics of the two-spin\nmagnetic correlations, with the physics of both the cen-\nter of mass and the relative degrees of freedom of two\nexcited spins of the magnetic material. Thus, it can be\nintroduced to study the spin valence bond states of the\nquantum magnets.\nOur paper is organized as follows. In Sec. II the the-\noretical formalism of the coincidence detection of cINS\nwill be provided. In Sec. III the coincidence probabilities\nof cINS for a ferromagnet and an antiferromagnet with\nlong-range magnetic order will be presented. Discussion\nof the experimental detection of cINS will be given in\nSec. IV, where a brief summary will also be provided.\nII. THEORETICAL FORMALISM FOR cINS\nIn this section we will establish the theoretical formal-\nism for the coincidence detection of cINS. First, we will\nreview the principle of the single-spin INS in Sec. IIA.\nWe will then provide the theoretical formalism for cINS\nin Sec. IIB.2\nA. Review of INS\nSuppose the incident neutrons have momentum qiand\nspinβiwith a spin distribution function P1(βi). The in-\ncident neutrons interact with the electron spins of the\ntarget magnetic material via the electron-neutron mag-\nnetic interaction\n/hatwideVs=/summationdisplay\nqiqfg(q)/hatwideσqfqi·S⊥(q), (2)\nwhereg(q)≡gF0(q), withgbeing an interaction con-\nstant andF0(q) being a magnetic form factor, and q=\nqf−qi, with/hatwideq=q\nq. The operator /hatwideσqfqiis defined for\nneutrons,\n/hatwideσqfqi=/summationdisplay\nβiβfd†\nqfβfσβfβidqiβi, (3)\nwheredqβandd†\nqβare the respective neutron annihila-\ntion and creation operators and σis the Pauli matrix.\nThe electron spin operator S(q) is defined by\nS(q) =/summationdisplay\nlSle−iq·Rl,Sl=/summationdisplay\nα1α2c†\nlα2Sα2α1clα1,(4)\nwhereclαandc†\nlαaretheannihilationandcreationopera-\ntors ofthe Wannier electrons at position Rl, respectively,\nandS=σ\n2isthe spin angularmomentum operator. Here\nwe assume that the material electrons which have a dom-\ninant interaction with the incident neutrons are the local\nWannier electrons. It is noted that S⊥(q) is defined as\nS⊥(q) =S(q)−/hatwideq(S(q)·/hatwideq). (5)\nA simple review of the electron-neutron magnetic inter-\naction/hatwideVsis given in Appendix A.\nOne incident neutron with momentum qican be scat-\ntered by the material electrons into the state with mo-\nmentum qf. The relevant scattering probability is de-\nfined as\nΓ(1)(qf,qi) =1\nZ/summationdisplay\nαββiβfe−βEαP1(βi)\n×|/an}b∇acketle{tΦβ|/hatwideS(1)(+∞,−∞)|Φα/an}b∇acket∇i}ht|2,(6)\nwhere the initial state |Φα/an}b∇acket∇i}ht=|Ψα;qiβi/an}b∇acket∇i}htand the final\nstate|Φβ/an}b∇acket∇i}ht=|Ψβ;qfβf/an}b∇acket∇i}htand|Ψα/an}b∇acket∇i}htand|Ψβ/an}b∇acket∇i}htare the elec-\ntron eigenstates whose eigenvalues are EαandEβ, re-\nspectively. /hatwideS(1)(+∞,−∞) is the first-order expansion of\nthe time-evolution Smatrix ofthe perturbation electron-\nneutron magnetic interaction /hatwideVsand is defined as\n/hatwideS(1)(+∞,−∞) =−i\n/planckover2pi1/integraldisplay+∞\n−∞dt/hatwideVI(t)Fθ(t),(7)\nwhere/hatwideVI(t) =eiH0t//planckover2pi1/hatwideVse−iH0t//planckover2pi1, withH0being the sum\noftheHamiltoniansofthematerialelectronsandtheneu-\ntrons.Fθ(t) defines the interaction perturbation time,\nFθ(t) =θ(t+∆t/2)−θ(t−∆t/2),(8)whereθ(t) is the step function.\nIt should be noted that in the above scattering prob-\nability, we have defined implicitly the initial and final\nstates by the density matrices as follows:\n/hatwidePI=1\nZ/summationdisplay\nαβie−βEαP1(βi)|Ψα;qiβi/an}b∇acket∇i}ht/an}b∇acketle{tβiqi;Ψα|,\n/hatwidePF=/summationdisplay\nββf|Ψβ;qfβf/an}b∇acket∇i}ht/an}b∇acketle{tβfqf;Ψβ|. (9)\nInthispaperwewillfocusonthecaseswheretheincident\nneutronsarethethermalneutronsinthe spinmixedstate\ndefined by\n/summationdisplay\nβiP1(βi)|βi/an}b∇acket∇i}ht/an}b∇acketle{tβi|=1\n2(| ↑/an}b∇acket∇i}ht/an}b∇acketle{t↑ |+| ↓/an}b∇acket∇i}ht/an}b∇acketle{t↓ |).(10)\nWe introduce an imaginary-time Green’s function\nG(q,τ) =−/summationtext\nij/an}b∇acketle{tTτSi(q,τ)S†\nj(q,0)/an}b∇acket∇i}ht(δij−/hatwideqi/hatwideqj). Its\ncorresponding spectrum function χ(q,E) is defined as\nχ(q,E) =−2 ImG(q,iνn→E+iδ+), which follows\nχ(q,E) =2π\nZ/summationdisplay\nαβije−βEα/an}b∇acketle{tΨα|S†\ni(q)|Ψβ/an}b∇acket∇i}ht/an}b∇acketle{tΨβ|Sj(q)|Ψα/an}b∇acket∇i}ht\n×(δij−/hatwideqi/hatwideqj)n−1\nB(E)δ(E+Eβ−Eα).(11)\nThe scattering probability can easily be shown to follow\nΓ(1)(qf,qi) =|g(q)|2∆t\n/planckover2pi1χ(q,E(1))nB(E(1)),(12)\nwhere the transfer momentum and energy are defined as\nq=qf−qi,E(1)=E(qf)−E(qi),(13)\nwithE(q) =(/planckover2pi1q)2\n2mn(mnis the neutron mass), and nB(E)\nistheBosedistributionfunction. Intheabovederivation,\nwe have assumed that the time interval ∆ tis large and\nsin2(ax)\nx2→πaδ(x) whena→+∞.\nLet us consider the scattering cross section. We define\nthe incident neutron flux by JI=nIvI, where the den-\nsitynI=1\nVI(VIis the renormalization volume for one\nneutron) and the velocity vI=/planckover2pi1qi\nmn. The scattering cross\nsection per scatter σfollows\nJIσ=1\nNm∆t/summationdisplay\nqfΓ(1)(qf,qi), (14)\nwhereNmis the number of scatter electrons in the in-\ncident neutron beam. The double-differential scattering\ncross section is shown to follow\nd2σ\ndΩdEf=(γRe)2\n2πNmqf\nqi|F0(q)|2χ(q,E(1))nB(E(1)),(15)\nwhereEfis the energy of the scattered neutrons, γ=\n1.91 is a constant for the neutron gyromagnetic ratio,\nandReis the classical electron radius, defined as\nRe=µ0e2\n4πme=e2\n4πε0mec2, (16)3\nwithµ0being the free-space permeability and ε0being\nthe vacuum permittivity. This double-differential cross\nsectionwe haveobtainedis the same asthat from Fermi’s\ngolden rule5–7. Physically, the scattering probability and\nthe scattering cross section of INS come from the con-\ntribution of the first-order perturbation of the electron-\nneutron magnetic interaction.\nB. Theoretical formalism for cINS\nIn this section we will present a coincidence technique,\ncoincidence inelastic neutron scattering, which we call\ncINS.Itisproposedforthedetectionofthetwo-spinmag-\nnetic correlations of the target magnetic material. The\nschematic diagram of cINS is shown in Fig. 1. There are\ntwo neutron sources which emit two neutrons with mo-\nmentaqi1andqi2. These two neutrons are incident on\nthe target magnetic material and interact with the elec-\ntron spins. The two incident neutrons are then scattered\noutside of the material into the states with momenta qf1\nandqf2. Two single-neutron detectors detect the two\nscattered neutrons, and a coincidence detector records\nthe coincidence counting probability when each of the\ntwo single-neutron detectors detects one single neutron\nsimultaneously.\nFIG. 1: (Color online) Schematic diagram of cINS. The two\nred dashed lines represent two incident neutrons, and the tw o\ngreen solid lines represent two scattered neutrons. D 1and\nD2are two single-neutron detectors, and D 12is a coincidence\ndetector which records one counting when D 1and D 2each\ndetect one single neutron simultaneously.\nThe coincidence counting probability of the two scat-\ntered neutrons is described by\nΓ(2)(qf1qf2,qi1qi2) =1\nZ/summationdisplay\nαββiβfe−βEαP2(βi1,βi2)\n×|/an}b∇acketle{tΦβ|/hatwideS(2)(+∞,−∞)|Φα/an}b∇acket∇i}ht|2,(17)\nwhere the initial state |Φα/an}b∇acket∇i}ht=|Ψα;qi1βi1qi2βi2/an}b∇acket∇i}htand the\nfinal state |Φβ/an}b∇acket∇i}ht=|Ψβ;qf1βf1qf2βf2/an}b∇acket∇i}ht.P2(βi1,βi2) de-\nfines the spin distribution function of the incident ther-\nmal neutrons. In the following, we will consider the cases\nwithP2(βi1,βi2) =P1(βi1)P1(βi2)./hatwideS(2)(+∞,−∞) is the\nsecond-order expansion of the time-evolution Smatrixand is defined by\n/hatwideS(2)(+∞,−∞)\n=1\n2!/parenleftbigg\n−i\n/planckover2pi1/parenrightbigg2/integraldisplay/integraldisplay+∞\n−∞dt1dt2Tt[/hatwideVI(t1)/hatwideVI(t2)]Fθ(t1,t2).\n(18)\nHere the time function Fθ(t1,t2) is defined as Fθ(t1,t2) =\nFθ(t1)Fθ(t2). Physically, the coincidence probability of\ncINS is determined by the second-order perturbation of\nthe electron-neutron magnetic interaction.\nFollowing the theoretical treatment for cARPES10, we\nintroduce the two-spin Bethe-Salpeter wave function de-\nfined in Eq. (1). With the two-spin Bethe-Salpeter wave\nfunction, we can show that the coincidence probability of\ncINS can be expressed as\nΓ(2)= Γ(2)\n1+Γ(2)\n2, (19)\nwhere\nΓ(2)\n1=1\nZ/summationdisplay\nαββiβfe−βEαP1(βi1)P1(βi2)\n×1\n/planckover2pi14/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay/integraldisplay+∞\n−∞dt1dt2Mαβ,1(t1,t2)Fθ(t1,t2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n,\nΓ(2)\n2=1\nZ/summationdisplay\nαββiβfe−βEαP1(βi1)P1(βi2)\n×1\n/planckover2pi14/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay/integraldisplay+∞\n−∞dt1dt2Mαβ,2(t1,t2)Fθ(t1,t2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n.\nHere the matrix elements Mαβ,1andMαβ,2are defined\nas\nMαβ,1=g(q1)g(q2)/summationdisplay\nijφ(ij)\nαβ(q1t1,q2t2)\n×σ(i)\nβf1βi1σ(j)\nβf2βi2ei(E(2)\n1t1+E(2)\n2t2)//planckover2pi1,\nMαβ,2=g(q1)g(q2)/summationdisplay\nijφ(ij)\nαβ(q1t1,q2t2)\n×σ(i)\nβf1βi2σ(j)\nβf2βi1ei(E(2)\n1t1+E(2)\n2t2)//planckover2pi1,\nwhere the transfer momenta are defined as\nq1=qf1−qi1,q2=qf2−qi2,\nq1=qf1−qi2,q2=qf2−qi1,(20)\nand the transfer energies are defined as\nE(2)\n1=E(qf1)−E(qi1),E(2)\n2=E(qf2)−E(qi2),\nE(2)\n1=E(qf1)−E(qi2),E(2)\n2=E(qf2)−E(qi1).(21)\nPhysically, there are two different classes of microscopic\nneutron scattering processes involved in the coincidence\nscattering. One is with the state changes of the two neu-\ntronsas|qi1βi1/an}b∇acket∇i}ht → |qf1βf1/an}b∇acket∇i}htand|qi2βi2/an}b∇acket∇i}ht → |qf2βf2/an}b∇acket∇i}ht, and4\nthe other one is with |qi1βi1/an}b∇acket∇i}ht → |qf2βf2/an}b∇acket∇i}htand|qi2βi2/an}b∇acket∇i}ht →\n|qf1βf1/an}b∇acket∇i}ht. The matrix elements Mαβ,1andMαβ,2and the\ncorresponding coincidence probabilities Γ(2)\n1and Γ(2)\n2de-\nscribe these two different classes of microscopic neutron\nscattering processes, respectively. It should be noted\nthat here we have ignored the quantum interference of\nthese two different scattering contributions as they come\nfrom different scattering channels of energy transfer with\nenergy-conservation-like resonance features at different\nenergies.\nWe define the center-of-mass time tc=1\n2(t1+t2)\nand the relative time tr=t1−t2and denote the two-\nspin Bethe-Salpeter wave function φ(ij)\nαβ(q1,q2;tc,tr) =\nφ(ij)\nαβ(q1t1,q2t2). We can introduce the Fourier transfor-\nmations of φ(ij)\nαβ(q1,q2;tc,tr) as follows:\nφ(ij)\nαβ(q1,q2;tc,tr)\n=/integraldisplay/integraldisplay+∞\n−∞dΩdω\n(2π)2φ(ij)\nαβ(q1,q2;Ω,ω)e−iΩtc−iωtr,\nφ(ij)\nαβ(q1,q2;Ω,ω)\n=/integraldisplay/integraldisplay+∞\n−∞dtcdtrφ(ij)\nαβ(q1,q2;tc,tr)eiΩtc+iωtr.\nFor the incident thermal neutrons in the spin mixed state\ndefined by Eq. (10), the coincidence probability is shown\nto follow\nΓ(2)=1\n/planckover2pi141\nZ/summationdisplay\nαβije−βEα\n×[C1/vextendsingle/vextendsingleφ(ij)\nαβ,1(q1,q2)/vextendsingle/vextendsingle2+C2/vextendsingle/vextendsingleφ(ij)\nαβ,2(q1,q2)/vextendsingle/vextendsingle2],(22)\nwhere the two factors are defined as\nC1=|g(q1)g(q2)|2,C2=|g(q1)g(q2)|2,(23)\nand the two wave functions φ(ij)\nαβ,1(q1,q2) and\nφ(ij)\nαβ,2(q1,q2) are defined as\nφ(ij)\nαβ,1(q1,q2)\n=/integraldisplay/integraldisplay+∞\n−∞dΩdω\n(2π)2φ(ij)\nαβ(q1,q2;Ω,ω)Y1(Ω,ω),(24)\nφ(ij)\nαβ,2(q1,q2)\n=/integraldisplay/integraldisplay+∞\n−∞dΩdω\n(2π)2φ(ij)\nαβ(q1,q2;Ω,ω)Y2(Ω,ω).(25)Here the functions Y1(Ω,ω) andY2(Ω,ω) are given by\nY1(Ω,ω) =sin[(E(2)\n1//planckover2pi1−Ω/2−ω)∆t/2]\n(E(2)\n1//planckover2pi1−Ω/2−ω)/2\n×sin[(E(2)\n2//planckover2pi1−Ω/2+ω)∆t/2]\n(E(2)\n2//planckover2pi1−Ω/2+ω)/2,(26)\nY2(Ω,ω) =sin[(E(2)\n1//planckover2pi1−Ω/2−ω)∆t/2]\n(E(2)\n1//planckover2pi1−Ω/2−ω)/2\n×sin[(E(2)\n2//planckover2pi1−Ω/2+ω)∆t/2]\n(E(2)\n2//planckover2pi1−Ω/2+ω)/2.(27)\nIn large, but finite, ∆ t, we can make the approxima-\ntion that/integraltext∆t/2\n−∆t/2dt2/integraltext∆t/2\n−∆t/2dt1→/integraltext∆t/2\n−∆t/2dtc/integraltext∆t/2\n−∆t/2dtr.\nIn this case the functions Y1(Ω,ω) andY2(Ω,ω) can be\napproximated as\nY1(Ω,ω) =sin[(Ω−E(2)\n1//planckover2pi1−E(2)\n2//planckover2pi1)∆t/2]\n(Ω−E(2)\n1//planckover2pi1���E(2)\n2//planckover2pi1)/2\n×sin[(ω−E(2)\n1/2/planckover2pi1+E(2)\n2/2/planckover2pi1)∆t/2]\n(ω−E(2)\n1/2/planckover2pi1+E(2)\n2/2/planckover2pi1)/2,(28)\nY2(Ω,ω) =sin[(Ω−E(2)\n1//planckover2pi1−E(2)\n2//planckover2pi1)∆t/2]\n(Ω−E(2)\n1//planckover2pi1−E(2)\n2//planckover2pi1)/2\n×sin[(ω−E(2)\n1/2/planckover2pi1+E(2)\n2/2/planckover2pi1)∆t/2]\n(ω−E(2)\n1/2/planckover2pi1+E(2)\n2/2/planckover2pi1)/2.(29)\nIn the limit with ∆ t→+∞, it can be shown that\nΓ(2)=1\n/planckover2pi141\nZ/summationdisplay\nαβije−βEα[C1/vextendsingle/vextendsingleφ(ij)\nαβ(q1,q2;Ω1,ω1)/vextendsingle/vextendsingle2+C2/vextendsingle/vextendsingleφ(ij)\nαβ(q1,q2;Ω2,ω2)/vextendsingle/vextendsingle2], (30)5\nwhere the transfer frequencies are defined as\nΩ1=1\n/planckover2pi1(E(2)\n1+E(2)\n2),ω1=1\n2/planckover2pi1(E(2)\n1−E(2)\n2),Ω2=1\n/planckover2pi1(E(2)\n1+E(2)\n2),ω2=1\n2/planckover2pi1(E(2)\n1−E(2)\n2). (31)\nThe coincidence probability Γ(2)in Eq. (30) shows that cINS can explicitly detect the frequency Bet he-Salpeter wave\nfunction, which describes the dynamical magnetic physics of the ta rget material with two-spin excitations involved.\nThis can be seen more clearly from the following spectrum expression of the frequency Bethe-Salpeter wave function:\nφ(ij)\nαβ(q1,q2;Ω,ω) = 2πδ[Ω+(Eβ−Eα)//planckover2pi1]φ(ij)\nαβ(q1,q2;ω), (32)\nwhereφ(ij)\nαβ(q1,q2;ω) follows\nφ(ij)\nαβ(q1,q2;ω) =/summationdisplay\nγ/bracketleftBigg\ni/an}b∇acketle{tΨβ|S(i)\n⊥(q1)|Ψγ/an}b∇acket∇i}ht/an}b∇acketle{tΨγ|S(j)\n⊥(q2)|Ψα/an}b∇acket∇i}ht\nω+iδ++(Eα+Eβ−2Eγ)/2/planckover2pi1−i/an}b∇acketle{tΨβ|S(j)\n⊥(q2)|Ψγ/an}b∇acket∇i}ht/an}b∇acketle{tΨγ|S(i)\n⊥(q1)|Ψα/an}b∇acket∇i}ht\nω−iδ+−(Eα+Eβ−2Eγ)/2/planckover2pi1/bracketrightBigg\n.(33)\nObviously, the frequency Bethe-Salpeter wave func-\ntion involves the following dynamical magnetic physics\nof two spins of the target magnetic material: (1)\nthe center-of-mass dynamics of the two spins de-\nscribed byδ[Ω+(Eβ−Eα)//planckover2pi1], which shows the trans-\nfer energy conservation with the center-of-mass de-\ngrees of freedom involved, and (2) the two-spin\nrelative dynamics φ(ij)\nαβ(q1,q2;ω), which has reso-\nnance structures peaked at ∓(Eα+Eβ−2Eγ)/2/planckover2pi1\nwith weights /an}b∇acketle{tΨβ|S(i)\n⊥(q1)|Ψγ/an}b∇acket∇i}ht/an}b∇acketle{tΨγ|S(j)\n⊥(q2)|Ψα/an}b∇acket∇i}htand\n/an}b∇acketle{tΨβ|S(j)\n⊥(q2)|Ψγ/an}b∇acket∇i}ht/an}b∇acketle{tΨγ|S(i)\n⊥(q1)|Ψα/an}b∇acket∇i}ht, respectively. There-\nfore, cINS can provide the momentum-resolved dynami-\ncal two-spin magnetic correlations of the target magnetic\nmaterial.\nIII. COINCIDENCE PROBABILITIES OF THE\nFERROMAGNET AND ANTIFERROMAGNET\nIn this section we will study the coincidence probabil-\nities of cINS for a ferromagnet and an antiferromagnet\nwhich have long-range magnetic order with well-defined\nmagnon excitations.\nProvided that (1) the two incident neutrons are in-\ndependent following a spin distribution function as\nP2(βi1,βi2) =P1(βi1)P1(βi2) and (2) the single-spin\nmagnetic excitations of the target material have well-\ndefined momenta and are decoupled from each other, the\ncoincidence probability of cINS has a simple product be-\nhavior, which can be expressed mathematically as\nΓ(2)= Γ(1)(qf1,qi1)·Γ(1)(qf2,qi2)\n+ Γ(1)(qf1,qi2)·Γ(1)(qf2,qi1).(34)\nThis is a general result which can be exactly proven from\nthe definitions of the scattering probability of INS and\nthe coincidence probability of cINS, Eq. (6) and (17).\nWe will consider localized spin magnetic systems with\na cubic crystal lattice, the Hamiltonians of which aredefined by\nH=J\n2/summationdisplay\nlδSl·Sl+δ, (35)\nwhereδ=±aex,±aey,±aez. The localized spins are\nin a low-temperature ordering state with the magnetic\nmoments ordered along the ezaxis.\nA. Ferromagnet\nLet us consider a ferromagnet with J <0. We in-\ntroduce the Holstein-Primakoff transformation, S+\nl=/radicalBig\n2S−a†\nlalal,S−\nl=a†\nl/radicalBig\n2S−a†\nlal,Sz\nl=S−a†\nlal, where\nalanda†\nlare the bosonic magnon operators. In linear\nspin-wave theory, the spin Hamiltonian can be approxi-\nmated as\nHFM=/summationdisplay\nkεka†\nkak, (36)\nwhereεk=|J|zS(1−γk), withγk=1\nz/summationtext\nδeik·δand\ncoordination number z= 6. Hereak=1√\nN/summationtext\nlale−ik·Rl.\nLet us first study the scattering probability of the\nsingle-spin INS. Suppose the incident thermal neutrons\nare in the spin mixed state defined by Eq. (10). It can\nbe shown from Eq. (12) that the scattering probability\nΓ(1)follows\nΓ(1)\nFM(qf,qi)\n=|g(q)|2∆t\n/planckover2pi1nB(E(1))/bracketleftBig\nχxx(q,E(1))(1−/hatwideq2\nx)\n+χyy(q,E(1))(1−/hatwideq2\ny)+χzz(q,E(1))(1−/hatwideq2\nz)/bracketrightBig\n,\n(37)6\nwhere the spin spectrum functions χii(q,E) are given by\nχxx(q,E) =χyy(q,E) =πNS[δ(E−εq)−δ(E+ε−q)],\nχzz(q,E) = 2π/summationdisplay\nk[nB(��k)−nB(εk+q)]δ(E+εk−εk+q).\n(38)\nHere the transfer momentum and energy, qandE(1),\nare defined as in Eq. (13). While the transverse spin\nflips lead to single-magnon peak structures in the scat-\ntering probability, the longitudinal spin fluctuations con-\ntribute magnon density fluctuations. Besides these in-\nelastic scattering contributions, there is one additional\nelastic scattering contribution from the magnon conden-\nsation, which gives\nΓ(1)\nFM,c=2π|g(q)|2∆t\n/planckover2pi1(NmFM)2δ(E(1))δq,0(1−/hatwideq2\nz),\n(39)\nwheremFM=1\nN/summationtext\nl/an}b∇acketle{tSz\nl/an}b∇acket∇i}htis the ordered spin magnetic\nmoment per site. It is noted that in experiment, Nis the\nnumber of local Wannier electron spins in the incident\nneutronbeam. Whenconsideringonlythesingle-magnon\ncontributions without that of the magnon density fluctu-\nations, the inelastic scattering probability of INS for the\nordered ferromagnet follows\nΓ(1)\nFM(qf,qi) =πNS|g(q)|2∆t\n/planckover2pi1nB(E(1))(1+/hatwideq2\nz)\n×[δ(E(1)−εq)−δ(E(1)+ε−q)].(40)\nNow let us study the coincidence probability of cINS\nfor the ordered ferromagnet. Suppose the two incident\nthermal neutrons with momenta qi1andqi2are scat-\ntered into the states with momenta qf1andqf2and the\nincident neutrons are in spin states with P2(βi1,βi2) =\nP1(βi1)P1(βi2) andP1(βi) defined in Eq. (10). Since\nthe magnons are well-defined single-spin excitations with\nthe momentum being a good quantum number, the coin-\ncidence probability of cINS for the ordered ferromagnet\nwith only contributions from the single-magnon excita-\ntions has a product behavior described by Eq. (34), i.e.,\nΓ(2)\nFM= Γ(1)\nFM(qf1,qi1)·Γ(1)\nFM(qf2,qi2)\n+ Γ(1)\nFM(qf1,qi2)·Γ(1)\nFM(qf2,qi1),(41)\nwhere the four Γ(1)\nFM(qf,qi)’s are the scattering prob-\nabilities of the single-magnon relevant INS defined inEq. (40). The magnon density fluctuations are not well-\ndefined excitations, and their contribution would break\ndown this simple product behavior.\nB. Antiferromagnet\nNow let us consider an antiferromagnet in a cubic\ncrystal lattice with long-range magnetic order. It has\na spin lattice Hamiltonian defined by Eq. (35) with\nJ >0. We introduce the spin rotation transformation\nasSx\nl=eiQ·RlSx\nl,Sy\nl=Sy\nl,Sz\nl=eiQ·RlSz\nl, where\nQ= (π/a,π/a,π/a ) is the characteristic antiferromag-\nnetic momentum. Introducing the Holstein-Primakoff\ntransformation for the new spin operators, the spin\nHamiltonian can be approximated in a linear spin-wave\ntheory as\nHAF=/summationdisplay\nk′\nψ†\nk/parenleftbigg\nA Bk\nBkA/parenrightbigg\nψk,ψk=/parenleftbiggak\na†\n−k/parenrightbigg\n,(42)\nwhereA=JzS,Bk=−JzSγk, andψkis a bosonic\nNambu spinor operator. Here the sum over kinvolves\neach pair ( k,−k) once. With the canonical transforma-\ntion\n/parenleftbiggak\na†\n−k/parenrightbigg\n=/parenleftbigg\nukvk\nvkuk/parenrightbigg/parenleftbiggβk\nβ†\n−k/parenrightbigg\n,(43)\nthe Hamiltonian can be diagonalized into the form\nHAF=/summationdisplay\nk′Ek(β†\nkβk+β†\n−kβ−k),(44)\nwhereEk=/radicalbig\nA2−B2\nk. Hereu2\nk=A+Ek\n2Ek,v2\nk=A−Ek\n2Ek,\nandukvk=−Bk\n2Ek.\nIt can easily be shown that the neutron scatter-\ning probability of INS for the ordered antiferromagnet\nΓ(1)\nAF(qf,qi) follows an expression similar to Eq. (37)\nfor Γ(1)\nFM(qf,qi), with the corresponding spin spectrum\nfunctionsχii(q,E) given by\nχxx(q,E) =χyy(q,E)\n=πNSA+Bq\nEq[δ(E−Eq)−δ(E+Eq)] (45)\nand\nχzz(q,E) = 2π/summationdisplay\nk/braceleftBig\n[C(1)\nkqδ(E+εk−εk+q+Q)−C(4)\nkqδ(E−εk+εk+q+Q)][nB(εk)−nB(εk+q+Q)]\n+ [C(2)\nkqδ(E−εk−εk+q+Q)−C(3)\nkqδ(E+εk+εk+q+Q)][1+nB(εk)+nB(εk+q+Q)]/bracerightBig\n.(46)\nHereC(1)\nkq=u2\nk+q+Qu2\nk+uk+q+Qvk+q+Qukvk,C(2)\nkq=u2\nk+q+Qv2\nk+uk+q+Qvk+q+Qukvk,C(3)\nkq=v2\nk+q+Qu2\nk+7\nuk+q+Qvk+q+Qukvk, andC(4)\nkq=v2\nk+q+Qv2\nk+uk+q+Qvk+q+Qukvk. Similar to the ordered ferromagnet, there is\nalso one additional elastic scattering contribution due to the magno n condensation,\nΓ(1)\nAF,c=2π|g(q)|2∆t\n/planckover2pi1(NmAF)2δ(E(1))δq,Q(1−/hatwideq2\nz), (47)\nwheremAF=1\nN/summationtext\nleiQ·Rl/an}b∇acketle{tSz\nl/an}b∇acket∇i}htis the ordered antiferromagnetic moment per site. Here the trans fer momentum and\nenergy,qandE(1), are also defined in Eq. (13). In the approximation with only the single -magnon contributions,\nthe inelastic scattering probability of INS for the ordered antiferr omagnet follows\nΓ(1)\nAF(qf,qi) =πNS|g(q)|2∆t\n/planckover2pi1A+Bq\nEqnB(E(1))(1+/hatwideq2\nz)[δ(E(1)−Eq)−δ(E(1)+Eq)]. (48)\nNow let us consider cINS with the thermal neutrons\nwhich have initial incident momenta qi1andqi2and fi-\nnal scattered momenta qf1andqf2. The incident neu-\ntrons are independent, with the spin state defined by Eq.\n(10). In linear spin-wave theory defined by the approxi-\nmateHamiltonian(42), theNambuspinoroperatorswith\ndifferent momenta are decoupled. This means that the\nsingle-magnon excitations in the ordered antiferromag-\nnet are decoupled. Therefore, in the linear spin-wave\ntheory with only contributions from the single-magnon\nexcitations, the conditions for the product behavior of\nthe coincidence probability in Eq. (34) are also satisfied\nin the ordered antiferromagnet. In this approximation\nthe coincidence probability of cINS for the ordered an-\ntiferromagnet follows a similar product behavior defined\nas\nΓ(2)\nAF= Γ(1)\nAF(qf1,qi1)·Γ(1)\nAF(qf2,qi2)\n+ Γ(1)\nAF(qf1,qi2)·Γ(1)\nAF(qf2,qi1),(49)\nwhere the four Γ(1)\nAF(qf,qi)’s are the scattering proba-\nbilities of the single-magnon relevant INS defined in Eq.\n(48).\nIV. DISCUSSION AND SUMMARY\nIn this paper we have proposed a coincidence tech-\nnique, cINS, which has two neutron sources and two neu-\ntron detectors, with an additional coincidence detector.\nThe two neutron sources emit two neutrons which are\nscattered by the electron spins of the magnetic material\nand are then detected by the two neutron detectors. The\ncoincidence detector records the coincidence probability\nofthe two scatteredneutrons, which givesinformation on\na two-spin Bethe-Salpeter wave function. This two-spin\nBethe-Salpeter wave function defines the momentum-\nresolved dynamical wave function of the magnetic ma-\nterial with two spins excited. Thus, cINS can explicitly\ndetect the two-spin magnetic correlations of the mag-\nnetic material. The coincidence probabilities of cINS for\na ferromagnet and an antiferromagnet with long-range\nmagnetic order have been calculated and show a prod-\nuct behavior contributed by the single-magnon relevantINSs. This trivial product behavior for the ordered ferro-\nmagnet and antiferromagnet is consistent with the mag-\nnetic properties dominated by the nearly free magnon\nexcitations, which have no intrinsic two-spin magnetic\ncorrelations.\nOn the experimental instrument of cINS, we remark\nthat the two incident neutrons can come from one neu-\ntron source. In this case the initial momenta of the two\nincident neutrons follow qi2=qi1+δq, withδq→0.\nThese two incident neutrons can be regarded equiva-\nlently to be emitted from two different neutron sources\nbut with nearly the same momenta. Thus, the theoret-\nical formalism for cINS with one neutron source can be\nsimilarly established following the one we established in\nSec. IIB for cINS with two neutron sources. There are\ntwo main challenges in the experimental realization of\ncINS. One is to develop a two-neutron coincidence detec-\ntor, and the other one is accurate control of the coinci-\ndence detection. The two-photon coincidence measure-\nment in modern quantum optics11and the coincidence\ndetection of the photoelectron and the Auger electron in\ndouble-photoemission spectroscopy12may provide a use-\nful guideline.\nThe cINS we have proposed is one potential technique\nto study novel magnetic correlations which are far be-\nyond the physics of the single-spin magnons. For exam-\nple, the long-sought quantum spin liquids1–4from strong\nfrustrationand quantum fluctuations show novelphysics,\nsuch as various spin valence bond states13–16and novel\nquantum criticality17. Experimental study of the spin\nvalence bond states by cINS would provide new insights\ninto quantum spin liquids. The various quantum mag-\nnetic materials with spin dimers, such as TlCuCl 318,\nSrCu2(BO3)219, and BaCuSi 2O620, could be the first fo-\ncus in a cINS experiment. Quantum spin liquid materials\nin triangular, honeycomb, kagome, and hyperkagome lat-\ntices (e.g., the materials reviewed in Ref. [4,21]) are also\ninteresting target materials for a cINS experiment.\nIn summary, we have proposed a coincidence tech-\nnique, cINS, which can explicitly detect the two-spin\nmagnetic correlations of magnetic materials. It can be\nintroduced to study the dynamical physics of the spin\nvalence bond states of quantum magnets.8\nACKNOWLEDGMENTS\nWe thank H. Shao and D. Z. Cao for invaluable discus-\nsions. This work was supported by the National Natural\nScience Foundation of China (Grants No. 11774299 and\nNo. 11874318) and the Natural Science Foundation of\nShandong Province (Grants No. ZR2017MA033 and No.\nZR2018MA043).\nAppendix A: Electron-neutron magnetic interaction\nLet us review the electron-neutron magnetic\ninteraction5–7. We define the neutron spin mag-\nnetic moment as µn=−γµNσ, whereγ= 1.91 is a\nconstant for the neutron gyromagnetic ratio, µN=e/planckover2pi1\n2mp\nis the nuclear magneton, with mpbeing the proton mass,\nandσis the Pauli matrix. We define the electron spin\nmagnetic moment as µs=−gsµBSand the electron\norbital magnetic moment as µl=−glµBL, where the\ngfactors are set as gs= 2 andgl= 1 andµB=e/planckover2pi1\n2meis the Bohr magneton. The spin angular momentum\noperator Shas eigenvalues ±1\n2, and the orbital angular\nmomentum operator is defined as L=1\n/planckover2pi1re×pe. Suppose\nthere is an electron at position rewhich can produce a\nmagnetic field at position rnas\nB=µ0\n4π∇×/bracketleftbigg\n(µs+µl)×R\nR3/bracketrightbigg\n,(A1)\nwhereµ0is the free-space permeability and R=rn−re.\nTheelectron-neutronmagneticinteractioncanbedefined\nbyV=−µn·B, which follows\nV=−µ0\n4πγµNµBσ·∇×/bracketleftbigg\n(gsS+glL)×R\nR3/bracketrightbigg\n.(A2)\nHere we have introduced the orbital angular momentum\nLto describe the orbital motions of the electrons7. It\nis more convenient in the study of the orbital motions\nof electrons in compounds with transition metal and/or\nrare earth atoms.\nLet us present the second quantization of the electron-\nneutron magnetic interaction. Introduce the single-\nneutronstates {|qβ/an}b∇acket∇i}ht}, whereqisthe neutronmomentum\nandβdefines the neutron spin, and the single-electron\nstates{|λ/an}b∇acket∇i}ht}, whereλinvolves the momentum, orbital,\nand spin degrees of freedom, etc. Let us introduce the\nfollowing identities:\n1 =1\nV1/integraldisplay\ndre|re/an}b∇acket∇i}ht/an}b∇acketle{tre|\nfor the electrons, and\n1 =1\nV2/integraldisplay\ndrn|rn/an}b∇acket∇i}ht/an}b∇acketle{trn|\nfor the neutrons. Here V1andV2are the renormalization\nvolumes for the single-electron and single-neutron states,respectively. The electron-neutron magnetic interaction\nin second quantization can be expressed as\n/hatwideV=/hatwideVs+/hatwideVl, (A3)\nwhere\n/hatwideVs=4πAs\nV2/summationdisplay\nqiqf/hatwideσqfqi·[/hatwideq×(Ds(q)×/hatwideq)],(A4)\n/hatwideVl=4πAl\nV2/summationdisplay\nqiqf/hatwideσqfqi·[/hatwideq×(Dl(q)×/hatwideq)].(A5)\nHere the momentum q=qf−qi, and/hatwideq=q\nq. It is noted\nthat/hatwideq×(D(q)×/hatwideq) can be reexpressed as D⊥(q):\nD⊥(q) =D(q)−/hatwideq(D(q)·/hatwideq). (A6)\nIn the electron-neutron magnetic interaction /hatwideV, the con-\nstantsAsandAlare defined as\nAs=−µ0\n4πγgsµNµB,Al=−µ0\n4πγglµNµB,(A7)\nand the operator /hatwideσqfqiis defined as\n/hatwideσqfqi=/summationdisplay\nβiβfd†\nqfβfσβfβidqiβi, (A8)\nwheredqβandd†\nqβare the annihilation and creation op-\nerators for the neutrons. The operators DsandDlin/hatwideV\nare defined as\nDs(q) =/summationdisplay\nλ1λ2c†\nλ2M(s)\nλ2λ1(q)cλ1,(A9)\nDl(q) =/summationdisplay\nλ1λ2c†\nλ2M(l)\nλ2λ1(q)cλ1,(A10)\nwherecλandc†\nλare the annihilation and creation oper-\nators for the electrons and\nM(s)\nλ2λ1(q) =1\nV1/integraldisplay\ndre[ψ∗\nλ2(re)Sψλ1(re)]e−iq·re,\nM(l)\nλ2λ1(q) =1\nV1/integraldisplay\ndre[ψ∗\nλ2(re)Lψλ1(re)]e−iq·re.\nHereψλ(re) is the single-electron wave function.\nLet us focus on the spin degrees of freedom of the elec-\ntrons and ignore the orbital ones. We consider the elec-\ntrons to be in the local Wannier states {|lα/an}b∇acket∇i}ht}with posi-\ntionRland spinα.Ds(q) can be approximately defined\nas\nDs(q) =F0(q)S(q), (A11)\nwhere the spin operator S(q) is defined as\nS(q) =/summationdisplay\nlSle−iq·Rl,Sl=/summationdisplay\nα1α2c†\nlα2Sα2α1clα1,(A12)9\nand the magnetic form factor F0(q) is given by\nF0(q) =1\nV1/integraldisplay\ndaψ∗\nl(a)ψl(a)e−iq·a,a=re−Rl.(A13)\nHere we have made an approximation to consider only\nthe on-site intraorbital integrals and ignore all the other\ncontributions. For the itinerant electrons in the Bloch\nstates{|kα/an}b∇acket∇i}ht}, the operator Ds(q) can be given by\nDs(q) =/summationdisplay\nk1k2Fk2k1(q)Sk2k1, (A14)\nwhere the spin operator is defined by\nSk2k1=/summationdisplay\nα1α2c†\nk2α2Sα2α1ck1α1,(A15)\nand the form factor Fk2k1(q) is given by\nFk2k1(q) =1\nV1/integraldisplay\ndreψ∗\nk2(re)ψk1(re)e−iq·re.(A16)\nHereψk(re) is the Bloch-state wave function. In the\napproximation with ψk(re) =eik·re,Ds(q) can be sim-\nplified as\nDs(q) =/summationdisplay\nkSk,k+q. (A17)\nIn summary, the electron-neutronmagneticinteraction\nwith only the spin degreesof freedom of the electrons can\nbe given as follows. For the local Wannier electrons,\n/hatwideVs=/summationdisplay\nqiqfg(q)/hatwideσqfqi·S⊥(q), (A18)\nwhereg(q)≡gF0(q), withg=4πAs\nV2, andS⊥(q) is the\nprojection of S(q) in the perpendicular plane normal to\nthe momentum qand is defined similarly to D⊥(q) in\nEq. (A6). For the itinerant Bloch electrons,\n/hatwideVs=/summationdisplay\nqiqfk1k2gk2k1(q)/hatwideσqfqi·Sk2k1,⊥,(A19)\nwheregk2k1(q)≡gFk2k1(q) andSk2k1,⊥is defined simi-\nlarly toD⊥(q) in Eq. (A6). It should be noted that the\nform factors F0(q) andFk2k1(q) have strong qdepen-\ndence.\nOne remark is that in the above electron-neutronmag-\nnetic interaction, the contributions from the spin and or-\nbital magnetic moments are independently derived. In\nthis case, the spin-orbit coupling is weak like for the elec-\ntrons of the transition metal atoms. In the case with\nstrong spin-orbit coupling such as that of the electrons\nof rare earth atoms, the total angular momentum Jis\nconserved. In this case we can introduce the total mag-\nnetic moment µJ=−g(JLS)µBJ, with the Land´ e gfac-\ntorg(JLS) defined following glL+gsS=g(JLS)J. A\nsimilar derivation can give us an electron-neutron mag-\nnetic interaction in this case. Another remark is that the\nDebye-Waller factor5,6from the crystal lattice effects is\nignored in our discussion on the neutron scattering prob-\nability of the inelastic neutron scattering.Appendix B: Calculations for scattering probability\nof INS\nLet us introduce the imaginary-time Green’s functions\nGij(q,τ) =−/an}b∇acketle{tTτSi(q,τ)S†\nj(q,0)/an}b∇acket∇i}htwithi,j=x,y,z.\nThe corresponding spectrum functions are defined as\nχij(q,E) =−2 ImGij(q,iνn→E+iδ+). Then we have\nG(q,τ) =/summationdisplay\nijGij(q,τ)(δij−/hatwideqi/hatwideqj),(B1)\nand\nχ(q,E) =/summationdisplay\nijχij(q,E)(δij−/hatwideqi/hatwideqj).(B2)\nFirst, let us consider the ferromagnet in a cubic crys-\ntal lattice with a long-range magnetic order. We intro-\nduce the imaginary-time Green’s function for the ferro-\nmagnetic magnons, Ga(q,τ) =−/an}b∇acketle{tTτak(τ)a†\nk(0)/an}b∇acket∇i}ht. Its fre-\nquency Fourier transformation is given by\nGa(k,iνn) =1\niνn−εk, (B3)\nwhere the magnon energy dispersion εkis defined in Eq.\n(36). It can be shown that in the linear spin-wave ap-\nproximation,\nGxx(q,iνn) =Gyy(q,iνn)\n=NS\n2[Ga(q,iνn)+Ga(−q,−iνn)] (B4)\nand\nGzz(q,iνn) =−1\nβ/summationdisplay\nk,iν1Ga(k+q,iν1+iνn)Ga(k,iν1).\n(B5)\nThe other Green’s functions follow\nGij(q,iνn) = 0,fori/ne}ationslash=j. (B6)\nFrom these results, we can obtain the spectrum functions\nχij(q,E) in Eq. (38) for the ordered ferromagnet.\nNow let us consider the antiferromagnet in a cubic\ncrystal lattice with a long-range magnetic order. We in-\ntroduce the imaginary-time Green’s function of a Nambu\nspinor operator,\nGψ(k,τ) =−/an}b∇acketle{tTτψk(τ)ψ†\nk(0)/an}b∇acket∇i}ht, (B7)\nwhereψkis defined in Eq. (42). It can be shown that\nthe frequency Green’s function follows\nGψ(k,iνn) =iνnτ3+A−Bkτ1\n(iνn)2−E2\nk,(B8)\nwhereAandBkare defined in Eq. (42) and the magnon\nenergyEkis given in Eq. (44). Here τi(i= 1,2,3) are\nthe Pauli matrices.10\nIt can be shown that\nGxx(q,iνn) =NS\n2Tr[Gψ(q+Q,iνn)+Gψ(q+Q,iνn)τ1],\nGyy(q,iνn) =NS\n2Tr[Gψ(q,iνn)−Gψ(q,iνn)τ1],(B9)and\nGzz(q,iνn) =−1\nβ/summationdisplay\nk,iν1[G(11)\nψ(k+q+Q,iν1+iνn)G(11)\nψ(k,iν1)+G(21)\nψ(k+q+Q,iν1+iνn)G(12)\nψ(k,iν1)].(B10)\nThe other Green’s functions Gij(q,iνn) = 0 for the cases\nwithi/ne}ationslash=j. With these results, we can obtain the spec-trum functions χij(q,E) in Eqs. 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Milošević\u0003,y,z\nyDepartement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen,\nBelgium\nzNANOlab Center of Excellence, University of Antwerp, Belgium\nE-mail: milorad.milosevic@uantwerpen.be\nAbstract\nDzyaloshinskii-Moriya interaction (DMI) is at heart of chiral magnetism and causes\nemergence of rich non-collinear and unique topological spin textures in magnetic ma-\nterials, including cycloids, helices, skyrmions and other. Here we show that strong\nintrinsic DMI lives in recently discovered van der Waals magnetic two-dimensional\n(2D) materials, due to the sizeable spin-orbit coupling on the non-magnetic ions. In\na perfect crystal, this intrinsic DMI remains hidden, but is released with any break of\npoint-inversion symmetry between magnetic ions, unavoidable at the sample edges, at\never-present structural defects, with any buckling of the material, or with non-uniform\nstrain on an uneven substrate. We demonstrate such release of latent chirality on an\narchetypal magnetic monolayer CrI 3, and discuss the plethora of realizable DMI pat-\nterns, their control by nanoengineering and tuning by external electric field, thereby\nopening novel routes in 2D magnetoelectronics.\n1arXiv:2009.03421v1  [cond-mat.mes-hall]  7 Sep 2020Keywords\nMagnetic chirality, 2D materials, spintronics, magnonics.\nRecent realizations of the 2D ferromagnetic materials, monolayer CrI 31and bilayer\nCr2Ge2Te6,2opened the gate for many magnetic 2D materials synthesized since. In spite\nof the Mermin-Wagner theorem, 2D magnetism in these materials is possible, even at finite\ntemperatures, due to the magnetic anisotropy stemming from the strong spin-orbit cou-\npling (SOC) on the non-magnetic ions. Although the phase transition temperatures are\nin general relatively low, 2D magnetism can be sustained in some materials even up to\nroom-temperature, such as reported for monolayer VSe 23and MnSe 2.4All the experimental\nevidence to date suggests that in pristine 2D magnetic materials the spins align parallel to\nthe easy-axis direction and hence show no apparent magnetic chirality.\nThe crucial ingredient for chiral magnetism to arise is the Dzyaloshinskii-Moriya interac-\ntion (DMI).5,6Such interaction appears in materials with strong spin-orbit coupling which\nlack point inversion symmetry between the magnetic ions, and favors orthogonal ordering\nof adjacent spins, thereby competing with the symmetric magnetic exchange that aligns the\nspins. This competition results in rich noncollinear spin textures, among which skyrmion lat-\ntices7–9and spin spirals.10Previously, DMI was found to emerge due to symmetry breaking\nin non-centrosymmetric bulk materials11–14and in ultrathin elemental ferromagnets (Fe, Co,\nNi) interfaced with a heavy-metal layer15–19or graphene.20However, DMI did not receive\nmuch attention in recently discovered two-dimensional (2D) magnetic materials.1,2,21Besides\nthe fact that such materials are still very new, they do preserve inversion symmetry, and\ntheir magnetic atoms (in latter case chromium) can hardly be sensitive to interfacing by a\nheavy-metal layer due to nonmagnetic layers in between (in latter case iodine) - hence it is\nnot intuitive that any sizeable DMI can be induced in them using the established routines.\nHowever, one quickly recalls that it is the strong spin-orbit coupling in the nonmagnetic\nligands in latter 2D materials that supports appearance of long-range magnetization in spite\n2of the Mermin-Wagner theorem.1,2,22,23If so, then one expects that spin-orbit coupling on\nthe non-magnetic ligands can also prompt the DMI to arise. It is definitely plausible that\neach non-magnetic ion contributes some DMI to a pair of magnetic ions, but the net DMI\nstemming from all nearest-neighbor non-magnetic ions will cancel out in a perfect lattice.\nHowever, once the point-inversion symmetry between the magnetic atoms in a 2D magnet\nis broken, this latentDMI should reveal itself.\nIntuitively, one does not expect such DMI to be large. However, very recently it was\npredicted that symmetry-breaking in the Janus (two-faced) forms of CrX 3(X= I, Br, and\nCl) and MnX 2(X=S and Se) can produce strong DMI, sufficient to obtain skyrmion-like\nspin textures, even at finite temperatures.24–26Still, Janus 2D materials are far from trivial\nto realize with the current experimental state-of-the-art. Instead, every synthesized or ex-\nfoliated 2D magnetic material, on a substrate, will exhibit enough structural deformations\nto warrant local release of latent DMI. Understanding and control thereof is the primary\nobjective of this Letter.\nSpecifically, we demonstrate the release of latent DMI in monolayer CrI 3, hidden by\nthe point-inversion symmetry between the nearest-neighbor Cr ions. We tackle examples\nof symmetry breaking that are ever present in experimental samples, such as lateral edges,\ninhomogeneous strain, buckling, vacancies, or substitutional atoms, and external manipula-\ntions such as the applied electric field.27–30Finally, we go on to quantify the link between the\nstructural and electronic manipulation of the DMI release, and provide guidelines on how\nchirality can be induced and tailored in 2D magnetic materials. Ability of such controlled\nrelease of latent DMI is of clear fundamental value, but also bears relevance to a spectrum\nof potential applications in magnonics and spintronics.31,32\nTo describe the magnetic interactions, we consider Heisenberg spin Hamiltonian H=\n1\n2P\ni;jSiJijSj+P\niSiAiiSj, where Si= (Sx\ni;Sy\ni;Sz\ni)is the spin vector of the ith site. Jij\nandAiiare3\u00023matrices describing the total magnetic exchange interaction between the\ndifferent sites and the single-ion anisotropy (SIA), respectively. Total exchange consists of\n3the symmetric and the antisymmetric (DMI) contribution. DMI energy contribution can\nbe written as EDM=P\n(i;j)Dij\u0001(Si\u0002Sj), where Dij= (Dx\nij;Dy\nij;Dz\nij)is the DMI vector\nbetween magnetic ions on ith andjth site. In this Letter, we will focus our attention\non the antisymmetric (DMI) exchange, as spin-aligning symmetric exchange and single-ion\nanisotropy were given enough attention in earlier works.22,23The DMI vector between the\nnearest-neighbor magnetic ions is extracted from the total exchange matrix between them\nusing theD\r\nij=1\n2\u0001(J\u000b\f\nij\u0000J\f\u000b\nij), where (\u000b;\f;\r )are cyclic Cartesian coordinates (x;y;z ),\n(y;z;x )and(z;x;y ). In order to obtain total exchange matrix elements, and consequently\ncalculate the profile of DMI vectors between the magnetic ions in different structures based\non monolayer CrI 3, we apply the Four-State Methodology (4SM)33–35based on mapping ab\ninitioenergies onto given Heisenberg spin Hamiltonian.\nMagnetic chirality at iodine vacancies\nIn either exfoliated or directly synthesized 2D materials, the structural imperfections such as\natom vacancies are unavoidable. In case of chromium halide monolayers, the vacancies in the\nouter halide layers are the most likely ones to appear, which obviously breaks the inversion\nsymmetry for the chromium atoms in the vicinity of the missing atom(s). Hence such a\nrealistic situation is clearly favorable for the appearance of DMI. We therefore start our\nconsideration from an isolated, single iodine vacancy in otherwise pristine CrI 3monolayer,\nas shown in Fig. 1a. We found that the arising DMI is largest at the Cr-Cr pair that misses\nthe iodine ligand, with the DMI vector orthogonal to the Cr-Cr bond.5The intensity of such\nDMI, created at an isolated single iodine vacancy, is surprisingly large - 4:64meV/Cr-Cr\npair - which is comparable to artificially created DMI at Co/Pt interfaces.19\nDuring the formation of the crystal the defects are often not isolated from each other,\nbut rather form a pattern, such as e.g. dislocation or grain boundary. Moreover, the recent\ndiscoveries report that ligand vacancies can be deliberately created in transition-metal com-\nplexes, such as monolayer WS 2.36Therefore, for comparison to the isolated vacancy case, we\n4Figure 1: (a)The released DMI at an isolated I vacancy (left), a substitution of Br atom\n(middle), and a substitution of Cl atom (right). (b)The profile of awaken DMI (red arrows)\nin case of a line of iodine vacancies in the monolayer CrI 3, top (above) and side view (below).\nOnly the strongest DMI vectors are shown. (c)Distribution of the DMI intensity in the\nvicinityofthedefectline. (d)DistributionoftheCartesiancomponentsofDMI,illuminating\nthe change in direction of the DMI vector around the defect.\nhave also investigated a line of iodine-vacancies in monolayer CrI 3. Due to computational\nlimitations, the considered line-defect was ideally parallel to the zig-zag chain of Cr atoms,\nas shown in Fig. 1b. Although we graphically present only the DMI arising on that chain of\nCr atoms (being dominant), the line-defect will awake weak yet non-negligible DMI in the\nbroader neighborhood as well. It exists in a narrow region, ranging less than 1 nm from the\nvacancies, and exhibits direction that deviates from being orthogonal to corresponding Cr-\nCr bond by the small angle. The spatial distribution of the DMI intensity, and its vectorial\ncomponents, are shown in Figs. 1b and 1c, respectively, for different Cr-Cr pairs sequen-\ntially away from the line-defect. The intensity of DMI for the pair missing one ligand is 5.57\nmeV/Cr-Cr pair; the pair with both ligands present, but situated on defect line experiences\nDMI of 2.48 meV/Cr-Cr pair. The DMI on the first-nearest pairs on the left and the right\n5side of the defect line falls to 0.44 and 0.25 meV/Cr-Cr pair, respectively, thus decays by\nan order of magnitude compared to the DMI on the defect line. The trend continues at the\nsecond-nearest pair on the right side, where DMI amounts to 0.1 meV/Cr-Cr pair, while at\nall the other pairs, further from the defect line, DMI can be considered dormant (as in the\npristine CrI 3monolayer). DMI liberated at iodine vacancies in monolayer CrI 3has intensity\ncomparable to the highest values known in artificial magnetic heterostructures. We note\nthat the isotropic exchange interaction increases twice at the defect site.\nMagnetic chirality caused by ligand substitution\nHaving understood the case of iodine vacancy, we next look at the DMI awaken by substi-\ntution of a foreign ligand to the iodine site. As two plausible cases, we chose substitution of\nI by Br or Cl, which are also halides, knowing that Cr-Br and Cr-Cl compounds also exhibit\nmagnetism.37,38As a first result, we note that substitutional (either Br or Cl) atom resides\nmuch closer to the Cr-Cr pair than iodine atom did, which already breaks the inversion\nsymmetry and suggests that DMI should appear. Indeed we found DMI on the Cr-Cr pair\nof 0.09 and 0.10 meV for Br and Cr substitution, respectively. As shown in Fig. 1a DMI\nvector is orthogonal to the Cr-Cr bond pointing the substituted atom with a very slightly\nrotation from the Cr-I-Cr-X plane (X=Br or Cl) through the in-plane direction. The DMI\nvalues here are consistent with recently reported values for Janus monolayers of Cr(I,Br) 3\nand Cr(I,Cl) 3, being 0.27 meV/Cr-Cr pair and 0.19 meV/Cr-Cr pair, respectively.24How-\never, they are 50 times smaller as compared to DMI at an isolated iodine vacancy. The\nsubstitution of a foreign ligand completes the octahedral coordination of the Cr pair and\nthereby recovers the exchange pathways as in pristine CrI 3, minimizing the DMI. Still, the\ndifferences in the structure as well as in the spin-orbit coupling of the participating atoms\nand the electronic structure of the resultant crystal strongly affect the magnetic interaction\nand cause DMI to appear.\n6Chirality at sample edges\nFigure 2: The profile of awaken DMI (red arrows) near the edge on the monolayer CrI 3, top\n(above) and side view (below).\nThe symmetry-breaking at the boundaries of any magnetic film is expected to cause the\nso-called boundary-induced chirality, completely independent of any internal DMI field.39,40\nIn the past, such contribution to DMI was derived using particular tensorial constructions\nwith respect to the point group of the system. In the case of 2D materials however, the\nDMI arising at the edges can be evaluated directly from first principles. With that goal,\nwe examined the zig-zag edge of a monolayer CrI 3nanoribbon. As shown in Fig. 2, we find\nthat the edge-induced DMI vectors are slightly tilted from the direction orthogonal to the\nCr-I-I-Cr plane, towards the out-of-plane direction of the monolayer. The intensity of the\nDMI vector between the Cr atoms at the edge is 0:87meV/atom, thus sizeable but several\ntimes smaller than in the case of iodine vacancy. Moreover, this DMI is truly localized at\nthe edge, and vanishes immediately away from the edge, as pristine symmetry is restored.\nThis is not surprising since the edge significantly affects the structural (and therefore the\nelectronic) surrounding only of the Cr-Cr pair nearest to it, where Cr atoms lack the third Cr\nnearest-neighbor. It is interesting to note that in case of a zigzag edge the 2-fold rotational\n7symmetry around horizontal direction is not removed, and this is reflected in the profile of\nDMI (see Fig. 2). In the case of an armchair edge, we do not expect the structure remains\nstable since there will be four I atoms dangling at the outermost Cr-Cr pair. That would\nresult in reconstruction or loss of I atom which is out of our scope in the present study.\nChirality awaken by local strain\nFigure 3: (a)Schematic representation of a monolayer CrI 3experiencing localized strain at\ntheatomically-thinstepbetweentheterracesofthesubstrate. (b)Topview(above)andside\nview (below) of the profile of the awaken DMI (red arrows) in case of locally applied strain\n(mimicking the situation depicted in (a)). (c)Linear dependence between the components\nof awaken DMI and the localized strain. (d)The DMI pattern expected in the case of non-\nuniform distribution of strain in monolayer CrI 3, as found in e.g. ripples depicted on the\nside.\nStrain, either applied intentionally or induced locally due to interaction with a rough or\nterraced substrate (see Fig. 3a), can strongly alter physical properties of a 2D crystal,41–45\nincluding magnetic ones, and is often used to manipulate those properties in a controllable\nfashion. However, biaxial uniform strain does not break the inversion symmetry of a mag-\nnetic monolayer and will not cause appearance of DMI. Although less obvious, the same\nholds for uniaxial strain. Still, non-uniform straining is a well documented and a well es-\ntablished technique to manipulate e.g. optical properties of 2D materials, related to exciton\nconfinement and single-photon emission, and can be employed to awaken DMI in magnetic\n2Dmaterials. ThesimplestscenariotoconsidertheDMIduetosymmetrybrokenbystrainis\nthe case of strain localized along an extended line, separating two parallel zig-zag chromium\nchains in a CrI 3monolayer. As we schematically show in Fig. 3a, such a case would be an\n8idealized approximation of a monolayer laid over an atomic step between the flat terraces of\nthe substrate, as often found at surfaces of e.g. SiC, MgO and other bulk crystals.\nThe blue dashed lines in Fig. 3b indicate the strained section of the monolayer CrI 3in\nour calculations. The released DMI vectors (shown by red arrows) are orthogonal to the\nbond between Cr atoms in the corresponding pair, pointing towards the strained section,\nand slightly tilted towards the in-plane of monolayer, compared to direction orthogonal to\nthe Cr-I-I-Cr plane. We examined the effects of both compressive and tensile strain, up to\n6% of the Cr-Cr distance in the pristine case. The obtained components of the DMI vector\nand its modulus are plotted as a function of applied strain in Fig. 3c. The maximal realized\nDMI, corresponding to the 6% of strain, amounted to 0.07 meV/atom. Nevertheless, it is\nquite remarkable (i) that DMI changes the direction with change of strain direction (stretch\nvs. compress), and (ii) that intensity of DMI and all its components change linearly with the\napplied strain, as shown in Fig. 3c. These properties add to the versatility of realizable DMI\npatterns and the resulting spin textures in 2D magnetic materials, where linear dependence\non strain is of great convenience for controlled manipulation. Bearing in mind the plethora\nof ways to generate nonuniform strain in 2D materials, our results suggest broad possibilities\nto control spin-chirality in a 2D system. For example, as depicted in Fig. 3d, a gradient of\nstrain experienced within a ripple of a 2D material would lead to a correspondingly varied\nDMI.\nDiscussion\nHaving considered a number of cases above to awake DMI, we attempt to quantify the\nemergent DMI by quantifying the break of the inversion symmetry using the electron density\naround the Cr-Cr pair. To this end, we define a quantity \u0018(~R0;VD)def=1\n2R\n~ r2VDj\u001a(~R0+~ r)\u0000\n\u001a(~R0\u0000~ r)jd3~ r,where~R0isthepositionvectorofthepointofinterest; VDstandsforthedomain\nof integration, which is chosen symmetrically around ~R0;\u001ais the valence electron charge\ndensity; and1\n2serves to avoid double-counting within the considered symmetric domain. We\n9tookVDas 0.2 Å width shells around the point of interest, and enumerated such rings with\nintegersn= 1;2;:::;nmaxaccording to the proximity of the ring to the Cr-Cr pair. One\nexpects that closer the symmetry breaking is to the Cr-Cr bond, it will yield larger influence\non the emerging DMI. Hence, in order to quantify the backbone of DMI in this study, we\nintroduce quantity \n(~R0;nmax) =Pnmax\nn=11\nn\u0018(~R0;n). By varying nmaxwe have determined\nthat the region that influences awakened DMI for a Cr-Cr pair of interest is bordered by\nthe first passive ligand atoms, i.e. the nearest ligands that are not bridging respective Cr\natoms. In Table 1, we listed the obtained \nvalues of the considered structures, next to the\nrespectively found jDj. We note that \nperfectly reflects the DMI in the system in case of\nstructural deformations of CrI 3, where the iodine ligands of the Cr-pair are complete yet\ndeformed for whatever reason. However, in case of vacancies or substitutions in the lattice,\na need for an additional quantifier arises due to the strong changes in the spin-orbit coupling\nand the electronic structure in the vicinity of the Cr-pair.\nTable 1: The calculated break of symmetry depending on the ground state electron charge\ndensity around the respective Cr-Cr pair, \n(~R0;nmax); modulus of the DMI vector, jDj; and\nthe ratio of DMI and isotopic exchange interaction, jDj\u000e\njJj.\nstructure pair \n(~R0;nmax)jDjjDj\u000e\njJj\n(e)(meV)\nSingle vacancy 1st pair 1.44 4.68 0.39\n\"inner\" pair 0.17 0.03 0.01\nLine vacancies 1st pair 1.69 5.57 0.48\n2nd pair 1.18 2.47 1.06\n\"inner\" pair 0.19 0.03 0.01\nEdge edge pair 0.59 0.87 0.15\n\"inner\" pair 0.18 0.00 0.00\nStrain 6% 1st pair 0.31 0.08 0.02\nStrain 2% 1st pair 0.13 0.03 0.01\nSub. Br 1st pair 0.74 0.09 0.025\nSub. Cl 1st pair 1.28 0.10 0.030\nPseudosub Br-I 1st pair 0.32 0.10 0.020\nAs briefly mentioned in the section of Br and Cl substitution, the DMI interaction orig-\ninates from the superexchange46between magnetic Cr cations, mediated by spin-orbit cou-\n10pling on the bridging ligand. In the case of CrI 3, the hopping occurs between the dorbitals\nof the nearest-neighbor Cr3+ions, through the conduction orbitals of bridging I atoms.\nTherefore, in order to clarify the DMI awaken in a system, one has to look at the electronic\nstructure of the system, reflecting the different available hopping channels.\nIn Fig. 4a, we show the electronic band structure of the four different cases, pristine, Br\nand Cl substitutions and single I vacancy. First of all, it is clear that the single substitution\nof I by Br (blue) and Cl (orange) does not change electronic structure of the crystalline\nmonolayer. One slight change is that the overall conduction band of Br and Cl is weakly\nshifted higher in energy. This can be associated with the overall decrease in the exchange\ninteraction since slightly higher energies are needed for the excitations. In addition, there\nare small splittings of bands at the high symmetry points which appear as degenerate bands\nin the pristine case. These small variations of the electronic band structure in both cases of\nBr and Cl substitution compared to the pristine case make it reasonable to expect the small\nDMI as we obtained, even though our parametric break of spatial charge symmetry, !, is\nremarkably large in these cases.\nOn the other hand, the band structure of the single iodine vacancy is completely dif-\nferent as compared to the pristine and substitution cases. Several occupied bands appear\nenergetically closer to the Fermi level, and one midgap band is created just above the Fermi\nlevel. In order to understand the spatial character of the midgap band we calculated the\npartial charge density at the \u0000point, as shown in Fig. 4b. Seemingly the absence of one I\nligand creates an additional channel on the remaining I ligand that is energetically (strongly)\nfavored for the exchange hopping from one Cr to another. Since that additional channel is\nfully asymmetric with respect to the Cr-Cr pair, it is the source of the large DMI seen in\nthe case of an iodine vacancy.\n11Figure 4: (a)The electronic band structure of four different cases, pristine, single Br and Cl\nsubstitution cases, and vacancy defect. (b)The partial charge density of the midgap state\nat\u0000which we label by P.\nChirality versus the electric field\nIt was recently shown that external out-of-plane electric field breaks the inversion symmetry\nin monolayer ferromagnets, and induces DMI - growing linearly with increasing applied\nelectric field.27–30However, the intensity of DMI in monolayer CrI 3is still under debate,\nsince the reported values strongly vary from one article to another, for example from 0.01 to\n0.4 meV/atom for electric field of 0.1 V/Å. It is likely that such large variation in reported\nDMI values stems from the choices of exchange-correlation functional approximations in\nrespective DFT calculations.30\nIn order to compare the results to DMI released at the structural defects presented so\nfar, we performed the calculations of DMI in a CrI 3monolayer exposed to electric field using\nthe consistent methodology throughout all the calculations. In Fig. 5a we show top and\n12Figure 5: (a)The profile of awaken DMI in case of externally applied electric field on the\nmonolayer CrI 3, top (above) and side view (below). (b)Linear dependence between the\ncomponents of awaken DMI and applied electric field. (c)The schematic representation of\nthe DMI in case of non-uniformly applied strain, together with the externally applied electric\nfield orthogonal to the monolayer. (d)The schematic representation of the DMI in case of\nthe edge influence and external electric field.\nside view of the DMI vectors induced in the CrI 3monolayer for the applied out-of-plane\nelectric field of 0.1, 0.2 and 0.4 V/Å, beyond which value the structure was no longer stable.\nNote that although the inversion symmetry is broken between the nearest-neighbor Cr atoms\nwhen electric field is introduced, the mirror plane orthogonal to their bond still defines the\nsymmetry operations. This results in the DMI vector orthogonal to the bond, as restricted\nby the Moriya’s symmetry rules.5This can be seen in Fig. 5b, where x-component of the\nDMI vector (parallel to the corresponding Cr-Cr bond) remains consistently zero, as the\nconsequence of symmetry rules. On the other hand, the yandzDMI components, as well\nas the total DMI intensity, grow linearly with the external electric field. The increase of\nelectric field by 0.1 V/Åresults in 0.04 meV gain in total DMI per Cr-pair. Moreover, DMI\ninduced by gating will change the sign with the change of polarity of the applied electric\nfield. Therefore, as was case with the locally applied strain, external electric field can be\nused as the turning knob to modify both magnitude and direction of the spin-chirality in a\nmagnetic 2D material.\nSince local strain and external electric field release comparable DMI, it is instructive\nto combine the inhomogeneous strain with the out-of-plane external electric field to achieve\n13richer patterns of possible chiral spin textures. In Fig. 5c we present a DMI profile that arises\nfrom nonuniform strain as shown in Fig. 3d when exposed to external electric field. One sees\nthat that DMI vectors induced in two manners ‘interfere’ in constructive and destructive way\nintermittently on the lattice, where DMI will be amplified by external electric field on one\ncheckerboard sublattice of Cr-Cr bonds and reduced on the other. In addition, with inclusion\nof the external electric field significant DMI will arise even on Cr-pairs where strain did not\ninduce any (depicted by black arrows in Fig. 5c).\nDMI pattern due to a joint effect of the sample edge and the applied electric field is\ndepicted in Fig. 5d. Similarly to the previous case, the electric field will intermittently\namplify and reduce the DMI vectors present at the edge, and release additional DMI further\nfrom the edge. However, here the influence of external field will be smaller compared to\nthe case of inhomogeneous strain, since DMI induced by the symmetry breaking on edge is\nsignificantly larger than the maximal one induced by electric field.\nFigure 6: (a)The released DMI versus the intrinsic electric field generated by the symmetry\nbreaking in monolayer CrI 3, for all investigated structures. (b)The geometric relation\nbetween the vectors of DMI and of the intrinsic electric field, for three studied structural\ndeviations from the pristine crystal.\nIntheprevioussectionswehaveshownhowdifferentwaysofinversionsymmetrybreaking\ninduce different profiles of DMI. Different scales for DMI values suggest that in case of\nphysically removed atoms from monolayer, inversion symmetry is ’more’ broken than in case\nof inhomogeneous periodic strain. Therefore, it would be very useful to quantify this degree\nof inversion symmetry breaking between nearest neighbor Cr atoms. In case of pristine\ncrystal, next to DMI in the point between Cr atoms, total crystal electric field is 0 as well,\n14due to inversion symmetry. Moreover, net-average of this electric field around that point\nis 0 as well. Next to this, introduction of electric field breaks the symmetry, as discussed.\nTherefore, it is reasonable to assume the vice versa: that when inversion symmetry is broken\nbetween Cr-Cr nearest neighbor pair, besides DMI, significant net crystal electric field will\nappear as well. We will use this field to quantify the break of inversion symmetry. In order to\ncalculate the crystal electric field in the point between two Cr atoms, for different structures,\nwe take into account the charge density of (valence) electrons as the !calculation.\nIn Fig. 6a we present the intensity of the crystal electric field between the Cr-Cr pairs\nof interest, for each structure under investigation, together with the corresponding DMI.\nIn Fig. 6b one can see the relative position of the two vectors. The results presented here\nnot only confirm that DMI arises with the break of inversion symmetry (non-zero electric\nfield), but also provide an additional information, about the degree of symmetry breaking\nwith different deviations from pristine monolayer. Applied strain, vacancy defects and edge\nof a monolayer can be understood as a building block for awakening DMI by breaking the\nsymmetry, each block having its particular characteristics. Locally applied strain breaks\nthe inversion symmetry on a smaller scale, just as external field. This is reflected in the\nelectric field between two nearest neighbor Cr atoms of order of magnitude 0.1 V/Å and the\nconsequence is DMI intensity of 0.01-0.1 meV/atom, tunable with the strain. Note that we\nreport the same DMI and electric field between Cr-Cr pair in case of 6% local strain and\n0.2 V/Åexternal electric field, applied to the monolayer CrI 3. Contrary to the local strain,\nor external electric field, edge or vacancy defects break the symmetry on the bigger scale.\nIn this case, the good scale for the electric field is 1 V/Å, and awaken DMI has value of\n1-6 meV/atom. In order to fully characterize each ’building block’ for releasing DMI, we\npresent the DMI and inner electric field vectors, between Cr-Cr pairs on Fig. 6 (d) for each\nstructural deviation from pristine crystal.\nOne aspect of the results presented here lies in the fact that size of electric field induced\nby different structural deviations, corresponds to the size of awaken DMI. This suggests\n15that calculated electric field in-between two Cr atoms is a good measure for the degree\nof symmetry breaking and therefore, it is reasonable to perform atomic-scale electric field\nmeasurements with each successful formation of magnetic monolayer. These measurements\nwillcontaininformationaboutthepositionanddegreeofthesymmetrybreakinamonolayer,\nand therefore implicitly, information about the position and magnitude of released DMI in\nthat monolayer.\nAnother aspect of our results comes from the fact that released DMI is restricted to only\nfew Å from the structural deviation that causes it, for each building block. This further\nsuggests that effect of two or more blocks combined can be considered as a superposition\nof individual effects, without significant ‘interaction’ of the blocks. Therefore, one can now\nestimate and even design many different profiles of DMI, as we did in a few examples, by\ncombining the building blocks presented here.1\nIn summary, we have brought out first guidelines on how to release DMI in 2D magnetic\nmaterials and tailor its textures further. Related to the amount of released DMI, we also\nquantifiedtheinversionsymmetrybreakinginmonolayerCrI 3, aftercalculatingtheemergent\nelectric field inside the crystal in cases of different structural deviations from the pristine\ncrystal. Moreover, we have compared the obtained results with the DMI awaken by external\nelectric field, and discussed the cumulative effects of different DMI sources. The employed\nprinciples in this work remain valid in any emergent magnetic 2D material, and our results\nprovide clear yet rich pathways towards design, control and characterisation of versatile\nDMI and spin textures and their use in spintronic and magnonic applications of magnetic\nmonolayers.\n1Though we restrict our analysis on typical example of monolayer ferromagnet, CrI 3, there is no reason\nwhy all conclusions from here could not be applied on monolayer antiferromagnets, such as e.g. FePS 3.\n16Acknowledgement\nThis work was supported by the Research Foundation-Flanders (FWO-Vlaanderen) and\nthe Special Research Funds of the University of Antwerp (TOPBOF). The computational\nresources and services used in this work were provided by the VSC (Flemish Supercomputer\nCenter), funded by Research Foundation-Flanders (FWO) and the Flemish Government –\ndepartment EWI.\nReferences\n(1) Huang, B.; Clark, G.; Navarro-Moratalla, E.; Klein, D. R.; Cheng, R.; Seyler, K. 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Bonell, and M. Jamet\nUniv. Grenoble Alpes, CEA, CNRS, Grenoble INP, IRIG-SPINTEC, 38000 Grenoble, France\n(Dated: September 10, 2020)\nImaging the magnetic con\fguration of thin-\flms has been a long-standing area of research. Since\na few years, the emergence of two-dimensional ferromagnetic materials calls for innovation in the\n\feld of magnetic imaging. As the magnetic moments are extremely small, standard techniques like\nSQUID, torque magnetometry, magnetic force microscopy and Kerr e\u000bect microscopy are challenging\nand often lead to the detection of parasitic magnetic contributions or spurious e\u000bects. In this work,\nwe report a new magnetic microscopy technique based on the combination of magnetic circular\ndichroism and Seebeck e\u000bect in semiconductor/ferromagnet bilayers. We implement this method\nwith perpendicularly magnetized (Co/Pt) multilayers sputtered on Ge (111). We further show that\nthe electrical detection of MCD is more sensitive than the Kerr magnetometry, especially in the ultra-\nthin \flm regime, which makes it particularly promising for the study of emergent two-dimensional\nferromagnetic materials.\nI. INTRODUCTION\nWith the recent emergence of two-dimensional\nferromagnets1{3, magnetic imaging techniques have to\nbe pushed to their ultimate detection limits to sense\nvery low magnetic moments and stray \felds. In this\nrespect, several advanced scanning magnetic probe mi-\ncroscopies have been successfully used to image the mag-\nnetic con\fguration of ultra-thin ferromagnets down to\nthe monolayer limit. For instance, magnetic force mi-\ncroscopy (MFM) and NV-center microscopy are sensitive\nto the magnetic stray \feld from the \flm by using a mag-\nnetic tip and a single NV spin in diamond respectively4,5.\nSpin-polarized scanning tunneling microscopy (SP-STM)\nis probing the unbalance between spin up and down den-\nsities at the Fermi level by tunneling magnetoresistance\nbetween the magnetic material and the atomically sharp\nmagnetic tip. It could be used to image magnetic do-\nmains in Fe 3GeTe 2at low temperature6. Electron mi-\ncroscopies like transmission electron microscopy in the\nLorentz mode or scanning electron microscopy with po-\nlarization analyzer (SEMPA) were also used to image the\nmagnetic domains and skyrmions in Fe 3GeTe 27,8. Both\nrely on the interaction between electrons and the mag-\nnetic \flm. Magneto-optical Kerr e\u000bect (MOKE) and\nphotoemission electron microscopy combined with x-ray\nmagnetic circular dichroism (XMCD-PEEM) relying on\nthe light-matter interaction were used either in scanning\nmode or in far \feld to probe the magnetic domains in\nseveral 2D ferromagnets9,10. Finally, the intrinsic semi-\nconducting properties of the ferromagnet itself could be\nused to image magnetic domains in CrBr 3taking ad-\nvantage of the optical selection rules for the absorption\nand emission of circularly polarized light11. Ultra-thin\n\flms are almost transparent for light and it is possible\nto use the substrate on which the material was grown\nor transfered to perform magnetic imaging. Indeed, the\ntransmitted light interacts with the ferromagnet through\nmagnetic circular dichroism and can be analyzed electri-\ncally by using the thermoelectric or photoelectric e\u000bectsin the semiconducting substrate. If the substrate exhibits\nstrong photoresponse, this last hybrid technique combin-\ning light and electrical measurements can be very sensi-\ntive to the magnetic state of the ultra-thin ferromagnet.\nBy scanning the light beam, the magnetic con\fguration\ncan be easily mapped at a submicrometer scale with high\nsignal-to-noise ratio.\nIn this work, we report the growth of ultra-thin per-\npendicularly magnetized electrodes on Ge (111). Ger-\nmanium exhibits strong thermoelectric and photoelec-\ntric responses12. In order to obtain perpendicular mag-\nnetic anisotropy (PMA), we grow (Co/Pt) multilayers\nthin \flms using magnetron sputtering. In these \flms,\nthe reduced symmetry and spin-orbit coupling at the\ninterface between Co and Pt are responsible for the\nPMA13,14. Moreover, it was shown that the PMA in-\ncreases with the number of repetitions (i.e. the number\nof interfaces).15,16. We probe the local magnetization ori-\nentation using simultaneously the anomalous Hall e\u000bect,\nKerr microscopy and a new original technique based on\nthe helicity dependence of the photovoltage in Ge by the\nmagnetic circular dichroism (MCD) in (Co/Pt). This\ntechnique relies on the Seebeck e\u000bect in Ge. We study\nthe (Co/Pt) thickness dependence of the Kerr e\u000bect and\nthe MCD signal by changing the number of (Co/Pt) rep-\netitions and we demonstrate that this MCD-based detec-\ntion becomes much more sensitive than the Kerr e\u000bect\nin the ultra-thin \flm regime, which is promising for the\nfuture investigation of the magnetic properties of two-\ndimensional ferromagnets.\nII. SAMPLE PREPARATION AND\nEXPERIMENTAL SETUP\nIn this study, we use a 2 mm- thick Ge/Si (111) \flm\ndeposited by low-energy plasma-enhanced chemical va-\npor deposition (LEPECVD)17. The deposition rate was\n\u00194 nm s\u00001and the substrate temperature was \fxed at\n500\u000eC. Post-growth annealing cycles have been used toarXiv:2009.03982v1  [cond-mat.mtrl-sci]  8 Sep 20202\nimprove the crystal quality. The Ge layer is non inten-\ntionally doped with a residual electron carrier concentra-\ntionn\u00192\u00021016cm\u00003as measured by Hall e\u000bect at room\ntemperature.\nThis lown-doped 2 mm-thick Ge/Si (111) substrate\nis subsequently cleaned in acetone and isopropanol in an\nultrasonic bath for 5 minutes to remove organic species.\nThen the substrate is dipped into a 50 % hydro\ruoric\nacid solution to remove the native Ge oxide and is trans-\nferred to the sputtering chamber. We do not heat the\nsubstrate during the growth as it promotes the chemi-\ncal reaction between Co and Ge atoms at the interface,\nwhich is detrimental for the magnetic properties. The\nchamber base pressure is in the 10\u00008mbar range. Af-\nter introducing the sample, we set the Ar pressure in the\nchamber to PAr\u00191:2\u000210\u00002mbar using a \rowmeter. A\n5 W DC power is applied to generate the plasma, giving\na deposition rate of 0.25 \u0017A/s for Co and 0.79 \u0017A/s for Pt\nas measured by a quartz microbalance. We start with the\ndeposition of a 0.5 nm-thick Co layer and end with a 1.8\nnm-thick Pt layer which also acts as a capping layer pre-\nventing Co oxidation under atmospheric conditions. In\nthis work, we grew (Co/Pt) nsamples where the (Co/Pt)\nbilayer is repeated from one to four times ( n= 1;2;3 and\n4).\nFigure 1. (color online) a) Sketch of the (Co/Pt) n/Ge (111)\nsample layout used for magnetic microscopy experiments. b)\nAnomalous Hall e\u000bect c) Magneto-optical Kerr microscopy d)\nHelicity dependent photovoltage due to MCD: the di\u000berence\nof transmitted power between \u001b+and\u001b\u0000light helicities re-\nsults in a di\u000berence of temperature distribution in the sample.\nThis di\u000berence is recorded electrically using the Seebeck ef-\nfect of Ge: the magnetic con\fguration \frst translates into a\nthermal information and then into an electrical one.\nWe then proceed with the de\fnition of 200 \u000250mm2\nHall bars in the (Co/Pt) n\flm. We \frst use the laser\nlithography technique to de\fne the conduction channel\nand we etch the (Co/Pt) n\flm using ion beam etch-\ning (IBE). Electrical contacts are lithographically de\fned\nand Ti(5 nm)/Au(120 nm) contacts are deposited by e-\nbeam evaporation. The \fnal device is sketched in Fig. 1a). The electrical contacts allow for magnetic charac-\nterizations by magnetotransport measurements and the\nchannel is large enough to perform Kerr microscopy. One\ncontact is not connected to the Hall bar, in order to mea-\nsure the voltage between the ferromagnetic \flm and the\nGe substrate and to detect a possible non-local spin sig-\nnal.\nAs shown in Fig. 1 b), a DC current IDCis applied in\nthe (Co/Pt) bar, the electrons are de\rected transversely\nas a consequence of the anomalous Hall e\u000bect (AHE). The\ntransverse resistance, de\fned as RAHE, is proportional to\nMz, the out-of-plane component of the magnetization.\nIn the meantime, we perform magneto-optical Kerr\ne\u000bect (MOKE) imaging of the magnetization. The\nsample is illuminated with a circularly polarized laser\nbeam, the circular polarization ( \u001b\u0006) is modulated at\nf= 42 kHz by using a photoelastic modulator (PEM).\nThe re\rected light is then analyzed by a polarizer and\nthe light intensity is recorded using a photodiode. The\nresulting photovoltage is demodulated at 2 !by a lock-\nin ampli\fer, to obtain the Kerr rotation \u0012k(see Fig. 1 c)).\nThe (Co/Pt) \flm being very thin, the circularly po-\nlarized light is partially transmitted through the \flm\nand electron-hole pairs are photogenerated in Ge. As\nthe (Co/Pt) magnetization is perpendicular, the left and\nright circularly polarized photons have di\u000berent trans-\nmission coe\u000ecients due to MCD18(see Fig. 1 d)). Due\nto light absorption, the Ge layer is locally heated at\nthe position of the laser spot and a Seebeck voltage\nVSeebeck\nDC develops between the Au electrodes in Fig. 1\nd):VSeebeck\nDC =V\u001b+=S\u0001T\u001b+for the\u001b+polarized light\nandVSeebeck\nDC =V\u001b\u0000=S\u0001T\u001b\u0000for the\u001b\u0000polarized light,\nSbeing the Seebeck coe\u000ecient of Ge and \u0001 T\u001b+(resp.\n\u0001T\u001b\u0000) the temperature di\u000berence between the Au elec-\ntrodes for the \u001b+(resp.\u001b\u0000) polarized light. Note that\nif the laser spot is exactly located in the middle of the\ntwo Au electrodes, the Seebeck voltage is zero for both\nhelicities. Since the \u001b+and\u001b\u0000polarized lights are dif-\nferently absorbed in Ge due to the MCD in the (Co/Pt)\nlayer,V\u001b+6=V\u001b\u0000and we detect a voltage VMCD at the\nPEM frequency as a combination of the Seebeck e\u000bect in\nGe and the MCD in (Co/Pt).\nThe DC and demodulated voltages VDCandVMCD are\nsimultaneously recorded with a nanovoltmer and lock-in\nampli\fer, respectively, while the magnetic \feld is swept.\nAlternatively, we can \fx the magnetic \feld and image\nthe sample magnetic con\fguration by scanning the laser\nbeam at normal incidence.\nIII. ANOMALOUS HALL EFFECT, KERR\nEFFECT AND ELECTRICAL DETECTION OF\nTHE MCD\nWe \frst focus on the (Co/Pt) 3sample, Fig. 2 a) shows\nthe sample re\rectivity recorded by scanning the laser3\nbeam on the microstructure. The (Co/Pt) Hall bar pat-\ntern is in green, the Au/Ti contacts in red and the Ge\nsubstrate in blue. The circularly polarized laser beam is\n\frst focused on the center of the Hall bar (at the position\nof the red spot). Fig. 2 b-d) show the magnetic signals\nfor a\u0006500 Oe magnetic \feld sweep, applied perpendicu-\nlarly to the \flm plane, recorded simultaneously using the\nthree aforementioned techniques. All the measurements\nare performed at room temperature.\nFigure 2. (color online) a) Two-dimensional re\rectivity map\nof the (Co/Pt) 3/Ge Hall bar, the red circle indicates the laser\nbeam position during the magnetic \feld sweep (applied per-\npendicularly to the sample plane). The black dashed line\ncorresponds to the line scan along xof Fig. 3 a-d). b) AHE\nhysteresis loop . c) MOKE hysteresis loop using a 100 % cir-\ncularly polarized red light ( \u0015= 661 nm) focused on the Hall\nbar center, the spot size is about 1.5 mm. d)VMCD hysteresis\nloop. The voltage is demodulated at the PEM frequency !\nand is measured between a Hall bar contact and the substrate,\na currentIDC= 100 mA is applied during the measurement.\nIn this geometry, the observation of a square hysteresis\nloop indicates that the (Co/Pt) 3sample magnetization\nis out-of-plane. For this n= 3 repetitions sample, the\ncoercive \feld is Bc\u0019160 Oe. We also note that the\nMCD signal is one order of magnitude larger than the\nKerr signal, so the technique looks interesting for ultra-\nthin ferromagnetic \flms where the Kerr signal amplitude\nis very small.\nThe anomalous Hall e\u000bect gives a macroscopic picture\nof the magnetization, whereas the MOKE and MCD tech-\nniques can be spatially resolved by scanning the sample\nwith the laser beam. We perform line scans along the\nxdirection. We \frst apply +500 Oe or \u0000500 Oe along\nzto saturate the \flm magnetization either up or down\nand then record the corresponding remanent state + Mr\nor\u0000Mrat zero \feld. Fig. 3 a) shows the sample re\rec-\ntivity, the (Co/Pt) \flm being more re\rective than Ge,\nit corresponds to the central area where the photodiode\nsignal is the largest. Fig. 3 b) reports the AHE line scans,\na weak spatial dependence of the signal is observed as a\nFigure 3. (color online) Line scans along the xdirection (black\ndashed line in Fig. 2 a)) of the remanent magnetic states + Mr\nalong +zin blue and \u0000Mralong\u0000zin red (B= 0 T). a)\nSample re\rectivity. b) Anomalous Hall e\u000bect c) VMCD and d)\nKerr angle.\nconsequence of the Seebeck e\u000bect that takes place due to\nthe scanning laser spot heating locally the Ge \flm (this\ncontribution can be removed by using an AC current and\na lock-in detection to measure the AHE). Fig. 3 c) and\nd) show the remanent magnetization measured by VMCD\nand Kerr e\u000bect, respectively. A clear contrast can be\nobserved in both cases and we con\frm the local nature\nof the MCD signal: when the laser beam directly illu-\nminates the Ge \flm, the VMCD signal vanishes. Again,\nwe note that the VMCD signal is more than one order of\nmagnitude larger than the Kerr e\u000bect signal. In order\nto better understand the nature of the VMCD signals, we\nthen performed large two-dimensional maps of the mag-\nnetic con\fguration.\nThe magnetization is \frst initialized in the + Mrre-\nmanent state by applying a +500 Oe external magnetic\n\feld along + z. Fig. 4 a) shows the sample re\rectivity,\nthe Hall bar contours are highlighted by a black dashed\nline. Fig. 4 b) and d) show the VMCD signal and the\nDC photovoltage VDC, respectively, using the contacts\ncon\fguration shown in Fig. 1a. We observe that the DC\nphotovoltage is positive when the laser beam scans the\ntop area (Y > 0mm) and negative in the bottom area\n(Y < 0mm). It corresponds to the Seebeck voltage in\nGe due to the temperature di\u000berence between the two\nelectrical contacts induced by the laser spot heating.\nInterestingly, we observe the same behavior for the\nVMCD signal (demodulated at the PEM frequency). By\nusing both the DC and MCD photovoltages, we can \frst\ncalculate\r, the MCD signal (in %) of the (Co/Pt) \flm:\n\r=VMCD=VDC\u00190:3%. This normalization can be\nperformed point-by-point, for each position of the laser\nbeam and results in a position-independent map of the\nmagnetic con\fguration. The DC photovoltage intensity4\nalso allows us to estimate the temperature gradient\nin the Ge channel using the Seebeck e\u000bect relation\nand the Seebeck coe\u000ecient of Ge ( S= 330 mV/K)12.\n\u0001T=VMax\nDC=S\u001936 K .\nFigure 4. (color online) Two-dimensional maps of the rema-\nnent magnetic states + Mralong +z(B= 0 T) for IDC= 0\nA. a) Sample re\rectivity. b) VMCD c) Kerr angle and d) DC\nphotovoltage.\nTo further understand how the VMCD signal is a\u000bected\nby the temperature distribution in Ge when scanning\nthe laser beam, we record hysteresis loops for di\u000berent\nvertical positions ( Y) of the laser spot on the Hall bar.\nFig. 5 a) shows that the hysteresis loop signal is reversed\nbetweenY > 0mm andY < 0mm while the Kerr e\u000bect\nis independent of the beam position (Fig. 5 b)). The\ndi\u000berence of signal between the two remanent states is\nplotted as a function of Yin Fig. 5 c), we clearly see the\nVMCD signal changing with the laser beam position.\nTheVMCD signal being geometry-dependent, it is\nnot suitable and reliable to perform magnetic imaging.\nSeveral approaches can be used to solve this problem.\nFirst, one can simply normalize the VMCD signal by\nthe DC photovoltage VDCto obtain an almost position-\nindependent measurement. However, in the region lo-\ncated in the middle of the two electrical contacts, the\nsensitivity of this technique vanishes.\nOne can optimize the contacts geometry to have an\nalmost uniform temperature in Ge at the level of the\nmagnetic microstructure to image regardless of the laser\nbeam position by patterning one contact close to the mi-\ncrostructure and a second one far away. This would opti-\nmize the Seebeck e\u000bect-based detection of the magnetic\ncircular dichroism. Moreover, using a material with a\nlarge Seebeck coe\u000ecient like Ge ( S= 330 mV/K) is nec-\nessary to obtain large signals.\nFigure 5. (color online) a) VMCD and b) Kerr angle hysteresis\nloops recorded for di\u000berent vertical positions of the laser beam\non the Hall bar. c) Di\u000berence of signal between the positive\nand negative remanent states from VMCD and Kerr angle as\na function of the position of the beam on the Hall bar.\nAn alternative technique consists in applying a bias\ncurrent through the Hall bar, along the MCD electrical\ndetection axis (along yhere). In this way, the charge\ncarriers photogenerated by the laser beam are drifting\nalong the applied bias electric \feld. However, due to\nMCD, the densities of photogenerated charge carriers\nfor\u001b+and\u001b\u0000polarized lights are di\u000berent giving rise\nto a modulated voltage Vdrift\nMCD at the PEM frequency.\nThen, the total VMCD signal contains both the Seebeck\nMCD voltage VSeebeck\nMCD and this drift component Vdrift\nMCD\nexcept that the Seebeck voltage is even (independent)\nwith respect to the bias current direction while the drift\ncomponent is odd. We report this type of measurement\nin Fig. 6. The magnetization is prepared in \u0000Mrstate\nby applying a\u0000500 Oe external magnetic \feld. A DC\nbias current is dynamically applied between the two\ndetection contacts and the even and odd components\nof theVMCD signal with respect to the current are\ncalculated and plotted as a function of the position of\nthe laser spot. Fig. 6 a) shows two-dimensional maps\nof the bias current-dependent voltage Vodd\nMCD for bias\ncurrents from 20 mA to 100 mA, the corresponding\npro\fles for X= 0mm are shown in Fig. 6 c). We observe\na clear spatial-independent VMCD signal, which varies\nlinearly with the bias current. The current-independent\ncomponent Veven\nMCD maps are reported in Fig. 6 b) and the\ncorresponding pro\fle for X= 0mm in Fig. 6 d), we \fnd\nagain the \fngerprint of the Seebeck e\u000bect-based MCD\ndetection.5\nFigure 6. (color online) a) Two-dimensional maps of the bias\ndependent (odd with I) component of the VMCD signal for\nbias currents from 20 mA to 100 mA. b) Corresponding bias\nindependent (even with I) component. c) Vodd\nMCD pro\fles at\nX= 0mm. d)Veven\nMCD pro\fles atX= 0mm.\nIn the following, we do not apply any bias current and\ntake advantage of the position of the disconnected con-\ntact far from the the Hall bar to maximize the Seebeck\ne\u000bect-based detection of MCD. By using this con\fgura-\ntion, the scanning area is far from the middle of the two\ndetection contacts and the Seebeck voltage ( i.e.VMCD\nsignal) is almost independent of the position of the laser\nbeam on the scanned area. This is necessary to have a\nreliable magnetic image of the (Co/Pt) microstructure.\nWe \frst investigate the dependence of the magnetic sig-\nnals as a function of the repetition number nof (Co/Pt)\nbilayers. Fig. 7 summarizes the results where the magne-\ntization is measured simultaneously using the VMCD and\nthe Kerr e\u000bect. The light beam is focused on the cen-\nter of each Hall bars as illustrated in Fig. 1 a). When\nsweeping the magnetic \feld perpendicularly to the \flm\nplane, hysteresis loops are observed, indicating that all\nthe \flms show PMA. We can also notice that the coer-\ncive \feld increases with the number of repetitions, as a\nconsequence of a larger magnetic anisotropy due to the\nincrease of the number of interfaces19,20. TheVMCD sig-\nnal is approximately one order of magnitude larger than\nthe Kerr signal, regardless the number of repetitions. We\nstress out the fact that the signal to noise ratio is also\nsigni\fcantly larger when using the VMCD technique, the\nlock-in detection parameters (\fltering and averaging) be-\ning the same for both techniques. We also observe that\nVMCD increases when decreasing the number of repeti-\ntions whereas the Kerr e\u000bect signal decreases as shown in\nFig. 7 c) and d). It con\frms the fact that this technique\nis very interesting to detect the magnetization of ultra-\nthin ferromagnets where the Kerr e\u000bect signal is barely\ndetectable using a conventional Si-based photodiode.\nIn order to better understand the thickness dependence\nof the MCD signal, we consider \u0015Land\u0015R, the absorp-\ntion length of the (Co/Pt) n\flm for left and right cir-cular helicities. We de\fne the average absorption length\nas:\u0015= (\u0015L+\u0015R)=2 and the contrast of absorption due\nto the MCD as: \u000e= (\u0015L\u0000\u0015R)=2. The transmitted\nlight intensity is expressed as IR(L)=I0:exp\u0000\n\u0000t=\u0015R(L)\u0001\nfor the right-handed (left-handed) circularly-polarized\nlight. The Seebeck voltage is given by the temperature\ndi\u000berence which is proportional to the light intensity:\nVSeebeck =S\u0001T=A:S:I , whereAis a constant of the\nmaterial. If we now assume that \u000e <<\u0015 , we obtain the\nfollowing expression for VMCD:\nVMCD =VL\u0000VR=A:S:(IL\u0000IR) =A:I0:S:\u000e:t\n\u00152exp\u0000t\n\u0015\n(1)\nThis relation shows that unlike the Kerr e\u000bect, the\nMCD signal has an optimum of sensitivity when t=\u0015. In\nour case, we can see that the VMCD signal is already in the\nexponential decrease regime, implying that the optimum\nof sensitivity is below n= 2 (equivalent to 4.6 nm). This\nalso indicates that the techniques will be most suited for\nferromagnetic metals, where the absorption length is in\nthe nanometer range.\nFigure 7. (color online) Hysteresis loops for di\u000berent (Co/Pt)\nrepetitions ( n= 2;3;4) simultaneously measured by a) the\nVMCD technique and b) the Kerr e\u000bect. The beam is focused\non the center of the Hall bar for each (Co/Pt) nsample and\nthe laser power is 650 mW. c) Summary of the two magnetic\nsignal dependence with the Co/Pt repetition number. d) Ra-\ntio between the VMCD and the Kerr e\u000bect signals as a function\nof the number of repetitions.\nIn order to further con\frm the nature of the VMCD\nsignal, we vary the incident light polarization. The PEM\nis used to control the light helicity, the retardation is\n\fxed at 0.25 \u0015. As shown in Fig. 8 a), we rotate the\nentrance polarizer with respect to the PEM axes. The\nrotation angle is de\fned as \u000b(see Fig. 8 a) and b)). The\nlight polarization is linear when the entrance polarizer\nand the PEM axis are aligned ( \u000b= 0\u000e, 90\u000e, 180\u000eand6\n270\u000e). The light polarization is circular for \u000b= 45\u000e\nand 225\u000e(right handed), and for \u000b= 135\u000eand 315\u000e\n(left handed).\nFigure 8. (color online) a) Schematic top view of the scan-\nning confocal setup, the light polarization is obtained by as-\nsociating a linear polarizer and a photoeleastic modulator. b)\nDe\fnition of the angle \u000b,xandydenote the PEM optical\naxis. c) Laser power dependence with the polarizer angle \u000b.\nd)VMCD signal normalized by the laser power, the top insets\nindicate the light polarization states.\nHere, we focus on the (Co/Pt) 4. A 1000 Oe exter-\nnal magnetic \feld is \frst applied to saturate the mag-\nnetization along the + zdirection, it is then turned to\nzero to measure the remanent magnetization state. The\nlaser beam is focused on the Hall bar center and the de-\npendence on the polarizer angle is recorded. The laser\nbeam is already polarized out of the optical \fber, so the\ntransmitted laser power is also a\u000bected by the polarizer\nrotation. The laser power is also recorded using a pow-\nermeter for each polarizer angle. As shown in Fig. 8 c),\nPlaser follows the Malus law:\nPlaser =P0cos2(\u000b\u0000\u000b0) (2)\nWhereP0is the nominal laser power, \u000bis the angle\nbetween the polarizer and the PEM optical axis and \u000b0\nis the angle between the initial laser beam polarization\nand the \frst polarizer (see Fig. 8 b)). The power depen-\ndence on the polarizer angle gives minima for \u000b= 20\u000e\nand 200\u000e, indicating that \u000b0= 110\u000e. In order to cor-\nrectly measure the VMCD signal, we have to normalize\nthe recorded VMCD byPlaser. The dependence of VMCD\non the polarizer angle is reported in the Fig. 8 d), the\ninset on top shows the incident light polarization state.\nWe observe a cos \u000bsin\u000bangular dependence: VMCD van-\nishes when \u000b= 0\u000e, 90\u000e, 180\u000eand 270\u000e, i.e. when the\nlight polarization is linear. It shows minima (maxima)\nfor\u000b= 45\u000eand 225\u000e(\u000b= 135\u000eand 315\u000e) for\u001b+and\u001b\u0000\nlight polarizations respectively. This result emphasizes\nthe fact that the detected voltage is due to the di\u000berentabsorption of circularly polarized light by the ferromag-\nnetic \flm, resulting in di\u000berent photovoltages in Ge for\nclockwise and counterclockwise light helicities.\nFigure 9. (color online) a) AHE hysteresis loop recorded with\na currentIDC= 100 mA. b) MOKE hysteresis loop using a\n100 % circularly polarized red light ( \u0015= 661 nm) focused\non the Hall bar center, the spot size is about 1.5 mm. c)\nVMCD hysteresis loop, the voltage is demodulated at the PEM\nfrequencyfand measured between two Hall bar contacts.\nFinally, to prove that the VMCD signal is related to a\nphotovoltage generated in Ge and not directly in the fer-\nromagnetic \flm, we have grown a (Co/Pt) 2\flm on a SiO 2\nsubstrate and patterned the same Hall bars. Again, the\nmagnetic properties are measured using simultaneously\nthe AHE,VMCD and the Kerr e\u000bect. As shown in Fig. 9,\nthe magnetic anisotropy is also perpendicular, the hys-\nteresis loop can be detected using the AHE or the Kerr\ne\u000bect but there is no VMCD signal. This result con\frms\nthat the measured photovoltage comes from Ge due to\nMCD in the (Co/Pt) \flm.\nIV. APPLICATION TO THE STUDY OF\nMAGNETIC DOMAIN WALL MOTION\nWe exploit the MCD detection technique to image mul-\ntidomain magnetic con\fgurations and the motion of do-\nmain walls. Here, we focus on the (Co/Pt) 3sample. We\nintroduce a magnetic domain wall in the Hall bar by ap-\nplying a speci\fc magnetic \feld sequence and use the two\nmagnetic microscopy techniques (Kerr e\u000bect and MCD\ndetection) to image the domain wall propagation. We\nrepeat the following \feld sequence: the magnetization is\n\frst saturated along + z, then a negative magnetic \feld\nBnuclis applied to nucleate domains and we image the\nmagnetic con\fgurations. The sequence is iterated by in-\ncreasingjBnucljin order to move the domain wall.\nThe magnetic con\fguration can be imaged simultane-\nously using the Kerr e\u000bect microscopy and the electrical\ndetection of the local magnetization based on the MCD7\nFigure 10. (color online) a) Re\rectivity maps. b) VMCD maps.\nc) Kerr e\u000bect maps for di\u000berent applied magnetic \felds. Be-\nfore each two-dimensional scan, a +500 Oe \feld is \frst applied\nto saturate the magnetization along the + zdirection, a pre-\ncise negative magnetic \feld value is then applied to nucleate\nand propagate a domain wall. We can see the domain wall\npropagating when increasing the magnitude of the magnetic\n\feld.\nin the (Co/Pt) \flm. Fig. 10 a) shows the re\rectivity\nof the sample for di\u000berent magnetic \feld intensities, the\n(Co/Pt) (resp. Ge) \flm corresponds to the red (resp.\nblue) color. Fig. 10 b) and c) show the Kerr e\u000bect and\nVMCD maps recorded for the di\u000berent applied magnetic\n\felds. ForB=\u0000109 Oe, the magnetization is still satu-\nrated and uniform on the Hall bar scanned area. Then,\nby iterating the magnetic \feld sequence, we see a domain\nwall propaating in the Hall cross, the magnetic domains\nare pointing toward + z(in red) and\u0000z(in blue). For\nB=\u0000119 Oe, corresponding to the box delimited by a\nblack dashed line in Fig. 10 ,) the domain wall is located\nin the middle of the Hall cross. By further increasing\nthe negative magnetic \feld, we observe the propagation\nof the wall along the \u0000ydirection. Interestingly, the do-\nmain wall (resp. its propagation) is perpendicular (resp.\nparallel) to the current applied in the Hall bar.\nHere, by imaging the two-dimensional magnetization\nmaps simultaneously with the two techniques, we con-\nclude about the very high sensitivity of the MCD detec-\ntion technique.\nV. CONCLUSION\nTo conclude, we have successfully grown perpendicu-\nlarly magnetized thin \flms on a Ge (111) substrate. Themagnetic properties of (Co/Pt) multilayers were investi-\ngated using the anomalous Hall e\u000bect, magneto-optical\nKerr e\u000bect microscopy and a new hybrid electro-optical\ntechnique based on the magnetic circular dichroism in\n(Co/Pt) and combining the thermoelectric and semicon-\nducting properties of Ge. Our study reveals that this hy-\nbrid technique shows several advantages for the magnetic\ncharacterization of ultra-thin \flms and could be gener-\nalized to a large variety of semiconducting (Si, GaAs...)\nand thermoelectric substrates (Bi, Bi 2Se3...).\nThe detection being electrical, it is particularly well\nsuited for future investigation of the magnetic properties\nof 2D materials, where the standard magnetic imaging\ntechniques are di\u000ecult to setup. Moreover, both the sig-\nnal and the signal-to-noise ratio are much larger than the\nKerr e\u000bect ones.\nWe showed that the electrical detection of the mag-\nnetic circular dichroism of (Co/Pt) originates from the\nSeebeck e\u000bect, as a result of the di\u000berence of thermal\ngradients between the two electrical contacts. We\ndemonstrated that the measurement geometry can be\noptimized in order to maximize this thermal contribution\nand obtain a uniform measurement by using strongly\nasymmetric contacts. Alternatively, by applying a bias\ncurrent parallel to the detection axis, one can suppress\nor enhance the total sensitivity of the technique by\ncombining the thermal and drift contributions. The drift\ncomponent can also be isolated by using its symmetries\nwith respect to the bias current in order to obtain a\nmeasurement almost independent of the geometry.\nFinally, we point out the fact that it is not necessary\nto connect electrically the ferromagnetic \flm, the two\ncontacts can simply be made on the semiconducting sub-\nstrate close to the ferromagnet. This feature added to\nthe high sensitivity of the technique in the ultra-thin \flm\nregime makes this technique an excellent alternative to\ntraditional magnetometry for the investigation of ferro-\nmagnetism in the emergent 2D ferromagnets grown (or\ntransfered) on semiconductors.\nVI. ACKNOWLEDGEMENTS\nThe authors acknowledge the \fnancial support from\nthe ANR projects ANR-16-CE24-0017 TOP RISE and\nANR-18-CE24-0007 MAGICVALLEY as well as Dr.\nCarlo Zucchetti and Dr. Federico Bottegoni from the Po-\nlitecnico di Milano (Italy) for setting up the microscope\nand for fruitful discussions.\n1B. Huang, G. Clark, E. 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Phys. 97, 10J109\n(2005)."    },    {        "title": "2009.06187v1.Voltage_driven_Magnetization_Switching_via_Dirac_Magnetic_Anisotropy_and_Spin__orbit_Torque_in_Topological_insulator_based_Magnetic_Heterostructures.pdf",        "content": "arXiv:2009.06187v1  [cond-mat.mtrl-sci]  14 Sep 2020Voltage-driven Magnetization Switching via Dirac Magneti c Anisotropy and Spin–orbit Torque in\nTopological-insulator-based Magnetic Heterostructures\nTakahiro Chiba1and Takashi Komine2\n1National Institute of Technology, Fukushima College, 30 Na gao,\nKamiarakawa, Taira, Iwaki, Fukushima 970-8034, Japan\n2Graduate School of Science and Engineering, Ibaraki Univer sity,\n4-12-1 Nakanarusawa, Hitachi, Ibaraki 316-8511, Japan\n(Dated: September 15, 2020)\nElectric-field control of magnetization dynamics is fundam entally and technologically important for future\nspintronic devices. Here, based on electric-field control o f both magnetic anisotropy and spin–orbit torque,\ntwo distinct methods are presented for switching the magnet ization in topological insulator (TI) /magnetic-TI\nhybrid systems. The magnetic anisotropy energy in magnetic TIs is formulated analytically as a function of the\nFermi energy, and it is confirmed that the out-of-plane magne tization is always favored for the partially occupied\nsurface band. Also proposed is a transistor-like device wit h the functionality of a nonvolatile magnetic memory\nthat uses voltage-driven writing and the (quantum) anomalo us Hall effect for readout. For the magnetization\nreversal, by using parameters of Cr-doped (Bi 1−xSbx)2Te3, the estimated source-drain current density and gate\nvoltage are of the orders of 104–105A/cm2and 0.1 V , respectively, below 20 K and the writing requires n o\nexternal magnetic field. Also discussed is the possibility o f magnetization switching by the proposed method in\nTI/ferromagnetic-insulator bilayers with the magnetic proxi mity effect.\nI. INTRODUCTION\nElectrical control of magnetism is essential for the next ge n-\neration of spintronic technologies, such as nonvolatile ma g-\nnetic memory, high-speed logic, and low-power data trans-\nmission [ 1]. In these technologies or devices, the mag-\nnetization direction of a nanomagnet is controlled by an\nelectrically driven torque rather than an external magnetic\nfield. A representative torque is the current-induced spin– orbit\ntorque (SOT) [ 2] in heavy-metal/ferromagnet heterostruc-\ntures, wherein the spin Hall e ffect in the heavy metal [ 3,4]\nand/or the Rashba–Edelstein e ffect (also known as the inverse\nspin-galvanic effect) at the interface [ 5] play crucial roles in\ngenerating the torque. Recently, several experiments have re-\nported a giant SOT e fficiency in topological insulator (TI)-\nbased magnetic heterostructures such as both TI /magnetic-TI\n[6] and TI/ferromagnetic-metal hybrid systems [ 7,8]. A TI\nhas a metallic surface state in which the spin and momen-\ntum are strongly correlated (known as spin–momentum lock-\ning) because of a strong spin–orbit interaction in the bulk\nstate [ 9,10], which is expected to lead to the giant SOT\n[11]. Indeed, magnetization reversal by SOT has been pro-\nposed theoretically [ 12–15] and demonstrated experimentally\nin magnetic TIs [ 6,16,17] as well as TI/ferromagnet bilay-\ners [18–23]. Remarkably, the critical current density required\nfor switching is of the order of 105A/cm2, which is much\nsmaller than the corresponding values (106–108A/cm2) for\nheavy-metal/ferromagnet heterostructures [ 3–5]. In particu-\nlar, the magnetization switching of magnetic TIs is more e ffi-\ncient: Yasuda et al. [17] succeeded in reducing the switching\ncurrent density by means of a current pulse injected paralle l\nto a bias magnetic field, whereas Fan et al. [16] realized mag-\nnetization reversal by means of a scanning gate voltage with a\nsmall constant current and in-plane magnetic field.\nAnother important method for controlling magnetic proper-\nties is the electric-field e ffect in magnets, such as controllingferromagnetism in dilute magnetic semiconductors [ 24], ma-\nnipulating magnetic moments in multiferroic materials [ 25],\nand changing the magnetic anisotropy in an ultrathin film of\nferromagnetic metal [ 26–28]. In particular, voltage control\nof magnetic anisotropy (VCMA) in ferromagnets promises\nenergy-efficient reversal of magnetization by means of what\nis known as voltage torque, which has been demonstrated\nby using a pulsed voltage under a constant-bias magnetic\nfield in a magnetic tunnel junction [ 29]. This approach is\nbased on the clocking scheme in which one first sets the fer-\nromagnet to an initial stable state under the application of\nan external bias and then inputs the signal voltage pulse to\ndetermine the final state. Recent experiments using heavy-\nmetal/ferromagnet/oxide heterostructures have demonstrated\nthat the critical current for SOT-driven switching of perpe n-\ndicular magnetization can be modulated by an electric field\nvia VCMA [ 30,31]. By contrast, the electric-field e ffect in\na magnetic TI [ 32,33] and a TI/ferromagnetic-insulator (FI)\nbilayer [ 34,35] has been investigated to date in terms of the\nvoltage-torque-driven magnetization dynamics. Note that Se-\nmenov et al. [34] demonstrated magnetization rotation be-\ntween the in-plane and out-of-plane directions by VCMA at\nthe TI/FI interface. Therefore, it becomes highly desirable to\ncontrol the magnetic anisotropy and SOT simultaneously by\nmeans of the electric field in TI-based magnetic heterostruc -\ntures, which may lead to magnetization switching that is mor e\nenergetically efficient.\nIn this paper, inspired by the SOT and VCMA approaches\nfor magnetization control, we combine them and present\ntwo distinct clocking methods for magnetization switching in\nTI/magnetic-TI hybrid systems. First, we model the current-\ninduced SOT and magnetic anisotropy energy (MAE) in TI-\nbased magnetic heterostructures as a function of the Fermi\nenergy to determine a stable magnetization direction at the\nelectrostatic equilibrium. Then we propose a transistor-l ike\ndevice with the functionality of a nonvolatile magnetic mem -\nory that uses (i) VCMA writing that requires no external mag-2\nnetic field and (ii) readout based on the anomalous Hall e ffect.\nFor the magnetization reversal, we estimate the source-dra in\ncurrent density and gate voltage. Finally, we show the switc h-\ning phase diagram for the input pulse width and voltages as a\nguide to realizing the proposed method of magnetization re-\nversal. We also discuss the possibility of using the propose d\nmethod for magnetization reversal in TI /FI bilayers with mag-\nnetic proximity [ 36–38] at the interface.\nII. MODEL\nWe begin this section by deriving the current-induced SOT\nin TI-based magnetic heterostructures by using the current -\nspin correspondence of two-dimensional (2D) Dirac electro ns\non the TI surface. Next, in the same system we formulate the\nMAE analytically to determine a stable magnetization direc -\ntion at the electrostatic equilibrium and to reveal the cont rolla-\nbility of the VCMA e ffect. To model the SOT and VCMA, we\nconsider 2D massless Dirac electrons on the TI surface, whic h\nis exchange coupled to the homogeneous localized moment of\na magnetic TI (or an attached FI as discussed in Sec. IV B ).\nWhen the surface electrons interact with the localized mo-\nment, they have an exchange interaction that can be modeled\nby a constant spin splitting ∆along the magnetization direc-\ntion with unit vector m=M/Ms(in which Mis the mag-\nnetization vector with the saturation magnetization Ms) [39].\nThen, the following 2D Dirac Hamiltonian provides a simple\nmodel for the electronic structure of the TI surface state:\nHk=/planckover2pi1vFˆσ·(k׈z)+∆ˆσ·m, (1)\nwhere h=2π/planckover2pi1is the Planck constant, vFis the Fermi ve-\nlocity of the Dirac electrons, ˆ σis the Pauli matrix operator\nfor the spin, and∆is the exchange interaction. For simplic-\nity, we ignore here the particle–hole asymmetry in the surfa ce\nbands. Introducing the polar angle θand azimuthal angle ϕ\nform=(cosϕsinθ,sinϕsinθ,cosθ), the energy dispersion\nof the Hamiltonian ( 1) can be expressed as\nEks=s/radicalBig\n(/planckover2pi1vFk)2+∆2−2/planckover2pi1vFk∆sinθsin(ϕ−ϕk), (2)\nwhere s=±corresponds to the upper and lower bands, and\ncosθk=∆ cosθ/|Eks|and tanϕk=ky/kxare the polar and\nazimuthal angles of the spinors on the Bloch sphere, respec-\ntively.\nA. Current–induced spin–orbit torque\nWe begin by discussing a current-induced SOT to the mag-\nnetization in TI-based magnetic heterostructures [ 12,14,15,\n40–43]. The SOT stems from the exchange interaction be-\ntween the magnetization and the electrically induced noneq ui-\nlibrium spin polarization µ(in units of m−2) [7,11], which can\nbe described by\nTSO=−γ∆µ\nMsd×m, (3)(a)\n(c)/g39 = 50 meV\n30 meV\n10 meV(d)\n/g39 = 10 meV\n30 meV\n50 meV(b)\n/g39 = 50 meV\n30 meV\n10 meV/g39 = 50 meV\n30 meV\n10 meV/g21/g39 \nFIG. 1. Current-induced nonequilibrium spin polarization ∆µ/d\nscaled by thickness dof ferromagnet as a function of EFfor different\nvalues of∆(with mz=1): (a) x-component∆µx/d; (b) y-component\n∆µy/d; (c)∆µx/dand (d)∆µy/datEF=95 meV (corresponding\ncarrier density∼1012cm−2) as functions of mzfor different values\nof∆. In these graphs, we use vF=4.0×105ms−1,d=10 nm, and\nEx=0.1 V/µm. The details of the calculations are given in the text.\nwhereγis the gyromagnetic ratio and dis the thickness of\nthe ferromagnetic layer (magnetic TI). In short, the SOT is\nobtained by calculating the electrically induced spin pola riza-\ntion on the TI surface.\nAs a characteristic feature of the Dirac Hamiltonian ( 1), the\nspin operator ˆ σis directly proportional to the velocity oper-\nator ˆv=∂Hk/(/planckover2pi1∂k)=vFˆz׈σdue to the spin-momentum\nlock. In this sense, we can identify the nonequilibrium spin\npolarization µwith the electric current Jon the TI surface,\nnamely\nµ=−1\nevFˆz×J, (4)\nwhere−e(e>0) is the electron charge. In the following,\nwe useµandJto denote the quantum statistical expectation\nvalues of ˆ σandˆj=−eˆv, respectively. Here, we empha-\nsize that the nonequilibrium spin polarization involves on ly\nin-plane spin components. Hence, the in-plane component of\nthe spin susceptibility corresponds to the electric conduc tivity\nvia Eq. ( 4). In the framework of Boltzmann transport the-\nory and the Kubo formula, previous studies [ 41,42,44–47]\nhave calculated the longitudinal and transverse (anomalou s\nHall) conductivities on magnetized TI surfaces by assuming\na short-range impurity potential with Gaussian correlatio ns\n/angbracketleftˆV(r1)ˆV(r2)/angbracketrightimp=nV2\n0δ(r1−r2) in which nis the impurity\nconcentration, V0is the scattering potential, and /angbracketleft···/angbracketright impindi-\ncates an ensemble average over randomly distributed impuri -\nties. For an electric field Ealong the TI surface, the driving3\nsheet current can be written as J=σLE+σAHˆz×Ewith [ 42]\nσL=e2\n2hEFτ\n/planckover2pi11−ξ2m2\nz\n1+3ξ2m2z,\nσAH=−4e2\nhξmz1+ξ2m2\nz/parenleftBig\n1+3ξ2m2z/parenrightBig2,(5)\nwhereτ=4/planckover2pi1(/planckover2pi1vF)2//parenleftBig\nnV2\n0EF/parenrightBig\nis the transport relaxation time\nof massless Dirac electrons within the Born approximation,\nξ= ∆/EFwith the Fermi energy EFmeasured from the\noriginal band-teaching (Dirac) point, and mzdenotes the z-\ncomponent of m. Note thatσAHin Eq. ( 5) is independent\nof the impurity parameters but diagrammatically contains t he\nside-jump and skew scattering contributions as well as the i n-\ntrinsic one associated with the Berry curvature of the surfa ce\nbands [ 44,46]. According to Eq. ( 4), the current-induced spin\npolarization for E=Exˆxis therefore\nµ=−1\nevF(−σAHE+σLˆz×E)≡µxmzˆx+µyˆy, (6)\nwhere\nµx=−4eEx\nhvF∆EF/parenleftBig\nE2\nF+∆2m2\nz/parenrightBig\n/parenleftBig\nE2\nF+3∆2m2z/parenrightBig2, (7)\nµy=−eEx\n2hvFEFτ\n/planckover2pi1E2\nF−∆2m2\nz\nE2\nF+3∆2m2z. (8)\nEquations ( 7) and ( 8) are substantially equivalent to the\ncurrent-induced nonequilibrium spin density that Ndiaye et al.\ncalculated directly by using the Kubo–Streda formula in-\nvolving the spin vertex correction [ 15]. Note thatµxorigi-\nnates from the magnetoelectric coupling (the so-called Che rn–\nSimons term) [ 12,39] that is proportional to the anomalous\nHall conductivity [see Eq. ( 6)]. Meanwhile,µystems from the\nRashba–Edelstein e ffect due to the spin-momentum locking\non the TI surface [ 40].\nFrom Eqs. ( 3) and ( 6), we finally obtain the form of SOT\narising from the TI surface [ 15,42] [see the Appendix for the\ncurrent expression of SOT], namely\nTSO=γ∆µx\nMsdmzm׈x+γ∆µy\nMsdm׈y. (9)\nThe first term contributes as a damping-like (DL) torque but\none that is quite di fferent from that of the spin Hall e ffect\nin traditional heavy-metal /ferromagnet heterostructures [ 3,4].\nIn fact, for the in-plane magnetization configuration ( mz=0),\nthis DL torque vanishes because of the absence of the mag-\nnetoelectric coupling via the anomalous Hall e ffect, whereas\nthe SOT driven by the spin Hall e ffect acts on the magneti-\nzation. Meanwhile, despite its origin, the second term acts\nas only a field-like (FL) torque. This feature is also di ffer-\nent from that of the Rashba–Edelstein e ffect in the usual 2D\nferromagnetic Rashba systems in which there might be both\nFL and DL contributions [ 49,50]. Figure 1(a) and (b) show\ntheEFdependence of the x- and y-components, respectively,of∆µ(in units of Jm−2) for different values of the surface\nband gap. For this calculation, ∆is used within the values\nreported experimentally in magnetically doped [ 51] and FI-\nattached [ 37,52] TIs. We also adopt n=1012cm−2and\nV0=0.2 keVÅ2as impurity parameters based on an analy-\nsis of the transport properties of a TI surface [ 53]. These im-\npurity parameters can reproduce the experimentally observ ed\nlongitudinal resistance ( ∼10 kΩ) in magnetic TIs. Remark-\nably, as seen in Fig. 1(a), even when the Fermi level is inside\nthe surface band gap ( EF<|∆|), the x-component survives as\n[12,41]\nµx=−eEx\n2hvF=−1\nevFσQAHEx, (10)\nwhereσQAH=e2/(2h)sgn( mz) characterizes the quantum\nanomalous Hall effect on the magnetized TI surface [ 48], re-\nflecting the topological nature of 2D massive Dirac electron s.\nBy contrast, because of the Rashba–Edelstein e ffect, the y-\ncomponent shown in Fig. 1(b) survives in only the metallic\nsurface states ( EF≥|∆|). Figure 1(c) and (d) show the mzde-\npendence of the x- and y-components, respectively, of ∆µfor\ndifferent values of the surface band gap. In these plots, we in-\nclude mzin the x-component of∆µ. Reflecting the anomalous\nHall effect on the magnetized TI surface, the x-component is\nodd upon magnetization reversal, whereas the y-component is\neven in magnetization reversal because it is proportional t oσL\nvia the Rashba–Edelstein e ffect.\nB. Dirac magnetic anisotropy\nHere, to evaluate the VCMA e ffect in TI-based magnetic\nheterostructures, we investigate the MAE associated with t he\nexchange interaction in Eq. ( 1). The MAE is defined as the\ndifference in the sums over occupied states of energy disper-\nsions ( 2) withθ=0 as the reference state [ 54], namely\nUMAE=occ./summationdisplay\nksEks(θ)−occ./summationdisplay\nksEks(θ=0). (11)\nExpanding Eq. ( 11) aroundθ≈0 leads to UMAE≈Kusin2θ,\nwhere the uniaxial magnetic anisotropy constant Ku(in units\nof Jm−2) is given by\nKu=−occ./summationdisplay\nkss(/planckover2pi1vFk)2∆2sin2(ϕ−ϕk)\n2/bracketleftBig\n(/planckover2pi1vFk)2+∆2/bracketrightBig3/2. (12)\nThe sign of Kuspecifies the type of MAE, namely perpen-\ndicular magnetic anisotropy (PMA, Ku>0) or easy-plane\nmagnetic anisotropy ( Ku<0). For the partially occupied en-\nergy bands, we have Ku>0; i.e., PMA is always favored by\nthe magnetization coupled with Dirac electrons on the TI sur -\nface. A qualitative understanding of the characteristic PM A\nis given by a gain of electronic free energy associated with\nthe exchange interaction between the Dirac electrons and lo -\ncalized moment. When the magnetization is along the out-\nof-plane direction, a surface band gap (2 ∆mz) emerges in the4\nmassless Dirac dispersion, which reduces the electron grou p\nvelocity (kinetic energy). Meanwhile, for the in-plane mag ne-\ntization orientation, the exchange interaction merely shi fts the\nsurface band in the k-space. In terms of the exchange inter-\naction maximizing the energy gain of the Dirac electron sys-\ntem, the case possessing the surface band gap is expected to\nbe more favorable with lower electronic free energy than tha t\nwith the shifted surface bands by an in-plane magnetization ,\nwhose scenario can be interpreted as being analogous to the\nPeierls transition in electron–lattice coupled systems [ 51].\nTo integrate Eq. ( 12), we assume hereinafter that the low-\nenergy Dirac Hamiltonian ( 1) is a valid description for k≤kc\nwith a momentum cut kc=/radicalbig\n∆2c−∆2/(/planckover2pi1vF) [55] in which 2∆c\nis the bulk band gap of TIs induced by the band inversion due\nto the spin–orbit interaction. We also define the Fermi wave\nvector kF=/radicalBig\nE2\nF−∆2/(/planckover2pi1vF), as shown in Fig. 2(a). Without\nloss of generality, we assume the case in which EFcrosses\nthe upper surface band, namely EF>∆. Then, we obtain the\nmagnetic anisotropy constant by integrating over the relev ant\nenergy range as\nKu=∆2\n8π(/planckover2pi1vF)2/bracketleftBigg\n∆c−EF−/parenleftBigg1\nEF−1\n∆c/parenrightBigg\n∆2/bracketrightBigg\n. (13)\nUp to the lowest order of ∆, Eq. ( 13) takes the simplest\nanalytic from of Ku= ∆2kc/(8π/planckover2pi1vF), which corresponds\nto the out-of-plane MAE for EF≤ |∆|derived earlier by\nTserkovnyak et al. [55]. Figure 2(b) shows the EFdepen-\ndence of Kufor different values of the bulk and surface band\ngaps. In this plot, ∆c=150 meV and∆c=100 meV corre-\nspond to the bulk band gaps for Bi 2−xSbxTe3−ySey(BSTS) and\n(Bi1−xSbx)2Te3(BST) [ 10], respectively. Because we assume\nthat the surface states have energy dispersions with partic le–\nhole symmetry, the magnetic anisotropy constant retains th e\nform of Eq. ( 13) in the case in which EFcrosses the lower\nsurface band, namely EF<−∆. Hence, Kuis maximum with\nthe form of Ku|EF=|∆|forEF≤|∆|and decreases apart from\nthe energy level of ±∆. The reason is that there are simply\nfewer active electrons for EF<−∆, while for EF>∆the en-\nergy decrease in the lower band is compensated partially by\nthe upper-band energy increase in conductive electrons.\nAt the end of this section, we compare the formulated\nMAE with a recent experiment employing Cr-doped BST\nthin films which is sandwiched by two di fferent dielectrics\nand hence has the Dirac electron systems on each interface\n[16]. The magnetic anisotropy field due to Eq. ( 13) is de-\nfined by BK=2Ku/(Msd). For the Cr-doped BST films\nwith 7 quintuple-layer ( d≈7 nm), a calculated result is\nBK=117 mT ( Ku/d=0.5 kJm−3) while an experiment\nreports BK≈570 mT without an electric gate in Ref. 16.\nIn this calculation, we choose the parameters for Cr-doped\nBST:∆c=100 meV ,∆ = 30 meV , vF=4.0×105ms−1,\nMs=8.5×103Am−1, and an electron/hole carrier density\n1.0/0.2×1012cm−2(corresponding to|EF|=98/51 meV) for\nthe each interface [ 16]. The difference of BKmight come from\ndisregarding the realistic particle–hole asymmetry induc ed by\nthe higher–order k-term of the energy dispersion which makes\na more sharp electronic density of states in the surface va-Eks\nkxEF\n/g21/g39 c\n/g21/g39 kFkcconduction band\nvalence band(a)(b)\n/g39 = 50 meV\n/g39 = 30 meV\n/g50/g21/g39 \nFIG. 2. (a) Schematic of massless (dashed line) and massive ( solid\nline) surface state dispersions at ky=0 in which EFdenotes the Fermi\nenergy measured from the Dirac point ( Eks=0) of the original mass-\nless surface bands, 2 ∆is the surface band gap due to an exchange\ninteraction, and 2∆cis the bulk band gap. kFandkccorrespond to the\nFermi wave vector and cuto ffwave vector, respectively. (b) Scaled\nmagnetic anisotropy energy Ku/das a function of EFfor different\nvalues of∆cand∆. Red and blue lines are for ∆c=150 meV and\n∆c=100 meV , respectively. In this plot, we use vF=4.0×105ms−1\nandd=10 nm.\nNBHOFUJD\u00015*NFUBM\nEJFMFDUSJDVGxz\ny\ndVS\nmdD\nlx\n5* \nFIG. 3. Schematic geometry (side view) of field-e ffect transistor\n(FET)-like device comprising a magnetic topological insul ator (TI)\nfilm (with thickness d) sandwiched by a nonmagnetic TI and a di-\nelectric attached to a top electric gate VGin which the Dirac electron\nsystem should appear on the top surface of the magnetic TI [ 17].dD\nis the thickness of the dielectric and lxis the length of the conduction\nchannel. VSis the voltage difference between the source and drain\nelectrodes. Current flows on the x–yplane depicted by a yellow line\nthat corresponds to the TI surface state. The arrows denote t he initial\n(red) and final (blue) magnetization directions in the magne tization\nreversal.\nlence band and enhances the hole–mediated Dirac PMA up to\nKu/d=8 kJm−3[34]. In this respect, our simple Dirac model\ncould not reflect the detail of the realistic surface band but\ncould capture the permissible magnitude of BK. Therefore,\nthe above comparison implies that the interfacial Dirac PMA\ngives a significant contribution to the magnetic anisotropy in\ndilute magnetic TIs.5\nNYN[\nNZTPVSDF\u0001\t7 4\nHBUF\u0001\t7 (\n(a) (b) (c)\n\u0012\n\u0013\u0014\u0015\nNBHOFUJD\u00015* NBHOFUJD\u00015* U4\nNYNZN[\nNYNZN[\ntmx,m y,m z\nVV\n\u0013\u0014 \u0012 \u0015\nFIG. 4. (a) Time evolution of each component of magnetizatio n with pulsed source voltage ( VS). The numerical calculation is performed with\nstatic VG=0.32 V , VS=1.5 V , and tS=1 ns. (b), (c) Corresponding magnetization switching traje ctories during duration of (b) 1 →2 and (c)\n3→4 in left upper panel. The vertical arrows denote the initial (red) and final (blue) magnetization directions in the magne tization reversal.\nThe green arrow on the surface of the magnetic TI (black cube) indicates the direction of the current-induced spin polari zation: ˆµ=µ/|µ|.\nThe spin–orbit torque (SOT) is active from 1(3) to the black s tar (⋆).\nIII. MAGNETIZATION SWITCHING\nTo demonstrate magnetization switching via SOT and\nVCMA, we propose a field-e ffect transistor (FET)-like de-\nvice with a magnetic TI film as a conduction channel layer\n[16] in which source-drain ( VS) and gate ( VG) voltages are\napplied, as shown in Fig. 3. To investigate the macroscopic\ndynamics of the magnetization in the device, we solve the\nLandau–Lifshitz–Gilbert (LLG) equation including the SOT\n(9), namely\ndm\ndt=−γm×Beff+αeffm×dm\ndt+TSO(VG), (14)\nwhere Beffis an effective magnetic field obtained by finite m\nfunctional derivatives of the total energy UM, namely Beff=\n−δUM/(Msδm), andαeffdenotes the effective Gilbert damping\nconstant. As discussed in Sec. II B, the interfacial Dirac PMA\ngives a significant contribution to the magnetic anisotropy in\nin dilute magnetic TIs. Because of the thin magnetic TI in\nFig.3, we assume that UMconsists of the MAE ( 13) and the\nmagnetostatic energy that generates a demagnetization fiel d,\ni.e.,\nUM=1\ndKu(VG)/parenleftBig\n1−m2\nz/parenrightBig\n+1\n2µ0M2\nsm2\nz, (15)\nwhereµ0is the permeability of free space. As discussed in\nSec. II, both MAE and SOT depend on the position of EF,\nwhich can be controlled electrically via (see the Appendix f or\ndetails)\nEF(VG)=/planckover2pi1vF/radicalBigg\n4π/parenleftBigg\nnint+∆2\n4π(/planckover2pi1vF)2+ǫ\nedDVG/parenrightBigg\n, (16)\nwhereǫis the permittivity of a dielectric of thickness dDand\nnintis the intrinsic carrier density at VG=0. In this study,for the dielectric layer with dD=20 nm, we adopt a typical\ninsulator, namely Al 2O3(for which the relative permittivity is\nǫ/ǫ0=9.7) [16,56].\nBecause the ferromagnetic Curie temperature ( Tc) of Cr-\ndoped BST is less than 35 K [ 57], we first consider magneti-\nzation switching at zero temperature. Influence of finite tem -\nperatures on the magnetization switching will be discussed in\nSec. IV C . The equilibrium magnetization direction with nei-\ntherVSnorVGis determined by minimizing Eq. ( 15) regard-\ning the polar angle θ. However, for simplicity, we assume that\nθ≈0◦at the electrostatic equilibrium. This assumption is per-\nmissible because hereinafter we consider a magnetic TI such\nas Cr-doped BST with a very small Ms=8.5×103Am−1\n[16], neglecting the demagnetizing field e ffect from the sec-\nond term in Eq. ( 15) at VG=0. For numerical simula-\ntion, Eq. ( 14) is solved by setting θ=1◦andϕ=0◦as\nthe initial condition for m(t=0). In the simulation, we\nchoose the parameters for Cr-doped BST as ∆c=100 meV ,\n∆= 30 meV , vF=4.0×105ms−1,d=7 nm, n=1012cm−2,\nV0=0.2 keVÅ2[53],nint≈0 cm−2(corresponding to\nEF(VG=0)=∆= 30 meV) [ 56,57],γ=1.76×1011T−1s−1,\nandαeff=0.1. Note that a large enhanced damping from 0.03\nto 0.12 due to the strong spin–orbit interaction of TIs has be en\nreported in TI/ferromagnetic-metal bilayers [ 58]. In addition,\nno modulation ofαeffis assumed during the duration of VGbe-\ncause we consider a thicker magnetic TI film ( d=7 nm) than\nthe ferromagnetic metal used in the magnetic tunnel junctio n\n[59].\nA. Switching via a source-drain current pulse JS\nWe investigate the magnetization switching via a source-\ndrain current pulse [ 17]. Under a static gate voltage [ 16], we\napply a step-like voltage pulse of width tSto the source elec-\ntrode, as shown in Fig. 4(a) (see the upper panel). The applied6\n(a) (b)\nNYN[\nNZ(c)\u0015 \u0012\n\u0013 \u0014\nNBHOFUJD\u00015* NBHOFUJD\u00015* U(\nNYNZN[\nNYNZN[\ntmx,m y,m z\nVV\n\u0013 \u0014 \u0012 \u0015TPVSDF\u0001\t7 4\nHBUF\u0001\t7 (\nFIG. 5. (a) Time evolution of each component of magnetizatio n with pulsed gate voltage VG. The numerical calculation is performed with\nVG=0.38 V , VS=0.3 V , and tG=6.7 ns. (b), (c) Corresponding magnetization switching traje ctories during (b) 1 →2 and (c) 2→3 in the\nleft upper panel. The red, blue, and green arrows and the blac k star have the same meanings as in Fig. 4.\ngate voltage can reduce the energy barrier for the magneti-\nzation reversal due to the VCMA e ffect in Eq. ( 13). The re-\nsulting time evolution of m(t) is shown in Fig. 4(a) with the\nconstant gate voltage turned on at t=0. The lower panel\nshows clearly that mzchanges its sign by the pulsed VSin-\nputs, demonstrating the out-of-plane magnetization switc h-\ning. Furthermore, applying subsequent pulses switches the\nmagnetization direction faithfully, and the change is inde pen-\ndent of the pulse’s sign. The estimated switching time be-\ntween points 1 and 2 (3 and 4) is ∼18 ns. In this simu-\nlation with static VG=0.32 V and VS=1.5 V , the mag-\nnetic anisotropy field BKand effective field due to the DL(FL)\nSOT BDL(FL)= ∆µx(y)/(Msd) are evaluated for mz=1 as\nBK=21 mT, BDL=6.0 mT, and BFL=17 mT, respec-\ntively. Note that the calculated magnetic anisotropy field i s\nBK=136 mT at VG=0. The estimated current density\n(corresponding to BDL(FL) ) is JS=1.9×105A/cm2(see\nSec. IV A for details), which is consistent with those of TI-\nbased magnetic heterostructures [ 6,17–20,22]. Figure 4(b)\nand (c) show the magnetization switching trajectories duri ng\nthe durations shown by numbers (1–4) in the left upper panel\nof Fig. 4(a). An equilibrium magnetization almost along ±ˆz\nis rotated steeply around the current-induced spin polariz ation\n(µ) by the pulsed SOT until the black star ( ⋆), after which\nBKgradually stabilizes the magnetization with oscillations\naround the easy axis.\nB. Switching via a pulsed gate voltage VG\nWe also investigate the magnetization switching via a\npulsed gate voltage. Under a constant source-drain bias, we\napply a step-like voltage pulse of width tGto the gate elec-\ntrode, as shown in Fig. 5(a) (see the upper panel). The source-\ndrain bias induces the spin polarization that takes the role of a\nconstant-bias magnetic field [ 29], while the pulsed gate volt-\nage reduces the energy barrier for the magnetization rever-\nsal via the VCMA e ffect during its duration. The resultingtime evolution is shown in Fig. 5(a) with the source-drain bias\nturned on at t=0. Figure 5(b) and (c) show the correspond-\ning magnetization switching trajectories in which an equil ib-\nrium magnetization at an initial state (1 or 3) is rotated ste eply\naround the bias effective magnetic field (proportional to µ)\nby the SOT until the black star, after which BK(=136 mT\natVG=0) aligns the magnetization direction with the easy\naxis by the Gilbert damping. As shown, the z-component\nof the magnetization changes sign by the pulsed VGinputs,\ndemonstrating the out-of-plane magnetization switching. The\nestimated switching time from the initial to final states is a l-\nmost the same as the pulse width tG∼6.7 ns. In this simu-\nlation with static VG=0.38 V and VS=0.3 V , we evaluate\nBK=3.0 mT, BDL=1.2 mT, and BFL=3.6 mT for mz=1.\nWe emphasize that the current density corresponding to the\nSOT is JS=4.1×104A/cm2, which is smaller than that of the\npulsed SOT method discussed above and those of TI-based\nmagnetic heterostructures reported to date [ 22].\nC. Switching phase diagram\nWe conclude this study with a guide for realizing the pro-\nposed magnetization switching methods. In particular, exp er-\nimenters may be interested in how the final-state solution of\nmzdepends on the input pulse width and voltages. In the fol-\nlowing plots, we use the parameters for Cr-doped BST. Fig-\nure6(a) shows the phase diagram of mzfor a source-drain\ncurrent pulse as a function of both tSandVG. The diagram\nis calculated up to VG≈0.39 V to which the Fermi level\nreaches the bottom of a bulk conduction band. The final-state\nsolution of mzoscillates rapidly depending on tSrather than\nVG, whereas the diagram has a threshold VGat the vicinity of\n0.32 V at which the SOT competes with the anisotropy field.\nIn Fig. 6(b), we also show the phase diagram of mzfor a\npulsed gate voltage as a function of both tGandVS. Clearly,\nthe final-state solution of mzoscillates depending on both tG\nandVS, whereas switching tends to succeed in the short pulse7\nN[(a)\n(b)N[\nFIG. 6. (a) Final-state diagram of mzatVS=1.5 V as function of\npulse duration time tSand gate voltage VG. (b) Final-state diagram of\nmzatVG=0.39 V as function of pulse duration time tGand source-\ndrain voltage VS.\nregion of sub-nanosecond order. Consequently, switching w ill\nbe achieved in the wide pulse duration between the nano-\nand sub-microsecond scales. In practice, the proposed de-\nvice would be mounted by combination with semiconductor\ndevices such as CMOS whereas the switching speed of VLSI\n(very large-scale integration) is not so fast at present. He nce,\ncontrol with a pulse width of a few nano-second is considered\nrealistic. Figure 6(b) shows that the controllability of the tG\npulse is better than the tSpulse because of the width of the\npulse. From the viewpoint of the speed, it can be expected\nthat the tGpulse has better compatibility with VLSI.\nIV . DISCUSSION\nA. Source-drain current and Hall voltage: FET and memory\noperations\nTo evaluate the magnitude of current density ( JS) realiz-\ning the magnetization switching in Sec. III, we calculate thesource-drain current flowing on the TI surface in Fig. 7(a). We\nalso calculate the magnitude of the output Hall voltage ( VH)\nto read out a direction of the out-of-plane magnetization by\nits sign [ 60]. According to Eq. ( 5) and Ex=VS/lx, the corre-\nsponding quantities can be written as\nJS=σL(VG)\ndVS\nlx, (17)\nVH=σAH(VG)\ndVS\nlxly, (18)\nwhich are plotted in Fig. 7(b) and (c), respectively. In these\nplots, we use the same parameters as for Cr-doped BST in\nSec. III. As a reminder, note again that nint≈0 cm−2(cor-\nresponding to EF(VG=0)= ∆ = 30 meV) is assumed\nfor the electrostatic equilibrium. Figure 7(d) shows the FET\noperation (on/off) of the proposed device. Clearly, we can\nswitch the source-drain current by a reasonable gate volt-\nage compared with modern FET devices. Therefore, com-\nbining this FET operation with the proposed magnetization-\nswitching method promises an FET with the functionality of\na nonvolatile magnetic memory [ 61]. The bit stored in this\ndevice is read out by measuring VHand determining its sign.\nSo, if EF(VG=0) is tuned within the surface band gap by the\nelement substitution [ 56], the quantum anomalous Hall e ffect\nmight allow the readout process to be free from energy dissi-\npation due to Joule heating.\nB. TI/FI bilayers\nWe discuss the possibility of magnetization switching in\nTI/FI bilayers with the magnetic proximity e ffect at the in-\nterface [ 36–38]. A device corresponding to Fig. 3is proposed\nby replacing the dielectric with an FI and the TI /magnetic-TI\nbilayer with a TI. In this case, it is easily shown that an ex-\nchange interaction between interface Dirac electrons and l o-\ncalized moments of the FI appears in the same form as that\nof the Dirac PMA given by Eq. ( 13) [34,55]. Then, the mag-\nnetization dynamics can be analyzed by using the LLG equa-\ntion ( 14) involving crystalline magnetic anisotropies ( KCMA)\nof the FI. Switching methods similar to those discussed in\nSec. IIIwill be achieved for a magnetically almost-isotropic FI\nwith small net magnetization (desirably KCMAd/Ku≪1 and\nMs/lessorsimilar105Am−1), reducing the demagnetizing field e ffect. The\npotential candidates for the FI layer are 2D van der Waals fer -\nromagnetic semiconductors [ 52,62] and rare-earth iron gar-\nnets [ 63–65] near their compensation point, where the intrin-\nsic magnetic anisotropy and magnetostatic field become infe -\nrior to those of the Dirac PMA. However, the compensation\npoints of the candidates are at a finite temperature, therefo re\nthe thermal excitation of the TI bulk state and the e ffective\nmagnetic field due to the thermal fluctuation of localized mo-\nments might affect the switching probability, which is beyond\nthe scope of the present investigation. An alternative migh t\nbe to use an insulating antiferromagnet with the A-type lay-\nered structure [ 66] that has no magnetostatic field because of\nthe tiny net magnetization. Note that a recent experiment re -8\n(b) (a)\nNBHOFUJD\u00015*lxly\nVS\nxzyJS\n(c)VH\n(d)\nPO P⒎ EF/g21/g39 74\u0001\t7\n \n   \u0012\u000f\u0016\n  \u0001\u0011\u000f\u00147(\u0001\t7\n \n  \u0011\u000f\u0011\u0011\u0012\n  \u0011\u000f\u0011\u0012\n  \u0011\u000f\u0011\u0016\n  \u0011\u000f\u0012\n  \u0011\u000f\u0013\n  \u0011\u000f\u0014\n  \u0011\u000f\u0014\u0019VJ\n7(\u0001\t7\n \n  \u0011\u000f\u0011\u0011\u0012\n  \u0011\u000f\u0011\u0012\n  \u0011\u000f\u0011\u0016\n  \u0011\u000f\u0012\n  \u0011\u000f\u0013\n  \u0011\u000f\u0014\n  \u0011\u000f\u0014\u0019\nVV\n\tN7\n \nVJ\nEF\nFIG. 7. (a) Top view of device proposed in Fig. 3(here the gate\nand dielectric layers are hidden), where lxandlyare the lengths of\nthe source-drain and Hall directions, respectively. (b) So urce-drain\ncurrent density JSof proposed device versus VSfor different values\nofVG. (c) Corresponding Hall voltage |VH|versus VSfor different\nvalues of VG. (d) Transfer characteristics obtained for di fferent VS,\nindicating the FET operation (on /off) of the proposed device. Insets\nshow the band pictures corresponding to the on /offstates. In these\nplots, we assume|mz|=1,d=7 nm, and lx=ly=10µm.\nports that the TI surface states on Bi 2Se3produce a PMA in\nthe attached soft ferrimagnet Y 3Fe5O12[67].\nHerebefore, we have focused only on the all-insulating sys-\ntems (i.e., TI and FI), whereas TI /ferromagnetic-metal(FM)\nbilayer systems are important for spintronic applications and\nexperiments. It is necessary to pay attentions for the direc t\napplication of our model to the TI /FM bilayer represented by\na pair of Bi 2Se3and Py because of the following two reasons.\nFirst, our 2D model cannot capture the three-dimensional\n(3D) nature of the transport in the TI /FM bilayer. Indeed, in\nBi2Se3/Py bilayers, most of the electric current shunt through\nthe Py layer and conductive bulk states of the Bi 2Se3layer,\nwhich reduces the portion of the current interacting with th e\nTI interface state. From this viewpoint, Fischer et al. [68]\nshow that in the FM layer spin-di ffusion transport perpendic-\nular to the interface plays a crucial role to generate the DL\ntorque. In contrast, based on a 3D tight-binding model of the\nTI/FM bilayer, Ghosh et al. [43] demonstrate that a large DL\nSOT is generated by the Berry curvature of the TI interface\nstate rather than the spin Hall e ffect of the bulk states. Sec-\nondly, orbital hybridization between the 3 dtransition metal\nand TI deforms the TI surface states, which shifts the Dirac\npoint to the lower energy and generates Rashba-like metal-\nlic bands across EF[69,70]. Besides, the hexagonal warping\neffect might be important for Bi 2Se3with a relatively large\nEFdue to its crystal symmetry. According to Li et al. [71],the Berry curvature for hexagonal warping bands involves no t\nonly out-of-plane magnetization components but also those of\nthe in-plane, which implies that the in-plane magnetizatio n\ncan contribute to the DL SOT. Note that the hexagonal warp-\ning term is important under threefold-rotational symmetry as\nthe Bi 2Se3crystal structure while it becomes small in bulk\ninsulating TIs (our focus) such as BSTS and Cr-doped BST\ndue to reduction of the symmetry by the elemental substitu-\ntion [ 72].\nC. Influence of finite temperatures\nFor the experimental probe of our proposal, one may be\ninterested in how finite temperatures a ffect the magnetization\nswitching. In our model, there are mainly three temperature\neffects: (i) temperature ( T)–dependence of physical quantities\nof TIs (σL(AH) andKu), (ii) the thermal excitation of the TI\nbulk states at finite temperatures, and (iii) a random magnet ic\nfield due to the thermal fluctuation of localized moments, po-\ntentially leading to a switching error. The cases (i) and (ii )\nattribute to electronic properties while the case (iii) is i n usual\ntreated by magnetization dynamics.\nRegarding the case (i), at low temperatures that satisfies\nkBT≪∆<EF(kBis the Boltzmann constant), we can re-\ngard the Fermi–Dirac distribution function f(E)≈Θ(E−EF)\nas the step function and then ignore T–dependence ofσL(AH)\nandKu. Indeed, when we set ∆= 30 meV and 30 meV ≤\nEF≤100 meV for the system, the above condition is satisfied\nbelow 30 K ( kBT≈2.6 meV) that is a temperature less than\nTcof magnetic TIs or FIs discussed in Sec. IV B . Regarding\nthe case (ii), based on the model of Ref. 53, we investigate the\ncontribution of the bulk states in terms of the electron dens ity\nby using\nRatio=nbdTI\nns, (19)\nwhere nbis the bulk electron density of the TI with thickness\ndTIandnsis the surface electron density. Here, we assume\nthat the bulk state of a TI thin film is a quantum well sys-\ntem along the zdirection, namely Eks,ℓ=/planckover2pi12/(2m∗)/parenleftBig\nk2\nx+k2\ny/parenrightBig\n+\nEz,ℓ(dTI)+∆ c+s∆bwith Ez,ℓ(dTI)=/planckover2pi12/(2m∗)ℓ2π2/d2, where\nℓis an integer, m∗is the effective electron mass, and ∆bis\nthe exchange energy in the bulk of magnetic TIs with spin in-\ndexs=±. The corresponding electron density is given by\nnb/s=/integraltext∞\n−∞dED b/s(E)f(E), where Db/s(E)=/summationtext\nksδ/parenleftbigE−Eb/s/parenrightbig\nis the bulk ( Eb=Eks,ℓ)/surface ( Es=Eksin Eq. ( 2)) den-\nsity of states. The computed result is shown in Fig. 8(a).\nConsequently, we find that Eq. ( 19) is less than 10 % below\n30 K for a Cr-doped BST thin film with EF=95 meV (cor-\nresponding carrier density ∼1012cm−2),∆b=30 meV , and\nm∗/m0=0.15 (m0is the free-electron mass) [ 73], which im-\nplies that the thermal excitation of the TI bulk states is neg li-\ngible below 30 K.\nRegarding the case (iii), we perform similar simulations\nincluding the random magnetic field at T=20 K with an9\nN [(a)\n(b)dTI  = 10 nm\n9\n83BUJP \u0012\u0011 \u000e\u0013 \n\u0012\u0011 \u000e\u0015 \n\u0012\u0011 \u000e\u0017 7\u0012\u0011 \nFIG. 8. (a) Ratio=nbdTI/nsas a function of Tfor different magnetic\nTI thickness dTI. (b) Final-state diagram of at T=20 K as function\nof pulse duration time tGwith VG=0.39 V and source-drain voltage\nVS. The influence of the thermal fluctuation emerges in a longer tG\nregion with lager VS.\nisotropic 3D Gaussian distribution [ 29]\nhth=/radicalBigg\n2kBTαeff\nVMsγ/parenleftBig\n1+α2\neff/parenrightBig\n∆t, (20)\nwhereV≈ lxlyd=10−18m3is the volume of the magnet,\n∆t=0.1 ps, and parameters for Cr-doped BST are used. First,\nwe investigate the e ffect of Eq. ( 20) on the pulsed JSinduced\nmagnetization switching and find that the random magnetic\nfield does not affect the switching diagram of Fig. 6(a) up\nto 20 K. We also test the case of the pulsed VGdriven mag-\nnetization switching and show the result in Fig. 8(b) for the\nextended input parameter ranges. As seen, the influence of\nEq. ( 20) emerges in the longer pulse region with lager VS, re-\nflecting randomness of the thermal fluctuation, whereas we\nconfirm that stable magnetization switching is achieved in t he\nwide parameter range. Note that the thermal stability of mag -\nnetKuV/(kBT)≈5.2×105atEF=30 meV ( VG=0) and\nT=20 K satisfies the required condition ( >60) for a non-\nvolatile memory.V . SUMMARY\nIn summary, we have presented two distinct methods for\nmagnetization switching by using electric-field control of the\nSOT and MAE in TI /magnetic-TI hybrid systems. We for-\nmulated analytically the uniaxial magnetic anisotropy in m ag-\nnetic TIs as a function of the Fermi energy and showed that the\nout-of-plane magnetization is always favored for the parti ally\noccupied surface band. We further proposed a transistor-li ke\ndevice with the functionality of a nonvolatile magnetic mem -\nory adopting (i) the VCMA writing method that requires no\nexternal magnetic field and (ii) read-out based on the anoma-\nlous Hall effect. For the magnetization reversal, by using pa-\nrameters of Cr-doped BST, the estimated source-drain curre nt\ndensity and gate voltage were of the orders of 104–105A/cm2\nand 0.1 V , respectively, below 20 K. As a conclusion of this\nstudy, we showed the switching phase diagram for the input\npulse width and voltages as a guide for realizing the propose d\nmagnetization-reversal method. We also discussed the pos-\nsibility of magnetization switching by the proposed method\nin TI/FI bilayers with the magnetic proximity e ffect. Simi-\nlar magnetization switching may be achieved by the FI layer\nwith 2D van der Waals ferromagnetic semiconductors or rare-\nearth iron garnets near their compensation point. However,\nthe compensation points of the FIs are at a finite temperature ,\nso the thermal excitation of the TI bulk state and the e ffective\nmagnetic field due to the thermal fluctuation of localized mo-\nments might affect the switching probability, which is beyond\nthe scope of the present investigation. Simultaneous contr ol\nof the magnetic anisotropy and SOT by an electric gate may\nlead to low-power memory and logic devices utilizing TIs.\nVI. ACKNOWLEDGMENTS\nThe authors thank Yohei Kota, Koji Kobayashi, Seiji Mi-\ntani, Jun’ichi Ieda, and Alejandro O. Leon for valuable dis-\ncussions. This work was supported by Grants-in-Aid for Sci-\nentific Research (Grant No. 20K15163 and No. 20H02196)\nfrom the JSPS.\nAppendix A: Current-expression of SOT\nWe rewrite the SOT in terms of a current density flow-\ning on the magnetized TI surface. According to Eq. ( 6) in\nthe main text, the current-induced spin polarization invol ving\nJS=σLE/d=(σL/d)(VS/lx)ˆx(in units of Am−2) is given by\nµ=−d\nevF(−θAHJS+ˆz×JS), (A1)\nwhere\nθAH≡σAH\nσL=8/planckover2pi1\nEFτ∆mzEF/parenleftBig\nE2\nF+∆2m2\nz/parenrightBig\n/parenleftBig\nE2\nF−∆2m2z/parenrightBig/parenleftBig\nE2\nF+3∆2m2z/parenrightBig (A2)10\nis an anomalous Hall angle. Inserting Eq. ( A1) into Eq. ( 3) in\nthe main text, we obstinate the current-expression of SOT\nTSO=γ∆\nevFMsm×θAHJS−γ∆\nevFMsm×(ˆz×JS). (A3)\nRecalling that the first term is responsible for the DL-torqu e\nassociated with a magnetoelectric coupling, one may expect\nthat a giant current-induced SOT is obtained by Eq. ( A2) in\nthe case of EF≈∆mz. However, JSthen becomes nearly\nzero because the relation between θAHandJSis tradeoff[see\nEq. ( 5) in the main text], and therefore in Eq. 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Naumov, Priyanka Manchanda and Pratibha Dev\nDepartment of Physics and Astronomy,\nHoward University, Washington, D.C. 20059, USA\n(Dated: November 29, 2021)\nAbstract\nExploring and controlling magnetism in two-dimensional (2D) layered magnetic crystals, as well\nas their inclusion in heterogeneous assemblies, provide an unprecedented opportunity for funda-\nmental science and technology. To date, however, there are only a few known intrinsic 2D magnets.\nHere we predict a novel 2D intrinsic magnet, yttrium sulphur selenide (YSSe), using \frst principles\ncalculations. The magnetism of this transition metal dichalcogenide originates from the partially-\n\flled 3p- and 4p-orbitals of the chalcogens, unlike other known intrinsic magnets where magnetism\narises from the partially-\flled 3 d- and 4f-orbitals. The unconventional magnetism in YSSe is a re-\nsult of a unique combination of its structural and electronic properties. We further show that a lack\nof mirror symmetry results in piezoelectric properties, while the broken space- and time-symmetry\nensures valley polarization.YSSe is a rare magnetic-piezoelectric material that can enable novel\nspintronics, valleytronics and quantum technologies.\n1arXiv:2009.07232v1  [cond-mat.mtrl-sci]  15 Sep 2020Until recently, long-ranged magnetic ordering was believed to be impossible in 2D lay-\nered materials due to its destruction by increased thermal agitations, in accordance with\nthe Mermin-Wagner Theorem. The discovery of long-ranged intrinsic magnetism in pristine\n2D materials|CrI 31and Cr 2Ge2Te62|showed that magnetism can survive in 2D due to the\npreferred alignment direction of the local moments (magnetic anisotropy), which counters\nthermal excitations. Magnetism in 2D crystals has inspired a renewed interest in the funda-\nmental physics of long-ranged magnetism in reduced dimensions. This discovery also o\u000bers\ntechnological opportunities, with potential uses in miniaturized and \rexible spintronics de-\nvices, as well as proximity-e\u000bect devices, such as the superconductor/semiconductor/magnet\nheterostructure that was proposed by Sau et al.3as the solid-state platforms for Majorana\nbound states. In spite of an active search for additional magnetic 2D crystals, there are very\nfew known intrinsic 2D magnets. Here we present our theoretical discovery of a yttrium-\nbased magnetic semiconductor, yttrium sulphur selenide (YSSe), which is a Janus transition\nmetal dichalcogenide. The word \\Janus\" is used to emphasize that the structure lacks mir-\nror symmetry, with two di\u000berent chalcogens (S and Se) on the two opposite faces. The\nmagnetism in YSSe is shown to originate from the unpaired electrons in 3p- and 4p-derived\nstates of the chalcogens. To the best of our knowledge, this is the \frst time an intrinsic\nmagnet is reported with moments contributed by 3p- and 4p-derived states, although these\norbitals are known to be involved in defect-induced magnetism in otherwise non-magnetic\nPtSe 24. In addition to the broken time-reversal symmetry, YSSe lacks inversion symmetry\nas well as mirror symmetry. The unique combination of properties, which ensure that YSSe\nis both a piezoelectric and a magnet, can potentially open doors to new applications.\nResults\nStructural, electronic and spin properties of YSSe monolayers. Our calculations\nare based on density-functional theory (DFT) within the generalized gradient approximation\n(GGA)5, using the Vienna Ab-Initio Simulation package (see Methods section). YSSe exists\nin two low-energy phases: (i) a semiconducting 1H-phase, which we \fnd to be magnetic,\nand (ii) a non-magnetic metallic 1T-phase. For the semiconducting 1H-phase, seen in Fig-\nure 1(a), the Y-Se bond is longer than the Y-S bond and the bond-angles { \u0012SYSand\u0012SeYSe\n{ are also di\u000berent, resulting in an asymmetrical structure with C3v-symmetry (broken trigo-\nnal prismatic symmetry). As shown in Figure 1(b), 1T-YSSe also has C3v-symmetry (broken\n2ac1H-YSSe\nb\ndY-S = 2.73 Åa = 4.13 ÅdY-Se =2.87 Å\n1H-YSSe: Side-view1H-YSSe: Top-viewdSe-S =2.92 Å\"#$%#$=\t92.220\"#%#$=\t129.250\"#%#= 98.450\ndY-Se =2.87 ÅdY-S = 2.69 Å1T-YSSe: Side-view\ndSe-S =3.75 Å1T-YSSe: Top-viewa = 4.10 ÅFIG. 1. Structural properties of YSSe monolayers: aSide- and top-views of 1H-YSSe. bSide-\nand top-views of 1T-YSSe. cPhonon dispersion for 1H-YSSe showing no negative frequencies,\ndemonstrating dynamical stability of the predicted structure.\noctahedral-symmetry) due to the lack of a mirror plane in a Janus TMD and anisotropic\nbonding. The non-magnetic, metallic 1T-phase is lower in energy by 9 :07meV as com-\npared to the 1H-phase, and both structures are likely to occur at room temperature. As\nwe are interested in the magnetic structure, we will concentrate on the 1H-phase, making\ncomparisons with the 1T-phase when needed.\nSince we are predicting a new material, we studied its inter-dependent structural, elec-\ntronic and spin properties. The calculated phonon dispersion for 1H-YSSe, shown in Fig-\nure 1(c), has no negative phonon branches, demonstrating dynamical stability of the pre-\ndicted structure. Furthermore, our ab initio dynamics simulations, performed at 300 Kfor\n10ps, show that the structure is thermodynamically stable. The cohesive energy of YSSe\n(4:722eV=atom ) in the 1H-phase is large, where the cohesive energy, ECis de\fned as the\nenergy required to separate a crystal into constituent isolated atoms. It is calculated using\nthe formula: EC=\u0000(EYSSe\u0000EY\u0000ES\u0000ESe)=3, withEXbeing the total energy of species\nX.\nThe electronic structure and spin properties of 1H-YSSe can be understood within Crys-\ntal Field Theory, taking the electronic con\fguration of yttrium ([Kr] 4 d15s2), which is a\ngroup-III element, into account. In contrast to the +4 formal oxidation state of the tran-\nsition elements in group-VI based TMDs, such as MoS 2, or the synthesized, non-magnetic\n3a\nb\nf1H-YSSe1H-YSSe\n1T-YSSe\n1H-YSSe\n1T-YSSec\ndeFIG. 2. Electronic properties of YSSe monolayers: aBand structure for 1H-YSSe showing\nnearly-dispersionless, spin polarized bands (highlighted) for the \flled majority-spin and the empty\nminority-spin channels. Also plotted are charge density isosurfaces (yellow and blue for positive\nand negative isovalues), showing that these bands originate in the 3p- and 4p-derived states of S\nand Se, with antibonding character. bProjected density of states plotted for the p-states of S and\nSe, along with d-states of Y in 1H-YSSe, showing sharp peaks around the Fermi level (reference\nenergy). cBand structure for the metallic 1T-YSSe. dandeProjected density of states for\n1H-YSSe and 1T-YSSe with components of p-orbitals of S and Se, showing the switching of the\norder of the pz-derived singlet state, and the doublet-states composed of the pxandpyof S and Se\nbetween the two phases.\nJanus structure, MoSSe6, yttrium adopts a +3 formal oxidation state in YSSe. This leaves\nthe S- and Se-atoms in YSSe under-coordinated, with a total of one unpaired electron, which\nis almost equally split between the two chalcogens due to their near-equal electronegativ-\nities,\u001f(\u001fS= 2:58 and\u001fSe= 2:55). Furthermore, due to the C3v-symmetry of 1H-YSSe,\nthe triply-degenerate p-orbitals of the chalcogens split into a doublet ( px,py) and a singlet\n(pz). In the case of 1H-YSSe, due to the smaller distance between the S and Se atoms\n[dSe\u0000S= 2:92\u0017A, see Figure 1(a)], the 3 pzand 4pzstates of S and Se, respectively, which\noverlap in space and energy, hybridize and form bonding A1and antibonding A\u0003\n1-states.\nThe antibonding A\u0003\n1-state in 1H-YSSe is closest to the valence band edge. This is in di-\n4rect contrast with the 1T-YSSe phase structure, where dSe\u0000S= 3:75\u0017A, resulting in smaller\nsplitting between the bonding A1and antibonding A\u0003\n1-states, with the latter being lower\nin energy than the doublet states, E. TheE-states are partially \flled and are composed\nof hybridized doublets, derived from yttrium's d-states and the p-derived ( px,py) states\nof S and Se. Figure 2(a) shows the resulting spin-resolved band structure of a 1H-YSSe\nmonolayer along the high-symmetry points of the Brillouin zone (\u0000- K-M-K0-\u0000). The band\nstructure shows spontaneous spin splitting with more electrons occupying one of the spin\nchannels (majority spin) versus the other (minority-spin channel) due to the exchange in-\nteraction. The large exchange splitting between majority and minority spin A\u0003\n1-states is due\nto quantum-con\fnement e\u000bects in the 2D-crystal, along with the structural attributes of\n1H-YSSe. The large covalent radius of a yttrium atom (1 :90\u0017A) results in a large separation\nbetween neighboring atoms on the two faces of the TMD ( dS\u0000S=dSe\u0000Se= 4:13\u0017A). The\nspatial localization, in turn, results in a large exchange interaction. This spatial localization\nis indicated by the nearly dispersionless A\u0003\n1majority- and minority-spin bands [highlighted\nin Figure 2(a)], with bandwidths of \u00180:12eVand\u00180:11eV, respectively. Figure 2(a) also\nshows the charge density plots of the A\u0003\n1-state for the majority and minority spin channels\nat the \u0000-point, with yellow (blue) corresponding to positive (negative) isosurfaces. The iso-\nsurface plots show that the pz-derived states of the chalcogens are very localized, retaining\nthe atomic-like character of p-orbitals even in the solid state. This localization and spin-\nsplitting is also evident in the projected density of states plotted in Figure 2(b). This also\nexplains why the 1H-phase of YSSe is semiconducting, while its 1T phase is metallic. 1H-\nYSSe is a magnetic semiconductor. It has a majority-spin A\u0003\n1-state that is completely \flled\nand a minority-spin A\u0003\n1-state that is completely empty. On the other hand, 1T-YSSe, with\nthe partially \flled E-states, is a (non-magnetic) metal as can be seen in its band structure\nplotted in Figure 2(c). Figures 2(d)-(e) plot the component-resolved projected density of\nstates, showing the switching of the order of the singlet state, composed of pzof S and Se,\nand the doublet, composed of pxandpyof S and Se, between the two phases. The electronic\nstructure of 1H-YSSe leads to its calculated properties, including: (i) the formation of a\nlocal magnetic moment of 1 \u0016Bper formula unit, most of which comes from S ( \u00180:45\u0016B)\nand Se (\u00180:56\u0016B), with a small counter-polarized moment coming from Y ( \u0018\u00000:02\u0016B),\nand (ii) an indirect band gap of 2.094 eV in the majority spin-channel, with the valence\nband maximum lying along the \u0000 \u0000Kdirection and the conduction band maximum at the\n5M-point, and a direct bandgap of 1.28 eV in the minority spin-channel at the \u0000-point.\nLong-ranged magnetism in 1H-YSSe. So far, we have shown that the 1H-YSSe has\na local magnetic moment. However, in 2D crystals, collective magnetism, i.e. long-ranged\nmagnetism, survives only if the crystals display a preferred alignment direction of the local\nmoments (magnetic anisotropy), which counters the destructive e\u000bects of thermal excita-\ntions. Magnetic anisotropy originates from spin-orbit coupling (SOC) and can be quanti\fed\nby calculating the magnetic anisotropy energy ( MAE ), which is de\fned as the di\u000berence\nin total energies of the structures with magnetizations that are parallel and perpendicular\nto the atomic plane: MAE =Ek\u0000E?. Here,EkandE?are obtained from noncollinear\nDFT calculations with fully-relativistic pseudopotentials that take SOC into account. We\n\fnd that 1H-YSSe has an easy magnetization plane (XY) with an MAE value of about\n\u0000122\u0016eV. Hence, YSSe is an XY-magnet with no energy barrier to the rotation of the mo-\nment in the atomic plane, thereby, exhibiting continuous O(2) spin symmetry. In accordance\nwith the Mermin-Wagner theorem, for 2D YSSe with an O(2) spin-symmetry, there should\nbe no long-ranged ordered state at \fnite temperatures. Nevertheless, the system is allowed\nto undergo a Berezinskii-Kosterlitz-Thouless (BKT) transition from a high-temperature dis-\nordered phase to a low-temperature, quasi-ordered phase. The critical temperature, TC,\nof the BKT transition in an XY-magnet can be estimated by combining the results from\nDFT calculations to those from Monte Carlo simulations of the XY model7,8, according to\nwhichTC= 0:89J=kB. Here,kBis the Boltzmann constant and Jis the exchange coupling\nbetween magnetic moments. Jcan be obtained from the di\u000berence between the total en-\nergies of YSSe with moments aligned in ferromagnetic (FM) and antiferromagnetic (AFM)\ncon\fgurations: \u0001 Emag=EAFM\u0000EFM= 8J\u00162, where\u0016is the magnetic moment per\nformula unit. We \fnd that \u0001 Emagis +2:90 meV per formula unit in a collinear (i.e. non-\nrelativistic) calculation, while the non-collinear calculations (including SOC) yield a \u0001 Emag\nof +3.07 meV per formula unit. The positive sign of the calculated \u0001 Emagmeans that FM\nordering is the preferred ordering of moments. Using the calculated values of \u0001 Emagin\nTC= 0:89\u0001Emag=8\u00162kB, we estimate the critical temperature to be TC= 3:74 K (without\nSOC) and a TC= 3:96 K with SOC e\u000bects taken into account.\nIn order to understand the small values of \u0001 Emagand hence, the exchange coupling ob-\nserved in our DFT calculations for YSSe, we turn to and expand the scope of the semiempir-\n6Energy (eV)\na\nbc\n-0.04-0.03-0.02-0.010.000.010.020.03Energy difference (eV) Valence band Conduction Band\n-K1Γ  K1K2-K2ΓValence BandConduction BandEnergy (eV)Energy (eV)\nEnergy Difference (eV)FIG. 3. E\u000bects of SOC: aBand structure along the paths indicated by red, blue and green lines\nin the Brillouin zone (BZ) shown on the left. The magnetization, M, is taken to be parallel to the\ny-direction, corresponding to the C 1h(C1) magnetic group. This lowering of symmetry results in\na larger irreducible Brillouin zone, with the K1andK2being distinct due to di\u000berent projections\nofK1!\u0000 andK2!\u0000 along M. The highlighted region of the band structure shows only the\ntopmost valence and lowest conduction bands. bSpin-resolved band structure for 1H-YSSe along\ndi\u000berent paths in the BZ for the case when the magnetization is directed along the ydirection.\nOnly the\u0006Syspin projections are shown since \u0006Sxand\u0006Szprojections are almost negligible.\nThe highlighted region of the spin-resolved band structure shows that the topmost valence band is\nmostly spin up and lowest conduction band is mostly spin down. The colors are used to encode the\nexpectation values of the projections. cThe magnetic anisotropy energy contributions calculated\nalong the k-lines in the BZ. Only the contributions corresponding to the highest valence and lowest\nconduction bands are presented.\nical Goodenough-Kanamori rules for superexchange interactions. The superexchange mech-\nanism refers to exchange coupling between the two nearest magnetic atoms with partially\n\flled orbitals, mediated through an intermediary, non-magnetic atom. Conventionally, the\nmagnetic ions are transition metal cations, bridged by an anion (e.g. O2\u0000in MnO). In\nYSSe, it is the S and Se atoms that have the partially \flled orbitals and carry the mag-\nnetic moments, and the exchange coupling between the magnetic anions is mediated by the\n7non-magnetic cation, Y. In spite of this role reversal, the superexchange mechanism can be\napplied to understand exchange coupling in YSSe. This is because, in principle, there is\nno physical law against: (i) the formation of local moments due to partially \flled localized\norbitals in an anion, and (ii) mediation of exchange through an intermediate non-magnetic\ncation. From the Goodenough-Kanamori rules, we know that superexchange maximally\nfavors AFM (FM) coupling if the anion-cation-anion angle is 180\u000e(90\u000e), maximizing the\nhopping between the magnetic ions. The small values of \u0001 Emagwith and without SOC are\na result of the competition between AFM coupling, which is favored between moments on\nS and Se [with \u0012SYSe = 129:25\u000ein Figure 1(a)], and FM coupling, which is favored between\nmoments on S (Se)-atoms on the same faces [with \u0012SYS= 98:45\u000eand\u0012SeYSe = 92:22\u000ein Fig-\nure 1(a)]. In order to prove that the proposed exchange mechanism is indeed dictating the\nlong-range magnetic behavior, and in turn, the calculated value of \u0001 Emag, we studied the ef-\nfect of hydrostatic strain on this quantity. The applied strain changed the angles between the\noverlapping orbitals, a\u000becting the \u0001 Emag-values in keeping with the Goodenough-Kanamori\nrules [see Supplementary Figure 1].\nWe further analyzed the e\u000bects of SOC by performing band structure calculations along\nseveral k-lines in the Brillouin zone (BZ). This band structure is plotted in Fig. 3(a), which\nalso shows the path followed in the BZ. Figure 3(a) corresponds to the case when the\nmagnetization direction, M, is parallel to the y-direction, corresponding to the C 1h(C1)\nmagnetic group9[see Supplementary Note I]. The enhanced highlighted view of the topmost\nvalence and lowest conduction bands [Fig. 3(a)] shows valley polarization in both bands.\nThe di\u000berence between the maxima along the K1!\u0000 and \u0000!\u0000K1directions in the\ntopmost valence band was found to be \u001840 meV. The lowest conduction band also shows\na similar di\u000berence in the maxima, but with opposite sign ( \u0018\u000038 meV). This asymmetry\nin the band structure relative to the \u0000-point comes from the interplay between the Rashba\nand exchange e\u000bects due to the broken space- and time-reversal symmetries10. In Fig. 3(b),\nwe plot spin-resolved band structure for 1H-YSSe, showing only the \u0006Syspin projections,\nas the\u0006Sxand\u0006Szprojections are almost negligible. Figure 3(b) also shows an enhanced\nhighlighted view of the topmost valence and lowest conduction bands, with colors encoding\nthe spin projections. One can see that the topmost valence and lowest conduction bands are\nessentially spin up and spin down bands. This, in turn, implies that the spin, S, is almost\nparallel to Min our case. This leads to a highly anisotropic Rashba spin-splitting, which is\n8maximal when kis perpendicular to spin S(orM) and almost zero when kis parallel to S\n(orM). Hence, the degree of the splitting in the maxima itself, depends on the component\nof the k-vector perpendicular to the magnetization direction ( MorS) as can be seen from\nthe Rashba Hamiltonian, HR=\u000bR(ez\u0002k)\u0001S, which describes an electron of spin Smoving\nwith momentum kunder the in\ruence of an electric \feld oriented along the z-axis ( ez)10.\nHere,\u000bRis the Rashba parameter. From the Rashba Hamiltonian, we \fnd the splitting\nshould be strongest along the K1!\u0000!\u0000K1line as it is perpendicular to M. This can\nbe seen in the highlighted topmost valence and lowest conduction bands plotted in Figs 3(a)\nand (b), which shows that the splitting in the maxima along the K1!\u0000 and \u0000!\u0000K1is\ngreater than the splitting in the maxima along the K2!\u0000 and \u0000!\u0000K2directions.\nIn order to understand the relatively small value of MAE (about \u0000122\u0016eV) exhibited\nby our system, we considered the di\u000berences in the Bloch eigenstates: \u0001 E=\u000f(k)(010)-\n\u000f(k)(001), calculated for two magnetic states with the magnetization Maligned in the\nyandz-directions [see Fig. 3(c)]. In this \fgure, only the energy di\u000berences involving the\nhighest valence and lowest conduction bands are shown and the k-lines are the same as those\nin Figs 3(a) and (b). Both the curves exhibit a counterbalanced trend between \u0000K1\u0000\u0000 and\n\u0000\u0000K1, within the K1\u0000K2region and between K2\u0000\u0000 and \u0000\u0000(\u0000K2). Moreover, the curves\nadditionally exhibit a counterbalanced trend with respect to each other: when the energy\ndi\u000berence for the valence band drops the corresponding di\u000berence for the conduction band\nrises and vice versa. Both these trends are explained by the (anisotropic) Rashba e\u000bect in\nthe presence of ferromagnetic ordering. The counterbalanced contribution to the MAE in\nthek-space, shown in Fig. 3(c), corresponds to the hidden \frst order perturbation. Hence,\nour system exhibits what is known as \\unconventional MAE contributions in the k-space\"11.\nWhen integrated over the entire BZ, the \frst order contribution to the MAE vanishes (being\nan odd function in the k-space), and the resulting MAE will be given by the relatively small\nsecond-order perturbation terms12, explaining the small value of MAE [see Supplementary\nNote II].\nPiezoelectric properties of 1H-YSSe. In addition to being a magnet, 1H-YSSe also\ndisplays piezoelectric properties owing to its noncentrosymmetric structure. The piezoelec-\ntric e\u000bect refers to the electromechanical coupling in which polarization changes in response\nto the applied strain. Due to the C3vsymmetry of YSSe, the piezoelectric tensor has only\ntwo independent components: e11ande13, associated with the in-plane and out-of-plane\n9d\na\nefbc\nin-plane            out-of-planeIn-plane polarizationIn-plane polarization\nFIG. 4. Strain and layer-thickness dependent properties of 1H-YSSe: aChanges in in-plane and\nout-of-plane polarization of YSSe under the application of uniaxial strain along the armchair direc-\ntion. bElectronic and ionic Berry phases (di\u000berent scales) corresponding to the in-plane electric\npolarizations of the 1H-YSSe monolayer as a function of uniaxial strain along the armchair direc-\ntion, showing exact cancellation of the two phases at 0% strain and the switching of their role as\nthe dominant contributor to the total phase as we go from lattice contraction to expansion. cTotal\nBerry phase, showing the best \ft curve (black line). dLowest energy con\fguration (AA) of a bi-\nlayer of YSSe, wherein the S- and Se-atoms within the two layers bond, resulting in a non-magnetic\nstructure. eSpin density plot (\u0001 \u001a=\u001aMajority\u0000\u001aMinority), showing that in the odd number of\nlayers, the structure retains a net magnetic moment. fMagnetic moment of a stack of YSSe as a\nfunction of layer-thickness, showing odd-even layer-dependence.\npolarizations, respectively. We calculated the polarization of YSSe using the Berry phase\nmethod13,14(see Methods section). We \fnd that in the unstrained state, there is no in-plane\npolarization (as required by C3vsymmetry), while there is a spontaneous polarization of\n0:524\u000210\u000012C/m along the symmetry-allowed C3-axis (z-direction), with the dipole mo-\nment directed from the Se to S atoms. Figure 4(a) shows the in-plane and out-of-plane\nelectric polarizations, as a function of the uniaxial strain along the armchair direction. Here\nwe de\fne the strain percentage in a system with lattice constant balong the armchair di-\nrection as: 100\u0002(b\u0000b0)=b0, withb0being the equilibrium lattice constant. The slopes of\n10the curves in Fig. 4(a) give e11= 0:165\u000210\u000010C/m (in-plane piezoelectric coe\u000ecient at\n0% strain), and e13=\u00000:174\u000210\u000010C/m (out-of-plane piezoelectric coe\u000ecient), respec-\ntively. The best-\ft curve for the highly nonlinear behaviour of the in-plane polarization\nis given by b1x+b2x2, where the linear coe\u000ecient b1\u0011e11. The quadratic coe\u000ecient,\nb2= 0:130\u000210\u000010C/m, is unusually large as compared to those calculated for other similar\n2D materials, such as MoSSe [see Supplementary Fig. 2].\nIn order to understand the origin of the non-linear behaviour of the in-plane polarization\nas a function of strain, we analyzed both the electronic and ionic contributions to the total\nBerry phase, \u001etot. Figure 4(b) shows the electronic and the ionic phases, which are of\nopposite signs, resulting in very small values for \u001etot. In fact, at 0% strain the values of the\nelectronic and ionic phases are exactly half of the polarization quantum each ( \u00000:5 and 0:5,\nrespectively), and they cancel each other exactly, giving \u001etot= 0. In going from 0% strain\nto +4% (corresponding to the lattice expansion along the armchair direction), the electronic\nphase changes by a mere 0 :665\u000210\u00002, while the ionic phase changes by \u00001:382\u000210\u00002.\nHence, the ionic contribution dominates the total phase in this region [see Fig. 4(b)]. In\nparticular, in the strain region between +2% and +4%, the electronic phase is almost a\nconstant. This means that the charge centers of the maximally-localized Wannier functions\npractically follow a homogeneous deformation15. At the same time, the ionic positions fail\nto follow a homogeneous deformation (i.e. they undergo internal distortions relative to the\nlattice). Hence, we observe an overwhelming dominance of the ionic contribution to the total\nphase. The situation changes in going from 0% to \u00004%, which corresponds to the lattice\ncontraction along the armchair direction. Here, in contrast to the case of lattice expansion,\nit is the change in the electronic phase (= \u00002:418\u000210\u00002) that dominates in the total phase\n(as compared to the ionic phase that changes by 2 :052\u000210\u00002). Thus, 0% strain serves as a\ncrossover point, across which the role of the dominant contributor to total phase is switched\nbetween the ionic and electronic phases. This e\u000bect, which is especially enhanced by the\nnear-cancellation of the electronic and ionic terms in the total phase, is the reason for the\nnonlinear piezoelectricity in the system.\nLayer-thickness dependent properties of 1H-YSSe. Lastly, we considered the in\ru-\nence of layer-thickness on the electronic structure properties of a YSSe monolayer. The\nmonolayers of YSSe can be stacked in di\u000berent registries such as: (i) AA-stacking, with the\nchalcogen of one layer over the chalcogen of the next layer, and (ii) AB stacking, with the\n11chalcogen in one layer over the metal atom of the second layer [see Supplementary Fig. 3].\nUnlike most TMDs, YSSe prefers AA stacking over AB, as this stacking allows YSSe to form\nbonds between the two layers, lowering its energy by about 1.35 eV [see Fig. 4(d)]. The re-\nsultant bilayer structure is non-magnetic. On the other hand, the trilayer structure is again\nmagnetic, with a moment of 1 \u0016B. In a trilayer, two of the layers form bonds, allowing the\nthird non-bonded layer to retain its magnetism. This can be seen in Fig. 4(e), which shows\nthe spin-density plot (\u0001 \u001a=\u001aMajority\u0000\u001aMinority) for a trilayer. We \fnd that YSSe shows\nthis interesting odd-even layer dependence of magnetic properties for all tested thicknesses\n[see Fig. 4(f)].\nDiscussion\nUsing \frst principles based-methods, we have predicted a new magnetic Janus TMD,\n1H-YSSe. Due to its unique structural, electronic and spin properties, 1H-YSSe displays\nan interesting and uncommon combination of magnetism and piezoelectricity in a single\nmaterial. Such materials are rare because conventional magnets have localized electrons in\npartially \flled 3 d-orbitals of transition metals (or the 4 f-orbitals of rare earth metals), while\npiezoelectric materials are insulators, usually consisting of d0ord10elements16. A material\nwhere both of these properties coexist is potentially useful in proximity e\u000bect devices, such\nas the superconductor/semiconductor/magnet heterostructures being explored as solid-state\nplatforms for Majorana bound states, where the latter can be tuned/controlled by external\nstimuli (e.g. electric \feld and/or strain) applied to the magnet. Furthermore, the broken\nspace- and time-symmetry ensures valley polarization in the topmost valence and the lowest\nconduction bands. These properties make YSSe a promising material for novel applications\nsuch as ultra-compact spintronics, valleytronics, and quantum devices.\nMethods\nWe carried out the spin-polarized density functional theory (DFT) calculations using\nthe projected augmented wave method (PAW)17,18as implemented in the Vienna Ab-Initio\nSimulation package. A Monkhorst-Pack19k-point mesh of size 16 \u000216\u00021 and a plane wave\nenergy cuto\u000b of 550 eV were used in the calculations. We employed the generalized gradi-\nent approximation of Perdew, Burke and Ernzerhof (PBE)20for the exchange-correlation\nfunctional. In the Supplement (Supplementary Fig. 4), we also provide the results of the\n12computationally-expensive hybrid functional, HSE0621,22, which includes a \fxed percentage\nof Hartree-Fock exchange in the exchange-correlation functional. This functional was used\nsolely for the semiconducting phase of YSSe for its greater accuracy in predicting band gaps\nof semiconductors. For the case of the metallic phase of YSSe, we used only PBE, as hybrid\nfunctionals are known to give erroneous results for metals23. The structure was optimized\nby ensuring that the forces on each atom are less than 0.001 eV/ \u0017A. We also ensured that the\ninteraction between a monolayer and its images is minimized by adding a vacuum layer of\n20\u0017A. The dynamical and thermodynamical stability of the YSSe monolayer was ensured by\ncalculating phonon dispersion using the \fnite displacement method24as implemented in the\nPhonopy code25and by performing ab-initio molecular dynamic simulations using canonical\nensembles.\nThe piezoelectric properties of YSSe were obtained via the Berry-phase approach13,14.\nTo calculate the Berry phases, we used the Ceperley-Alder functional26and Hartwigsen-\nGoedecker-Hutter pseudopotentials27as implemented in ABINIT package28. The chosen\npseudopotentials are characterized by 6, 3 and 6 valence electrons, for S, Y, and Se ions,\nrespectively. Strictly speaking, the existence of magnetic ordering in the system can addi-\ntionally induce a spontaneous in-plane polarization via spin-orbit coupling, as in the case of\nterbium manganite29. This magneto-electric coupling, however, was found to be negligibly\nsmall. 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The computational sup-\nport is provided by the Extreme Science and Engineering Discovery Environment (XSEDE)\nunder Project PHY180014, which is supported by National Science Foundation grant num-\nber ACI-1548562. For three-dimensional visualization of crystals and volumetric data, use\nof VESTA 3 software is acknowledged.\nAuthor Contributions\nP.D. conceived and designed the project. P.K., I.N. and P.D. equally contributed to the \frst\nprinciples calculations performed to determine the properties of YSSe monolayers. P.M.\ndetermined properties of multiple layers of YSSe. All authors were involved in the analysis\nof results and discussions. P.K. I.N. and P.D. wrote the manuscript. All authors reviewed\nand contributed to the \fnal revision of the manuscript.\nAdditional information\nCompeting Interests\nThe author declares no competing \fnancial interests.\n16Correspondence\nCorrespondence and requests for materials should be addressed to P.D. (email: prati-\nbha.dev@howard.edu).\n17"    },    {        "title": "2010.02455v2.An_ultra_stable_1_5_tesla_permanent_magnet_assembly_for_qubit_experiments_at_cryogenic_temperatures.pdf",        "content": "An ultra-stable 1.5 tesla permanent magnet assembly for qubit experiments\nat cryogenic temperatures\nC. Adambukulam,a)V. K. Sewani, H. G. Stemp, S. Asaad,b)M. T. Mądzik,c)A. Morello, and A. Lauchtd)\nCentre for Quantum Computation & Communication Technology, School of Electrical\nEngineering & Telecommunications, UNSW Sydney, New South Wales 2052,\nAustralia\n(Dated: August 12, 2021)\nMagneticfieldsareastandardtoolinthetoolboxofeveryphysicist, andarerequiredforthecharacterizationof\nmaterials,aswellasthepolarizationofspinsinnuclearmagneticresonanceorelectronparamagneticresonance\nexperiments. Quite often a static magnetic field of sufficiently large, but fixed magnitude is suitable for these\ntasks. Here we present a permanent magnet assembly that can achieve magnetic field strengths of up to 1.5 T\nover an air gap length of 7 mm. The assembly is based on a Halbach array of neodymium (NdFeB) magnets,\nwith the inclusion of the soft magnetic material Supermendur to boost the magnetic field strength inside the\nair gap. We present the design, simulationand characterization of thepermanent magnet assembly, measuring\nan outstanding magnetic field stability with a drift rate, |D|<2.8ppb/h. Our measurements demonstrate\nthat this assembly can be used for spin qubit experiments inside a dilution refrigerator, successfully replacing\nthe more expensive and bulky superconducting solenoids.\nPACS numbers: 07.55.Db, 75.50.Ww, 03.67.Lx, 76.30.-v, 76.60.-k\nI. INTRODUCTION\nExperiments that require strong magnetic fields usu-\nally rely on superconducting solenoids, which are large,\nexpensive, and require both a stabilized current source\nand cryogenic temperatures for operation. However,\nmany experiments only need a static magnetic field of\norder of magnitude 1 T, that is “ set-and-forget ”, as is the\ncase for spin-based quantum computation with electron\nspin states. With a g-factor ofg≈2and a gyromagnetic\nratio ofγe≈28 GHz/T, electron spins in gate-defined\nquantum dots or donors in silicon1–3require only mod-\nerate magnetic fields of B= 0.5−1.5T to achieve a\nZeeman splitting γeBthat is larger than the thermal en-\nergykBTat typical dilution refrigerator temperatures of\nT < 100mK. This then allows for the read-out and ini-\ntialization of the spin state via spin-dependent tunneling\nto and from a thermally-broadened electron reservoir4,5\nor via a spin relaxation process.6,7Alternative spin ini-\ntialization and readout methods, like singlet-triplet ini-\ntialization for quantum dots7–10or optical cycling for\ncolour centers in diamond and silicon carbide,11,12do\nnot require γeB/greatermuchkBT, and can therefore work at\nhigher temperatures7,11or with quantum systems that\nhave smaller g-factors like electron spins in GaAs quan-\ntum dots9or hole spins in Ge quantum dots.10Neverthe-\nless, these systems still require external magnetic fields\nfor selective addressability of individual quantum bits10\na)Corresponding Author: c.adambukulam@unsw.edu.au\nb)Currently at Niels Bohr Institute, University of Copenhagen,\nBlegdamsvej 172100 Copenhagen, Denmark.\nc)Currently at QuTech, Delft University of Technology, 2826 CJ\nDelft, The Netherlands.\nd)a.laucht@unsw.edu.auor to provide a well-defined quantization axis. In any\ncase, magnetic fields of B= 0.3−1.5T are usually suf-\nficient for most quantum computation experiments.\nIn this article we present a permanent magnet assem-\nbly that can be used for experiments that require mag-\nnetic fields between 0.35T and 1.5T for samples with\nfootprints of several square millimeters. To achieve these\nhigh fields, we employ a design based on Halbach ar-\nrays with strong neodymium (NdFeB) permanent mag-\nnets, and use the soft magnetic material Supermendur to\nboost the magnetic field strength even further. The dif-\nferent components are held in place by a copper box with\nindividual compartments and brass lids. One such per-\nmanent magnet assembly can be constructed for around\n$750USD, made up of $100USD for the neodymium\nmagnets, $150USD for the Supermendur pieces and\n$500USD for the machined copper box and brass lids.\nIn Sec. II we show detailed sketches of the design, and\nuse simulations in Sec. III to verify that fields of up to\n1.5Tmagnitudecanbeachieved. Furthermore, thefields\nare tunable by varying the length of the Supermendur\npieces and, therefore, the length of the air gap. We then\ncharacterise the assembly as a whole, by first presenting\ncryogenic measurements in Sec. IVA to show that the\nspin reorientation transition (SRT) of neodymium does\nnot cause any issues, and comparing the cool-down times\nof a cryogen-free BlueFors dilution refrigerator with a\npermanent magnet assembly to that of a superconduct-\ning magnet. Next, using a single donor spin qubit in\nsilicon - in Sec. IVB - we demonstrate that spin qubit\nexperiments can indeed be carried out in the permanent\nmagnet assembly. Monitoring the resonance frequency of\nthe spin qubit allows us to show the long-term stability\nof the magnetic field is limited by magnetic field drift,\n|D|<2.8ppb/h at the base temperature of our dilution\nrefrigerator. Moreover, this drift is at least an order ofarXiv:2010.02455v2  [physics.ins-det]  11 Aug 20212\nFigure 1. (a) Schematic of the permanent magnet assembly. N and S, and I and II denote north and south, and the two\npermanent magnet shapes, respectively. (b) The magnet assembly as used in an experiment. The magnets are housed in a\ncopper enclosure 1/circlecopyrtwith the enclosure lids 2/circlecopyrtand brackets 3/circlecopyrtused to keep the magnets and Supermendur aligned and in\nplace. The device 4/circlecopyrtis housed in its own copper enclosure 5/circlecopyrtwith DC and MW electrical access provided by DC lines 6/circlecopyrt\n(interfaced with MMCX connectors on the back of 5/circlecopyrt) and a coplanar waveguide (interfaced with a 2.92 mm K connector 7/circlecopyrt)\non a printed circuit board 8/circlecopyrt. Thermal contact, in a dilution refrigerator, is ensured by bolting 1/circlecopyrtto the mixing chamber\nplate with the base plate 9/circlecopyrt. (c) Cross-section of the magnet assembly in the yz-plane, at the position of the device 4/circlecopyrt.\nmagnitude better than that of a commercially available\nsuperconductingmagnetfittedwithastate-of-the-artlow\ndrift rate option.\nOur proposed design can, therefore, replace traditional\nsuperconducting magnets at a significantly lower cost,\nin experiments where moderate magnetic field gradients\ncan be tolerated. This is for example, the case for\nSi/Ge quantum dot spin qubits which are routinely con-\ntrolled via electrically-driven spin resonance inside mag-\nneticfieldgradientsof ∼1000T/mmgeneratedbymicro-\nmagnets.13Moreover, we outline a method to nullify the\nmagnetic field gradients, greatly expanding the applica-\nbility of our proposed solution. Additionally, dilution\nrefrigerators can typically only accommodate a single su-\nperconducting magnet. Due to the proposed design’s\nlight weight, relatively small size, and low stray fields,\nseveral magnet assemblies can be mounted in the same\nfridge and can be used alongside a superconducting mag-\nnet. Thisallowssimultaneousmeasurementofseveralde-\nvices and reduces cool-down times. Further advantages\nare the stable magnetic field with only a few percent shift\nin field between room temperature and mK, and the in-\nherent absence of quenching when cryo-cooling fails.II. MAGNET ASSEMBLY\nWe base our magnet assembly (see FIG. 1) on the de-\nsignproposedin Ref.14, a variationof thelinearHalbach\narray.15–17The major point of difference between the de-\nsign in Ref. 14 and the assembly described in this article\nis the partial replacement of the center magnets with the\nsoft magnetic material, Supermendur (as discussed be-\nlow) which are then raised out of the magnet plane to\nprovide simple electrical access. A single linear Halbach\narrayisanarrangementofmagnetsconstructedsuchthat\nthe magnetic field is cancelled on one side of the array\nand enhanced on the other.18In the referenced design,\nas in our design, two Halbach arrays are placed back-to-\nback. In this configuration, the magnetic field outside of\nthe array is cancelled and a large and confined magnetic\nfield is generated between the two arrays, and specifically\nbetween the two center magnets of the assembly. In the\ndesign proposed in Ref. 14, this magnetic field would be\nat the center of the air gap, which is where devices would\nneed to be mounted. This is a relatively confined space,\nwhere it would be difficult to easily fit and route the nec-\nessary electrical and microwave connections to operate\nour devices. This poses difficulties when integrating the3\nFigure 2. (a) Magnitude of the total magnetic field strength |B|along anxz-cross-section of the magnet assembly, with the\nlocation of the qubit taken to be the origin. The dashed line shows the outline of the assembly, the 80 mTcontour marks the\ncritical field of lead and the grey regions denote supermendur. (inset) Magnitudes of Bz(left) andBr=/radicalbig\nB2x+B2y(right)\nmagnetic field components in the air gap. The desired spin quantization axis is the z-axis. (b) The norm of the magnetic field\ngradient|∇B|. (c) The magnetic field strength at the origin as function of air gap length. The experimental data point is from\nthe measurement shown in FIG. 4(a). (d) The norm of the magnetic field gradient as a function of air gap length.\nassembly into our experimental setup, as to mitigate this\nissue would require extending the air gap length until the\nfield was impracticably weak or the assembly impracti-\ncably large. However, when the Supermendur pieces are\nraised,alargemagneticfieldisproducedbetweenthemat\nthe surface of the assembly, where devices can be placed,\nwith enough space to fit necessary electrical connections.\nIn addition, a level of control over the magnetic field\nstrength at the air gap can be attained by the use of\nbrass spacers. The spacers, in conjunction with smaller\nSupermendur pieces, allow the air gap length and hence,\nthe magnetic field produced by the assembly itself, to be\nselected at its time of construction. In a more recent,\nhowever untested, version of the magnet assembly, we\nuse Supermendur pieces with threaded holes to realize a\ncontinuously tunable air gap, that can be tuned by the\nturn of a screw.\nJudicious selection of the magnetic material used is es-\nsential in ensuring that the designed assembly produces\nMagnet Size (mm) Elevation (mm)\nI 20×30×60 0\nII 10×30×10 0\nSupermendur 10×25×50 13.5\nTable I. The magnet dimensions (formatted as x×y×z)\nand elevation from the bottom of the array (in the negative\ny-direction).a sufficiently large magnetic field while minimising its\ndimensions. As NdFeB magnets are the strongest class\nof available permanent magnets,19,20N52 grade NdFeB\nmagnets with a typical residual magnetization of approx-\nimately 1.45 Tare used in the assembly. However, at\ntemperatures below 135 K, it becomes energetically fa-\nvorable for the magnetic moments in the easy axis of\nthe Nd 2Fe14B phase of the magnet to tilt from its room\ntemperature axis of [001]. At 4.2K, the easy axis cants\n±30.6◦towards the [110]axis.21,22This is the well doc-\numented spin reorientation transition (SRT) which has\nthe effect of reducing the magnetization of the NdFeB\npermanent magnet. Consequently, the SRT lowers the\nmagnetic field produced by NdFeB magnet assemblies at\ncryogenic temperatures.23,24Nonetheless, it is expected\nthat the magnetization of N52 NdFeB magnets remains\nlarge even after the SRT. It should be noted that, follow-\ning the SRT, although the magnitude of the magnetiza-\ntion will decrease, we expect that due to effect of aver-\naging the tilt over the large number of Nd 2Fe14B grains,\nthe direction of magnetization should remain the same\nregardless of temperature (see Appendix).\nThe inclusion of a soft magnetic material with a low\nresidual magnetization and a high saturation magnetiza-\ntion can be used to further strengthen the magnetic field\nproduced by the assembly. As mentioned above, the cen-\nter permanent magnets are partially replaced with Su-\npermendur, a soft magnetic alloy of 2%vanadium, 49%4\niron and 49%cobalt that has a saturation magnetization\nof approximately 2.4 T.25That is∼1T greater than the\nresidual magnetization of NdFeB. Hence, the magnetic\nfield produced by the assembly can be enhanced by plac-\ning saturated supermendur near the air gap. To that\nend, the center magnets are partially replaced by Super-\nmendur which, given its low saturation induction,25is\nsaturated near the air gap. Thus, the Supermendur pro-\nvides a major contribution to, and hence substantially\nincreases, the magnetic field in the air gap. In addition,\nbased on the negligible decrease in saturation magneti-\nzation observed for Supermendur at 4.2 K,26we assume\nthat the cryogenic and room temperature behavior of Su-\npermendur are similar.\nAs a consequence of the highly confined magnetic field\nin conjunction with the use of strong magnetic materi-\nals, the magnet assembly is compact with a footprint of\n55×87 mm. For reference, the diameter of the mix-\ning chamber plates of Bluefors LD and XLD series dilu-\ntion refrigerators are 290 mmand500 mm, respectively.\nThus, we estimate that at least four magnet assemblies\ncanbemountedverticallytoonesideofthemixingcham-\nber plate of an LD series dilution refrigerator [that is,\nmounted with the z-axis perpendicular to the mixing\nchamber plate surface with the point of contact being the\nbase plate of the assembly - 9/circlecopyrtin FIG. 1(b)]. If both, the\ntop and the bottom surfaces of the mixing chamber plate\nare fitted with magnet assemblies, that number rises to\neight and would further increase for an XLD series dilu-\ntion refrigerator. This enables multiple experiments to\nbe run in the same dilution refrigerator, with the upper\nbound on the number of experiments set by the cabling\nand necessary electronics and not by the bore of a super-\nconducting solenoid.\nIII. SIMULATION\nThe magnet assembly was simulated in Radia, a\nboundary integral method magnetostatics solver for Wol-\nfram Mathematica, developed by the European Syn-\nchrotron Radiation Facility.27,28Given the large coerciv-\nity of NdFeB magnets at cryogenic temperatures,23we\nmodelthepermanentmagnetsasinsensitivetoanyexter-\nnalmagneticfieldappliedtothemagnetandwitharesid-\nual magnetization of 1.24 T. This is the room tempera-\nture residual magnetization rotated by 30.6◦and then\nprojected onto the room temperature easy axis. While\nthis model is not an entirely accurate representation of\nan NdFeB magnet, especially given the rising magnitude\nof the residual magnetization below the SRT,21it suf-\nfices to estimate the worst case behaviour of the mag-\nnets. We model the Supermendur as a non-linear mag-\nnetic material based on its room temperature magnetiza-\ntion curve. Finally, to simulate the effect of varying the\nair gap length, we correct our estimate of the magnetiza-\ntion of the NdFeB magnets at millikelvin temperatures\nfrom the electron spin resonance frequency in a silicon-\nFigure 3. (a) Temperature dependence of the magnetic field\ngenerated by the magnet assembly, with the NdFeB spin re-\norientation transition (SRT) indicated. (b) Pulse tube (PT,\nleft) and mixing chamber (MXC, right) temperature during\nthe two cool-down stages of a dilution refrigerator equipped\nwith either a superconducting solenoid or a permanent mag-\nnet assembly.\nbased single-atom spin qubit device29–31mounted inside\nthe permanent magnet assembly (see Sec. IVB).\nIn FIG. 2 we plot the results of our magnetic field\nsimulations for an air gap of 7 mm(the assembly is with-\nout brass spacers). The main panel in FIG. 2(a) shows\n|B|the magnitude of the magnetic field strength for the\nwhole magnet assembly, plotted for the xz-cross-section\nat they-position of the qubit location. The bottom in-\nsets are zoom-ins to the qubit region, showing the Bzand\nBr=/radicalBig\nB2x+B2ycomponents with respect to the qubit\nquantization axis. At the expected qubit position, we\ncomputeamagneticfluxdensityof 1.44 Tthatisstrongly\naligned to the z-axis and strongly confined to the air gap.\nThe high magnetic flux density ensures that the Zeeman\nsplitting of the electron spin is larger than the thermal\nenergy at dilution refrigerator temperatures, a prerequi-\nsite for many spin initialization and readout schemes.4–7\nIn addition, the strong magnetic field confinement to the\ninside of the enclosure would, in the future, allow for\ntotal electromagnetic isolation of the qubit by enclosing\nthe assembly and device inside a superconducting shield.\nIn FIG. 2(a), we mark the 80 mTiso-field line indicat-\ning where the stray field falls below the critical field of\nlead, to approximate the dimensions required by such a\nshielding enclosure. The outline of the magnet assembly5\nis indicated by the dashed line.\nInFIG.2(b)weplotthenormofthemagneticfieldgra-\ndient which we compute as |∇B|=/radicalBig/summationtext\ni,j(∂iBj)2where\ni,j∈{x,y,z}. We find|∇B|∼18mT/mm at the qubit\nlocation. This is considerably greater than that which\ncan be achieved by shimmed superconducting solenoids\n<1µT/mm32or even∼5µT/mm - the value simu-\nlated for the superconducting solenoid used in our setup\nwhen generating a 1.55T magnetic field.33A more de-\ntailed inspection reveals that the only non-zero gradient\ncomponents are ∂yBz=∂zBy≈−12.6mT/mm. Nullifi-\ncation of this gradient term can be achieved by lowering\nthe elevation of the Supermendur pieces from 13.5mm\nto2.5mm, thereby, introducing mirror symmetry in the\nxz-plane. Samples must then be placed, at the centre of\nthe assembly (at an elevation of 15mm from the bot-\ntom of the array). However, this approach comes with\nthe disadvantage of difficult electrical access. While the\nmoderate magnetic field gradient would be detrimental\nto the performance of spatially extended qubit ensem-\nbles and spatially separated qubits, they are not detri-\nmental to single donor spins in silicon. In our setup a\nmagnetic field gradient can be transduced into magnetic\nnoise on the qubit if the magnet moves with respect to\nthe sample stage.13,32,33This is to be expected in com-\nmon cryomagnetic systems, where the sample plate and\nthe magnet are connected at very distant flanges - as is\nthe case with superconducting solenoids. The matter can\nbe particularly severe in cryogen-free refrigerators due to\nthe vibrations produced by the pulse-tube coolers. How-\never, in the case of the permanent magnet assembly, the\nmagnet and sample form part of the same compact and\nrigid structure. Relative motion between the magnet as-\nsembly and the sample is suppressed, making the field\ngradient inconsequential for qubit coherence.\nFinally, in FIG. 2(c) and FIG. 2(d) we show how the\nmagneticfieldstrengthandgradientatthequbitposition\nvary when the size of the air gap length is increased. This\ncan be accomplished for example by shortening the Su-\npermendur pieces and inserting brass spacers near the air\ngaptokeeptheminplace. Thesimulationsshowthatthe\nfield strength can be adjusted from 1.44 Tto0.36 Twhen\nthe air gap length is varied from 7mm to 50mm. The\nupper limit to the produced magnetic field is ∼2.74T\nand is the sum of the saturation magnetization of Super-\nmendur and the magnetic field provided by the rest of\nthe assembly. This upper limit is achieved when the air\ngap length approaches 0mm. In order to ensure a large\nenough sample volume to comfortably mount devices, we\nlimit the minimum air gap length to 7mm. The air gap\nlength can be extended - by shortening the Supermendur\npieces - until no Supermendur remains. At this point the\nassembly produces ∼0.35T. This provides a large range\nof magnetic fields for qubit experiments, albeit without\nthe possibility for in-situ tunability of the assembly in its\ncurrent form.\nFigure 4. (a) Coherent nuclear magnetic resonance (NMR)\nspectrum of the ionized31P+donor nuclear spin performed\nwith a permanent magnet assembly. (b) Free induction de-\ncayT∗\n2(left) and spin echo THahn\n2(right) times for ionized\n31P+donor nuclear spins from various devices of the latest\nbatches.30The measurements were performed with either a\npermanent magnet assembly or a superconducting electro-\nmagnet and were performed on different devices leading to\nthe spread in the measured T∗\n2andTHahn\n2values.\nIV. EXPERIMENTS\nA. Cryogenic Behaviour of the Assembly\nTo measure the temperature response of the magnet\nassembly, a Cryomagnetics Inc. HSU-1 calibrated Hall\neffect sensor is placed in the air gap and the assembly\nis installed inside a dilution refrigerator. The Hall volt-\nage is measured during warm up of the fridge. FIG. 3(a)\nshows the temperature response of the assembly with the\nNdFeB spin reorientation transition (SRT) visible. We\nattribute the relatively small change in magnetic field\nover the temperature range to the saturated Superme-\nndur, which produces the bulk of the magnetic field at\nthe air gap and which is insensitive to fluctuations in the\nNdFeB magnetization. Overall, the magnetic field inside\nthe air gap changes by only ∼6%over the whole tem-\nperature range from 300 Kdown to mK,34making it an\ninteresting option for wide-range temperature dependent\nstudies and measurements.\nNext, we measure the cool-down time of a BlueFors6\nBF-LD400 dilution refrigerator with either a permanent\nmagnet assembly or an American Magnetics 5-1-1 T su-\nperconducting vector magnet mounted. We observe a\n∼14 himprovement in the time taken for the pre-cool\nstep from room temperature to 4 K[see FIG. 3(b) - left\npanel]. This is expected as the mass of the superconduct-\ning solenoid ( 38.1 kg) is considerably larger than that of\nthe assembly ( 2.83 kg). When using the superconduct-\ning magnet, the cooldown time from 4K to 30mK is\n∼8h shorter than that of the permanent magnet assem-\nbly. As the superconducting solenoid operates at 4 Kit is\nnot further cooled during the second cool-down step from\n4 KtomK, however, the magnet assembly is cooled to\nmKalong with the sample. This, along with the weaker\ncooling powers at lower temperatures, accounts for the\nlonger cooling time in the second cool-down step when\nthe magnet assembly is used [see FIG. 3(b) - right panel].\nNonetheless, there is a ∼6h improvement in total cool-\ndown time when using the permanent magnet assembly.\nThere is one further distinct advantage of the perma-\nnent magnet assembly compared to a superconducting\nmagnet. Superconducting magnets may quench and raise\nthecryostattemperaturebytensofkelvin. Thissituation\nis particularly common in cryogen-free dilution refrigera-\ntors that are critically dependent on a chilled water sup-\nply. The helium compressor of the pulse tube will shut\noff within tens of seconds of any interruption to the cool-\ning water resulting in the overheating and quenching of\nthe superconducting solenoid. This is not an issue with\nthe permanent magnet assembly.\nB. Spin Qubit Experiments\nAs a final test for the permanent magnet assembly, we\nuse it to provide the magnetic field for spin qubit ex-\nperiments with ion implanted, single31P donors in iso-\ntopically enriched28Si (see Appendix for more details).\nThe physical systems and devices have been described in\nample detail elsewhere,5,29,31and we simply use them to\nprobe the magnetic field and its stability.\nIn FIG. 4(a) we show a coherent nuclear magnetic\nresonance (NMR) spectrum of the ionized31P+nuclear\nspin. In this charge configuration, the nucleus has no\nhyperfine coupling. The NMR frequency is thus given\nbyνNMR =γnB, whereγn=−17.23 MHz/Tis the nu-\nclear gyromagnetic ratio of31P. By performing repeated,\nnon-destructive single-shot readout while varying the fre-\nquency of an RF magnetic field, we can measure the z-\ncomponent of the applied magnetic field to a high degree\nof precision. We extract Bz= 0.984082±0.000004T,\nwhich is in good agreement with our simulations, as\nshown in FIG. 2(c).\nFigure 5. (a) Shift in the NMR frequency of a31P+nuclear\nspin measured over the course of ∼13h. The measurement\nwas performed using the permanent magnet assembly. The\ndata is fit to a line and a drift rate of −1.2±0.8ppb/h is\nextracted. A 10-point rolling average filtered version of NMR\nshift (light grey curve) has been provided to make the drift\nin the NMR frequency shift more apparent. (b) The over-\nlapped Allan deviation and its 95%confidence interval (CI)\nof the data presented in (a) as function of integration time, τ.\n(c) The long term shift in the resonance frequency of a31P+\nnuclear spin and a31P0electron spin in the permanent mag-\nnet assembly (orange circles) and a superconducting solenoid\n(blue squares), respectively. (d) The shift in NMR frequency\nand mixing chamber (MXC) temperature as function of time.\nAfter timet= 4h, the MXC heater is turned on, resulting in\nthe rise in MXC temperature. At t= 51h, the MXC heater\nis turned off and the temperature returns to the dilution re-\nfrigerator base temperature.\nWe try to gauge possible fluctuations in the magnetic\nfield produced by the assembly by conducting coherence\ntime measurements on the nuclear spin. Unfortunately,\nwewerenotabletocomparethesamedeviceanddonorin\nboth a superconducting solenoid and a permanent mag-\nnet assembly. Hence, we show in FIG. 4(b) the free in-7\nduction decay coherence times T∗\n2and Hahn echo coher-\nence times THahn\n2for a few devices belonging to similar,\nrecent fabrication batches.30While the outcome of this\ncomparison is certainly not conclusive, the data suggests\nthat the permanent magnet assembly does not worsen\nthe qubit coherence, and possibly improves it compared\nto a superconducting solenoid. A more conclusive test\nof the possible fluctuations of the permanent magnet as-\nsembly would be to perform spin noise spectroscopy, as\nperformed in Ref. 29 using the electron spin of a sin-\ngle31P donor. In that system, however, the frequency\nfluctuations due to magnetic field instability would be\ncompounded with other effects, caused by, for example,\nresidual coupling to29Si nuclei or electric field noise af-\nfecting the electron g-factor or electron-nuclear hyper-\nfine coupling.35For this reason, we have chosen to uti-\nlize the31P nucleus as our magnetic field probe. How-\never, because of the much longer ( ∼3orders of mag-\nnitude) coherence time of the nuclei compared to elec-\ntrons, nuclear spin noise spectroscopy would take an un-\nmanageable amount of time and has indeed never been\nattempted.\nFinally, we attempt to determine the long term stabil-\nity of the permanent magnet assembly using the NMR\nfrequency of a31P+nuclear spin. Previously we at-\ntempted to gain insight into magnetic field fluctuations\nthat occur in the 1Hz to 20kHz regime; which strongly\naffect qubit coherence and therefore may be detected by\nthe free induction decay or spin echo. Here, we measure\nthemagneticfielddriftandfluctuationsoverthecourseof\nhours and days. To gauge the magnetic field stability, we\ncompute the overlapped Allan deviation36as presented\nin FIG. 5(b) from the shift in NMR resonance frequency\nshown in FIG. 5(a). We measure a minimum Allan devi-\nation on the order of ∼10−9when the integration time\nτ∼1500s.\nTo investigate the drift, we fit the NMR frequency\ntrace produced in the FIG. 5(a) to a line and extract\na drift rate, D=−1.2±0.8ppb/h. Additionally, we\nmeasure the shift in NMR frequency over the course of\n17days [see FIG. 5(c)] but were unable to resolve any\ndrift beyond the fitting error. Thus, we set an upper\nboundonthemagneticfielddriftrateto |D|<2.8ppb/h.\nA possible source of the magnetic field instability is the\ndrift and fluctuations of the temperature of the perma-\nnent magnet assembly. To that end we measure the shift\nin the NMR frequency [see FIG. 5(d)] when the MXC\nheater is turned on and the mixing chamber plate tem-\nperature rises. We observe a ∼40ppm shift in the NMR\nfrequency when the MXC temperature reaches ∼60mK.\nThe significant lag in the NMR shift compared to the\nchange in the MXC temperature suggests that the effect\nis indeed due to the very slow change in the magnetic\nfield while the permanent magnet assembly reaches ther-\nmal equilibrium. In principle, it is conceivable that some\nmechanisms internal to the device (paramagnetic spins,\nPauli magnetism in electron gases) might cause minus-\ncule magnetic fields which are also temperature depen-dant, but we would expect such effects to react much\nmore quickly to temperature change at the mixing cham-\nber. In any case, the results presented here provide am-\nple evidence that the normal temperature stability of a\ndilution refrigerator results in remarkably low magnetic\nfield drift rates. In previous measurements with stan-\ndard superconducting solenoids in a liquid helium bath\nand operated in persistent mode, we noticed a decay of\n∼15ppm/h when the driving current through the leads\nwas off, and∼1.5ppm/h when the nominal driving cur-\nrent was still fed through the leads.29Using an Amer-\nican Magnetics superconducting magnet with low drift\nrate option for the persistent mode switch reduces the\nmagnetic field decay to ∼40ppb/h [see FIG. 5(c)]. This\ncorresponds to a change in the 40GHz resonance fre-\nquency of the electron spin qubit of about 1 MHz in a\nmonth. Thedrift rateof thepermanent magnetassembly\nis over an order of magnitude lower than the drift mea-\nsured with the state-of-the-art superconducting solenoid,\nand compares favourably to the magnetic field drift spec-\nified for commercially available NMR spectrometers of\n∼10ppb/h37andthe∼1ppb/hthatarepracticallypos-\nsible.38Moreover, the excellent magnetic field stability is\na strong indication that the use of permanent magnet\nassemblies like the one described here can be a key en-\nabling technology to achieve the best possible long-term\nstability and for qubits whose energy splitting depends\nupon a magnetic field.\nV. CONCLUSION\nIn conclusion, we have presented a permanent mag-\nnet assembly based on neodymium (NdFeB) magnets\nthat can provide magnetic field strengths of up to 1.5 T\nover an air gap of 7 mmlength. The assembly works\nfor a wide temperature range from 300 KtomKtem-\nperature with only a few percent variation in magnetic\nfield strength. This makes it ideal for a wide variety\nof experiments, including spin qubits, nuclear magnetic\nresonance, and electron paramagnetic resonance spec-\ntroscopy, where medium-high fields with good stability\nare required. Furthermore, with a production price of\n∼$750USD, a size of 138×87×55mm, and a weight\nof2.83kg, the assembly is much cheaper and more com-\npact than superconducting solenoids, and allows several\nexperiments to be run in parallel on the same mixing\nchamber plate of a dilution refrigerator.\nACKNOWLEDGMENTS\nWe acknowledge the team involved in the fabrication\nof the donor spin qubit devices used for the field stability\ntest: Fay E. Hudson, Kohei M. Itoh, David N. Jamieson,\nA. Melwin Jakob, Brett C. Johnson, Jeffrey C. McCal-\nlum, and Andrew S. Dzurak. The research was funded by\nthe Australian Research Council Centre of Excellence for8\nQuantum Computation and Communication Technology\n(Grant No. CE170100012) and the US Army Research\nOffice (Contract No. W911NF-17-1-0200). We acknowl-\nedge the support of the Australian National Fabrication\nFacility (ANFF). A.L. and C.A. acknowledge support\nthrough the UNSW Scientia Program. The views and\nconclusions contained in this document are those of the\nauthorsandshouldnotbeinterpretedasrepresentingthe\nofficial policies, either expressed or implied, of the ARO\nor the US Government. The US Government is autho-\nrized to reproduce and distribute reprints for government\npurposes notwithstanding any copyright notation herein.\nDATA AVAILABILITY STATEMENT\nThe data that support the findings of this study\nare available from the corresponding authors upon\nreasonable request.\nAPPENDIX\nA. Cryogenic Magnetization of NdFeB Magnets\nAs a quick test to check whether the magnetization\ndirection of the NdFeB magnets experienced a change in\nmagnetization direction when cooled, a Hall effect sensor\nwas placed at the center of three orthogonal surfaces of\na sample magnet and the magnetic field was recorded at\nroom temperature and at 1.5K (see Tab. II). While this\nmeasurement cannot be used to determine the cryogenic\nresponse of the magnet, it serves to demonstrate that no\nsignificant change in magnetization axis occurs.\nSurface B at 300 K(T) B at 1.5 K(T)\n/angbracketleftx/angbracketright 0.00723 0.00663\n/angbracketlefty/angbracketright 0.576 0.449\n/angbracketleftz/angbracketright 0.0653 0.0737\nTable II. Surface magnetic flux density of a 20×30×60mm\nN55 NdFeB magnet magnetized along the y-direction, at\n300 Kand1.5 K. While the magnitude reduces by ≈20%,\nthe magnetization direction does not change. Note, that the\nNdFeB magnets of this size that are used in the assembly are\nactually magnetized in the x-direction.\nB. Device Information\nThe devices used in these experiments consist of ion\nimplanted31P donors in isotopically enriched28Si with\na29Si concentration of ∼800ppm. For measurements\nwith samples mounted to a superconducting magnet, P+\n2\nmolecular ions were implanted such that approximately\nfour donor pairs are created in a 90×90nm2window.Measurements performed on samples mounted to a per-\nmanent magnet assembly were implanted with P+ions\nto yield a donor concentration of 1.25×1012donors/cm2.\nWe estimate a mean donor separation of 8nm and 4nm\nfor the former and latter samples, respectively.\nC. Measuring the Magnetic Field with a Single Nuclear\nSpin\nThe shift in NMR frequency presented in FIG. 5(a)\nwas measured by side-of-fringe spin magnetometry.39,40\nIn such a scheme, following the initialization of the spin\ntoxi=|⇑/angbracketright/|⇓/angbracketrightby measurement of Sz, the spin is put\ninto the state (in the rotating frame and up to a global\nphase)\n|ψ/angbracketright=1√\n2/parenleftbig\n|⇓/angbracketright±i|⇑/angbracketright/parenrightbig\n(C1)\nby aπ\n2rotation about the x−axis. The spin is then\nleft to precesses in the xy-plane of the Bloch sphere for\na fixed time, τ. If there is a detuning, ∆ =ω−ω0where\nωis the NMR frequency and ω0is the frequency of the\nrotating frame, then following the time τthe spin is in\nthe state\n|ψ/angbracketright=1√\n2/parenleftbig\nei∆τ\n2|⇓/angbracketright±ie−i∆τ\n2|⇑/angbracketright/parenrightbig\n.(C2)\nA subsequentπ\n2rotation about the y−axisis used to\nput the spin - up to a global phase - to either,\n|ψ/angbracketright= cos/parenleftbigg∆τ\n2−π\n4/parenrightbigg\n|⇓/angbracketright−isin/parenleftbigg∆τ\n2−π\n4/parenrightbigg\n|⇑/angbracketright,(C3)\nif the initial state xi=|⇑/angbracketrightor\n|ψ/angbracketright=isin/parenleftbigg∆τ\n2−π\n4/parenrightbigg\n|⇓/angbracketright− cos/parenleftbigg∆τ\n2−π\n4/parenrightbigg\n|⇑/angbracketright,(C4)\nifxi=|⇓/angbracketright. We find that the spin flip probability, Pflip\n- that is the probability of measuring xm=|⇑/angbracketrightgiven\nxi=|⇓/angbracketright, or vice versa - is given by\nPflip=1\n2+1\n2sin ∆τ, (C5)\nor equivalently,\n∆≈2Pflip−1\nτ. (C6)\nFrom Eq. C6, it is apparent that the sensitivity is im-\nproved with increasing τ. However, in practice we are\nlimited by decoherence which adds a factor of e−χ(τ)-9\nwhereχis the coherence function - to the denomina-\ntor of Eq. C6. As we measure T∗\n2= 51.1±2.5ms and\nχ(τ) = (τ/T∗\n2)2, we setτ= 25ms as a compromise.\nIt can be shown41that the uncertainty in Pflipis given\nby\n1\n2C√\nN, (C7)\nwhereCis dimensionless efficiency parameter, depen-\ndent on the spin readout fidelity and Nis the number of\nrepetitions performed. For our31P+nuclear spin, C∼1\nandN= 300, resulting in an error in ∆of0.23Hz or\n14ppb.\nIn FIG. 5(c), the magnetic field of the permanent mag-\nnet assembly was probed by measuring the NMR spec-\ntrum with the error in this measurement arising from\nthe uncertainty in determining its centre frequency. The\nNMR linewidth is, in this measurement, is ∼307Hz,\nin contrast to the ∼4.25kHz linewidth presented in\nFIG. 4(a) which was measured using a higher RF power.\nIn both NMR spectra, the linewidth is power broadened.\nThe fitting error - specifically the length of the 95%con-\nfidence interval - ranges from 13.5Hz to 28.8Hz, corre-\nsponding to an accuracy between 0.80ppm and 1.7ppm\nat∼16.961MHz.\nTo measure the magnetic field drift of the supercon-\nducting solenoid, a Ramsey interferometry scheme was\nperformed on a31P0electron spin. Here, instead of using\na rotation about xfollowed by a fixed τand subsequent\nrotation about y, both rotations are about x. The spin\nflip probability is then\nPflip=1\n2+1\n2e−χ(τ)cos ∆τ. (C8)\nBy varying τand measuring Pflip, the resulting data\nmay be fit to Eq. C8 and |∆|can be determined. To\ndetermine the sign of ∆, two different values of ω0may\nbe used. 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Ferrari\nINO-CNR BEC Center and Dipartimento di Fisica, Universit\u0012 a di Trento,\nand Trento Institute for Fundamental Physics and Applications, INFN, 38123 Povo, Italy.\n(Dated: May 16, 2023)\nA rich variety of physical e\u000bects in spin dy-\nnamics arises at the interface between di\u000berent\nmagnetic materials [1]. Engineered systems with\ninterlaced magnetic structures have been used to\nimplement spin transistors, memories and other\nspintronic devices [2, 3]. However, experiments\nin solid state systems can be di\u000ecult to inter-\npret because of disorder and losses. Here, we\nrealize analogues of magnetic junctions using a\ncoherently-coupled mixture of ultracold bosonic\ngases. The spatial inhomogeneity of the atomic\ngas makes the system change its behavior from\nregions with oscillating magnetization | resem-\nbling a magnetic material in the presence of an ex-\nternal transverse \feld | to regions with a de\fned\nmagnetization, as in magnetic materials with a\nferromagnetic anisotropy stronger than external\n\felds. Starting from a far-from-equilibrium fully\npolarized state, magnetic interfaces rapidly form.\nAt the interfaces, we observe the formation of\nshort-wavelength magnetic waves. They are gen-\nerated by a quantum torque contribution to the\nspin current and produce strong spatial anticorre-\nlations in the magnetization. Our results estab-\nlish ultracold gases as a platform for the study\nof far-from-equilibrium spin dynamics in regimes\nthat are not easily accessible in solid-state sys-\ntems.\nThe local magnetization in a magnetic material evolves\ndepending on three ingredients: the external magnetic\n\feld, the nonlinear ferromagnetic anisotropy and the in-\nhomogeneity of the magnetization itself. The evolution\ncan be described by using the well known Landau-Lifshitz\nequation (LLE) [4, 5]. When a large external magnetic\n\feld is applied, all spins precess around it. For a vanish-\ning \feld amplitude, the spins precess around a preferen-\ntial spatial direction characteristic of the material itself,\ngiven by the magnetic anisotropy. In real materials, this\nanisotropy is usually very small compared to the low-\nest technically achievable uniform external \feld [5]. The\nterm of the LLE incorporating the inhomogeneity { deriv-\nable from the Heisenberg exchange term { becomes par-\nticularly relevant in the presence of magnetic interfaces.\nIn the absence of such an exchange term, the LLE re-\nduces to the Josephson equations for Bose-Einstein con-\ndensates [6, 7]. Similarly to a magnet in an external \feld,\nJosephson equations have di\u000berent dynamical regimes:either the system oscillates between two states (preces-\nsion around external \feld) or it is self-trapped in one of\nthem (precession around the dominant anisotropy direc-\ntion).\nIn our experiment, we coherently couple two hyper-\n\fne states,jF;mFi=j1;\u00061i, of a near-zero-temperature\nBose-Einstein condensate of23Na atoms, trapped in an\nelongated harmonic potential (see Methods); Fis the to-\ntal atomic angular momentum and mFis its projection\nalong the quantization axis z. This system is equiva-\nlent to a magnetic material with nonuniform magnetic\nanisotropy in the presence of an external transverse \feld,\nas sketched in Fig. 1. In the analogy, the e\u000bective exter-\nnal magnetic \feld is represented by the electromagnetic\nradiation that coherently couples the spin states, while\nthe ferromagnetic anisotropy is due to the nonsymmet-\nric interatomic interactions and varies spatially thanks\nto the density inhomogeneity of the sample. The inter-\natomic interaction energy of sodium mixtures, together\nwith the low magnetic \feld noise of our magnetically\nshielded system, allows us to tune the coupling strength\nbelow the interaction energy. This corresponds, in mag-\nnetic materials, to an external \feld below the magnetic\nanisotropy. If all spins are initially aligned along zand a\n\feld is suddenly applied, they start precessing, as illus-\ntrated by the vector on the Bloch sphere (Fig. 1b). The\nprecession happens around a direction that depends on\nthe local properties. While standard magnetic materi-\nals quickly align along one axis (grey vector in Fig. 1c)\nbecause of dissipation, our nondissipative atomic system\nallows to study a longer dynamical evolution.\nThe atomic gas can be described with a two-component\norder parameter \t( x) = ( +1; \u00001)T, where \u000bis the\nmacroscopic wave function of the Bose-Einstein con-\ndensate in the state \u000b=\u00061. The tight con\fnement\nalong two spatial directions makes the spatial dynam-\nics be essentially one-dimensional along the xdirection\n(see Methods). Therefore, the state of the system is\nfully described by the density matrix (\t\u0003\n\t)(x) =\nf \u0003\n\u000b(x) \f(x)g\u000b;\f=\u00061. The density matrix is composed\nof a scalar part, n= Tr(\t\u0003\n\t), corresponding to the\ntotal density of the condensate, and of the spin-density\ns= Tr( \u001b\t\u0003\n\t), withjsj=nand\u001brepresenting the\nvector of Pauli matrices. Hereafter vector quantities are\nde\fned on the Bloch sphere.\nIn general the dynamics is described (see Methods) by\ncoupled di\u000berential equations for n,sand the velocity\n\feldv=j=n, wherejis the atom density current [8].arXiv:2011.04271v2  [cond-mat.quant-gas]  15 May 20232\nFIG. 1. Analogy between a coherently-coupled atomic\nmixture and a magnetic heterostructure. a , Sketch of\nthe trapped ultracold atomic mixture in the two hyper\fne\nstatesj1;+1i(blue) andj1;\u00001i(red), coupled via coherent\nradiation with strength \n R.b, Local evolution of the system,\nrepresented on the Bloch sphere in the center (where inter-\nactions\u0014szexceed the coupling \n R) and in the tails (where\nthe coupling exceeds the interaction term). c, Pictorial view\nof the magnetic analogue. The material has a spatially vary-\ning ferromagnetic anisotropy \r, smaller or larger than the\nexternal magnetic \feld B, respectively on the external or in-\nternal regions. The equilibrium magnetization of the material\n(grey arrow) follows the dominating e\u000bect between intrinsic\nanisotropy and external \feld. Magnetic interfaces are present\nbetween these regions. Black arrows in panel brepresent the\ncontributions of the di\u000berent physical quantities both for the\natomic system (top) and for its magnetic analogue (bottom).\nSince the total atom number is a conserved quantity, n\nsatis\fes the continuity equation, with the purely advec-\ntive current j: _n+@xj=0. The equation of motion of\nsre\rects the possibility of twisting the spin and the ab-\nsence of spin conservation. Both features are due to the\ncombination of the coherent Rabi coupling and the lack\nof SU(2) symmetry of the non-driven system. The Rabi\ncoupling is described by the linear transverse \feld \n R^x.\nThe lack of SU(2) symmetry leads to a nonlinear \feld\n\u0014sz^z, with\u0014proportional to the di\u000berence between intra-\nand intercomponent interactions, \u000eg, and including the\ne\u000bect of the dimensional reduction (see Methods). Thespin equation of motion can be written as\n_s+@xjs=H(s)\u0002s; (1)\nwhere we introduce the e\u000bective magnetic \feld H=\n\nR^x+\u0014sz^z. The spin current reads\njs=vs+~\n2mn@xs\u0002s: (2)\nThe \frst term is the spin advection. The spatial deriva-\ntive of the second one is the quantum torque , which\ndepends explicitly on ~=m. Remarkably, the quantum\ntorque originates as a pure quantum e\u000bect, vanishing\nwhen ~is set to zero, or equivalently when the mass of\nthe atoms is in\fnite (see classical analogue in Ref.[9]).\nThe equation of motion for the spin density, Eq. (1), in\nthe absence of spin advection, is equivalent (see Methods)\nto a non-dissipative LLE [8]. Therefore, if the density\nand velocity dynamics can be neglected, the dynamics\nof a coherently-coupled Bose gas mimics the magnetiza-\ntion dynamics in a magnetic sample, where the quantum\ntorque plays the role of the exchange term. Since the\nquantum torque depends on the curvature of s, it plays a\ncrucial role in the presence of magnetic interfaces. Often\nin literature the e\u000bective \feld in the LLE includes the\ntorque as well [5], which in the magnetic context is due\nto the exchange interaction.\nTaking advantage of the absence of dissipative terms in\nEq. (1), we study the long time dynamics of systems with\nfar-from-equilibrium initial con\fgurations. Before dis-\ncussing the actual experimental con\fguration (Fig. 1a), it\nis useful to consider the dynamics in the simple spatially\nhomogeneous case. Equation (1) reduces to _s=H(s)\u0002s,\ni.e., equivalent to the Josephson equations for weakly-\ninteracting Bose gases [7]. These equations are usually\nwritten in terms of the relative magnetization Z=sz=n\nand the relative phase \u001e= arctan(sy=sx), which in\nFig. 1b correspond to the projection of the quantum state\non thezaxis of the Bloch sphere, and its equatorial angle,\nrespectively, s=n(p\n1\u0000Z2cos\u001e;p\n1\u0000Z2sin\u001e;Z).\nWith an initially fully polarized state sz=\u0000n, two\ndi\u000berent dynamical regimes are possible: (i) for \n R>\nj\u0014n=2j, the magnetization oscillates between sz=\u0006n\nwith a frequency \n (see the dynamics in C and D in\nFig. 2b and associated continuous line in insets), a.k.a.\nJosephson oscillations; (ii) for \n R<j\u0014n=2j, the system\nenters the so-called self-trapped regime [7, 10], where the\nspin precesses such that \u0000n\u0014sz(t)\u0014max(sz)<0. In\nthe self-trapped regime, Znever changes sign (see the\ndynamics in A and B in Fig. 2b). Interestingly, the pre-\ncession frequency { not only the amplitude { drastically\nchanges across the transition, as shown in Fig. 2a, with\na softening of the precession frequency at the transition\npoint.\nA harmonically trapped Bose-Einstein condensate in\nthe Thomas-Fermi approximation shows a density reduc-3\nFIG. 2. Quantum torque e\u000bect at the interface. a ,\nMain local oscillation frequency \n of the relative magnetiza-\ntionZas a function of the position in the gas. The softening\nof the precession frequency at xcmarks the transition be-\ntween the two dynamical regimes. The insets show the time\nevolution of Z(t) for four di\u000berent spatial points A-D (dashed\nlines). b, Spatial variation of the magnetic anisotropy. The\nred line marks the critical value. c-d, Time evolution of the\nmagnetization across the cloud according to LLE without ( c)\nand with ( d) quantum torque contribution. Four di\u000berent\nspatial points are considered: deep in the self-trapped regime\n(A), weakly self-trapped (B), weakly oscillating regime (C)\nand deep in the oscillating regime (D). Continuous and dashed\nlines in the inset of acorrespond to the time evolution with-\nout and with the quantum torque term, respectively. Note\nthat in B and C, the local dynamics is disrupted due to the\nexcitation of short wavelength magnetic waves. e-g, Pro\fles\nofZ,jjsj=ncs, with 2mc2\ns=~\u0014n, andj@xjsj=jH\u0002sjmaxcal-\nculated for \n Rt=2\u0019=0.7, 0.8, 0.9, 1, 1.5 around the interface\nbreaking (black to gray).tion from the center outwards, n(x) =n0(1\u0000x2=R2\nx)\n(black line in Fig. 2b). Thanks to this inhomogeneity,\nwe can realize systems that present simultaneously both\nbehaviors in spatially di\u000berent regions. If the central den-\nsityn0is large enough for the system to be locally in the\nself-trapped regime, there exists a position xcthat sepa-\nrates this region from the low-density one, where Rabi-\nlike oscillations of the magnetization occur. The sharp\ntransition from one regime to the other creates an inter-\nface.\nWhen the system is prepared in an initially fully po-\nlarised statej1;\u00001i, after about 1 =\nR, the formation of\nmagnetic interfaces takes place at xc. In the absence of\nspin current, the interfaces would stand between the two\ndynamically di\u000berent regions, which would follow stan-\ndard Josephson dynamics. However, quantum torque be-\ncomes relevant at the interfaces and breaks the Joseph-\nson dynamics. Figures 2c and 2d show the theoretical\nevolution of the magnetization throughout the sample,\nrespectively without and with the quantum torque term.\nFigure 2e-g compare the pro\fles of Z,jsand@xjs\nnear the interface for 5 consecutive times (from black\nto gray). They clearly illustrate that, while the magne-\ntization gradient rises at the interface, the spin current\nrapidly increases and its spatial derivative peaks. When\nthe quantum torque term becomes of the order of jH\u0002sj,\nit counteracts the magnetization gradient forming mag-\nnetic waves with short wavelength.\nOther systems of atomic mixtures have been studied\nin this context, but none observed both regions in a\nsingle sample and the sharp interface separating them.\nEither self-trapped or oscillating regimes were observed\nin \"zero\"-dimensional (single mode) systems both with\nRabi- [10] and tunnel-coupling [11]. Complex dynamics\nwas observed in elongated, inhomogeneous BECs in the\npresence of Rabi coupling [12, 13]. Double-well poten-\ntials were also used to investigate Josephson dynamics\nboth in bosonic [14{17] and fermionic [18, 19] gases.\nFigure 3a shows the experimental measurement of the\nmagnetization. We let the system evolve in the pres-\nence of coherent coupling with \n R= 0:33j\u0014n0jfor a\nvariable time tand we separately image the two spin\npopulations. This allows us to extract the local mag-\nnetization (see Methods). For each experimental run,\nwe integrate the magnetization of the elongated atomic\nsample in the radial directions and obtain Z(x). Com-\nbining the measurements of Z(x) at di\u000berent times t,\nwe reconstruct the full dynamics. As predicted by the-\nory, the system spatially explores two completely di\u000ber-\nent regimes, depending on the ratio between the driv-\ning frequency \n Randj\u0014n(x)j. At short time, we ob-\nserve the creation of two magnetic interfaces at positions\n\u0006xc, where the condition \n R=\u0014n(xc)=2 is matched.\nThe magnetization in the central region slightly oscil-\nlates, never changing sign, while in the outer part atoms\nundergo full oscillations at a frequency close to \n R. As4\n200\n 150\n 100\n 50\n 0 50 100 150 200\nx (m)\n0\n1\n2\n3\n4Rt/2\na\nc\ncs\n1\n01\nZ\n200\n 150\n 100\n 50\n 0 50 100 150 200\nx (m)\n0\n1\n2\n3\n4\n5\n6Rt/2\nb\nc\ncs\n1\n01\nZ200\n 150\n 100\n 50\n 0 50 100 150 20001\nni/n Rt=2\nc\n01n/n0\n200\n 150\n 100\n 50\n 0 50 100 150 20001\nni/n Rt=2\nf\n01n/n0200\n 150\n 100\n 50\n 0 50 100 150 20001\nni/n Rt=3\nd\n01n/n0\n200\n 150\n 100\n 50\n 0 50 100 150 20001\nni/n Rt=3\ng\n01n/n0200\n 150\n 100\n 50\n 0 50 100 150 200\nx (m)\n01\nni/n Rt=5\ne\n01n/n0\n200\n 150\n 100\n 50\n 0 50 100 150 200\nx (m)\n01\nni/n Rt=5\nh\n01n/n0experimental data\n1D hydrodynamics\nFIG. 3. Evolution of the magnetization. a , Measured magnetization along the non-uniform 1D gas as a function of time in\nthe presence of a small coupling (\n R= 2\u0019\u0002118 Hz = 0:33j\u0014n0j). Panel bshows the full 1D hydrodynamic simulation of the\nsame system. At the interface, around the critical position xc(black bars), we observe a net change in the behavior from the\noscillating to the self-trapped regime. The width of the bars is evaluated considering the uncertainties in nandRx. Between\nthe two regions, a new one, characterized by a strong \ructuation of the magnetization, emerges and grows. The \ructuating\nregion expands in the self-trapped region with a constant velocity (continuous line), estimated to be v= 1:8(2)cs(experimental\ndata) andv= 1:4(1)cs(numerical data). The dashed and dot-dashed lines show the slope corresponding to the local spin sound\ncsand sound velocity c.c-e, Measured density pro\fle of each component (red, blue), normalized to the local total density\n(gray), for \n Rt= 2\u0019, 3\u0019and 5\u0019.f-h, Same as c-e, but evaluated numerically. Notice that the experimental pro\fles are taken\nafter a short expansion (see Methods) that enhances the total density modulation.\ntime goes on, the interfaces break, the self-trapped region\nbecomes smaller and smaller and strongly \ructuating re-\ngions are created and grow in size. In Fig 3b we report\nthe corresponding results obtained with the full hydro-\ndynamic simulation, that we numerically implement by\nsolving the equation of motion for \t, represented by two\ncoupled Gross-Pitaevskii equations [20]. Figure 3f-h and\nExtendedDataFig.1 show that the density and the su-\nper\ruid velocity terms are irrelevant for the dynamics.\nTherefore for our protocol, the LLE provides a very good\ndescription of the system dynamics. We impute the dif-\nference between experimental data and simulation to a\npossible mismatch between the actual parameters in the\nexperiment and those used in the simulations. Deep in\nboth the oscillating and self-trapped regions one can see\na smooth spatial variation. In the strongly \ructuating\nregion, instead, the magnetization varies on a very small\nlength scale, as visible also in the measured (Fig. 3c-e)\nand numerically simulated (Fig. 3f-h) density pro\fles.\nIn order to characterize the evolution of the strongly\n\ructuating region, we estimate the wave front speed as itpropagates in the inner region with almost uniform den-\nsity. Note that the strongly inhomogeneous density and\nthe increasing role of thermal component in the outer re-\ngions make it harder to study the wave front propagating\noutwards and to compare it with the zero-temperature\nsimulations. We identify the edge of the \ructuating re-\ngion by calculating the standard deviation of the mag-\nnetization in 5 µm-wide windows, which exhibits a jump\nfrom a low to a nonzero value as the self-trapped regime\nis consumed by the \ructuating regions. We observe that\nthe front propagates with a rather constant speed as in-\ndicated by the solid line in Fig. 3a. The propagation\nspeedv= 3:1(1) mm/s (solid) is larger than the lo-\ncal spin sound velocity cs'1:7 mm=s (dashed), while\nit remains well below the local density sound velocity\nc'9:2 mm=s (dot-dashed). We estimate the two veloc-\nities from 2 mc2\ns=~\u0014n, andmc2=~\u0014ng= (g\u0000g12), re-\nspectively, where gandg12are the interaction constants\n(see Methods).\nFigure 4a shows the correlations of the magnetiza-\ntionCZZ(\u0001x) =R\nZ(x)Z(x+ \u0001x)=R\nZ(x)2far from5\nFIG. 4. Correlation of the magnetization. a, Spatial\nautocorrelation of the magnetization for di\u000berent regions of\nthe sample during the evolution, indicated by the rectangles\nin the inset. In the self-trapped (A) and oscillating (B) re-\ngions, the long-range order of the sample is maintained. In the\nexcited region (C-E), the coherence has a minimum at around\n5-6µm. At longer times the minimum becomes negative. Er-\nror bars corresponds to s.d. of the correlation at di\u000berent\ntimes averaged in the rectangles. b,Same as a) but for the\nmagnetization reported in the inset, that was obtained from a\ndi\u000berent dataset (\n = 2 \u0019\u000262 Hz = 0:28jkn0j) and extended\nto longer times.\nthe breaking point, calculated on the data reported in\nFig. 3a. The integral is performed in the areas enclosed\nin the rectangles shown in the insets. While in the self-\ntrapped (A) and oscillating (B) regions the system main-\ntains correlations over a long spatial scale, the \ructuating\nregions (C) show a strong decay. On longer timescales\n(D-E), the autocorrelation shows a negative minimum\naround 5-6 µm. Its position does not depend on the time\nin the evolution, nor on the position where correlations\nare calculated. On long time scales the function stabilizes\nto a well de\fned shape showing anticorrelations. Figure\n4b shows the same analysis on a di\u000berent dataset (inset)\nextending to longer times, where we do not observe any\nfurther evolution of the autocorrelation.\nThe continuous lines are \fts to the data in regions E\nwith the function f(x) = cos(\u0019x=L 1) exp(\u0000(x2=2L2\n2)).\nFor panel b) the best-\ft parameters are L1= 7:1(1)µm\nandL2= 4:0(1)µm. Such a behavior of the magnetic\ncorrelations indicates a well-de\fned size of the magnetic\ndomains ('L1) that is larger than the spin healing length\n\u0018s=~=p2mn\u000eg\u00190:8µm and the Rabi healing length\n\u0018R=p\n~=m\nR\u00192:6µm.\nThe observed size of the magnetic domains and the\ncomparable inward and outward wave front propagation,\nvisible in the simulation, trigger interesting questions\nabout the origin of the spin structures. The density\npro\fles (Fig. 3c-h), the fact that the speed of the mag-netic wave front is larger than typical Landau spin crit-\nical velocity (Fig. 4), and the spin correlations in the\ndownstream, suggest that these excitations are closely\nrelated to magnetic shock waves [21{23]. Shock waves\nhave been studied in single component ultracold atomic\ngases [24, 25] and very recently, in the presence of spin-\norbit coupling [26]. However, the LLE in the presence\nof both transverse magnetic \feld and anisotropy are not\nintegrable and have been shown to present a chaotic be-\nhavior [27]. Therefore our protocol could excite a new\nkind of magnetic shock waves with a chaotic character,\nleading to a turbulent behavior of the magnetization that\nmight have connections with a spin glass [28]. Even on\nthe pure theoretical side, such kind of waves have never\nbeen studied and thus deserve further analysis.\nMETHODS\nOne dimensional super\ruid spinor hydrodynamics\nand LLE\nIn this section we brie\ry explain how the spinor su-\nper\ruid hydrodynamics can, in certain situations, be re-\nduced to the Landau-Lifshitz equations.\nAt zero temperature, a two-component Bose gas com-\nposed of atoms of mass mis described by super\ruid\nspinor hydrodynamics, that is equivalent to the Gross-\nPitaevskii description, which provides the dynamics of\nthe total density n, the super\ruid velocity vand the\nspins. The hydrodynamics equations can be written as\n(see, e.g., [8]):\n_n+ div(nv) = 0; (3)\nm_v+ div\u0010\nmv2\n2+\u0016+sz\nnh+V\u0000~2r2pn\n2mpn+~2jrsj2\n8mn2\u0011\n= 0;(4)\n_s+P\n\u000b@\u000b(js;\u000b) =H(s)\u0002s; (5)\nwhere\u000b=x;y;z indicates here the three spatial direc-\ntions.\nThe \frst equation is the standard continuity equation\nand the second one is the Euler equation for the super-\n\ruid velocity, where the chemical potential \u0016and its spin\ncounterpart hare given by\n\u0016[n;sz] =1\n2(g+g12)n+~\nR\n2n\nn2\u0000s2zsx\nh[n;sz] =1\n2(g\u0000g12)sz\u0000~\nR\n2sz\nn2\u0000s2zsx;(6)\nwithgandg12the intra- and intercomponent interac-\ntion strengths, respectively, and \n Rthe Rabi coupling\nstrength. Let us also remind here that for a spinor su-\nper\ruid, the velocity is generally not irrotational. Equa-\ntion (5) is the most important one since it determines\nthe spin dynamics. It shows that the total spin is not a\nconserved quantity as long as the e\u000bective magnetic \feld6\nH(s) = (\n R;0;(g\u0000g12)sz) is non-zero. The spin current\nis a tensor which contains two contributions:\njs;\u000b=v\u000bs+~\n2m\u0010\n@\u000bs\u0002s\nn\u0011\n; (7)\nwhere the \frst term corresponds to the classical spin ad-\nvection and the second is the current due to the spin-twist\nand at the origin of the quantum torque term in Eq. (5)\ndiscussed in the main text.\nIf the dynamics is such that the density and the su-\nper\ruid velocity are constant, the system is described\nonly by Eq. (5). The latter is formally equivalent to the\nso-called Landau-Lifshitz equation (LLE) for the mag-\nnetisation dynamics in magnetic materials (see, e.g., [5]).\nFor the geometry of the experiment, we also have that\nthe spin dynamics is e\u000bectively one-dimensional, which,\nformally corresponds to have the possibility of writing\ns(r;t) =f(x;r?)s(x;t); (8)\nwithxthe axial coordinate, r?the transverse coordinates\n(irrelevant for the spin dynamics) and f(x;r?) = (1\u0000\nx2=R2\nx\u0000r2\n?=R2\nr)=(\u0019R2\nr) a function taking into account\nthe Thomas-Fermi density pro\fle, with radii RxandRr.\nSubstituting the Ansatz (8) into Eq. (5) and integrating\noverr?, one obtains the one-dimensional LLE Eq. (2) of\nthe main text with\n~\u0014= (g\u0000g12)R\ndr?f2(0;r?)R\ndr?f(0;r?)=2\n3g\u0000g12\n\u0019R2r:(9)\nExperimental procedure\nIn the experiment, we initially create a polar-\nized Bose-Einstein condensate of 5 \u0002105atoms in\nj1;\u00001i. The atoms are trapped in an elongated\noptical trap [29] with radial and axial frequencies\nf!x;!rg=2\u0019=f10(1);1006(1)gHz, with Thomas-Fermi\nradiiRrandRxequal to 2:2µm and 210 µm, respectively.\nA stable magnetic \feld of 1 :3018 G is applied along the\nzaxis to lift the degeneracy of the Zeeman substates.\nWe stabilize the mixture against spin-changing collisions\nby lifting the energy of the j1;0istate through a mi-\ncrowave \feld blue-detuned respect to the j1;0i!j 2;0i\ntransition. The high stability and uniformity of the \feld\nis ensured by the presence of a 4-layer magnetic shield\n[30] around the main vacuum cell, that suppresses the\n\feld \ructuations by more than 5 orders of magnitude\ndown to 2\u0016G on the timescale of the full experimental\nsequence. At the used magnetic \feld, this corresponds to\nan energy \ructuation of about \u0019h\u00023 Hz. We suddenly\nswitch on a microwave coupling between j1;\u00001iand\nj1;+1i, by means of a two-photon transition, detuned\nby \u0001 from the intermediate state j2;0i. The strength of\nthe e\u000bective Rabi coupling \n Ris tuned by changing \u0001.The coupled states possess the peculiar feature that the\nintracomponent coupling constants g\u00001=g+1=g, while\nthe interspecies one g+1;\u00001is 7% smaller than g[31].\nThis leads to miscibility [32, 33]. Our initial condition\ns= (0;0;\u0000n) corresponds, in the presence of Rabi cou-\npling, to a highly excited state, far from the equilibrium\npoint s= (n;0;0). After a variable evolution time, we\n\frst image the density distribution of state j1;\u00001i, then\nof statej1;+1i. Both absorption pictures are taken after\na short expansion time, respectively 2 ms and 3 ms after\nthe switch o\u000b of coupling and trapping potential. We\nexpand the sample in order to increase the signal to noise\nratio of the measured magnetization, while the di\u000berent\nexpansion times are due to technical limitations. Density\n\ructuations in the region of strongly \ructuating mag-\nnetization are enhanced by the di\u000berent expansion times.\nIn order to determine the one dimensional interaction\nparameter\u0014, we measured [34] the plasma oscillation fre-\nquency!p, i.e., the frequency of the small magnetic \ruc-\ntuations around the ground state s= (n;0;0). The lat-\nter can be indeed measured with high accuracy and can\nbe directly compared with the simple analytical expres-\nsion!p=p\n\nR(\nR+\u0014n). By measuring the plasma\nfrequency in the central slice of the trap we extract \u0014n0.\nThe obtained value is in good agreement with the one re-\nsulting from the atom number and trapping frequencies.\nWe checked that with such a value at hand we can re-\nproduce the experimental spatial dependent plasma fre-\nquency simply by using the Thomas-Fermi one dimen-\nsional density n(x) =n0(1\u0000x2=R2\nx).\nACKNOWLEDGEMENTS\nWe thank F. Dalfovo for his critical reading of the\nmanuscipt and M. Oberthaler, G. Consolo, D. Go, E.\nMendive-Tapia and N. Pavlo\u000b for fruitful discussions.\nWe acknowledge funding from INFN through the FISH\nproject, from the European Union's Horizon 2020 Pro-\ngramme through the NAQUAS project of QuantERA\nERA-NET Cofund in Quantum Technologies (Grant\nAgreement No. 731473) and from Provincia Autonoma\ndi Trento. We thank the BEC Center, the Q@TN initia-\ntive and QuTip.\nAUTHORS' CONTRIBUTION\nA.Recati and G.F. conceived the project. A.F.\nperformed the experiment. A.F. and A.Z. analyzed the\nexperimental data. D.T. set up the experiment control.\nA.G., A.Recati and A.Roy developed the theory and\nperformed the corresponding numerical simulations.\nA.Recati, A.Z., and G.L. wrote the manuscript. All\nauthors contributed to the discussion and interpretation7\nof the results.\n\u0003Correspondence to\ngiacomo.lamporesi@ino.cnr.it\nalessio.recati@ino.cnr.it\nalessandro.zenesini@ino.cnr.it\nyCurrent address: CNR Nanotec, Institute of Nanotech-\nnology, via Monteroni, 73100, Lecce, Italy.\nzCurrent address: Institute for Quantum Electronics,\nETH Z urich, Switzerland.\nCOMPETING INTERESTS\nThe authors declare no competing interests.\nDATA AND CODE AVAILABILITY\nThe data that support the \fndings of this study are\navailable from the corresponding author upon reasonable\nrequest.\n[1] H. Zabel and S. D. 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Lamporesi, and G. Ferrari, Phys. Rev. A\n104, 023326 (2021).8\n0\n5\n10\n15\n20t (ms)an/n (SH)\nb|j| (SH)\n0\n5\n10\n15\n20t (ms)cZ (SH)\nd|js| (SH)\n130\n 125\n 120\n 115\n 110\nx (m)\n0\n5\n10\n15\n20t (ms)eZ (LLE)\n130\n 125\n 120\n 115\n 110\nx (m)\nf|2mnxs×s| (LLE)\n-30%0%30%\n1\n01\n0100020003000\n0100020003000\n1\n01\n0100020003000\nFIG. 5. ExtendedDataFig.1 Comparison between spin hydrodynamic (SH) and LLE simulation. Total density\nrelative modulation ( a), modulus of the total current ( b), magnetization ( c) and modulus of the spin current ( d) simulated\naccording to the SH around the interface before and after its breaking. The magnetization resulting from LLE ( e) shows a\nvery good agreement, accordingly to the negligible role of the density modulation in the dynamics. f, Contribution of the spin\ncurrent term related to the quantum torque. Also in this case, panel fshows that the second term in Eq. 2 dominates over the\n\frst one, hence the quantum torque drives the dynamics. The colorscale units of plots b,d,f is atoms/ms."    },    {        "title": "2011.07088v1.Metal_organic_Frameworks__Possible_New_Two_Dimensional_Magnetic_and_Topological_materials.pdf",        "content": "1 \n Metal -organic Frameworks: Possible New T wo-Dimensional  Magnetic and \nTopological materials  \nJie Li  and Ruqian Wu * \nDepartment of Physics and Astronomy, University of California, Irvine, California  92697 -4575, \nUSA. \n \n \n \nFinding  new two -dimensional (2D) materials with novel quantum properties is highly desirable \nfor technological innovations . In this work,  we studied  a series of metal -organic framework s \n(MOFs) with different metal core s and discovered various  attractive propertie s, such as \nroom -temperature magnetic ordering, strong perpendicular  magnetic anisotrop y, huge topological \nband gap ( >200meV) , and excellent spin -filtering performance. As many MOFs have been \nsuccessfully synthesized in experiments, our results  suggest real istic new 2D functional materials \nfor the design of spintronic nanodevices.  \n \n   \n \n \n \n \n \n* E-mail: wur@uci.edu.  2 \n 1. Introduction  \nEver since the successful synthesis  of graphene ,1 two-dimensional (2D) materials have attracted \ntremendous research interest . A large verity of 2D materials have been predicted, fabricated and \ncharacterized , such as h exagonal boron nitride  (h-BN),2 silicone ,3 germanene,4 transition metal \ndichalcogenides,5 black phosphorous ,6 Cr2Ge2Te6,7 and CrI 3.8 They all  have exotic  quantum \nproperties such as the quantum spin Hall effect  (QSHE),  quantum anomalous Hall effect (QAHE) , \nvalley -polarized anomalous Hall effect  (VAHE) , high carrier  mobility , or stable  low-dimensional  \nferromagnetic order ing. Search ing for functional 2D materials with  emergent physical properties \nis currently at the forefront of research activities in several subareas, including condensed mater \nphysics, chemistry, nanoscience, and materials science.  \nAmong potential 2D materials in this realm, m etal-organic framework s (MOF s) are \nparticularly attractive as they have various advantages  such as  easy fabrication  and manipulation , \nhigh mechanical flexibility , and low cost . Many  stable MOF s haves been predicted  through  \ntheoretical studies , such as t riphenyl -lead based topological 2D materials ,9-10 nickel bis complex \nπ-nanosheet,11 π-conjugated covalent -organic frameworks ,12 and many of them  have already been \nsynthesized in experiment s. For example , Shi et al. synthesized a series of MOF s on the Au(111) \nsurface by positing tripyridyl ligands with transition metal atoms .13 Pawin et al. synthesized a n \nanthracenedicarbonitrile  based coordination network on the Cu(111) surface .14 Koudia  et al. \nsynthesized 2D transition -metal phthalocyanine -based MOF s by annealing Pc molecules on the \nAg(111) surface.15 Abdurakhmanova et al. synthesized and explored TM -TCNQ networks.16-18 \nObviously, these  successes  have paved the way for the design of new 2D functional  MOF s with \ndistinct properties  that are desired for a pplications . 3 \n In this letter, we theoretical ly predict  a series of new 2D MOF s with a simple hexagonal \nlattice based on transition metal tris (dithiolene) complexes  which have been recently synthesized \nin experiment s.19-20 We explore their functionalit ies such as magneti c ordering and topological \nfeature s through systematic  ab-initio calculation s. As we  adjust the species of the metal core s, \nthese  2D MOF s manifest various  attractive properties such as high Curie temperature , strong  \nout-of-plane magnetic ani sotrop y, sizeable  topological band gap ( >200meV) and excellent \nspin-filtering performance . These findings suggest the suitability  of these  2D MOF materials for \nthe design of innovative  nano -device s.  \n \n2. Calculation methods  \nAll density functional theory ( DFT ) calculations were  carried out using the Vienna ab -initio \nsimulation package (VASP)  with the projector augmented wave (PAW) method  was adopted for \nthe interaction between valence electrons and ionic cores ,21-22 and t he energy cutoff for the plane \nwave basis expansio n was set to 700 eV .  The spin -polarized generalized -gradient approximation \n(GGA) with the functional developed by  Perdew -Burke -Ernzerhof (PBE)  was choosed for the \nexchange -correlation functional .23 The vdW  correction  (DFT -D3) was included for  the description \nof dispersion forces.24 To sample the two -dimensional Brillouin zone, we used a 9×9 k-grid mesh. \nAll atoms  were fully relaxed using the conjugated gradient method for the energy minimization \nuntil the force on each atom became smaller than  0.01 eV/Å.  The edge states of 2D TM -Hex were \ncalculated by TB model based on the maximally localized Wannier functions (MLWFs) as \nimplemented in Wannier90 code 25 and Wannier tools code.26 \n 4 \n 3. Results and discussion  \nAs shown in Fig. 1(a),  these 2D MOF materials  take a honeycomb lattice and each sublattice site \nhas a molecule with three ligands around its  vanadyl core. The ligand  size can be change d, for \nexample, among structures  Ⅰ, Ⅱ and Ⅲ as depicted in Fig. S1.19-20 In this wor k, we designed  two \n2D me tal-organic hexagonal lattice (M -Hex) with different porosity as shown in Fig. 1(a) and Fig. \nS2 (label as M -Hex-Ⅰ and M -Hex-Ⅱ, respectively). We tested  different  transition metal core s, \nincluding all  3d, 4d and 5d transition metal atoms . The 2D planar  structure  was found to maintain \nwell after the structural relaxation procedures. The optimized  lattice constants of a few selected \nsystems are shown in Fig. S3.  \n \n \n \n \n \n \n \n \n \n \n \n  \nFig. 1. (a)  The top and side (inset) views of schematic structure of the predicted 2D  M-Hex-Ⅰ (the \ndashed square shows the supercell).  (b) The top and side (inset) views of spin density  of Mn -Hex-Ⅰ \n(a) \n(b) \nC \n S \n TM \n5 \n (red and blue color represent  positive and negative spin densities , respectively.) .  \n \nTo further confirm their thermal and dynamic  stability, we t ook Mn (Mn -Hex-I/II), Re \n(Re-Hex-I) and Os (Os -Hex-I) as  examples  and performed phonon and molecular dynamics \nsimulations . The corresponding phonon bands are shown in Fig. S4 in the Supplementary \nInformation. The absence of imaginary frequency branch indicates that these systems are \ndynamically stable. Furthermore, we kept them at 300K for 10ps (5000 steps) through  ab initio \nmolecular dynamics (AIMD) simulatio ns with a 5 5 supercell. The AIMD results show that  the \ntotal energ ies fluctuate around the ir equilibrium values without noticeable  sudden change (see Fig. \nS5 in the Supplementary Information) . As no structure destruction was observed , we believe  that \nthese lattices are  thermally stable at least up to room temperature. In the following, we separate \nour discussions  in two parts : systems with  either sustainable magnetization  (3d core atoms ) or \nwith strong spin orbit coupling (4d and 5d core atoms ). The former  focuses on the magnetic \nproperties and the latter focuses on the topological properties.  \nFrom our DFT calculations,  only (V, Cr, Mn, Fe) -Hex-Ⅰ(Ⅱ) MOFs  are magnetic , as shown in \nFig. S3 in the Supplementary Information . For instance, each Mn atom  in the Mn-Hex-I/II lattice  \nhas a  charge state of +3 and a spin magnetic moment of 2.0 u B. As seen from the spin density in \nFig. 1(b), the adjacent ligands also have an antiparallel spin polarization but the magnet moments \nare negligible, < 0.01 u B/atom. Importantly , Mn(V, Fe) -Hex-Ⅰ(Ⅱ) lattices prefer  ferromagnetic  \n(FM) coupling . The corresponding exchange parameters, Ji, are obtained by mapping the DFT \ntotal energies of different magnetic configurations  (see Fig. S6 in the Supplementary Information) \nto the classical Heisenberg  Hamiltonian:  \nܪ=ܪ−ܬଵ∑ܵழ,வ∙ܵ−ܬଶ∑ܵழ,வ∙ܵ                   (1) 6 \n where  J1, and J2 represent the nearest and the next nearest neighbor  exchange interaction s, \nrespectively, (see Table S  in the Supplementary Information).  Here, we only consider J1 and J2, \nsince the third nearest neighbor  exchange interaction s are negligible  due to the large  distance.  As \nshown in Table S Ⅰ, the  exchange  parameters are unexpectedly large for the porous netwo rks, \nindicating possible high Curie temperatures for them.  In addition , Mn-Hex-I (II) also have \nout-of-plane spin orientation and large magnetic anisotropic energy (MAE) of 0.7 5 (0.86) meV.  \nUsing the torque method, the corresponding Fermi  level dependent total and spin channel \ndecomposed MAEs are shown in Fig. S7 in the Supplementary Information. One may see that \ntheir large MAEs in a broad energy range around the actual Fermi level are mostly from the \ncross -spin contribution s. With the DFT results of exchange and magnetic anisotropy energies, we \ncalculate  their T c by using the renormalized spin -wave theory (RSWT ).27-28 The renormalized \nmagnetization (M(T)/M(0))  as a function of temperature T  is shown in  Fig. 2(a), and the Curie \ntemperature is determined by the location where  the renormalized magnetization drops  zero. One \nmay see that these systems have exceedingly high  Curie temperatures  compared to  other  2D vdW \nmagnetic layers  (45K , 66K and 160 K for CrI 3, Cr2Ge2Te6 and CrOCl, respectively) .7,8,29 \nEspecially , Tc of Mn-Hex-I is as high as  ~323K. Even  though  the exchange interaction s and Curie \ntemperature decrease as the ligands become longer , the Curie temperature of Mn-Hex-II is still as \nhigh as 160 K . The possibility of having magnetic ordering  up to  such high temperature s in 2D \nstructures  make s them  very attractive for  spintronic application s.  \nTo explor e the mechanism of strong  exchange  coupling and electronic properties  in MOFs , \nthe projected density of states (PDOS) of Mn-Hex-I are given in Fig. 2(b) , with projections to \ndifferent atoms (upper panel) and different Slater orbitals (lower panel) . First, the MOF is 7 \n half-metallic, with a large band gap in the majority spin channel.  Strong hybrization occurs \namong Mn, C and S orbitals, as manifested by the large broadenings of the DOS peaks. T he \ninvolvement of delocalized π electrons of C and S atoms certainly mediate the exchange  coupling \nbetween transition metal atoms along the lig ands, as was discussed by several authors for similar \nsystems .30-33 Even for  M-Hex-Ⅱ, there are still hybridized states around the Fermi level so the π  \nelectrons  still effectively assist the exchange coupling in the MOFs , as shown  by the charge \ndistribution of states in the range of ±0.2 eV around  the Fermi level in Fig. S 8 in the \nSupplementary Information. Note that these states around the Fermi level is 100% spin polarized \nin the FM state, and hence may mediate the long -range exchange interacti ons in MOFs, much like \nan “extended” direct exchange mechanism. In contrast, the hybridization is much weakened in the \nAFM phase, as the ݀௫௬/௫మି௬మ and ݀௫௭/௬௭ peaks skink their widths (see the dashed lines in the \nlower panel  of Fig. 2(b)) . This is  an indication that the Mn -ligand hybridization is much reduced \nand the hopping between adjacent Mn atoms becomes forbidden, which is unfavorable in energy.  \n \n \n \n \n \n \n \n \nFig. 2. (a) The renormalized magnetization as a function of temperature T  for V(Fe, Mn) -Hex-I(Ⅱ), \nrespectively. (b) The corresponding  PDOS  of Mn -Hex-I(II). (Top) the total DOS and the PDOS of \n(a) \n (b) 8 \n C, S and Mn atoms  of Mn -Hex-I in the FM state , the dash lines are the total DOS of Mn -Hex-II. \n(Down) the PDOS of d orbitals of Mn atoms of Mn -Hex-I in the FM state  (solid lines) and AFM \nstate (dash lines).  \n \n  As we also look for possible nontrivial topological properties from these honeycomb \nMOFs, t he band structures of Mn-Hex-I without and with spin orbit coupling (SOC) are given in \nFig. 3.  Indeed, Mn-Hex-I has the Dirac cones at the K and K` points, but there is another band \nthat cross the Fermi level in the vicinity around the Γ point. Therefore, Mn-Hex-I is a half-metal \nwith a large band gap in the majority spin channel , 1.07 eV (0.42 eV for Mn-Hex-II). As the FM \nordering may sustain at high temperature, one possible use of these exotic porous 2D magnetic \nmonolayers is making spin filters. To this end, we investigated  two proto typical Mn -Hex-I \nnanoribbons, i.e., with zigzag edges (4 zigzag chains in width) or armchair edges (3 armchair \nchains in width).  As shown in Fig. S9 in the Supplementary Information , only the spin down \nchannel of these ribbons open s for electron conduction in a broad energy range and hence they \ncan be excellent  for spin-filtering applications . Furthermore, the  armchair  Mn-Hex-I nanoribbon  \nhas a small gap (<0.1 eV) in the majority spin channel and is expected to be highly responsive to \ngating voltages and can be used for switching as well.  Note tha t Mn-Hex-I has two other \ntopological nontrivial gaps (13.9  meV and 24.4  meV) as shown in the dash box es in Fig. 3(b) , \nwhich are relatively large compared to several  previous predictions of MOFs ,34-36 but they are too \nfar away from the Fermi level . Interestingly, Mn -Hex-II has the Dirac cones right at the Fermi \nlevel, with a SOC induce gap of 3 meV (cf. Fig. S10). Although this gap is small, it might be \nworthwhile to try for the realization of the quantum anomalous Hall effect.  \nAll MOFs with 5d core s are nonmagnetic and thus it is interesting to see if they can manifest 9 \n QSHE , as they have strong SOC at the cores. From the band structures without and with SOC in \nFig. 4(a), we found that Os -Hex-I has a topological ly nontrivial band gap right at the Fermi level . \nWithout the  SOC  term in calculations , the highest valence band and the lowest conduction band \ntouch each other around  the Γ point. With SOC, a huge gap of about 225.6  meV open s and the \nFermi  level naturally lies in  the gap, a desirable  feature  for topological material s. We also \ncalculated the corresponding band structures  by switching off  SOC s of carbon , sulfur and osmium \natom s. As shown in Fig. S1 1, the gap change s to 7 meV without SOC of Os and to  231 meV  \nwithout SOC of C and S , respectively. Obviously, the large gap mostly results  from the SOC of \nOs. There are also Dirac cones  for Os-Hex-I, like graphene, but they are at -0.5 eV. These Dirac \ncones are accessible when Os is replaced by Re, which  reduces  one electron from the unit cell, as \nshown in Fig. S1 2 in the Supporting Information . \n   \n \n \n \n \n \n \n \nFig. 3. (a) (b) The band structures of Mn -Hex-I without and with SOC, respectively.  \n \nTo verify if the SOC induced band gap s of Os-Hex-I and Re -Hex-I are topologically \nnontrivial , their Z2 numbers and edge states were calculated. As shown in Fig. 4(b), we may see \n(a) \n (b) 10 \n that Z2=1 for Os-Hex-I by counting the positive and negative n -field numbers over half of the \ntorus. This undoubtedly  indicate s that Os-Hex-I is a strong 2D top ological insulator . Similarly, \nRe-Hex-I is a strong  TI materia l as well, as shown in Fig. S1 2 in the Supplementary Information . \nAs the band structure of Re -Hex-I is similar with that of graphene but with a much larger \ntopological band gap ( 80 meV), it offers a new platform for studies of QSHE in honeycomb \nlattice.  Obviously, the high structural stability and large topological gaps of these MOFs are very \npromising for practical applications and deserve experimental verifications.  \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 4. (a) The band structure  of Os -Hex-I without and with SOC, respectively.  Color bar \nindicates the atom -projected weights.  (b) The n -field configuration with red solid, blue hollow \ncircles and blank boxes denoting n= -1, n=1 and n=0. ( c) The corresponding 1D band structure  \nand edge states.  (d) SOC -induced band gap as a function of in -plane strain . \n(b) \n(c) \n 0 \n 4 \nOs \nC&S  \n(a) \n(d) 11 \n  \nAs further evidence, we constructed an Os-Hex-I nanoribbon (~100nm in width) and see if it \nhas the topologically protected edge states. To handle large number of atoms in the system, we \nused a tight -binding ( TB) model with parameters obtained by fitting the DFT band structure of 2D \nOs-Hex-I. In the TB model, we used s and p (s and d)  orbitals as the bas is for C and S (Os) atoms. \nThe TB band structure (see Fig. S1 3 in the supporting Information) agree with DFT band \nstructure very well, showing the high  quality of our fitting . In Fig.  4(c), o ne may see two bands \nfrom edge states across the Fermi level. They connect the bulk conduction and valence band s and \npresent in the  bulk-like band gap. Again, this supports the conclusion that Os-Hex-I MOF is a \nstrong 2D topological insulator  and can be used for the realization of the quantum spin Hall \neffect . \nFlex is a common way for the manipulation of  electronic propert ies of 2D materials , \nespecially for engineering  the topological phases .37-39 Here,  we considered the effect  of a biaxial \nstrain  on the band structure  of Os-Hex-I (see in Fig. 4(d)). One may see that Os -Hex-I can hold a \nlarge gap ( > 210 meV) with a similar band structures  in a reasonable range of strain (-3% ~3%).  \n \n4. Conclusion s \nIn summary, we propose d several  new 2D  functional  materials  based on honeycomb \nmetal -organic framework s, and investigated their structural , magnetic, electronic,  and topological \nproperties. DFT calculations show ed that MOFs with Mn has a strong magnetization up to room \ntemperature and can be used for  switching and  spin-filtering applications. 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Lett. , 2012, 109, \n016801.  \n "    },    {        "title": "2012.04258v3.Intrinsic_magnetic_topological_insulators.pdf",        "content": "1 \n \n Intrinsic magn etic topological insulators   \n \nPinyuan Wang1, Jun Ge1, Jiaheng Li2,3, Yanzhao Liu1, Yong Xu2,3,4, and Jian Wang*1,2,5,6 \n1International Center for Quantum Materials, School of Physics, Peking University, Beijing \n100871, China  \n2State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua \nUniversity, Beijing 100084, China  \n3Frontier Science Center for Quantum Information, Beijing 100084, China  \n4RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351 -0198, Japan  \n5CAS Center for Excellence in Topological Quantum Computation, University of Chinese \nAcademy of Sciences, Beijing 100190, China  \n6Beijing Academy of Quantum Information Sciences, Beiji ng 100193, China  \n*Correspondence: jianwangphysics@pku.edu.cn  \nABSTRACT  \nIntroducing magnetism into topological insulators  breaks  time-reversal symmetry , and the \nmagnetic exchange interaction can  open  a gap in the otherwise gapless  topological surface states . \nThis allows  various novel topological quantum states to be generated, including the quantum \nanomalous Hall effect (QAHE) and axion insulator states . Magnetic doping and magnetic \nproximity are viewed as being useful  means of explor ing the interaction between topology and \nmagnetism . However, the inhomogeneity of magnetic doping  leads to complicated  magnetic \norder ing and small exchange gap s, and c onsequently  the observed QAHE appears only at ultralow \ntemperatures.  Therefore , intrinsic magnetic topological insulators  are highly desired  for increasing \nthe QAHE working temperature  and for investigati ng topological quantum phenomena  further . \nThe realization and characterization of such systems are essential for both fundamental physics \nand potential  technical revolutions. This review  summarize s recent research progress in intrinsic \nmagnetic topological insulators , focusing mainly on the antiferromagnetic  topological insulator  \nMnBi 2Te4 and its family  of materials . \nI. INTRODUCTION  \nBefore the concept of topology was introduced into condensed matter physics , symmetry and \nsymmetry breaking  were  believed to govern the phase transitions of condensed matter  [1]. The \ndiscovery of the integer quantum Hall effect (IQHE)  [2] paved a new way for classifying matter \nand understanding phase transition s, and since then  various topological systems have been  \nrecognized in condensed matter physics  [1–17]. In 1980, von Klitzing  et al. [2] discovered that \nwhen a two -dimensional (2D) electron gas is subjected to a strong magnetic field, the longitudinal \nresistance becomes zero while the Hall resistance shows a quantized plateau w ith a height of h/νe2, \nwhere h is Planck’s constant,  ν is the filling factor , and e is the electron charge.  Later, it was \nrevealed that the filling factor  ν is actually a  topological invariant, which is an integral invariant \ndefined at the Brillouin zone and is insensitive to the geometry of the system  [2]. This topological 2 \n \n invariant is the well-known Thouless –Kohmoto –Nightingale –Nijs invariant  [3], also known as the \nChern number, which is expressed by the integral of the Berry curvature in the 2D Brillouin zone  \n[1], i.e.,  𝐶=1\n2𝜋∑∫𝑑2𝑘𝐹𝑚 𝑚 , where the Berry curvature 𝐹𝑚=∇𝑘×𝐴𝑚 is the curl of the Berry \nconnection of the  mth occupied band  𝐴𝑚. In 2D systems, the Chern number can be used to \nclassify whether the system is topologically trivial  (C = 0) or nontrivial  (C ≠ 0). For ordinary \ninsulators, their energy bands are topologically equivalent to those of a vacuum , and so these \nsystems share the same topological invariants as a vacuum and are topologically trivial. However, \nthere exist topologically nontrivial systems that are topologically inequivalent to ordinary \ninsulators . In 2D systems with broken time-reversal  symmetry  and C ≠ 0, gapless edge states \nemerge at the boundary between topologically nontrivial materials and ordinary insulators \n(vacuum) because of  the distinct band topology  [1]. The electrons in these edge states can \ntransport only unidirectional ly along the boundary and are called chiral edge states  [1], wherein \nelastic backscattering by impurities is forbidden . Therefore , the topological  edge  states can \ntransport with little dissipation  [1,15] , thereby offering potential for generating new device s or \nconcepts for low -power -consumption electronics and spintronics  [4,5] . \nSymmetry plays a key role in  materials  that have topologically nontrivial  properties , i.e. , \ntopological materials , and the t opology of certain systems is protected by certain symmetry, such \nas time-reversal symmetry  [6,17] , lattice symmetry  [18], and electron –hole symmetry  [15,16] . For \nmaterials with time-reversal symmetry, the Chern number must be zero  [6,7] . In 2005, Kane  and \nMele  [6,7] proposed classifying time-reversal invariant system s by another type of topological \ninvariant  known as the Z 2 invarian t. Here, the terminology “Z 2” comes from the fact that  this \ninvariant can take only two values, i.e., 0 or 1, depending on the parity of the intersection numbers \nbetween the edge states and the Fermi level. Therefore, 2D time -reversal invariant insulators can  \nbe divided into two categories : an insulator with Z2 invar iant equal to  0 is considered to be \ntopologically trivial, in which the edge states intersect with the Fermi level an even number of \ntimes ; by contrast, an insulator with Z2  invaria nt equal to 1 is identified as a topological insulator \n(TI)[1,6,7] , in which the edge states in tersect with the Fermi level an odd number of times and \nalways exist in the bulk gap . Currently , it is claimed that HgTe/CdTe quantum wells  [9,10] , \nInAs/GaSb quantum wells  [19], and some single -layer transition -metal dichalcogenides  [20–22] \nare 2D TIs. Known as helical states, t he edge states of a 2D TI  comprise a pair of \ncounter -propagating chiral states with spin-momentum locking, wherein  electrons that transport in \nopposite directions carry opposite spins ; this feature comes from time -reversal symmetry , which \nforbids electron backscattering . When the bulk state is an insulating  one, the transport signal is \ncontributed mainly by the helical states and give s rise to the so -called quantum spin Hall effect \n(QSHE ) [6–10]. The transport evidence of QSHE in 2D TI is the quantized longitudinal resistance, \nwhich has been observed in HgTe/CdT e quantum wells  [10], InAs/GaSb quantum wells  [19], and \nmonolayer WTe 2 [22]. However, the longitudinal -resistance plateau in 2D TIs is imperfect , and 3 \n \n some observations are inconsistent with  theoretical prediction s [23,24] . The Z2 classification can \nalso be extended to three -dimensional (3D)  systems , with four Z 2 invariants  being used to \ndistinguish  ordinary insulator s from “weak” and “strong” 3D TIs [11]. A weak 3D TI can be \nregarded as being stacked 2D TIs  whose gapless surface states are easily destroyed . A strong 3D \nTI is more robust , with gapless surface states  occurring  at all the surface s. These surface states \nexhibit a helical spin texture where in the spin is locked to  the momentum  [1]. Backscattering for \n2D surface states is only forbidden for 180°  but not for slighly different angles . At present, the \nmost representative strong TIs are materials in the Bi2Te3 family [13], whose large bulk energy \ngap and gapless surface states have been confirmed by angle -resolved photoelectron spectroscopy \n(ARPES)  [12,14] . \n \nFigure 1 . Schematic of  quantum anomalous Hall effect  (QAHE ) in a magnetic  topological \ninsulator  (TI). Illustrated are the exchange gap in the topological surface states of the TI induced \nby magnetism (denoted by the red arrows) and the resultant chiral edge state. When the Fermi \nlevel is tuned  to the exchan ge gap, electronic transport is dominated by the edge state , which \nresults in the QAHE  (TRS , time-reversal symmetry) . Adapted from [25]. \nIntroducing magnetism  into a TI thin film breaks the time-reversal symmetry , and the \nmagnetic exchange interaction can  open  a gap in the otherwise gapless surface states  (Figure 1(a) ). \nAccordingly, the film exhibits a chiral edge state carrying a quantized Hall conductance with a \nvalue  of e2/h, and this is known as the quantum anomalous Hall effect ( QAHE ) [25]. Unlike the \nIQHE, the QAHE  require s no  external magnetic fiel d and so offers great potential for \nlow-power -consumption electronics.  \nMagnetic doping  [25,26]  and magnetic proximity  [17] are expected to be useful  ways to \n4 \n \n introduce magnetism into TIs and realize the QAHE.  Chang et al. [25] were the first to realize the \nQAHE experimentally in  a Cr-doped (Bi,Sb) 2Te3 TI film at ultralow temperatures down to 30  mK. \nHowever,  in a magnetic ally doped TI film, the random distribution of magnetic impurities leads to \nmany  regions without ferromagnetic  (FM)  order ing even below the Curie temperature  [27]; the \nelectrons in the edge states may scatter with the surface states and the bulk band of the non -FM \nregions [27], resulting in the overall transport properties of the film deviating from quantization , \nand so the QAHE can be observed only at ultralow temperatures  [25,28] . A method that offers to \nsolve this problem is using a magnetic proximity structure  [17], in which the TI film is \nsandwiched between two FM insulators. However, despite  much exploration and effort in that \ndirection, the QAHE is yet to be detected experimentally  using th at approach. The coupling \nbetween the magnetic and topological states is usually weak and depends sensitively on the \ninterface properties, making  experimental realization of the QAHE challenging.  \nAs a new and efficient means of incorporat ing magnetism into T Is, intrinsic magnetic TIs \noffer a large magnetic exchange gap and homogeneous surface magnetic order ing [29–30], which \nare highly desir ous for investigati ng the QAHE  further  and potential application s in \nlow-power -consumption electronics.  Also , intrinsic magnetic TIs provide an ideal platform for \nstudying novel topological states  such as axion insulator states  [17]. Even though magnetic ally \ndoped heterostructures have been used to construct axion insulator states  [31], convincing \nevidence for axion states is still needed , and the extremely low transition temperature and the \ncomplicat ed preparation of heterostructure s might  not meet the requirement s for further \nexploration.  In intrinsic magnetic T Is, although magnetism can break the time-reversal symmetry,  \nsome systems with a certain magnetic order ing can retain an “ equival ent” time-reversal  symmetry  \nthat combin es time-reversal operation  and other symmetry operations such as 𝒯Cn and 𝒯τ, where  \n𝒯 is the time-reversal operation, Cn is the rotation operation, and τ is the fractional lattice \ntranslation operation. This equivalent time-reversal symmetry provides additional constraints on \nthe bulk band and protects the Z2 topological properties of magnetic TIs. An example is that the \n𝒯τ symmetry operation  in the antiferro magnetic  (AFM)  TI MnBi 2Te4 (MBT ) necessarily \nquantizes the bulk “axion angle” to π, which is very important for form ing an axion insulator  \n[29,32] . \nHowever, despite AFM  TIs having been pr oposed  in 2010  [33], there  were no experimentally \nrealizable candidate  materials until the proposal  of the AFM TI phase in MBT  [29,32,34 –36]. \nRienks et al. [30] reported that Mn  doping in Bi 2Te3 films results in the spontaneous formation of \nstoichiometric MBT  septuple layers  (SLs) rather than the chemical substitut ion of Bi by Mn  \n[37,38] . These MBT SLs interpolate between  Bi2Te3 quintuple layers to form natural Bi2Te3/MBT \nheterostructures.  Excitingly, a large topological gap of around 90 meV was observed in this \nheterostructure, indicating the  potential  for intrinsic magnetic TIs to realize the QAHE at higher \ntemperatures  [28,30] . The AFM TI phase of single -crystal  MBT  was soon identified  \nexperimentally  [36]. MBT is a layered material with the 𝑅3̅𝑚 space group. Monolayer MBT 5 \n \n comprises a Te–Bi–Te–Mn–Te–Bi–Te SL , which can be viewed as intercalating an MnTe bilayer \ninto a Bi2Te3 quintuple layer  [29,35,3 9] (Figure 2(a)). Theoretical calculations show that the \nground state of MBT  holds FM order ing within the SL and AFM order ing between neighboring \nSLs [32,34,36] . In the ground state , the magnetic moments of the Mn atoms are aligned with an \nout-of-plane easy axis  (z-axis) and are opposite between adjacent layers, which is labeled as the \nAFM -z state. The magnetic properties are provided mainly by the 3 d states of Mn , while the \ntopological ly nontrivial properties are dominated by the p states of Bi  and Te[30]. The long -range \nFM intralayer coupling, AFM interlayer coupling , and magnetic anisotropy with an out -of-plane \neasy axis have been confirmed by measurements using inelastic neutron scattering [40-42]. \nBand -structure calculations  have show n that various exotic topological phases may emerge in \nMBT, including quantum anomalous Hall  insulator s (QAHI s), axion insulator s, and QSH E \ninsulator s in thin films , as well as 3D QAHI s, Weyl semi -metal s (WSM s), Dirac semi -metal s \n(DSM s), and AFM TI s in bulk, as shown in Figure 2(b)[29]. These 2D or 3D topological states \ncould be manipulated by spatial dimensions and magnetic configurations . Also , the characteristics \nof van  der Waals materials  mean that MBT  is easily  exfoliat ed int o thin flakes , which is \nconvenient for gate modulation  and fabricat ion of  various heterostructures.  Therefore, intrinsic \nmagnetic TI  MBT  is an ideal system for investigating emergent rich topological phenomena . \n \nFigure 2 . Crystal structures and calculated topological properties of M nBi2Te4 (MBT ): (a) the \ndirections of the magnetic moments in each layer are represented by arrows ; τ denotes the \nhalf-lattice translation operation ; (b) various magnetic configurations ( FM, ferromagnetism; AFM , \nantiferromagnetism; PM, paramagnetism) and corresponding topological states of thin -film and \nbulk MBT  (WSM , Weyl semi -metal; DSM , Dirac semi -metal; AI , axion insulator ); (c) calculated \nband structures of bulk AFM MBT ; (d) the top surface states open an exchange energy gap  \ninduced by magnetic moments under the AFM -z configuration, while all side surface states remain \n6 \n \n gapless  due to  equivalent time -reversal symmetry.  Adapted from  [29]. \nII. AFM AND FM PHASES OF MBT  \nWhen exfoliated into thin films, the upper and lower surfaces of few -SL MBT  have either the \nsame or opposite magnetic moments according to the parity of the number of layers,  thereby \ndetermin ing the topological properties of the  syste m. Also , given that the interlayer AFM \ninteraction in MBT is weak, the interlayer magnetic sequence of MBT can be modulated into \nout-of-plane FM b y applying a moderate out -of-plane magnetic field , whereby  MBT beco mes the \nsimplest magnetic WSM  with only one pair of Weyl points  [29,32,4 3]. In the following, the \nelectronic properties of the MBT system are discussed by considering the number of layers, \ntunable magnetic configurations , and other parameters.  \nA. AFM phase  \nIn the ground magnetic configuration (AFM -z), the equivalent time -reversal symmetry S =𝒯τ \nis maintained, where τ denotes the half -magnetic -lattice translation along the z-axis connecting \nnearest -neighbor SLs with opposite magnetic moments . The existence of the equivalent \ntime-reversal symmetry  means that AFM -z MBT can be classified into the Z 2 topology.  The \ncalculated band structures and related topological invariants show  that bulk AFM -z MBT  is an \nAFM TI [29,32,34] . The bulk band opens a topologically nontrivial energy gap after considering \nspin–orbit coupling  (Figure 2(c)). Despite the side surface states remain ing gapless,  the top \nsurface state s open an exchange energy gap below the Né el  temperature  (~25 K) because of  the \nout-of-plane net magnetic moments (Figure 2(d)) [36]. \nFew-SL MBT  retains the characteristics of interlayer AFM  order ing (Figure 3). Therefore, the \noverall magnetic properties and the corresponding topological properties of few-SL MBT  change \nwith the number of layers . Table  I gives the calculated  thickness -dependent magnetic anisotropy \nenergies  (MAEs) and the energy difference between the AFM and FM phase s (Δ𝐸𝐴/𝐹=𝐸𝐴𝐹𝑀 −\n𝐸𝐹𝑀) in MBT.  All the MAEs are positive, indicating an out -of-plane easy axis  in good agreement \nwith experime nts. The positive Δ𝐸𝐴/𝐹 of monolayer MBT indicates that the FM phase is the \nlow-energy state , and thus  monolayer  MBT  is a 2D FM material  [44]. Furthermore, the calculated \nband structures indicate that monolayer MBT is topologically trivial (Figure 3(a)). Even -layer \nMBT  is a fully compensated antiferromagnet  (cAFM) , in which  the magnetic moment s of each \nlayer are completely compensated  because of  the AFM nature . However,  odd-layer  MBT with \nthree or more layers is  an uncompensated antiferromagnet  (uAFM)  exhibiting net magnetization . \nTable  II gives the topological phases and band gaps of monolayer and multi -layer MBT . \nOdd-layer MBT thin films have Chern insulator states of | C| = 1 (Figure 3(c)) . The Chern number \nof even -layer MBT is zero, but it is topologically nontrivial . The upper and lower surface s of \neven-layer MBT  contribute half quantum Hall conductance with opposite signs , giving zero Hall \nconductance . This configuratio n is known as the zero plateau  quantum anomalous Hall (ZPQAH) \neffect  state or axion insulator state  [32] (Figure 3(b)(d) ).  \n 7 \n \n  \nFigure 3 . Calculated band structures of few -SL AFM MBT : (a) single -layer MBT is topological ly \ntrivial , whereas (b) two or (d) more even layers of MBT open a topological ly nontrivial gap of \naround 100  meV that exhibits ZPQAH , and (c) three or more odd layers of MBT show QAHE.  \nAdapted from  [34]. \nTABLE  I. Thickness dependence of magnetism of MBT films [34]. 𝑇𝑐 represents the Curie \ntemperature for monolayer MBT and the Néel temperature for multilayer MBT.  The numbers in \nbrackets indicate the error bar . \nThickness  \n(no. of SLs)  Δ𝐸𝐴/𝐹 \n[meV/(Mn pair)]  Ordering  MAE  \n[meV/Mn]  𝑇𝑐 [K] \n1 14.77  FM 0.125  12(1)  \n2 −1.22 cAFM  0.236  24.4(1)  \n3 −1.63 uAFM  0.215   \n4 −1.92 cAFM  0.210   \n5 −2.00 uAFM  0.205   \n6 −2.05 cAFM    \n7 −2.09 uAFM    \n∝ (bulk ) −2.80 cAFM  0.225  25.42(1)  \n  \n8 \n \n TABLE  II. Thickness dependence of the MBT films ’ topology and band gap size [34]. \nThickness (no. of SLs)   Topology   Band gap [meV]  \n1  Trivial   321 \n2  ZPQAH   107 \n3  QAH   66 \n4  ZPQAH   97 \n5  QAH   77 \n6   ZPQAH   87 \n7  QAH   85 \n∝ (bulk）  3D AFM TI   225 \n \nFigure 4. Topological p hase transition of bulk FM MBT under external magnetic field : (a) phase \ntransition between type -I WSMs  (solid squares)  and type -II WSMs  (open squares ); Weyl points \nevolve in the kx–kz plane as the direction of the magnetic field (black arrows) rotates from the \nz-axis to the x-axis; calculated band structures of (b) bulk and (c) surface states of FM MBT with \nmagnetic orientation angles of 10 ° (upper) and 50 ° (lower ); (d) band dispersion  around Weyl \npoints in out-of-plane direction under  distinct  magnetic configurations with polar angle θ = 0°, 20°, \n40°, 60° , and 80° ; the Fermi level  is denoted  by the dashed line , and the tilted Weyl cone  becomes \nupright gradually with increasing θ. Adapted from  [43]. \n  \n9 \n \n TABLE  III. Distinct symmetries possessed by bulk MBT with different magnetic configurations \n[43]. \n 𝒫 𝒫𝒯 𝑀𝑥 𝐶3𝑧 𝑆=𝒯𝜏, \nAFM -z ✓ ✓ × ✓ ✓ \nAFM -x ✓ ✓ ✓ × ✓ \nFM-z ✓ × × ✓ × \nFM-x ✓ × ✓ × × \n \nB. FM phase  \nUnder a moderate magnetic field, AFM MBT  can be transformed into FM order ing, and \naccordingly the topological properties also change. The FM -z phase of the bulk material is \npredicted to be the type-II WSM  phase . As given in Table  III, unlike the time-reversal invariant \nWSMs with even pairs of Weyl points, this mag netic WSM breaks time-reversal or equivalent \ntime-reversal symmetry and host s only a pair of Weyl points. Therefore, FM -z MBT is claimed to \nbe the simplest WSM, which is advantageous for future experimental studies of Weyl physics . \nNontrivial transport phenomen a including negat ive longitudinal magnetoresistance  [45], large \nintrinsic anomalous Hall effect  [46], and large anomalous Nernst effect  [47] can be  expected  in \nthis WSM  phase. The pair of Weyl points is located on the Γ –Z axis , which is protected by \nthreefold rotational symmetry C3z. When the external magnetic field is rotated, the bulk system  \nchanges continuously from the FM-z phase to the FM-x phase  (shown in Fig ure 4), and t he pair of \nWeyl points  deviate s from the Γ–Z axis to become general k points  in the Brillouin zone because \nof the broken C3z symmetry . When the polar angle θ of the external magnetic field is such that \n10° < θ < 20°, the system changes from type -II WSM to type -I WSM  (Figure 4(a)). When θ = 90° , \nthe Weyl points meet and annihilate each other , making  the FM-x phase  into a trivial FM insulator . \nFigures  4(b) and 4(c) show the calculated  bulk band structures and Fermi arc of MBT with θ = 10° \n(upper) and 50 ° (lower) , which are typical of type -II and type -I Weyl points , respectively . The \nband dispersions of the Weyl points  in the out -of-plane direction are shown in Fig ure 4(d), which  \nalso clearly demonstrates  the evolution from  type-II to type -I WSMs  with different magnetic \norientation s. \n 10 \n \n  \nFigure  5. QAHE at zero magnetic field in a five -SL MBT flake.  𝑅𝑦𝑥 and 𝑅𝑥𝑥 versus magnetic \nfield a s measured at 1.4  K. A nearly quantized Hall resistance of  𝑅𝑦𝑥=0.97ℎ\n𝑒2 is observed , and \n𝑅𝑥𝑥 reaches 0.061ℎ\n𝑒2. An even better Hall plateau of 0.998ℎ \n𝑒2 is obtained as the magnetic field is \nincreased up to 2.5  T. Adapted from [48]. \nIII. EXPERIMENTAL OBSERVATION OF CHERN INSULATOR WITH HIGH \nWORKING TEMPERATURE AND HIGH CHERN NUMBER  \nIn previous studies on Chern insulator states (or the QAHE) in magnetic ally doped TI films,  it \nwas necessary to measure at ultralow temperature s because  the working temperature of the QAHE \nis very low  [25]. Moreover, only C = 1 (one chiral edge state) can be realized  in magnetic ally \ndoped TI films.  Therefore , the key issue in the QAHE field  is searching for Chern insulators with  \nhigher working temperature and high Chern number ( C > 1), motivated by both fundamental \nresearch interest and potential application s in low -power -consumption electronics and spintronics \nphysics . \nIn a five -SL MBT thin flake,  Deng  et al. [49] reported the nearly quantized Hall resistance of \n𝑅𝑦𝑥=0.97ℎ\n𝑒2 at 1.6  K under zero magnetic field, which reache d a better plateau of 0.998ℎ\n𝑒2 \nupon increasing the perpendicular magnetic field to 2.5  T (Fig ure 5). Adopting the quantization \ncriterion of 𝑅𝑦𝑥~0.97ℎ\n𝑒2, a quantization temperature of 6.5  K was obtained  under an external \nmagnetic field of 7.6  T, which was higher than the previous re cord of around 2  K by selective \ndoping  in TI. \nGe et al. [50] reported C = 1 Chern insulator states with much higher working temperatures \n[Figs.  6(a)–6(d)] . In a seven -SL MBT device, Ge et al. observed a well -quantized Hall plateau of \n11 \n \n 𝑅𝑦𝑥=0.98ℎ\n𝑒2 at 1.9  K by applying a small back gate of 6. 5 V, along with a nearly vanishing \nresistance of 𝑅𝑥𝑥=0.012ℎ\n𝑒2, which is a hallmark of  the C = 1 Chern insulator state  (Figure. \n6(a)(b)) . Surpr isingly, the quantized  Hall plateau shrank slowly with increasing temperature and \nsurvived at temperature s as high as 45 K (𝑅𝑦𝑥~0.904ℎ\n𝑒2 , as shown in F igure. 6(a)), which  is the \nhighe st working temperature  to date for Chern insulator states or the QAHE . Ge et al. observed a  \nsimilar high-working -temperature Chern insulator state in a n eight -SL MBT device, where the \nHall resistance quantization plateau remained around 0.97ℎ\n𝑒2 above 30  K and reach ed 0.997ℎ\n𝑒2 \nat 1.9 (Figure. 6(c)(d)). In both samples , the working temperature s of the Chern insulator states \nwere higher than the Né el temperature, indicat ing that the interlayer AFM coupling is irrelevant \nfor the topological properties of FM MBT and the FM order ing is key for quantiz ing the Hall \nresistance. A moderate out -of-plane magnetic field  causes the quantization to occur above the \nNé el temperature  by aligning the magnetic moments , and making the FM ordering more robust  \nwould allow the Q AHE to be realize d at temperatures above that  of liquid  nitrogen.  \nMore interestingly, Ge et al. [50] detected  a Hall resistance plateau of 0.99ℎ\n2𝑒2 with a \nvanishing 𝑅𝑥𝑥~0.004ℎ\n2𝑒2 in 10-SL MBT devices  at 2 K and −15 T by applying a back gate \nvoltage of −58 V ≤ Vbg ≤ −10 V (Figure  7(a)), indicat ing a high-Chern -number Chern insulator \nstate with C = 2. Figures  7(b) and 7(c) show the temperature evolution of the C = 2 Chern \ninsulator states. With increasing temperature, the Hall resistance plateau remain ed around  \n0.97ℎ\n2𝑒2 at 13  K and 0.964ℎ\n2𝑒2 at 15 K, while 𝑅𝑥𝑥 remained below 0.026ℎ\n2𝑒2 at 13  K and \n0.032ℎ\n2𝑒2 at 15 K. Ge et al. also detected  the C = 2 state in nine-SL MBT devices.  Surp risingly, \neven for high-Chern -number Chern insulator states , the quantization  temperature is much higher \nthan liquid -helium  temperature s, suggesting  a different mechanism from  that of magnetically \ndoped TI films and  poten tial application s in low -energy -dissipation electronics.  From Landauer –\nBüttiker theory  [51], although the chiral edge current in a QAHI is dissipationless, there is contact \nresistance between the electrode and the chiral edge channel. In two -terminal devices, t his contact \nresistance would cause a limited longitudinal resistance with a minimum value  of h/(Ce2), thereby \nconstrain ing the development of low -power -consumption electronics utilizing the ballistic \ntransport of  chiral edge states. One way to solve this problem is to se ek the QAHE with large \nChern number, which could reduce the contact resistance by  a factor of  1/C. Undoubtedly, the \nobservation of the C = 2 Chern insulator state provides a good starting point for achieving this \ngoal.  \nA fundamental issue is the physical origin of the observed high -Chern -number Chern 12 \n \n insulator states. In the absence of a magnetic field, bulk MBT is an AFM TI in which the side \nsurface states remain gapless while the top and bottom surface states are gapped by magnetic \nmoment s. For MBT thin films, the gapped top and bottom surface states carry a half quantum Hall \nconductance of opposite or identical signs, leading to C = 0 or 1. Chern insulator states with C > 1 \nare impossible physically with TI thin films . However, by applying a moderate magnetic field, \nbulk MBT become s an FM WSM  and thin-film FM MBT exhibits a  Chern -insulator band \nstructure  when the thickness is within the quantum confined regime , wherein the energy gap \ndecreases and the Chern number may increase as the film gets thicker  [43,52] . In other words, as a \nquantized system , thin -film FM MBT  could have high Chern number  (C > 1) with increasing \nthickness . This unusual feature can be attributed to  the combination of the quantum confinement \neffect and  the WSM characteristics of the MBT FM phase , which paves a new way to realize \nhigh-Chern -number Chern insulators  (Figure 7(d))  [53,54]. For N-SL FM MBT, its electronic \nstates can be viewed as the quantum -well states of a n FM WSM and possess a finite band gap due \nto quantum confinement effects. The anomalous Hall conductance of such a system would be \nestimated by 𝜎𝑥𝑦 ≈𝑐0𝑁𝑘𝑤𝑒2\n𝜋ℎ, where 𝑐0 is the out -of-plane thickness of each SL , N is the number \nof SLs , and 𝑘𝑤 is half the distance between the Weyl point s at the Brillouin zone  (Figure 7(e)) [50]. \nFor a gapped 2D topological thin film,  𝜎𝑥𝑦 must take quantized values as 𝜎𝑥𝑦 =𝐶(𝑁)𝑒2\nℎ, \ntherefore the Chern number is determined approximately by 𝐶(𝑁)~𝑐0𝑁𝑘𝑤\n𝜋. Upon  increas ing the  \nlayer number N by 𝜋/|𝑐0𝑘𝑤| (around four SLs according to  first-principles calculation s), the \nChern number increases by 1  (Figure 7(f)) , which is consistent with the experimental observations. \nThe experimental discovery of the C = 2 Chern insulat or state in MBT devices also offers indirect \nevidence for the magnetic WSM phase in  FM-z MBT.  \nTopological edge states in the QAHE can maintain the quantum characteristics of electrons \n(such as no dissipation) at the macro scopic  scale  and may be used to design and construct \nelectronic devices based on new principles. The high -working -temperature and  \nhigh-Chern -number Chern insulator states discovered in MBT indicate that if the proper intrinsic \nmagnetic topological materials and external parameters are chosen,  then there is hope  of realiz ing \nthe QAHE at temperatures above that of liquid  nitrogen  and perhaps even at room temperature, \nwhich would be a milestone  breakthrough in  practical application s of the QAHE.  13 \n \n  \nFigure 6. High -temperature Chern insulator s in (a), (b) seven -SL and (c), (d) eight -SL MBT  \ndevice s: (a), (b) temperature dependence of C = 1 Chern insulator states in a seven -SL MBT \ndevice ; the nearly quantized Hall resistance plateau re mains at temperature s as high as 45  K; (c), \n(d) temperature dependence of Ryx and Rxx versus magnetic field in an eight -SL MBT device ; a \nwell-defined quantized Hall resistance plateau survives up to 30  K. Adapted from  [50]. \n  \n14 \n \n  \nFigure 7. High -Chern -number Chern insulator states with C = 2 in 10-SL MBT device : (a) Rxx and \nRyx as function s of back gate voltage  at 2 K and −15 T, featured by a Hall resistance plateau of \nh/2e2 and vanishing Rxx at back gate voltage of − 58 V ≤ Vbg ≤ −10 V, indicating a Chern insulator  \nwith Chern number C = 2; (b), (c) temperature dependence of high -Chern -number Chern insulator \nstates in 10 -SL MBT device ; Ryx and Rxx at different temperatures from 2  K to 15  K are shown as \nfunction s of the magnetic field  strength; a Hall resistance plateau of 0.97 h/2e2 survives to 13 K; (d) \nschematic of high -Chern -number Chern insulator states with two chiral edge states across the band \ngap; gray and green  indicate adjacent MBT SLs ; (e) illustration of band structure of bulk FM \nMBT, which is a magnetic WSM ; the distance between the Weyl points at the Brillouin zone is \n2𝑘𝑤, and the Chern number jump s at the positions of the Weyl points ; (f) calculated Chern \nnumber as a function of film thickness.  Adapted from  [50]. \nIV. EXPERIMENTAL OBSERVATION OF AXION INSULATOR  \nIn the context of quantum chromodynamics, considering the axion field introduce s additional \nterms in Maxwell’s equations  that couple the electric and magnetic field s. Interest ingly, by using \neffective topological field theory, it is found that the electromagnetic response of a 3D TI includes \nthe additional term  𝑆=𝜃\n4𝜋2𝛼∫𝑑4𝑥𝑬⋅𝑩 [17], which is similar to  Maxwell’s equation s having \nadditional axion  terms. Here, 𝑬 and 𝑩 are the electromagnetic fields,  𝛼 is the fine -structure \nconstant, and the so -called axion angle 𝜃 is a dimensionless pseudoscalar parameter that is \ndefined modulo 2π  [32]. Also known as the axion term,  the extra term couple s the magnetic  and \nelectric field s and leads to  the topological magnetoelectric effect (TME)  [32]. From t he effective \naction of a 3D TI, the current response term  is 𝑗=𝑗𝑓𝑟𝑒𝑒 +𝛼\n4𝜋2∇ 𝜃 ×𝑬−𝛼𝜃\n4𝜋2𝜕𝑩\n𝜕𝑡. Note that the \n15 \n \n second term therein is the Hall current (perpendicular to the electric field) and  is due entirely to \nthe axion term, which is the result of the nontrivial topology of the system. If the energy gap is \nfully opened on the surface and the bulk-state axion angle is strictly quantized to π , then the axion \nterm will be the  only current contribution to the stable system . Integrating along the surface \nnormal direction, we obtain the half quantum conductance σxy = e2/2h contributed by each surface , \nwhere the sign is determined by the direction of the surface magnetic moment s. When the \ndirections of the magnetic moments are opposite between the top and bottom surfaces, t he current \ndirections of the axi on terms are also opposite (considering the local coordinates of the upper and \nlower surfaces) . Therefore , although  a certain surface has half -integer Hall conductance, the net \nHall current of the system becomes zero because of  the cancellation of the up and down surfaces , \nand so ZPQAH can be observed  [34]. Furthermore, a circulat ing Hall current is formed when \napplying an electric field , as shown in Fig ure 8. Such a circulation is equivalent to the surface \nmagnetizing current induced by a quant ized magnetization 𝑴=−(𝑛+1\n2)𝑒2\nℎ𝑐𝑬 wherein 𝑛 is an \ninteger. Similarly, a charge polarization can be induced by applying a magnetic field. When a \nmagnetic field is turned on  slowly , the induced electric field will generate a Hall current parallel to \nthe magnetic field and thus charge s will accumulate on the top  and bottom surfaces, which is \nequivalent to a quantized polarization 𝑷=(𝑛+1\n2)𝑒2\nℎ𝑐𝑩. Such striking induction between electric \nand magnetic fields is  known as the TME , which is direct evidence for axion insulator states . \nHowever, the strict requirements of instrument accuracy and sample quality  make it quite \nchallenging to observ e the TME . Therefore , the research effort  to date has been focused on  \ndetecting the zero Hall platform  as evidence for the axion insulator phase [31,5 5]. \nTo realize the axion insulator states , three conditions must be satisfied  [17,32] : (i) all the \ntopological surface states of the system must be fully gapped ; (ii) there must be a certain \nsymmetry to quantize the axion angle 𝜃; (iii) the Fermi surface must fall into the energy gaps of \nthe bulk state and all the surface states. FM/TI/FM heterostructures have been used to construct \naxion insulator s, such as V-doped (Bi,Sb) 2Te3/(Bi,Sb) 2Te3/Cr-doped (Bi,Sb) 2Te3 heterostructures  \n[31]. The principle is to make use of the different coercive fields of the magnetic films on the top \nand bottom FM  sides  of the heterostructure , thereby making the magnetic moment directions of \nthe upper and lower surfaces of the system opposite under a specific external magnetic field. \nHowever, the working temperature of the system is of the order of millikelvin s and prepa ring the \nheterojunction is complicated,  thereby not favor ing wide spread investigations.  16 \n \n  \nFigure 8 . Edge states of axion insulator and QAHI . Directions of magnetic moments are denoted \nby red arrows and the Hall currents are represented by blue arrows.  Each gapped surface \ncontributes a half-quantized Hall conductance, the signs of which are opposite in axion insulator \nstates and the same in QAHI  states. In axion insulator states, an electric field can induce a \nquantized circulating Hall current that is equivalent to the surface magnetizing current induced by \na quantized magnetization. Adapted from  [29]. \nAs an intrinsic magnetic TI, MBT is expected to overcome these difficulties . Its ordered \nstructure makes MBT more advantageous for avoid ing disorde r effects  such as band -gap \nfluctuation  and superparamagnetism  [32]. The large band gap facilitates the observation of axion \ninsulator states at higher temperature . The AFM -z phase of even -layer MBT thin films is an \nintrinsic axion insulator  because the top an d bottom surface states are natural ly gapped by \nopposite magnetic moments . Liu et al. [55] reported a large longitudinal resist ivity with a weak \nmagnetic field at 1.6  K by applying a gate voltage of 25 V (Figure 9(a)) . More interestingly, the \nHall resistivity  remained zero in a field range of −3.5 T < B < 3.5  T. Figure  9(b) shows (i) the \nzero field longitudinal resistivity  and (ii) the slope of the weak -field Hall resistivity  versus the \nmagnetic field  strength  as functions of the gate voltage 𝑉𝑔. For 𝑉𝑔 = 22–34 V, the longitudinal \nresistivity  exhibits insulating behavior while the slope of the Hall resistivity exhibits a zero plateau, \nwhich is different from conventional insulators. With increasing magnetic field, both the \nlongitudinal  and Hall resistivities undergo a sharp transition. The Hall resistivity  reaches a plateau \nof 0.984ℎ\n𝑒2 with a longitudinal resistivity  of 0.018ℎ\n𝑒2 at −9 T (Figure 9(a)) . As discussed above, \nthe zero Hall plateau indicates the appearance of the axion insulator phase , while the quantized \nHall plateau and vanishing longitudinal resistance are characteristics of Chern insulator states with \nChern number C =1. Therefore, a magnetic -field-driven phase transition between axion  and Chern \ninsulator states  is observed . \nRecently, two independent  studies reported unusual layer -dependent magnetic properties in \nMBT devices  [56,57], observ ing anomalous magnetic hysteresis loops in polar reflective magnetic \ncircular dichroism spectroscopy (RMCD) results  [56] and anomalous Hall curves  in even -layer \nMBT [57]. More intriguingly,  Ovchinnikov et al. [57] found that the hysteresis loop near zero \nmagnetic field vanish ed for an odd -layer device in the AFM state. Actually, most device samples \n17 \n \n in experiments are inevitably affected by various factors during device fabrication process and \nmeasurements, including sample quality and oxidation. Consequently, the properties of devices \ndepend sensitively on the specific experimental conditions and might vary between different \nsamples.  \n \nFigure 9. Magnetic -field-driven  axion -insulator –Chern -insulator transition in a six-SL MBT  \ndevice : (a) longitudinal  and Hall resistivities  versus magnetic field  strength  at various \ntemperatures with gate voltage 𝑉𝑔=25 V; (b) gate dependence of axion insulator  state. Adapted \nfrom  [55]. \nV. MBT -RELATED SYSTEMS  \nA. Heterostructures  comprising MBT  and magnetic monolayer materials  \nConstructing a hetero structure  from MBT thin flakes and other 2D magnetic materials has \nbeen proposed as a promising way to stabilize the surface moments of MBT and realize \nhigh-temperature QAHE.  Fu et al. [58] calculated the structural characteristics of an MBT –CrI 3 \nheterostructure . They found that the CrI 3 monolayer had strong FM exchange interaction with the \nMBT surface layer, effectively increasing the stability of magnetic order ing in the MBT surface \nlayer. The CrI 3 monolayer induced an exchange bias as large as 40 meV , which is much larger \nthan the N éel temperature of MBT ( around  25 K, which correspond s to an energy of around  \n2 meV  [36]). Moreover,  from band -structure calculations, the proximity with the CrI 3 monolayer  \nhad little effect on the band topology of MBT, as shown in Figure 10(a).  Therefore,  \nheterostructure s of MBT thin films and CrI 3 monolayer s could help to realize the QAHE at higher \ntemperature s. \nB. MBT –superconductor heterostructures  \nThe realization of Majorana  fermions in condensed matter systems, including Majorana  zero \nenergy mode s and Majorana  chiral edge states, has been of great concern  [4,15,16,5 9–62]. \n18 \n \n Majorana zero energy modes  can be used to construct quantum bits  and have important \napplications in topological quantum computation  [59]. One-dimensional  (1D)  Majorana chiral \nedge state s are topological edge state s of 2D p+ip chiral superconductors.  Fu and Kane  [60] \nproposed  heterostructure s comprising FM magnet s, TIs, and s -wave superconductors to form \nequivalent p+ip superconductors , wherein t he possible 1D Majorana  chiral edge states can be  \nobtained . Furthermore, Qi et al. [16] suggested that a chiral topological superconductor carrying a \nMajorana chiral state might be promising in a QAHI in proximity to a n s-wave superconductor . \nBased on this proposal, He et al. [63] claimed to realize Majorana chiral edge states characterized \nby an e2/2h conductance plateau  in Cr -doped (Bi,Sb) 2Te3 thin films in proximity with an Nb \nsuperconductor , but the results are question ably extrinsic  [64–67]. Therefore, it is important to \nobtain equivalent p+ip superconductors by using intrinsic magnetic topological system . Peng et al. \n[68] proposed a realization of chiral Majorana states using a heterostructure comprising MBT and \nan s-wave superconductor . In such a  hetero structure , the superconduc ting proximity  effec t can \nopen  a superconducting gap  on the MBT side surface . This energy gap and the magnetic exchange \nenergy gap of the upper and lower surfaces are topologically distinct , and the hinge of the \nheterostructure will exhibit 1D  Majorana  chiral edge states  (as shown in Figure 10(b)). \nC. MBT family  \nIn addition to MBT, a large family of van der Waals materials that can be expressed as \nMnBi 2Te4(Bi 2Te3)n (n = 1,2,3 …) also have intrinsic magnetic topological bands  [69–80]. In these \nmaterials, the MBT layers are separated by n layers of Bi 2Te3, as shown in Fig ure 10(c). The \ninterlayer AFM interactions decrease quickly as the distance between MBT layers increase s, so \nthe magnetic and topological properties are highly tunable by the number n of Bi 2Te3 layers . For \nlarger n, the AFM coupling is weaker and FM ground states are favored. When n = 1 or 2, \nMnBi 4Te7 and MnBi 6Te10 retain interlayer AFM coupling with Né el temperature s of 13  K and \n11 K, respectively  [77,78]. Transport  measure ments , ARPES experiments , and band calculations \nhave shown that AFM MnBi 4Te7 is an AFM TI below the Né el temperature , with gapless side \nsurface states similar to MBT . However,  the top and bottom surface states are associated with the \nMBT or Bi 2Te3 termination. The energy gap s of MnBi 4Te7 top surface states with MBT \ntermination are much smaller than those with Bi 2Te3 termination  [78]. For n = 2, MnBi 6Te10 is an \nAFM TI below  the Né el temperature,  having the full topological bulk gap with gapless side \nsurface states and gapped top  and bottom surface states  [69]. More inte restingly, the FM state \nemerge s when applying a magnetic field as weak as 0. 1 T and is preserved when the magnetic \nfield is lowered to zero at 2  K. The band structures in the FM state show a gap of around 0.15 eV. \nTheoretical calculation s show that bulk FM MnBi 6Te10 is a high-order TI with Z 4 = 2, equivalent  \nto an axion insulator with θ = π [69]. Moreover, theoretical calculations of few -layer FM \nMnBi 6Te10 show that two, three , and four layers of MnBi 6Te10 ([Bi 2Te3]–[MnBi 2Te4]–[Bi2Te3]–…) \nare intrinsic Chern insulators  with C = 1 at zero magnetic field  [69]. For n = 3, experimental 19 \n \n transport results show that MnBi 8Te13 falls into the FM phase below 10.5  K, and first -principles \ncalculations and ARPES measurements further demonstrated that MnBi 8Te13 is an intrinsic FM \naxion insulator with Z 4 = 2 [70]. Interlayer exchange interaction between neighboring MBT SLs \nbecomes too weak whe n the layer number n of intercalated Bi2Te3 layers is increased further. \nBelow the critical temperature, the MBT SL s can be regarded as being magnetic ally independent \nfrom  each other, and the magnetization of the SLs become s disordered along the z-axis. \nThe nontrivial topology of the (MnBi 2Te4)(Bi 2Te3)n family is connected strongly with the \ncorresponding magnetic  properties in this series , thereby allow ing us to  regulate the strength of the \ninterlayer AFM coupling in particular and so control the topological states. For large n, small or \neven zero external magnetic field is required to achieve Chern insulator stat es because of the \nweaker interlayer AFM coupling.  This fact could also help to realiz e a topological superconductor \nhosting Majorana fermions based on the MBT family [68]. \nIn addition to the (MnBi 2Te4)(Bi 2Te3)n family, the intrinsic van der Waals material \nMn 2Bi2Te5—as a sister compound of MBT —has also been predicted to be a topological AFM \nmaterial,  one that could host the long -sought dynamical axion field [81]. Unlike  AFM MBT , in \nwhich a certain symmetry fixes the axion field θ to a quantized value ( θ = π), in Mn 2Bi2Te5 θ \nbecomes a bulk dynamical field from magnetic fluctuations and varies from zero to π because of  \nthe breaking of both time -reversal and inversion symmetries [81]. Such a dynamical axion field \ncould induce novel phenomena, such as axionic polariton s [82] and nonlinear electromagnetic \neffect s induced by axion instability  [83]. A similar dynamical axion insulator state was also \npredicted in an (MnBi 2Te4)2(Bi 2Te3) heterostructure [84]. The dynamical axion insulator states \nrealized in these MBT -related systems may pave the way to a new generation of axion -based \ndevices. The MBT  family thus provides a feasible and highly tuna ble 2D platform for exploring \nnovel quantum topological phases  [48,50,55] , low-power -consumption electronics , spintronics  \n[4.5] , and topological quantum computing  [59]. \nD. Chemically substituted MBT  \nAnother route to regulating the properties of MBT is element  substitution  [52,85 -87]. Both \nmagnetic and carrier properties are tunable by substituting Sb for Bi , and thus topological effects \ncan be controlled by changing the proportion of Sb atoms in Mn(Sb xBi1−x)2Te4 [85,86]. Yan et al. \n[85] reported that with increasing Sb content in Mn(Sb xBi1−x)2Te4, the Né el temperature decrease d \nslightly  from 24  K for MnBi 2Te4 to 19  K for MnSb 2Te4, while the critical magnetic field  strengths  \nrequired for  spin-flop transition and  moment saturation decreased dramatically . The above results  \nsuggest that both the interlayer AFM coupling and magnetic anisotropy  weaken with increasing \nSb content for x < 0.86. Besides,  Sb substitution can also control the Fermi level . With  increasing  \nx, a crossover from n-type to p-type conducting behavior is observed in bulk Mn(Sb xBi1−x)2Te4 \naround x ≈ 0.63 [85]. A critical  issue affecting the experimental observation of the \nhigh-temperature QAHE in MBT is the bulk n-type conductivity,  usually induced by high 20 \n \n densities of Mn  and Bi antisite defects  [88]. Sb substitution could modulate  the carrier \nconcentration  and tune the Fermi level to the bulk gap [85]. Another strategy to lower the Fermi \nlevel  is to grow higher -quality MBT single crystals under  Te-rich condition s [88]. \nTheoretical  calculations predict that MnSb 2Te4 is a topologically triv ial AFM material  \n[32,8 6], but experiments report both AFM  (TN~19 K) [85] and FM  (TC~25 K) [89] MnSb 2Te4. \nFurthermore, Wimmer et al. [87] conducted detailed experimental explorations combining ARPES , \nscanning tunneling microscopy ( STM ), magnetometry , and x -ray magnetic circular dichroism in \nepitaxial MnSb 2Te4. They observed that  MnSb 2Te4 exhibits robust out -of-plane FM order ing with \na Curie temperature of 45 –50 K, and t hrough ARPES and STM  they uncovered a Dirac point near \nthe Fermi level with out -of-plane spin polarization and a magnetic ally induced gap of around \n17 meV that vanishes  at the Curie temperature. These are typical features of an FM TI. By \nconducting further  density -functional theoretical calculation s and x-ray diffraction  experiments, \nWimmer et al. concluded that the FM and topologically nontrivial properties of M nSb 2Te4 \noriginate from the antisite mixing of Mn and Sb.  The fact that the Curie temperature in MnSb 2Te4 \nis much higher than the N éel temperature in MBT  may pave a new way to realizing the \nhigh-temperature QAHE.   \n  21 \n \n  \nFigure 10 . MBT heterojunctions and MBT family : (a) calculated band structures of one-SL to \nsix-SL MBT thin films in proximity to a CrI 3 layer ; the blue lines represent MBT bands and the \nred lines represent Cr-eg bands ; in the insets, the MBT layers and CrI 3 monolayer are denoted by \npurple and blue layers , respectively ; the results show that  the CrI 3 monolayer  induces an exchange \nbias of around 40  meV, and the band topology of MBT is little affected ; adapted from [58]; (b) \nMajorana hinge mode s at the edge of the interface  between an AFM  TI (e.g., MBT ) and an s -wave \nsuperconductor ; the Majorana modes are indicated by blue and red arrows , and t he easy axis of \nAFM is  in the z direction ; adapted from [68]; (c) crystal structures of MBT family ; the common \nstructural characteristic is that one MBT SL is sandwiched by n Bi2Te3 quintuple layers ; shown are \nthe cases of  n = 0 for MBT, n = 1 for  MnBi 4Te7, and n = 2 for MnBi 6Te10. \n  \n22 \n \n VI. PUZZLE OF SURFACE STATES OF MBT  \nARPES experiments have purportedly shown the exchange gap of surface states induced by \nthe out -of-plane magnetic moments below the Né el  temperature  as having values  from tens of \nmillielectronvolts to 200 meV  [36,90,91]. However, it is puzzling that the observed surface energy \ngaps vary little with temperature. Even above the Né el temperature, a limited “energy gap” can \nstill be detected  [91], so the energy gap might not originate from the magnetic order. On the \ncontrary, some high-resolution  ARPES experiments have reported the  top and bottom surfaces as \nhaving a  gapless surface st ate [86,92–96] with a small evolution with temperature (Fig ure 11(a)). \nThe above result s are obviously inconsistent with the theoretical predictions  for the surface gap \nopened by the out -of-plane magnetic moments . Li et al. [93] attributed the gapless states to the \nlimited interaction between the Mn-3d state that provides the magnetism and the Bi/Te -p orbital \nstate that dominates the topological properties . Using resonance ARPES technology , the Mn -3d \nstate was found to be located at 4 eV below the Fermi energy and is negligible in the energy range \nin which the nontrivial topology arises  [93]. Therefore , Li et al. claim that it may be difficult for \nmagnetism to open the exchange energy gap of the surface state . However, this conclusion is yet \nto be supported by first-principles calculation s and contradict s the Chern insulator phase and the \naxion insulator states observed by transport  measurements  [48,50,55]. Besides, a large magnetic \ngap of around  90 meV at the Dirac point in  Bi2Te3/MBT heterostructures has been reported, which  \nvanishes  above the Curie temperature  [30]. This result, on the other hand, indicates that the \ninteraction between  magnetism and topology  in MBT may be  strong enough to induce a n \nobservable magnetic gap , and the absence of the magnetic  gap in previous experiments  calls for \nalternative explanations.  \n  23 \n \n  \nFigure 1 1. (a) Gapless surface states observed by ARPES at various temperatures.  Adapted from \n[92]. (b) Magnetic -moment fluctuation model, which considers that the magnetic -moment \nfluctuation forms a series of disordered domains: boundary states can be formed between \ndisordered domains with  different moments , and these edge states  provide the density of electronic \nstates at the Dirac point . Adapted from  [97]. \nChen et al. [92] showed  that the observation of gapless surface states can be explained by  \nspatial ly dependent magnetic moment s on the surface . As shown in Fig ure 11(b), the AFM \ncoupling between the surface layer and the underlying layer may be weaker than the AFM \ncoupling of the bulk state, which leads to  fluctuation of the magnetic moment on the surface, \nforming a series of magnetic -moment domains with the same or opposite directions. The opposite \nmagnetic moments induce gapped surface states with gaps of opposite sign, so gapless edge states \ncrossing the energy gap appear between the domain. These edge states provide the electron \ndensity of states for the observation of the gapless surface state in the ARPES experiment. A \nsimilar explanation  is also proposed  [52], in which the directions of surface magnetic moments  \nchange in different terraces (domains) . Consequen tly, topological edge states occur at the step \nedges of different terraces . The afore mention ed hypotheses can be justified further by more \nspatially resolved spectroscopy measurements , such as scanning tunneling spectroscopy  and \npoint -contact technology . Recently , a point -contact experiment reported a surface gap of around  \n50 meV, which vanished as the sample became paramagnetic  with increasing temperature  [98]. \nBecause  point -contact technology studies the local properties of samples, this result indicates the \nexistence  of a local magnetic gap at the MBT surface.  \n  \n24 \n \n VII. OUTLOOK  \nThe intrinsic magnetic TIs provide a promising platform to explore the interplay between \ntopology and magnetism, which could open up new ways for novel fundamental physics as well as \ngive rise to potential technical revolutions. Theo retical studies have predicted several intrinsic \nmagnetic TIs, such as the MBT family, EuSn 2As2, and EuIn 2As2 [36,93,9 9]. Among these \nmaterials, MBT has attracted the most attention and is believed to have the greatest potenti al, this \nbeing  because  MBT brings hope for many novel topological quantum states, such as Chern \ninsulator states  and axion insulator states  [29,32,34,4 8-50,55]. At moderate magnetic fields, MBT \nthin flakes translate into FM WSM states with only one pair of Weyl points, in which \nhigh-working -temperature and high -Chern -number Chern insulators are realized  [50]. Besides , 2D \nheterostructures based on the MBT family could offer unprecedented opportunities to discover \nnew fundamental physics of topological fermions such as the long -sought Majorana fermions  [68]. \nAs a van der Waals magnet, MBT also paves an ideal avenue to exploring novel magnetic phases  \nsuch as the skyrmion phase  [40,44]. \nNotably, the QAHE with high working temperature can be expected in MBT. The key factor \nthat limits the working temperature of the QAHE in MBT is the low magnetic -ordering \ntemperature, which is restricted by the strong magnetic fluctuation caused by weak ma gnetic \ninteraction in MBT. Applying a moderate external magnetic field can effectively increase the \nanisotropy of MBT, suppress the magnetic fluctuation, and thus increase the effective \nmagnetic -ordering temperature [50]. Another way to realize the high -temperature QAHE in MBT \nis to strengthen the interlayer coupling in MBT by magnetic doping. It is predicted that V -doped \nMBT is FM with a high Curie temperature of aro und 45 K [100] . However, magnetic doping may \nintroduce additional magnetic inhomogeneity and thus cause more scattering that may destroy the \nQAHE. Another way to improv e the magnetic properties of MBT is chemical  substitution . It is \nreported that MnSb 2Te4 with slight Mn/Sb antisite mixing  is an FM TI with a Curie temperature of \n45–50 K [87]. Constructing heterostructures of MBT with magnetic monolayer materials is also \nexpected to be a promising route to suppressing the magnetic fluctuation. Magnetic monolayer \nmaterials may effectively increase the stability of the magnetic ordering in the MBT  surface layer \nand thus raise the QAHE working temperature [58]. Once the QAHE is realized above the \nliquid -nitrogen temperature, a milestone breakthrough for the pra ctical application of topological \nmaterials can be expected, which would stimulate research interest significantly in the fields of \nphysics, materials, and information technology.  \n \n \n \n \n \n \n 25 \n \n  \n \nAcknowledgements  \nThis work was financially supported by the Beijing  Natural Science Foundation (Grant No. \nZ180010),  the National Key R&D Program of China (2018YFA0305600, 2017YFA0303302), the \nNational Natural Science Foundation of China (Grant No. 11888101, Grant No. 11774008 ), the \nStrategic Priority Research Program of Chinese Academy of Sciences (XDB28000000), the Open \nResearch Fund Program of the State Key Laboratory of Low -Dimensional Quantum Physics, \nTsinghua University (Grant No. KF202001).  \n \nCompeting Interests  \nThe authors declare no competing interests.  \n \nReferences :  \n[1] Hasan, M. Z., & Kane, C. L. (2010). 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Alloying V in MnBi 2Te4 for Robust Ferromagnetic Coupling \nand Quantum Anomalous Hall Effect. arX iv preprint arXiv:2004.04862.  \n \n "    },    {        "title": "2012.05059v1.Optimization_of_a_NdFeB_permanent_magnet_configuration_for_in_vivo_drug_delivery_experiments.pdf",        "content": "Optimization of a NdFeB permanent magnet configuration for \nin-vivo drug delivery experiments  \n \n \nA.Omel yanchik1,4, G.Lamura2, D.Peddis3,4, and F.Canepa2,4* \n \n1 Immanuel Kant Baltic Federal  University, Kaliningrad, Russian Federation  \n2 CNR -SPIN, Corso Perrone 24, I -16152, Genova , Italy  \n3 Istituto di Struttura della Materia – CNR, Monterotondo Scalo, RM, Italy  \n4 Department of Chemistry and Industrial Chemistry Università di Genova, Genova , Italy  \n \n \n \nAbstract  \n \nWe propose a new concept of magnetic focusing for targeting and \naccumulation of functionalized superparamagnetic nanoparticles in \nliving organs through composite configurations of different \npermanent magnets. The proposed setups  fulfill  two fundamental \nrequirements for in vivo  experiments: 1) reduced size of the magnets \nto best focusing on  small areas representing the targeted organs of \nmice  and rats and 2) maximization of the  magnetic driving force  \nacting on the magnetic nanoparticles dispersed in blood. To this aim , \nseveral  configurations  of permanent magnets organized with \ndifferent degree s of symmetry  have been tested. The product 𝐵∙\n𝑔𝑟𝑎𝑑(𝐵) proportional to the magnetic force has been experimen tally \nmeasured , over a wide area (20  20 mm2), at a distance corresponding \nto the hypothetical distance of the mouse  organ from the magnets. A \nnon-symmetric configuration of mixed shape permanent magnets \nresulted in particularly promising to achieve the best performance s \nfor further in vivo  experiments.  \n \nKeywords: permanent magnets , magnetic field , target drug delivery , magnetic driving force . \n \n* Corresponding author.  \nE-mail address: fabio.canepa@unige.it  \n \n \n \n \n \n© 2020 . This manuscript version is made available under the CC -BY-NC-ND \n4.0 license http://creativecommons.org/licenses/by -nc-nd/4.0/  \n \n \n  \n1. Introduction  \n \nIn the last years , functionalized monodomain Magnetic NanoP articles  (MNPs)  as carriers for \nmagnetic drug  delivery  therapy  have been extensively studied as a promising less destructive \nalternative to typical chemiotherapic protocols  against several types of cancer  [1–3]. Special \nattention was given to side -specific targeting of stem cells enhanced with gene therapy [4–6] \nwhich invoked interests in the fundamental studies of influences o f the magnetic field and \nMNPs on the viability and manipulation of the cell`s  behavior [7–9]. This approach is very \npromising for example for engraftment of cells of the cardiovascular system after surgery.  \n \nHowever, while exhaust ive researches on the MNPs synthesis  and functionalization for drug \ndelivery have been carried out  [10,11] , relatively  few studies have been performed on the optimal \nmagnetic field and magnetic gradient parameters required to transport and accumulate the \nnanoparticles in the targeted organ  [12]. While the achievement of the high -value gradient \nmagnetic field at low -dimension is the relative ly well-developed field [8,13,14] , the upscaling of \nthose configurations to the scale of human organs is a complicated task because of the fast \nattenuation with the increase of the distance.  \n \nTypically, t he production of a magnetic field can be achiev ed by the use of 1) a superconducting \nmagnet, 2) an electromagnet or 3) a configuration of permanent magnets. Some results  \nconcerning the development of a system for magnetic drug delivery  have been performed by \nthe use of a superconducting magnet  [15,16] , for example, a supercondu cting magnet of \nmagnetic resonance imaging device [17]. However, even if its use allows to obtain very high \nvalues of the magnetic field and also to its gradient in the flow direction, the complex \nexperimental configuration and the quite expensive maintenance costs , prevent  the use of a \nsuperconducting magnet for in vivo  experiments. Electromagnetic systems for magnetic drug \ndelivery were mainly developed from a theoretical point of view, by 3D designs and simulation \nmethods , using appropriate software as Comsol Multiphysics [18] or Finite Element Model \n(FEM)  [19]. \n \nAt present, the most feasible way to obtain a useful magnetic field for in vitro  and in vivo  drug \ndelivery experiments is the use of a  single  or a suitable configuration of permanent magnets.  \nOn one side, theoretical simulations have been performed on a n ideal  system of MNPs  \ndispersed in blood  using different configuration s of permanent magnets  [20–25]. On the other \nside, different in vitro  experiments have been carried out using a single magnet or a \nconfiguration of multiple magnets separated by non-magnetic materials [26]. In several studies  \nin vivo  experiments of magnetic drug delivery and targeting to a specific organ have been \nperformed  using a suitable configuration of NdFeB permanent magnets  [27–31]. For exa mple, \nthe configuration of permanent magnets was used to radially symmetric cell deposition in \nvessels of mice [6,32] . In those studies, combined action magnetic nanoparticles and fields \nreinforced gene and cell therapy of vessels after irreversible damage caused for example by \nmechanical denudation. In most of the related works, attention was devoted to the magnetic \nfield produced by them, but not to the gradient required or the magnetic driving force \n(proportional to their product  𝐵∙𝑔𝑟𝑎𝑑(𝐵) ) to focalize  the functionalized MNPs to the specific \norgan and no attention was devoted to the relative dimensions of the magnets with respect to \nthe targe ted organ  as well . \n \nSo, there is a lack of information regarding  the choice of the permanent magnets geometry and \ntheir relative dimensions with respect to the mice used for in vivo  tests and  the direct 2D \nmapping  of the magnetic field of different permanent magnets configurations  obtained at \ndifferent heights as well as calculations concerning the relative gradients  and the magnetic \nforce .  \nSince the in vivo tests require a lot of time (from 2 up to 4 hours) in or der to ensure the \naccumulation of functionalized MNPs in the targeted organ, the small dimensions of the rats \nused for the experiments prevent the use of large (and heavy) permanent magnets.  So, the \nutilization of a light system of small permanent magnets in a non-metallic thin structure can \nsignificantly increase the test efficiency.  \n \nTherefore, t his paper aim s to present a preliminary experimental study on three  different \npermanent magnets configurations useful for in vivo  magnetic drug delivery , compared  to the \nresults achieved by the use of a single magnet with the same grade . The target is to maximiz e \nthe magnetic driving force over a surface area of about 1 cm2, (i.e. a value comparable to the \ntypical surface of an organ of a mouse as hearth, liver or lungs ). The studied configurations of \nmagnets were designed for in vivo  experiments where magnets will be placed on the skin of \nmice without surgery to enhance the accumulation of MNPs in a specific organ.  The magnetic \nmapping was performed by the direct measurement of the magnetic induction on the surfac e \nof the system  and at the distance of Z = 4 mm from it, i.e. the typical distance between a mouse \norgan and the skin.  A configuration that creates the strongest product in regions of interest \nwas selected further in vivo experiment.  \n \n2. Experimental details  \n \nAll the commercial  Nd 2Fe14B permanent magnets were from HKCM  Engineering e.K.  (DE) , with \nthe same N52 quality grade.  For the magnetic configurations, t wo different magnet s geometries \nwere used: a cubic geometry with l = 5 mm and a cylindrical one with 3 3 mm size (diameter  \nlength), with easy magnetization direction axially oriented. All magnets were Ni coated.  \nDetailed characterization of the magnetic properties of cylind rical magnets of the same grade \nwas reported in our previous work [33]. At room temperature remanent magnetic induction of \nsingle magnet was maximal among a set of commercially available  magnets and reached about \n14 kiloGauss (kG).  \n \nFor the magnetic induction  (𝐵⃗ ) measurement s, a Lake Shore gaussmeter  model 475 DSP \ncoupled with a n axial Lake Shore 400 HSE Hall  probe (Lake Shore Cryotronics Inc., USA)  was \nused. Thus the instrument gives the measurement of the axial component of the induction \nfield 𝐵⃗  vector. The sensitivity of the probe was better than 5 milli Gauss (mG) , while the  \nmagnetic induction  range was  up to 35 kG . A nominal active area of sensing element  reported \nby the manufacturer was about of 1 mm in diameter. The stem for Hall probe was made from \nfiberglass epoxy and had a diameter of 5 mm. The precise positioning of the Hall probe was \nachieved by a homemade positioner setup with a 3 -axial coordinatio n with the error of 0.05 \nmm.  The stem was therefore fixed vertically respect to the magnet plane. By considering a \ncoordinate system x,y,z with its origin on the magnets flat surfaces, the value returned by the \ninstrument is the z component of the inductio n field as a function of the in -plane coordinate s \nat a fixed 𝑧̅ height  𝐵𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 =𝐵𝑧(𝑥,𝑦,𝑧̅). \n \nAt first step,  the magnetic induction produced by a single cylindrical magnet (diameter and \nheight  equal to 3 mm) and by different configurations of several magnets were mapped by \nmeasuring 𝐵𝑧(𝑥,𝑦,𝑧̅) over a  10×10 mm2 surface by 1  mm step at  fixed height  𝑧̅ = 4 mm  from the \nmagnet surface . At each x -y step of the magnetic induction  map, the in-plane gradient \ncomponents were calculated with the following equations:  {(𝑔𝑟𝑎𝑑(𝐵𝑧(𝑥,𝑦,𝑧̅)))𝑥=(∆𝐵𝑧(𝑥,𝑦,𝑧)\n∆𝑥)\n𝑦,𝑧=𝑐𝑜𝑠𝑡.\n(𝑔𝑟𝑎𝑑(𝐵𝑧(𝑥,𝑦,𝑧̅)))𝑦=(∆𝐵𝑧(𝑥,𝑦,𝑧)\n∆𝑦)\n𝑥,𝑧=𝑐𝑜𝑠𝑡. . \n \nBesides singe magnet, t hree  magnetic configurations were adopted:  \n \na) Three cylindrical magnets  were  inserted in a Teflon disk of 13 mm diameter, 3 mm \nthickness , used as support , at the vertices of an equilateral triangle of 4 mm side . All \nthe magnets present ed the same polarization direction  (Fig. 1A). \nb) Four cylindrical magnets inserted in a squared Teflon disk (20 mm side, 3 mm \nthickness) at the vertices of a square of 5 mm side. Yet all the magnets were set with \nthe same  polarization direction  (Fig. 1B). \nc) A mixed configuration  fixed  again in a squared Teflon disk  (20 mm side, 5 mm \nthickness) : two cubic magnets  at two vertices of an ideal square of 5 mm side on the \nsame  diagonal and six cylindrical magnets , three by three , fixed in the remanent two \nvertices along the other diagonal . In this case, the polarity of magnets was changed  \nstarting from mutually opposite polarization of cube and cylinder magnets to the same \norientation  of all magnets  (see Fig.1C). \n \n \nFigure 1 . Visualization three cylindrical magnet system (a), four cylindrical magnet system (b), and mixed \nsystem of cubic and cylinder magnets (c).  \n \n \nIn all cases, the weight of the magnet set -up is less than 10 grams  in order to prevent any \nphysical impediment to  the rats and to allow an easy fixing to the selected position during in \nvivo experiments.  \n \n3. Results  and discussion  \n \n3.1. Magnetic assumptions  \n \nThe magnetization of MNPs in the superparamagnetic limit at low field is a linear function of \nthe applied magnetic field as 𝑀=𝜒𝑁𝑃𝐵, where NP is the initial mass magnetic susceptibility \n[34]. Roughly, in the general case of  iron based  MNPs  (e.g. spin iron oxide, iron)  with size below \n10 nm (the most studied case for biomedical applications) this is l argely verified for fields below \n10 kGauss. Thus, in field range considered in this study (hundreds of gausses)  a MNP with mass \nmNP in a dynamic fluid with fixed flow rate, is subjected to a magnetic force (𝐹��)𝑥,𝑦 whose in -\nplane components (𝐹𝑀)𝑥 and (𝐹𝑀)𝑦 are defined here below : \n \n(𝐹𝑀)𝑥,𝑦=𝑚𝑁𝑃𝜒𝑁𝑃𝐵𝑧(𝑥,𝑦,𝑧̅)∙[𝑔𝑟𝑎𝑑(𝐵𝑧(𝑥,𝑦,𝑧̅))]𝑥,𝑦 (1) \n \nwhere , 𝐵𝑧(𝑥,𝑦,𝑧̅) is the value of the  z component of the magnetic induction at a fixed distance  \n𝑧̅ from the magnet and 𝑔𝑟𝑎𝑑(𝐵𝑧(𝑥,𝑦,𝑧̅))=√[𝑔𝑟𝑎𝑑(𝐵𝑧(𝑥,𝑦,𝑧̅))]𝑥2+[𝑔𝑟𝑎𝑑(𝐵𝑧(𝑥,𝑦,𝑧̅))]𝑦2 is \nthe intensity of in-plane gradient of 𝐵𝑧  measured at a fixed 𝑧̅ . In the rest of the article we will \nrefer simply to 𝐵∙𝑔𝑟𝑎𝑑(𝐵). Typically , the diamagnetic contribution of the fluid can be \nneglected , as well as that ascribed to the organic coating . So, it is evident that the request of a \nstrong force acting on a MNP needs a large value of the product ∙𝑔𝑟𝑎𝑑(𝐵). This means that \nnot only B must be the maximum at the 𝑧̅ distance from the target, but also i ts gradient along \nx and y directions . For this very reason , in the following paragraphs we will present the \nmeasurements of the induction field, the in -plane gradient and their products for several \nmagnet configurations.  \n \n3.2. Single  cylindrical  magnet  \nThe measurements of the magnetic induction obtained , at 𝑧̅ = 4 mm,  for a single cylindrical  \nmagnet are reported in Fig. 2A. In Fig. 2B, we report the calculated gradient for each step  (1 mm)  \nof the 2D surface map  displayed  in Fig. 2A. The l ength of arrows is proportional to the \nmagnitude of gradient while the orientation of arrows shows the direction to increase of the \nmagnetic induction magnitude. We subsequently calculated  𝐵∙𝑔𝑟𝑎𝑑(𝐵) map as  presented in \nFig.2C, where the color grade rep resents the intensity of the magnetic force acting on MNPs . \nIn particular, t he dark red circular crown  represent s the area where  the maximum  \naccumulation of MNPs  is expected to be localized . \n \n02468100246810B, GY (mm)\nX (mm)-300-250-200-150-100-50050100150200250300\n \na) \n0 2 4 6 8 100246810\n Y (mm)\nX (mm)0102030405060grad(B), G/mm \nb) \n0 2 4 6 8 100246810Y (mm)\nX (mm)0.02.0x1034.0x1036.0x1038.0x1031.0x104Bgrad(B), G2/mm \nc) \nFigure 2. Map of magnetic induction (a), the gradient of magnetic induction (b) and value of the product \n(c) for a single cylindrical magnet at the surface under the magnet ( 𝑧̅ = 4 mm).  The black line represents \nthe magnet 's real dimension and shape.  \n \n3.3. Configuration s of three and four cylindrical  magnets  \n3.3.1. Three cylindrical magnets  \nAn i mprovement of  the attracting force  acting on the MNP (see eq. 1)  can be achieved \nincreasing the product  𝐵∙𝑔𝑟𝑎𝑑(𝐵) playing with the  spatial magnetic induction distribution \ngenerated by  several magnets  such as the three -magnet set -up. This configuration has also the \nadvantage to increase the surface area on which such product is effective . The 2D-map of· for \nthe present configuration at fix ed h eight 𝑧̅ = 4 mm is presented in Fig. 3A. To estimate the \nspatial distribution of the product, we calculated 𝐵∙𝑔𝑟𝑎𝑑(𝐵) averaged over a 10 ×10 mm2 \nRegion Of Interest (ROI). The value of the product averaged over ROI and the maximum value \nof the product achieved in a single point are reported in Table 1 . In order to maximize the \ninduction gradient , we considered two magnets UP and one DOWN  set-up as well. The \nproduct 𝐵∙𝑔𝑟𝑎𝑑(𝐵)   was less satisfactory for this configuration and therefore  they are not \npresented here.  \n \n3.3.2.  Four cylindrical magnets  \nIn Fig. 3B, the magnetic pattern  (at 𝑧̅  = 4 mm)  of the system of four magnets distributed in \ncorners of the Teflon square  is presented. Also , in this case, an alternative configuration, formed by coupled UP and DOWN magnets along the diagonals of the square  was taken into \naccount . As in the  previous case of t hree magnets system, a reduced value of the product            \n𝐵∙𝑔𝑟𝑎𝑑(𝐵)  were obtained  (not shown) . Notable that the symmetrical configuration of \nmagnets demonstrates a slightly asymmetric pattern of the magnetic gradient probably \nbecause of the different properties of each  individual magnet s even from the same bunch.  \n \nA comparison of the  triangular and squared set -ups demonstrate s that 𝐵∙𝑔𝑟𝑎𝑑(𝐵)  is higher \nover one single spot  in the former , while in the latter it results distributed over longer \ndistance s, thus allow ing to achieve a  higher area where the magnetic force can  attract  MNPs . \nAlso,  for the four magnets configuration , the best averaged product over ROI and the \nmaximum value in a single point are reported in the same Table 1.  \n \n0246810121416182002468101214161820Y (mm)\nX (mm)0.05.0x1031.0x1041.5x1042.0x1042.5x1043.0x1043.5x1044.0x104Bgrad(B), G2/mm\n \na) \n0246810121416182002468101214161820Y (mm)\nX (mm)0.05.0x1031.0x1041.5x1042.0x1042.5x1043.0x1043.5x1044.0x104Bgrad(B), G2/mm \nb) \nFigure 3. Map of the product for systems of three (a) and four (b) cylindrical magnets all up at the surface \nunder the magnet ( 𝑧̅ = 4 mm).  The black lines represent the magnet 's real dimensions and  shapes.  \n \n3.4. Configurations of mixed systems of cylindrical and cubic  magnets  \n \nIn the  above  configurations of small cylindrical magnets (3 3 mm), the contribution of \n𝑔𝑟𝑎𝑑(𝐵) to the magnetic  force of eq.(1) is small  and therefore  FM is mainly due to the magnetic \ninduction B. Therefore, for further improvement of FM, a combination of cylindrical and  cubic \nmagnets was designed and realized : the results concerning the level 𝑧̅ = 4 mm are reported i n \nFig.4. We measured all the possible configurations  resulting by  changing the magnet polarity  \nat that fixed distance . In Fig. 4A the 𝐵∙𝑔𝑟𝑎𝑑(𝐵)  2D-map is plotted when the polarity of cubic \nmagnets was opposite to cylindrical ones. In such a case opposite oriented magnets suppress  \nthe induction of each other : as a result , this configuration exhibits a low value of  B and \nconsequently  𝐵∙𝑔𝑟𝑎𝑑(𝐵). Turning one of the sextets of cylindrical magnets, a higher value of \nFM was achieved (Fig. 4B). A similar good result was obtained also with  all magnets polariz ed \nin the same direction  (Fig. 4C). But in this case, in  the center of the system, B has a plateau and \nthus 𝑔𝑟𝑎𝑑(𝐵) assumes very low intensit ies thus decreasing consequently FM as well. We note \nthat the value of ROI parameter was  smaller than what was obtained  when  one of the sextets \nmagnet groups was turned opposite ( Fig.4B and Table 1).  \n   \nFigure 4. 2D-Map of B (upper panel)  and 𝐵∙𝑔𝑟𝑎𝑑(𝐵) (bottom panel)  at 𝑧̅ = 4 mm  for systems of mixed \ncubic and cylindrical magnets with the opposite polarity of cubic magnets to cylindrical  ones  (a), one of \nthe sextets of cylindrical magnets  with  opposite polarity (b) and all magnets polarized in the same direction \n(c). The black lin es represent the magnet 's real  dimensions and  shapes.  \n \nTable 1 . Maximum  value of 𝐵∙𝑔𝑟𝑎𝑑(𝐵) and its average value over ROI at 𝑧̅ = 4 mm.  \n \nConfiguration  𝑩∙𝐠𝐫𝐚𝐝(𝐁), G2/mm  \nMaximum value in a \nsingle point  Average over ROI of \n10×10 mm2 square  \n3 cylinder magnets (Fig. 3A) 3.7 104 1.7 104 \n4 cylinder magnets (Fig. 3B) 2.7 104 2.0 104 \nMixed 2UP/2DN (Fig. 4A) 3.7 104 2.9 104 \nMixed 3UP/1DN (Fig. 4B) 1.2 105 9.7 104 \nMixed 4UP (Fig. 4C) 1.6 105 8.1 104 \n \nFrom the data reported in Table 1, adopting the configuration of the magnets  reported in \nFig.4B, the average value of the product 𝐵∙𝑔𝑟𝑎𝑑(𝐵) =9.7×104 G2/mm can be obtained over \nan area of 1010 mm2. Mouse heart dimensions are about 10 4.2 mm2 [35], while mouse liver \npresents a larger volume, around 22 cm3 [36]. These values  suggest that the above magnets \nconfiguration s can be successfully used for in vivo experiments of magnetic drug delivery and \ntargeting.  \nConclusions  \nDespite that the physics behind the propagation of magnetic fields in the space is well known, \nthe way of the shaping of magnets or their systems to satisfy the requirements of certain bi o-\napplication is relatively missed in literature. Here the new insights on the magnetic profile \nproduced by different configurations of small permanent magnets are presented. The choice \nof dimensions and the adopted geometries are strictly related to the dim ensions of the mice \norgans typically used in in vivo  experiments. Our results give evidence that  the magnetic force \nis maximized by a subtle balance between the polarity configuration and the geometric shape \nof the used magnets. In particular, we found tha t, over the typical mice organ size, the average \nproduct of 𝐵∙𝑔𝑟𝑎𝑑(𝐵) attains 105 G2/mm with a mixed magnets configuration with one of the \ncylindrical magnet groups with opposite polarity respect to the others.  In vivo  experiments are \nin progress in or der to validate our experimental achievements.   \n \nDeclaration of Competing Interest  \nThe authors declare that they have no known competing financial interests or personal \nrelationships that could have appeared to influence the work reported in this paper.  \n \nCRediT authorship contribution statement  \nAlexander Omelyanchik: Visualization, Investigation, Writing - Original draft preparation; \nGianrico Lamura: Investigation, Methodology, Conceptualization, Writing - Original draft \npreparation, Writing - Reviewing and Editing, Data curation; Davide Peddis: Writing - \nReviewing and Editing, Conceptualization, Methodology, Data curation, Supervision; Fabio \nCanepa: Writing - Reviewing and Editing, Conceptualization, Methodology, Data curation, \nSupervision.  \n  REFERENCES  \n[1] M.A. Agotegaray, V.L. 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Hallas\nDepartment of Physics and Astronomy and Quantum Matter Institute,\nUniversity of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada\n(Dated: May 5, 2021)\nPyrochlore lattices, which are found in two important classes of materials { the A2B2X7pyrochlore\nfamily and the AB2X4spinel family { are the quintessential 3-dimensional frustrated lattice archi-\ntecture. While historically oxides ( X= O) have played the starring role in this \feld, the past decade\nhas seen materials synthesis breakthroughs that have lead to the emergence of \ruoride ( X= F) and\nchalcogenide ( X= S, Se) pyrochlore lattice materials. In this Research Update, we summarize recent\nprogress in understanding the magnetically frustrated ground states in three families of non-oxide\npyrochlore lattice materials: (i) 3 d-transition metal \ruoride pyrochlores, (ii) rare earth chalcogenide\nspinels, and (iii) chromium chalcogenide spinels with a breathing pyrochlore lattice. We highlight\nhow the change of anion can modify the single ion spin anisotropy due to crystal electric \feld e\u000bects,\nstabilize the incorporation of new magnetic elements, and dramatically alter the exchange pathways\nand thereby lead to new magnetic ground states. We also consider a range of future directions {\nmaterials with the potential to de\fne the next decade of research in frustrated magnetism.\nI. INTRODUCTION\nIn the frustrated magnetism community, interest in\npyrochlore oxides \rourished following the 1997 discovery\nof the spin ice state in Ho 2Ti2O7[1,2], and shortly there-\nafter in 1999, the discovery of a putative spin liquid state\nin Tb 2Ti2O7[3]. Since that time, a detailed characteriza-\ntion of the magnetic ground state of almost every known\noxide pyrochlore has been undertaken [ 4], including in\nrecent years, pyrochlores that can only be formed under\nhigh pressure conditions [ 5]. Furthermore, the magnetic\nground states of these pyrochlores have been poked and\nprodded in every imaginable way, from extreme applied\npressure [ 6] to high magnetic \felds [ 7] to exacting studies\nof the e\u000bect of o\u000b-stoichiometry [ 8] and most recently\ntheir growth as epitaxial thin \flms [ 9]. This enormous\nbody of research has revealed that the magnetic ground\nstates of many oxide pyrochlores are exceedingly fragile,\nwith very few materials sharing the same magnetic phe-\nnomenology. The origin of these diverse magnetic states\nis, \frst and foremost, the frustrated lattice architecture\nof the corner-sharing tetrahedral pyrochlore lattice but\nthe second most important factor is the nature of the\nmagnetic ion itself, both its single ion properties and its\ninteractions with neighbors.\nPyrochlore oxides, with the general formula A2B2O7,\ntake their name from a naturally occurring mineral. This\nname is also bestowed upon the lattice of corner-sharing\ntetrahedra, known as the pyrochlore lattice, which both\ntheAandBatoms form in the eponymous family of\nmaterials. Pyrochlore oxides have extensive chemical ver-\nsatility [ 10], which is another factor that contributes to\nthe diversity in their magnetic states. The spinel oxides,\nAB2O4, which also take their name from a naturally occur-\nring mineral, provide a second structural platform to study\nfrustrated magnetism on the pyrochlore lattice. However,\nin the spinel structure, only Bforms a pyrochlore lattice\nwhileAmakes up a diamond lattice. Like the pyrochlore\noxides, spinel oxides have a signi\fcant chemical versatility,yielding a large number of compounds [ 11,12]. However,\nmany of these materials have magnetic Asites, which\nspoils the simple frustration of the pyrochlore Bsite.\nNonetheless, there are many compounds in the spinel\nfamily with geometric-frustration driven physics due to\nboth orbital and magnetic degrees of freedom [13, 14].\nWhile oxide pyrochlore lattice materials have domi-\nnated the \feld for multiple decades, recent advances in\nmaterials synthesis have brought several non-oxide fam-\nilies to the forefront, including \ruoride pyrochlores and\nchalcogenide spinels. The chemical and structural modi-\n\fcations that accompany this replacement of the anion\nin turn leads to di\u000berent local environments and hence\ndi\u000berent single ion anisotropies, di\u000berent relative energy\nscales, and new magnetic ions on the pyrochlore lattice,\nall of which have lead to signi\fcant new physics.\nIn this Research Update we outline what's been ac-\ncomplished with nonoxide pyrochlores, primarily over the\npast \fve years, and discuss challenges and opportunities\nfor the future. In Section II, we introduce the \ruoride\npyrochlores which have the unique characteristic of host-\ning late 3dmagnetic transition metals, which cannot be\nachieved in oxide pyrochlores. In Section III, we discuss\nthe rare earth chalcogenide spinels, which have distinct\nproperties from their oxide pyrochlore counterparts due\nto the di\u000berent local environment and the reduced crystal\n\feld energy scale. In Section IV, we turn to the chromium\nchalcogenide spinels with an emphasis on the breathing\npyrochlore spinels that have alternating small and large\ntetrahedra, breaking the degeneracy of the pristine py-\nrochlore lattice but unlocking new, exotic magnetic states.\nThe magnetic properties of the materials discussed in\nSec. II - IV are summarized in Table II. Finally, in Sec-\ntion V, we turn to future directions: known materials\nthat as of now have been the subjects of relatively few\ninvestigations or places where altogether unknown, new\nmaterials might be lurking.arXiv:2012.10060v2  [cond-mat.str-el]  3 May 20212\nA= Cd, Mg\nA =Li, Cu, Ag\nAA’Cr4X8\nAR2X4\nA’= Al, Ga, In\nCr\nX= S, SeR = Dy –Yb\nX= S, SeA = Na\nA’ = Ca, Sr\nB = Mn –Ni\nF(a) (b) (c) AA’B2F7\nFIG. 1. The crystal structures of three families of non-oxide materials with pyrochlore (corner sharing tetrahedral) sublattices.\n(a) The \ruoride pyrochlores, AA0B2F7, where the B-site pyrochlore sublattice, highlighted in yellow, is occupied by a 3 d\ntransition metal. (b) The rare earth chalcogenide spinels, AR2X4, where the R-site pyrochlore sublattice, highlighted in orange,\nis occupied by a rare earth. (c) The breathing pyrochlore chromium spinels, AA0Cr4X8, where the Cr pyrochlore sublattice has\nalternating small (red) and large (teal) tetrahedra. The top set of panels show the unit cell for each structure and the lower set\nof panels show the local anion environment at the magnetic site.\nTABLE I. Crystal structure details, including the Wycko\u000b site\n(Wyck.), point group symmetry (P.G.), atomic coordinates\n(x,y,z), and oxidation state (Ox.), for the \ruoride pyrochlores,\nAA0B2F7, the chalcogenide spinels, AR2X4, and the breathing\npyrochlore chalcogenides, AA0Cr4X8.\nFluoride Pyrochlores AA0B2F7\nSpace Group Fd \u00163m\nWyck. P.G. x y z Ox.\nA=A016d D 3d 0.5 0.5 0.5 +1/+2\nB 16c D 3d 0 0 0 +2\nF1 8 b T d 0.375 0.375 0.375 \u00001\nF2 48 f C 2vx 0.125 0.125 \u00001\nChalcogenide Spinels AR2X4\nSpace Group Fd \u00163m\nWyck. P.G. x y z Ox.\nA 8a T d 0.125 0.125 0.125 +2\nR 16d D 3d 0.5 0.5 0.5 +3\nX 32e C 3vx x x \u00002\nBreathing Pyrochlores AA0Cr4X8\nSpace Group F \u001643m\nWyck. P.G. x y z Ox.\nA 4c T d 0.25 0.25 0.25 +1\nA04a T d 0 0 0 +3\nCr 16 e C 3vx1x1x1 +3\nX1 16e C 3vx2x2x2 \u00002\nX2 16e C 3vx3x3x3 \u00002II. FLUORIDE PYROCHLORES\nWhile the late 3 dtransition metals, from Fe to Cu, are\namong the most important elements in studies of quantum\nmagnetism, one may lament that they are entirely un-\nrepresented among the pyrochlore oxides, A2B2O7. The\nreason for this is that charge neutrality and structural\ntolerance factors cannot be simultaneously satis\fed for\nany combination of AandBcations that includes a late\n3dtransition metal [ 10]. In recent years, this obstacle\nhas been sidestepped by replacing divalent oxygen, O2\u0000\nwith monovalent \ruorine, F\u0000, allowing pyrochlores with\nB= Mn2+, Fe2+, Co2+and Ni2+to be synthesized for\nthe \frst time Fig. 3(a)) [ 15{17]. However, this triumph\ncomes at a cost: overall charge neutrality requires an\ne\u000bective 1.5+ valence state on the A-site, which neces-\nsitates the deliberate introduction of chemical disorder.\nThe resulting chemical formula of the pyrochlore \ruorides\nisAA0B2F7, whereA= Na+andA0= Ca2+, Sr2+are\nrespectively alkali (Group 1) and alkaline earth (Group\n2) metals in equal proportions, randomly distributed over\ntheAsublattice, as shown in Figure 1(a) with crystallo-\ngraphic details provided in Table I.\nRelated \ruoride pyrochlores in polycrystalline form\nwere \frst reported in 1970 [ 18], but investigations of\ntheir magnetism have only recently begun following their\ngrowth as large single crystals using the \roating zone\ntechnique [ 15]. In contrast to the much studied rare earth\npyrochlores, where the typical energy scale of the exchange3\ninteractions is 1 meV or smaller, the exchange interactions\nfor magnets based on 3 dtransition metals are an order of\nmagnitude or more larger. Therefore, whereas studying\nthe correlated ground states of rare earth pyrochlores\nrequires extremely low temperatures, well below 1 K, the\ncorrelated states of the \ruoride pyrochlores can be stud-\nied at more easily accessible temperatures. All \ruoride\npyrochlores that have been characterized to date share\ntwo key attributes: (i) they are highly frustrated and (ii)\nthat frustration is quenched at low temperatures by a\nspin freezing transition due to exchange disorder wrought\nby the intrinsic chemical disorder. The spin freezing tran-\nsitions occur at a temperature, Tfcommensurate with\nthe energy scale of the bond disorder, which is given by\n\u0001=kB=Tfp\n3=8\u0019O(100) K [19], while correlations onset\nat signi\fcantly higher temperatures, j\u0012CWj\u0019O(102) K.\nThus, much of the experimental e\u000bort on \ruoride py-\nrochlores has centered on characterizing their correlated\nstates below their Curie-Weiss temperatures, j\u0012CWj, and\nabove their spin freezing temperatures, Tf.\nA. Cobalt Fluoride Pyrochlores\nThe cobalt-based \ruoride pyrochlore, NaCaCo 2F7, was\n\frst reported in 2014 [ 15] and its sister compound,\nNaSrCo 2F7, was subsequently reported in 2015 [ 20].\nWhile more experimental studies have focused on the for-\nmer compound, all available data suggests that the proper-\nties of these two compounds are largely indistinguishable.\nIt is interesting to note that this strongly contrasts with\nrare earth oxide pyrochlores, where the choice of the\nnon-magnetic B-site can dramatically modify the mag-\nnetic ground state [ 4]. Both NaCaCo 2F7(\u0012CW=\u0000140 K,\nTf= 2:4 K) and NaSrCo 2F7(\u0012CW=\u0000127 K,Tf= 3:0 K)\nhave strongly frustrated antiferromagnetic interactions,\nwith frustration indices of 58 and 42, respectively [ 15,20].\nThe spin freezing transitions in these materials, which are\nmarked by a frequency dependent cusp in ac susceptibility,\noccur at a temperature commensurate with the strength\nof the chemical disorder. Thus, the slight increase in the\nfreezing temperature of NaSrCo 2F7can be understood as\noriginating from enhanced bond disorder due to the larger\ndi\u000berence in ionic radii between Na ( r= 1:18\u0017A) and Sr\n(r= 1:26\u0017A), as compared to Na and Ca ( r= 1:12\u0017A).\nThe single ion properties of the Co2+ions in these com-\npounds are quite interesting in-and-of-themselves. This\nwas initially hinted by a Curie-Weiss analysis of their\nmagnetic susceptibility data, which revealed a param-\nagnetic moment around 6 \u0016B/Co2+[15,20], dramati-\ncally enhanced from the spin-only value for high spin\nCo2+in a pseudo-octahedral environment ( S=3=2,\n\u0016calc= 3:87\u0016B). Instead, the observed moment is much\ncloser to the value expected with a full orbital contribution\n(J=9=2,\u0016calc= 6:63\u0016B). A subsequent inelastic neutron\nscattering study of the crystal \feld excitations in both\nNaCaCo 2F7and NaSrCo 2F7con\frmed the intermediate\nspin-orbit coupling, which also leads to strong deviationsfrom Heisenberg type moments [ 21]. Instead, the Co2+\nmoments in these materials have relatively strong XY\nanisotropy, preferentially lying in the plane perpendicular\nto the local [111] direction, which is the axis that con-\nnects adjacent tetrahedra. This type of anisotropy is also\nfound in the erbium and ytterbium families of rare earth\noxide pyrochlores [ 22], as well as possibly the dysprosium\nchalcogenide spinels [ 23]. An open question surrounds the\ncomplete absence of anisotropy in magnetization measure-\nments for NaCaCo 2F7[24], which does not conform with\nthe expected behavior of an XY antiferromagnet [25].\nAlthough the ultimate ground states of NaCaCo 2F7\nand NaSrCo 2F7are frozen glass-like states, they are not\ncharacterized by random spin con\fgurations. Correlations\nwith a typical length scale of a single tetrahedron set in\nat temperatures as high as 200 K and inter-tetrahedra\ncorrelations develop around 50 K, as observed by neutron\nscattering, NMR, and ESR measurements [ 24,26,27].\nElastic neutron scattering measurements below the spin\nfreezing transition, shown in Fig. 2(a), show that the\nfrozen state is made up of short-range ordered clusters\nwith antiferromagnetic XY spin con\fgurations (\u0000 5in the\nlanguage of irreducible representations, shown in Fig. 5\n(b)) with a correlation length of 16 \u0017A [28]. Two unique,\nnearly degenerate ordered states exist within \u0000 5, and\nthe spin freezing transition preempts the selection of a\nunique ordered state in the cobalt pyrochlores. While\nthe low energy spin excitations are consistent with a\ncontinuous degree of rotational symmetry in the local XY\nplane for these \u0000 5clusters, at higher energies a di\u000berent\ntype of spin excitation emerges. This higher energy spin\nexcitation can be rather accurately captured by collinear\nantiferromagnetic correlations on individual tetrahedra,\nbreaking the local XY anisotropy [ 28]. The presence of\nthese two distinct excitations is suggestive of a frustration\nbetween intra- and inter-tetrahedral correlations.\nAnother cobalt-based pyrochlore lattice material is re-\nalized in the carbonate Na 3Co(CO 3)2Cl, which forms\nin theFd\u00163space group [ 29]. Unlike the \ruorides,\nNa3Co(CO 3)2Cl is a fully site ordered material with no\ninherent disorder. The energy scale of this material's spin\ncorrelations, which is roughly parameterized by the Curie-\nWeiss temperature \u0012CW=\u000034 K [ 30], is reduced from the\n\ruorides by nearly an order of magnitude. This is at least\npartially attributable to the larger Co-Co bond distance,\n4.9\u0017Ain Na 3Co(CO 3)2Cl as compared to 3.7 \u0017A in the\n\ruorides. This material has a particularly complex phase\nbehavior, \frst entering a short-range correlated state at\n17 K, exhibiting a spin-glass like transition at Tf= 4:5 K,\nbefore \fnally proceeding through an apparent long range\nordering transition at TN= 1:5 K [ 30]. The ordered state\nis an all-in/all-out antiferromagnet, shown in Fig. 5 (a),\nwhich implies the Co2+moments in this material have\nIsing local anisotropy unlike the XY anisotropy observed\nin the \ruorides. Future studies should con\frm that this\ncomplex phase behavior is intrinsic and not the result of\nsample inhomogeneity or phase separation.4\nFIG. 2. Elastic neutron scattering measurements of (a)\nNaCaCo 2F7and (b) NaCaNi 2F7below their spin freezing tran-\nsitions,Tf, reproduced from Ref. [ 28] and [ 31], respectively.\nThe elastic scattering of NaCaCo 2F7consists of broadened\nmagnetic Bragg peaks and zigzag di\u000buse scattering. The for-\nmer originates from short range antiferromagnetic XY order,\nand the latter originates from low energy excitations in the XY\nplane. The di\u000buse scattering for NaCaNi 2F7is well described\nby the Heisenberg Hamiltonian. Sharp pinch points, for exam-\nple near (1 1 3), indicate there is zero average magnetization\nper tetrahedra.\nB. Nickel Fluoride Pyrochlores\nThe nickel \ruoride pyrochlore NaCaNi 2F7(\u0012CW=\n\u0000129 K,Tf= 3:6 K) is another highly frustrated an-\ntiferromagnet, with a frustration index of 36 [ 16]. The\ne\u000bective moment derived from Curie-Weiss \fts, 3 :7\u0016B,\nis larger than what is typically observed for the spin-\nonly moment of Ni2+(S= 1,\u0016calc= 2:83\u0016B) but falls\nsigni\fcantly short of what would be expected for a full\norbital contribution ( J= 1,\u0016calc= 5:59\u0016B) [16]. Thus,\nspin-orbit coupling, while not negligible, is less signi\fcant\nthan in the case of the cobalt analogs, placing NaCaNi 2F7closer to the Heisenberg limit. This picture is further\nvalidated by polarized neutron scattering measurements,\nwhere it was observed that the scattering in the spin \rip\nand non-spin \rip channels is nearly identical, indicating\nthe interactions are highly isotropic [31].\nWhile it is true that bond disorder induces a spin freez-\ning in each of the \ruoride pyrochlores, irrespective of the\nmagnetic ion involved, their magnetic correlations do in\nfact di\u000ber substantially. In the case of the cobalt variants,\nspin freezing appears to preempt long-range magnetic\norder, while in the case of NaCaNi 2F7, there is no in-\ndication that the compound is close to magnetic order.\nElastic neutron scattering measurements in the frozen\nstate, shown in Fig. 2(b), reveal structured di\u000buse scat-\ntering with a correlation length of 6 \u0017A [31], close to the\nnext nearest neighbor bond distance. Fits to the neutron\nscattering data show that NaCaNi 2F7is very close to\nan exact realization of the nearest neighbor Heisenberg\nantiferromagnet with S= 1, where exchange anisotropy,\nnext nearest neighbor exchange, and bond disorder all rep-\nresent small corrections [ 31]. Sharp pinch point features,\nas famously observed in the dipolar spin ices, indicate\nthat there is zero net magnetization on each tetrahedron\n(Fig. 2(b)) [ 31,32]. This observation in particular is sug-\ngestive of the classical Heisenberg pyrochlore spin liquid\n\frst studied by Villain [ 33], where the ground state man-\nifold is comprised of all states with zero magnetization\nper tetrahedron, yielding a macroscopic degeneracy. This\nmay be consistent with the residual entropy of 0 :176R\nper mol Ni observed in NaCaNi 2F7, which is only 16% of\nthe expected entropy release for Rln (3) [31].\nDespite ultimately undergoing a spin freezing transi-\ntion, quantum e\u000bects appear to be quite signi\fcant in\nNaCaNi 2F7. Neutron scattering measurements show that\nat low temperatures, 90% of the magnetic scattering oc-\ncurs inelastically, dramatically exceeding the semiclassical\nexpectation of 50% for S= 1 [ 31]. In contrast, the cobalt\nvariants agree well with this semiclassical picture where\nthe observed 30% elastic approaches the expected 33%\nforJ=1=2[28]. Further evidence for the importance\nof quantum \ructuations in NaCaNi 2F7comes from \u0016SR\nmeasurements, which show the observed static \feld in\nthe frozen state is an order of magnitude smaller than\nexpected, perhaps signalling that the Ni moments are not\nfully frozen [34].\nC. Manganese, Iron, and Future Prospects\nThe \ruoride pyrochlore family is rounded out by a trio\nof compounds, NaSrMn 2F7(\u0012CW=\u000090 K,Tf= 2:5 K),\nNaSrFe 2F7(\u0012CW=\u000098 K,Tf= 3:7 K), and NaCaFe 2F7\n(\u0012CW=\u000073 K,Tf= 3:9 K) [ 17]. Following the familial\ntrend, all three are highly frustrated antiferromagnets\nwith frustration indices of 36, 26, and 19, respectively. In\nall three, the calculated paramagnetic moment is slightly\nlarger than what would be expected for high spin 3 d5\nMn2+(S=5=2) and high spin 3 d6Fe2+(S= 2), re-5\nH\nLi\nNa\nK\nRb\nCsBe\nMg\nCa\nSr\nBaSc\nY\nLaTi\nZr\nHfV\nNb\nTaCr\nMo\nWMn\nTc\nReFe\nRu\nOsCo\nRh\nIrNi\nPd\nPtCu\nAg\nAuZn\nCd\nHgB\nAl\nGa\nIn\nTlC\nSi\nGe\nSn\nPbN\nP\nAs\nSb\nBiO\nS\nSe\nTe\nPoF\nCl\nBr\nI\nAtNe\nAr\nKr\nXe\nRnHe\nCe\nThPr\nPaNd\nUPm\nNpSm\nPuEu\nAmGd\nCmTb\nBkDy\nCfHo\nEsEr\nFmTm\nMdYb\nNoLu\nLr(A3+)2(B4+)2O7\n(A2+)2(B5+)2O7\n(A1.5+)2(B2+)2F7H\nLi\nNa\nK\nRb\nCsBe\nMg\nCa\nSr\nBaSc\nY\nLaTi\nZr\nHfV\nNb\nTaCr\nMo\nWMn\nTc\nReFe\nRu\nOsCo\nRh\nIrNi\nPd\nPtCu\nAg\nAuZn\nCd\nHgB\nAl\nGa\nIn\nTlC\nSi\nGe\nSn\nPbN\nP\nAs\nSb\nBiO\nS\nSe\nTe\nPoF\nCl\nBr\nI\nAtNe\nAr\nKr\nXe\nRnHe\nCe\nThPr\nPaNd\nUPm\nNpSm\nPuEu\nAmGd\nCmTb\nBkDy\nCfHo\nEsEr\nFmTm\nMdYb\nNoLu\nLr(A2+)(B3+)2O4\n(A2+)(B3+)2(S/Se) 4(a) (b) Pyrochlore s Spinels\nFIG. 3. The various metals that are known to form (a) oxide and \ruoride pyrochlores ( A3+\n2B4+\n2O7,A2+\n2B5+\n2O7, andA1:5+\n2B2+\n2F7)\nand (b) oxide and chalcogenide spinel ( A2+B3+\n2O4andA2+B3+\n2(S/Se) 4). Of particular note, pyrochlores, in panel (a), with\nlate 3dtransition metals can only be formed as \ruorides, and spinels, in panel (b), with rare earth ions can only be formed as\nsul\fdes or selenides.\nspectively [ 17]. Similar to Ni2+(S= 1), these three\ncompounds are all likely good realizations of Heisenberg\nspins on the pyrochlore lattice. However, due to their\nmore classical nature (larger Svalues) they will make\nfor an interesting comparison with NaCaNi 2F7in terms\nof how well they conform to the classical spin liquid\nstate expected for an antiferromagnetic Heisenberg py-\nrochlore [ 35]. To that end, detailed neutron spectroscopic\nstudies on these Mn and Fe variants will be of the utmost\ninterest. One might also expect that it may be possible to\nextend this series to include other transition metal ions;\nwhile the strong Jahn-Teller e\u000bect in copper renders it an\nunlikely candidate, chromium may be a promising avenue\nto explore.\nTwo related Mn and Fe based compounds that bear\nmentioning here are Na 3Mn(CO 3)2Cl and FeF 3, based\non Mn2+and Fe3+, respectively. These two ions have\nthe same electronic con\fguration, 3 d5(S=5=2), and\nboth compounds are good realizations of the classical\nHeisenberg pyrochlore antiferromagnet. Na 3Mn(CO 3)2Cl\nshares the same crystal structure as the aforementioned\nNa3Co(CO 3)2Cl and, unlike the \ruorides, does not pos-\nsess intrinsic disorder. It does not magnetically order\nor undergo a spin freezing transition down to at least\n0.5 K, which when coupled with a Curie-Weiss tempera-\nture of\u000041 K yields a large frustration index of at least\nf= 80 [ 36]. Below 1 K, Na 3Mn(CO 3)2Cl exhibits a\nrather sharp increase in its heat capacity that continues\ndown to the lowest measured temperatures, 0.5 K, and\nthe nature of this feature remains unresolved [36].\nFinally, FeF 3with space group Fd\u00163mis a unique ma-\nterial with no isomorphic compounds. The pyrochlore\nlattice of Fe3+is built from a network of corner shar-\ning FeF 6octahedra [ 37]. Compared to other materials\ndiscussed in this section, all of which are relatively re-\ncently characterized, magnetic studies of FeF 3date backto 1986 [ 37,38]. This material ultimately magnetically\norders close to 20 K in an antiferromagnetic all-in/all-out\nstructure (Fig. 5 (a))[ 38,39]. However, intense frustration\nis evidenced by a lack of Curie-Weiss behavior in suscepti-\nbility up to at least 300 K and structured magnetic di\u000buse\nscattering up to at least 100 K [ 38,40]. Further studies of\nFeF 3and Na 3Mn(CO 3)2Cl would be facilitated by their\ngrowth in large single crystal form, which represents a\nsigni\fcant challenge in both cases.\nIII. RARE EARTH CHALCOGENIDE SPINELS\nMuch like structural tolerance factors forbid the forma-\ntion of oxide pyrochlores with late 3 dtransition metals,\nthe large ionic radius of the 4 frare earth elements does\nnot allow them to be accommodated by the oxide spinel\nstructure. Given the diversity of magnetic states that can\nbe unlocked in the rare earth R2B2O7pyrochlores sim-\nply through replacement of the non-magnetic B-site [ 41],\nthe more signi\fcant shift to the spinel structure is an\nimportant prospect to search for new and exotic magnetic\nstates. It is only through the replacement of oxygen by\na larger chalcogen, sulfur or selenium, that rare earth\nelements can be incorporated into the spinel structure\n(Fig. 3(b)) { and even then, it is only the heaviest rare\nearths, which have the smallest ionic radii due to the\nlanthanide contraction. In this family, compounds of the\nformAR2X4, withX= S and Se, are stable for A= Cd\nandR= Dy\u0000Yb, [ 42] as well as for the slightly larger\nA= Mg, with R= Ho\u0000Yb [43], \frst reported in 1964\nand 1965, respectively. Remarkably, replacing the Asite\nby any larger alkaline earth metal (Ca, Sr, or Ba) gives an\nassortment of orthorhombic and rhombohedral structure\ntypes [43].\nMany of the distinctive properties of the rare earth6\nchalcogenide spinels are directly attributable to the mod-\ni\fed local environment. In this structure, with crystal-\nlographic details provided in Table I, the rare earth sits\nin a distorted octahedral environment (Fig. 1(b)) as op-\nposed to the pseudo-cubic coordination of the rare earth\nsite in oxide pyrochlores. This leads, in many cases, to\nan inversion of the single ion anisotropy ( e.g.ions that\nare Ising-like in the oxide pyrochlores are XY-like in the\nchalcogenide spinels). Furthermore, the coordinating an-\nions are between 10 and 30% farther away than in the\npyrochlore oxides and chalcogens are themselves less elec-\ntronegative than oxygen. Both of these factors serve to\nlower the energy scale of the crystal \feld splitting and\nthis leads to more complex ground state manifolds than\nthe well-isolated ground state doublets generally found in\nthe rare earth pyrochlore oxides. Taken altogether, the\nchalcogenide spinels exhibit entirely di\u000berent crystal elec-\ntric \feld (CEF) ground states than their oxide pyrochlore\ncounterparts, as highlighted with erbium in Fig. 4.\nThe crystal \feld states of the chalcogenide spinels were\n\frst studied in the 1970's [ 44{48]. The accuracy of these\nstudies was limited by their assumption of a perfect octa-\nhedral environment and also because magnetization data\nalone was used for the re\fnements of the CEF Hamil-\ntonian. More recent studies have used inelastic neutron\nscattering to directly measure the CEF levels and have\nfound that although the ligands form a near perfect octa-\nhedra, the \feld from the next nearest neighbor creates a\nsigni\fcant anisotropy along the local [111] directions [ 49{\n51]. The other major omission of the works prior to the\nnew millennia is that they did not take into account the\nimportance of geometric frustration, which explains their\n\fndings that none of these materials magnetically order\nabove 2 K [ 44{46,52]. The importance of geometric frus-\ntration in these chalcogenide spinels was not considered\nuntil 2005 [ 53]. The combination of strong anisotropy,\ncomplex crystal \feld ground states, and intense geomet-\nric magnetic frustration yields a collection of remarkable\nmagnetic states in the rare earth chalcogenide spinels.\nA. Erbium Chalcogenide Spinels\nIn the erbium chalcogenide spinels, AEr2X4(A= Cd\nor Mg and X= S or Se), the pyrochlore sublattice is\noccupied by Er3+, which is a Kramers ion with a total\nangular momentum of J=15=2and an expected param-\nagnetic moment of \u0016calc= 9:6\u0016B, which agrees well with\nexperiment [ 53]. While erbium-based oxide pyrochlores\nhave local XY anisotropy of varying strength [ 54], the\nchange in the local environment in the spinels yields per-\nfectly Ising moments [ 49,50,55], constrained to lie along\nthe local [111] directions (Fig. 4). However, the \frst ex-\ncited CEF levels are found at energies as low as 4 meV\nabove the ground state [ 49,50] and as a result, their\nin\ruence cannot be entirely disregarded. In particular,\ncoupling between the ground and excited states through\nso-called virtual CEF \ructuations becomes appreciableat these energy scales [ 56], which can disrupt the other-\nwise perfect Ising anisotropy. Studies thus far suggest\nthat theAEr2X4compounds, with varying AandXions,\nhave broadly similar properties, which are to \frst order\ndictated by the strong Ising anisotropy.\nThe earliest studies on the erbium spinels predicted that\nthese compound would order between 4 and 10 K, based\non the the onset of short range order seen in magnetiza-\ntion and M ossbauer spectroscopy measurements [52, 57].\nHowever, it wasn't until 2010 that the nature of the\nshort range order was understood [ 55]. The very large\ne\u000bective moments in the erbium spinels combined with\ntheir Ising anisotropy, gives rise to long range dipolar\ninteractions that produce a ferromagnetic coupling [ 58],\nsigni\fcantly larger in magnitude than the antiferromag-\nnetic near neighbour exchange [ 49,50]. Thus, the erbium\nspinels possess all the requisite ingredients for spin ice\nphysics and that is indeed what has been found. This\nwas \frst uncovered in CdEr 2Se4, where heat capacity\nmeasurements showed the absence of long-range magnetic\norder but the presence of a broad heat capacity anomaly\ncentered below 1 K [ 55]. The remnant entropy associated\nwith this heat capacity anomaly approaches the Paul-\ning value, associated with the macroscopic degeneracy\nof ice-rules con\fgurations [ 2,59]. The spin ice state of\nCdEr 2Se4ultimately drops out of thermal equilibrium,\nleading to a spin freezing transition at Tf= 0:6 K [ 55].\nA M ossbauer study on CdEr 2S4showed that the spin\n\ructuations are signi\fcantly slower than in the erbium\noxide pyrochlores [ 60], a likely consequence of the the\nlarger energy cost to \rip an Ising moment.\nThe spin ice states observed in the erbium spinels have\nintriguing di\u000berences with the well known holmium and\ndysprosium oxide pyrochlores, in which the spin ice state\nwas \frst discovered [ 61]. A careful analysis of the heat\ncapacity of MgEr 2Se4, as well as the previously pub-\nlished data in CdEr 2Se4, shows that, in fact, the remnant\nentropy is about 20% lower than the expected Pauling\nentropy of a spin ice [ 50]. In the same compound, Monte\nCarlo simulations with the dipolar spin ice model cap-\nture the qualitative features of the heat capacity and the\nmagnetic di\u000buse scattering [ 49], but fail to quantitatively\ndescribe the data [ 50]. AC susceptibility studies suggest\nthat the monopole quasiparticle excitations in the ground\nstate of CdEr 2Se4have three orders of magnitude higher\nmobility than in the prototypical spin ice Dy 2Ti2O7[49].\nThese results suggest that there are additional interac-\ntions beyond the normal dipolar ice model, allowing both\nfaster monopole dynamics and reducing the number of\nstates in the degenerate ground state manifold. It is likely\nthat these additional interactions stem from the much\nsmaller energy gap between ground state and \frst exited\nCEF levels [49, 50].7\nFIG. 4. The crystal \feld splitting of Er3+in the pyrochlore\noxide Er 2Ti2O7[54] compared to the chalcogenide spinels\nCdEr 2S4[49] and MgEr 2Se4[50]. Due to the reduction in anion\nelectronegativity, the overall magnitude of the crystal \feld\nsplitting decreases signi\fcantly going from X= O toX= S\nand Se. The change in the local environment also produces\nan inversion of the anisotropy: Er 3+in the pyrochlore oxide\nhas XY anisotropy while in the chalcogenide spinels it has\nIsing anisotropy (perpendicular and parallel to the local [111],\nrespectively).\nB. Ytterbium Chalcogenide Spinels\nThe ytterbium spinels, AYb2X4(A= Cd or Mg and\nX= S or Se), have a pyrochlore network of Yb3+, which\nhas a Hund's rules total angular moment of J=7=2and\na paramagnetic moment of \u0016calc= 4:5\u0016B. In contrast to\ntheir erbium counterparts, the ytterbium spinels do have a\nwell-isolated CEF ground state, a Kramers pseudo-spin-1=2\ndoublet which is separated by approximately 25 meV from\nthe \frst excited state. There are con\ricting reports on\nthe spin anisotropy of the Yb3+moments, which can only\npartially be explained by material dependent parameters\n(the speci\fc AandXions). While one study found that\nthe in- and out-of-plane components are nearly equal [ 62]\nand thus, Heisenberg-like, others have reported that it\nis weakly [ 63], or moderately Ising-like [ 51]. An exact\nidenti\fcation of the ground state anisotropy is di\u000ecult\nbecause direct measurements of the CEF energy levels via\nneutron scattering leads to an underconstrained problem.\nAn additional constraint can be added by considering the\nmagnetization as a function of applied \feld [ 51], which\nled to the most Ising-like result. However the accuracy\nof this method is debatable, as the strength of short\nrange correlations at low temperature may confound the\fts [63]. Electron paramagnetic resonance spectroscopy\nmeasurements on magnetically dilute samples could help\nresolve this question.\nAll four ytterbium spinels have long-range antiferro-\nmagnetic ordering transitions between 1 and 2 K [ 62].\nNeutron di\u000braction has only been performed in the two\nA= Cd compounds, both of which order into the \u0000 5\nirreducible representation [ 63,64], which is pictured in\nFig 5(b), where linear combinations of the basis vectors\n 2and 3trace out the local XY plane. The ordered mo-\nment is signi\fcantly reduced compared to the calculated\nvalue for the CEF ground state doublet, indicating that\nstrong quantum \ructuations are at play [ 63,64]. Sponta-\nneous muon precession is observed in CdYb 2X4[64] but\nnot in MgYb 2S4[62], suggesting that a di\u000berent ground\nstate might be found in the Mg analogs, or they may have\nstronger disorder. Within the ordered state, gapless spin\nexcitations are observed in both electron spin resonance\nmeasurements [ 65] and heat capacity [ 62]. It is interesting\nto note that there are several order-by-disorder mecha-\nnisms through which the \u0000 5degeneracy can be lifted,\nwhich would open a small energy gap that as of now has\nnot been detected [ 66]. One of these mechanisms, virtual\nCEF \ructuations, can likely be excluded, as the splitting\nto the \frst excited CEF level is rather large. Muon spin\nrelaxation studies show that there are persistent spin\ndynamics well below the ordering temperature [ 64], a fea-\nture that is apparently ubiquitous in rare earth pyrochlore\nlattice materials, irrespective of anion [67{69].\nAlthough the moments of the ytterbium spinels are\nnot in and of themselves XY-like, there is tantalizing\nevidence to suggest shared phenomenology with the XY\noxide pyrochlores, whose properties are dictated by in-\ntense phase competition. The \frst indication comes from\nheat capacity, where it is observed that the sharp lambda-\nlike ordering anomalies are preceded in temperature by\na broad anomaly, such that only 30% of the entropy\nrelease occurs below TN[62,64]. This two-stage order-\ning has become synonymous with phase competition [ 22].\nMore direct evidence of competing phases comes from\nthe low magnetic \felds required to destabilize their mag-\nnetic ground states, as was \frst observed in CdYb 2S4\nwith electron spin resonance [ 65]. At moderate \felds\nthere is a crossover from the XY antiferromagnetic \u0000 5\nstate (Fig. 5 (b)) to a spin-ice like ferromagnetic \u0000 9state\n(Fig. 5 (d)) [ 63]. Theoretical modeling further reinforces\nthe possible importance of phase competition in this fam-\nily of compounds, showing that CdYb 2Se4lies at a classi-\ncal phase boundary separating the splayed ferromagnetic\nand antiferromagnetic \u0000 5phase [ 70]. However, inelastic\nneutron scattering reveals weakly dispersive spin excita-\ntions within its ordered state that do not, at face value,\nresemble the overdamped spin excitations found in the\nytterbium oxide pyrochlore family [ 71]. Future works\nshould clarify the role of phase competition and order-by-\ndisorder e\u000bects in the Yb spinels, for which single crystal\nsamples will be crucial.8\n(a)\nΓ3ψ1\n(b)\nΓ5ψ2\n Γ5ψ3\n(c)\nΓ7ψ4\n(d)\nΓ9ψ7\n Γ9ψ8\n Γ9SF\n1\nFIG. 5. The irreducible representations (\u0000) and their basis vectors (  ), for allowed k= 0 magnetic orders on the pyrochlore\nlattice. (a) \u0000 3is commonly known as the all-in all-out structure. (b) \u0000 5is usually found in compounds with XY anisotropy\nand is composed of linear combinations of  2and 3. (c) \u0000 7, often referred to as the Palmer-Chalker state, has a threefold\nbasis, where  5and 6are equivalent to  4rotated about the primary crystal axis. (d) \u0000 9has a sixfold basis, where we show\ntwo representative basis vectors,  7and 8. The other four basis states (  9, 10, 11, and 12) are related to  7and 8by a\nsimple rotation to an equivalent cubic direction. The linear combinations of  7and 8forms the ice-like splayed ferromagnet\nstate (SF) shown. Other linear combinations can give ferromagnetic states with the moment canted away from the local [111].\nC. Holmium Chalcogenide Spinels\nThe holmium chalcogenide spinels are still poorly un-\nderstood due to their complex single ion properties. The\nHo3+cation that occupies the pyrochlore lattice has a\ntotal angular momentum of J= 8 (\u0016calc= 10:6\u0016B) and\nis hence a non-Kramers ion where a ground state dou-\nblet is no longer guaranteed. The CEF ground state was\ninitially suggested to be a non-magnetic singlet [ 46], due\nto low temperature susceptibility measurements show-\ning temperature independent paramagnetism in CdHo 2S4.\nMore recently, direct measurement of the CEF levels in\nMgHo 2Se4via inelastic neutron scattering reveals a non-\nKramers doublet ground state, in close proximity to an\nexcited doublet and singlet below 1 meV, and yet another\nsinglet below 3 meV [ 51]. This complicated CEF scheme\nwith many low lying levels bares some resemblance to\nthe terbium oxide pyrochlores, whose ground states are\nformed by two closely spaced doublets. In the terbium\npyrochlores, the competing energy scales of single ion\ne\u000bects and the ion-ion interactions leads to rich and com-\nplex physics that has eluded description by a microscopic\ntheory [ 72{75]. In the holmium spinels the lowest lying\nexcited CEF levels are even more numerous and closely\nspaced than the 1.5 meV level in the terbium pyrochlores,\nand thus we can expect similarly rich physics.\nThe magnetic ground state of the holmium chalco-\ngenides has only been studied in detail in one com-\npound, CdHo 2Se4, which has a magnetic transition atTC= 0:87 K [ 68]. However, as is the case for the terbium\npyrochlores, this ordering transition appears unconven-\ntional; despite the very sharp anomaly in heat capacity,\nmagnetic susceptibility only shows a broad in\rection at\nTC. Muon spin relaxation measurements indicate signi\f-\ncant \ructuations in the ordered state that persist in the\nparamagnetic regime above TCwith a \ructuation rate\nthat is two orders of magnitude slower than expected [ 68].\nGiven the non-Kramers nature of Ho3+and its complex\nCEF ground state, this e\u000bect may be related to a muon-\ninduced local distortion rather than an intrinsic origin [ 76].\nNeutron scattering studies of the static and dynamic prop-\nerties of the holmium chalcogenide spinels below TCwill\nbe illuminating.\nD. Dysprosium, Thulium, and Future Prospects\nThere are additional members of the rare earth chalco-\ngenide spinel family with either thulium or dysprosium\non the pyrochlore sublattice, for which experimental mea-\nsurements are still in their infancy. However, what is\nknown coupled with theoretical predictions suggests rich\nphysics that warrants further research.\nMgTm 2Se4has a spin singlet CEF ground state with a\nsecond singlet level within 1 meV [ 51]. This pair of nar-\nrowly separated singlets is analogous to a regular Kramers\ndoublet that is split by an external magnetic \feld, equiv-\nalent to an Ising system split by a transverse \feld [ 77].9\nThis is a particularly interesting case because, for the\nlocal symmetry of the pyrochlore lattice (where the Ising\nmoments point along eight equivalent local [111] direc-\ntions and hence are not colinear), such a transverse \feld\nwould be impossible to reproduce with the application\nof an external \feld. The exact ground state would then\ndepend on the relative strength of the singlet-singlet CEF\nlevel spacing and the nearest neighbor exchange, with\nthe most interesting scenario being where they are of a\ncomparable magnitude, possibly leading to a spin liquid\nstate [ 77]. While experimental studies of these compounds\nare limited, we know that there is no magnetic order in\nCdTm 2S4down to 2 K [ 53], or in MgTm 2Se4down to 0.4\nK [78].\nThe CEF ground state of the dysprosium spinels has\nnot been directly determined. However, a scaling analysis\nof the CEF parameters for the isostructural Er compound\nsuggest that CdDy 2Se4would have XY spin anisotropy,\nalthough with a relatively small CEF separation of only\n1.6 meV [ 23]. Even disregarding the e\u000bects of the low-\nlying excited CEF levels, CdDy 2Se4has been proposed\nas a candidate for quantum order-by-disorder or a U(1)\nquantum spin liquid [ 23]. If the exchange interaction in\nCdDy 2Se4favor ordering along the local [1 1 1] directions,\nthen the expected large in-plane component of the spin\nanisotropy would allow for strong quantum \ructuations\nand could be a candidate for a quantum spin ice state [ 79].\nChalcogenide spinels with rare earths larger than dys-\nprosium have not been shown to be stable under regular\nsynthesis conditions. However preparation of compounds\nwith other rare earth elements in the Bsite may be\npossible under high pressure conditions. High pressure\nmethods have been successfully applied to expanding the\nfamily of oxide pyrochlores [ 5] but have not yet, to our\nknowledge, been tested with other anions.\nAlthough we focused here on compounds where only\nthe pyrochlore sublattice is occupied by a magnetic ion,\nthere exist chalcogenide rare earth spinels with either\nMn or Fe on the A-site. These materials have either\nsurprisingly low [ 80], or no observed [ 80{83] magnetic\nordering transitions down to liquid helium temperatures,\npointing to the importance of geometric frustration. In\nfact, of the chalcogenides spinels with magnetic ions on\nboth sites, it is only those with signi\fcant site mixing\nand disorder that have been observed to magnetically\norder [ 84{86]. The nature of the magnetic frustration in\nthis family is still poorly understood, and the inclusion\nof a magnetic atom on the A-site adds an entirely new\nlayer of complexity that is beyond the scope of pyrochlore-\nlattice frustration.\nGiven the highly anisotropic nature of rare earth mag-\nnetism, many of the most illuminating experiments, in-\ncluding inelastic neutron scattering, are best accomplished\nwith single crystal samples. Thus far, the rare earth\nchalcogenide spinels have only been produced in poly-\ncrystalline form. Flux crystal growth is a promising av-\nenue as related chalcogenide compounds have been suc-\ncessfully grown with arsenic and antimony chalcogenide\ruxes [ 87], or with cadmium chloride \rux [ 88]. A second\npossible route is vapor transport methods. Similar cad-\nmium chalcogenide spinels have been successfully grown\nwith iodine or cadmium chloride as a transport agent [ 88].\nStudies to \fnd suitable single crystal growth methods are\ndemanding, but such samples would open up many new\navenues of research.\nIV. CHROMIUM CHALCOGENIDE SPINELS\nIn contrast to the other materials discussed in this re-\nview, the family of chromium spinels ACr2X4(A= Cd,\nCo, Cu, Fe, Mn, Zn; X= S, Se), where chromium oc-\ncupies the pyrochlore sublattice, have been extensively\nstudied dating back well over 50 years [ 88]. The chromium\nspinels are unique in that, despite being a family of com-\npounds with only one magnetic ion, they span a wide\nrange of magnetic phenomena. The Cr3+ion is an ex-\ncellent example of an isotropic S= 3=2 moment with\nseveral competing exchange interactions that are highly\ndependent on lattice spacing [ 89]. This gives a near\ncontinuum of dominant interactions, from the geomet-\nrically frustrated antiferromagnetic chromium oxides such\nas ZnCr 2O4(\u0012CW=\u0000398 K [ 90]), to the bond frus-\ntrated ZnCr 2S4(\u0012CW= 7:9 K [ 91]), to the ferromagnetic\nHgCr 2Se4(\u0012CW= 200 K [ 92]). Within this large range of\ninteraction strengths, we \fnd spin-spiral [ 93], phase coex-\nistence [ 94], and compounds with ferromagnetic spin \ruc-\ntuations preceding antiferromagnetic order [ 95]. Beyond\ntheir magnetic ground states, the physics of chromium\nspinels extend far beyond the scope of this review includ-\ning semiconductors [ 92], multiferroics [ 96], Chern semimet-\nals [97], and colossal magnetoresistance [ 98]. We will not\nattempt to be complete on this topic and will instead\nrefer the reader to other comprehensive reviews [99].\nA. Breathing pyrochlore chromium spinels\nIn the past few years, there has been a surge of in-\nterest in materials with so-called breathing pyrochlore\nlattices, in which the corner sharing tetrahedral network\nalternates between large and small tetrahedra. The prop-\nerties of these materials are often understood in terms of\ntheir breathing ratios, Br=d0=dwheredandd0are the\nbond lengths within the small and large tetrahedra, respec-\ntively. The breathing ratio is a good proxy for the relative\nstrength of their inter- and intra-tetrahedron exchange\ncouplings,JandJ0(Fig. 6(a)). The non-interacting limit\noccurs when J\u001dJ0, resulting in a scenario of decou-\npled small tetrahedra, as realized in Ba 3Yb2Zn5O11with\nBr= 1:9, where the ground state appears to be a nonmag-\nnetic singlet [ 104{106]. Conversely, when the breathing\nratio is closer to unity, as is the case for the chromium\nspinel oxides LiInCr 4O8and LiGaCr 4O8withBr\u00191:05,\nintra-tetrahedral interactions are su\u000eciently strong to\nyield collective magnetic behavior [ 107,108]. The ex-10\nTABLE II. An overview of the magnetic properties of the materials discussed in this review. For each family and magnetic\nion, one representative compound is chosen. Details include the angular momentum ( Sfor transition metals, and Jfor rare\nearths), the expected moment size ( \u0016calc), the paramagnetic moment determined from Curie-Weiss \ftting ( \u0016e\u000b), the single-ion\nanisotropy, the Curie-Weiss temperature ( \u0012CW), transition temperatures, and a brief description of the magnetic ground state.\nFluoride Pyrochlores and Related Materials\nS \u0016a\ncalc(\u0016B)\u0016e\u000b(\u0016B) Anisotropy \u0012CW(K) Transition temp. (K) Ground state\nNaCaNi 2F7[16] 1 2.83 3.7 Heisenberg \u0000129 T f= 3.6 Spin glass\nNaCaCo 2F7[15] 3/2 3.87 6.1 XY [21] \u0000139 T f= 2.4 Spin glass\nNa3Co(CO 3)2Cl [30] 3/2 3.87 5.3 Isingb\u000034 T N= 1.5 AFMc(All-in/all-out)\nNaSrFe 2F7[17] 2 4.90 5.94 Heisenberg \u000098 T f= 3.7 Spin glass\nNaSrMn 2F7[17] 5/2 5.92 6.25 Heisenberg \u000090 T f= 2.5 Spin glass\nNa3Mn(CO 3)2Cl [36] 5/2 5.92 5.97 Heisenberg \u000041 No order to 0.5 K\nFeF3[39] 5/2 5.92 n/adIsingbn/adTN= 21.8 AFM (All-in/all-out)\nRare Earth Chalcogenide Spinels\nJ \u0016 calc(\u0016B)\u0016e\u000b(\u0016B) Anisotropy \u0012CW(K) Transition temp. (K) Ground state\nCdDy 2Se4[53] 15/2 10.65 10.76 XY [23] \u00007:6 Unknown\nCdHo 2S4[68] 8 10.8 10.8 See text \u00003:6 T N= 0:9 Unknown\nCdEr 2Se4[55] 15/2 9.58 9.6 [53] Ising \u00001:2 T f= 0:6 Spin ice\nCdTm 2S4[53] 6 7.56 7.58 Singlet [51] \u000011:8 Spin singlet\nCdYb 2S4[62] 7/2 4.53 4.41 [53] See text \u000010 T N= 1:8 AFM (\u0000 5order)\nBreathing Pyrochlore Chalcogenide Spinels\nS \u0016a\ncalc(\u0016B)\u0016e\u000b(\u0016B) Anisotropy \u0012CW(K) Transition temp. (K) Ground state\nLiGaCr 4S8[100] 3/2 3.86 3.96 Heisenberg 19.5 T f= 10 Cluster glass\nCuInCr 4Se8[101] 3/2 3.86 3.58 Heisenberg 100 Sample dep.\nCuInCr 4S8[101] 3/2 3.86 3.83 Heisenberg \u000077 T N= 40 AFM\nCuGaCr 4S8[102] 3/2 3.86 Heisenberg T N= 4:2 AFM (Incomm. spiral)\nCuGaCr 4S8[103] 3/2 3.86 Heisenberg 142 T N= 17 AFM (Incomm. spiral)\naFor transition metal compounds, the calculated moment is the spin only contribution.bIsing spins are implied by the\nall-in/all-out ordered ground state of these compounds.cAntiferromagnet (AFM).dFeF3does not exhibit Curie-Weiss like\nsusceptibility up to the highest measured temperatures.\npected magnetic ground state in the interacting limit is\ndictated by the signs of JandJ0, under the assump-\ntion that further neighbour exchange is negligible [ 109].\nIn LiInCr 4O8and LiGaCr 4O8, the intratetrahedral ( J\nandJ0) exchange couplings are both antiferromagnetic\n(Fig. 6(b)). These highly frustrated materials exhibit\nstrong spin-lattice coupling and are only able to achieve\nlong range magnetic order after \frst undergoing a magne-\ntostructural tetragonal lattice distortion [107, 110, 111].\nNon-oxide breathing pyrochlores can be found amongst\na site-ordered structural derivative of the chromium\nchalcogenide spinels. These materials have the chemi-\ncal formula AA0Cr4X8(A= Li, Cu, Ag; A0= Al, Ga, In;\nX= S, Se) where AandA0are, respectively, monovalent\nand trivalent nonmagnetic metals. For certain combi-\nnations ofAandA0, there is no site ordering and the\ntwo cations are randomly distributed over the diamond\nsublattice of the spinel structure with space group Fd\u00163m\n(e.g. Ag+/Ga3+withX= S [ 112,113]). However, for\nmost combinations of AandA0, there is a zinc-blende\nordering of the two nonmagnetic metals, each of which\nforms its own face-centered cubic sublattice, lowering the\nspace group symmetry to F\u001643m(e.g. Cu+/In3+with\nX= S, Se [ 112{114]), as shown in Figure 1(c), and withcrystallographic details provided in Table I. Magnetic\nsusceptibility measurements on these materials \fnd a\nparamagnetic moment that agrees well with the expected\nspin-only value ( S=3=2) for Cr3+in a pseudo-octahedral\nenvironment [ 100,115,116]. While the \frst reports of\nthese compounds date back more than 50 years, interest in\ntheir magnetic ground states has recently been reignited.\nThe replacement of X= O withX= S or Se in the\nbreathing pyrochlore spinels has three major e\u000bects (i) fer-\nromagnetic superexchange is enhanced due to the nearly\n90\u000eCr{X{Cr bond angles, such that either or both Jand\nJ0can be ferromagnetic; (ii) the overall magnitude of the\nJandJ0exchange couplings decrease; and (iii) the second\nand third nearest neighbour exchange couplings ( J2,J3a,\nJ3b) become non-negligible (see Fig. 6(b)) [ 107,117,118].\nThe net result is that an entirely distinct set of magnetic\nground states are expected for these breathing chalco-\ngenide spinels. Similar to the oxide analogs, spin-lattice\ncoupling remains signi\fcant, as evidenced by an extended\nregion of negative thermal expansion in LiGaCr 4S8coin-\nciding with the temperature interval over which magnetic\ncorrelations develop [ 100]. However, no structural distor-\ntions have been detected.\nIn terms of the magnetic properties, one particularly11\nJ (K)60\n-30030J       J’      J3a\nJ3bJ2J’J J3a(a)\n(b)\nFIG. 6. (a) Near neighbor exchange couplings in the breath-\ning pyrochlore chromium spinels, where the chromium atoms\nare given by the black spheres and the chalcogen is given by\nthe white sphere. The alternating small and large tetrahedra\nare indicated in red and teal with intratetrahedral exchange\ncouplingsJandJ0, respectively. The second and third nearest\nneighbor exchange couplings are J2,J3a, andJ3b. (b) This bar\ngraph shows the magnitude and sign of the leading exchange\ncoupling constants, J,J0, andJ3adetermined from DFT cal-\nculations in Ref. [ 117] for four di\u000berent breathing pyrochlore\ncompounds. Here positive values of Jrepresent antiferromag-\nnetic couplings and negative values of Jare ferromagnetic.\ninteresting compound among the chromium chalcogenide\nbreathing pyrochlores is LiGaCr 4S8. While the earliest\nstudies of this material suggested that a heat capacity\nanomaly near 14 K marked an antiferromagnetic ordering\ntransition [ 112,116], more recent studies have found that\nit is instead associated with a spin freezing transition [ 100].\nBoth DFT calculations and \fts to di\u000buse neutron scat-\ntering data reveal that both JandJ0for this material\nare ferromagnetic and that the origin of frustration is\nstrong antiferromagnetic third nearest neighbor coupling,J3a(Fig. 6(b)) [ 117,118]. The resulting so-called cluster\nglass state is a form of emergent frustration in which\nferromagnetic tetrahedral clusters can be mapped onto\nan antiferromagnetically-coupled FCC lattice.\nCuInCr 4Se8, also with ferromagnetic JandJ0has been\npredicted to possess an incommensurate spiral magnetic\norder emerging out of a chiral spin liquid state [ 117].\nExperimentally, signi\fcant sample dependence has been\nobserved in this material, with non-stoichiometric samples\nadopting a spin glass-like state while the magnetic ground\nstate of stoichiometric samples remains unknown [ 102,112,\n119]. Neutron di\u000braction has revealed long range antiferro-\nmagnetic order in CuInCr 4S8[101,120], CuGaCr 4S8[102],\nand AgInCr 4S8[103], with the latter two adopting incom-\nmensurate spiral order states. A number of the X= S\nbreathing pyrochlore spinels have multiple magnetization\nplateaus with and without hysteresis suggestive of complex\nphase diagrams due to the delicate balance of compet-\ning near and further neighbour interactions [ 116,121].\nGrowth of any of these compounds in single crystal form\nwould be a signi\fcant advancement that would enable a\nbetter understanding of their interesting magnetic states.\nV. FUTURE DIRECTIONS\nIn this report, we have focused on the families of non-\noxide pyrochlore lattice materials that have received the\nmost detailed investigations to date, namely the \ruoride\npyrochlores, the rare earth chalcogenide spinels, and the\nbreathing pyrochlore chromium chalcogenides. We have\nalso brie\ry touched on a handful of related compounds\nincluding Na 3M(CO 3)2Cl (M= Co, Mn) and FeF 3. Our\nfocus in this \fnal section shifts to material families that\nwe hope will receive a similar level of investigation in the\nyears to come. Some of these are materials that have\nbeen synthesized, in some cases decades earlier, but have\nnever received a detailed investigation of their magnetic\nproperties. Others are families of compounds where we\nbelieve new materials lurk, remaining to be discovered.\nAn example of a family that has been long known\nbut has received relatively little attention is the ABB0F6\nfamily (A= Group I and B= 3dtransition metal), which\nbelong to the same structure type as the so-called \f-\npyrochlore oxides, AOs2O6(space group Fd\u00163m). The\n\f-pyrochlore \ruorides have a pyrochlore sublattice that is\noccupied by a random distribution of the B2+andB03+\natoms. The original reports of this family, from 1967,\ninclude more than two dozen compounds [ 122]. Of these,\nonly one compound, CsNiCrF 6, appears to have had a\ndetailed investigation into its magnetic properties, which\nrevealed Coulomb phase behavior for the magnetic, charge,\nand lattice degrees of freedom [ 61,123,124]. However, like\nthe disordered A-site pyrochlore \ruorides, this material\nenters a frozen glassy state at low temperatures due to\nchemical disorder. There are no known sul\fde or selenide\nanalogs of the \f-pyrochlores but they could conceivably\nexist withB= Os or some other 5 dtransition metal.12\nAnother interesting path towards discovering new ma-\nterials is to search for non-oxides analogs of known oxide\npyrochlore derivatives. For example, the \\tripod kagome\"\nstructure,R3Sb3A2O14(space group R\u00163m), whereRis\na magnetic rare earth and A= Mg or Zn, has emerged\nas a new family of magnetically frustrated materials just\nwithin the last few years [ 125,126]. This structure is\nobtained by doubling the pyrochlore formula unit and\ndiluting the two pyrochlore sublattices by one quarter.\nWhen viewed along the the [111] crystallographic direc-\ntion, the pyrochlore structure can be visualized as alter-\nnating triangular and kagome layers; this non-magnetic\ndilution selectively substitutes on the triangular sublat-\ntice, leaving the magnetic kagome layers intact. Despite\nthe newness of this topic, it already apparent that these\ntripod kagome materials can exhibit rich magnetic ground\nstates [ 127,128]. The natural question then is whether\nthis pyrochlore derivative structure could be stabilized\nwith \ruoride and chalcogenide anions. A \ruorinated\nversion of this structure, M2+\n3B2+\n3A1+\n2F14(M= 3dtran-\nsition metal, B= Group 2, and A= Group 1), would\nhave no inherent chemical disorder, unlike the \ruoride\npyrochlores.\nOther interesting prospects exist in the domain of mixed\nanion systems, where the mixture of anions allows dif-\nferent metal valences to be accessed and hence di\u000berent\nmagnetic states. There are, for example, \ruorosul\fde\nand oxy\ruoride pyrochlores of the type A2B2F6O and\nA2B2F6S wherein the anions are site-ordered such that\nthe one sulfur or one oxygen fully occupies the 48 fWyck-\no\u000b site [ 129]. An NMR study on Hg 2Cu2F6S, a rare\nexample of a Cu2+pyrochlore lattice material, found that\nthe nearest neighbor coupling was on the order of 100 K,\nyet there is no magnetic order down to 2 K, indicating\nstrong frustration [ 130]. Low temperature topochemi-\ncal anion substitution reactions may prove a promising\nroute to synthesizing new mixed anion pyrochlore lattice\nmaterials, as has been shown in oxynitride molybdatepyrochlores [131].\nFinally, it should be emphasized that the materials\nwe have focused our attention on here are all insula-\ntors. While this is a foregone conclusion in the case of\nthe \ruoride pyrochlores due to the strongly electroneg-\native \ruoride anion, it need not be so in the case of\nthe sul\fde and selenide spinels based on transition met-\nals with partially \flled d-shells. One such example is\nCuIr 2S4, which at room temperature is a paramagnetic\nmetal with a pyrochlore lattice that is occupied by a mix-\nture of non-magnetic Ir3+and magnetic Ir4+[132,133].\nAtTMI= 230 K, CuIr 2S4undergoes a metal-to-insulator\ntransition accompanied by a structural distortion that\ndisrupts the perfect pyrochlore sublattice and results in a\nspin dimerized state, where orbital degrees of freedom and\nmagnetic frustration may play a signi\fcant role [ 134]. The\nchalcogenide spinels are therefore a promising platform to\nstudy the interplay of magnetic frustration and itinerant\nelectronic degrees of freedom, which is an emerging topic\nthat is likely to deliver entirely new exotic states.\nNon-oxide pyrochlore lattice materials are fertile ground\nfor new and unusual magnetically frustrated states of mat-\nter. While materials breakthroughs over the past decade\nhave lead to signi\fcant progress, and many interesting\n\fndings, there is still much more to be done.\nACKNOWLEDGMENTS\nThis work was supported by the the Natural Sciences\nand Engineering Research Council of Canada and the\nCIFAR Azrieli Global Scholars program. This research\nwas undertaken thanks in part to funding from the Canada\nFirst Research Excellence Fund, Quantum Materials and\nFuture Technologies Program. Crystal structure \fgures\nwere made using VESTA [135].\n[1]M. J. Harris, S. T. Bramwell, D. F. McMorrow, T. H.\nZeiske, and K. W. Godfrey, \\Geometrical frustration\nin the ferromagnetic pyrochlore Ho 2Ti2O7,\" Physical\nReview Letters 79, 2554 (1997).\n[2]Arthur P Ramirez, A Hayashi, RJ Cava, R Siddharthan,\nand BS Shastry, \\Zero-point entropy in `spin ice',\" Na-\nture399, 333{335 (1999).\n[3]J.S. Gardner, S.R. Dunsiger, B.D. Gaulin, M.J.P. Gin-\ngras, J.E. Greedan, R.F. Kie\r, M.D. Lumsden, W.A.\nMacFarlane, N.P. 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Garrity1, Nirmal J. Ghimire3,4, Naween Anand5, Francesca \nTavazza1 \n1 Materials Science and Engineering Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899, USA.  \n2 Theiss Research, La Jolla, CA 92037, USA.  \n3. Department of Physics and Astronomy, George Mason University, Fairfax, VA 22030, USA.  \n4. Quantum Science and Engineering Center, George Mason University, Fairfax, VA 22030, USA . \n5. Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA.  \nAbstract  \nMagnetic topological insulators and semi -metals have a variety of properties that make them \nattractive for applications including spintronics and quantum computation , but very few high -\nquality candidate materials are known . In this work, w e use systematic high -throughput density \nfunctional theory calculations to identify magnetic topological materials  from the  ≈40000 three -\ndimensional materials in the JARVIS -DFT database  (https://jarvis.nist.gov/jarvisdft) . First, we \nscreen materials with net magnetic moment  > 0.5 μB  and spin -orbit spillage > 0. 25, resulting in  \n25 insulating and 564 metallic candidates. The spillage act s as a signature of spin-orbit induced \nband -inversion. Then , we carry out  calculations of  Wannier charge centers, Chern number s, \nanomalous Hall conductivit ies, surface bandstructure s, and Fermi -surface s to determine \ninteresting topological characteristics  of the screened compounds. We also train machine learning \nmodel s for predicting the spillage, bandgap s, and magnetic moment s of new compounds,  to further \naccelerate the screening process. We experimentally synthesize and characterize a few candidate \nmaterials  to support our theoretical predictions .  \n 2 \n Corresponding author : kamal.choudhary@nist.gov  \nIntroduction  \nThe interplay of topology1,2 and electronic band  structures  in non -magnetic materials has led to \nseveral new material categories,  most notably  topological insulators (TI)1,3, Dirac  semi -metals, and  \nbroken -inversion Weyl semimetals  (SM)4,5, topological crystalline insulators6 , nodal  line semi -\nmetals7,8. However, many potentially usef ul quantum effects9-13, like anomalous Hall conductivity  \n(AHC) , are only possible in topological materials with broken time reversal symmetry (TRS) , \nincluding exotic phases such as Chern insulators9,14, magnetic axion insulators9,15, and magnetic \nsemimetals16. Experiments su ch as anomalous Hall conductivity17, spin -Seebeck18, spin -torque \nferromagnetic resonance19, and angle -resolved phot oemission spectroscopy (ARPES)20, Fourier \ntransform scanning tunneling spectroscopy (FT -STS)21, and Shubnikov de  Haas (SdH) oscillations \ncan be useful for analyzing the topological behavior. Only a few such materials are reported  \nexperimentally, and many of those materials are limited to very low temperatures or have trivial \nbands that overlap with the topological b and features, limiting their utility. There is a significant \nopportunity to find more robust magnetic topological materials and to further our understand ing \nof the underlying mechanisms leading to their topological properties .  \nA common feature of many topological materials classes is the presence of spin -orbit induced band \ninversion, where the inclusion of spin -orbit coupling in a calculation causes the character of the \noccupied wavefunctions at a k -point to change. S pin-orbit spillage  (SOS)22-24 is a method to \nmeasure this band inversion by comparing the wavefunction s with and without spin-orbit coupling \n(SOC ). SOS is based on density functional theory (DFT) calculations  based wavefunction analysis  \nand has been proven to be a useful technique for finding topological materials . Previous studies22-\n24 have looked at three -dimensional ( 3D) non-magnetic materials as well as two-dimensional ( 2D) 3 \n mate rials with and without magnetism . Due to its ease of calculation, without any need for \nsymmetry analysis or dense k -point interpolation, the SOS is an excellent tool for identifying \ncandidate materials to many topological phases. Advantages of the spillage  technique include that \nit can apply to materials with low or no symmetries, including disordered or defective materials, \nand that it can identify the fundamental  driver of topological behavior, the band inversion, even if \nthe exact topological classification a material will depend on detailed features like the exact \nmagnetic ordering , spin-direction , or sample thickness . After identifying high spillage materials, \nfurther analysis is necessary to iden tify the specific topological phases that may arise from the \nband inversion.  \nStoichiometric magnetic topological insulators ( MTIs ) are very rare. MnBi 2Te325,26, an \nantiferromagnetic TI , is one of the most studied and  well-characterized examples of a 3D MTI,  \nand thin films of MnBi 2Te3 exhibit  quantized AHC26. Several m agnetic semimetals (MSM) , such \nas CuMnAs, Fe 3GeTe 2, LaCl, EuCd 2As2 have been reported as well16. Recently there have been \nseveral efforts to systematically identify topological materials , especially for the non -magnetic \nsystems23,24,27 -29 . The spin orbit spillage  technique has been successfully used to identify \nthousands of 3D non -magnetic insulators, semi -metals23 as well as 2D non-magnetic and magnetic  \ninsulators and semimetals such as VAg(PSe 3)2, ZrFeCl 6, MnSe and TiCl 324. Identification of MTIs \nand MSMs has been developed  by topological quantum chemistry groups30,31 in which \nwavefunction symmetry indicators are used to identify topological materials.   \nIn this work, we screen for 3D magnetic topological insulators (MTI) and semimetals using the \nSOS technique. We then analyze the resulting high -spillage materials using con ventional Wannier \ntight-binding Hamiltonian -based techniques to calculate Chern numbers, anomalous Hall \nconductivities , Berry -curvatures , and Fermi -surface s, as well as to local band crossings.  Starting 4 \n with crystal structures optimized using the OptB88vdW32 van der Walls functional, we first \nidentify materials using the Perdew -Burke -Ernzerhof (PBE)33 generalized gradient approximation \n(GGA) functional, and then carry out Strongly Constrained and Appropriately Normed (SCAN)34 \nmeta -GGA functional calculations  of a subset of materia ls. \nWhile our DFT -based computational screening is relatively efficient, it is still computationally \nexpensive when applied to a set of thousands of materials. To further accelerate the identification \nand characterization process, we develop classification machine learning models for metals/non -\nmetals, magnetic/non -magnetic and high -spillage/low -spillage materials, which acting together \ncan screen topological materials in different classes. Specifically, we use JARVIS -ML based \nclassical force -field inspired descriptors (CFID)35 and gradient boosting decision tree (GBDT) for \ndeveloping the ML models. CFID based models have been successfully been used for developing \nmore than 25 high -accurate ML property p rediction models36. Using this approach , we can first \npredict topological materials using ML, then confirm with SOS and Wannier tight-binding \napproaches . The selected materials  can be promising for experimental synthes is and \ncharacterizations. All the data and models generated through this work are publicly distributed \nthrough JARVIS -DFT23,24,36, JARVIS -WTB37 and JARVIS -ML webapps36. We also share  the \ncomputational tools and workflows developed for this work through JARVIS -Tools open access \nsoftware to enhance the reproducibility and transparency of our work.  As spillage is a \ncomputational screen ing technique for topological materials, there are many experimental \ntechniques to delineate topological characteristics such as ARPES, SHE, and QHE. In this paper, \nwe use some of these techniques to support the findings of spillage based two screened materials.  \nThis paper is organized as follows: first we show the screening strategy for high -spillage magnetic \nmaterials and present statistical anal ysis of some of their properties. Next, we show bandstructures 5 \n and k -point dependent spillage for a few example candidate materials to illustrate the strategy. \nAfter that we further analyze selected insulating and metallic band structures with Wannier tigh t-\nbinding approaches. Then, we analyze the periodic table distribution trends and develop machine \nlearning classification models to accelerate the identification processes. Finally, we  show \nexperimental characterization s of a few candidate materials . \n \nResu lts and discussion  \n \n  \nFig. 1  Flow -chart for screening high -spillage materials and analysis. a) flowchart  for screening, \nb) spillage distribution analysis for all the materials under investigation, c) pie chart showing high \nspillage insulators and metals , d) magnetic moment distribution for high -spillage materials, e) \nPBE vs SCAN bandgaps.   \n \nA flow chart for screening magnetic topological materials is shown in Fig. 1a. First, we screen for  \nmaterials with  net magnetic moment (>0.5 μB)  in the ferromagnetic phase,  which lead s to 8651 \ncandidates out of 39315 materials in the JARVIS -DFT database. T hen we look for materials that \n6 \n are reasonably stable and are likely to display topological band inversion by screening for materials \nthat: a) are less than  0.5 eV/atom  above the convex hull38, b) have small non-SOC bandgaps (<1.5 \neV), and c) have at least one atom with high atomic mass (M>64 ). This results in 4734 remaining \nmaterials . We have computed  the spin-orbit spillage ( SOS) with PBE+SOC  for 1745 materials  \n(prioritizing the calculations of the number of atoms  in unit cell  less than 20). Next , we perform  \nWannier tight -binding Hamiltonian (WTBH)  calculations  with high quality (MaxDiff<0.1 eV)37 \nto predict topological invariants, surface bandstructure s, Fermi -surface s, and anomalous Hall \nconductivity. So far, we have obtained high -quality WTBHs  for 146 candidate materials. T o study \nthe effects of exchange -correlation,  we run (SCAN)34 meta -GGA functional  calculations for high -\nspillage materials.  Note that it may be difficult to carry out high dense k -point DFT calculations \nwith SOC for thousands of materials , so after the WTBH generation, we carry our hig h-density k -\npoint calculation Wannier TB models to find if the bandgap truly exists.  Most of the materials \nstudied in this work come from experimentally  determined structures from the inorganic crystal \nstructure database (ICSD)  39.   \nIn Fig. 1b we show the spillage distribution of the materials investig ated in this work.  As the \nspillage can be related to the number of band -inverted electrons  at a k -point , we observe spike s at \ninteger numbers22-24.  Spin-orbit coupling can also change the mixing between different orbitals, \nrather than pure band inversion, which results in fractional spillage amounts.  As shown in Fig. 1b \nusing a sp illage threshold of 0.25 for screening eliminates 5 1 % of  materials , leaving  25 insulating \nand 564 metallic candidate materials with high spillage  and non -zero magnetic moment. Similarly, \nin our previous works for 3D non -magnetic and 2D materials22-24, spillage technique was shown \nto discard more than 50  % candidates in the initial screening steps  also. A material with non -zero \nspillage is a candidate topological material  and we choose a threshold of 0.25 to narrow down the 7 \n options. In Fig. 1c we show the pie chart for high spillage insulating and metallic materials \ndistribution . This suggests that magnetic topological insulators (MTI) are far rarer than \nsemimetals. In the later sections, we discuss with examples some of the insulating and metallic \nhigh-spillage materials and characterize them using Wannier tight -binding Hamiltonian approach \nalso. Next in Fig 1d, we observe that the magnetic moment of the systems could be up  to 6 μB \nwith mostly integer or close to integer values for the magnetic moments.  Due to the large \ncomputational expense of searching for magnetic ground states,  we only considered ferromagnetic \nspin configuration i.e., all spins of the system in a fixed direction. We expect that many of high -\nspillage materials that we find to be ferromagnetic may turn out to have lower energies in the  anti-\nferromagnetic or ferri -magnetic configurations . In Fig. 1e, we compare the bandgaps of the \nmaterials with PBE+SOC and SCAN+SOC for 65 high-spillage materials. Recently the SCAN \nfunctional has been proposed as the functional to solve the bandgap and high correlated system \nissues which can be important for magnetic topological materials. SCAN has been shown to predict \nbandgaps and magnetic moments better than LDA, LDA+U, and PBE in many cases40-42. We \nobserve that SCAN+SOC bands are very close or in some cases slightly higher than PBE+SOC \nbandgaps  for most of the materials . However, for some systems , it can be up  to 10 times larger  \nsuch as  for LiVH 2OF 5 (JVASP -47705). Some of the materials that are metallic in PBE turns into \ninsulating in SCAN predictions  (for example, LiMnAsO 4(JVASP -55805), Li 4Fe3CoO 8 (JVASP -\n42538)) , which indicates that  magnetic metals found to be h igh spillage using PBE may in fact be \nsmall gap  topological insulators . We provide more detailed PBE vs. SCAN comparison s in the \nsupplementary information (Table S 1). 8 \n   \n \nFig. 2 Examples of bandstructure and k -dependent spin -orbit spillage plots for a few selected \ncandidate materials  with PBE+SOC . Bandstructures are shown in a) Mn 2Sb (JVASP -15693 ), b) \nNaMnTe 2 (JVASP -16806 ), c) Rb3Ga (JVASP -38248, d) CoSI  (JVASP -78508), i) Mn 3Sn (JVASP -\n1820 9), j) Sc 3In (JVASP -17472), k) Sr 3Cr (JVASP -37600), l) Mn 3Ge (JVASP -78840), q) \nNaRuO 2(JVASP -8122),  r) CoNb 3S6 (JVASP -21459),  s) Y3Sn (JVASP -37701 ), t) CaMnBi 2 (JVASP -\n18532). The red and blue lines show SOC and non -SOC bandstructures respectively.  The k -\ndependent spillage is shown in (e), (f), (g), (h), (m), (n), (o) , (p), (u), (v), (w) and (x)  respectively.   \n \n9 \n In Fig. 2 we show the non-spin orbit and spin -orbit bandstructures for a few screened insulating \nand semi -metallic systems  along with corresponding spin -orbit spillage  plots such as a) Mn 2Sb \n(JVASP -15693), b) NaMnTe 2 (JVASP -16806), c) Rb 3Ga (JVASP -38248, d) CoSI (JVASP -\n78508), i) Mn 3Sn (JVASP -18209), j) Sc 3In (JVASP -17472), k) Sr 3Cr(JVASP -37600), l) Mn 3Ge \n(JVASP -78840),  q) NaRuO 2(JVASP -8122), r) CoNb 3S6 (JVASP -21459),  s) Y 3Sn (JVASP -\n37701), t) CaMnBi 2 (JVASP -18532). The red and blue lines show SOC and non -SOC \nbandstructures respectively.  The k -dependent spil lage is shown in (e), (f), (g), (h), (m), (n), (o), \n(p), (u), (v), (w) and (x) respectively.  Such bandstructures and spillage plots for 11483 materials  \n(including 2D and 3D magnetic and non -magnetic systems)  are distributed through the JARVIS -\nDFT website  along with several other materials properties such as crystal structure, heat of \nformation, elastic, piezoelectric, dielectric, and thermoelectric constants . In all  the cases , the \nspillage is higher than 0.25 and the magnetic moment s in the ferromagnetic configuration for these \nsystems are more than  1 μB. The NaRuO 2 shows a PBE+SOC gap of 56 meV while other materials \nare metallic . We note that in some cases, the magnetic ordering or magnetic moment can change \nsignificantly when adding SOC to a calculation, resulting in a high spillage value without any \ndirect relation to band inversion. Hence, it is important to further analyze the candidate mat erials  \nby directly computing topological behavior, and we show examples of this analysis for NaRuO 2 \nand Y 3Sn below.  \nIn our earlier work37, we created  a database of  automat ically generated  WTBH , which we use \nhere to analyze  topological behavior and support our findings from the spillage -based screening . \nThe accuracy of the WTBH is evaluated based on the MaxDiff criteria37 which compares the \nmaximum band -energy difference between DFT and WTB on k -points within and beyond  our DFT \ncalculations  k-points . We set a MaxDiff  (maximum energy difference at all k -points between 10 \n Wannier and DFT bands)  value of 0.1 eV as the tolerance for a good -quality WTBH. Out of all \nthe spillage -based  candidate materials we observe  at least 146 high of them have low MaxDi ff. \nFor the systems with high spillage and high -quality WTBH, we predict Wannier charge center s, \nsurface bandstructure s, and anomalous Hall conductivity for the insulating cases and AHC, Fermi -\nsurfaces and node plots for the metallic cases. Our Wannier database is available at \nhttps://jarvis.nist.gov/jarviswt b/ with interactive features.  We provide heat of formation, \nspacegroup, convex hull and other important details for each material in the corresponding \nwebpage (such as  https ://www.ctcms.nist.gov/~knc6/static/JARVIS -DFT/JVASP -8122.xml )as \nwell as in the supplementary information (Table S2) These webpages can also be downloaded as \nXML documents containing raw data for replotting or analysis by the users.  \nWe identify NaRuO 2 as a candidate 3D Chern insulator through the above systematic screening \nprocess  based on PBE+SOC and SCAN+SOC . NaRuO 2 is a trigonal system, belonging to R 3̅m \nspacegroup.  The heat of formation of the system is negative ( -1.293 eV/atom ) suggesting the \nsyste m should be thermodynamically favorable . Also, the system has a formation energy that is \n0.089 eV/atom  above the convex hull , suggesting that the system is slightly unstable but in a range \nwhere is may be synthesizable , and it has in fact been synthesized  experimentally43. We observe \nthat th is material is metallic with out SOC  (Fig. 2a), but as we turn on SOC, a gap opens  at the B \nand X  point s, which result s in high spillage of 0.56 . At least 18 materials show bandgap opening \ndue to inclusion of spin -orbit coupling.  Next, w e calculate the Chern number using  the Wannier \ncharge center s as shown in Fig. 3a and b . We observe gapless charge centers , indicating that the \nmaterial is a 3D Chern insulator.  The Chern number of four planes i.e., k1=0.0; k 1=0.5; k 2=0.0; \nk2=0.5 (k3=0.0; k 3=0.5 and k 2=0.0; k 2=0.5 remaining the same ); where k 1, k2, k3 is in fractional \nunits is determined  as -2. In Fig. 3c we see a conducting channel in the (001) surface suggesting 11 \n that the material is conducting at its surface , but the bulk is insulating even though the time reversal \nis broken in the system. The Chern number is directly proportional to the anomalous Hall \nconductivity which is an experimentally me asured quantity.  For a 3D Chern material, AHC is \ncalculated as 𝐶3𝑏3𝑒2\n2𝜋ℎ which turns out to be 1540 ohm-1cm-1, which is what we find using  Wannier \ncalculation -based  quantity in the Fig. 3d.  In this case the AHC in Fig. 3d is quantized which can \nbe leveraged for precise quantum control  from the perspective of  building devices.  In addition, we \nanalyzed this material using SCAN+SOC , and we find that the band structure is very similar to the \nPBE+ SOC result,  and the topological properties are the same (see t he supplementary information \nFig. S1 ).  \n \n12 \n Fig. 3 Wannier -charge center , surface bandstructure and anomalous hall conductivity for NaRuO 2 \n(JVASP -8122)  with PBE+SOC . a) W annier charge centers (WCC)  for k 1=0.0, b) k 1=0.5, c ) (001)  \nsurface bandstructure, d) AHC plot for the compound.  \n \n \nIn Fig. 4 we show the analysis of an example candidate topological metal Y 3Sn. Y3Sn crystallizes \nin P6 3mmc space group and hexagonal system, has negative formation energy ( -0.43 eV/atom) and \n0.1 eV/atom energy above convex hull, suggesting that it should be experimenta lly synthesizable.  \nThe bandstructures in Fig. 1 s show multiple band crossings  for this system and has a spillage of \n0.25. We plot the Fermi surface of this system in Fig. 4a which shows several conducting F ermi-\nchannels represented by deep blue spots . The lighter colors indicate that there are not bands at the \nFermi level.  This material belongs to the Kagome lattice  and such Fermi -surfaces have recently \ngained interest due to unique  nodal line  like features  44,45. The (001) surface for this material also \nshows multiple bands crossing Fermi -level, which is shown in Fig. 4c. We observe several nodes \nin this material as  shown in Fig. 4c with color coded energy level values. Energy levels with  null \nvalue or blue color represents bands at Fermi level. The calculated anomalous Hall conductivity \nof this system is shown in Fig. 4d. The AHC is not quantized such as NaRuO 2, but still has a non -\nzero value at zero field which can be due to the topological features of the bandstructure . The \nSCAN+SOC and PBE+SOC bandstructure comparison for this system is also shown in the \nsupplementary section (Fig. S 2), which shows shifts in energy for several bands . 13 \n   \nFig. 4 Analysis  for Y3Sn (JVASP -37701) as a  candidate semi -metal  with PBE+SOC . a) Fermi -\nsurface, b) (001) surface bandstructure, c) nodal points/ lines, d) anomalous Hall conductivity.  \nNext, i n Fig. 5a, we show the likelihood that a compound containing a given element has a high -\nspillage  for the 473 4 materials screened from step a . More specifically, for every compound \ncontaining a given element, we calculate the percentage that have a spillage greater than 0. 25. \nConsistent with known TMs, we observe that materials containing the elements such as Mn, R e, \nFe, Ir, Pt, Bi and Pb are by far the likeliest ones to have high spillage. To contribute to SOC -\ninduced band i nversion, an element must both have significant SOC and contribute to bands \nlocated near the Fermi level, which favors heavy elements with moderate electronegativity.  We \nuse similar analysis for materials for thermoelectrics , solar cells, elastic constants  etc. We can see \nsome basic trends in the data but we inten d to move towards more machine -learning prediction \nbased on ML. To  further accelerate the screening  of magnetic topological  materials we train three \n14 \n classification  models using classical force -field inspired descriptors ( CFID )35 descriptors to predict \nthe spillage, magnetic moment and bandgaps , based on  data from  the JARVIS -DFT database.  The \nCFID descriptors  provide a complete set of structural chemical feature s (1557 for each material)  \nwhich we use  with the Gradient Boosting Decision Tree ( GBDT ) algorithm as implemented in \nLightGBM46 to train high accuracy ML models. The accuracy of the classification can be measured \nin terms of Receiver Operating Characteristic ( ROC ) Area Under Curve ( AUC ), which is 0.8 1 for \nspillage, and 0.97 for both the magnetic and bandgap models  (using  a 90 % to 10 % train test \nstrategy) . The ROC AUC is 0 .5 for a random model, and 1.0 for a perfect model . The models \ntrained for this work  have ROC AUC greater than 0.81 , signifying useful predictive power . The \ngradient boosting algorithm allows for feature importance to be extracted after training the model. \nSome of the high -importance descriptors of the ML models are : unfilled d-orbitals, and \nelectronegativity which is intuitively reasonable . After training the ML models, we apply them on \n1399770  material s from JARVIS, AFLOW47, Materials -Project (MP)48 and Open Quantum \nMaterials Database (OQMD)49 to find 7721 0 likely high-spillage materials  using machine \nlearning . The ML screened materials can then be subjected to  the DFT workflow used in this work \n(see Fig. 1a ) to further accelerate the search for magnetic topological materials.  The ML models \nare distributed through  the JARVIS -ML webapp. 15 \n  \n \nFig. 5 Periodic table trends and c lassification model receiver operating characteristics (ROC) \ncurves. a) periodic table trends of compounds with high -spillage values. The elements in a material \nare weighed 1 or 0 if the material has high or low -values. Then the percentage probability of \nfinding the element in a high -value material is calculated. b ) For high/low spillage model \n(threshold 0. 25), c) high/low magnetic moment (threshold 0.5 μ B), d) Metals/non -metals based on \nelectronic bandgaps (threshold 0.05 eV).   \n \n \nNext, we discuss experimental results that support  some of our theoretical findings. The AHE was \nfirst observed in ferromagnets where its origin lies in the interplay between spin –orbit coupling \n(SOC) and magnetization.  Berry phase calculations have been proven accurate to predict SOC -\ninduced intrinsic AHE in ferroma gnets including Weyl (semi)metals, non -collinear \nantiferromagnets, non -coplanar magnets, and other nontrivial spin textures.  In Fig. 6a , we show  \nthe experimental anomalous Hall conductivity as a function of magnetic field at 23  K, 25 K and \n16 \n 23K  for CoNb 3S6. A large anomalous Hall conductivity at 23K takes the value 27Ω−1cm−1, which \nis a signature of experimental non -trivial band topology.  Corresponding computational non -SOC, \nSOC bandstructures for this system, which has a maximum spillage val ue of 0.5 are shown in Fig. \n2t. In Fig. 6 b, we sh ow the spin -pumping ferromagnetic resonance ( SP-FMR) measurements by \nutilizing the inverse spin Hall effect (ISHE). In ISHE, a pure spin current 𝐽⃗𝑆 gets converted to a \ncharge current 𝐽⃗𝐶 due to spin dependent asymmetric scattering phenomena. For spin pumping FMR \nmeasurements, Mn 3Ge (100 nm)/  Permalloy  (Py) (10 nm), Pt (10 nm) /Py (10 nm) and Py (10 nm) \nsamples were prepared on sapphire substrate. A Pt device was also fabricated and analyzed because \nit provides an ideal benchmark for ISHE comparison. Fig. 6b shows the  comparison between the  \nISHE charge curren t (𝑉𝐼𝑆𝐻𝐸 /𝑅𝑒𝑞) for all three devices, where Req is the total device resistance \nacross the contact pads. Resistance values Req for all devices were measured at room temperature \nin four -probe configuration. As expected, the Py single layer device is unaffected by ISHE, and \nthus Vsp is entirely antisymmetric. On the other hand, the peak 𝑉𝐼𝑆𝐻𝐸 /𝑅𝑒𝑞 value of the Mn 3Ge/Py \ndevice  is significant ly larger than that of the Pt/Py device. The ratio of spin -Hall angles \n𝜃𝑆𝐻𝑀𝑛3𝐺𝑒/𝜃𝑆𝐻𝑃𝑡 is estimated to be around 8 ± 2. The larger spin -Hall angle of Mn 3Ge is a result  of \nnon-trivial band -topology which is consistent with the spil lage signature.  \n 17 \n   \nFig. 6  Experimental measurements of some of the candidate materials.  a) anomalous Hall effect \nof CoNb 3S6, b) comparison of inverse spin -Hall signal (symmetric component) among measured \ndevices for Mn 3Ge. Right Inset: Optical image of the spin -pumping FMR device. Left Inset: Linear \nfit to the resonance linewidth (ΔH) at various resonance frequencies.  \n \nIn summary, we have demonstrated the applicability of spin -orbit spillage, machine learning and \nexperimental techniques to identify and c haracterize magnetic topological materials. We have also \nshown several remarkable trends in the topological chemistry with statistical analysis and periodic \ntable distribution plots. Because we employ a high -throughput approach to screen a large database, \nwe employ several assumptions , including  assuming a ferromagnetic spin ordering and not \nperforming detailed analysis of the dynamic or thermodynamic stability of our candidate materials . \nDetailed investigation of each material is out of the scope of this p aper and will be under taken in \nfuture  work . We have made our datasets and tools publicly available to enhance the reproducibility \nand transparency of our work. We believe that our work can be of great help to  guide future \ncomputational or experimental efforts to discover and characterize new magnetic topological \nmaterials.  \nMethods  \n18 \n Density functional theory:  DFT calculations were carried out using the Vienna Ab -initio \nsimulation package (VASP)50,51 software using the workflow59 given on our Github page \n(https://github.com/usnistgov/jarvis ). We use the OptB88vdW functional32, which gives accurate \nlattice parameters for both vdW and non -vdW (3D -bulk) solids52. We optimize the crystal -\nstructures of the bulk and monolayer phases using VASP with OptB88vdW. The initial screening \nstep for <1.5 eV bandgap materials is done with OptB88vdW bandgaps from the JARVIS -DFT \ndatabase. Because SOC is not  currently implemented for OptB88vdW in VASP, we carry out spin -\npolarized PBE and spin -orbit PBE calculations in order to calculate the spillage for each material.  \nSuch an approach has been validated by Refs. 23,53. The  crystal structure was optimized until the \nforces on the ions were less than 0.01 eV/Å and energy less than 10-6 eV. We use Wannier9054 and \nWannier -tools55 to perform the Wannier -based evaluation of topological invariants.  \nAs introduced in Ref.56, we calculate the spin -orbit spillage, 𝜂(𝐤), given by the following equation:  \n𝜂(𝐤)=𝑛𝑜𝑐𝑐(𝐤)−Tr(𝑃𝑃̃)                                                                                                            (1) \nwhere,  \n𝑃(𝐤)=∑ |𝜓𝑛𝐤⟩⟨𝜓𝑛𝐤|𝑛𝑜𝑐𝑐(𝐤)\n𝑛=1  is the projector onto the occupied wavefunctions without SOC, and \n𝑃̃ is the same projector with SOC f or band n and k -point k. We use a k-dependent occupancy \n𝑛𝑜𝑐𝑐(𝐤) of the non -spin-orbit calculation so that we can treat metals, which have varying number \nof occupied electrons at each k -point23. Here, ‘Tr’ denotes trace over the occupied bands. We can \nwrite the spillage equivalently as:  \n𝜂(𝐤)=𝑛𝑜𝑐𝑐(𝐤)−∑ |𝑀𝑚𝑛(𝐤)|2 𝑛𝑜𝑐𝑐(𝐤)\n𝑚,𝑛=1               (2) 19 \n where 𝑀𝑚𝑛(𝐤)= 〈𝜓𝑚𝐤|𝜓̃𝑛𝐤〉 is the overlap between occupied Bloch functions with and without \nSOC at the same wave vector k. If the SOC does not change the character of the occupied \nwavefunctions, the spillage will be near zero, while band inversion will result in a large spillage.  \nAfter spillage calculations, we ru n Wannier based Chern and Z 2-index calculations for these \nmaterials.  \nThe Chern number, C is calculated over the Brillouin zone, BZ, as:  \n𝐶=1\n2𝜋∑∫𝑑2𝒌𝛺𝑛 𝑛           (3) \n𝛺𝑛(𝒌)=−Im〈∇𝒌𝑢𝑛𝒌|×|∇𝒌𝑢𝑛𝒌〉=∑2Im〈𝜓𝑛𝑘|𝑣̂𝑥|𝜓𝑚𝑘〉〈𝜓𝑚𝑘|𝑣̂𝑦|𝜓𝑛𝑘〉\n(ꞷ𝑚−ꞷ𝑛)2 𝑚≠𝑛    (4) \nHere,  𝛺𝑛is the Berry curvature , unk being the periodic part of the Bloch wave in the nth band, 𝐸𝑛=\nћꞷ𝑛, vx and vy are velocity operators. The Berry curvature as a function of k is given by:  \n𝛺(𝒌)=∑∫𝑓𝑛𝑘𝛺𝑛(𝒌)𝑛          (5) \nThen, the intrinsic anomalous Hall conductivity (AHC)  𝜎𝑥𝑦 is given by:  \n𝜎𝑥𝑦=−𝑒2\nћ∫𝑑3𝒌\n(2𝜋)3𝛺(𝒌)         (6) \nIn addition to searching for gapped phases, we also search for Dirac and Weyl semimetals by \nnumerically searching for band crossings between the highest o ccupied and lowest unoccupied \nband, using the algorithm from WannierTools55. This search for crossings can be performed \nefficiently because it takes advantage of Wannier -based band interpolation. In an ideal case, the \nband crossings will be the only points at the Fermi level; however, in most cases, we find additional \ntrivial metallic states at the Fermi level. The surface spectrum was calculated by using the Wanni er \nfunctions and the iterative Green’s function method57,58 .  20 \n Starting from ~40000 materials in the JARVIS -DFT database, we screened for materials with \nmagne tic moment >0.5 μB and having heavy elements (atomic weight≥65) and bandgaps <1.5 \neV. After carrying out spin -orbit spillage calculations on them, we broadly classify them into \ninsulators and semimetals with non -vanishing and vanishing electronic bandgap s. For materials \nwith high spillage,  we run Wannier calculations to calculate the Chern number, anomalous hall \nconductivity, surface bandstructures and Fermi -surfaces. We also run SCAN functional based \ncalculations on the high spillage materials to check t he changes in bandgaps and magnetic \nmoments. So far, we have calculated 11483 SOSs for both magnetic/non -magnetic, metallic/non -\nmetallic systems.  \nMachine learning model:  \nThe machine -learning models are trained using classical force -field inspired descripto rs (CFID) \ndescriptors and supervise machine learning techniques using gradient boosting techniques in the \nLightGBM46 package59.  The CFID gives a unique representation of a material using structural \n(such as radial, angle and dihedral distributions), chemical, and charge descriptors. The CFID \nprovides 1557 descriptors for each material. We use ‘VarianceThreshold’ and ‘S tandardScaler’ \npreprocessing techniques available in scikit -learn before applying the ML technique to remove \nlow-variance descriptors and standardize the descriptor set. We use DFT data for developing \nmachine learning models for high/low spillage (threshol d 0.5), high/low magnetic moment \n(threshold 0.5 μB), high/low bandgap (threshold 0.0 eV) to further accelerate the screening process  \n The CFID has been recently used to develop several high -accuracy ML models for material \nproperties such as formation energ ies, bandgaps, refractive index, bulk and shear modulus and \nexfoliation energies k -points, cut -offs, and solar -cell efficiencies. The accuracy of the model is 21 \n evaluated based on area under curve (AUC) for the receiver operating characteristic (ROC). We \nprovide a sample script for the ML training in the supplementary information.  \nExperiment al details:  \nCoNb 3S6:  \nSingle crystals of CoNb 3S6 were grown by chemical vapor transport using iodine as the transport \nagent59. First, a polycrystalline sample was prepared by heating stoichiometric amounts of cobalt \npowder (Alfa Aesar 99.998 %), niobium powder (Johnson Matthey Electronics 99.8 %), and sulfur \npieces (Alfa Aesar 99.9995 %) in an evacuated silica ampoule at 900°C for 5 days. Subsequently, \n2g of the powder was loaded together with 0.5g of iodine in a fused silica tube of 14mm inner \ndiameter. The tube was evacuated and sealed under vacuum. The ampoule of 11cm length was \nloaded in a horizontal tube furnace in whic h the temperature of the hot zone was kept at 950°C \nand that of the cold zone was ≈850°C for 7 days. Several CoNb 3S6 crystals formed with a distinct, \nwell-faceted flat plate -like morphology. The crystals of CoNb 3S6 were examined by single crystal \nX-ray d iffraction at room temperature. Compositional analysis was done using an energy \ndispersive X -ray spectroscopy (EDS) at the Electron Microscopy Center, ANL.  \nTransport measurements were performed on a quantum design PPMS following a conventional 4 -\nprobe meth od. Au wires of 25μm diameter were attached to the sample with Epotek H20E silver \nepoxy. An electric current of 1mA was used for the transport measurements. The following \nmethod was adopted for the contact misalignment correction in Hall effect measureme nts. The \nHall resistance was measured at H=0 by decreasing the field from the positive magnetic field \n(RH+), where H represents the external magnetic field. Again, the Hall resistance was measured \nat H=0 by increasing the field from negative magnetic f ield (RH−). Average of the absolute value 22 \n of (RH+) and (RH−) was then subtracted from the measured Hall resistance. The conventional \nantisymmetrization method was also used for the Hall resistance measured at 28K (above TN) and \nat 2K (where no anomalous Hall effect was observed), which gave same result as obtained from \nthe former method.  \nMn 3Ge: \nIn ISHE, a pure spin current 𝐽⃗𝑆 gets converted to a charge current 𝐽⃗𝐶 due to spin dependent \nasymmetric scattering phenomena59. To maximize the ISHE signal, the  external magnetic field is \napplied along [ 11̅00] and dc voltage is measured along [ 112̅0] directions. An optical image of the \nspin-pumping device is shown in Fig. 6b . For spin pumping FMR measurements, (i) Mn 3Ge (100 \nnm)/ Py  (10 nm) and (ii) Pt  (10 nm) Py  (10 nm) (iii) Py  (10 nm) samples were prepared on sapphire \nsubstrate. They were fabricated into 1000  μm×200  μm bars by photolithography and ion milling. \nCoplanar waveguides (CPW) with 170 -nm thick Ti  (20 nm)/Au  (150 nm) were subsequently \nfabrica ted. Using ICP -CVD method, an additional SiN (150 nm) layer is deposited between CPW \nand the sample for electric isolation. The microwave frequencies were tuned between 10 GHz to \n18 GHz with varying power (12 dBm - 18 dBm) while magnetic field was swept be tween -0.4 T \nto 0.4 T along the CPW axis. Measurements were performed at room temperature and field \nresolution of 2 mT was adopted throughout.  \n \n \nData availability  \nJARVIS -related data is available at the JARVIS -API ( http://jarvis.nist.gov ), and JARVIS -DFT \n(https://jarvis.nist.gov/jarvisdft/ ) webpages.  23 \n Code  availability  \nPython -language based codes with examples are available at JARV IS-tools page: \nhttps://github.com/usnistgov/jarvis . \nContributions  \nK.C. designed the computational workflows, carried out high -throughput calculations, analysis, \nand developed the websites. K.F.G helped in  developing the workflow and analysis  of the data . \nN.J.G. performed the experiments for CoNb 3S6. 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Comp Phys Comm  224, 405 -416 (2018).  \n56 Liu, J. & Vanderbilt, D. Spin -orbit spillage as a measure of band inversion in insulators.  Phys Rev B   \n90, 125133 (2014).  \n57 Marzari, N. & Vanderbilt, D. Maximally localized generalized Wannier functions for composite \nenergy bands. Phys Rev B  56, 12847 (1997).  26 \n 58 Souza, I., Marzari, N. & Vanderbilt, D. Maximally localized Wannier fun ctions for entangled energy \nbands. Phys Rev B  65, 035109 (2001).  \n59          Please note commercial software is identified to specify procedures. Such identification does not \nimply recommendation by National Institute of Standards and Technology (NIST).  \n \nSupplementary information: High -throughput search for magnetic \ntopological materials using spin -orbit spillage, machine -learning and \nexperiments  \nKamal Choudhary1,2, Kevin F. Garrity1, Nirmal J. Ghimire3,4, Naween Anand5, Francesca \nTavazza1 \n1 Materials Science and Engineering Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899, USA.  \n2 Theiss Research, La Jolla, CA 92037, USA.  \n3. Department of Physics and Astronomy, George Mason University, Fairfax, VA 22 030, USA.  \n4. Quantum Science and Engineering Center, George Mason University, Fairfax, VA 22030, USA.  \n5. Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA.  \n \n \n \n \nTable S1 Bandgap comparison for PBE+SOC and SCAN+SOC functio nals with 1000/atom k -\npoint settings.  \nJID PBE+SOC(eV)  SCAN+SOC(eV)  \nJVASP -44705  0.02  1.2337  \nJVASP -2817  0.37  1.3048  \nJVASP -12648  0.27  1.1202  \nJVASP -10484  0.13  0.9678  \nJVASP -57388  0.14  0.91  \nJVASP -16408  0.28  0.5995  \nJVASP -55805  0.09  0.35  \nJVASP -21389  0.21  0.4312  \nJVASP -42538  0.03  0.2225  \nJVASP -8385  0.2 0.3273  \nJVASP -44991  0.04  0.116  \nJVASP -49890  0.01  0.072  \nJVASP -43466  0.03  0.0865  27 \n JVASP -82132  0.03  0.0528  \nJVASP -26528  0.35  0.3722  \nJVASP -81597  0.53  0.5518  \nJVASP -8384  0.01  0.0266  \nJVASP -16409  0.3 0.3104  \nJVASP -52135  0 0.0046  \nJVASP -59757  0 0 \nJVASP -17989  0 0 \nJVASP -43095  0 0 \nJVASP -17898  0 0 \nJVASP -16012  0 0 \nJVASP -59630  0 0 \nJVASP -55644  0 0 \nJVASP -8538  0 0 \nJVASP -59509  0 0 \nJVASP -26231  0 0 \nJVASP -39287  0 0 \nJVASP -26527  0 0 \nJVASP -38244  0 0 \nJVASP -38246  0 0 \nJVASP -38201  0 0 \nJVASP -38157  0 0 \nJVASP -38248  0 0 \nJVASP -17268  0 0 \nJVASP -37701  0 0 \nJVASP -45925  0 0 \nJVASP -34486  0 0 \nJVASP -81304  0 0 \nJVASP -76959  0 0 \nJVASP -80606  0 0 \nJVASP -81275  0 0 \nJVASP -81033  0.01  0 \nJVASP -76869  0.06  0.048  \nJVASP -80151  0.03  0.0142  \nJVASP -24841  0.04  0.0205  \nJVASP -16201  0.02  0 \nJVASP -18368  0.02  0 \nJVASP -44742  0.02  0 \nJVASP -76813  0.02  0 \nJVASP -77062  0.17  0.1493  \nJVASP -50689  0.03  0 \nJVASP -17265  0.39  0.3555  \nJVASP -79685  0.29  0.2462  28 \n JVASP -22442  0.16  0.1138  \nJVASP -81240  0.15  0.103  \nJVASP -8122  0.11  0.056  \nJVASP -21125  0.2 0.1449  \nJVASP -76930  0.16  0.1028  \nJVASP -21502  0.06  0 \nJVASP -26681  0.18  0.1136  \nJVASP -21417  0.07  0 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. S1 PBE+SOC and SCAN+SOC bandstructures for NaRuO 2. \n29 \n  \nFig. S2 PBE+SOC and SCAN+SOC bandstructures for Y 3Sn. \n \n \n \n \nTable S2 High -spillage material candidates.  \nJID Formula  Spillage  Gap Magmom  Ehull  \nJVASP -53555  MnIn2Te4  0.828  0 4.963  0 \nJVASP -53226  Ca5MnPb3  7.203  0 9.001  0 \nJVASP -15074  YCo5  0.376  0 6.796  0 \nJVASP -14699  MnTe  1.038  0 8.073  0.012265  \nJVASP -60253  UTe3  3.323  0 4.89  0 \nJVASP -84812  RbMnAs  0.44  0 3.772  0 \nJVASP -53303  MnTe  0.293  0 13.39  0.061846  \nJVASP -43433  Li2NbFe3O8  2.936  0 3 0 \nJVASP -60477  RuCl3  0.264  0 1.985  0 \nJVASP -43643  Li2CoSnO4  0.322  0 2.789  0.003284  \nJVASP -43846  MnP2WO8  1.539  0 6 0 \nJVASP -16176  Li2NbF6  0.432  0 1 0 \nJVASP -18583  TiAgHg2  1.104  0 1.649  0.234241  \nJVASP -18620  Mn2Au5  0.282  0 8.357  0.0225  \nJVASP -48582  Li6Mn5SbO2  5.953  0 21.945  0 \nJVASP -49957  Li2RhO3  0.459  0 1.826  0 \nJVASP -14962  PuFe2  4.719  0 3.591  0 \nJVASP -84911  RbMnAs  0.367  0 3.967  0 \nJVASP -18519  SrMnBi2  1.428  0 7.883  0 \nJVASP -18467  K2MoCl6  0.263  0 2 0 \n30 \n JVASP -18551  RbMnSe2  0.302  0 4 0 \nJVASP -18468  K2OsBr6  0.934  0 1.994  0 \nJVASP -14964  Mn3Rh  0.815  0 0.903  0.059416  \nJVASP -14584  CrPt3  2.521  0 2.791  0 \nJVASP -15006  CrSb2  0.545  0 4.192  0.205306  \nJVASP -44114  Li2V3SnO8  1.008  0 0.988  0.038036  \nJVASP -43542  Li3YNi2O6  2.944  0 1.998  0 \nJVASP -15562  RbFeS2  0.278  0 3.13  0 \nJVASP -48117  Li2Fe3TeO8  3.823  0 2.025  0.003622  \nJVASP -60294  UIN 1.201  0.03  4 0 \nJVASP -43107  Li2Fe3SnO8  0.252  0 4 0.031039  \nJVASP -14926  VSb 1.02  0 2.891  0.123297  \nJVASP -20094  Co3Mo  1.016  0 0.9 0 \nJVASP -54386  NbFeO4  0.523  0 2.015  0 \nJVASP -18625  VAu2  2.102  0 3.782  0.022372  \nJVASP -15649  MnCuSb  1.015  0 4.296  0.187604  \nJVASP -14711  FeAs  0.919  0 1.923  0.002932  \nJVASP -14928  MnNbSi  1.02  0 2.14  0 \nJVASP -11974  NbCo3  1.094  0 11.578  0.078038  \nJVASP -87133  UTe2  2.541  0 4.916  0 \nJVASP -43362  Li2Fe3TeO8  0.469  0 9.606  0 \nJVASP -15567  MnCo2Sb  0.304  0 5.793  0.041562  \nJVASP -43661  Li4Mn3Co3Sn2O6  4.974  0 8.016  0 \nJVASP -15104  Mn3ZnC  0.551  0 6.801  0.009448  \nJVASP -15237  HfGaCo2  4.98  0 0.876  0 \nJVASP -42681  Ti2Co3Te3O6  3.082  0 9.495  0 \nJVASP -19910  CoPt  1.378  0 4.447  0 \nJVASP -14889  RuF3  1.989  0 1.975  0 \nJVASP -50360  Bi2PO6  0.329  0 0.737  0 \nJVASP -52375  HfMn6Sn6  0.391  0 13.037  0.009234  \nJVASP -54439  FeB2W2  0.645  0 3.35  0 \nJVASP -43466  Li2Fe3SbO8  5.063  0.03  3 0.012412  \nJVASP -18924  FePt3  1.348  0 4.197  0 \nJVASP -21561  Nb4CrS8  1.024  0 4.931  0 \nJVASP -14934  CrCuSe2  0.296  0 3 0.254504  \nJVASP -43125  Cr3SbP4O6  3.53  0.12  9 0 \nJVASP -42156  Li4Fe3SbO8  2.151  0 8.995  0 \nJVASP -48191  Li2P2WO8  1.162  0 4 0.007233  \nJVASP -50895  PrTlO3  0.721  0 2 0 \nJVASP -18525  Rb2WBr6  0.569  0 2 0 \nJVASP -52305  Ga2CuO4  0.393  0 1.989  0 \nJVASP -16908  TiBi2O6  1.081  0 1.891  0.252842  \nJVASP -16201  MgRhF6  0.902  0.02  0.991  0 31 \n JVASP -53381  Cr4Cu3Se8  0.629  0 11.097  0.037619  \nJVASP -53248  Co2B6W2  0.537  0 17.019  0 \nJVASP -47467  TaFeO4  0.69  0 2 0.016797  \nJVASP -19524  UI3 2.823  0.04  6 0 \nJVASP -50977  Li2CuBiO4  0.594  0 0.723  0 \nJVASP -15326  CaMn2Sb2  0.517  0 7.131  0 \nJVASP -16161  ScFeGe  0.446  0 4.543  0 \nJVASP -42352  HfCrO4  2.027  0 8 0 \nJVASP -15617  SrCo2P2  0.266  0 0.591  0 \nJVASP -53515  Mn2SbTe  0.333  0 15.725  0.298565  \nJVASP -16001  K2RhF6  1.338  0.13  0.994  0 \nJVASP -6145  US3 1.769  0 4 0 \nJVASP -60459  UTe3  1.707  0 3.409  0 \nJVASP -18528  Tl2WCl6  1.043  0 2 0 \nJVASP -53265  AlFe2Mo  0.389  0 0.789  0 \nJVASP -50093  TiNbO4  0.654  0 0.533  0 \nJVASP -43389  Li4MnW3O2  1.119  0.42  4.99  0 \nJVASP -52989  Sr3Mn4O2  0.462  0 9.994  0 \nJVASP -43935  Li4Mn3V3Sn2O6  2.408  0 14.756  0 \nJVASP -43390  Li4Cr5SbO2  7.516  0 3 0 \nJVASP -43694  CdFeO3  0.395  0 12.87  0 \nJVASP -15511  FeAgS2  0.276  0 2.056  0.054059  \nJVASP -726 CrS2  0.308  0 2.047  0 \nJVASP -14679  VPt3  1.294  0 1.357  0.014238  \nJVASP -19895  CoPt3  1.687  0 2.833  0 \nJVASP -15663  MnSbAu  0.904  0 4.559  0.151837  \nJVASP -53544  Cr4Cu3Te8  1.05  0 11.359  0 \nJVASP -47523  Co3SbO8  0.28  0 3.696  0 \nJVASP -42538  Ta2CrNO5  2.788  0.03  6 0 \nJVASP -6097  VCl3  0.973  0 4 0 \nJVASP -50922  CuAuO2  0.456  0 0.51  0.012338  \nJVASP -20076  FeSe  0.421  0 4.509  0.198073  \nJVASP -18532  CaMn2Bi2  1.171  0 8.652  0.264587  \nJVASP -49890  Ba4IrO6  2.44  0.01  1.651  0 \nJVASP -43616  LiFeSbO4  0.354  0 8.13  0.052363  \nJVASP -51442  GaFe2Co  0.735  0 5.679  0.07515  \nJVASP -15583  MnAlAu2  0.289  0 3.785  0 \nJVASP -20110  MnPt3  2.412  0 4.245  0 \nJVASP -6766  HfFeCl6  1.042  0.02  4 0 \nJVASP -15211  FeSiRu2  0.394  0 3.686  0 \nJVASP -50689  Li4Ti3Cu3Te2O6  1.026  0.03  2.771  0 \nJVASP -1867  FeAgO2  0.75  0 1.032  0.000927  \nJVASP -16213  MnNbGe  0.381  0 3.711  0 32 \n JVASP -14419  CoI2  0.852  0 2.409  0 \nJVASP -13600  ZrFeCl6  1.015  0.03  4 0 \nJVASP -57304  Sr3Fe2Cu2Se2O5  1.014  0 7.634  0 \nJVASP -19831  MnSb  1.005  0 6.273  0.0527  \nJVASP -56996  TaFe2  2.098  0 2.382  0 \nJVASP -58164  Mg2TiIrO6  2.476  0 1.21  0 \nJVASP -56764  Tl4CrI6  0.295  0 8 0.00395  \nJVASP -58491  V4ZnS8  1.181  0 1.181  0 \nJVASP -57228  Sr2Mn3Bi2O2  0.516  0 11.681  0 \nJVASP -57886  BaCr4O8  0.354  0 9.992  0 \nJVASP -56321  RbTiBr3  0.347  0 2.621  0 \nJVASP -15423  TlCrTe2  0.401  0 3.054  0 \nJVASP -49947  Na2IrO3  1.294  0 1.798  0 \nJVASP -57036  Sr2CoCl2O2  0.421  0 2.108  0 \nJVASP -58165  Ca2TiIrO6  2.291  0 0.82  0 \nJVASP -57153  FeBiO3  0.322  0 2.811  0.100464  \nJVASP -54622  GaFe2Ni  0.255  0 4.509  0.08455  \nJVASP -15424  MnRh2Pb  0.881  0 4.674  0 \nJVASP -51466  NiRh2O4  0.359  0 3.986  8.42E -05 \nJVASP -58166  TiZn2IrO6  1.96  0 1.126  0 \nJVASP -35684  Mn3Sn  0.477  0 1.01  0.216904  \nJVASP -58167  Mg2SnIrO6  1.657  0 0.865  0 \nJVASP -21107  Mn3P6Pd20  1.106  0 14.047  0 \nJVASP -57905  BaTi4O7  0.408  0 3.473  0.119577  \nJVASP -15256  FeCuPt2  1.662  0 3.93  0 \nJVASP -54890  RbW3Cl9  0.579  0.06  1.616  0 \nJVASP -58177  Mg2MnIrO6  1.523  0 4.194  0 \nJVASP -59902  ZnCr2S4  0.303  0 12 0.158825  \nJVASP -58503  ZnFe4S8  1.05  0 6.979  0 \nJVASP -35299  MnCoSn  1.013  0 8.783  0.348821  \nJVASP -59607  YMnGe  0.411  0 11.184  0 \nJVASP -58179  Mg2CoIrO6  1.673  0 2.154  0 \nJVASP -15693  Mn2Sb  0.5 0 11.488  0.333028  \nJVASP -15730  ThCo2Si2  1.149  0 0.97  0 \nJVASP -51048  FeCu2SnS4  0.313  0 3.874  0.006128  \nJVASP -15428  KCo2Se2  1.006  0 2.002  0 \nJVASP -56922  MnAsRh  0.347  0 9.719  0 \nJVASP -51894  FeBiO3  2.269  0 13.908  0.005442  \nJVASP -57388  MnBiAsO5  3.258  0.14  10 0 \nJVASP -57951  YCu3Sn4O2  1.146  0 2.015  0 \nJVASP -58180  Zn2CoIrO6  1.813  0 1.382  0 \nJVASP -54898  SrCo2As2  0.471  0 0.846  0 \nJVASP -16042  TaF3  0.497  0 1.99  0.448904  33 \n JVASP -18575  Rb2WCl6  0.591  0 2 0 \nJVASP -58507  Ca2CuIrO6  2.84  0 3.717  0 \nJVASP -49616  ZnFeO3  0.9 0 8.717  0.063702  \nJVASP -57848  Ba6Ru3Cl2O2  0.513  0 3.365  0 \nJVASP -55136  Fe3PtN  1.033  0 7.954  0 \nJVASP -57242  Sr2Fe2S2OF2  0.472  0 7.526  0 \nJVASP -59737  Ta2CuO6  0.354  0 1.997  0 \nJVASP -21401  NaSr3IrO6  3.184  0.23  3.893  0 \nJVASP -54949  Co2Sn  1.118  0 2.772  0.219942  \nJVASP -19286  Y2Fe2O7  3.858  0 8.012  0 \nJVASP -57319  Ba2YFe3O8  0.506  0 4.796  0 \nJVASP -8616  Sr2CoBr2O2  0.282  0 2.128  0 \nJVASP -55156  Fe3RhN  1.011  0 8.232  0 \nJVASP -59673  Y2Ru2O7  2.262  0 1.468  0 \nJVASP -57430  Ba2Mn2Bi2O  2.292  0 19.266  0 \nJVASP -15076  FePd  0.28  0 6.495  0.024811  \nJVASP -15057  Fe3Se4  0.506  0 4.125  0.079199  \nJVASP -60098  YFeO3  0.805  0 24.747  0.037703  \nJVASP -8096  Cr2FeSe4  0.613  0 5.997  0 \nJVASP -58040  Ba2TlFe2O7  1.942  0 7.098  0 \nJVASP -54955  KMnAg3C6N6  1.008  0 1 0.166123  \nJVASP -14753  HfZn2  1.052  0 1.177  0.002744  \nJVASP -55644  KFeBr3  0.339  0 16 0 \nJVASP -38339  RbAuO3  1.701  0 1.329  0.414073  \nJVASP -58042  Ba2TlCo2O7  0.317  0 5.352  0 \nJVASP -8618  CrPbO3  1.021  0 2 0 \nJVASP -5314  UCl5  1.152  0.41  2 0 \nJVASP -59564  V2CdO4  1.025  0 8 0 \nJVASP -58337  KY2Ti2S2O5  0.326  0 0.917  0 \nJVASP -59847  YMn3Se2ClO8  0.419  0 18.612  0 \nJVASP -60099  YCoO3  1.365  0 11.449  0.139314  \nJVASP -36101  Er2Co3Ge5  1.248  0 0.916  0 \nJVASP -57577  TiFeBi2O6  0.284  0 4.001  0 \nJVASP -56803  Ba3Fe3Se7  0.484  0 19.803  0 \nJVASP -8619  SrCrO3  0.639  0 1.973  0 \nJVASP -57326  Ni2Ag3O4  1.039  0 2.738  0.02373  \nJVASP -56962  CrGaCo2  1.008  0 3.035  0 \nJVASP -58227  BaMgCo4O8  0.411  0 11.524  0 \nJVASP -7958  MnAsRh  0.32  0 8.557  0.087241  \nJVASP -59570  Ba3Cr2O8  0.409  0 2 0 \nJVASP -8018  MnIr  2.046  0 2.495  0 \nJVASP -37845  CuSeO4  0.455  0 1.95  6.24E -05 \nJVASP -58095  YCoO3  1.009  0 3.761  0.163071  34 \n JVASP -7892  BaY2NiO5  1.009  0 1.252  0 \nJVASP -57868  CrSn2  0.29  0 1.647  0.181985  \nJVASP -60101  YMoO3  0.837  0 10.18  0.053661  \nJVASP -12919  CuPtF6  0.631  0 1.999  0 \nJVASP -59682  RbHgN3O6  1.012  0 4 0.343516  \nJVASP -56072  MnBi  2.039  0 6.924  0.277582  \nJVASP -21191  Ca2NiIrO6  1.31  0 1.549  0 \nJVASP -59496  Cd2Re2O7  2.491  0 1.746  0 \nJVASP -42985  Li5Nb2V5O2  1.047  0 3.417  0 \nJVASP -59573  Sr3NiIrO6  2.902  0 4.471  0 \nJVASP -59637  Mn4Ge6Ir7  1.728  0 16.86  0 \nJVASP -41091  Ta2FeOs  0.945  0 1.545  0 \nJVASP -55190  Rb2FeI4  0.373  0 8 0.000214  \nJVASP -60102  YVO3  1.014  0 10.626  0.087396  \nJVASP -8623  FeBiO3  0.323  0 3.258  0.041491  \nJVASP -56304  TlCoCl3  1.032  0 5.985  0 \nJVASP -21588  Sr2ScIrO6  3.617  0 3.152  0 \nJVASP -58461  BaFe4O8  0.266  0 10.253  0 \nJVASP -2817  K4IrO4  0.632  0.37  2.88  0 \nJVASP -14308  Ta2InCuTe4  0.508  0 2.969  0.336609  \nJVASP -13174  Ta2CrO6  0.451  0 7.356  0 \nJVASP -56818  MnSbPt  0.854  0 3.913  0.372015  \nJVASP -59580  BaVO3  0.653  0 1.154  0 \nJVASP -49659  Nb2Co2O9  1.003  0 3.882  0.15346  \nJVASP -8538  CeNiSb2  2.223  0 0.546  0 \nJVASP -58250  YCo4B  0.544  0 6.681  0 \nJVASP -59704  Fe2CuGe2  0.393  0 0.508  0 \nJVASP -59773  Mn3GeIr  0.859  0 38.663  0.138852  \nJVASP -54797  CrCoPt2  1.687  0 1.199  0 \nJVASP -58411  Rb2CoSe2  1.034  0 5.996  0 \nJVASP -59709  In2CoS4  0.547  0 5.991  0 \nJVASP -55805  LiMnAsO4  8.433  0.09  20 0 \nJVASP -59509  Ta3Co3C  1.468  0 4.005  0 \nJVASP -59587  YMnO3  1.006  0 7.998  0 \nJVASP -8240  YFeO3  0.618  0 2.557  0.197367  \nJVASP -41131  Hf2CoIr  1.023  0 0.69  0 \nJVASP -59648  Mn3SiIr  2.239  0 25.834  0.052054  \nJVASP -21327  Zn2BiWO6  1.488  0 1.15  0 \nJVASP -20932  Cr2CuSe4  0.45  0 10.336  0 \nJVASP -58257  FeRh2S4  1.214  0 6.105  0 \nJVASP -7856  YCrO3  1.228  0 2.978  0.144098  \nJVASP -59882  YCo2S4  3.012  0 1.785  0.142833  \nJVASP -21582  MnNb3S6  1.016  0 8.337  0 35 \n JVASP -57296  SrFe2Se4O2  1.933  0 5.999  0 \nJVASP -8323  CoBi2O6  2.093  0 1.784  0.180329  \nJVASP -59593  NiRh2O4  0.322  0 3.986  0 \nJVASP -20447  FePt  0.413  0 3.243  0 \nJVASP -38273  KRhO3  0.349  0 1.784  0 \nJVASP -21693  HfMn2  2.318  0 2.938  0 \nJVASP -45533  YFeO3  1.32  0 11.999  0 \nJVASP -7858  TlFeO2  1.084  0 5.29  0 \nJVASP -21125  Sr3LiIrO6  4.647  0.2 3.869  0 \nJVASP -21071  Cr2HgSe4  1.029  0 11.956  0 \nJVASP -20414  Co3W  0.255  0 0.833  0.096612  \nJVASP -26876  CeH2  0.973  0 0.763  0 \nJVASP -38639  Ni3Au  1.081  0 1.538  0.088389  \nJVASP -21589  Ba2ScIrO6  2.197  0 3.073  0 \nJVASP -59107  Zr6Co23  0.415  0 25.261  0 \nJVASP -41448  TmUTc2  2.133  0 0.877  0 \nJVASP -20397  ZrMn2  0.264  0 2.878  0.000832  \nJVASP -8633  VPbO3  0.328  0 1 0 \nJVASP -8737  MnPd3  0.372  0 9.241  0 \nJVASP -26681  Na3OsO5  1.26  0.18  2.702  0 \nJVASP -8433  BaMnGe  0.329  0 7.109  0 \nJVASP -12965  Mn3As2  0.392  0 14.986  0.095392  \nJVASP -26231  Co2B6Mo2  0.444  0 17.539  0 \nJVASP -20780  Rb2O3  0.942  0 4 0 \nJVASP -8249  YCrO3  1.23  0 2.978  0.143851  \nJVASP -39263  FeAg3  0.397  0 2.97  0.356336  \nJVASP -24357  Sr3CaIrO6  2.334  0 1.645  0 \nJVASP -41360  HfScCo2  1.189  0 0.652  0 \nJVASP -22381  Sr4IrO6  2.288  0 1.604  0 \nJVASP -22441  Na3Cd2IrO6  1.503  0.23  1.831  0 \nJVASP -22442  Ba3NaIrO6  1.853  0.16  3.946  0 \nJVASP -49567  MgMo6O6  2.044  0 2.361  0 \nJVASP -39364  Ni3Hg  1.068  0 0.883  0.285456  \nJVASP -25704  NbVF6  0.466  0 4 0 \nJVASP -15904  MnSbIr  1.029  0 3.103  0 \nJVASP -8204  NiPt  2.669  0 1.828  2.00E -05 \nJVASP -26047  Co2Re2B6  0.768  0 17.729  0 \nJVASP -8335  ZnCr2N2  0.288  0 5.855  0.382623  \nJVASP -39275  FeAu3  0.315  0 3.117  0.168094  \nJVASP -8122  NaRuO2  0.496  0.11  1 0.021784  \nJVASP -26796  Li2MoF6  1.034  0 4 0 \nJVASP -7922  BaMn2P2  0.342  0 2.791  0 \nJVASP -7868  FeAgO2  0.766  0 2.083  0 36 \n JVASP -38211  Rb3Pb  1.125  0 0.912  0 \nJVASP -38310  RbInO3  0.298  0 2 0.284021  \nJVASP -39279  FePbO3  0.298  0 3.441  0 \nJVASP -7869  Sr2MnO4  0.283  0 3 0 \nJVASP -26074  Rb3Fe2Se4  1.094  0 30.015  0 \nJVASP -16563  MnSnIr  0.672  0 3.447  0.344516  \nJVASP -7702  TiAu  1.023  0 0.934  0.113969  \nJVASP -38264  MnTl3  1.228  0 8.367  0.456432  \nJVASP -38317  RbF3  0.411  0 2 0.161013  \nJVASP -20801  Cr2Te3  1.454  0 24.218  0 \nJVASP -21389  Sr3CuPtO6  2.174  0.21  1.919  0 \nJVASP -14273  TiCdHg2  0.529  0 0.835  0.175958  \nJVASP -15949  ZnFe3C  0.295  0 4.078  0 \nJVASP -38319  RbMnO3  0.387  0 1.908  0 \nJVASP -22401  Sr3MnN3  1.01  0 2.486  0 \nJVASP -38492  K3Rh  1.073  0 0.501  0.477367  \nJVASP -19159  YFe4Cu3O2  0.819  0 9.021  0 \nJVASP -15951  ZnCo3C  1.005  0 0.927  0 \nJVASP -8510  Cr2As  0.86  0 3.255  0.162059  \nJVASP -39287  K3Sn  0.424  0 0.912  0 \nJVASP -38226  MnSn3  0.302  0 7.979  0.339145  \nJVASP -21417  Sr3ZnIrO6  2.193  0.07  1.436  0 \nJVASP -38325  RbNiO3  0.383  0 2.666  0 \nJVASP -26526  Ba2Fe2S2OF2  0.314  0 7.827  0 \nJVASP -8301  MgMoF6  1.013  0 2 0 \nJVASP -15918  LiTiTe2  1.022  0 0.843  0 \nJVASP -26527  SrCrF6  0.284  0 2 0 \nJVASP -21422  Ba2CaOsO6  0.603  0 1.822  0 \nJVASP -38183  Rb3Tl  1.317  0 1.175  0.035357  \nJVASP -8567  Li2FeBr4  0.852  0 4 0 \nJVASP -8609  Sr2CoCl2O2  0.414  0 2.115  0.000768  \nJVASP -16654  CoCu2SnSe4  0.616  0 2.458  0 \nJVASP -26528  Rb2IrF6  1.404  0.35  0.996  0 \nJVASP -27213  Ni2Hg2OF6  1.058  0 8.165  0 \nJVASP -37387  Th3U  2.582  0 2.336  0.250054  \nJVASP -15391  TlCo2Se2  0.426  0 1.794  0 \nJVASP -16407  LiRhF6  0.561  0 1.999  0 \nJVASP -21502  Li2RhF6  2.022  0.06  1.993  0 \nJVASP -16367  MnAlPt  1.054  0 6.85  0 \nJVASP -27661  Zn2Fe3O8  0.465  0 12 0 \nJVASP -38614  MgPbO3  0.391  0 2 0.092507  \nJVASP -16368  MnAlPt2  1.057  0 4.176  0 \nJVASP -38289  RbSrO3  0.331  0 3 0 37 \n JVASP -16408  LiIrF6  1.158  0.28  2 0 \nJVASP -26823  Ba3NiIr2O9  3.343  0 9.097  0 \nJVASP -19598  V3Te4  0.92  0 6.915  0.008259  \nJVASP -16409  K2IrF6  1.396  0.3 0.995  0 \nJVASP -22415  Sr3MgIrO6  1.99  0.12  1.588  0 \nJVASP -16823  CrTe4Au  0.542  0 2.872  0 \nJVASP -14403  Mn2Sb  0.5 0 11.488  0.333028  \nJVASP -38341  RbAgO3  1.28  0 1.663  0.39356  \nJVASP -8383  YWF5  1.5 0 3.498  0 \nJVASP -17460  Ba2Mn3As2O2  3.292  0 3.371  0 \nJVASP -858 Co 0.531  0 3.153  5.10E -06 \nJVASP -55303  Mn3TeO6  0.305  0 29.848  0.054318  \nJVASP -17641  CaMnGe  0.313  0 4.563  0 \nJVASP -55470  Co2Mo3O8  7.451  0 0.623  0 \nJVASP -19739  TiHg  1.407  0 1.372  0.148755  \nJVASP -17315  CrAsRh  0.259  0 10.625  0.000782  \nJVASP -8384  YNiF5  0.372  0.01  2 0 \nJVASP -38244  Rb3In  0.418  0 1.284  0.094258  \nJVASP -39452  RuAu3  1.302  0 1.719  0.368509  \nJVASP -17265  BaIrF6  1.356  0.39  0.997  0 \nJVASP -17509  RhO2F6  3.993  0 4.048  4.00E -06 \nJVASP -8349  ZnCoF6  0.321  0 1 0 \nJVASP -19704  NbF4  0.254  0 0.682  0 \nJVASP -17643  CoNiSn  1.014  0 1.788  0.007715  \nJVASP -39453  RuAu3  1.487  0 3.489  0.397432  \nJVASP -38515  KAgO3  0.91  0 1.549  0.394317  \nJVASP -37981  CoTeO3  0.544  0 0.729  0.307307  \nJVASP -16338  FeTe  0.654  0 3.87  0.207063  \nJVASP -37453  TaTiFe2  1.042  0 0.83  0 \nJVASP -8385  YCoF5  1.175  0.2 3 0 \nJVASP -12407  TiFe6Ge6  1.016  0 8.704  0 \nJVASP -38248  Rb3Ga  0.469  0 1.472  0.112694  \nJVASP -17730  FeSn  1.009  0 3.584  0.176643  \nJVASP -16719  YFe2B2  0.429  0 1.027  0 \nJVASP -47356  Li2Si2WO7  0.887  0 4 0 \nJVASP -12287  ZrMnGe  0.267  0 8.375  0 \nJVASP -12608  Sr3Co2Cl2O5  0.693  0 3.49  0 \nJVASP -17854  Mn3SnC  0.392  0 3.33  0.009175  \nJVASP -17646  RbMnAs  0.255  0 3.881  0 \nJVASP -37314  SrAlO3  2.995  0 0.526  0 \nJVASP -36857  MnAuO2  0.374  0 3.984  0 \nJVASP -46728  Li4Mn3SbP4O6  3.224  0.09  15.761  0 \nJVASP -21188  Ca2FeIrO6  0.982  0 5.586  0 38 \n JVASP -17462  Sr2Mn3As2O2  3.371  0 7.453  0 \nJVASP -18123  FePd3  0.402  0 4.169  0 \nJVASP -19789  CrSb  0.378  0 5.604  0.153325  \nJVASP -17168  MoPt3  1.132  0 1.472  0.15261  \nJVASP -37028  TiInFe2  2.041  0 0.907  0 \nJVASP -16723  CrIr3  1.093  0 1.237  0 \nJVASP -10854  YFe2O4  0.834  0 14 0.213896  \nJVASP -37407  TePdO3  0.99  0 1.019  0 \nJVASP -37144  NbFe3  1.008  0 5.132  0 \nJVASP -49613  Y2Co2O7  0.266  0 1.903  0 \nJVASP -44414  Li2Mn3WO8  1.054  0 16.292  0 \nJVASP -46240  CoBi2O6  4.185  0 0.549  0 \nJVASP -12612  Ba2UCoO6  1.058  0 3.011  0 \nJVASP -18049  CeB6  0.88  0 0.721  0 \nJVASP -15970  Cr3SnN  1.006  0 1.687  0 \nJVASP -16474  TiCdHg2  0.529  0 0.835  0.175958  \nJVASP -44705  MnSb4O2  1.432  0.02  3 0 \nJVASP -10855  ZnFe2O4  3.993  0 4.001  5.50E -05 \nJVASP -37189  Mn3Ga  0.754  0 3.11  0.095038  \nJVASP -11508  TiBi2O6  1.145  0 3.857  0.332215  \nJVASP -11584  Sr4MgFe2S2O6  0.343  0 8.37  0 \nJVASP -37701  Y3Sn  0.286  0 1.596  0.106332  \nJVASP -8357  AlWF5  1.143  0 3.97  0 \nJVASP -8414  BaYCoCuO5  0.557  0 2.252  0 \nJVASP -49615  YMoO3  0.705  0 3.257  0.001451  \nJVASP -44418  Na3CrBAsO7  6.077  1.09  6 0 \nJVASP -17654  MnGaPt  0.463  0 3.336  0.246664  \nJVASP -37190  MnGaFeCo  1.018  0 3.068  0 \nJVASP -17478  Sc3In  1.013  0 2.465  0 \nJVASP -45802  Li8Cr3TeO2  3.222  0 5.932  0 \nJVASP -27678  Mn4ZnCu3O2  1.061  0 9.079  0 \nJVASP -18328  NaMnBi  0.348  0 8.108  0.057047  \nJVASP -11811  Ba2Mn2Sb2O  0.444  0 19.229  0 \nJVASP -18131  FeCu2Sn  0.881  0 2.688  0.263135  \nJVASP -12355  Zr2Fe3Ge  0.336  0 7.794  0 \nJVASP -38645  Ni3Au  1.248  0 3.103  0.097749  \nJVASP -11692  RbFeMo2O8  0.7 0 1 0 \nJVASP -37208  SiPdO3  1.016  0 1.954  0 \nJVASP -44507  Li2MnBAsO7  2.041  1.17  6 0 \nJVASP -10857  ZnCr2Se4  1.012  0 11.98  0 \nJVASP -17618  Mn2GaCo  2.034  0 2.019  0 \nJVASP -37946  Co3Bi  1.067  0 2.899  0.443311  \nJVASP -18209  Mn3Sn  0.789  0 10.047  0.202535  39 \n JVASP -18104  VCo2Sn  0.411  0 2.813  0 \nJVASP -17328  Mn3ZnN  1.005  0 4.201  0 \nJVASP -38764  ZnFeRh2  0.52  0 4.221  0 \nJVASP -52119  MnReO4  2.649  0 7.645  1.75E -06 \nJVASP -17828  ZnFeSb  0.582  0 2.538  0.474698  \nJVASP -18136  NdCoSi  0.936  0 0.511  0 \nJVASP -37600  Sr3Cr  1.006  0 4.864  0.495398  \nJVASP -8416  BaYVCuO5  3.021  0 1.571  0 \nJVASP -44720  P2WO7  2.354  0 4 0.008388  \nJVASP -37040  TiAu  1.722  0 1.708  0.04604  \nJVASP -18366  K2RuCl6  1.011  0 2 0 \nJVASP -42952  Li4Co3TeO8  0.945  0 9.012  0 \nJVASP -19792  Fe3Pt  1.068  0 8.229  0.057277  \nJVASP -46260  Li2Ni3BiO8  2.085  0 2.988  0 \nJVASP -37424  Ta2Be2O5  1.309  0 0.702  0 \nJVASP -10656  ZnMo2O4  0.664  0 3.853  0.156794  \nJVASP -36884  TlFeF3  0.305  0 4 0 \nJVASP -16803  MnAu  0.38  0 4.067  0.121371  \nJVASP -44512  Li4Mn5NbO2  0.993  0 2.81  0 \nJVASP -44654  LiZnFe0O6  8.443  0 15.74  0 \nJVASP -16853  Ni3Pt  1.307  0 2.199  0 \nJVASP -16804  MnAu  1.04  0 4.069  0.121112  \nJVASP -36885  TlCoF3  0.297  0 3 0 \nJVASP -17300  BaMn2As2  0.35  0 3.707  0 \nJVASP -17454  Sr2CoO4  0.271  0 1.953  0 \nJVASP -18213  MnCu2SnSe4  0.296  0 4.636  0 \nJVASP -16854  CoCu2Sn  0.4 0 0.955  0.222465  \nJVASP -17624  PuFe2Si2  2.996  0 4.569  0 \nJVASP -18302  VGaCo2  0.415  0 1.943  5.55E -05 \nJVASP -16806  NaMnTe2  1.044  0 4.052  0.093483  \nJVASP -18368  K2OsCl6  1.079  0.02  1.999  0 \nJVASP -11590  Sr4CaFe2S2O6  0.482  0 8.477  0 \nJVASP -37044  TiFe2As  0.274  0 1.008  0 \nJVASP -45852  Li2Nb2Fe3O0  1.282  0.08  8.067  0 \nJVASP -17625  Mn3GeC  0.333  0 2.997  0 \nJVASP -44518  Li5Nb2Fe5O2  5.969  0 15 0.043384  \nJVASP -18303  MnTePd  0.49  0 4.838  0 \nJVASP -20640  FePt  0.413  0 3.243  0 \nJVASP -17790  Mn2Sb  1.03  0 2.746  0.124949  \nJVASP -17237  FeNiPt2  0.661  0 4.629  0 \nJVASP -18214  BaMn2Ge2  1.008  0 4.419  0 \nJVASP -34926  Sr5Bi3  1.492  0 1.639  0 \nJVASP -17471  Sr2Mn3Sb2O2  3.303  0 8.403  0 40 \n JVASP -36950  CaTcO3  0.45  0 1.819  0 \nJVASP -9317  YFeW2O8  4.021  0 3.593  0 \nJVASP -18255  InFe2CuSe4  1.021  0 7.611  0 \nJVASP -46173  HfFeO3  1.518  0 16 0 \nJVASP -18349  TiI3 0.969  0 0.704  0.003149  \nJVASP -45854  Li3Nb2Fe3O0  2.947  0 5.002  0.026501  \nJVASP -17457  Ba2Mn3Sb2O2  1.028  0 9.098  0 \nJVASP -11697  LiFeAs2O7  0.353  0 4.997  0 \nJVASP -17394  MnCu2Sb  0.752  0 3.86  0.300664  \nJVASP -18174  Co2As  1.064  0 4.734  0.27925  \nJVASP -44438  Li2Co3SnO8  0.326  0 1.631  0 \nJVASP -17718  Mn2Ge  0.344  0 8.937  0.088172  \nJVASP -19978  NbF3  0.308  0 2 0.08838  \nJVASP -37051  Ti2GaFe  0.856  0 0.982  0 \nJVASP -44593  Li4Mn3Sn5O6  3.986  0 12.94  0 \nJVASP -19388  Ca2FeSbO6  0.337  0 8.905  0 \nJVASP -44528  Li2Co3TeO8  1.45  0 0.962  0 \nJVASP -11093  ZnFe4S8  0.621  0 6.639  0.033962  \nJVASP -34927  Ba5Bi3  2.017  0 1.848  0 \nJVASP -11592  Sr4MgCo2S2O6  1.067  0 5.16  0 \nJVASP -17303  CdRhF6  0.767  0.09  0.992  0 \nJVASP -17637  MnGaNi2  0.298  0 4.013  0.00848  \nJVASP -45210  NbV3O8  4.062  0 7.962  0 \nJVASP -18176  Rb2RhF6  1.013  0.19  0.996  0 \nJVASP -46361  TaFeO4  2.096  0 4.012  0 \nJVASP -34315  CuSeO4  0.457  0 1.95  0 \nJVASP -20585  ZrMn2  0.264  0 2.878  0.000832  \nJVASP -17721  MnSnPd2  0.25  0 4.136  0 \nJVASP -9209  Ba2YCo3O7  1.016  0 6.49  0 \nJVASP -18028  FeTe  0.283  0 4.155  0.104371  \nJVASP -36838  NiAuO2  0.432  0 1.669  0 \nJVASP -18220  Mn2CoSn  0.357  0 1.885  0.150639  \nJVASP -8486  PuNi5  3.485  0 3.303  0 \nJVASP -12626  InFeO3  0.624  0 5.654  0 \nJVASP -20602  Co3W  0.255  0 0.833  0.096613  \nJVASP -17304  HgRhF6  0.701  0 0.981  0 \nJVASP -44617  Li2Fe3SnO8  0.503  0 4.156  0.085692  \nJVASP -44742  Li5Ni5Sn2O2  5.359  0.02  8.956  0 \nJVASP -46857  Li2Co3SnO8  0.27  0 0.859  0.020608  \nJVASP -44461  VBiO3  0.756  0 8 0.014917  \nJVASP -19396  Ca2SbMoO6  0.43  0 2.121  0 \nJVASP -12627  Sr3Fe2Cl2O5  0.288  0 7.522  0 \nJVASP -18082  MnSnPt  0.399  0 3.63  0 41 \n JVASP -17803  FeSb2  0.727  0 1.72  0 \nJVASP -46861  FeSbO4  0.268  0 5.77  0 \nJVASP -42911  LiMnSbO4  4.231  0 9.477  0 \nJVASP -10907  AlBi3O9  0.588  0 5.951  0.027749  \nJVASP -18193  FeCo2Ge  0.291  0 5.279  0 \nJVASP -43066  CoSbO4  0.357  0 0.776  0 \nJVASP -44991  FeSb4O2  2.071  0.04  2 0 \nJVASP -44750  Li2Ni3WO8  1.136  0 3.997  0 \nJVASP -34319  TlV3Cr2S8  2.865  0 6.571  0 \nJVASP -45016  LiFeSnO4  0.456  0 2 0 \nJVASP -12631  CdFe2O4  1.621  0 4.006  0 \nJVASP -10484  Ba2SrIrO6  0.824  0.13  2.617  0 \nJVASP -9343  VW2O8  0.773  0 0.894  0 \nJVASP -9897  Mg2CrWO6  1.216  0 3.683  0 \nJVASP -12217  ZnFe2O4  6.588  0 4.001  4.93E -06 \nJVASP -9466  Ba2TlNi2O7  0.465  0 3.151  0 \nJVASP -34389  FeSnF6  0.503  0 4.039  0 \nJVASP -9216  Ba2YNi3O7  0.288  0 0.603  0 \nJVASP -12231  KFeMo2O8  0.529  0 1 0 \nJVASP -12637  FeMoClO4  2.006  0 4.497  0 \nJVASP -11536  YV2O4  0.506  0 9.995  0.094075  \nJVASP -10731  ZnCo4O8  0.253  0 1.777  0.01025  \nJVASP -9469  Ba2YTlV2O7  2.984  0 3.127  0 \nJVASP -9640  YCu2O4  1.012  0 5.939  0.335261  \nJVASP -9527  VZnSF5  1.25  0 3.998  0 \nJVASP -9203  Ba2AlNi3O8  1.083  0 1.341  0 \nJVASP -9676  ZnCr2Se4  1.012  0 11.98  1.06E -05 \nJVASP -9196  Ba2AlCr3O8  3.056  0 9 0 \nJVASP -34404  NbCrF6  3.008  0 2.115  0 \nJVASP -52140  Ba3Ti3O8  0.38  0 1.571  0.016329  \nJVASP -34347  BaFeF4  0.924  0 8 0 \nJVASP -9363  BaZnFe4O8  1.021  0 18.898  0 \nJVASP -11670  Sr2CoMoO6  0.312  0 2.993  0 \nJVASP -9258  Sr2AlTlV2O7  3.397  0 4 0 \nJVASP -11731  MnPt3O6  0.276  0 4.87  0 \nJVASP -9962  V2ZnO4  0.342  0 6.627  0.117976  \nJVASP -9185  Ba2YV3O8  1.012  0 2.24  0 \nJVASP -12643  Ba2UMnO6  1.931  0 5 0 \nJVASP -34878  K2Zr7Cl8  0.894  0 1.885  0 \nJVASP -34479  Ba4Fe2S4I5  0.512  0 0.957  0 \nJVASP -10839  YMnO3  1.006  0 7.998  1.11E -05 \nJVASP -9366  BaCaCo4O8  0.802  0 7.42  0 \nJVASP -9966  ZnFe2O4  0.529  0 12.329  0.144093  42 \n JVASP -34752  KCdN3O6  0.544  0 4.001  0.377445  \nJVASP -12648  MnSnB2O6  0.926  0.27  5 0 \nJVASP -34485  ZrCoF6  0.875  0.18  3 0 \nJVASP -9267  Sr2YTlV2O7  1.041  0 2.878  0 \nJVASP -24743  Fe3W3N  1.934  0 4.38  0 \nJVASP -34425  Ba6Ru2PtCl2O2  4.102  0 5.99  0 \nJVASP -9187  Ba2YCo3O8  1.025  0 2.726  0 \nJVASP -9269  Ba2TlBi2O7  1.109  0 0.555  0 \nJVASP -78840  Mn3Ge  3.009  0 1.039  0.000129  \nJVASP -9491  Sr2FeCuSO3  0.348  0 8.436  0 \nJVASP -9541  YFeO3  1.005  0 8.072  0.059007  \nJVASP -34429  ZrFeF6  1.004  0 4 0.013753  \nJVASP -10342  Ge2MoO6  0.277  0 3.999  0 \nJVASP -52113  Sr2CoReO6  0.407  0 2.047  0 \nJVASP -24596  RbCoCl3  0.309  0 5.992  0 \nJVASP -9494  ZnCoPO5  1.109  0 2.002  0 \nJVASP -79568  MnGaFe2  1.942  0 2.075  0 \nJVASP -78684  MnSnPt  0.274  0 3.406  0.210315  \nJVASP -79239  Mn3Pt  0.791  0 1.908  0.093006  \nJVASP -9495  Sr2CoSO3  1.033  0 4.155  0 \nJVASP -79241  MnGePd2  0.428  0 4.112  0 \nJVASP -79574  NbFe3  0.886  0 3.23  0.06218  \nJVASP -9931  ZnFe2O4  5.142  0 4.001  0 \nJVASP -9496  Sr2MnSO3  0.286  0 6.149  0 \nJVASP -79576  MnGaFe2  0.319  0 6.399  0.037086  \nJVASP -78859  NiBrO  1.006  0 1.299  0 \nJVASP -79583  Mn2CuGe  0.438  0 0.669  0.107469  \nJVASP -79200  VGaCo2  0.4 0 1.943  0 \nJVASP -24841  Y6OsI0  1.144  0.04  1.177  0 \nJVASP -80740  Ti2GaFe  0.849  0 0.982  8.37E -05 \nJVASP -79431  MgMnPt2  1.202  0 4.386  0 \nJVASP -79586  VFeCoAs  1.004  0 2.944  0 \nJVASP -16934  YCoO3  1.005  0 1.171  0.25161  \nJVASP -78508  CoSI  0.685  0 1.969  0.44561  \nJVASP -78280  Mn2Sb  1.068  0 1 0.143192  \nJVASP -79206  FeSe  0.421  0 4.509  0.198133  \nJVASP -79916  NbZnCo2  0.537  0 0.74  0 \nJVASP -78380  BaN 0.442  0 1 0.268514  \nJVASP -78429  NaSe  0.252  0 0.915  0.374696  \nJVASP -79435  Mn2CoSn  0.729  0 1.883  0.150843  \nJVASP -78470  RbSe  0.288  0 0.998  0.455465  \nJVASP -78658  NiPt  2.665  0 1.828  0 \nJVASP -79097  Mg3Re  1.082  0 0.51  0.361552  43 \n JVASP -79593  Fe3Pt  1.068  0 8.23  0.057144  \nJVASP -80098  MnTe  1.038  0 8.073  0.012304  \nJVASP -78434  KSe 0.252  0 0.996  0.421998  \nJVASP -79500  VFeCoGe  0.629  0 1.954  0 \nJVASP -82132  MnBi2Te4  1.368  0.03  4.998  0 \nJVASP -80251  CrSnRh2  0.297  0 2.89  0.238679  \nJVASP -78832  MnBi  2.039  0 6.925  0.277638  \nJVASP -79457  CrInNi2  0.459  0 3.557  0.082176  \nJVASP -79604  VGaFeCo  0.413  0 0.952  0 \nJVASP -79562  GaFeNi2  0.339  0 3.035  0.004254  \n \n \n \n "    },    {        "title": "2102.03791v1.Recent_progress_and_challenges_in_magnetic_tunnel_junctions_with_2D_materials_for_spintronic_applications.pdf",        "content": " \n Recent progress and challenges in magnetic tunnel junctions \nwith 2D materials  for spintronic applications  \n \nLishu zhangab, Jun Zhoub, Hui Li*a, Lei Shen*c, Yuan Ping Feng*bd \n \n \na Key Laboratory for Liquid -Solid Structural Evolution and Processing of Materials, \nMinistry of Education, Shandong University, Jinan 250061, China  \nb Department of Physics, National University of Singapore, 2 Science Drive 3, \nSingapore 117542, Singapore  \nc Department of Mechanical Engineering, National University o f Singapore, 9 \nEngineering Drive 1, Singapore 117542, Singapore  \nd Center for Advanced 2D Materials, National University of Singapore, 6 Science Drive \n2, Singapore 117546, Singapore  \n \n \n* Corresponding author: lihuilmy@hotmail.com  (H.L.) ; shenlei@nus.edu.sg  (L.S.);  \nphyfyp@nus.edu.sg  (Y .F.); \n   \n Abstract  \nAs Moore’s law is gradually losing its effectiveness, developing alternative high -speed \nand low -energy -consuming information technology with post -silicon advanced \nmaterials is urgently needed. The successful application  of tunneling magnetoresistance \n(TMR)  in magnetic tunnel junctions (MTJs)  has given rise to a tremendous economic \nimpact on magnetic informatics, including MRAM, radio -frequency sensors, \nmicrowave generators and neuromorphic computing networks. The emergence  of two-\ndimensional ( 2D) materials brings opportunities for MTJs based on 2D materials which \nhave many attractive characters and advantages . Especially, the recently discovered \nintrinsic 2D ferromagnetic materials with high spin -polarization hold the promise for \nnext-generation nanoscale MTJs. With the development of advanced 2D materials, \nmany efforts on MTJs with 2D materials  have been made both theoretically and \nexperimentally. Various 2D materials, such as semi -metallic graphene, insulating h -BN, \nsemiconducting MoS 2, magnetic  semiconducting CrI 3, magnetic metallic Fe 3GeTe 2 and \nsome other recently emerged 2D materials are discussed as the electrodes and/or central \nscattering materials of MTJs in this review. We discuss  the fundamental and main  issues \nfacing MTJs, and review the  current progress made with 2D MTJs , briefly comment on \nwork with some specific 2D materials,  and highlight how they address the current \nchallenges in MTJs , and finally offer an outlook and perspective of 2D MTJs .  \nKeywords  \nMagnetic tunnel junctions, spintronics, 2D materials, spin -orbit torque, \nmagnetoresistance  \n   \n Contents  \n1. Introduction  ................................ ................................ ................................ .............  4 \n1.1 Spintronics  ................................ ................................ ................................ .... 7 \n1.2 MTJs  ................................ ................................ ................................ ...........  10 \n2. Current status and issues of MTJs with 3D materials  ................................ ...........  13 \n3. 2D materials  ................................ ................................ ................................ ..........  16 \n3.1 Theoretical prediction and experimental synthesis  ................................ .........  17 \n3.2 Advantages of 2D materials in solving current 3D MTJs problems  ...............  18 \n4.  2D -materials -based MTJs  ................................ ................................ .....................  19 \n4.1 Target ing high spin -polarization  ................................ ................................ ..... 19 \n4.2 Targeting effective spin -injection  ................................ ................................ ... 27 \n4.3 Targeting spin -manipulation  ................................ ................................ ...........  33 \n4.4 Targeting stability ................................ ................................ ............................  38 \n5. Summary and perspectives  ................................ ................................ ......................  40 \nAcknowledgements  ................................ ................................ ................................ ...... 42 \nNotes and references  ................................ ................................ ................................ .... 43 \n \n \n   \n 1. Introduction  \nThe conventional silicon -based metal -oxide -semiconductor devices which work \non the manipulation of charges (one degree of freedom of electrons) will come to an \nend in the near future due to more fundamental issues . Exploring advanced information \ntechnology  with high -speed operation and low -energy consumption to replace existing \nsilicon -based technologies is urgently demanded. So far, many  strategies have been \nproposed, like nanoelectronics,1-4 molecular electronics,5 spintronics,6-9 and quantum \ninformation technologies10-12. Amo ng them, spintronics has exhibited tremendous \npotential and thus attracted lots of attentions. Spintronics which is based on \nmanipulation of spins (another degree of freedom of electrons) have promise to \nintegrate memory technology at the heart of informat ion processing units such as \nclassical and neuromorphic, which would be a big change in how architectures are \ndesigned towards in -memory computing (currently memory and logic are separated \nlayers that need to communicate).  In addition, spintronics is more compatible with \nconventional electronics, compared to other strategies, so that many techniques applied \nin traditional electronics can be extended to spintronics.  \nEven though information is processed using spin, it is desirable to manipulate spin \nor switch  magnetization using electrical means with  magnetoresistive materials. \nMagnetoresistive devices are constructed based on the magnetoresistive  materials \nwhich in general exhibit a change in resistance with the application of a magnetic field. \nEarly in its development, magnetoresistance devices ma de use of the anisotropic \nmagnetoresistance (AMR) effect . Recently, such devices are mainly based on  giant \nmagnetoresistance (GMR) effect , or tunneling magnetoresistance (TMR) . The \ndiscovery of magnetic tunnel junctions (MTJs) at room temperature  dramatically \nincreased the storage density .13, 14 If a MTJ has TMR over 100%, it can be used to make \nnot only magnetic field sensors15 and reading heads of hard drives,16 but also \nmagnetoresistive random access memories (MRAM)17. The more recent demonstration \nof spin transfer torque (STT) 18 and spin -orbit torque (SOT) effect 19, 20 makes MTJs \nmore valuable for manufacturing multitudinous spintronics, including MRAM,21-23  \n radio -frequency sensors,24, 25 microwave generators26 and even artificial neuromorphic \nnetworks27. STT allows  change of the magnetization direction of a material by a spin -\npolarized current. By passing a current through a thick magnetic layer (the “fixed \nlayer”), one can produce a spin -polarized current. If this spin -polarized cu rrent is \ndirected into a second, thinner magnetic layer (the “free layer”), the angular momentum \nof charge carriers (such as electrons) can be transferred to this layer, changing its \nmagnetization orientation. This can be used to excite oscillations or eve n flip the \norientation of the magnet. T hus, the different resistive states , a low -resistance parallel  \n(P) magnetic configuration and high -resistance state , anti-parallel  (AP) magnetic \nconfiguration), can be realized to represent  the ‘0’ and ‘1’ state in the STT memory \nrespectively.28-30 Thanks to voltage -depended switching ability for STT principle, the \ndifferent logic gates can be reconstructed by the same MTJ structures with only one \nsingle -cycle operation. This promising computing har dware application proves MTJs \nhave great potential to be applied in more aspects in the future.  SOT is another approach \nof magnetization switching, an interconversion of charge and spin current, which is a \npromising phenomenon that can be used to improve t he performance of MRAM \ndevices21-23. The SOT effect has been achieved in heavy metal 31 and 2D topological \ninsulator (TI) systems32. It essentially requires two functional layers, namely, one \nferromagnetic (FM) and one nonmagnetic layer with large sp in-orbit coupling (SOC). \nThe latter is to accumulate spin charges and inject it into the adjacent FM layer. The \nspin current then exerts a torque on the magnetic moment of the FM layer and revert it \nwith an angle. The switching efficiency strongly depends on the strength of SOT. \nFurthermore, a large spin polarization in the nonmagnetic layer is necessary for efficient \nspin injection. Thus, heavy metals and 2D TIs with large spin Hall angles, such as Bi 33 \nand 2D -TI α-Sn,34 are used for SOT switching. Much progress has been made in SOT \nmagnetic switching with heavy metals 35, 36. Recently, TIs, such as Bi 2Se3, have \nattracted attention due to their spin -momentum locking property which in principle is \nable to achieve an efficient SOT switching, even though the switching process needs to \nbe further  improv ed in terms of the switching hysteresis and its completeness.37  \n Advantages  MTJs devices include  low-power consumption with high processing \nspeed, non -volatility, metal –oxide –semiconductor technology compatibility and high \nintegration density.38,39 The most common material s used to fabricate  MTJs are \nferromagnetic metals an d alloys such as Fe and CoFeB, Heusler alloys and dielectrics \nlike MgO and AlO x.40-43 What cannot be ignored is that remarkable breakthrough has \nalso been achieved in 2D materials synthesis and MTJs begin to be created based on \nthem.44 Due to its low dimensionality and quantum nature, the us e of 2D materials add s \nmany unique features into MTJs such as flexibility, and extremely high scaling.45, 46 \nFirst-principles calculations based on density  functional theory (DFT) have been \nthe most  widely used method  in theoretical studies of  2D materials and  their \napplications in devices such as MTJs. It can be said that the rational design of high -\nperforming MTJs would always face challenges in experiment s without proper  \ntheoretical guid ance, such as in the development  of (Co)Fe/MgO/(Co)Fe MTJs . \nCompared to  other theoretical and computational methods, the  first-principles method \ndoes not require empirical  parameters and experimental inputs, which makes it an ideal \nmethod for studying new materials  and their heterostructures . DFT calculations are also \nvaluable in  predict ing materials and device behaviors under  extreme conditions that are \ndifficult to achieve experimentally.  \nDFT was proposed by Hohenberg and Kohn  in 1964 .47 And at the very next year \nKohn and Sham launched its primary fulfillment.48 DFT has become a convincing \nquantum simulation method in exploring the electronic structure of many systems \nthrough the use of the electron density as the fundamental vari able instead of the \nelectron wave function. The first MTJ based on Fe/MgO/Fe was proposed by DFT \ncalculations,49 and was subsequently demonstrated experiment ally40, 50. Now, the \nFe/MgO/Fe -based MTJs are the main components in the reading head of the hard disk \nin personal computers. Besides, in searching for the magnetism of 2D materials  and \npredicting new structures, first -principles calculations has also been used as a powerful \ntool for providing theoretical guides for experimental exploration.  \nEven though DFT has been very successful in studying and predicting new  \n materials, it is still a computationally expensive method. Despite of significant \nimprovements in recent decades, it is still difficult to incorporate all of the experimental \n\"real world\" subtlety. Concerning calculation of transport properties, many systems \nbeyond MgO have been initially predicted by DFT to lead to high spin polarizations \n(with similar symmetry arguments), but experimentally their performance so far failed \nto match that of  MgO. As a r esult, direct comparison between computational prediction \nand experimental measurement for quantities such as MR ratio is non -trivial. \nConsidering also the fact that there have been many computational studies on MTJs but \na limited experimental realization of the predicted structures, here we mainly focus on \ncomputational works in this review , but also discuss available related experimental \nworks . We begin with a brief introduction to spintronics and conventional MTJs.  And \nthen we discuss the current status on MTJs , highlighting  the problems  encountered and \nchallenges . This is followed by  the advantage s of 2D materials in solving th ose MTJs \nchallenges. The rest of this review is organized as follows. In Sec. 4 , we review the \nrecent progress  of 2D-materials -based MTJs . This is organized into four subsections  \nbased on the key issues in 2D MTJs , i.e., target ing at spin polarization ( Sec. 4.1 ), spin \ninjection ( Sec. 4.2 ), spin manipulation (Sec. 4.3 ) and stability ( Sec. 4.4 ), respectively . \nWe finally conclu de and offer an outlook and perspectives in Sec. 5. \n \n1.1 Spintronics  \nConventional  electronic devices have one thing in common, that is, they rely on \nthe electronic transport in semiconductor materials such as silicon. With the size and \nfunction of silicon -based electronic devices reaching the limit, further downscaling of \nsilicon based electronic devices becomes impossible . New concepts are required for \nfuture electronic devices which should also meet certain requirements such as low \npower operation . To this respect, it is noted that  the energy scale of spin dynamics is \ntypically many orders of magnitude smaller than that of charge dynamics, and low \npower electronics operation can thus be achieved in spintronic devices.  Spintronics has \nbecome as a rapidly developing field  under this b ackground. Spintronics is based on  \n the manipulation of spin of electrons to store, encode and transmit data.  \nIn spintronics, i nformation is first marked as up spin or down spin; and the spin -\ncarrying electrons are transported along a path; and finally at a final point, the spin \ninformation is read. The conduction electrons’ spin orientation needs to  sustain for \nseveral nanoseconds, in order for them  to be used in electrical circuit and chip. \nTransporting current through a ferromagnetic material and transmit ting the spin-\npolarized electron s to the receiver is a common method to generate spin -polarized \ncurrent. The successful implementation of  spintronic devices and circuits ( Fig. 1 ) relies \non the realization  of six elementary functionalities: spin –orbital torque, spin detection, \nspin transport, spin manipulation, spin -optical interaction and single spin  device . Spin–\norbital torque  is induced through the spin –orbit al interaction in FM/heavy metal  \nbilayers by flowing an in -plane electrical current.51 The spin detection include s \ndetectio n of circularly polarized light,52 transient Kerr/Faraday linearly polarized light \nrotation,53 spin Hall voltage,54, 55 electric resistance change,56 and tunneling‐induced \nluminescence microscopy ,57 which has been the most convenient method from device \nperspective and applications till date.  Spin transport is expected  to offer low -loss spin \nchannels,  which can provide long -distance propagation of spin signals and enabl e more \noperations  of spin signals .58, 59 Spin manipulation is required to achieve more \nfunctionality of spintronic device s like that in electronic devices .60-62 Spin-optical \ninteraction s include spin -Hall effects in inhomogeneous media and at optical interfaces, \nspin-dependent effects in nonparaxial (focused or scattered) fields, spin -controlled \nshaping of light using anisotropic structured interfaces (metasurfaces ), and robust spin -\ndirectional coupling via evanescent near fields.63 The single -spin, singlet, and polarized \nphases of a quantum  dot allow different currents to flow through the dot.  The spin state \nof the dot  is controlled either by adding electrons or by tuning the magnetic field, and \nthus a prototype single -spin transistor  is produced.  64, 65 \n At present, a variety of spin electronic devices  based on different mechanisms  of \nspintronics  have been studied and designed . Several applications are also highlighted \nin Figure 1. In a SOT-driven device, the heavy metal induces strong SOC and thus  \n generate a SOT -driven  switching. In a spin logic device, b y defining bistable \nmagnetizations of electrodes  along the easy axis as the input  logic (‘1’ and ‘0’) and the  \nas-detected current as the logic output , Boolean operation s can be achieved . The spin \nelectrons transport from one FM layer to another FM layer , by passing through a barrier, \nhigh and low spin current can be achieved by the magnetic alignment of the two \nelectrodes . As such, this kind of device s, called MTJs , can be used as a memory device \nto store information even under the power -off state (nonvolatile) . Spin field effect \ntransistors  (FET)  was proposed by Datta and Das  first.66 It is based on manipulation of \nelectron spin during transport driven by an electric field in semiconductors. This device \nworks similarly to  a charge -based  transistor. A spin current is injected into the channel \nmateri al from a FM electrode (source), in which spin polarization is electrically \nmanipulated by a gate voltage (or other means), and finally spin polarization is detected \nat the drain. In a spin light emitting diode ( LED ), when the spin -polarized electron  is \ninjected, it recombines with a hole and emits a circularly polarized photon which is \nused to as sess the polarization of the injected spin . When a  spin polarized electron is \ninjected into a quantum dot, the spin state  of the quantum dot can be changed  and can \nbe controlled by a gate voltage. All these spintronic devices have lower power \nconsumption, lower cost, more stable and excellent performance in high -capacity \nstorage  than traditional electronic devices whose operation principle is only based on \ncharge . Therefore , spintronic devices will play a great role in the next generation of \nelectronic information science and technology.  \nIn the past several years, three important  focus area s of spintronics research  have \nbeen explored by scientists: 1) fabricating nanoscale structures including new magnetic \nmaterials, hybrid heterostructures, and functional materials; 2) studying the spin effect \nincluding spin injection, transport and detection; and 3) improving the performance of \nMTJ -based devices.  \n  \n  \nFig. 1 Overview of  some spintronics  devices . Key aspects , including physical effects, \nelementary  functionality and applications, are schematically illustrated. The elementary \nfunctionalities include spin –orbital torque , spin detection, spin transport , spin \nmanipulation , spin -optical interaction and single spin . The corresponding applications  \nof these functionalities  includ e the SOT device, spin logic device, MTJ, spin FET, spin \nLED, and spin quantum dot  device . In this review, we focus on the magnetic tunnel \njunction.  \n \n1.2 MTJs  \nUnder the well -developed knowledge on how to manipulate spins,67-74 one can \ngener ate state -of-the-art spintronics devices with desired  properties. Thus, it is vital to \nexplore the application possibilities of spintronic effects in order to achieve more \npromising spintronics devices. Such electronic devices have made a big impact on \ncomputer technology through achieving higher and higher information storage in hard \ndesk drives as well as faster and faster reading speed of data in RAMs. MTJ is one of \n \n the most important form s in spintronics applications as mentioned in Fig.1 . In this \nsection, we will introduce MTJs . \nA basic MTJ consists of two ferromagnetic layers separated by a thin insulating \nlayer, as schematically shown in Figs. 2 (a) and (c). The tunneling conductance or \nresistance of such a  device depends on whether the  magnetizations of the two electrodes \nare parallel or antiparallel. If RP and RAP are the resistance  in the parallel and antiparallel  \nstate, the tunneling magnetoresistance (TMR) ratio is given by75 \nTMR=𝑅𝐴𝑃−𝑅𝑃\n𝑅𝑃=2𝑃1𝑃2\n1−𝑃1𝑃2 \nwhere P1 and P2 are the spin polarization  of the two electrodes . The origin of TMR \narises from different density of states (DOS) for spin up and spin down electrons as \nshown in Figs. 2 (c) and (d). Because electron spins are preserved during the transport, \neach type of spin can only tunnel into the subband of the same  spin. Therefore, the \ntunnel current is high (or resistance is low) when the magnetization s of the two \nelectrodes are parallel due to the match ing DOS on both sides [ Fig.2(b) ], and that in \nthe antiparallel state is low (or resistance is high) [ Fig.2(d) ], even though this may \nchange depending on spin selection at the interface. It is worth noting that  although it \nis insignificant for small TMR , for large negative TMR , the resistance variation is \nsometimes normalized preferentially by RAP, so as to obtain the same value as positive \nTMR , and thus being comparable in absolute value.76 \nTMR at room temperature wa s first demonstrated by Miyazaki 77 and Moodera 78. \nImmediately after that the TMR ratio was risen rapidly to 81% in a Co 0.4Fe0.4B0.2 (3)/Al \n(0.6) -Ox/Co 0.4Fe0.4B0.2 (2.5) (thickness in nm) MTJ at room temperature.79 \nSubsequently , TMR ratio as large as 604% was achieved  in MgO based MTJ,  \nCo0.2Fe0.6B0.2 (6)/MgO (2.1)/Co 0.2Fe0.6B0.2 (4) (thickness in nm) at room temperature.80 \nThe dramatic increase in MR ratio, compared to that of its predecessor, GMR devices, \nled to the domination of MTJs in magnetic data storage industry .  \nThe first successful application of MTJ was demonstrated in computer read head \ntechnology with Al 2O3 barrier and MgO barrier MTJs. The magnetic recording density  \n in the hard disk drive increased considerably compared to  traditional devices.81-\n85Another MTJ application is to develop the MRAM  which exceeds the density of \nDynamic RAM (DRAM), speed of static RAM (SRAM) and non -volatility of flash \nmemory. Moreover, these nanoelectronics generate less heat and operate  at lower power \nconsumption.  \n \nFig. 2 (a-b) Schematic diagram of  MTJs in P config uration and corresponding band \ndiagram . (c-d) Schematic diagram of  MTJs in AP configuration and corresponding band \ndiagram . (e-g) The exampled applications for MTJs, i.e., the read head, microwave \noscillator and MRAM.  \n \nBecause there is no or almost no interlayer coupling between the two \nferromagnetic layers in MTJs, only a small external magnetic field is needed to reverse \nthe magnetization direction of one ferromagnetic layer, thus realizing a huge change in \ntunneling r esistance. Therefore, MTJs have much higher magnetic field sensitivity than \nmetal multilayer films. At the same time, MTJs have high resistivity, low energy \nconsumption and stable performance.  All in all , MTJs act as one of the most important \nspintronics  applications , which can use as  a key component in many spintronics devices, \n \n such as,  the read head  of hard disk drives , microwave oscillator  and MRAM , as shown \nin Figs. 2 (e -g). Recent studies have demonstrated that magnetization in the free layer \nof an MTJ can be switched by STT or SOC , even though in practice  it is not so easy \ndue to  highly spin polarized current density  required . \nTo fabricate an MTJ with a giant TMR is crucial for practical applications. With \nthe development of nanotechnology, there are more and more ways to construct junction \nstructures . For the preparation methods in the laboratory, methods such as molecular \nbeam epitaxy (MBE),86 magnetron sputtering,87 electron beam evaporation88 and \nchemical vapour deposition (CVD),89 are often used . In industry, method s used to \nprepare micron, sub -micron and nano mag netic tunnel junction, magnetic tunnel \njunction array, TMR magnetic read -out head and MRAM include lithography, electron \nbeam exposure, ion beam etching, chemical reaction etching, focused ion beam etching, \netc. Among them, lithography combined with ion be am etching is the preferred process \nwith low cost and mass production in micromachining process. Generally, all MTJs \nconsisted of FM layers and an insulator layer. The most common ways to fabricate the \nFM layer in the past years is sputter deposition (magn etron sputtering and ion beam \ndeposition).90-92 The magnetic alignment and thickness are the key parts of MTJ \nfabrication in the experiment. A better method to fabricate insulating layer alway s keeps \nforging ahead. For example, ion beam oxidation,93 glow discharge,94, 95 plasma,96 \natomic -oxygen exposure97 and ultraviolet -stimulated oxygen exposure98 have been \nused as alternate ways for the insulator -layer deposition. In terms of preparation and \nprocessing, the issues about the control of the oxide barrier and the interfaces, the \nshielding tolerance, the thermal stability, and the robustness of the lifetime of the device \nneed to be solved urgently.  \n \n2. Current status and issues  of MTJs  with 3D materials  \nWith the rapid development of MTJs , the TMR value of MTJs  increases rapidly in \nthe last  few years, and quickly  approaches  the theoretical value. However, at present, \ndespite of extensive studies and much progress has been made, there are still  many  \n problems  and challenges  that need to be understood and addressed to improve the \nefficiency , performance and stability of MTJs . For example,  one of the important  issues \nis to control the quality of the interface between ferromagnetic layer and barrier layer .76 \nThe effect of the interface bonding on magnetoresistive properties  must be considered ,99, \n100 because it determines the effectiveness of transmission of electrons with different \norbital properties (and/or symmetry) through the interface, and electrons with different \norbital properties carry unequal spin polarization.101 Therefore, the interface bonding \nhas profound effects on conductance.102 For this aspect, layered 2D materials which is \nconnected by van der Waals (vdW) force without chemical bonding can avoid these \nproblems. And also, it is known that a long spin lifetime in the  nonmagnetic (NM) \nmaterials  and an efficient polarization of the injected spin are required . Fert et al.100 \nshow that introducing a spin dependent interface resistance at the FM/NM interfaces \ncan solve the prob lem of  the conductivity mismatch between FM and NM materials. \nThey find a significant magnetoresistance can be obtained if the junction resistance at \nthe FM/NM and NM/FM interfaces is chosen in a relatively narrow range depending \non the resistivity, spin d iffusion length and thickness of NM.  However, introducing 3D \ntunneling barrier such as Al 2O3 indeed has effect on improve s pin injection efficiency103, \n104, but new issues such as pinholes and clusters emerge105. And using 2D materials like \nh-BN which owns w ell defined interface contact  and l ess defects  to act as tunneling \nbarrier can avoid these new issues perfectly106. In addition, the spin -dependent \nelectronic structure of electrodes,107 the symmetry selection rules that are known to \ncontrol TMR in MTJs with electrodes and crystalline tunnel barriers,50 the role of the \ntunnel bar rier layer and its electronic structure49, 108 are all important issues and deserve \nserious attention . Moreover, the diffusion and oxidation process of elements in barrier \nlayer is another puzzling issue in the growth process. Taking  CoFeB/MgO/ CoFeB \nMTJs  as an example , Burton et al . found that B atoms in the crystalline CoFeB \nelectrodes tend to migrate to the interface which leads to a decrease  in the  TMR  ratio \ndue to a significant suppress ion of the majority -channel conductance  through states of \n∆1 symmetry .109 Similarly, the diffusion and oxidation process also have  an influence  \n on tunneling process .110 Another thing is the tunneling mechanism of barrier layer .111 \nMgO systems show much larger TMR than that of traditional AlO x systems.112-114 \nHowever, the tunneling mechanism in single -crystalline or textured MgO barriers is \nquite different from traditional AlO x amorphous barrier materials.111 In addition, \nperpendicular magnetic anisotropy (PMA)  of out-of-plane  magnetized MTJs  aroused a \nlot of attention, because  in-plane anisotropy only yields a typical anisotropy field in the \n100–200 Oe due to shape anisotropy, while PMA can yield an effective anisotropy field \nof several kOe.115 In order to realize the miniaturization of devices, it is urgent to \ndevelop processing technology on atomic scale. However, controlling thickness of \nmetal oxide tunnel barriers is hard and challenging.116, 117 2D materials have the natural \nadvantage of ultra -thin to atomic scale, which can avoid some troubles in processing. \nAnd it also need s to develop device s working at room -temperature. But most of the \nexisting FM materials own too low Curie temperature  (TC) caused the  need to search \nfor room temperature FM . Last but not least , the origin  and the influence of different \nlayer  thicknesses on the transport properties  are also important. Understanding and \nresolve these issues  will greatly promote the progress of MTJs  in theor etical  and \npractical application s. The discovery of 2D materials  and their heterostructures  \nprovides a new playground for MTJs. Some of the 2D mat erials may offer promising \nroutes to resolve some of these issues with their unique properties like sharp interfaces , \nnatural and tunable van de Waals insulating gap , layer -by-layer control of the thickness, \nhigh PMA , the potential for a diffusion barrier (thermal stability), and even provides \nthe possibility of new functionalities such as spin filtering.  Under this background, this \nreview  discusses  and summarizes  around the following  four main  problems , as shown \nin Fig. 3 : (1) spin polarization, (2) spin injection, (3) spin manipulation, and (4) spin \nstabilit ies.  \n  \n  \nFig.3 Typical problem -oriented designs via  MTJs.  \n3. 2D materials  \nWhen thin down the layered materials to their physical limits, they exhibit novel \nproperties which is different from their bulk counterpart . Thus, these materials are \nspecifically referred to as “2D materials”. In other words, 2D materials refer to \nmaterials in which electrons can only move freely in two dimensions (in a plane) on the \nnon-nanometer scale (1 -100 nm). 2D materials emerged  with t he successful separation \nof graphene, a single atomic sheet  of carbon atoms with a bonding length of 1.42Å , by \nGeim’s team in Manchester University in 2004.118 Graphene is also regarded as the \nmost widely studied 2D material. It has been well -known that the pristine graphene is \na unique 2D hexagonal structure with zero -bandgap and semi -metallic property, which \nis an important allotrope o f carbon.  Due to 2D materials’ various crystal structures and \nphysical properties, many other 2D materials beyond graphene are also undergoing a \nlot of research work, including semiconductors (e.g., transition metal dichalcogenides \n \n such as MoS 2), insulator s (e.g., h -BN), superconductors (e.g., NbSe 2), and magnets (e.g., \nFe3GeTe 2). These great advances have expanded 2D nano devices.   \nThe research on MTJs has been committed to continuously improve the high TMR \nratio all the time . However, in the process of further improving the performance and \nreducing the size, more and more challenges emerge.  Some of the  2D materials may \noffer promising routes to resolve some of these issues with their unique properties , and \neven provides the possibility of new functionali ties such as spin filtering.   \n3.1 Theoretical predict ion and experimental synthes is  \nRecently , more and more 2D materials were predicted theoretically and some of \nthem have been  synthesized experimentally . Take 2Dmatpedia  database119 as an \nexample, this database contains 6 ,351 2D materials  at presen t, of which  1500  materials \nare magnetic  based on the spin -polarized DFT calculations  (the total magnetic moment \nof the material is large  than 0.5 μB). The known 2D magnetic materials are summarized \nin Fig. 4  120 and they are divided into different  categories according to whether they are \nsynthesized experimentally or predicted theoretically and their properties . By \ncombining  2D materials with different properties  from this table  as he terojunctions, \nrich 2D MTJs can be constructed . \n In order to make 2D magnetic devices , the long -range magnetic order  in 2D \nmagnetic materials  is necessary.  It is known that  adjacent atomic moment s or spins  are \ncoupled through a n exchange interaction  in a lattice , leading  the magnetic order  of \nmaterials.121 The magneti sm depends  on the lattice dimensionality or crystal structure, \nas well as the spin dimensionality  of the system. The uniaxial a nisotropy  is able to \nsustain long -range  magnetic ordering, which has been experimentally observed in \nseveral  magnetic 2D materials recently. It is worth noting that 2D Ising kind of  behavior \nhas been reported in neutron scattering experiments in layered  materials much earlier.122 \nThe existence of magnetism down to monolayers in several magnetic 2D materials has \nbeen established very  recently . Intrinsic magnetic order in 2D layered Cr2Ge2Te6 was \nfound at low temperatures  in 2017.123, 124 Almost the same time , the ferromagnetic orde r  \n was also found  in monolayer CrI 3 up to 45 K.  Subsequently, several magnetic materials  \nincluding VSe 2 and Fe 3GeTe 2 have been found at room temperature.  These findings \noffer  a new opportunity to manipulate the spin -based devices efficiently in the future .125, \n126 \n \nFig. 4 2D magnetic material s library . In this diagram , the gray line below lists \ntheoretically predicted vdW ferromagnets (left), half  metals (center), and multiferroics \n(right) , respectively. And others above are 2D magnetic materials have been \nexperimentally confirmed. Among them are bulk ferromagnetic vdW crystals  (green -\ncolored),  bulk antiferromagnets (orange -colored), bulk multiferroics  (yellow -colored)  \nand a-RuCl 3 (a proxim ate Kitaev quantum spin  liquid)127 is colored by purple.  \nReproduced with permission from Gong  et al. , Science 363, 6428  (2019) . Copyright \n2019  The American Association for the Advancement of Science . \n3.2 Advantages of 2D materials in solving current 3D MTJ s problems  \nIn Sec. 2 , the potential problems of 3D MTJs when they are scaling down to the \nnano level are introduced. Various 2D materials  with natural monolayers are now \navailable through the large scale d CVD  growth. It is thus naturally to seek high -\nperforming , flexible and stable MTJs tunnel barriers based on 2D materials  and their  \nheterojunctions.  2D materials provide a reliable solution to the problems in the \nmanufactu ring of high -performance MTJs through the layer -by-layer control of the \nthickness, sharp interfaces, and high PMA. In this direction, low resistance area \nproducts, strong exchange couplings across the interface, and high TMR  in MTJs were \npredicted  and synthesized . \n \n 4.  2D -materials -based MTJs  \nAs discussed above , 2D materials are expected to  offer solutions to some of the \nchallenges when further improving the performance and reducing the size  of MTJs.  In \nthis section, the current development status o f 2D MTJs around the current problems in \nthe development of conventional MTJs will be review ed. Following  each problem and \nchallenge, possible solutions and future development directions  are discussed . \n4.1 Target ing high  spin-polarization  \nThe generation, transport and detection of spin current in MTJs are three key parts \nto integrate spin into existing electronics successfully.  In this section, we focus on the \ncurrent works  which target at improv ing spin polarization  in MTJs with 2D materials . \nThe s pin polarization is the most important factor for governing TMR  performance as \na high spin -polarized current is essential for high magnetoresistance . It is known that \nthe subtle offset  between two spin channels  causes the  net spin polarization, which is \ngreatly affected  by atomic, electronic and magnetic structures of the system. A \nstraightforward way to have high spin polarization is to use half-metallic  magnetic \nmaterials , in which the Fermi level only crosses one spin cha nnel, resulting in 100% \nspin polarization. Thus, half -metallic materials become good  candidates for MTJ \ndevice s. Using them as electrodes , 100% spin -polarized currents  under a bias voltage  \nmay be generated in MTJs  with high TMR.  However, the MTJs with half -metallic -\nmaterials do not show very high TMR as expected from the materials point of view \n[cite reference here]. It is because of the complicated geometry structure in MTJ devices \nmade of several different materials and nonequi librium electronic structure under bias.   \nFrom the experimental aspect, with the fast development of CVD technology \nrecently,  MTJ s using 2D materials such as  MoS 2128, graphene129, 130 and boron n itride  \n(BN)131 as nonmagnetic spacer have been fabricated  successfully. Their van de Waals \ninterface is expected to overcome the disordered interface between two 3D bulk \nmaterials.  However, t heir reported magnetoresistance  is quite low , which are  \n undesirable. This was attribute d to the use of permalloy electrodes,  such as Fe, Ni and \nCo, injecting current with a relatively  low spin polarization.  In addition, the inherent \nproperties of these materials also hinder the performance of MTJ s. Taking  the C o-Fe \nsystem as an example,  current -induced switching in FeCoB/MgO requires intense \ncurrent densities  to overcome the large Fe Gilbert damping.132 Thus, it is important to \nsearch  different 2D materials stacks  with other electrodes , which can offer better \nopportunities to implement such new technologies. High spin-polarized Heusler alloys, \na large family of ternary compounds,133 appear as promising candidate s of electrodes . \nFor example,  most  early researches on M oS2-MTJs with permalloy  electrodes show  \nrelatively low TMR.  Adopting the electrodes of Fe 3Si, a Heusler alloy with a lower \nGilbert damping parameter and a higher saturation magnetization , Rotjanapittayakul  et \nal.134 reported a large TMR in Fe 3Si-MoS 2 MJTs. It is because of the  similar lattice to \nMoS 2, small Gilbert damping and high Curie temperature of Fe 3Si. A small Gilbert \ndamping parameter leads to a potentially low critical current density for STT switching. \nThe Tc of Fe 3Si is large, above 800K, and the spin -polarization at low temperature \n(~45%)135 is larger than that of  Co (~34%) and Ni (~11%)136. In addition , Wu et al.137 \npreform a research on the ferromagnetic Fe 3O4 electrodes in  Fe3O4/MoS 2/Fe 3O4 MTJs. \nA clear large TMR phenomenon appears below 200K temperature. They  also \nperformed first -principles calculations and found that Fe 3O4 keeps high spin-polarized \nelectron band at the interface of the MoS 2 and Fe 3O4. This calculation provide s a clear \nand deep physical explanation on how the TMR phenomenon appears  in their \nexperim ent.  \nBy using  the DFT combining with non-equilibrium Green function (NEGF) \nmethod , 80% spin injection efficiency (SIE)  and 300% magnetoresistance ratio are \npredicted in multiple 2D  barrier layers on the performance of MTJ s. This is an effective \nmean to improve spin polarization in such junctions.  There are also some work s in the \nsame  direction. For example, Zhang et al.138 investigated the vertical transport across \nM/MoS 2/M (M = Co and Ni) MTJs with MoS 2 layer numbers N = 1, 3, and 5 . Their \nresults revealed that the thinner junctions a re metallic because of the strong coupling  \n between MoS 2 and the ferromagnets, and the junctions with thick er MoS 2 begin to show \ntunneling effects. A higher MR is achieved by increasing the number of interlayers. In \ntheir junctions, both positive (63.86%) a nd negative MR ( -70.85%) can be obtained. A \nsimilar model based on their prediction was later developed experimentally.  Galbiati et \nal.139 reported the fabrication of NiFe/MoS 2/Co devices with mechanically exfoliated \nmultilayer MoS 2 using an in-situ fabrication protocol that allows high -quality \nnonoxidized  interfaces to be maintained between the ferromagnetic electrodes and the \n2D layer. Their devices display a MR  ratio up to 94%. Beyond interfaces and material \nquality, they suggested that spin -current depolarization could explain the limited MR. \nThis point s to a possible path towards the realization of larger spin signals in MoS 2-\nbased MTJs.  \nBesides the number of layers  of MoS 2, the l ength  of scattering region  can affect \nthe MR ratio.  By employing a three -band tight -binding model combined with the NEGF \nmeth od, Jin et al. ,140 studied the spin -dependent electron transport in a zigzag \nmonolayer MoS 2 with ferromagnetic electrodes. Their results reveal that the \nconductance shows a quantized oscillating phenomenon in the P configuration, while \nthe conductance exhib its a zero platform in a large -energy region in the AP \nconfiguration . In addition, the length of the central part of the structure has a certain \ninfluence on the MR ratio. It is found that as the length of the middle region increased, \nthe MR ratio decreased gradually. However, this prediction has not been approved by \nexperiments yet. It hoped that the experimental reported would appear in the future.   \nBeyond non-magnetic MoS 2, 2D intrinsic magnetic materials play a n important  \nrole in the spin polarization of MTJs . The reported molecular beam epitaxial growth of \n2D magnetic materials for Fe 3GeTe 2, VSe 2, MnSe x, and Cr 2Ge2Te6 opens new \npossibility for MR devices.  They  are magnetic conductors or insulators , which  provide  \ndiverse application perspectives. For example, magnetic insulators are ideal for central \ntunneling  layer in MTJ s. And CrI 3 is a typical example of 2D magnetic insulat ing \nmaterials that have emerged in recent years. Atomically thin CrI 3 flakes were fabricated \nrecently by mechanical exfoliation of bulk crystals onto oxidized silicon substrates.124  \n In CrI 3 flakes, intrinsic ferromagnetism and out -of-plane magnetization is observed. \nInterestingly, it is FM in the monolayer but becomes AFM in bilayer and back to FM in \nboth trilayer and bulk.141 And n -layer CrI 3 in the high -temperature phase exhibits inter -\nlayer AFM coupling, which provides natural pinning layer for CrI 3. Yan et al.142 studied \nthe electron trans port properties of CrI 3/BN/n -CrI 3 (n=1, 2, 3, 4) MTJs as shown in Figs. \n5 (a-d), and found the n=3 MTJ shows a fully polarized spin current with ~3600% TMR \nratio when at the equilibrium state. More interestingly, the odd -even effect appears due \nto the difference of the number of pinning layers.  The usage of different number of CrI 3 \npinning layers  greatly regulates the spin polarization . \nAs CrI 3 is a semiconductor which can serve as  a spin-filter  tunnel barrier  when \nsandwiched between graphene electrodes , Song et al. performed an experimental work  \nfor improving spin polarization in MTJs by increas ing the thickness of CrI 3 layers (Figs. \n5 (e-g)).143 The TMR of this spin-filter MTJs (sf -MTJs)  can be drastically enhanced \nwith the increase of the thickness of CrI 3 layers , which is correspondi ng with the former  \ntheoretical result s.142 When the thickness of CrI 3 increases to four layers, the TMR ratio \ncan reach 19 ,000% at low temperature.  The four -layer CrI 3 MTJ points to the potential \nfor using layered antiferromagnets for engineering multiple magnetoresistance states in \nan individual multiple -spin-filter MTJ. The low TC of CrI 3 (around 50 K ) limits its \npractical device application. It is urgent to find intrinsic 2D magnetic materials with \nhigh Curie or Neel temperature for room -temperature MTJ devices.  \nA similar theoretical investigation of a 2D spin filter and spin -filter MTJs \nconsisting of atomically thin Fe 3GeTe 2 was also reported.144 The models and the main \ndata are shown in Figs. 6 (a-d). By the DFT -NEGF method, the TMR effect is obtained  \nin single/double -layer Fe 3GeTe 2-hBN-Fe3GeTe 2 heterostructures. For heterostructures \nconsisting of single - and double -layer Fe 3GeTe 2, the calculated MR ratio is 183% and \n252% at zero  bias, respectively. The Fe 3GeTe 2 MTJ in the P state  shows a spin \npolarization of more than 75%.  \nBesides the use as a spin-filter barrier, the metallic nature of Fe 3GeTe 2 also enables \nitself to be used as magnetic electrode s in vdW MTJs  to provide high spin polarization ,  \n which has advantages over insulating CrI 3 that is used as a spin -filter barrier  only. On \nthe one hand , a large magnetic field up to 1 T is required in CrI 3-based MTJs  to switch \nthe antiferromagnetic ground state to ferromagnetic.145, 146 On the other hand , CrI 3-\nbased MTJs are volatile, i.e., the magnetic field needs to be maintained to preserve the \nferromagnetic order, while Fe 3GeTe 2-based MTJs are nonvolatile due to two stable P \nand AP magnetization configurations  still appear  under the  absence of applied field.  For \nthis 2D metal with room -temperature ferromagnetism ,147 Li et al.148 studied the spin -\ndependent electron transport across vdW MTJs consisted of a graphene or h -BN spacer \nlayer with Fe 3GeTe 2 ferromagnetic electrodes (see Figs. 6 (e-f)). The authors found the \nresistance changes by thousands of percent from the P to AP state for both (graphene \nand h -BN spacers). The two different electronic structures of conducting channels in \nFe3GeTe 2 arouse a remarkable TMR effect. The authors further argued that the  strain \nand interfacial distance are two main factors that can influence this gi ant TMR ratio. \nAlso using Fe3GeTe 2 as ferromagnetic electrodes , Zhang et al .149 designed \nFe3GeTe 2|InSe|Fe 3GeTe 2 MTJs and found TMR reached ~700% in this kind of MTJs. \nBy analyzing both complex band structure of the barrier and band structure of the \nelectrode, the origin of the considerable TMR  is disclosed , as shown in Figs. 6 (h-i). \n \n \n \n Fig. 5 (a) Top and side views of bilayer CrI 3 in the high -temperature phase. (b) The \nmodel structures of Cu/CrI 3/BN/n -CrI 3/Cu (n=3 , and 2 of them are the AFM pinning \nlayer ). (c) TMR  versus bias voltage of Cu/CrI 3/BN/n -CrI 3/Cu, (d) SIE versus bias \nvoltage of Cu/CrI 3/BN/n -CrI 3/Cu i n P and AP configurations.  (a-d) Reproduced with \npermission  from Yan et al. , Phys. Chem. Chem. Phys. 22, 26 (2020) . Copyright 2020 \nRoyal Society of Chemistry.  (e) Schematic of magnetic states in bilayer CrI 3. The left \npanel:  layered -antiferromagnetic state, which suppresses the tunneling current at zero \nmagnetic field. The m iddle and right  panel:  fully spin -polarized states with out -of-plane \nand in -plane magnetizations, which do not suppress it. ( f) Schematic of 2D sf -MTJ. ( g) \nTunneling current of a bilayer CrI 3 sf-MTJ at selected magnetic fields. The t op inset : \noptical microscope image of the device with scale bar  of 5 μm. The red dashed line \nshows the position of the bilayer CrI 3. The b ottom  panel:  schematic of the magnetic \nconfiguration for each I t-V curve.  (e-g) Reproduced with permission  from Song et al., \nScience 360, 1214 ( 2018) . Copyright 2018  The American Association for the \nAdvancement of Science.  \n \n \nFig. 6 (a) The model of single -layer Fe 3GeTe 2 sandwiched between two Cu electrodes. \n(b) Spin-involved I -V curves and TMR  of current varying with bias voltage of the \nmodel  corresponding to (a) . (c) The model of double -layer Fe 3GeTe 2 sandwiched \nbetween two Cu electrodes. (d) Spin-involved I -V curves an d TMR  of current varying \n \n with bias voltage of th e model  corresponding to ( c). (a-d) Reproduced with permission  \nfrom Lin et al., Adv. Electron. Mater. 6, 1900968 ( 2020) . Copyright 2020 WILEY ‐VCH.  \n(e) The magnetic junction structures of Fe 3GeTe 2/h-BN/Fe 3GeTe 2. (f) Electron \ntunneling probability in MTJs distribution in 2D brillouin region . (e-f) Reproduced with \npermission  from Li et al. , Nano Lett. 19, 5133 ( 2019) . Copyright 2019 American \nChemical Society.  (g)The structure of Fe3GeTe 2|InSe|Fe 3GeTe 2 MTJ.  (h) Complex band \nstructure for the InSe part  (left panel) and band structure of Fe3GeTe 2 (right panel).  (g-\nh) Reproduced with permission  from Zhang et al.  J. Phys. Chem. C 124, 27429 ( 2020).  \nCopyright 20 20 American Chemical Society.  \n \nIn the studies of the influence of the  number of  layers  on the  MTJ performance , it \nturns out that the number of layers of the central barrier layer has a great influence on \nthe value of TMR (see m ore details in Table 1). \n \nTable 1 Central and electrodes materials, defined TMR and TMR ratio  with the function \nof the number of spin -filter tunnel barrier layers.  \nCentral \nmaterials  Electrode  \nmaterials  TMR _max  Defined  TMR  Remarks  Reference  \nMnSe 2/1-layer \nh-BN/MnSe 2 Ir 222.12%  𝑇𝑀𝑅=𝐺𝑃−𝐺𝐴𝑃\nG𝐴𝑃 G is conductance  150 \nRu 419.94%  \nMnSe 2/2-layer \nh-BN/MnSe 2 725.07%  \nVSe 2/1-layer h -\nBN/VSe 2 Ir 254.48%  \nRu 183.55 % \nVSe 2/2-layer h -\nBN/VSe 2 199.15%  \n1-Layer MoS 2 Co(fcc)  63.86%  𝑇𝑀𝑅=𝐺𝑃−𝐺𝐴𝑃\nG𝐴𝑃 G is conductance  138 \nCo(hcp)  33.40%  \nNi(fcc)  5.30%   \n 2-Layer MoS 2 Co(fcc)  58.80%  \nCo(hcp)  55.60%  \nNi(fcc)  −13.79%  \n3-Layer MoS 2 Co(fcc)  −70.85%  \nCo(hcp)  55.91%  \nNi (fcc)  55.91%  \n1-Layer MoS 2 Fe3Si (100)  109.44%  𝑇𝑀𝑅=𝐺𝑃−𝐺𝐴𝑃\nG𝐴𝑃 G is conductance  151 \n3-Layer MoS 2 306.95%  \n5-Layer MoS 2 278.87%  \n7-Layer MoS 2 154.56%  \n9-Layer MoS 2 121.63%  \nCrI3/h-BN/1 -\nlayer CrI 3 Cu (111)  2900%  𝑇𝑀𝑅=𝐼𝑃−𝐼𝐴𝑃\nI𝐴𝑃 I is current  142 \nCrI3/h-BN/2 -\nlayer CrI 3 1800%  \nCrI3/h-BN/3 -\nlayer CrI 3 3600%  \nCrI3/h-BN/4 -\nlayer CrI 3 ~0% \n1-Layer \nFe3GeTe 2/h-\nBN/1-layer \nFe3GeTe 2 Cu 183%  𝑇𝑀𝑅=1/𝐼𝐴𝑃−1/𝐼𝑃\n1/I𝑃 I is current  144 \n2-Layer \nFe3GeTe 2/h-\nBN/2-layer \nFe3GeTe 2 289%  \n1-Layer \nFe3GeTe 2/h-Cu 78% P = (D up-\nDdown)/(D up+D down \n BN/1-layer \nFe3GeTe 2 𝑇𝑀𝑅𝐽𝑢𝑙𝑙𝑖𝑒𝑟𝑒\n=1/𝐼𝐴𝑃−1/𝐼𝑃\n1/I𝑃=2𝑃2\n1−𝑃2 ) is the spin \npolarization of \nelectrode; D up and \nDdown are the spin -\nup and spin -down \ndensity of states in \nthe Fermi level of \nelectrode  2-Layer \nFe3GeTe 2/h-\nBN/2-layer \nFe3GeTe 2 520%  \n1-Layer \nFe3GeTe 2/h-\nBN/1-layer \nFe3GeTe 2 2-Layer \nFe3GeTe 2 160%  𝑇𝑀𝑅𝑠𝑐𝑎𝑡𝑡𝑒𝑟𝑖𝑛𝑔\n=1/𝐼𝐴𝑃−1/𝐼𝑃\n1/I𝑃\n=(D𝑢𝑝−𝐷𝑑𝑜𝑤𝑛)2\n2D𝑢𝑝𝐷𝑑𝑜𝑤𝑛 T is transmission  \n2-Layer \nFe3GeTe 2/h-\nBN/2-layer \nFe3GeTe 2 215%  \nh-BN/2 -layer \nCrI 3/TaSe 2/h-\nBN Graphite  240% 𝑇𝑀𝑅=𝑅𝐴𝑃−𝑅𝑃\nR𝑃 R is resistance  19 \n2-layer \nCrI 3/TaSe 2 40% \n \n4.2 Target ing effective  spin-injection  \nNote that in conventional MTJ s, the electrodes usually adopt a kind of  metal \nmaterial , while the central scattering region is composed of a n insulator  material. Thus,  \nthe electrodes and the central scattering region are usually composed of different \nmaterials . The contact between them plays a significant role in determining the \ntransport properties.152 Unfortunately, due to lattice  mismatch and conductivity   \n mismatch, the contact  between metallic electrodes and the insulating barrier  is usually \npoor, resulting in low spin injection , which weakens the device performance.153, 154 If \nthese problem s are not solved well , even in the off state, electrons from one side of the \nelectrode have possibil ity of tunneling through the central scattering region to the other \nside of the electrode, resulting in the leakage of current.155 Accordingly, it is difficult to \nfabricate high -performance devices based on the present configuration in experiments. \nAs such, resolving the contact problem  and/or spin injection into the central  region  is \nimportant for dev eloping high performance MTJs. On the other hand, t he parcel on the \nBN would effectively reduce  the spin mismatch problem . In th is section, we will review \nthe recently reported 2D MTJs which target at improving spin injection.  \nTo resolve lattice mismatch between electrodes and the barrier layer  for high spin \ninjection , the most directly mean is to adopt the same material.  One 2D material can \nachieve both metal and insulator with different phase, for example MoS 2 in H phase \nand T  phase. By using this mean, Zhou et al. 156 constructed a kind of graphene -based \nMTJ  as shown in Figs. 7(a-b). They systematically studied the transport properties of \nthe zigzag graphene nanoribbon (ZGNR) using DFT -NEGF  method . Remarkably, a 100% \nSIE and a giant TMR of up to 107 is predicted in their designed graphene -based MTJ \ndevice, which shows much better perf ormance than that of traditional 3D MTJs.  \nBesides choos ing the same material, select ing two materials from the same family \ncan also  minimize  the mismatch problem . Inspired by the experimental synthesis of the \nmagnetic layered crystal of Mn 2GaC, its 2D counterpart of the half -metallic Mn 2CF2 \nMXene layer can serve as the magnetic electrode for MTJs. Ti 2CO 2 MXene can be \nchosen as the tunneling barrier, which is one of the most studied MXenes in both \nexperiments and theories.157-159 Balcı et al.160 designed an MXene -based MTJ, as shown \nin Figs. 7 (c-e). The highlight of their work is the electrodes and barrier layer materials \nthey chose are from the same family, which avoiding the lattice mismatch problem. And \nalso, the band gap of Ti 2CO 2 barrier layer is almost the same as the half -metallic gap of \nMn 2CF2 electrodes. Both the barrier and the electrodes have a common carbon  layer \nthat contributes the most to the transmission. Based on above, the proposed MTJ is  \n match both in structure and electronic structure. The proposed MTJ also exhibit s TMR \nwith a peak value up to 106 and average values above 103 within the bias of ±1 V . And \nthen, Ünal Özden Akkuş et al .161 further studied a similar work that investigated \ncharacteristics of a Ti 2CT 2 (T=O or F) MXene -based device which consists of \nsemiconducting Ti 2CO 2 and Ti 2CF2 metallic electrodes. Their DFT -NEGF transport \ncalculations suggest that the device, made of a Ti 2CT 2CO 2 semiconductor and two \nTi2CF2 metallic electrodes, shows field -effect transistor characteristics when the \nsemiconducting part is longer than about 6 nm.  It is found that devices with larger \ntunneling barrier width should have a much better response to the gate voltage.  \n \n \n Fig. 7 (a) The structure s of the device  where  in-plane gate voltages are applied to the \nelectrode regions in the same direction (PP) and opposite directions (PN). The length \nof the central scattering region  without  no gate voltage is applied, is denoted by L.  (b) \nSpin-down transmission coefficient  (left panel) , spin injection efficiency ( middle panel ), \nand TMR  (right panel)  as a function of L  for the PP configuration . Reproduced with \npermission  from Zhou et al. Phys. Rev. Appl. 13, 044006 (2020) . Copyright 20 20 \nAmerican Phys ical Society.  (c) Top view and side view of Mn 2CF2 unit cell. ( d) The top \nand side view s of an MXene -based MTJ (Mn 2CF2/Ti2CO 2/Mn 2CF2). (e) I-V curves and \nTMR of the MTJ in P and AP configurations. Reproduced with permission  from Balcı \net al.  ACS Appl. Mater. Interfaces 11, 3609 (2018) . Copyright 2019 American Chemical \nSociety.  \n \nAnother  effective way to  solve the conductivity  mismatch problem and  improve \nspin injection  in MTJs is the introduction of  h-BN, as discussed in Sec. 2 . The insulating \nh-BN is proposed as an ideal covalent spacer for MTJs, which provides a higher MR \nratio and stronger exchange coupling at the interface to remove the conductivity \nmismatch between the metal leads and the FM layer. Many works in both theories  and \nexperiment s have proved that h -BN based MTJs have a good TMR performance. Such \nconclusion has great impact on the h -BN integration pathway. Begin with the t heoretical \nwork s, Qiu et al.,162 proposed an effective method to control the spin current in a  \nvertical MTJ by combining the strong spin filtering effect of graphene/ferromagnet \ninterface with the resonant tunneling effect of graphene/h -BN/graphene vdW \nheterostructure , in which  Ni(111) is used as electrodes. Their theoretical results reveal \nthat wh en the electronic spectra of spin electrons in two graphene layers are aligned, \nthe spin resonance would appear which results in a negative differential resistance \n(NDR) effect. By studying the similar structure with a periodic density functional \nmethod in  conjunction with Julliere’s model, Sahoo et al .163 constructed \nNi/BN/graphene/BN/Ni and Ni/graphene/Ni tunnel devices. It is found that the former, \nthe graphene/h -BN multi -tunnel junction, has a much higher TM R than the latter. The  \n underlying mechanism was explained by Wu et al.164 that the minority -spin-transport \nchannel of graphene can be strongly suppressed by the insulating h -BN barrier, which \novercomes the spin -conductance mismatch between Ni and graphene,  resulting in a \nhigh spin -injection efficiency.  Another example of using h -BN as the tunnel barrier, is \nMnSe 2/h-BN/MnSe 2.150 The schematic diagram of the proposed structure is shown in \nFig. 8. The monolayer T -MnSe 2 is selected as the ferromagnetic layer,165 and Ir and Ru \nare employed as metal electrodes, which have a smaller lattice mismatch with XSe 2 \ncompared with conventionally well -used electr ode metals (such as Au, Ag and Cu). It \nis found that such vertically vdW MTJ has a large TMR of 725%.150 This is due to a \nlarge transmission in majority channel in the P magnetic configuration, while it is \nsuppressed in the AP magnetic configuration, as shown in Figs. 8 (b) and (e). The \nauthors take two approaches for improving TMR : one is choosing suitable electrodes \nand another is  finitely  increasing the number of layers of the h -BN barrier.  The former \nminimizes the lattice mismatc h and the latter involving BN solves the spin mismatch \nproblem.   \nAs discussed above , introducing extra  2D BN layer has advantages in  improv ing \nTMR  than 3D tunneling barriers like Al 2O3, which has been supported in experiments  \nsuccessfully106. And many efforts have been done  in the actual manufacture of BN-\nbased MTJs  to improve TMR further. For example, a TMR ratio around 0.3 -0.5% can \nbe obtained by performing  wet transfer on ferromagnet,166 while h -BN is exfoliated on \nperforat ed membranes the TMR ratio is 1%,167 and the TMR can  research  up to 6% \ngrown  by large area CVD on Fe directly.168  \nBesides using calculations to predict the conclusions like above discussions, u sing \ncalculations to explain experiments is also important and necessary due to  calculations \ncan give some physica l insights . Liking t he research  on different ferromagnets to \ndevelop h -BN MTJs , it is particularly important to using calculations to understand the \ninterfacial effect on the MR performance of MTJs. For example, Piquemal -Banci et \nal.169 fabricated two h -BN-based MTJs with different FM electrodes, Co/h -BN/Co and \nCo /h -BN/Fe MTJs. In these two MTJs, h -BN is grown directly by CVD on pre - \n patterned Co and Fe stripes. The TMR ratio in these two MTJs are 12% and 50%, \nrespectively.  By using calculations method, the expected strong dependence of the h -\nBN electronic properties on the coupling with the FM electrode is further investigated.  \nThis calculations part gives an explanation of how h-BN improves TMR in their \nexperimental conclusion.  \nWhen  h-BN acts as the tunneling barrier, the FM layers can be various beyond Fe \nand Co. Like CrI 3 which is introduced above , its cousin CrBr 3 monolayer also show s \nFM character.141 And CrX 3 heterojunctions have been reported having large MR \nrecently.143 Inspired by this, Pan et al .170 systematically investigated the structura l, \nmagnetic and spin -transport properties of CrX 3/h-BN/CrX 3 (X = Br, I) MTJs.  The \nbarrier layer is h -BN and CrX 3 are used as the ferromagnetic layer.  Metal Au, Ag, Al, \nand Pt are chosen as electrodes. Taking the AgCrBr 3/h-BN/CrBr 3/Ag MTJ ( Fig. 8 (f)) \nas an example, the large TMR effect can be up to 1 ,565% in this series of MTJs. The \ndiagram of the band alignment and the  k-resolved transmission spectra of MTJ are \nshown in Figs. 8 (g-k). \n \n \nFig. 8 (a) The structure of Ru/MnSe 2/h-BN/MnSe 2/Ru MTJs. (b -e) k||=(ka, kb) \n \n dependent transmission spectra of MTJs based on MnSe 2 for majority -spin and \nminority -spin states in P and AP configurations, respectively. (a-e) Reproduced with \npermission  from Pan et al. , Chin. Phys. B 28, 107504 (2019) . Copyright 2019 CPB . (f) \nThe structure of CrBr 3/h-BN/CrBr 3 MTJs.  White colored atoms on the left and right \nsides are silver electrodes. ( g) The diagram of the band alignment of this MTJ. ( h-k) \nThe k -resolved transmission spectra of MTJ for majority - and minority -spin states in P \nand AP configurations.  (f-k) Reproduced with permission  from Pan et al ., Nanoscale 10, \n22196 (2018) . Copyright 2018 Royal Society of Chemistry.  \n4.3 Target ing spin-manipulation  \nManipulating spin is another powerful mean not only to improve TMR , but also \nto realize different functionalities of spintronic devices . There are many ways for spin \nmanipulation, such as adding other vdW  materials to induce local phase transition, \nintroducing SOT, making use of interlayer interaction , doping FM , and so on . In this \nsection, we  mainly review recent works which target spin manipulation  of MTJs . \nFor the aspect of spin manipulation through  phase transition  induced by an \nadjacent material, Begunovich  et al. 171 proposed a n ultrathin MTJ s based on vanadium \nditelluride monolayers with graphene as a tunnel barrier . Both trigonal prismatic (H -\nphase) and octahedral (T -phase) VTe 2 were considered in their study . The authors found \nthat the introduction of graphene makes the electronic characteristic of 2D T -VTe 2 \nchangeable from the metal to half -metal phase, making  T-VTe 2 a promising candidate \nfor MTJ applic ations.  Although several possible structures are considered , the one \nwhich follows the framework of Julliere model shows the highest TMR ratio up to \n220%.  \nInserting other vdW “heavy” materials may introduce SOT -driven operation s. \nCombining NEGF with noncollinear  density functional theory (ncDFT) methods, Dolui \net al.19 constructed bilayer -CrI 3/monolayer -TaSe 2 vdW lateral heterostructure as shown \nin Fig. 9 (a) , in which bulk non -magnetic metal electrodes are required in practice for \ngenerating in -plane charge current.  They found that the AFM –FM nonequilibrium  \n phase transition can be induced by the SOT in this MT Js where the unpolarized charge \ncurrent is injected parallel to the interface. The 1H phase monolayer of metallic TaSe 2 \nis chosen because of a small lattice mismatch  (0.1% ) to CrI 3 and inversion asymmetry.  \nBy introduc ing another  heavy metal  of WS 2, Zollner  et al .36 designed \nCr2Ge2Te6/graphene/WS 2 vdW  MTJs  (see Fig. 9  (b)), where SOC , valley -Zeeman and \nRashba splitting , and exchange  coupling  can be obtained.  \n \nFig. 9 (a) The geometric structure of the CrI 3/TaSe 2 SOT vdW MTJ. (a) Reproduced \nwith permission  from Dolui et al.,  Nano Lett. 20, 2288 (2020) . Copyright 2020 \nAmerican Chemical Society.  (b) The structure of the Cr2Ge2Te6/graphene/WS 2 vdW \nMTJ. (b) Reproduced with permission  from Zollner et al. , Phys. Rev. Res 2, 4 (2020) . \nCopyright 2020 American Phys ical Society.  (c) The structure of the VSe 2/MoS 2 SOT \nvdW MTJ.  (d) TMR and SHC of VSe 2/MoS 2 SOT vdW MTJ . (c-d) Reproduced with \npermission  from Zhou et al ., ACS Appl. Mater. Interfaces 11, 17647 (2019).  Copyright \n2019 American Chemical Society.  The key feature of this type of SOT -MTJs is that the \nwrite charge current horizontally flows a heavy non -magnetic 2D material, while the \n \n read spin current vertically flows a 2D vdW MTJ.  \n \nAs for enhancing TMR by making use of interlayer interaction, many efforts are \nalso done. To understand the inside mechanism  better, calculations method is needed. \nFor example, Heath et al.172 shed important insights from an atomistic viewpoint on the \nunderlying mechanism governing the spin transport in graphene/CrI 3 spin-filter \ntunneling junctions  by a combin ed first-principles and quantum bal listic transport \ncalculation,  as shown in Fig. 10. The calculated electronic structure s reveal that \ntunneling is the dominant transport mechanism in these heterostructures. The tunneling \neffect boosts differentiate intermed iate metamagnetic states presenting in the switching \nprocess.  This is manifested in an increase in TMR for energy above the Fermi level due \nto enhancement of Bloch states near the edge of the conduction band of CrI 3. \n \nFig. 10 Atomic and electronic structures of graphene/CrI 3/graphene MTJs. (a) Atomic \nstructures of graphene/CrI 3/graphene MTJs. (b) Band diagrams and corresponding band \nstructures in trilayer graphene/trilayer CrI 3/trilayer graphene junctions for various states. \n(c) TMR as a function of Fermi level for both trilayer (TMR↑↑↑ and TMR↑↑↓) and \nbilayer (TMR↑↑) systems. Reproduced with permission  from Heath et al., Phys. Rev. B \n101, 195439 (2020) . Copyright 2020 American Physical Society.  \n     \n \n In order to manipulate spin in non -magnetic 2D materials, one can dope them  by \ncharge transfer from FM metal s or proximity -induced spin splitting in themselves . For \nexample,  Asshoff et al.167 fabricated vertical graphene -based devices where ultimately \nclean graphene –FM interfaces were obtained by depositing the FM metals (FM=Co \nand FM'=Ni 0.8Fe0.2 alloy) on the two sides of a suspended graphene membrane, \nthereby preventing oxidation, minimizing the number of fabrication steps and limiting \nthe exposure of the devices to solvents during preparation.  Such kind of treatment \nimprove s the perform ance of MTJs.  \nApplying  a finite external bias voltage  has been proved to be an effective method \nto manipulate spin  transport . For example, Chen  et al.173 theoretically investigated the \nnonequilibrium spin injection and spin -polarized transport in monolayer black \nphosphorus (MBP) with ferromagnetic Ni contacts. The top and side views of their \nmodel are shown in Fig. 1 1. In this study, they explored the SIE, TMR ratio, spin -\npolarized curre nts, charge currents and transmission coefficients as a function of bias \nvoltage. Furthermore, they studied two different contact structures where MBP is \ncontacted by Ni(111) and Ni(100). Both structures are predicted to have great spin -\npolarized transport  performance.173 The Ni(100)/MBP/Ni(100) MTJ has the superior \nproperties of the SIE (~60%) and TMR ratio (40%), which maintains almost a constant \nvalue against the bias voltage.  \n \nFig. 11 (a) The structure of Ni(111)/MBP/Ni(111) MTJ in the top view. (b) The structure \nof Ni(100)/MBP/Ni(100) MTJ in the top view. (c -d) The side view of \n \n Ni(111)/MBP/Ni(111) MTJ and Ni(100)/MBP/Ni(100) MTJ respectively. (e -g) The top \nand side view of MBP. Ni and P atoms are yellow  and pink, respectively. (h -i) I-V \ncurves, TMR and SIE of Ni(111) /MBP/Ni(111) and Ni(100)/MBP/Ni(100) MTJ, \nrespectively. Reproduced with permission  from Chen et al., Phys. Chem. Chem. Phys. \n18, 1601 (2016) . Copyright 2016 Royal Society of Chemistry.  \n \nHydrostatic pressure can be used for continuous control of  interlayer coupling by \ninterlayer spacing in vdW crystals , and then tuning the spin interaction and transport . \nFor example, experiment ally, Song et al.174 demonstrated the changes of magnetic order \nby pressure in 2D magnet CrI 3. The MTJ  structure  is composed of bilayer/trilayer CrI 3 \nsandwiched by top and bottom multilayer graphene contacts, and h -BN encapsulate s \nthe whole MTJ in order to avoid sample degradation. Figure 12 shows the structure of \na bilayer CrI 3 MTJ. It is found that the interlayer magnetic coupling can be doubled by \na hydrostatic pressure.174  \n  \nFig. 1 2 Stacking -order determined 2D magnetism and tunneling measurements of \nbilayer CrI 3 under pressure. Reproduced with permission  from Song et al., Nature \nmaterials, 18, 12 (2019) . Copyright 2019 Springer Nature.  \n4.4 Target ing stability  \nBesides  the target  high-performance  of MTJs, one should consider the  structural \nand magnetic stabilities  in the practical application.  These practical problem s includ e \nwhether the MTJs can operate at room temperature , and whether the  2D materials used \nin MTJs can be  successful synthesized . Target at these problems, many efforts have \nbeen made  in 2D -materials -based MTJs . In this section, we would review the recently \nwork around the efforts which is target at the stability problems.  \nIn order to achieve 2D -materials -based MTJs, the first requi red is the 2D materials \nare stable in room -temperature. Thus, thermodynamic stability , dynamic stability , and  \n \n mechanical stability  should be assessed. In practical efforts, i nert 2D materials like BN \nare often used to wrap reactive 2D materials  like black phosphorene .175 \nRoom -temperature working devices also require magnetic stability. However, \nmost of 2D FM material s discovered to date suffer from low TC. As a result, the MTJ s \nconstructed by these materials  only works at low temperatures , such as CrI 3-based \nMTJs.  High temperature working MTJ s can be achieved by using 2D FM materials \nwhich  have been predicted/discovered to have high TC. Recently, m onolayer VSe 2 is \nreported to be a room -temperature ferromagnetic 2D material experimentally.176 Thus, \na VSe 2/MoS 2 vdW MTJ was theoretically designed by Zhou et al,20 as shown  in Figs. \n9 (c-d). They proposed a concept of SOT vdW MTJs, which can achieve both reading \nand writing functions  at room -temperature . Their NEGF results show a TMR up to \n846%. This proposed SOT vdW MTJs based on VSe 2/MoS 2 give 2D MTJs a new \nopportunity  for many magnetic -field-free device applications, which can work in room -\ntemperature . Later on, more XSe 2 (X= Mn, V) based MTJ s 150 with 300 K  worki ng \ntemperature are proposed . The search for room temperature FM materials  is always  \ndemanding  for 2D MTJs . Recently, Y ang et al .177 design ed excellent ultrathin spin \nfilters by using half -metal 2D Cr2NO 2 (see Fig. 13 ), which has a TC of 566 K, based on \nfirst-principles calculations.  The half-metal feature with  100% spin polarization of \nCr2NO 2 guarantee  a giant TMR up to 6,000%.   \n \n \n Fig. 13  The structure of Au/bilay -Cr2NO 2/Au. Reproduced with permission  from Yang \net al., Matter 1, 1304 (2019).  Copyright 2019 Elsevier . \n \nPerpendicular MTJs with out -of-plane interfacial magnetization have many \nadvantages, such as high thermal stability, infinite endurance, and fast with low -power \nswitching. They are the base to construct advanced non -volatile memory device which \ncan be built as non -von Neumann computing parad igms to overcome power bottleneck. \nSome  perpendicular MTJs have been proposed theoretically,178 and then was made in \nthe experiment41. From the materials point of view, it is important and necessary  to \ndiscover more 2D magnetic materials with out -of-plane magnetic anisotropy, which can \nbe used as building blocks for perpendicular MTJs.  \n5. Summary and perspectives  \nSince the discovery  of TMR in MTJs , the MTJ devices have shown a great  and \nprofound  impact on spintronics applications , including  the hard disk driver,  MRAM, \nradio -frequency sensors, microwave generators and neuromorphic computing networks. \nIn this review, we first start from four  main  issues of  conventional  MTJs, and then \nreview the current progress of 2D -materials -based  MTJs and h ow they address the \nproblems  of conventional MTJs followed by brief comments  on the new  scientific \nproblems and technical challenges in 2D MTJs .  \nIn the past decade, the emergence of 2D (magnetic) materials has brought fresh \nblood to the family of MTJs and its spintronics  applications . The good performance of \nelectron transport properties presented by graphene, h -BN, MoS 2, CrI 3, Fe 3GeTe 2 and \ntheir vdW heterostructures has led to the prediction and d emonstration of perfect spin \nfiltering and large TMR ratio, as elaborated in this review. A TMJ usually is a stack of \ntwo or more materials in terms of a FM -NM-FM configuration. The various ways of \nstacking of concerned 2D materials result in 2D MTJs with diverse structures, which \nare categorized as lateral and vertical vdW MTJs. The 2D materials can be the magnetic  \n electrodes and/or the central insulating materials. Many factors have been reported to \ninfluence the performance of 2D MTJs, such as the thickn ess, strain, interface and the \nnumber of layers of 2D materials, which have been discussed in this review.   \nFor the computation al aspect, the combination of DFT with the NEGF method is \none of the most common technique to study the MTJs’ transport properties. The main \nchallenge of this method is the over -estimated TMR ratio. It is because under the \nframework of DFT the complicated in terfacial configuration, such as disorder, and the \ntemperature effect cannot be described properly due to the demand in computational \nresources. Some advanced methods in calculations of transport properties of MTJs are \nneeded. In 2008, Ke et al .179 developed a nonequilibrium vertex correction (NVC) \ntheory to handle the configurational aver age of random disorder at the density matrix \nlevel so that disorder effects to nonlinear and nonequilibrium quantum transport can be \ncalculated from atomic first principles in a self -consistent and efficient manner. \nRecently, Starikov et al.180 described a DFT -based two -terminal scattering formalism \nthat includes SOC and spin noncollinearity. An implementation using tight -binding \nmuffin -tin orbitals combined with extensive use of sparse matrix techniques allows a \nwide variety of inhomogeneous structures to be flexibly modelled with various types \nof disorder including  temperature induced lattice and spin disorder. The low stability \nand TC of 2D magnetic materials, such as CrI 3, ham per the practical applications of \nMTJ devices.  \nThe future prospect in 2D MTJs’ development might be keeping pace closely with \nthe emergence of novel stable 2D magnetic materials with room -temperature Tc. \nHowever, the conventional DFT  method is not quick enough  to discover more  \ndemanded 2D materials with high Tc and out -of-plane magnetic anisotropy.  Recently, \nhigh-throughput computations have been carried out to screen high -Tc 2D materials . \nFor example, Jiang et al.181 investigated the electronic and magnetic properties of 22 \nmonolayer 2D m aterials with layered bulk phases. They showed that monolayer \nstructures of CrSI, CrSCl, and CrSeBr have notably high TC (>500 K) and favorable  \nformation energies, making these ferromagnetic materials feasible for experimental  \n synthesis.  Besides discovering of new 2D magnetic materials using high -throughput \ncalculations, theoretically rational design of novel systems such as mixed dimensional \nheterostructures and mixed dimensional vdW heterostructures certainly provide a stable \ngeometrical and magnetic structure as well as give the additional spin injection and \nspin-state control tools through the spin –orbit combined with the Coulomb effect, \nimpro ving the MTJ device performance.  \nIn recent years, m ore and more researches combine experimental work with \ncomputational work.  One should design a calculation model that can better reflect the \nactual situation of the experiment, instead of being too ideali stic. The calculation should \ntake full account of the actual experimental conditions as far as possible, and not be too \nunconstrained.  \nAs for experimental aspect, the key research direction to develop device mainly \nfocuses on integration engineering and th e understanding of 2D ferromagnet growth \nprocesses. At present, the research for this aspect is still at an early stage though some \nprogress has already been made. In fact, some integration parameters are not understood \nclearly. Various methods are applied  to growth 2D materials. Many parameters such as \n2D materials’ quality, crystallinity, phase, surface chemical state, thickness and the \ninteraction at the interface would have great impact on the performance of MTJ devices. \nIn addition to focus on improvin g TMR ratio  of 2D MTJs , the device lifetime, \nmechanical stress, thermal stability and switching current are also important and have \nresearch value in the experiment. Finally, with the implementation of large -scale growth \nof advanced 2D materials through ch emical vapor deposition methods, it is indubitable \nto expect a real promise exists for a new generation of 2D MTJs on a large scale.  \nAcknowledgements  \nThe authors would like to acknowledge the support from the National Natural Science \nFoundation of China (Grant No. 51671114 and No. U1806219), MOE Singapore \n(MOE2019 -T2-2-030, R -144-000-413-114, R -256-000-651-114 and R -265-000-691- \n 114). This work is also supported by the Special Funding in the Project of the Taishan \nScholar Construction Engineering.  \n \nData availability  \nData sharing is not applicable to this article as no new data were  created or analyzed in \nthis study.  \nNotes and references  \nThe authors declare no conflict of interest.  \n \n1 D. Akinwande, N. Petrone, and J. Hone, Two -dimensional flexible nanoelectronics, Nature \ncommunications 5, 1 (2014).  \n2 W. Lu, and C. M. 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Interfaces 10, 39032 (2018).  \n "    },    {        "title": "2102.04183v2.Transverse_thermoelectric_generation_using_magnetic_materials.pdf",        "content": "1 Transverse thermoelectric gene ration using magnetic materials \n \nKen-ichi Uchida1-3,a), Weinan Zhou1, and Yuya Sakuraba1,4,a)  \n AFFILIATIONS \n1 National Institute for Materials Science, Tsukuba 305-0047, Jap an \n2 Institute for Materi als Research, Tohoku University, Sendai 980 -8577, Japan \n3 Center for Spintronics Resear ch Network, Tohoku University, Sen dai 980-8577, Japan \n4 PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan \na) Authors to whom correspondence should be addressed: UCHIDA.Keni chi@nims.go.jp and \nSAKURABA.Yuya@nims.go.jp  ABSTRACT The transverse thermoelectric e ffect refers to the conversion o f a temperature gradient into a \ntransverse charge current, or vice versa, which appears in a co nductor under a magnetic field \nor in a magnetic material with spontaneous magnetization. Among  such phenomena, the \nanomalous Nernst effect in magne tic materials has been receivin g increased attention from the \nviewpoints of fundamental physics and thermoelectric applicatio ns owing to the rapid \ndevelopment of spin caloritroni cs and topological materials sci ence. In this research trend, a \nconceptually different transve rse thermoelectric conversion phe nomenon appearing in \nthermoelectric/magnetic hybrid mat erials has been demonstrated,  enabling the generation of a \nlarge transverse thermopower. Here, we review the recent progre ss in fundamental and applied \nstudies on the transverse thermoelectric generation using magne tic materials. We anticipate that \nthis perspective will further stimulate research activities on the transverse thermoelectric \ngeneration and lead to the develo pment of next-generation therm al energy harvesting and heat-\nflux sensing technologies.   2 Thermoelectric generation technologies have been considered as future independent \npower sources for Internet of Things applications because elect ricity can be generated from \nwaste heat.1-3 Existing thermoelectric devices are based on the Seebeck effec t owing to its \nrelatively high thermoelectric conversion efficiency. However, because the Seebeck effect is \nthe longitudinal thermoelectric effect in which a charge curren t is generated in the direction \nparallel to the applied temperat ure gradient [Fig. 1(a)], the t hermoelectric modules have \ncomplicated structures. As shown in Fig. 1(b), to increase the thermoelectric output, many pairs \nof two different thermoelectric materials must be connected in series and the materials must be \nlengthened in the direction of the  temperature gradient. This c omplicated structure limits the \ndurability and flexibility of thermoelectric devices and hinder s their wider applications. \nOne approach to overcome this problem is to use transverse ther moelectric effects. As \nthey generate a charge current in the direction perpendicular t o a temperature gradient, the \nthermoelectric generation is possible simply by forming a mater ial onto a heat source surface. \nBy utilizing this feature, the output voltage (power) can be en hanced by elongating the device \nlength (enlarging the device area) perpendicular to the tempera ture gradient without \nconstructing three-dimensional serial junctions. The thermoelec tric modules based on \ntransverse thermoelectric effects have a simple structure in wh ich conductors are connected in \nseries along the heat source surface, enabling the efficient us e of waste heat, reducing the cost \nof modules, and improving their durability and flexibility. Thu s, transverse thermoelectric \neffects have the potential to sol ve the problems associated wit h conventional Seebeck devices.  \nOne of the fundamental transverse thermoelectric effects is the  Nernst effect. The \nordinary Nernst effect (ONE), discovered by W. Nernst and A. V .  Ettingshausen in the late 19th \ncentury,4 refers to the generation of a charge current in the direction perpendicular to both a \ntemperature gradient and externa l magnetic field applied to a c onductor. Although ONE in \nsemiconductors and semimetals e xhibit a substantially large tra nsverse thermopower, it has not \nbeen applied in practice because its operation requires large m agnetic fields.5-8 In contrast, in \nmagnetic materials with spontaneous magnetization, the Nernst e ffect appears even in the \nabsence of magnetic fields owing to the spin-orbit interaction,  which is called the anomalous \nNernst effect (ANE)9-55 [Fig. 1(c)]. Therefore, th e transverse thermopower S in a magnetic \nmaterial under a magnetic field is described as the summation o f the contributions proportional \nto the magnitude of t he magnetic field H and magnetization M:  \nS = QH0H + QM0M,          ( 1 )  \nwhere 0 is the vacuum permeability and QH(M) is the proportionality factor of each term. 3 Equation (1) shows that the anomalous Nernst coefficient SANE (= QM0Ms with  Ms being the \nsaturation magnetization) can be extracted by extrapolating the  H dependence of S from the \nhigh field region, in which M is saturated, to zero field. When the total transverse thermop ower \nis dominated by ANE, the H dependence of S follows the M-H curve of the magnetic material. \nBased on Eq. (1), SANE is often compared in terms of M to discuss the scaling behavior.25,36 \nHowever, since ANE and related tr ansport properties are estimat ed for uniformly magnetized \nmaterials, it is natural to compare the coefficients in terms o f Ms, not M. Importantly, as shown \nin Fig. 2(a), SANE is not correlated with Ms in many materials including Fe, Ni, and simple binary \nalloys.  \nThe charge current density jc,ANE driven by ANE is expressed as \njc,ANE = SANE(m × T) ,         ( 2 )  \nwhere  is the longitudinal electrical conductivity, m is the unit vector of magnetization, and \nT is the temperature gradient applied to the magnetic material. When m aligns perpendicular \nto T, ANE works as a thermoelectric generator. In an open circuit c ondition with m and T \nrespectively being along the z and x directions, the following equation holds for the y direction: \njc,ANE + E = 0, where E is the electric field appearing due to the charge accumulation  induced \nby jc,ANE. Thus, one obtains  \nE = SANE(T × m) ,          ( 3 )  \nand the transverse thermoelectric voltage Vy = EyLy becomes observable in ANE experiments, \nwhere Ey and Ly are the magnitude of E and the length of the material along the y direction, \nrespectively. Here, the E direction is opposite to the direction of the ANE-driven elect ric field \nEANE (= jc,ANE/), which may confuse the definition of SANE, although Eqs. (2) and (3) are based \non the same argument (note also  that the definition, or sign, o f SANE is sometimes different in \ndifferent papers). Equation (3) in dicates that the thermoelectr ic output of ANE can be actively \ncontrolled through the manipulation of m. Owing to this feature and the aforementioned simple \ndevice structure, ANE is expected to realize simple and versati le thermal energy harvesting or \nheat-flux sensing applic ations [Fig. 1(d)].  \nAlthough primary studies on ANE were conducted a long time ago and research \nactivities were quite limited, ANE has received renewed attenti o n  i n  t h e  f i e l d  o f  s p i n  \ncaloritronics.56,57 In the early days of spin caloritronics, many experiments were  performed to \ndistinguish ANE from the thermo -spin conversion called the spin  Seebeck effect because these \nphenomena exhibit a similar symmetry to each other.58-62 Through such activities, spin \ncaloritronics has dramatically promoted ANE studies from the vi ewpoints of both fundamental 4 physics and thermoelectric appli cations. In particular, since t he observation of giant ANE in \nmagnetic topological materials in 2018,31 physics and materials science studies have further \naccelerated and become a trend in condensed matter physics. Lar ge ANE has also been \nobserved in various materials , including Heusler compounds,31,32,36,44,47,51,53 ferromagnetic \nbinary alloys,20,40,46,48,49 and multilayer films,21,55 of which SANE is an order of magnitude larger \nthan that of Fe.12,40,46 In parallel with the works on ANE, the anomalous Ettingshausen  effect, \nthe Onsager reciprocal of ANE, has also been observed in variou s magnetic materials in both \nbulk and film forms by means of a ctive thermal detection techni ques.63-74 Such measurements \nreveal that rare-earth permanent  magnets can be promising candi date materials for transverse \nthermoelectric conversion.70,74 More recently, not only plain magnetic materials but also hybr id \nstructures comprising thermoelectric and magnetic materials hav e begun to be used, and a \nconceptually different effect called the Seebeck-driven transve rse thermoelectric generation \n(STTG) has been demonstrated.75-77 In light of these research activ ities, in this perspective, we  \nreview recent studies on transver se thermoelectric generation u sing magnetic materials and \ndiscuss its potential applications. Here, we focus on ANE and S TTG; although the spin Seebeck \neffect also enables transverse thermoelectric generation, it is  not covered in this perspective.  \n To realize practical applications of ANE, it is necessary to f ind and develop magnetic \nmaterials with large  SANE. Design guidelines for materials suitable for ANE can be obtai ned by \nseparating SANE into two components: \n SANE = xxxy  AHExx,        ( 4 )  \nwhere xx, AHE, and xx (xy) are the longitudinal electrical resistivity, anomalous Hall \nresistivity, and diagonal (off-d iagonal) component of the therm oelectric conductivity tensor, \nrespectively. The second term on the right-hand side of Eq. (4) , AHExx, appears as a \nconsequence of the fact that t he longitudinal carrier flow indu ced by the Seebeck effect is bent \ndue to the anomalous Hall effect (AHE).78 The first term, xxxy, is usually regarded as intrinsic \nANE because xy directly converts T into a transverse electric field. A recent trend in \nimproving  SANE is to find materials with large xy, in which the Berry curvature of the electronic \nbands near the Fermi level plays an important role. Materials w ith topological band structures \nshow large xy values due to the Berry curvature. The resultant SANE of 6-8 VK1 in Co-based \nHeusler compounds is the current record high at room temperatur e.31,36,47,51,53 The large SANE in \nSmCo 5-type magnets is also believed to be dominated by large xy due to the intrinsic \nmechanism.70,74 Importantly, when ANE originates from electronic band structur es, SANE and \nxy have no correlation with the saturation magnetization (Fig. 2) . Another strategy for 5 enhancing ANE is to optimize magnetic multilayer structures; AN E in alternately stacked \nferromagnetic metal/nonmagnetic metal multilayers is enhanced b y increasing the number of \ninterfaces per unit volume.21,55 Although the enhancement of ANE in multilayers seems to be a \nuniversal behavior, the microsc opic mechanism responsible for t his phenomenon has not been \nclarified so far. As shown here , both the bulk transport proper ties and interface engineering are \nimportant for obtaining large ANE. Although ANE studies have pr ogressed rapidly in recent \nyears, the obtained  SANE values are still smaller than 10 VK1, which is 1-2 orders of \nmagnitude smaller than the Seebeck coefficients of thermoelectr ic materials in practical use. \nTherefore, further breakthrough developments in physics and mat erials science are needed for \nthe applications of transverse t hermoelectric generation. As a part of such efforts, STTG was \nproposed, which will be discussed later.  \nIn general, ANE can be characterized by measuring the magnetic field dependence of a \ntransverse thermopower in magnetic materials. Following Eq. (3) , the ANE-induced \nthermopower exhibits the odd dependence on the m direction. In contrast to isotropic bulk \nmaterials, ANE measurements for thin films are performed in two  different configurations \nbecause of the huge difference between the in-plane and out-of- plane dimensions.60-62 One \nconfiguration is the in-plane magnetized (IM) configuration, wh ere the magnetic field H (T) \nis applied along the in-plane (out -of-plane) direction [inset o f Fig. 3(a)]. The other configuration \nis the perpendicularly magnetized (PM) configuration , where H (T) is applied along the out-\nof-plane (in-plane) direction [i nset of Fig. 3(b)]. Figures 3(a ) and 3(b) show an  example of the \nexperimental results of ANE measured in the IM and PM configura tions, respectively, for the \ntwo 50-nm-thick Co 2MnGa thin films with different composition ratios: Co 53.0Mn 23.8Ga23.2 \n(referred to as Co 2MnGa-A) and Co 41.4Mn 27.9Ga30.7 (referred to as Co 2MnGa-B). The \nCo2MnGa-A and Co 2MnGa-B films were epitaxially grown on single-crystalline MgO ( 001) \nsubstrates. Because of the strong demagnetization field in the out-of-plane direction, the \nremanent magnetization is stabilized in the in-plane direction,  resulting in large ANE voltage \neven in the absence of an external magnetic field in the IM con figuration [Fig. 3(a)], whereas \nno ANE voltage is generated at zer o field in the PM configurati on [Fig. 3(b)]. As depicted in \nFig. 1(d), the IM configuration is  suitable for thermal energy harvesting and heat flux sensing \nbecause ANE-based thermoelectric generation is realized simply by forming films onto heat \nsources. Large ANE voltage owing to the remanent magnetization in the IM  configuration is \nalso preferable for pr actical applications. In contrast, in the IM configuration, it is difficult to \nestimate SANE quantitatively because the temperature difference between the top and bottom of 6 the thin films has to be quantified. Therefore, for the quantit ative estimation of SANE, the PM \nconfiguration is  widely used, although it often r equires a large magnetic field  to align m of \nfilms along the out-of-plane direct ion to overcom e the strong d emagnetization field. In the PM \nconfiguration, the magnitude and distribution of T can be exactly measured by several \nexperimental techniques, such as t hermometers attached on hot a nd cold sides of the substrate \nor heat baths,20,21,34,54 on-chip thin-film thermometers grown on the substrate,32,44,48 and an \ninfrared camera.40,46,47,53,55,65 Thus, Ey can be normalized by the temperature gradient xT along \nthe x direction, enabling the estimation of SANE by taking the zero-field intercept of the magnetic \nfield dependence of Ey/xT [see Eq. (1)].  \nIn Figs. 3(a) and 3(b), one can see the large difference in the  ANE-induced thermopower \nbetween the Co 2MnGa-A and Co 2MnGa-B films, although the X-ray diffraction (XRD) patterns \nfor these films show almost identical crystal structure and L21-type atomic orderin g [Fig. 3(c)]. \nOne of the reasons for such a dramatic variation of ANE in almo st the same materials is the \nlarge difference in the first term of Eq. (4), xxxy, through the intrinsic xy. To obtain the \nintrinsic αxy from the anomalous Hall conductivity xy (22\nAHE AHE xx  ) originating from \nthe Berry curvature, the followi ng expression is often used in first-principles calculations: \n  1\nxy xyfdeT                             (5)  \nwhere  B 1e x p 1fk T   is the Fermi distribution function with e (e > 0), , μ, and \nkB respectively being th e electron charge, ene rgy, chemical poten tial, and Boltzmann constant. \nFigure 3(e) shows the  F dependence of xy calculated for stoichiometric Co 2MnGa using \nEq. (5) based on xy shown in Fig. 3(d), where F is the Fermi energy. The xy value shows a \nsteep change around the Fermi level,  =F, because of the peak of xy near the Fermi level. \nThis calculation suggests that the large difference in the ANE- induced thermopower between \nthe Co 2MnGa-A and Co 2MnGa-B films is mainly attributed to a different position of th e Fermi \nlevel caused by their composition difference, which was directl y proven by photoemission \nspectroscopy in a previous study.53 Namely, the Co 2MnGa-A film has the Co-rich composition \ngiving an additional valence electron compared to the stoichiom etric case, which shifts the \nFermi level upward by approximate ly 0.07 eV corresponding the p eak position of the theoretical \nxy. Therefore, large xy of 3.3Am1K1 was experimentally obtained in the Co 2MnGa-A film, \nwhich is close to the theoretical xy of 4.2 Am1K1 at  F = 0.07 eV, whereas the Co 2MnGa-\nB film exhibits much smaller xy of 0.8 Am1K1 because of its lower valence electron number 7 caused by the Co-poor composition ratio. As indicated by this r esult, it is impor tant to tune the \nFermi level to obtain the theoretically predicted highest xy and the resultant large SANE in \nvarious materials. It is worth mentioning here that the contrib ution of extrinsic mechanisms, \nsuch as skew scattering79 and side jump,80 on ANE has been disregarded in the previous studies \nand thus not been clarified so far, which might cause a disagre ement between the experimental \nand theoretically calculated intrinsic values in the magnitude and sign of xy. For example, as \nshown in Fig. 3(e), the sign of xy in Co 2MnGa becomes negative by slightly reducing the \nvalence electron number. In the  experiment, however, positive xy was always obtained in the \nCo2MnGa-B film and other Co 2MnGa films having a valence el ectron number lower than the \nstoichiometry.53 Weischenberg et al. calculated both the intrinsic and side-jump contributions \non ANE in bcc Fe, hcp Co, fcc Ni, and L1 0-FePt and claimed their equal importance.17 Thus, \nthe elucidation of extrinsic con tributions to ANE in various ma terials is important to explore \nmaterials with large SANE.  \nToward the thermoelectric applic ations based on ANE, not only p hysics and materials \nscience studies but also device engineering is indispensable. A s described above, one of the \nadvantages of ANE-based thermoel ectric applications is the simp le device structure, in which \nthe thermoelectric voltage is ea sily enlarged by elongating the  total length of a magnetic wire \ngrown/attached on a heat source surface. Here, we present an ex perimental demonstration of \nlarge thermoelectric voltage generation in the ANE-based module  in the IM configuration [Fig. \n1(d)]. Figure 4(a) shows the magnetic field dependence of Vy f o r  a  l a t e r a l  C o 2MnGa-Au \nthermopile device, in which 50 Co 2MnGa wires with a length of 10 mm, width of 50 m, and \nthickness of 1 m are connected in series throu gh Au wires with a length of 10 mm, width of \n50 m, and thickness of 400 nm. Thus, the total length of the Co 2MnGa wires reaches 500 mm. \nThe wires were fabricated on an area of 10  10 mm2. It is clearly found that the Vy signal in \nthe Co 2MnGa-Au thermopile is two orders  of magnitude larger than that in the non-patterned \nCo2MnGa films shown in Fig. 3(a), which confirms the usefulness of  t h e  s i m p l e  l a t e r a l  \nthermopile device for ANE. A promising appl ication of such ANE- based thermopiles is a heat \nflux sensor with high flexibility and low thermal resistance.46 As shown in Fig. 4(b), the ANE \nsignal linearly increases with the heat flux density jq flowing across the thermopile device, \nwhich works as a heat flux sensor. The slope of the jq dependence of Vy, which gives a sensitivity \nof jq, was estimated to be 0.110 VW1m2 at 0H = 0.10 T, where the magnetization of the \nCo2MnGa wires aligns along the field direction. This sensitivity i s an order of magnitude larger \nthan that in the prototypical Fe 81Al19/Au thermopiles, which mainl y originates from the longer 8 total length of the magnetic wire and larger SANE in Co 2MnGa than in Fe 81Al19.46 However, it is \nimportant to mention that the sensitivity at zero field reduces  to 0.016 VW1m2 because of the \nsmall remanent magnetization of the narrow Co 2MnGa wires, which is caused by an increase \nin the demagnetization field a nd possible magnetic domain forma tion. Therefore, to preserve \nlarge transverse thermoelectri c voltage at zero field in elonga ted narrow magnetic wires, \nmagnetic materials with small mag netization and/or large uniaxi al magnetic anisotropy in the \nin-plane direction are preferable. For a practical usage of ANE -based heat flux sensors, the \nsensitivity of >1 VW1m2 is desired. Although this sensitivity value is comparable to o r \nsmaller than that of conventional Seebeck-based heat flux senso rs, ANE has strong advantages \nin flexibility and low thermal r esistance, extending applicatio ns of heat flux sensors.  \n Up to now, we have outlined the  recent activities on ANE. In 2 021, we proposed and \ndemonstrated the transverse ther moelectric generation different  from ANE. This is named \nSTTG.75 Although ANE appears in a plain magnetic material, STTG appear s in hybrid \nstructures comprising thermoelectric and magnetic materials and  originates from the \ncombination of the Seebeck effect in the former and AHE in the latter, which is inspired by the \nsecond term on the right-hand side of Eq. (4). Figure 5(a) show s a schematic of the typical \nstructure used to demonstrate STTG. When T is applied to a closed circuit comprising \nthermoelectric and magnetic materials, a charge current is indu ced by the difference in the \nSeebeck coefficients of the mater ials. This charge current is i n turn converted into a transverse \nelectric field owing to AHE in the magnetic material. Based on our phenomenological \ncalculation, the transverse therm opower in the hybrid structure  shown in Fig. 5(a) is expressed \nas follows:75,77 \nAHE\ntot ANE TE M\nTE MSS SSr\n ,        ( 6 )  \nwhere TE(M) and STE(M) are the longitudinal resistivity  and Seebeck coefficient of th e \nthermoelectric (magnetic) material, respectively, and   MT E T E T EM Myy xx zz rL L L L L L  is the size \nratio determined by the length of  the thermoelectric (magnetic)  material ,,\nTE(M)xyzL  along the x, y, \nand z directions. The second term on the right-hand side of Eq. (6) represents the STTG \ncontribution and can reach the order of 100 VK1 by optimizing the combination of the \nthermoelectric and magnetic materials as well as their dimensio ns. In fact, our experiments \nshow that the Co 2MnGa/n(p)-type Si hybrid material exhibits a transverse thermop ower of 82.3 \nVK1 (−41.0 VK1), which is one order of magnit ude larger than t he record-high SANE value \nand is quantitatively consistent  with the prediction based on E q. (6) [Fig. 5(b)]. STTG appears 9 in the absence of magnetic fields when the magnetic material la yer has a finite coercive force \nand remanent magneti zation, as demonstrated  by the measurements  using a L10-FePt/n-type Si \nhybrid material [Fig. 5(b)]. The se results confirm the usefulne ss and potential of STTG. \nHowever, at present, such a la rge thermopower is obtained only when the combination of \nthermoelectric slabs and magnetic films, i.e., the system with large r values, is used. For \nthermoelectric power generation (heat sensing) applications, it  is necessary to realize large \nSTTG in all-slab (all-film) hybrid materials with reasonable r values. Because STTG provides \nhigh flexibility for designing its performance based on a vast number of studies on the Seebeck \neffect and AHE, there is still pl enty of scope for the improvem ent.  \nFinally, we would like to highlight an aspect that should be co nsidered in ANE and \nSTTG experiments. When these phenomena are applied in practice,  it is preferable to utilize the \nsynergistic effects of both. However, when the transport proper ties related to ANE and STTG \nare investigated quantitatively,  they must be cl early and caref ully separated us ing well-defined \nsample systems. The contaminati on of the STTG contribution in A NE experiments may occur \naccidentally, for example, when sample films are formed on dope d Si substrates, which are \noften used in thermoelectrics a nd spin caloritronics. We verifi ed that, even if thermally oxidized \nSi substrates are used, unintende d electrical connections betwe en thin films and doped Si may \nbe formed through the side surfaces of the substrates, leading to significant errors in ANE results. \nSuch an electrical connection can be avoided by patterning thin  films using shadow masks or \nlithography techniques.   In this perspective, we have reviewed the progresses on the tr ansverse thermoelectric \ngeneration using magnetic materi als, with a particular focus on  our recent activities. Systematic \nstudies on ANE have clarified its mechanism and potential for n ext-generation thermoelectric \ntechnologies. By utilizing the emerging phenomena such as STTG,  studies on transverse \nthermoelectric conversion will become more active in the near f uture; not only plain magnetic \nmaterials but also hybrid or composite materials will be key sy stems, where interface \nengineering is also important. For practical applications of th e transverse thermoelectric \ngeneration, it is necessary to drive ANE and STTG in the absenc e of magnetic fields. Thus, \nsearching and developing magneti c materials with large anomalou s Nernst coefficients, large \nanomalous Hall angle, and high ma gnetic anisotropy are indispen sable. The previous study \nsuggests that magnetic material s with high magnetic anisotropy exhibit large ANE.\n20 T o  \nunderstand the origin of this be havior and confirm its universa lity, one has to perform more \nsystematic experiments using vari ous materials, where a high-th roughput screeni ng method for \nANE materials will be a powerful tool.81 Although we have discussed ANE and STTG mainly 10 in terms of the transverse therm opower, the reduction of the th ermal conductivity of magnetic \nmaterials is also important for improving the efficiency of the  transverse thermoelectric \ngeneration because the figure of  merit for ANE and STTG is inve rsely proportional to the \nthermal conductivity in the same manner as the Seebeck effect.62,75,77 To reduce the thermal \nconductivity, phonon engineering t echniques and nanostructuring  materials will be \neffective,82,83 while they have rarely been used in spin caloritronics.84 Therefore, for the further \ndevelopment of transverse thermo electric generation technologie s, the interdisciplinary fusion \namong spin caloritronics, nanoscale materials science, and ther mal engineering is desired. \n \nThe authors thank R. Iguchi, K. Masuda, A. Miura, Y. Miura, and  K. Yamamoto for \nvaluable discussions and M. Isomura, N. Kojima, B. Masaoka, and  R. Tateishi for technical \nsupports. This work was partia lly supported by CREST “Creation of Innovative Core \nTechnologies for Nano-enabled Ther mal Management” (JPMJCR17I1) and PRESTO \n“Scientific Innovation for Energy Harvesting Technology” (JPMJP R17R5) from JST, Japan; \nMitou challenge 2050 (P14004) from NEDO, Japan; Grant-in-Aid fo r Scientific Research (S) \n(18H05246) from JSPS KAKENHI, Japa n; and the NEC Corporation.   \n DATA AVAILABILITY \nThe data that support the findings of this study are available from the corresponding \nauthor upon reasonable request.  REFERENCES \n1 S. Twaha, J. Zhu, Y. Yan, and B. Li , Renew. Sust. Energy Rev. 65, 698 (2016). \n2 I. Petsagkourakis, K. Tybrandt , X. Crispin, I. Ohkubo, N. Satoh , and T. Mori, Sci. Tech. \nAdv. Mater. 19, 836 (2018).  \n3 M. Haras and T. Skotnicki, Nano Energy 54, 461 (2018). \n4 A. V. 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Smit, Physica 21, 877 (1955).  \n80 L. Berger, Phys. Rev. B 2, 4559 (1970).  \n81 R. Modak, K. Goto, S. Ueda, Y. Miura, K. Uchida, and Y. Sakurab a, APL Mater. 9, \n031105 (2021). \n82 A. J. Minnich, M. S. Dresselhau s, Z. F. Ren, and G. Chen, Energ y Environ. Sci., 2, 466 \n(2009).  \n83 M. Nomura, J. Shiomi, T. Shi ga, and R. Anufriev, Jpn. J. Appl. Phys. 57, 080101 (2018). \n84 A. Miura, T. Kikkawa, R. Iguchi , K. Uchida, E. Saitoh, and J. S hiomi, Phys. Rev. Mater. \n1, 014601 (2017). \n  15 \n \nFIG. 1.  Schematics of (a) the  Seebeck effect, (b) the thermoelectric m odule based on the Seebeck effect, (c) \nANE, and (d) the thermoelectric module based on ANE. ET, and M denote the electric field induced by \nthe Seebeck effect or ANE, temperature gradient, and magnetizat ion vector, respectively.  \n  16  \nFIG. 2.  Saturation magnetization Ms dependence of the absolute values of (a) the anomalous Nernst \ncoefficient | SANE| and (b) the transverse the rmoelectric conductivity | xy| for various magnetic materials. The \ninset of (a) shows the double-logarithmic Ms dependence of | SANE|. 0 is the vacuum permeability. x in the \nlegends denotes the composition ratio of the magnetic materials . Although ANE has been measured in a wide \ntemperature range, only the data measured at room temperature a re shown in this figure. The | SANE| and |xy| \nv a l u e s  f o r  t h e  m a t e r i a l s  w i t h  a n  a s t e r i s k  w e r e  e s t i m a t e d  t h r o u g h the measurements of the anomalous \nEttingshausen effect and the Onsager reciprocal relation. The | xy| value in Ref. 36 is corrected following the \ninformation in Ref. 51.  \n  \n17  \nFIG. 3.  (a) Magnetic field 0Hx dependence of the transverse thermoelectric voltage Vy and the transverse \nelectric field Ey in the Co 2MnGa-A and Co 2MnGa-B films at room temperature in the IM configuration, \nmeasured when the magnetic field and temperature gradient were applied along the x and z directions, \nrespectively. A heater power of 160 mW was applied to the top h eat bath to generate the temperature gradient \nalong the z direction zT. (b) Magnetic field 0Hz dependence of Ey normalized by the temperature gradient \nxT in the Co 2MnGa-A and Co 2MnGa-B films at room temperature in the PM configuration, measu red when \nthe magnetic field and temperature gradient were applied along the z and x directions, respectively. xT was \nmeasured with an infrared camera by coating the sample surface with a black ink having high infrared \nemissivity. (c) XRD patterns for the out-of-plane direction (  = 0°) and the <111> direction (  = 54.7°) of \nthe Co 2MnGa-A and Co 2MnGa-B films. The peaks denoted by  asterisks arise from the dif fractions from the \nMgO substrates. (d),(e) Intrinsic xy and xy for the stoichiometric Co 2MnGa as a function of the energy \ndifference from th e Fermi energy   F, obtained from the first-principles calculations.53 \n  \n18  \nFIG. 4.  (a) 0Hx dependence of Vy in the Co 2MnGa-Au thermopile device, in which 50 Co 2MnGa wires are \nelectrically connected in series through Au wires, for various values of jq. jq denotes the magnitude of the \nheat flux density jq along the z direction. The Co 2MnGa and Au wires align along the y direction. (b) jq \ndependence of Vy in the Co 2MnGa-Au thermopile device at the saturation magnetization ( 0Hx = 0.10 T) and \nremanent magnetization ( 0Hx = 0 T) states.  \n \n \n \nFIG. 5.  (a) Schematic of STTG in a clos ed circuit comprising thermoele ctric and magnetic materials \nelectrically connected at both ends. (b) Stot or SANE values at room temperature for the Co 2MnGa/n-type Si, \nCo2MnGa/p-type Si, Co 2MnGa/non-doped Si, and FePt/n-type Si hybrid materials used in Ref. 75. When Stot \n(SANE) in the hybrid materials was measured, the thermoelectric and magnetic layers were electrically \nconnected (disconnected). The Stot a n d  SANE values were estimated by extrapolating the magnetic field \ndependence of the transverse thermopower from the high field re gion to zero field.  \n"    },    {        "title": "2102.13325v1.Distinctive_magnetic_properties_of_CrI3_and_CrBr3_monolayers_caused_by_spin_orbit_coupling.pdf",        "content": "Distinctive magnetic properties of CrI 3and CrBr 3monolayers\ncaused by spin-orbit coupling\nC. Bacaksiz,1, 2, 3, 4D.\u0014Sabani,1, 2R. M. Menezes,1, 2, 5and M. V. Milo\u0014 sevi\u0013 c1, 2,\u0003\n1Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium\n2NANOlab Center of Excellence, University of Antwerp, Belgium\n3Bremen Center for Computational Material Science (BCCMS), Bremen D-28359, Germany\n4Computational Science Research Center, Beijing and Computational\nScience and Applied Research Institute Shenzhen, Shenzhen, China\n5Departamento de F\u0013 \u0010sica, Universidade Federal de Pernambuco,\nCidade Universit\u0013 aria, 50670-901, Recife-PE, Brazil\n(Dated: March 1, 2021)\nAfter the discovery of magnetism in monolayer CrI 3, the magnetic properties of di\u000berent 2D ma-\nterials from the chromium-trihalide family are intuitively assumed to be similar, yielding magnetic\nanisotropy from the spin-orbit coupling on halide ligands. Here we reveal signi\fcant di\u000berences\nbetween the CrI 3and CrBr 3magnetic monolayers in their magnetic anisotropy, resulting Curie\ntemperature, hysteresis in external magnetic \feld, and evolution of magnetism with strain, all pre-\ndominantly attributed to distinctly di\u000berent interplay of atomic contributions to spin-orbit coupling\nin two materials.\nPACS numbers: Valid PACS appear here\nI. INTRODUCTION\nIn the family of two-dimensional (2D) materials, ex-\nhibiting a range of exciting and advanced properties,1the\nintrinsic magnetism was long evasive. The \frst pathway\nto realize the magnetic 2D crystal has been exfoliation\nfrom layered bulk magnets. Nearly a decade after real-\nization of graphene, exfoliated monolayer FePSe 32,3and\nCrSiTe 34were reported to have long-range magnetic or-\nder, indirectly demonstrated via Raman and conductiv-\nity measurements, respectively. However, the \feld of 2D\nmagnetism has truly boomed only after the premiere di-\nrect evidence of two-dimensional (2D) ferromagnetism, in\nmonolayer CrI 35and bilayer Cr 2Ge2Te6,6that attracted\nmuch attention of the scienti\fc community. Since then,\na number of new 2D magnets were synthesized and uti-\nlized in di\u000berent heterostructures,7including monolayer\nCrBr 38and CrCl 3,9other members of the Cr-trihalide\nfamily next to CrI 3.\nAlthough the number of 2D magnets exfoliated either\nfrom magnetic or nonmagnetic10{12bulk counterparts is\nincreasingly large, the physics behind the magnetism in\nthese materials is common. Namely, the magnetic mo-\nment originates from unpaired d-electron of the transi-\ntion metal atoms, but the stability of magnetization at\n\fnite temperature comes as the consequence of magnetic\nanisotropy, lifting the restrictions stipulated by Mermin-\nWagner theorem.13There are two possible sources of\nthe magnetic anisotropy in 2D materials: one is the\nanisotropy of the magnetic ion due to the character and\nsymmetry of bond coordination with the non-magnetic\natoms, known as single-ion anisotropy (SIA); the other is\nthe anisotropy in magnetic exchange interaction between\nthe magnetic atoms within the crystal. In each case, the\nspin-orbit coupling (SOC) is a direct responsible for the\narising magnetic anisotropy.14Before the experimental realization of monolayer CrI 3,\nthe chromium-halides were predicted to be ferromagnetic\nin the monolayer form,15where by including SOC in\nthe consideration, magnetic anisotropy energies (MAEs)\nwere calculated.16After the actual exfoliation of CrI 3, a\nmore detailed study on the chromium-halides reported\nthe tunability of MAE by strain.17Other studies dis-\ncussed accurate calculation of the critical temperature of\nCrI3and other 2D magnetic materials using spin-wave\ntheory and Monte-Carlo (MC) simulations on top of ab\ninitio results,18{20and were mostly focused on the proper\ndescription of the magnetic interactions. Very recent\nworks then considered adsorption and substitution of for-\neign atoms, such as hydrogen or oxygen,21{23to manip-\nulate the exchange interaction in order to increase the\ncritical temperature.\nIt is now well established that the origin of the mag-\nnetic anisotropy in monolayer CrI 3, for both SIA and\nanisotropy of the exchange interactions, is associated\nwith the SOC of iodine atoms rather than chromium\nones.24{27As a consequence one can intuitively predict\nthat the magnetic anisotropy of monolayer CrBr 3should\nbe lower as compared to that in CrI 3due to di\u000berence\nin SOC between I and Br. As a corroboration to these\npredictions, recent experiment established the Curie tem-\nperature of CrBr 3as 21 K,8signi\fcantly lower than one\nof monolayer CrI 3(TC= 45 K).5However, the extent\nand manner of how individual atomic contributions to\nSOC a\u000bect the magnetic anisotropy in di\u000berent mono-\nlayer magnets remained unaddressed to date. Therefore,\nin this article we perform a thorough comparison between\nseemingly similar monolayer magnets, CrI 3and CrBr 3, in\norder to provide a comprehensive (qualitative and quan-\ntitative) understanding of the atomically-resolved e\u000bects\nof spin-orbit coupling on magnetism. Speci\fcally, we ex-\nplore the magnetic anisotropy, variation of magnetismarXiv:2102.13325v1  [cond-mat.mes-hall]  26 Feb 20212\nFIG. 1: Schematic representation of the structure of monolayer CrI 3(a) and CrBr 3(d). Panels (b) and (e) show the di\u000berence\nbetween the charge distribution after crystallization and the total charge distribution of bare atoms, which then indicates\nthe bonding and anti-bonding charges in the two materials. (c) and (f) panels show the density of states of two materials,\ndecomposed according to the atomic orbitals. Subscripts x0,y0andz0in the orbitals denote the local coordinates of the\ncorresponding atoms.\nunder biaxial strain, critical temperature and response\nto the magnetic \feld, and link the di\u000berences in SOC in\ntwo similar materials to their rather dissimilar emergent\nmagnetic properties.\nThe paper is organized as follows. In Sec. II we provide\ndetails of our computational methodology. The struc-\ntural and electronic properties, the magnetic properties,\nand the e\u000bect of strain are presented and discussed in\nSec. III A, III B, and III C, respectively. In Sec. III D,\nthe temperature-dependent magnetism and magnetiza-\ntion reversal of the two materials in external magnetic\n\feld are discussed. Sec. IV summarizes our \fndings.\nII. COMPUTATIONAL METHODOLOGY\nIn order to investigate the structural, electronic\nand magnetic properties we use calculations based on\ndensity functional theory (DFT). To obtain magnetic\nparameters, we employed four-state energy mapping\nmethodology.28,29When di\u000berent magnetic con\fgura-\ntions are examined, the magnetic moments are con-\nstrained in desired directions, in order to prevent their\nrelaxation into the ground state con\fguration (or any\nstable con\fguration other than desired one) during the\nself-consistent procedure. Heisenberg spin Hamiltonian\nis considered in the form:\nH=1\n2X\ni;jSiJijSj+X\niSiAiiSi; (1)\nwhereSi= (Sx\ni;Sy\ni;Sz\ni) is a vector. JijandAiiare 3\u0002\n3 matrices describing the magnetic exchange interaction\nbetween the di\u000berent sites and the single-ion anisotropy\n(SIA), respectively.\nWe used the Vienna ab initio simulation package\nVASP30{32which solves the Kohn-Sham equations itera-\ntively using a plane-wave basis set. To describe electronexchange and correlation, the Perdew-Burke-Ernzerhof\n(PBE) form of the generalized gradient approximation\n(GGA)33was adopted. The spin-orbit coupling (SOC)\nwas included in all calculations, as VASP includes SOC\nvia following added term34to DFT Hamiltonian:\nHSO=\u0016h2\n(2mec2)2\u0012\n1\u0000V(r)\n2mec2\u0013\u00002dV(r)\ndr~ \u001b\u0001~L; (2)\nwhere~ \u001b= (\u001bx;\u001by;\u001bz) stands for 2\u00022 Pauli spin ma-\ntrices,~L=~ r\u0002~ pis the angular momentum, and V(r)\nis the spherical part of the electron potential. The van\nder Waals (vdW) forces were taken into account using\nthe DFT-D2 method of Grimme.35In order to calcu-\nlate charge transfer between the atoms we employed the\nBader charge technique.36\nThe kinetic energy cut-o\u000b of the plane-wave basis set\nwas 600 eV and energy convergence criterion was 10\u00006\neV in the ground-state calculations. Gaussian smearing\nof 0.01 eV was used and the pressures on the unit cell\nwere decreased to a value lower than 1.0 kbar in all three\ndirections. On-site Coulomb repulsion37parameter, U,\nwas taken as 4 eV for magnetic Cr atom.38{40To avoid in-\nteractions between periodically repeating monolayers in\nvertical direction, our calculations were performed with\nsu\u000eciently large vacuum space of \u001813\u0017A.\nFor subsequent considerations of the temperature-\ndependent magnetization and the critical temperature\n(TC), we performed spin dynamics simulations based\non stochastic Landau-Lifshitz-Gilbert (LLG) equation,\nusing simulation package Spirit ,41adapted to accom-\nmodate the anisotropic interactions of our Hamiltonian\n[Eq. (1)]. 50\u000250 supercell of the spin lattice was con-\nsidered. The spin system is initialized in random con-\n\fguration at high temperature and then cooled down\nwith the temperature steps of 0.125 K. To obtain equilib-\nrium magnetization took relaxation over 105time steps3\n(with time step \u0001 t= 1 fs) for each temperature. For\nthe \feld-dependent calculations, the Zeeman term HB=\ng\u0016BP\niSi\u0001Bhas been included in Eq. (1), where Bis the\napplied magnetic \feld; g= 2 is the g-factor for S= 3=2;\n\u0016Bthe Bohr magneton and \u0016=gS\u0016B= 3\u0016Bis the\nmagnetic moment of Cr atoms.\nTABLE I: Structural and electronic parameters of monolayer\nCrI3and CrBr 3. The charge transfer per atom ( \u001aCrand\u001aX)\nwas calculated using Bader charge technique.\ndba \u0012 \u001a Cr=\u001aXEg\n(\u0017A) ( \u0017A) (\u000e) (e\u0000/e\u0000) (eV)\nCrI32.78 6.94 92.06 1.10/ \u00000:37 0.55\nCrBr 32.55 6.40 92.90 1.33/ \u00000:44 1.59\nIII. RESULTS AND DISCUSSION\nA. Structural and electronic properties\nWe start from the structural properties of monolayer\nCrX 3, where X stands for ligand I and Br atoms. CrX 3\ncrystallizes in the trigonal P31mspace group. The\nhexagonal planar lattice of Cr atoms is sandwiched be-\ntween triangular planar lattices of ligand atoms as shown\nin Figs. 1 (a) and (d) for respective materials. One\nCr atom bonds to 6 ligand atoms and each ligand atom\nbonds to 2 Cr atoms. The structures have 3-fold in-plane\nsymmetry, where the triangular lattices of ligand atoms\nare 180\u000erotated with respect to each other, which cre-\nates octahedral coordination around each Cr atom. As\nlisted in Table I, the Cr-I and Cr-Br bond lengths are\nfound to be 2.78 \u0017A and 2.55 \u0017A, respectively, which is di-\nrectly proportional with the radius of the bonding orbital\nof the ligand, I-5 pand Br-4p. Consequently, the lattice\nconstants of monolayer CrI 3and CrBr 3are also di\u000berent,\n6.94 \u0017A and 6.40 \u0017A, respectively, consistent with the ex-\nperimentally measured 6.95 \u0017A42and 6.50 \u0017A.43Note that\nboth monolayers exhibit slightly larger lattice constant\nthan their bulk or few-layer counterparts. The covalent\nbonding character consists of 1.10 e\u0000and 1.33e\u0000dona-\ntion of Cr and 0.37 e\u0000and 0.44e\u0000gain of I and Br atoms,\nrespectively. This bonding charge is visualized in Figs. 1\n(b) and (e), as obtained by subtracting bare atom charge\ndistributions from the charge distribution of the crystals.\nCharges obviously accumulate to the bonding sites and\naround Cr atoms. Observed charge accumulation around\neach ligand clearly shows three 'charge clouds' that can\nbe divided in two groups, representing two types of bond-\ning between Cr and ligand atoms: the facial ( \u001b) bonding,\nrepresented by two charge clouds facing the Cr atoms,\nand the lateral ( \u0019) bonding, represented by the third\ncharge cloud, in direction orthogonal to the Cr-X-Cr-X\nplane.\nTo depict the energetic arrangements of the orbital\nstates, we calculated the partial density of states for twoconsidered materials in Figs. 1 (c) and (f). Both materi-\nals are semiconductors, in which the valence band max-\nimum is dominated by p-orbital of the ligand. On the\nother hand, d-orbital of Cr resides much deeper in the\nvalence band, with similar energetic delocalization for\nboth CrI 3and CrBr 3.dx0y0,dx0z0anddy0z0are degen-\nerate, and are plotted together. dz02anddx02\u0000y02orbitals\nare also degenerate and appear at the conduction band.\npx0andpy0orbitals are degenerate as well, and domi-\nnate the valence band maximum. pz0orbital mostly re-\nsides in the middle of the valence band. One should note\nthat in order to obtain orbital states compatible with\nthe octahedral coordination, the general coordinates are\nrotated as suggested by Rassekh et al. .21Therefore we\nconsiderxy-plane andz-axis of the general Cartesian co-\nordinates aligned with the local coordinates, where the\nCr-X-Cr-X plane is considered as x0y0-plane, and z0axis\nis orthogonal to that plane. Orbital decomposition shows\nthat both materials exhibit very similar orbital delocal-\nizations. This is also visible in the charge density varia-\ntions shown in Figs. 1 (b) and (e). Purple regions show\nthe depletion of charges of isolated atoms through the\ncharge density of the crystal. It is obvious that the or-\nbitals laying along the Cr-X bonds exhibit most delocal-\nization. We understand from the charge depletion and\nthe orbital decomposition of DOS that dx0y0,dx0z0and\ndy0z0orbitals are localized and do not show any variation\nfrom their single-atom form. dz02anddx02\u0000y02orbitals\nform aspd2hybridization with the px0andpy0orbitals\nas expected for an octahedral coordination. Beside these\nsimilarities, the particular di\u000berence between CrI 3and\nCrBr 3is found between 4 pof Br and 5 pof I, leading to\nenergetic di\u000berences of the spd2hybridization. Since 4 p\norbital of Br is more con\fned as compared to 5 pof I, the\nstates of CrBr 3are shifted to higher energies. Therefore,\nthe band gap values are di\u000berent, found as 0.55 eV and\n1.59 eV for CrI 3and CrBr 3, respectively.\nB. Magnetic properties\n1. Spin-orbit coupling and magnetic anisotropy energy\nIt is already known that monolayer CrI 3exhibits fer-\nromagnetism at \fnite temperature due to the magnetic\nanisotropy originating from SOC on I atoms. It is natural\nto assume that a similar mechanism is responsible for fer-\nromagnetism in CrBr 3, as proposed in previous studies.17\nIn what follows we validate that assumption, but more\nimportantly, we analyze deeper the di\u000berences in individ-\nual atomic contribution to the total SOC in monolayer\nCrI3and CrBr 3, and the consequences thereof.\nFirst of all, we calculate the total energy leading to\nmagnetic anisotropy energy (MAE) and the contributed\nSOC energy (\u0001 ESO), relative to the respective ground-\nstate energies, depending on the spin direction angle\n(\u0012) with respect to the out-of plane direction. In other\nwords, MAE( \u0012) =E(\u0012)\u0000E(\u0012= 0\u000e) and \u0001ESO(\u0012) =4\nFIG. 2: (a) Magnetic anisotropy energy (MAE) and SOC\nenergy as a function of the spin alignment angle with the\nout-of-plane direction. (b) Schematic representation of the\ntop view of the structure. Atoms are enumerated in order\nto track the variation of the corresponding SOC energies for\nspin-alignment in x-direction ( \u0012= 90\u000eand\u001e= 0\u000e) (c), and\niny-direction ( \u0012= 90\u000eand\u001e= 90\u000e) (d). The energy of\nthe out-of-plane spin alignment, the ground state, is set to\n0 eV. Dashed lines represent total \u0001 ESO, while square dots\n(connected by solid lines) represent contributions of each atom\nto the total \u0001 ESO.\nESO(\u0012)\u0000ESO(\u0012= 0\u000e), where angle \u0012is measured from\nz-axis (see Fig. 2(b)). MAE is found to be 0.67 (0.11)\nmeV for for CrI 3(CrBr 3) favoring the out-of-plane di-\nrection. In the literature, the reported MAE values vary\nfrom\u00180:69(0:18) meV to\u00181:48(0:30) meV for CrI 3\n(CrBr 3) depending on the approximations used.16,17,44\nHowever, MAE of CrI 3is 4-5 times larger than that of\nCrBr 3in each previous report, as is in the present study.\nAs shown in Fig. 2(a), MAE( \u0012) and \u0001ESO(\u0012) exhibit\nsinusoidal functional behavior. It is further remarkable\nthat MAE( \u0012) is not equal to \u0001 ESO(\u0012), which indicates\nthat the collective rotation of the spins allows further re-\nlaxation of the spatial part of the wave function, to a\nlower energy. The ratio between MAE( \u0012) and \u0001ESO(\u0012)\nis around 0.5 for either monolayer, at each angle \u0012.\nFurthermore, we calculate the atomic contributions\nto the total SOC energies for the spins pointing in x-\n(\u0012= 90\u000eand\u001e= 0\u000e) andy-direction ( \u0012= 90\u000eand\n\u001e= 90\u000e), as shown in Figs. 2 (c) and (d). The total\ncontribution from Cr atoms is signi\fcantly lower than\nthat from ligand atoms, which con\frms results of previ-\nous works.24,25More importantly, the atomic contribu-\ntions of both I and Br vary depending on the direction\nof the bond coordination of the ligand with respect to\nthe spin direction. To be more direct, in Fig. 2(c) the\nligands X 1, X2, X5, and X 6energetically prefer the spin\npointing in the z-direction, while X 3and X 4prefer the\nspin inx-direction. On the other hand, in Fig. 2(d),the contributions from X 1, X2, X5, and X 6are almost\nzero, however, X 3and X 4prefer the spins aligned with\nz-direction. Brie\ry, the contribution ~ \u001b\u0001~Ldepends on the\ndirection of the orbital angular momentum of the ligand\nLx0,Ly0, which lay on the respective Cr-X bonding direc-\ntions aspx0andpy0. There is interplay with the dz02and\ndx02\u0000y02orbitals through the spd2hybridization. On the\nother hand, Lz0is orthogonal to Cr-X-Cr-X plane. These\nresults not only con\frm the previous works suggesting\nthe main contribution of MAE originates from SOC of\nthe ligand atoms, but also reveal that the main contri-\nbutions are stemming from the bonding orbitals. Previ-\nously, Lado et al. reported MAE of monolayer CrI 3as a\nfunction of SOC strength on Cr and I (ligand) separately\nand concluded that the SOC on ligand is the dominant\nfactor on the MAE.24Here we further reveal that not\nevery ligand has positive contribution to MAE - those\nwith a bond with Cr atom aligned with magnetization\ndirection of interest, will have negative (or no) contribu-\ntion. The apparent di\u000berence between CrI 3and CrBr 3,\non the other hand, is a direct consequence of the relative\nstrength of the SOC of 5 porbital of the I atom compared\nto the 4porbital of the Br atom.\n2. Magnetic exchange interaction and SIA\nTo characterize the magnetic properties of the two\nmonolayers under study, we calculate the magnetic ex-\nchange interactions matrix ( J12) in the \frst nearest-\nneighbor (NN) approximation, using a 2 \u00022 supercell\nshown in Fig. 3. Due to the threefold in-plane symmetry\nof CrX 3, once the exchange matrix of one of the three\n\frst NN pairs is obtained, such as for the pair (1-2), the\nmatrices of other \frst NN pairs, (2-3) and (2-5), can be\ncalculated by rotation operation on the exchange matrix\nof pair (1-2) around the out-of-plane axis. The so ob-\ntained pairwise results are listed in Table II. Depending\non the considered coordinate axes, the exchange matrix of\na pair can change. Here (1-2) pair lays on the x-axis, con-\nsequently (2-3) and (2-5) pairs make 60\u000eand\u000060\u000eangle\nwith thex-axis. Each pair of CrI 3and CrBr 3has a sym-\nmetric exchange matrix, due to preserved inversion sym-\nmetry, hence no Dzyaloshinskii-Moriya interaction (DMI)\nis present.29All diagonal elements show that both mono-\nlayer materials exhibit ferromagnetic interaction in the\n\frst NN consideration. As an e\u000bective exchange interac-\ntion for Cr, the mean of the matrices of (1-2), (2-3), and\n(2-5) pairs is listed in Table II as hJi, and is identical\nfor each magnetic site in the respective monolayer. In\nthis form of representation, it is clearly seen that CrI 3\nexhibits stronger ferromagnetic exchange as compared to\nCrBr 3. It is also rather remarkable that the out-of-plane\nexchange anisotropy, \u0001 = hJixx\u0000hJizz=hJiyy\u0000hJizz\nof CrI 3is 0.22 meV, much larger than 0.04 meV of CrBr 3.\nFinally, we also calculated SIA, which as a result of three-\nfold symmetry is represented by a single parameter, Azz\nii,\nout of nine elements of the Aiimatrix.Azz\niiof CrI 3is5\nTABLE II: Magnetic exchange parameters of monolayer CrI 3and CrBr 3.Jxx,Jyy, andJzzare diagonal elements, and\nJxy=Jyx,Jxz=Jzx,Jyz=Jzyare o\u000b-diagonal elements of the exchange matrix. The mean value hJiof the exchange\nparameters is also given. Out-of-plane anisotropy \u0001 is calculated as hJxxi\u0000hJzzi.Aiiis SIA parameter, same for each Cr\nsite. MAE and ESOare magnetic anisotropy and total SOC energies, respectively.\npairJxxJyyJzzJxy=JyxJxz=JzxJyz=Jzy\u0001Azz\nii MAE \u0001 ESO\n(i-j) (meV) (meV) (meV) (meV) (meV) (meV) (meV) (meV) (meV/f.u.) (meV/f.u.)\nCrI3 -0.22 -0.07 0.67 1.21\n(1-2) -5.10 -3.72 -4.63 0.00 0.00 0.83\n(2-3) -4.07 -4.76 -4.63 -0.60 0.72 -0.42\n(2-5) -4.07 -4.76 -4.63 0.60 -0.72 -0.42\nhJi-4.41 -4.41 -4.63 0.00 0.00 0.00\nCrBr 3 -0.04 -0.01 0.11 0.22\n(1-2) -3.45 -3.29 -3.42 0.00 0.00 0.09\n(2-3) -3.33 -3.41 -3.42 -0.07 0.08 -0.05\n(2-5) -3.33 -3.41 -3.42 0.07 -0.08 -0.05\nhJi-3.37 -3.37 -3.42 0.00 0.00 0.00\nfound to be\u00000:07 meV, much larger than that of CrBr 3,\n\u00000:01 meV. Both monolayers exhibit negative SIA, which\nindicates the energetic preference of the spin alignment\nin the out-of-plane direction. It is worth to mention that\nXuet al. previously calculated 3 \u00023 exchange inter-\naction using di\u000berent set of calculation parameters for\nCrI3and therefore reported di\u000berent values,25however,\nour results are qualitatively consistent with their ones.\nNamely, they found the ferromagnetic diagonal elements\nwith anisotropy favoring the out-of-plane direction, sym-\nmetric o\u000b-diagonal elements, and SIA parameter of Cr\natoms favoring out-of-plane as well. They also analysed\nthe exchange and SIA parameters as a function of SOC\nstrength and concluded the main contribution to those\nparameters comes from the SOC in I atoms as Lado et\nal.24did.\nIn order to reveal the e\u000bect of SOC from another per-\nspective, we next calculated the exchange matrix and SIA\nparameter in the non-collinear scheme without SOC. As\nexpected, the exchange matrix for both monolayers is di-\nagonal, with equal diagonal elements. SIA parameter is\nzero, which is also expected. The results con\frm that\nthe origin of the magnetic anisotropy, consequently the\norigin of the magnetization of these monolayer materi-\nals, is the spin-orbit interaction. However the compari-\nson between exchange parameters with and without SOC\nfurther indicates the di\u000berence between CrI 3and CrBr 3.\nThe exchange parameter of CrI 3without SOC is found to\nbe\u00004:53 meV which is between hJixx=hJiyy=\u00004:41\nmeV andhJizz=\u00004:63 meV when SOC is included.\nFor CrBr 3, on the other hand, it is found to be \u00003:38\nmeV which is almost equal to the in-plane exchange\nhJixx=hJiyy=\u00003:37 meV. Therefore, one understands\nthat the SOC in CrI 3not only enhances the exchange in-\nteraction of the out-of-plane spin components but also\nreduces the interaction of the in-plane components. For\nCrBr 3, the SOC contributes to the interaction of out-of-\nplane spins only.C. E\u000bect of strain\nThe responses of monolayer CrI 3and CrBr 3to biaxial\nstrain are further revealing the in\ruence of SOC on the\nmagnetic exchange interaction. Principally, the struc-\ntural changes upon straining are similar for CrI 3and\nCrBr 3. Cr-Cr distance changes according to the degree\nof strain applied ( \u00065%), however, the Cr-X bond length\nvaries\u0018\u00061%. Most of the changes therefore accumulate\nto the Cr-X-Cr (X = I, Br) angle, varying \u0018\u00065%. Such\nchanges in the angle cause distortion in the octahedral\ncoordination, as illustrated in Fig. 4(a). Therefore, it is\nexpected that strain induces the interaction of two spd2\nbonds of one ligand due to bong-angle modi\fcation and\nleads to modi\fcation of the SOC contribution in both Cr\nand the ligand.\nWe next calculated the total energy di\u000berence and the\ntotal SOC energy di\u000berence between out-of plane and in-\nplane spin alignments under biaxial strain. As shown\nin Figs. 4 (b) and (c), ESOof monolayer CrI 3[orange\ncurve in Fig. 4(b)] linearly increases under increasing\ntensile strain, which is similar to its behavior in CrBr 3\n[green curve in Fig. 4(c)]. However, the behavior of\nthe two materials under compressive strain is completely\nFIG. 3: 2\u00022 supercell where Cr atoms are indexed consis-\ntently with Table II. The equivalent Cr-Cr bonds are shown\nwith the same color.6\nFIG. 4: (a) Schematic illustration of the distortion caused on the octahedral unit of the material under tensile and compressive\nstrain. Panels (b) and (c) show total SOC energy and MAE as a function of biaxial strain for CrI 3and CrBr 3, respectively. The\natomic contributions are also plotted. (d) and (e) show the charge density variation (compared to bare atoms) in the strained\ncrystal of CrI 3and CrBr 3, respectively. One notes the overlap of the bonding charges under compressive strain, zoomed out\nfor facilitated visualization, and absent for the tensile strain.\ndi\u000berent. CrI 3exhibits parabolic increase with increas-\ning compressive strain, while CrBr 3maintains linear be-\nhavior and decreases with increasing compressive strain!\nThe atomic contributions reveal the source of these be-\nhaviors. As shown by magenta curve in Fig. 4(b), io-\ndine contribution dominates the behavior for both com-\npressive and tensile strain. For CrBr 3in Fig. 4(c), in\ncase of the compressive strain, the contributions from Cr\nand Br are almost equal but have opposite sign, meaning\nthat Br atoms favor to have spin in out-of-plane direction\nwhile Cr atoms prefer in-plane spin direction. For tensile\nstrain, Br contribution dominates and both Cr and Br\nprefer out-of-plane spin alignment. Since the structural\nchanges of both monolayers under strain are similar, one\nexpects similar variation of Cr energies, however, Cr of\nCrI3exhibits three times larger energy variation as com-\npared to Cr of CrBr 3. This indicates that the anisotropy\nrelated with Cr also originates from the bonding electrons\nrather than Cr-only electrons. In Figs. 4 (d) and (e) we\nshow the charge density variation for 5% and \u00005% strain\nin both CrI 3and CrBr 3. It is clearly seen that in case\nof compressive\u00005% strain the bonding charges of two\ndi\u000berent bonds of one ligand overlap. This reveals the\norigin of the interaction under compressive strain. For\n5% tensile strain, the bonding charges are clearly sepa-\nrated and exhibit no overlap.\nOur results for behavior of MAE are generally in good\nagreement with results reported by Ref.17. In case of\nCrBr 3we also obtain that MAE is smallest in case of\u00005% strain and it is linearly growing, reaching the max-\nimum for +5% strain. In case of CrI 3, however, our re-\nsults agree with Ref.17 only for the compressive strain -\nwith increased compressive strain, anisotropy is growing.\nOn the other hand, with tensile strain, we report com-\npletely di\u000berent behavior of MAE in CrI 3, compared to\nRef.17. There, anisotropy was decreasing with increased\ntensile strain, while in our study, the behavior is di\u000berent\n- anisotropy grows with increased tensile strain.\nThe variations of exchange parameters and SIA un-\nder biaxial strain are also examined based on the con-\nsiderations related to Table II and Fig. 3. The mean\nvalueshJiof the exchange matrices of three NNs are\nplotted as a function of strain in Fig. 5(a). The fer-\nromagnetic exchange interaction increases with the ten-\nsile strain and decreases with compressive strain for both\nCrI3and CrBr 3, for all components. The variation on the\ncurves of CrI 3is much larger such that the compressive\nstrain almost equalizes CrI 3and CrBr 3in terms of ex-\nchange interaction, and the di\u000berence between two mono-\nlayers increases with tensile strain. These results agree\nwith Ref.17 only in case of compressive strain. Namely,\nboth our study and mentioned previous work suggest that\nwith compressive strain, two materials become less FM.\nHowever in case of tensile strain, our results suggest that\nmaterials are more FM than in pristine case, while in\nRef.17, the opposite was suggested. In Fig. 5(b), the\nanisotropy between in-plane and out-of-plane exchange\nparameters, \u0001, is plotted as a function of strain. \u0001 of7\nFIG. 5: The magnetic exchange parameters (a), out-of-plane\nanisotropy (b), and SIA parameters (c) of monolayer CrI 3and\nCrBr 3as a function of the biaxial strain.\nCrI3increases for increasing either tensile or compres-\nsive strain. It is important to note that the behavior\nof \u0001 is consistent with the behavior of MAE in Fig. 4,\nsince most of the contribution to MAE comes from the\nanisotropy of the exchange interaction of the \frst NN.\nFor CrBr 3, the anisotropy slightly increases (decreases)\nunder tensile (compressive) strain, which is also consis-\ntent with the behavior of MAE. In Fig. 5(c), SIA is\npresented as a function of strain. SIA in CrI 3exhibits\na gradually slowing decrease (in absolute value) when\nmoving from compressive to tensile strain. In CrBr 3on\nthe other hand, SIA exhibits opposite behavior to that\nof CrI 3. It is signi\fcant that for compressive strain be-\nyond\u00003% SIA parameter becomes positive, indicating\npreference for in-plane direction of magnetization. No-\ntice that SIA is an order of magnitude smaller than the\nexchange parameter, therefore its contribution to overall\nmagnetic properties is limited. However, MAE of CrBr 3\nunder high compressive strain is of the same order as SIA,\nindicating that the drop of MAE is due to the decrease\nof SIA parameter, while \u0001 stays almost constant.D. Temperature-dependent magnetization and\nhysteresis in applied magnetic \feld\nThe above-indicated di\u000berences in the strength of the\nmagnetic exchange parameters, exchange anisotropy, and\nSIA parameter between two magnetic monolayers un-\nder investigation can be monitored via temperature-\ndependent magnetization and hysteretic behavior in ap-\nplied magnetic \feld, which are both readily experimen-\ntally accessible. Having obtained all the parameters to\nconstruct the Heisenberg spin Hamiltonian in Eq. (1) for\nstrained monolayers, we next calculate the temperature-\ndependent magnetization of CrI 3and CrBr 3and the cor-\nresponding TCwhere the magnetic phase transits from\nparamagnetic to ferromagnetic state. We used stochas-\ntic LLG simulation to obtain the temperature-dependent\nmagnetization M z/Ms(T), where M zis the out-of-plane\nmagnetization and M sis the saturation magnetization.\nAs shown in Figs. 6 (a) and (b), the obtained TCof\nthe unstrained monolayers of CrI 3and CrBr 3of 56 K\nand 38 K is reasonably close to the experimentally ob-\ntained values of 45 K and 21 K, respectively. These\nTCvalues are also consistent with those found in previ-\nous works,18,19obtained using renormalization spin-wave\ntheory combined with classical MC calculations47. Fur-\nther, as shown in Fig. 6(c), TCdecreases (increases) un-\nder compressive (tensile) strain, mainly due to the previ-\nously described strong variation of the exchange param-\neters with strain. In case of CrBr 3, contrary to CrI 3, the\nin\ruence of \u0001 and SIA is very small since the variation\nof those parameters under strain can be considered neg-\nligible. One should note that the behavior of TCas a\nfunction of strain presented here is completely di\u000berent\nthan that reported in Ref.17 where TCis calculated using\nmean-\feld theory. Such disagreement is hardly a surprise\nsince in Ref.17 a single exchange parameter is used to de-\ntermineTCin absence of information on anisotropy. Fur-\nther we note that two monolayer materials have almost\nequalTCat compressive\u00005% strain. In the case of ten-\nsile strain on the other hand, the di\u000berence between TCof\nCrI3and CrBr 3increases with strain. CrBr 3reaches the\nsaturation of TC\u001945 K at around 3% strain while CrI 3\nexhibitsTC\u001974 K at 5% strain and tends to further\nincreasedTCwith further straining.\nThe behavior of M z/Ms(T) curves is also illustrative\nof the di\u000berence between the two materials. In gen-\neral, all curves are well \ftted by the functional behav-\nior (1\u0000T=T C)\f, where\fis the critical exponent. In\nFig. 6(d), obtained exponents \fare plotted as a function\nof strain, together with \fvalues from di\u000berent available\nmodels for comparison. In our results, the critical expo-\nnent of monolayer CrI 3at all strains can be approximated\nby\f\u00190:24. This value suggests that strong out-of-plane\nanisotropy of CrI 3separates its behavior from those ex-\npected by 3D models and sets it closer to the 2D limit.\nOur value of \fis also comparable with the value of \u00180:26\nobtained in Ref.45 for bulk CrI 3since the intralayer in-\nteractions dominate the magnetic behavior even in bulk8\nFIG. 6: The temperature-dependent magnetization\nMz/Ms(T) for di\u000berent amounts of strain applied to mono-\nlayer CrI 3(a) and CrBr 3(b). Simulation data are \ftted by\n(1\u0000T=T C)\f, where\fis the critical exponent and TCthe Curie\ntemperature. (c) and (d) panels plot the thereby obtained TC\nand\fas a function of strain, respectively. For comparison,\n\fvalues from di\u000berent models7are shown as dashed lines in\n(d).\nCrI3.48CrBr 3with\f\u00190:3, on the other hand, is much\ncloser to the 3D Ising value ( \f= 0:3226) due to weak out-\nof-plane anisotropy. Our result is validated by (albeit at\nthe lower margin of) \f= 0:4\u00060:1 recently obtained from\nmagnetization measurements by Kim et al. .8\nFinally, motivated by its accessibility by advanced\nmagnetometry,8we calculated the hysteretic behavior of\nthe monolayers under magnetic \feld Btilted by angle \u0012B\nfrom thez-axis, for both CrI 3and CrBr 3.\nIn the simulations, the spin system is initialized at high\napplied \feld, where all the spins are aligned to the mag-\nnetic \feld direction. The \feld is then looped in steps\nof 0:01 T and 0 :05 T for CrBr 3and CrI 3respectively,\nwhere the magnetization is relaxed by 5 \u0002104time steps\nfor each value of \feld. Dipole-dipole interactions have\nbeen included in the simulations. In Fig. 7(a,b), we\nshow the obtained \u0001M z=jMz(!)\u0000Mz( )jas a color-\nmap plot where shades of red (CrI 3) and blue (CrBr 3)\ncolor show the deviation of Mz(B) hysteresis loop from\nthe rectangular shape. In Figs. 7 (c) and (d), we plot\nthe corresponding hysteresis curves for selected tilt an-\ngles of the applied magnetic \feld. One clearly sees that\nhysteresis loops evolve from sharp rectangular to an oval\nform as the tilt angle is increased, as was recently shown\nexperimentally by Kim et al. for the case of monolayer\nCrBr 3.\nIn absence of dipolar interactions, the behavior of mag-\nnetic spins in external \feld is captured by minimization ofthe energy ETOT(\u0012) =\u0000MAE cos(2\u0012)\u0000EBcos(\u0012\u0000\u0012B),\nwhereEBis the energy associated with the magnetic\n\feld, and grows linearly with increasing B. The value of\nthe critical, 'switching' \feld for the given angle \u0012B, scaled\nto the switching \feld for \u0012B= 0\u000e[\u0000 =Bcr(\u0012B)=Bcr(0\u000e)]\nis then analytically obtained from the condition that the\nenergy extrema coalesce, leading to equation\n(\u00002\u00001)3\n\u00004=\u000027\n4sin2(2\u0012B): (3)\nThis functional dependence of critical \feld on the tilt\nangle\u0012Bperfectly reproduces the numerically calculated\nswitching \feld, shown by dashed lines in Figs. 7(a,b).\nNotably, the switching \feld in absence of dipolar interac-\ntions (\u0000(\u0012B)) shows symmetric behavior with respect to\n\u0012B= 45\u000e. However, with dipolar interactions included,\nthat symmetry is broken in case of CrBr 3, and switch-\ning \feld for \u0012B= 0\u000ebecomes signi\fcantly lower than\nfor\u0012B!90\u000e. The latter was indeed validated experi-\nmentally, in Ref.8. However, dipolar interactions cause\nno changes in the critical \feld of CrI 3for any\u0012B, which\nis another important distinction between two materials\nthat could be veri\fed by Hall micromagnetometry.\nOne should however note that in our considerations\nwe do not involve \fnite size e\u000bects nor demagnetization,\nor temperature \ructuations, likely playing an important\nrole in experiment (next to the ever-present defects in\nthe monolayers, that can facilitate magnetic reversal lo-\ncally). Our simulations explore primarily the e\u000bect of the\nmicroscopic parameters on the apparent magnetic behav-\nior, with a goal of capturing the intrinsic di\u000berences be-\ntween CrI 3and CrBr 3. As a consequence, the switching\n\felds in our simulations are signi\fcantly larger than ex-\nperimentally reported values of \u00190.15T5for CrI 3and\u0019\n0.03T8for CrBr 3, for\u0012B= 0. Having said that, our sim-\nulations capture the large ratio between switching \felds\nof the two monolayer materials (approximately 8 and 5\nfor simulation and experiment, respectively).\nTo feature yet another experimentally veri\fable di\u000ber-\nence between monolayer CrI 3and CrBr 3, we also cal-\nculated the hysteresis curves for the strained magnetic\nmonolayers under out-of-plane magnetic \feld. We recall\nthat MAE and SIA, plotted in Fig. 4(b,c) and Fig. 5(c),\nrespectively, showed distinctively di\u000berent behavior in\ntwo materials when under strain. As shown in Figs. 7 (e)\nand (f), the width of the hysteresis loop strongly changes\nwith the strain. However, while the switching magnetic\n\feld of monolayer CrI 3increases under both compressive\nand tensile strain, the switching \feld of monolayer CrBr 3\nisreduced by compressive strain while the tensile strain\nincreases the switching \feld stronger than was the case\nin CrI 3(measured with respect to the unstrained case).\nThis behavior can thus be mapped on the behavior of\nexchange anisotropy and SIA under strain, all rooted in\nthe known di\u000berence in spin-orbit coupling between the\ntwo materials.9\nFIG. 7: The height of the hysteresis loop \u0001M z=jMz(!)\u0000Mz( )jas a function of the external magnetic \feld Btilted by\nangle\u0012Bfrom the out-of-plane direction, for (a) CrI 3and (b) CrBr 3. Panels (c) and (d) show the corresponding hysteresis\ncurves obtained for indicated angles \u0012B. Panels (e) and (f) highlight the di\u000berence in the hysteretic response of two materials\nwhen strained (here \u0012B= 0).\nIV. CONCLUSIONS\nWe compared two of the very \frst ferromagnetic 2D\nmaterials, monolayer CrI 3and CrBr 3, belonging to the\nsame chromium-trihalide CrX 3family. Although very\nsimilar qualitatively in structural and electronic proper-\nties (with some quantitative di\u000berences such as lattice\nconstant, bond length, and electronic band gap), these\nmaterials exhibit strong di\u000berences in magnetic proper-\nties and their behavior with external stimuli. We at-\ntribute these di\u000berences to the spin-orbit interaction not\nonly on ligand atoms (X=I, Br) and but also at the bond-\ning orbitals. We show that the energetic preference of\nthe direction of spin of a ligand directly depends on the\nbonding direction of that particular ligand relative to the\nspin direction. That means spin-orbit coupling (SOC) en-\nergy contribution of an individual ligand can be predicted\nqualitatively via its coordination and spin direction.\nWe also present the magnetic exchange parameters for\nthe two monolayers. The mean exchange interaction in\nCrI3is larger that one of CrBr 3, but the di\u000berence be-\ntween their out-of-plane anisotropy values (\u0001) as well as\nbetween single-ion anisotropies (SIA) are more than sig-\nni\fcant. We also clearly demonstrated that the origin ofboth the out-of-plane anisotropy and the SIA is the spin-\norbit interaction, since our analogous analysis without\nSOC yielded no exchange anisotropy and no SIA.\nBy applying biaxial strain, we revealed further di\u000ber-\nences between monolayer CrI 3and CrBr 3. The strain\nmostly changes the Cr-X-Cr angle instead of the bond\nlength. That results in signi\fcant structural distortion\nof the octahedral units of the monolayers. Magnetic\nanisotropy energy (MAE) of CrI 3increases under either\ncompressive and tensile strain while MAE of CrBr 3lin-\nearly increases (decreases) under increasing tensile (com-\npressive) strain. This di\u000berence in the variation of MAE\nis re\rected on the corresponding changes in the out-of-\nplane anisotropy of the exchange parameters and the SIA\nparameter. With such di\u000berences in obtained parameters\nfor the two materials, we calculated the temperature-\ndependent magnetization for pristine and the strained\nmonolayers, to reveal much stronger variation with strain\nofTCin CrI 3than in CrBr 3. The found critical expo-\nnent of our M(T) data places CrBr 3virtually in the 3D\nregime, owing to its low out-of-plane anisotropy, contrary\nto the strong 2D character of CrI 3.\nFor facilitated direct observation of the reported dif-\nferences between monolayer chromium-trihalides, and as10\na direct probe of their magnetic anisotropy, fostered by\nspin-orbit coupling, we also calculated the behavior of\nhysteretic magnetization loops as a function of the tilt\nangle between the applied \feld and the monolayer plane.\nWe revealed that magnetic behavior of CrBr 3is far more\na\u000bected by dipolar interactions than is the case in CrI 3,\nbut also that the behavior of the switching \feld with\nstrain is entirely di\u000berent in two materials, analogously\nto previously observed di\u000berences in MAE and SIA as\na function of strain. These \fndings are yet another\nproof that even subtle di\u000berences in atomic contributions\nto spin-orbit coupling between two akin materials can\nlead to rather dissimilar magnetic properties, and can\nbe broadly tuned by gating, straining and heterostruc-\nturing of the 2D material. Although sourced in proper-\nties at atomistic scale, these di\u000berences can clearly man-\nifest in macroscopic observables and are veri\fable exper-imentally (owing to e.g. recent advances in Hall mag-\nnetometry). Therefore, tailored solutions for spatially\nengineered spin-orbit coupling in magnetic monolayers\npresent an attractive roadmap towards advanced spin-\ntronic and magnonic nanocircuitry.\nAcknowledgments\nThis work was supported by the Research Foundation-\nFlanders (FWO-Vlaanderen) and the Special Research\nFunds of the University of Antwerp (TOPBOF). 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Wu, andC.\nGao, Science 366, 983 (2019).\n44R. Albaridy, A. Manchonand, and U. Schwingenschl ogl, J.Phys.: Condens. Matter 32355702 (2020).\n45Y. Liu and C. Petrovic, Phys. Rev. B 97, 014420 (2018).\n46S. Bramwell and P. Holdsworth, Journal of Physics: Con-\ndensed Matter 5, L 53 (1993).\n47Note that the feeding parameters in those works were\nobtained by DFT calculations with slightly di\u000berent\nparametrization than used in the present work, such as\nU= 3:5 eV instead of U= 4 eV.\n48It is interesting to mention that the critical exponent of\nthe 2D XY model is estimated as \f\u00190:2346from the ex-\nperimental observations on several layered magnets such as\nBaNi 2(PO 4)2, Rb 2CrCI 4and K 2CuF 4, which is very close\nto\fof CrI 3. We believe this to be a coincidence, since\n2D XY model is based on the easy-plane anisotropy con-\ntrary to the out-of-plane easy-axis of CrI 3. It is also known\nthat 2D XY model does not exhibit standard spontaneous\nphase transition, which should deviate from the behavior\nwe obtained in our simulations at around TC."    },    {        "title": "2103.06187v1.Search_and_Structural_Featurization_of_Magnetically_Frustrated_Kagomé_Lattices.pdf",        "content": "Search and Structural Featurization of\nMagnetically Frustrated Kagomé Lattices\nVanessa Meschke,†Prashun Gorai,‡Vladan Stevanović,‡and Eric S. Toberer\u0003,†\n†Department of Physics, Golden, CO 80401\n‡Department of Metallurgical and Materials Engineering, Golden, CO 80401\nE-mail: etoberer@mines.edu\nPhone: 303-273-3171\nAbstract\nWe have searched nearly 40,000 inorganic solids in the Inorganic Crystal Structural\nDatabase to identify compounds containing a transition metal or rare earth kagomé\nsublattice, a geometrically magnetically frustrated lattice, ultimately identifying \u0018500\nmaterials. A broad analysis of the chemical and structural trends of these materials\nshows three types of kagom�� sheet stacking and several classes of magnetic complex-\nity. Following the search and classification, we rapidly screen the magnetic properties\nof a subset of the materials using density functional theory to eliminate those that\nare unlikely to exhibit magnetic frustration. From the results of our computational\nscreening, we rediscover six materials that have previously been explored for their low\ntemperature magnetic behavior, albeit showing symmetry breaking distortions, spin\nglass behavior, or magnetic ordering. However, all are materials with antiferromag-\nnetic behavior, which we correctly predict. Finally, we also report three materials that\nappear to be unexplored for their magnetic properties.\n1arXiv:2103.06187v1  [cond-mat.mtrl-sci]  10 Mar 20211 Introduction\nGeometric magnetic frustration is often explored for the unique quantum phenomena it\nmay create, ranging from enabling superconducting states1,2to generating a quantum\nspin liquid (QSL)3. While superconductivity needs no introduction, we’ll briefly introduce\nQSLs, though more thorough reviews of QSLs are available for the reader3,4. QSLs are a\nnovel phase of matter that, despite the presence of strong magnetic interactions, do not\nexhibit magnetic ordering down to 0K5. QSLs have yet to be proven to exist in real ma-\nterials, but recent work has extensively focused on the characterization of materials with\nmagnetically frustrated lattices3,6and applying advancements in computational methods\nto better describe and predict the spin interactions of QSLs7. After several decades of effort\non both the experimental and theoretical fronts, few of the explored QSLs remain candi-\ndates. The lack of remaining QSL candidates arises from the relative ease of disproving\na material to be a QSL as compared to the challenge of proving its existence, yielding a\nsmall and ever-narrowing search space for this exotic phase of matter. As such, the lack\nof candidate compounds limits both the ability to discover materials with exotic magnetic\nproperties and learn from their aggregate behavior. The opportunity is thus for solid state\nchemistry to discover new materials that expand the structural landscape of possible QSLs.\nHerein, we ( i) comprehensively identify materials with magnetic kagomé sublattices, ( ii)\nreview and explore their associated properties, and ( iii) and identify candidates for further\nexploration.\nWhile we choose to only explore the kagomé lattice in this work, there are numer-\nous crystal lattices that exhibit geometric magnetic frustration. For example, the triangu-\nlar lattice in 2D and the pyrochlore and hyperkagomé lattices in 3D all make arranging\nspins antiferromagnetically impossible. Additionally, other exchange-frustrated lattices,\nnamely the square8–10and honeycomb lattices11–15also exist. However, the square and\nhoneycomb lattices’ more restrictive requirements for J i/Jjratios13,16,17and dependence\n2of magnetic interactions up to third nearest neighbors18,19complicate a high-throughput\nsearch for frustration. As such, the kagomé lattice makes an excellent initial choice for a\nlarge scale search for new candidate QSLs due to its relative simplicity in spin coupling\nterms and its previous exploration in QSLs3,20.\nIn addition to its relative simplicity in coupling requirements, the kagomé lattice was\nselected for screening due to its relation to one of the strongest candidate QSLs to date:\nherbertsmithite (ZnCu 3(OH) 6Cl2)6,21. The crystal structure of herbertsmithite, with its\nCu2+kagomé sublattice highlighted in blue, is shown in Figure 1. Herbertsmithite has yet\nto be disproven for QSL behavior22with experimental measurements showing no magnetic\nordering down to temperatures as low as 50mK.23,24\nFigure 1 The crystal structure of herbertsmithite with the Cu2+kagomé sublattice highlighted in blue.\nOn the left is one kagomé plane as seen when viewing the structure down the axis perpendicular to the\nplanes. On the right is the staggering of the kagomé planes for a single unit cell of herbertsmithite.\nMotivated by the ease of eliminating candidate QSLs and the challenge of generating\nnew materials with frustrated lattices, we seek to define the structure space of existing\nmaterials with kagomé sublattices to accelerate the search for new QSLs. While Karigerasi,\nWagner, and Shoemaker25have compiled a complete list of known 2D frustrated quantum\nmagnets that can be searched, filtered, and output using a web-based frontend accessible\nfrom the Illinois Data Bank, the paper does not supply a categorization or overview of the\nmagnetic or structural properties of the kagomés. Additionally, other reviews of QSLs3,5\ndo an excellent job of outlining the challenges of this search and summarizing previously\nexplored candidates, but they fail to provide a comprehensive assessment of all known\n3kagomés to date.\nWe begin with a structural search for compounds that are symmetry-ideal and nearly\nideal kagomé sublattices from the Inorganic Crystal Structure Database (ICSD). Armed\nwith this information, we explore the prototypes of materials with kagomé sublattices and\npropose a nomenclature for the describing these sublattices, focusing on structural trends\nfor the magnetic species. After performing this classification, we generate a complete list\nof known kagomé prototypes regardless of their performance to date. Within a subset\nof this list, we rule out materials as candidate QSLs both by examining the literature for\npreviously investigated kagomés and analysis of spin-polarized density functional theory\ncalculations. The final list of candidate materials can inspire more rigorous calculations\nand synthetic efforts as well as provide parent compounds for chemical mutations.\n2 Methods\nThe initial dataset was collected by examining stoichiometric, ordered crystal structures\ndocumented in the ICSD for the geometric features of the kagomé lattice. In particular,\nthe search screened for materials with 4-fold coordinated transition metal or rare earth\natoms at unique Wyckoff positions. At each of these unique Wyckoff positions, materials\nwere required to have equal bond distances between all neighbors and 2 each of 60, 120,\nand 180 degree bond angles (see Figure 2). Additionally, all triangles lining the hexagon of\nthe kagomé lattice needed to be equilateral. A tolerance of .3 Å was allowed on the bond\ndistances and 5ofor the bond angles. Finally, an additional search agnostic of Wyckoff\nposition was also performed with the same geometric requirements as listed above. These\ngeometric search constraints also included the hyperkagomé lattice in the search results.\nThe initial search yielded 497 unique compounds with magnetic atoms forming a\nkagomé sublattice at at least one Wyckoff position. Details of structural and chemical\ntrends for these materials is detailed in the results section. A subset of 87 of the 497\ncompounds that contained no transition metal or rare earth atoms beyond the kagomé\n4Figure 2 Geometric features of the kagomé lattice: four fold coordination highlighted in turquoise, equal\nside lengths of all triangles, and the two sets of 60oand 120obond angles.\nsublattice were selected for magnetic property screening using density functional theory\n(DFT). While DFT will not produce highly accurate magnetic properties for correlated ma-\nterials, it is an efficient way to screen large datasets for non-magnetic compounds and poor\ncandidate QSLs.\nThe magnetic moments of the calculation subset were calculated by performing spin-\npolarized calculations with GGA functionals and an automatic k-mesh using the Vienna Ab\nInitio Simulation Package (VASP). The high-throughput DFT calculations were performed\nusing PyLada26, a Python framework for the organizing and managing high-throughput\nfirst-principles calculations. Using Pylada, all spin configurations for supercells of the com-\npounds were enumerated and their properties calculated.\nTo eliminate compounds with little potential as QSLs, the magnitude of the magnetic\nmoments on the kagomé atoms of each of the compounds was first assessed. If a com-\npound’s kagomé sublattice showed no magnetic moment in any of its spin configurations,\nthat compound was eliminated from further screening due to its predicted lack of magnetic\nproperties. Following examination of the magnitude of the magnetic moments, the orien-\ntation of the moments in the lowest energy spin configuration for each compound was\nassessed. If a compound had only ferromagnetic (FM) interactions in its lowest energy\nspin configuration, it was eliminated from further assessment as it is unlikely a material\nthat is predicted to be FM in DFT would be a candidate QSL. Finally, compounds with ma-\njority antiferromagnetic (AFM) interactions in their lowest energy spin configuration were\n5most thoroughly examined for the energy differences between the lowest energy spin con-\nfiguration and the FM configuration. While DFT is not the most accurate tool to compute\nthe magnetic properties of a material, it is one of the only tools that provides a feasible\nmeans to rapidly screen even a fraction of the compounds in this dataset. More typical\ntools for predicting magnetic behavior, such as dynamic mean field theory, are simply too\ncomputationally expensive for a dataset of this size and for materials that may have more\nthan 70 atoms in the unit cell.\nThe magnitude of the energy differences between spin configurations for structures\nwith majority AFM interactions in their lowest energy spin configurations was investigated\nto approximate the strength of the coupling of the spins in the structure. While compu-\ntational methods to determine the coupling coefficients for magnetic materials from DFT\nexist27,28, they typically require using a spin Hamiltonian and accurately tuning the U pa-\nrameter in the Hubbard model28to produce accurate results. Additionally, the four-state\nmethod7,29,30also exists to compute the exchange coefficients, though it relies on knowing\nfour precise magnetic configurations of the system to accurately map the energies of these\nconfigurations to the Hamiltonian for the system. As such, we instead examine the energy\ndifferences between the lowest energy AFM configuration and the FM configuration as a\nproxy to this coupling term to screen for compounds with interacting spins.\n3 Results and discussion\nTo form the original dataset, \u001840,000 fully ordered, stoichiometric structures from the\nICSD were screened for structural features of the kagomé lattice as detailed in the Methods.\nFrom this search, 497 compounds were found with a kagomé sublattice at a minimum of\none Wyckoff position in the structure. A full list of these materials is detailed in a table\nin the Supplemental Information and the list is included as a CSV. Since the goal of this\nresearch is to build a comprehensive list of all known materials with a kagomé sublattice,\nthe search included elements that are typically non-magnetic, such as zinc and cadmium.\n6The chemical diversity of the elements forming the kagomé sublattice is highlighted as\na heat map of the periodic table in Figure 3. In terms of the counts of each element in\nthe heatmap, the relatively weak showing of the more costly transition metals such as\niridium and platinum is expected, especially considering their lack of representation in\nthe ICSD. However, the comparatively high number of rhodium containing compounds is\nsurprising, but appears to be due to three separate works31–33that explore a total of 24\nvariants of rhodium kagomé sublattices. The large counts of iron, cobalt, and nickel are\nmore expected given their prevalence in the ICSD.\nFigure 3 A heat map of the elements forming the kagomé sublattice in the 497 materials from the\nsearch. All transition metals and rare earths were considered in the search, so typically non-magnetic\nelements such as zinc and cadmium are shown in this heat map.\nVenturing beyond the elements forming the kagomé lattice, the compounds from our\nsearch can be categorized based on the elements surrounding the kagomé sublattice. Ini-\ntially, the most apparent differentiation of these compounds are those that have additional\ntransition metal or rare earth elements outside of the kagomé sublattice and those that\ndo not, which we will designate as complex and simple kagomé compounds respectively.\nComplex kagomé compounds can be further divided into two subcategories. First, if the\ntransition metal or rare earth atoms of the same species as the kagomé forming atom also\nappear outside of the kagomé sublattice, we will refer to these compounds as intrinsically\ncomplex kagomés. Intrinsically complex kagomé compounds are unlikely to be QSLs due\nto the high likelihood of magnetic interactions between kagomé and non-kagomé atoms of\nthe same species, breaking the kagomé sublattice’s frustration. Alternatively, the transition\n7metal or rare earth atoms outside the kagomé sublattice may be a different species than the\nkagomé forming atom. We refer to these materials as extrinsically complex kagomés, and\nmaterials such as herbertsmithite, where zinc sits outside the Cu2+lattice, and YFe 6Ge6,\nwhere yttrium atoms are outside the iron kagomé lattice, fall into this category. Extrin-\nsically complex kagomé compounds may very well be candidate QSLs, especially if the\natoms outside the kagomé sublattice carry no magnetic moment. A total 413 of the 497\ncompounds in the dataset are complex kagomés, with 123 of those being intrinsically com-\nplex. The remainder are simple kagomés.\nNext examining the structural complexity of the 497 compounds in the data set, a\nplethora of crystal structure prototypes emerged. In total, 130 unique structure proto-\ntypes appeared, with 81 of those prototypes unique to a single compound in the dataset.\nWithin the prototypes, MgFe 6Ge6was the most frequently appearing prototype, account-\ning for 13% of the compounds’ prototypes in this dataset. Beyond MgFe 6Ge6, the CeCo 4B,\nCo3GdB 2, Th 2Zn17(filled), Ni 3Pb2S2, and ErIr 3B2prototypes also make strong appearances,\nin total comprising an additional 28% of the appearing prototypes.\nWithin these prototypes, three alignments of the kagomé sheets were observed when\nviewing the kagomés down the axis perpendicular to the planes. An example of each of\nthe prototypes demonstrating the different alignments is shown in Figure 4, and simplified\nexamples of each are shown in the supplemental. First, the atoms of the kagomé sheets\nmay align when looking down the direction perpendicular to the planes, which we refer\nto as aligned planes. Examples of prototypes with aligned planes would be the MgFe 6Ge6\nand Co 3GdB 2prototypes. Aligned kagomé planes make up 310 of the 497 compounds in\nthe dataset. If the kagomé planes are not aligned, the planes stack with lateral translation\n(shear) between the layers. The most common shear is a (1\n3;1\n3) translation of the kagomé\nlattice vectors (e.g. the Ni 3Pb2S2prototype). This shear keeps the overall six-fold rota-\ntional symmetry of the individual kagomé plane when extended into a stack. We refer\nto this stack as a symmetric shear, and it accounts for the majority of the sheared stacks\n8observed in this data set and appears in 171 of the compounds. The final type of shear\nbetween planes is a (1\n2,0) shear and is displayed by only the Cs 2Pt3S4and K 2Pd3S4pro-\ntotypes. This shear does not preserve the six-fold axial symmetry and is referred to as\nasymmetric shear. Asymmetric shear is a fairly rare occurrence in this dataset, appearing\nin only 16 compounds. An example of each of the shears is shown in Figure 4 for a rep-\nresentative prototype of each category with its kagomé sublattice highlighted in blue, and\na simplified graphic visualizing the three stacking categories is given in the supplemental.\nIn addition to these alignment trends in kagomé plane stacking, all kagomé planes were\nfound to either be equally spaced in a compound or to group in sets of two close planes\nwith larger distances between the grouped planes. The plane alignment and spacing is\nincluded as data in a CSV.\nFigure 4 Representative prototype for each type of kagomé plane alignment. From left to right, the\nkagomé planes are aligned in MgFe 6Ge6, symmetrically sheared for Ni 3Pb2S2, and asymmetrically sheared\nfor Cs 2Pt3S4. In each prototype, the kagomé sublattice is highlighted in blue.\nFurther examining the structural trends in this our dataset, we next examine the spe-\ncific Wyckoff positions that form the kagomé sublattice. Examples of each of the following\nWyckoff position dependence for kagomé sublattice formation are given in the supplemen-\ntal. In this data set, we found that kagomés are either formed by a single Wyckoff position\nor multiple Wyckoff positions. 392 compounds were formed from only one Wyckoff po-\nsition for all planes. For the remaining compounds whose kagomés planes are composed\nof multiple Wyckoff positions, two the different Wyckoff positions either resided in differ-\nent planes or the same plane. For example, if multiple Wyckoff positions correspond to\n9separate planes of the kagomé sublattice, one Wyckoff position may form a kagomé layer\nat z = 0, with another Wyckoff position forms the kagomé layer at z = 0.5 for kagomé\nplanes that stack in the z direction. This is the case for the Ce 3Co11B4prototype. Alterna-\ntively, multiple Wyckoff positions may exist in a single kagomé plane, and all planes are\nconsist of multiple Wyckoff positions. For example, materials in the ErIr 3B2prototype ex-\nhibit this type of combined-Wyckoff kagomé plane. For compounds whose kagomé planes\nare formed from multiple Wyckoff positions, 56 had different Wyckoff positions at differ-\nent layers, whereas 49 compounds had kagomé where each plane consisted of multiple\nWyckoff positions.\nMoving beyond the structural and chemical trends of the full data set, the magnetic\nproperties of a subset of compounds were also investigated to begin building insights to\nthe connections between structure, chemistry, and magnetic behavior. A subset of 87 com-\npounds containing no transition metals or rare earths beyond the kagomé-forming species\nwere selected for an initial screening with DFT and a literature review to validate predic-\ntions of magnetic behavior. The compounds in this set spanned a diverse set of prototypes\nand transition metal or rare earth atoms forming the kagomé sublattice, each of which\nare featured Figure 5, and a full list of these compounds is included as a table in the sup-\nplemental and as a CSV. For Figure 5, a bin for each prototype is listed along the y axis\nof the heat map, and the options for transition metal or rare earth atoms that form the\nkagomé sublattice is on the x axis. Each box of the heat map details how many times a\ngiven prototype appeared in the calculation set with the corresponding transition metal or\nrare earth atom forming its kagomé sublattice. For example, the CoSn prototype appeared\nthree times in the calculation set, which is noted on the histogram to the right of the heat\nmap. For the three times the CoSn prototype appeared, it had 3 different transition met-\nals forming its kagomé sublattice: iron, cobalt, and rhodium. The histogram to the right\ndetails how many times each prototype appeared in the calculation set. Considering the\nstructural trends, most protoypes are unique to a specific compound in this dataset, and\n10most kagomé sublattices are formed from Fe, Ni, Cu, and Au. For one of the most well rep-\nresented prototype, Ni 3Pb2S2, a historical bias in the data is present due to the thorough\nexploration of the shandites’ half-metallic ferromagnetic behavior34–37.\nFigure 5 A heat map of which prototype and transition metal or rare earth atom combinations appeared\nfor the compounds investigated with DFT. Each box of the heat map details how many times a given\nprototype appeared in the calculation set with the corresponding transition metal or rare earth atom\nforming its kagomé sublattice.\nGoing beyond the trends in kagomé atom type, Figure 6 details the comparisons of\nthe distances between nearest neighbor kagomé atoms, which will be referred to as the\nkagomé bond distance, and the distances between the planes of the kagomés for each\nprototype. A separate version of this plot for all compounds in the dataset is included in the\nsupplemental. The range of both kagomé bond distance and planar spacing of the ‘Unique\nPrototype’ bin particularly highlights the diversity in structural trends for compounds with\na kagomé sublattice and shows promise for the discovery of new kagomé sublattices given\ntheir ability to exist around many additional types of atoms between their layers. For the\nentire dataset, the mean kagomé-kagomé bond length is 3.05Å( s: 0.52 Å) and the average\ndistance between kagomé planes is 5.69 Å( s: 1.48 Å). However, the ‘Unique Prototype’\nbin displays a much large deviation in both bond lengths and planar spacings. If the\nunique prototype bin is excluded, more consistency in the kagomé bond distance is found\n(avg: 2.92Å , s: 0.32Å), and the distance separating the kagomé planes varies between\nprototypes and within a given prototype (avg: 5.47Å, s: 1.19Å).\n11Figure 6 Comparison of the kagomé bond distance (red) and planar spacing (black) for the calculation\nset. The values of the bond distances and planar spacings are binned in the histogram to the right of\nthe scatter plot.\nHaving established structural and chemical trends in the compounds found, we move\nforward with computational screening and a literature review of the 87 compounds in\nthe calculation set. To investigate the quality of these materials as candidate QSLs, the\nmagnitude and orientation of the magnetic moments and the energy of the various spin\nconfigurations were collected from the DFT calculations. For the 87 compounds studied,\n62 were found to be non-magnetic with DFT. These 62 compounds were mostly composed\nof the shandites in the Ni 3Pb2S2prototype group as well as many of the uniquely appearing\nprototypes. The remaining 25 compounds had magnetic moments greater than 0.5 mBper\natom in the structure and are classified more thoroughly below.\nTo assess the overall validity of these magnetic moment calculations, calculation re-\nsults were compared with experimental findings in the literature. Overall, our calculations\nfind good agreement with experiment. For example, the shandite Co 3Sn2S2has been ex-\nperimentally found to be ferromagnetic below 175K,38and its Ni analog, Ni 3Sn2S2, is\nnon-magnetic,36both of which are confirmed by our calculations. Additionally, KV 3Ge2O9\nis reported AFM in literature,39a result that is also confirmed by our calculations, though\nwith a small energy differences between spin configurations in our calculations.\n12For the 25 materials with sizable magnetic moments, 10 had a ferromagnetic (FM) in\nplane and between plane spin configuration as their lowest energy. Of the remaining 15\ncompounds, 3 compounds had FM interactions in-plane with antiferromagnetic (AFM) in-\nteractions between planes. The final 12 compounds had AFM interactions as their lowest\nenergy configurations, and of these 12 compounds, five had small (< 5meV/kagomé form-\ning atom) differences in energy between the lowest energy configuration with mostly AFM\ninteractions and the FM configuration. The remaining 7 compounds with large energy\ndifferences between spin configurations and ground state that shows mostly AFM inter-\nactions are explored more thoroughly below. Additionally, we highlight two compounds\n(Na 2Mn3Cl8and Cu 3Pb(AsO 4)2(OH) 2) whose energy differences are smaller than this cut\noff but appear to have no literature regarding their magnetic properties.\n3.1 Jarosites: AFe3(OH) 6(SO 4)2(A= K, Na, H 3O)\nOur dataset contained four compounds with the KAl 3(SO 4)2(OH) 6prototype: KFe 3(OH) 6(SO 4)2,\nNaFe 3(OH) 6(SO 4)2, (H 3O)Fe 3(OH) 6(SO 4)2, and KCr 3(SO 4)2(OH) 6. KCr 3(OH) 6(SO 4)2is\na mineral known as alunite, which we predicted to have mostly AFM interactions in\nits lowest energy configuration. While KCr 3(OH) 6(SO 4)2showed relatively small energy\ndifferences between its lowest energy and its FM configuration (4 meV/Cr atom), the\nremaining three members of this prototype showed more promise as candidate QSLs.\nKFe 3(OH) 6(SO 4)2, NaFe 3(OH) 6(SO 4)2, and (H 3O)Fe 3(OH) 6(SO 4)2) are compounds belong-\ning to the mineral group of jarosites, and we will focus the rest of this section on this class\nof materials.\nJarosites are a naturally occurring hydrous sulfate mineral with a composition follow-\ning the formula AFe 3(OH) 6(SO 4)2, (A = Na, K, Rb, H 3O, Pb, or other metals or molecules),\nwhere the Fe atoms form the kagomé sublattice in the crystal40. Previous investigations\ninto both naturally occurring and synthetic jarosites has shown they typically exist with\nlarge concentrations of vacancies on the kagomé sublattice40,41. Despite this large frac-\n13tion of vacancies, jarosites are considered to be nearly ideal Heisenberg antiferromagnets,\nespecially when the vacancy concentration is minimized42.\nThe original jarosite for which this group of minerals derives its name has the for-\nmula KFe 3(OH) 6(SO 4)2, where the Fe atoms form the kagomé sublattice. From DFT, we\nfind jaroiste’s lowest energy Ising spin configuration, which mostly consists of AFM in-\nteractions, and the FM configuration to have an energy difference of 90 meV/Fe atom\nwith a magnetic moment of 4.2 mBper Fe atom. Next we examine a variation on jarosite\nwhere the K atoms surrounding the kagomé lattice are swapped for Na, creating natro-\njarosite (NaFe 3(OH) 6(SO 4)2). We predict natrojarosite to have a magnetic moment of 4.2\nmBper Fe atom. Additionally, the lowest energy configuration with mostly AFM interac-\ntions and FM spin configurations varied by 132 meV/Fe atom for natrojarosite. Finally,\nhydronium jarosite ((H 3O)Fe 3(OH) 6(SO 4)2) showed similar results to its previous min-\neral family members, showing a magnetic moment of 4.2 mBper Fe atom and an energy\ndifference of 105 meV per Fe atom from the mostly-AFM to the FM spin configurations.\nMost jarosites have been experimentally shown to undergo long range magnetic order-\ning between 50-65K. In particular, KFe 3(OH) 6(SO 4)2orders at 65 K43and NaFe 3(OH) 6(SO 4)2\nat 50 K44. Hydronium jarosite, however, displays a break from the ordering trend and has\nbeen shown to be spin glass with a glass transition temperature of 15K45. These experi-\nmental results eliminate all of the jarosites for QSL behavior. However, future investiga-\ntions into the jarosites could model Wills’44,46approach to taking advantage of the large\nvacancy concentration on the Fe sites and investigate the magnetic properties of doped\njarosites to further illuminate their magnetic behavior .\n3.2 Cs 2KMn 3F12\nFrom DFT, Cs 2KMn 3F12shows promise as a QSL candidate with a magnetic moment of\n3.7mB/Mn atom and a difference of 200 meV/Mn atom between the mostly-AFM and FM\nspin configurations. This compound and those similar in their chemistry and structure\n14have been previously investigated for its magnetic properties. While compounds such\nas Cs 2LiMn 3F12and Cs 2NaMn 3F12order at 2.1 K and 2.5 K, respectively, the ordering\ntemperature of Cs 2KMn 3F12appears to order around 7 K47,48. However, it is unknown\nif the ordering in this sample is long-range or short range and the exact nature of the\nordering should be investigated experimentally.\n3.3 Cs 2SnCu 3F12\nSimilar in composition to Cs 2KMn 3F12but quite different in structure, our predictions ini-\ntially show Cs 2SnCu 3F12to be a candidate for QSL behavior. With a magnetic moment of\n0.77 mB/Cu atom and an energy difference of 12 meV per Cu atom between the mostly\nAFM and the FM states, Cs 2SnCu 3F12appears promising from our predictions. However,\nliterature shows that Cs 2SnCu 3F12experiences long range magnetic ordering at 17K and\nstructural distortions are reported around 185 K49,50, experimentally eliminating it as a\ncandidate QSL.\n3.4 Na 2Ti3Cl8and Na 2Mn3Cl8\nIn calculation, Na 2Ti3Cl8showed promise for a QSL with a magnetic moment of 1.7 mBper\nTi atom in the structure and an energy difference of 390 meV/Ti atom in the structure from\nthe lowest energy, mostly-AFM configuration to the FM configuration. Despite its compu-\ntational promise, this material has been previously investigated for its magnetic properties\nand has been shown to undergo a Peierl’s-type distortion at 200K, forming Ti trimers,51\nexperimentally eliminating it as a candidate QSL. Calculations were also performed on the\nMn analog of this compound, Na 2Mn3Cl8, which was found to have a magnetic moment\nof 4.5 mBper Mn atom. However, Na 2Mn3Cl8shows very small energy differences (1.6\nmeV/Mn atom) between the mostly-AFM and FM spin configurations, with the AFM be-\ning lower. Magnetic measurements of Na 2Mn3Cl8have been made previously, finding no\ndistortions or magnetic ordering down to 100 K52. However, this compound should be\nfurther investigated experimentally to determine if the distortions observed in Na 2Ti3Cl8\n15persist with swaps of the kagomé-forming atom and to determine an ordering temperature\nif one exists.\n3.5 Corkite, PbFe 3(SO 4)(PO 4)(OH) 6\nSimilar in structure to the jarosites, we also find the compound corkite (PbFe 3(SO 4)(PO 4)(OH) 6)\nto have a spin configuration with mostly AFM interactions as its lowest energy state in cal-\nculation. With a magnetic moment of 4.2 mBper Fe atom and an energy difference of\n80 meV per Fe atom between the lowest energy, mostly-AFM and FM spin configurations,\ncorkite shows a remarkable similarity to the jarosites in DFT.\nCorkite is another naturally occurring mineral that can be found in acidic mine runoff53.\nWhile corkite’s structure has been extensively studied for determining the location of its\nphosphate and sulfate groups and to cope with the challenges of creating a high quality\ncrystalline sample53–55, little about its properties are reported in the literature other than\nits light brown color and its potential application in identifying economically valuable ores\nfor mining56. Magnetic measurements of corkite should be made to determine if this min-\neral orders as the jarosites do.\n3.6 Bayldonite, Cu 3Pb(AsO 4)2(OH) 2\nThough it has a smaller energy difference between its spin configurations than previously\nhighlighted compounds in this section, we choose to highlight Cu 3Pb(AsO 4)2(OH) 2be-\ncause its lack of magnetic data in the literature. Cu 3Pb(AsO 4)2(OH) 2, also known by the\nmineral name bayldonite, was shown to have a configuration dominated by AFM inter-\nactions as its lowest energy configuration with a magnetic moment of roughly 0.7 mB/Cu\natom and large energy differences between the lowest energy and the FM spin configura-\ntions (on the order of 4.2 meV/Cu atom). To the best of our knowledge, this compound has\nnot been investigated for its magnetic properties and should be explored experimentally\nto learn more about its magnetic behavior.\nIn the subset of materials DFT was performed on, we analyzed the proclivity for the\n16transition metal or rare earth elements produce a magnetic moment as a function of the\nprototype each appeared in. Our findings are summarized in Figure 7 by giving a percent-\nage of each transition metal or rare earth in each prototype that had a magnetic moment\nfrom DFT. As expected, certain transition metals and rare earths are more prone to dis-\nFigure 7 Percent of elements with a magnetic moment for each prototype for the calculation subset of\nkagomés. Ni, Pd, Pt, Ag, and Au never produced moments regardless of prototypes, while Cr, Mn, Fe,\nand Co always did.\nplaying a magnetic moment from calculation. Cr, Mn, Fe, and Co were always predicted to\nhave a magnetic moment, while Ni, Pd, Pt, Ag, and Au never did. On the more ambiguous\nend of the spectrum are Ti, Cu, and Rh, which only produced moments when appearing\nin specific prototypes, each of which were expected given the charge counting. Given the\nsmall number of compounds investigated in each prototype, determining which prototypes\nare most likely to produce a magnetic properties from this dataset is unwise. Regardless,\neach of these prototypes can be manipulated to expand the possible structure space avail-\nable for candidate QSLs. By swapping the various transition metal or rare earths forming\nthe kagomé lattice, trends between structure, chemistry, and magnetism can be further\nelucidated while providing more options to search for candidate QSLs.\n4 Conclusion\nDespite hundreds of kagomé sublattice-containing compounds in known materials, there\nwas a dearth of knowledge on their identification as well as structural and magnetic prop-\n17erties. By creating this dataset, we have been able to show that the kagomé sublattices\nappear in a chemically and structurally diverse set of materials. We report kagomé sublat-\ntices formed from 30 different transition metal or rare earths and 130 unique prototypes\nwith variety of spacings between kagomé atoms both in plane and out of plane, as well as\nthree different trends for stacking of the kagomé planes.\nAdditionally, we demonstrate of a method of computationally screening materials for\ntheir magnetic properties, particularly magnetic frustration, which is necessary for en-\nhancing the search for candidate QSLs. As is elucidated by this search, there is a wide\nchemical and structural space for kagomé sublattices to exist in and it is nowhere near\nfully explored. Finally, we predict nine candidate materials for possible QSL behavior\nfrom the results of our calculations. Of these nine materials, six (the jarosites, Na 2Ti3Cl8,\nCs2KMn 3F12, and Cs 2SnCu 3F12) have been experimentally eliminated as candidate QSLs,\nleaving Na 2Mn3Cl8, corkite, and bayldonite as promising candidate QSLs to be investigated\nexperimentally following this work.\nAcknowledgement\nV. Meschke acknowledges this material is based upon work supported by the National Sci-\nence Foundation Graduate Research Fellowship Program under Grant No. 1646713. EST\nacknowledges NSF award 1555340. The research was performed using computational\nresources sponsored by the Department of Energy’s Office of Energy Efficiency and Renew-\nable Energy and located at the National Renewable Energy Laboratory.\nConflicts of Interest\nThere are no conflicts to declare.\n18Supporting Information Available\nThe following files are available free of charge as supporting information:\n• calculation_set.xls: Spreadsheet listing the compounds DFT was used to screen mag-\nnetic properties on, as well as some key results of the calculations.\n• full_kagome_dataset.xls: Spreadsheet listing all found kagomé compounds from the\nsearch of the ICSD.\nNotes and references\n(1) Si, Q.; Abrahams, E. Strong Correlations and Magnetic Frustration in the High TcIron\nPnictides. Phys. Rev. Lett. 2008 ,101, 076401.\n(2) Hanawa, M.; Muraoka, Y.; Tayama, T.; Sakakibara, T.; Yamaura, J.; Hiroi, Z. Super-\nconductivity at 1 K in Cd2Re2O7.Phys. Rev. Lett. 2001 ,87, 187001.\n(3) Norman, M. R. 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In the past twenty years, a plethora of new\nartificial ferromagnetic materials have been found by introducing defects into non-magnetic host materials.\nIn contrast to the intrinsic ferromagnetic materials, they offer an outstanding degree of material engineering\nfreedom, provided one finds a type of defect to functionalize every possible host material to add magnetism\nto its intrinsic properties. Still, one controversial question remains: Are these materials really technologi-\ncally relevant ferromagnets? To answer this question, in this work the emergence of a ferromagnetic phase\nupon ion irradiation is systematically investigated both theoretically and experimentally. Quantitative pre-\ndictions are validated against experimental data from the literature of SiC hosts irradiated with high energy\nNe ions and own experiments on low energy Ar ion irradiation of TiO 2hosts. In the high energy regime, a\nbulk magnetic phase emerges, which is limited by host lattice amorphization, whereas at low ion energies an\nultrathin magnetic layer forms at the surface and evolves into full magnetic percolation. Lowering the ion\nenergy, the magnetic layer thickness reduces down to a bilayer, where a perpendicular magnetic anisotropy\nappears due to magnetic surface states.\nI. INTRODUCTION\nMagnetic materials play a major role in many spintronic and other technological applica-\ntions [1] such as magnetic storage [2], logic devices [3], magnetic field sensors and magnetic\nrandom access memory [4–6]. Materials with strong intrinsic ferromagnetic (FM) order above\nroom temperature, such as the transition metals Fe, Ni or Co and their alloys, are rather unusual\namong the magnetic materials known today [7] and there is still the need for new functional mate-\nrials with magnetic order above room temperature. In the past two decades, a method of creating\nartificial ferromagnetic materials has emerged and a multitude of so-called defect-induced fer-\nromagnets were reported [8–10]. Since the first prediction of an artificial ferromagnetic material\nwith transition temperature above 300 K, based on Mn doped ZnO appeared twenty years ago [11],\nthe field has substantially evolved. First, it was realized that doping with magnetic impurities was\nnot at all necessary in order to induce a robust FM order in the non-magnetic host matrix, rather\nall kinds of lattice defects were at the origin of the measured magnetic signals [12–15]. This\n\u0003lukas.botsch@uni-leipzig.de\n2realization promised great possibilities to construct new functional magnetic materials, as any\nnon-magnetic material could potentially host a certain kind of defect, turning it into an artificial\nferromagnet. The hunt was on and the result was a plethora of reports ranging from oxide, nitride,\ncarbon-based, 2D van der Waals and many more materials showing signals of ferromagnetism\nupon introducing all kinds of defects [8–10]. One of the most promising and versatile methods for\nintroducing these defects is the irradiation with non-magnetic ions [16], owing to the availability\nof ion sources ranging over the whole periodic table and energies from few eV to hundreds of\nMeV .\nAlthough many experiments were accompanied by theoretical studies, such as electronic struc-\nture calculations based on density functional theory (DFT), the search was mostly guided by blind\ntrial and error and a brute force approach. It is therefore not very surprising that most of the\nreported materials only showed very tiny magnetic signals, which soon led to debates about the\nnature of the effect [17, 18] and raised the question of whether this route could eventually lead\nto a robust magnetic order above room temperature, comparable with intrinsic ferromagnets. Fur-\nthermore, the measurement of the magnetization of such artificial ferromagnetic samples turns out\nto be quite difficult due to the inherent uncertainty of the magnetic volume, leading to largely\nunderestimated values in the literature. Considering the enormous amount of host material candi-\ndates and lattice defects, a more systematic search method and better selection criteria are highly\nneeded.\nIn this work, we present a systematic investigation of the emergence of such artificial ferro-\nmagnetic phases, both theoretically and experimentally. We first propose a scheme for the com-\nputational discovery of candidate materials that can be created by ion irradiation. The scheme is\nbased on first principle calculations, guided by experimental constraints, automatically restricting\nthe potential defects to those accessible experimentally and can readily be implemented for high\nthroughput material discovery. We further provide a method to determine the defect distribution\ncreated within the host lattices, allowing to obtain accurate magnetization values.\nTwo ion energy regimes are then investigated, namely at high energies \u0015100keV and at\nlow energies\u00141keV . The main physical processes governing the emerging FM phase in these\nregimes are identified and validated using high energy experimental data found in the literature\nand own low energy experiments. The predictions of the scheme are compared to experimental\nmagnetization data of SiC samples irradiated with high energy Ne ions, found in the literature.\nThe comparison confirms the role of host lattice amorphization as a limiting factor in the magnetic\n3percolation at high ion energy.\nWe then present own systematic experimental results, showing the emergence of an artificial\nFM phase in TiO 2hosts, upon irradiation with low energy ( \u00141keV) Ar ions. As predicted by\nour simulations, the magnetic percolation is most affected by sputtering processes at the surface\nin the low ion energy regime. The FM phase emerges in an ultrathin region beneath the surface,\nwhose thickness varies from a few atomic layers down to a magnetic bilayer, depending on the\nion energy. In these ultrathin films, the emerging FM phase reaches the full magnetic percolation\nlimit, where it spans over the complete sample surface area. The magnetic anisotropy is also\ninvestigated, showing a switch from in-plane to out-of-plane easy magnetization direction as the\nion energy and the resulting magnetic layer thickness decreases. This phenomenon is explained\nby the contribution of the surface to the magnetocrystalline anisotropy.\nII. COMPUTATIONAL METHODS\nMost of the theoretical work related to artificial ferromagnets has so far been devoted to un-\nderstanding the origin of the magnetic signals observed experimentally in different host systems.\nGuided by experimental intuition, a considerable computational effort was undertaken to identify\npossible defect structures, that carry non-zero magnetic moment and could explain the FM signals\nmeasured in nominally non-magnetic host systems upon ion irradiation. The method of choice are\nspin-polarized electronic structure calculations performed on different levels of DFT, which yield\nthe magnetic ground state of the defective systems and can provide an estimate of the magnetic\npercolation threshold. This is the threshold density of defects needed in a certain host matrix for\na long ranged ordered FM phase to emerge. Most studies rely on supercell methods to model\nsystems with different defect concentrations. As the number of possible defect structures that\ncould potentially yield a magnetic ground state in each single host matrix is enormous, studies are\nusually limited to investigating simple point defects, such as vacancies or interstitials. Obviously,\nan exhaustive search through all possible defect structures is not practical and a better method is\nneeded.\nConsidering artificial ferromagnets created by ion irradiation, a much better starting point\nwould be to only consider those defect structures that are experimentally accessible, i.e. that are\nlikely to be created by the impact of energetic ions on the host matrix. Calculations of irradiation\ndamage have been a standard tool in the context of accelerator physics for a long time. Molec-\n4MD+CA\nDFTDefect candidates\nD1% D2%\nArtificial FM\nCandidateIon irradiation:\nType\nEnergy\nInputs\nHost\nStructure\nMagnetic\nmomentPercolation\nlimit\nMCFIG. 1. Computational scheme for the prediction of artificial functional magnetic materials. It takes as\ninputs the atomic structure of a host material and the ion type and energy range of the ion irradiation. In a\nfirst step, molecular dynamics (MD) simulations and cluster analysis (CA) algorithms yield possible defect\nstructures and their corresponding creation probabilities, likely to be formed during the ion irradiation\nprocess. The resulting defective structures are used as input for DFT electronic structure calculations,\ngiving the magnetic ground state and the percolation limit for each defect type. Finally, using the results of\nall calculations and applying Monte Carlo (MC) methods, a magnetic phase diagram is constructed, which\nindicates the irradiation parameters likely to create an artificial FM.\nular dynamics (MD) methods, taking into account different levels of interatomic potentials, exist\nand yield accurate simulations of the damage resulting from collision cascades in a wide range of\nenergies [19]. The resulting damage structures, calculated in large host systems of several thou-\nsand atoms, can then be decomposed into smaller units of equivalent defects using cluster analysis\n(CA) techniques. These simulations not only yield structures of potential defect complexes arising\nin the collision cascades, beyond the simple vacancy or interstitial, but can also give statistical\ninformation about their creation probabilities.\nTaking the resulting structures of the defective host material as input for spin-polarized DFT\n5calculations, allows us to save substantial computational effort and gives much more realistic\nresults. Building on these remarks, we propose the computational scheme depicted in Figure 1,\ntaking as input the atomic structure of a host material, the type and energy of the ion irradiation,\nfrom which potential defective structures and their creation probabilities are obtained using MD\nsimulations and CA algorithms. DFT electronic structure calculations are then performed for the\nresulting defective structures and their magnetic ground state is determined. For defects yielding\na non-zero magnetic moment, the percolation limit is estimated. Finally, taking into account ion\nenergy loss in the irradiated host material and the defect formation probabilities predicted by the\nMD simulations, a magnetic phase diagram can be constructed using Monte Carlo (MC) methods.\nFrom this phase diagram, quantitative predictions of the total magnetic moment, the magnetic\nvolume and magnetization can be extracted.\nIII. HIGH ENERGY ION IRRADIATION IN SiC\nTo validate the predictions of the computational scheme, we first calculate the magnetic phase\ndiagram of 6H-SiC, resulting from the irradiation with high energy ( Eion= 140 keV) Ne ions and\ncompare the predicted magnetization values with those measured experimentally and published\nby Li and coworkers [20]. In the following sections we aim to give a step-by-step example of\nthe calculations involved in constructing the magnetic phase diagram and extracting quantitative\npredictions for the emerging FM phase.\nA. Molecular Dynamics Collision Cascade Simulations\nTo find the defects produced in 6H-SiC resulting from high energy ion irradiation, we per-\nformed a total of 9600 collision cascade simulations in 6H-SiC using the LAMMPS MD code [21–\n23]. The interatomic interactions were modeled using Tersoff/ZBL empirical potentials, as de-\nscribed by Devanathan et al. [24] and used previously for similar simulations [25]. To compute the\ncollision cascades, systems of 20\u000220\u000220unit cells (corresponding to 96000 atoms) with periodic\nboundary conditions were constructed. The unit cell parameters were set as a=b= 3:095 ˚A,\nc= 15:185 ˚A,\u000b=\f= 90\u000e,\r= 120\u000e. Twelve initial structures were equilibrated in the micro-\ncanonical (NVE) ensemble for 10-21 ps, respectively, with a timestep of 1 fs at T= 300 K. One Si\natom and one C atom located at the center of the simulation cells were selected as primary knock-\n660°(001)FIG. 2. PKA directions sampled in the conical cut of the unit sphere, with aperture 60\u000eand axis along (001)\ncrystal direction. Blue dots indicate the 10 initial directions used in the collision cascade simulations.\non atoms (PKA). Their initial kinetic energy was then set to values in the range 5 eV-200 eV by\nfixing the initial velocity along 10 different directions sampled from a cone with main axis along\nthe (001) crystal direction and an aperture of 60\u000e(see Figure 2).\nAfter setting the initial kinetic energy of the PKA atom, the systems were let to evolve in three\nphases, with timesteps of 0.01 fs, 0.1 fs and 1 fs for 0.1 ps, 1 ps and 10 ps, respectively, in order\nto capture the whole ballistic dynamics of the collision cascades.\nThe resulting collision cascade trajectories were then analyzed using the Ovito library. Single\npoint defects (vacancies, interstitials, antisites) were identified using a Wigner-Seitz decompo-\nsition of the initial and final structure. Defect clusters were then identified using a clustering\nalgorithm with a length cutoff of 2.5 ˚A, yielding the number of defect types created during each\nof the simulated collision cascades. The large degree of statistical sampling (120 simulations per\nPKA type and energy) allowed us to determine average defect creation rates, which are shown\nin Figure 3 for the C and Si PKAs in the energy range 5 eV to 200 eV . We find a displacement\nthreshold of Ed= 25 eV andEd= 40 eV for the C and Si PKA, respectively, in agreement with\nprevious reports [25].\nThe most prevalent defects are the C interstitial (C I), C vacancy (C V), Si vacancy (Si V), Si\nantisite (Si C) and the di-vacancy (Si V+ C V). We note that the creation rates shown in Figure 3 are\ngiven as average number of defects created per PKA event of a certain kinetic energy and type.\nIn general, incident ions collide with one or more lattice ions, transferring some of their kinetic\nenergy to the PKA. For a complete picture of the defect creation process, we need to determine\n70\n50\n100\n150\n200\nC PKA energy (eV)\n0.0\n0.5\n1.0\n1.5\n2.0Defects / PKA\nC PKA\nCI\nCV\nSiV+ C V\nSiC\nCV+ C I\nOthers\n0\n50\n100\n150\n200\nSi PKA energy (eV)\n0.0\n0.5\n1.0\n1.5Defects / PKA\nSi PKA\nCI\nSiV\nSiC\nCV\nSiI\nOthersFIG. 3. Defect creation rates in 6H-SiC obtained from collision cascades simulated by molecular dynamics\nmethods for C and Si PKAs in the energy range 5 eV to 200 eV . The five most prevalent defect types are\nshown, all other defects are grouped together. C I: carbon interstitial; C V: carbon vacancy; Si I: silicon\ninterstitial; Si V: silicon vacancy; Si C: silicon antisite.\nthe energy and spacial distribution of PKA events, created by energetic ions. But first, we need to\nidentify those defects that carry non-zero magnetic moment and determine the sign and range of\ntheir exchange interactions.\nB. Spin properties of the di-vacancy and Si vacancy in 6H-SiC\nThe spin properties of both the di-vacancy and the Si-vacancy in SiC have been extensively\ninvestigated due to their potential use as spin-defect qubits, both theoretically [26–31] and exper-\nimentally [20, 28, 29, 31–36]. According to spin-density functional theory calculations, the spin\nstates and interactions strongly depend on the charge state of the defects [26, 27, 29, 34]. While\nthe negatively charged Si vacancy has a spin-3/2 ground state, the neutral and higher charge states\n8are not spin-polarized [26, 33]. The neutral and negatively charged di-vacancies have a spin-1\nground state [27, 29, 37]. In both defects, the calculated spin densities indicate that the polariza-\ntion originates from localized states at the C atoms surrounding the Si vacancy.\nWang et al. investigated the exchange coupling between charged di-vacancies as a function of\nthe defect distance [29] and found that the negatively charged di-vacancies couple ferromagneti-\ncally at distances 10-18.5 ˚A, corresponding to a percolation threshold of \u00181at.%.\nC. Binary Collision Monte Carlo Simulations\nOut of all defects predicted in the collision cascade simulations, the Si and di-vacancy are\nthe most promising candidates to induce a long-ranged magnetically ordered phase in 6H-SiC, as\nthey both carry non-zero magnetic moment, tend to couple ferromagnetically and both Si and C\nvacancies are created with a high probability (see Figure 3) in the collision cascades.\nIn order to relate the defect creation rates obtained from the MD collision cascade simulations\nto ion irradiation experiments, we need to determine the energy and spacial distribution of PKA\nevents resulting from the collision of high energy ions with the SiC lattice atoms. We obtain\nthis using SRIM [38, 39], a widely used binary collision Monte Carlo code, that simulates the\nstopping and range of ions in matter by assuming the target material is amorphous. In the full\ncollision simulation mode, SRIM takes as input the number of ions to simulate, their type and\nkinetic energy, the target material composition and density, the displacement threshold of each\ntarget atom type and outputs a list of PKA collision events, including the transferred kinetic energy\nand the position.\nTo be able to compare our computational predictions to the experimental data published by Li\nand coworkers [20], we performed SRIM simulations of 105Ne+ions atEion= 140 keV , incident\non amorphous SiC, setting the displacement thresholds of the C atoms to Ed= 25 eV and that\nof Si toEd= 40 eV , according to our MD simulations (see Figure 3). We analyzed the resulting\nPKA events using a histogram method with a bin size of 5 eV for the PKA energy, matching\nthe energy resolution of our MD simulations. The PKA energy and depth (along the irradiation\naxis) distribution is shown for the two PKA types in Figure 4. In both cases, the maximum of the\ndistribution lies around 40-50 eV and at a depth of 160 nm below the surface. The maximum range\nof the Ne+ions is 260 nm. We note that, even though the initial kinetic energy of the ions was set\nto 140 keV , more than 90 % of the PKA events occur at an energy below 200 eV , indicating that\n90100200300\nDepth (nm)050100150200PKA Energy (eV)C PKA / ion\n0100200300\nDepth (nm)050100150200PKA Energy (eV)Si PKA / ion\n0.000.050.100.150.200.250.300.35\n0.00.10.20.30.4FIG. 4. PKA event distribution for 6H-SiC irradiated with Ne ions at Eion= 140 keV , calculated by binary\ncollision Monte Carlo simulations using SRIM. The number of resulting primary knockon atom (PKA)\nevents per incident ion is shown on the color scale (right axis) for C and Si PKAs as a function of the depth\nalong the (001) crystal direction (horizontal axis) and the PKA energy (left axis).\nthe ions transfer most of their kinetic energy to the electronic system of the target material before\nundergoing binary collisions with the target nuclei.\nD. Quantitative predictions and experimental validation\nCombining the results of the MD and SRIM simulations, we calculated the densities of all\ndefect types found in the MD simulations, as a function of ion fluence in the range 0- 5\u00021014cm\u00002\n(see Figure 5). Pairs of single C and Si vacancies, created in close proximity were counted towards\nthe di-vacancy density (Figure 5(b)) and the degree of amorphization (Figure 5(a)) was defined as\nthe total number of defects per lattice atom. As the creation rate of C vacancies is significantly\nlarger than that of Si vacancies, the resulting density of isolated Si vacancies is negligible and we\nwill therefore only take into account the di-vacancy defect in our further discussion.\nTaking into account a percolation threshold of 1 at.% for the di-vacancy defect [29], we identify\nthe threshold ion fluence required to induce long-ranged FM order along the irradiation direction,\nas indicated in Figure 5 by the lower white line. Below this line, at low ion fluences, the defect\ndensity is too low to create an artificial FM phase and the material is paramagnetic. Above the\nline, at high enough fluences, a magnetic percolation transition occurs and a FM phase emerges.\n10100\n200\n300\nDepth (nm)\n0\n1\n2\n3\n4\n5Fluence (1014cm1)\nFMamorphous\nPM\n0\n100\n200\n300\nDepth (nm)\n0\n1\n2\n3\n4\n5Fluence (1014cm1)\n(b)\nFMamorphous\nPM(a)\n020406080\namorphization (%)\n024681012\ndi-Vacancies (at. %)SiCFIG. 5. Magnetic phase diagram of 6H-SiC, irradiated with Ne ions at Eion= 140 keV . FM: ferromagnetic,\nPM: paramagnetic. The degree of amorphization (a) and concentration of di-vacancies (b) are shown on the\ncolor scales (right axis) and as a function of depth along the (001) crystal direction (horizontal axis) and the\nirradiation fluence (left axis).\nAs the spin polarization of the di-vacancy defect originates from the C atoms surrounding the Si\nvacancy, it appears plausible that a high degree of amorphization could destroy the FM phase. We\nhave therefore calculated a second threshold fluence, at which the degree of amorphization along\nthe irradiation direction reaches 50 %, as indicated by the upper white line in Figure 5. Above this\n110.0\n2.5\n5.0\n7.5\n10.0\nIrradiation fluence (1014cm2)\n0\n5\n10\n15\n20Msat(A m2kg1)\n(a)\nSiC\n0\n1\n2\n3\n4\nIrradiation Fluence (1013cm2)\n0\n20\n40\n60\n80\n100\n120msat(nA m2)\n(b)\nCeO 2FIG. 6. (a) Saturation magnetization of the ferromagnetic signal as a function of the ion irradiation fluence\nin a 6H-SiC single crystal sample irradiated with Ne+ions atEion= 140 keV (data reproduced from [20]);\n(b) Saturation magnetic moment of a CeO 2bulk sample, irradiated with Xe+ions atEion= 200 MeV (data\nreproduced from [40]). The magnetic properties were measured after each irradiation step. The solid line\nin (a) show the calculated magnetization as described in the text; in (b) the line is just a guide for the eye.\nAll measurements were done at room temperature.\nthreshold, at least two of the four C atoms surrounding the Si vacancy are displaced on average.\nThese two boundaries define a volume, in which a FM phase emerges. By integrating over all\ndefect magnetic moments in this volume, we obtain the saturation magnetization of the FM phase.\nThis is shown as a solid line in Figure 6(a), as a function of the irradiation fluence. Up to a fluence\nof2:5\u00021014cm\u00002, the magnetization increases linearly, as more defects are created. At higher\nfluences, the amorphization threshold is reached in a large portion of the magnetic volume and the\nmagnetization rapidly decreases.\nIn order to compare our theoretical predictions with the experimental data published by Li et\nal. [20], we calculated the saturation magnetization from the total magnetic moment data mea-\nsured using SQUID magnetometry by using the sample area and the depth of the magnetic phase\nresulting from our simulations. The resulting magnetization values are indicated in Figure 6(a)\n12as bullets. The magnetization observed experimentally first increases with the Ne ion fluence and\ndecreases at large enough fluences, matching our theoretical predictions quantitatively.\nThe authors suspected that the amorphization could play a role in the disappearance of the FM\nsignal in their samples at larger fluences [20]. Our simulations confirm that the amorphization of\nthe host lattice is the major limiting factor for the development of a dense FM phase in SiC. This\nmight also be the case in other materials, where artificial FM phases have been created using ion\nirradiation. Detailed experimental data showing the evolution of such a FM phase as a function of\nthe ion fluence is scarce. Shimizu et al. measured the magnetic properties of CeO 2single crystals\nirradiated with Xe+ions atEion= 200 MeV using SQUID magnetometry [40]. Figure 6(b) shows\nthe evolution of the total saturation magnetic moment of an emerging FM phase in CeO 2, as a\nfunction of the ion fluence. There, the magnetic moment follows the same qualitative trend: It first\nincreases rather linearly and at a fluence >2\u00021013cm\u00002, it decreases.\nIV . LOW ENERGY ION IRRADIATION IN ANATASE TiO 2\nMost experimental investigations of artificial FM phases emerging upon ion irradiation reported\nin the literature were performed at high ion energies \u0015100keV . This results in a rather large ion\npenetration depth and as we showed in the previous section, the evolution of the artificial FM\nphase is limited by the amorphization of the host lattice. In this section, we present results of our\nsystematic experimental investigation of a FM phase emerging in anatase TiO 2hosts upon low\nenergy Ar+ion irradiation ( Eion\u00141keV).\nA. Experimental Methods\nAmorphous TiO 2thin films were grown on SrTiO 2substrates by ion beam sputter deposi-\ntion [41]. Here a beam of low-energy Ar ions is directed onto a Ti target. Due to momentum and\nenergy transfer, target particles get sputtered and condense on a substrate. Additionally oxygen\nbackground gas was provided such that TiO 2thin films were formed. Ion energy, ion current, and\nion incidence angle were 1000 eV , 7 mA, and 30\u000e, respectively. The substrates were placed at a\npolar emission angle of 40\u000erelative to the target normal. The volumetric flow rate of Ar primary\ngas was 3.5 sccm and of O 2background gas 2.0 sccm, which resulted in a total working pressure\nof6\u000210\u00003Pa. More details are given in Ref. [42]. The films have a thickness of 40 nm. After\n1320 30 40 50 60 70 80 90 100\n2  (°)\n100101102103104105Intensity (a.u.)\nSTO (100)STO (200)\nSTO (300)Anatase TiO2 (004)\nKβ\nLαFIG. 7. X-ray diffraction measurements of the crystallized anatase TiO 2films grown on SrTiO 3(100)\nsubstrates. The data shown here were obtained from sample S1000 after the last irradiation step.\nannealing at 500\u000eC in air for 1h, the films crystallize in the anatase phase with the film surface\nnormal along the (001) crystal direction, as confirmed by XRD measurements (see Figure 7).\nThree thin film samples with a surface area of 5\u00025mm2were selected for the ion irradiation\nexperiments. Each sample was then irradiated with Ar+ions atEion= 200 eV (“S200”), 500 eV\n(“S500”) and 1000 eV (“S1000”), respectively, using a custom-built DC plasma chamber. The ion\ncurrent was measured through a gold frame surrounding the sample during the irradiation process\nand was used to calculate the fluence.\nThe magnetic properties of each sample were characterized before and after each irradiation\nstep using a commercial SQUID magnetometer (Quantum Design MPMS XL) using the recipro-\ncating sample operation (RSO) mode. In order to ensure the comparability of the results after each\nirradiation step, great care has been taken to reduce as much as possible any source of contamina-\ntion to the samples and the same protocol and schedule was maintained throughout the experiment.\nThe samples were clamped in a plastic straw (as shown in Figure 10(b) in [43]), allowing to mea-\nsure the magnetic responses to a magnetic field applied perpendicular and parallel to the film sur-\nface. The total magnetic moment of the sample was recovered from the raw SQUID voltage signal\nusing a point dipole approximation. The finite sample size has been corrected for using methods\ndescribed in Ref. [43], by applying a correction factor to the total moment (in-plane: 0.969296,\nperpendicular: 1.015966). For magnetic hysteresis loop measurements, m(B), the magnetic field\nwas first reduced from 0.1 T to nominally zero in oscillating mode, followed by a magnet quench\nto minimize the remanent field. In addition, the solenoid hysteresis was corrected for using a cal-\n14ibration measurement, as described in detail in [43]. For temperature dependent measurements,\nm(T), the sample temperature was first set to T= 380 K, followed by the same magnetic field\nreset procedure. After cooling the sample down to T= 2K, a constant magnetic field was set and\nthe zero field cooled (ZFC) curve was measured while heating up to T= 380 K followed by the\nfield cooled (FC) measurement. At each temperature step, we waited for 60 seconds to allow the\nsample to reach thermal equilibrium prior to the measurement.\nTo recover the relevant magnetic hysteresis parameters from the experimental data, we use the\nfollowing model:\nm(B) =\u001fB+2ms\n\u0019arctan\u0014B\u0006Bc\nBctan\u0012\u0019mr\n2ms\u0013\u0015\n; (1)\nwhere\u001faccounts for a linear contribution to the susceptibility, including a diamagnetic response\nfrom the substrate and a paramagnetic response at room temperature and moderate fields, ms,Bc\nandmrare the saturation moment, coercive field and remanent moment of a hysteretic response.\nAfter the measurements of sample S500, and before measuring the other two samples, we had\nto adjust the “SQUID tuning parameters” (drive power and frequency) as the SQUID was slightly\ndetuned, resulting in a significant reduction of the signal noise.\nB. Magnetic phase diagram\nTo understand the emergence of an artificial FM phase in TiO 2due to low energy ion irradia-\ntion, we used the same computational scheme as for the high energy case, outlined in the previous\nsection. Robinson et al. [44] performed detailed molecular dynamics simulations of collision cas-\ncades in anatase TiO 2, due to low energy ion irradiation and calculated the probability of resulting\ndefect structures, as shown in Figure 2, for collision cascades resulting from Ti and O primary\nknock-on atoms (PKA), using a very similar method to ours. At PKA energies near the displace-\nment threshold, Ed= 39 eV for Ti PKAs and Ed= 19 eV for O, the primary defects created\nare the di-Frenkel pair (dFP; 40% of Ti PKAs), consisting of two Ti atoms displaced into inter-\nstitial sites leaving behind two vacancies and the oxygen vacancy (O v; 50% of O PKAs) (see\nFigure 2(a,c)).\nBoth the dFP [45] and O v[46] defects in anatase TiO 2have been investigated previously using\nDFT calculations and found to carry a magnetic moment of 2\u0016Band1\u0016B, respectively. When their\nconcentration reaches \u00185% in the bulk, the magnetic moments start to couple ferromagnetically\n15dFPOIOV cluster other010203040Formation probability (%)(a) Ti PKA, EdEPKAEd+20 eV\nOV+TiVclusterOIOI+TiITiIother0510152025Formation probability (%)(b) Ti PKA, 5 eV  EPKA  200 eV\nOVOIOV+2OI01020304050Formation probability (%)(c) O PKA, EdEPKAEd+20 eV\nOVOIOV+2OIcluster other010203040Formation probability (%)(d) O PKA, 5 eV  EPKA  200 eV\nFIG. 8. Defect formation probabilities at primary knockon (PKA) energies EPKA near the displacement\nthresholdEd(a,c) and in the range 5\u0000200eV (b,d) for Titanium (a, b) and Oxygen (c, d) PKAs. Complexes\ncontaining more than four defects are categorized as “cluster”, the group labeled “other” contains defects\nwith less than 5% formation probability. Data taken from Ref. [44].\nand undergo a magnetic percolation transition. A long-ranged ordered phase emerges and persists\neven above room temperature [45]. According to reports from Stiller et al., dFP defects are mostly\nresponsible for the emergence of a FM phase in anatase TiO 2irradiated with low energy Ar+\nions [45]. The O vdefect was found to play a role in magnetic TiO 2doped with Cu [46].\nUsing SRIM simulations for Ar+ions atEion= 200 eV ,500eV and 1000 eV , and taking into\naccount the defect creation probabilities (Figure 8), a magnetic phase diagram can be constructed.\nAt low energies, the sample volume affected by the incident ions is much smaller than in the\nhigh energy case and the effect of surface sputtering has to be taken into account. SRIM allows\nto calculate the sputtering rate. We find values of 1.1, 1.5 and 2.3 nm / ( 1016ions / cm2) at ion\nenergies of 200 eV , 500 eV and 1000 eV , respectively.\nFigure 9 shows the dFP density and degree of amorphization as a function of the Ar ion fluence\nfor the three ion energies considered in this study. Due to sputtering, the amorphization of the\n160123\nDepth (nm)0.00.20.40.60.81.0Fluence (1016 cm1)\n(a)   Eion=200 eV\n0.46 nm\nFMsurface\nBulk\n0123\nDepth (nm)0.00.20.40.60.81.0Fluence (1016 cm1)\n(b) \nFMsurface\nBulk\n0123456\nDepth (nm)0.00.20.40.60.81.0Fluence (1016 cm1)\n(c)   Eion=500 eV\n1.20 nm\nFMsurface\nBulk\n0123456\nDepth (nm)0.00.20.40.60.81.0Fluence (1016 cm1)\n(d)\nFMsurface\nBulk\n0.02.55.07.510.0\nDepth (nm)0.00.20.40.60.81.0Fluence (1016 cm1)\n(e)   Eion=1000 eV\n1.99 nm\nFMsurface\nBulk\n0.02.55.07.510.0\nDepth (nm)0.00.20.40.60.81.0Fluence (1016 cm1)\n(f)\nFMsurface\nBulk\n0102030\nAmorphization (%)\n 0510152025\ndFP (at. %)\n0204060\nAmorphization (%)\n 0510152025\ndFP (at. %)\n020406080\nAmorphization (%)\n 0510152025\ndFP (at. %)FIG. 9. Magnetic phase diagram for the artificial FM created in anatase TiO 2by Ar+ions atEion= 200 eV\n(a,b),Eion= 500 eV (c,d) and Eion= 1000 eV (e,f). The color scale (right axes) shows the degree of\namorphization (a,c,e) and the density of dFP defects (b,d,f), along the irradiation direction (lower axes)\nand as a function of the irradiation fluence (left axes). The solid white lines separate the regions with high\nenough defect concentration to form a ferromagnetic phase (FM) from those with low defect concentration\nforming a paramagnetic phase (PM) and indicates the percolation transition. The dashed white lines indicate\nthe shift of the sample surface due to sputtering. At fluences >4\u00021015cm\u00002, the thickness of the FM\nphase is stable, as indicated by the arrows in (a,c,e). The visible steps in the color maps are due to the\ndiscretization of the depth, which has a step size of 0:25c= 2:4˚A, corresponding to the anatase layer\nspacing in the (001) crystal direction.\nhost lattice has a much smaller effect, as the thin amorphous surface layer is constantly removed.\n17This is indicated by a dashed line in Figure 9. The solid line indicates the magnetic percolation\ntransition between unordered (paramagnetic) and FM phases.\nAt fluences >4\u00021015cm\u00002, the defect creation and sputtering processes are at equilibrium.\nThe volume and defect densities of the emerging FM phase stay constant over the whole fluence\nrange. The equilibrium volume depends on the ion energy, as indicated by the thickness of the\nFM regions along the irradiation direction in Figure 9. At Eion= 200 eV , the emerging FM layer\ngrows to an equilibrium thickness dFM= 4:6˚A, corresponding to 0:48c(c= 9:51˚A, the Anatase\nlattice constant). At Eion= 500 eV we find an equilibrium thickness dFM= 12:0˚A (= 1:26c) and\natEion= 1000 eV ,dFM= 19:9˚A (= 2:09c). The anatase unit cell consists of four layers, stacked\nalong the (001) crystal direction. Therefore, the emerging FM phase is expected to be restricted to\nthe first 2, 4 and 8 layers of the host lattice, respectively.\nC. Experimental observation of the emerging FM phase in Anatase TiO 2\nFigure 10(a) shows two typical hysteresis curves, measured at room temperature and after sam-\nple S1000 had been irradiated with a fluence of 0:6\u00021016cm\u00002(blue) and 8:7\u00021016cm\u00002\n(orange). After subtracting the linear diamagnetic background (Figure 10(b)), a hysteretic FM sig-\nnal clearly appears, showing a magnetic moment at saturation msatthat increases with ion fluence\n(hysteresis curves at all irradiation fluences are shown in Figures S1-S3 in the supporting infor-\nmation). Figure 10(c) shows the zero-field cooled (ZFC) and field cooled (FC) curves, measured\nat an irradiation fluence of 18:3\u00021016cm\u00002, in the temperature range 2-380 K and at an applied\nmagnetic field of B= 0:05T. The opening between the ZFC and FC curves is a clear sign of a\nFM phase.\nBy systematically measuring the magnetic hysteresis of the samples as a function of the irradi-\nation fluence, we can gain some insight into the evolution of the emerging FM phase. Figure 11(a)\nshows the total magnetic moment at saturation at T= 300 K,msat, after subtracting the linear dia-\nmagnetic background, as a function of the irradiation fluence, for the three samples S200, S500 and\nS1000. The values of msathave been obtained by fitting the hysteresis curves with Equation (1).\nThe background signal m0, measured in each sample before any irradiation was subtracted. The\nmagnetization Msatwas calculated from the measured total moment msattaking into account the\nequilibrium volume of the magnetic phase predicted from the phase diagram (Figure 9).\nWe first observe that the moment at saturation increases with increasing total irradiation flu-\n180.2\n0.1\n0.0\n0.1\n0.2\nMagnetic Field (T)\n10\n5\n0\n5\n10Magnetic Moment (nA m2)\n(a)\n0.6 × 1016cm2\n8.7 × 1016cm2\n0.2\n0.1\n0.0\n0.1\n0.2\nMagnetic Field (T)\n5.0\n2.5\n0.0\n2.5\n5.0Magnetic Moment (nA m2)\n(b)\n0\n100\n200\n300\nTemperature (K)\n1.6\n1.4\n1.2\n1.0\n0.8\nMagnetic Moment (nA m2)\n(c)\nB = 0.05 T\n18.3 x 1016cm-2\nZFCFCFIG. 10. (a,b) TiO 2thin film irradiated with 0:6\u00021016cm\u00002(blue) and 8:7\u00021016cm\u00002(orange) Ar+ions\natEion= 1000 eV . (a) Hysteresis loop showing total magnetic moment as a function of applied magnetic\nfield atT= 300 K. (b) Hysteresis loop after subtracting a linear diamagnetic background. The magnetic\nfield was applied parallel to the film surface. (c) Zero-field cooled (ZFC) / field cooled (FC) curve, measured\natB= 0:05T for sample S1000 at an irradiation fluence 18:3\u00021016cm\u00002.\nence and saturates at high fluences. This is expected, as increasing the irradiation fluence also\nincreases the density of dFP defect complexes until the defect creation and sputtering processes\nreach equilibrium, in agreement with the phase diagram (Figure 9). The saturation magnetization\nof all three samples reaches values of the order of 35 Am2kg\u00001, which corresponds to a mean dFP\nconcentration of\u001817at. % or one dFP per unit cell.\nBy numerically integrating the dFP concentration found in our calculations over the volume of\nthe FM phase in Figure 9 and taking into account the magnetic moment ( 2\u0016B) of each dFP defect,\nwe can calculate the expected magnetization and total moment of the samples as a function of the\nirradiation fluence. The result is shown in Figure 11 as solid lines. The oscillations are numerical\nartifacts due to the discretization of the phase diagram. At the lowest ion energy (sample S200),\nour predictions show excellent agreement with the experimental data. The predicted equilibrium\nmagnetization at high irradiation fluences of all three samples also agrees very well with our\nmeasurements. The evolution of the magnetization of sample S1000 to the equilibrium value, on\nthe other hand, does not agree well with our calculations, that predict a much steeper approach to\nequilibrium. In fact, it appears like the saturation moment of sample S1000 first follows the same\nevolution as sample S200 (Figure 11) and then slowly increases to the equilibrium value.\n19051015\nIrradiation Fluence (1016 cm2)\n02468msatm0 (nA m2)\n(a)\nTiO2\nS200\nS500\nS1000\n051015\nIrradiation Fluence (1016 cm2)\n010203040MsatM0 (A m2 kg1)\n(b)\nS200\nS500\nS1000FIG. 11. Magnetic moment msat(a) and magnetization Msat(b) at saturation of the ferromagnetic signal,\nmeasured at room temperature as a function of the ion irradiation fluence, in the three TiO 2thin film sam-\nples, S200 (\u000f), S500 ( \u0004) and S1000 ( \u0007). The background signal, m0(M0) measured before any irradiation\nhas been subtracted from the experimental data. The solid lines show the magnetic moment (magnetization)\nvalues predicted from our theoretical calculations (Figure 9).\nD. The magnetic percolation process\nWe have seen in Figure 9 that the FM phase emerges in an ultrathin region of thickness between\n2 and 8 anatase layers. To better understand the evolution of these ultrathin FM layers, it is\ninstructive to take a closer look at the magnetic percolation process, i.e. the transition from isolated\nlocal magnetic moments to a long-ranged ordered phase upon increasing the defect density.\nA system of dilute defects in a host lattice that interact magnetically on a finite length scale can\nbe described using the framework of percolation theory [47, 48]. In the site-percolation model,\nsites on the host lattice can be occupied by a defect and one can associate a probability pwith\nthe occupation of a site. At p= 0, no defects are present in the system and at p= 1, all sites\n20are occupied by a defect. This probability is naturally related to the defect density in the system\nthrough the density of possible sites that can host a defect.\nThe length scale of the magnetic interaction, namely the exchange coupling, determines\nwhether two occupied sites are linked: Two occupied sites that are close enough to each other\nto interact ferromagnetically through the exchange interaction form a percolation domain. The\nmagnetic moments associated to each site belonging to one percolation domain are correlated,\nwhile those associated to different percolation domains behave independently.\nThe site-percolation model describes a second order geometrical phase transition, that occurs\nat a critical occupation probability pc. Atp<pc, in the subcritical regime, small isolated percola-\ntion domains form. The microscopic magnetic moments within each of these percolation domains\nare aligned parallel to each other due to the ferromagnetic coupling, but their orientation is arbi-\ntrary. The summed magnetic moment of each domain can be treated as one large paramagnetic\ncenter and the system is in a super-paramagnetic phase. At p > pc, in the supercritical regime,\none large percolation domain exists that spans over the whole system volume. This domain is\ncalled the percolation continent and exhibits the features of a ferromagnet, such as a spontaneous\nmagnetization.\nAt criticality, when p\u0018pc, the percolation continent emerges and the defective host system\nbecomes ferromagnetic. As with all second order phase transitions, the percolation transition can\nbe described by an order parameter, namely the probability P1, that a random defect belongs to\nthe percolation continent. Near criticality, the evolution of the order parameter follows a power\nlaw of the form\nP1/(p\u0000pc)\f; (2)\nwith a universal critical exponent \f, that only depends on the dimensionality of the system.\nWe simulate the magnetic percolation process in a grid of 200\u0002200anatase TiO 2unit cells\nand a thickness of 1, 2 and 3 layers, enforcing periodic conditions on the lateral boundaries. For\ncomparison, we perform the same simulations in a 200\u0002200\u0002200unit cell grid, enforcing periodic\nboundary conditions in all three directions for a bulk system. Magnetic dFP defects are randomly\ndistributed throughout the grids, varying the total defect concentration. Only nearest neighbor\ninteractions between dFP defects are taken into account, such that two nearest neighbor cells, each\ncontaining a dFP defect, interact ferromagnetically and form a percolation domain. Figure 12(a)\nshows the size of the largest of the domains, the percolation continent, normalized to the total size\nof the grid. Figure 12(b) shows the number of independent percolation domains within the grid,\n21TABLE I. Critical exponents \fobtained by fitting Equation (2) to the data shown in Figure 12(a).\n# layers \f\n1 0:210\u00060:007\n2 0:220\u00060:006\n3 0:259\u00060:007\nbulk 0:417\u00060:004\nas a function of the dFP defect concentration, normalized to the total number of cells in the grids.\nVarying the defect density, we see three regimes: at low concentration ( <5at. % in the mono-\nlayer), the sample is paramagnetic and the percolation domains are small (see Figure 12(c)). In\nthe intermediate regime (5-9 at. % in the monolayer), the magnetic dFP defects start to interact\nand the domains grow rapidly in size (see Figure 12(d)). At the percolation threshold, 9 at. %,\n7.5 at.%, 6 at.% and 5 at. % in the mono-, bi-, trilayer and the bulk system, respectively, the\ndomains merge and the percolation continent grows rapidly until spanning over the whole grid\n(see Figure 12(e)). We note that at a dFP density of 17 at. %, which we found at equilibrium in\nFigure 11, the percolation continent has fully evolved and the order parameter P1= 1.\nTable I shows the critical exponents obtained by fitting the curves shown in Figure 12(a) to\nEquation (2). Experimentally, the magnetic percolation transition can be obtained by taking the\nremanent magnetic moment at zero applied magnetic field and at high temperatures. Indeed, at ion\nfluences below the percolation transition, the samples are expected to be (super-)paramagnetic and\nno remanence is expected above the blocking temperature. Near the percolation transition, when\nthe percolation continent forms at a critical fluence fc, the remanent magnetic moment should\nfollow the same critical behavior as the order parameter P1.\nFigure 13 shows the evolution of the remanent magnetic moment mremmeasured in samples\nS200 and S1000 as a function of the irradiation fluence f. By fitting the data to a power law\n(mrem/(f\u0000f0)\f), we find critical fluences f0= 5\u00021015cm\u00002and2:5\u00021016cm\u00002for\nsamples S200 and S1000, respectively. The resulting critical exponents are \f= 0:22\u00060:04and\n0:42\u00060:07, respectively. Comparing these exponents to the theoretical values (Table I), we see that\nthe remanence observed in sample S1000 follows the critical behavior of a bulk 3D percolation\ntransition, while sample S200 follows the critical behavior of a magnetic bilayer system. These\nresults match very well with the aforementioned thickness of the FM phases (see Figure 9), that\n22051015\ndFP Concentration (at. %)0.000.250.500.751.00Max Domain size / # Cells(a)\n(c) (d)(e)200x200x1\n200x200x2\n200x200x3\nbulk\n051015\ndFP Concentration (at. %)0.000.020.040.060.080.100.12# Domains / # Cells(b)\n(c)\n(d)\n(e)0.03.46.810.2\nx (nm)10.2\n6.8\n3.4\n0.0y (nm)\n(c)\n0.03.46.810.2\nx (nm)10.2\n6.8\n3.4\n0.0y (nm)\n(d)\n0.06.713.420.0\nx (nm)20.0\n13.4\n6.7\n0.0y (nm)\n(e)FIG. 12. Magnetic percolation process in 200\u0002200\u0002Z(Z= 1;2;3and bulk) unit cell TiO 2slabs\nfor varying dFP concentrations. Nearest neighbor cells containing a dFP interact and form percolation\ndomains. Varying the dFP concentration, three regimes can be seen: at low concentration ( <5at. % in\nthe monolayer), the sample is paramagnetic and the percolation domains are small. In the intermediate\nregime (5-9 at. % in the monolayer), the magnetic dFP defects start to interact and the domains grow\nrapidly in size. At the percolation threshold, 9 at. %, 7.5 at.%, 6 at.% and 5 at. % in the mono-, bi-, trilayer\nand the bulk system, respectively, the domains merge and form a large percolation continent. The size of\nthe percolation continent is shown in (a) as a function of the dFP concentration; (b) shows the number\nof percolation domains (normalized by the total number of cells). (c)-(e) Examples of monolayer grids,\nin which individual domains are color-coded. Black cells correspond to non-magnetic cells, that do not\ncontain any dFP defect. Regions of the same color correspond to percolation domains, in which each cell\nhas at least one nearest neighbor cell containing a dFP. The defect concentration was set to 2.5 at. % (c),\n7.5 at. % (d) and 10 at. % (e). These three examples are marked by blue dots in (a) and (b). The dashed\nlines in (a) correspond to the fits of Equation (2).\n23024681012\nIrradiation Fluence (1016 cm2)\n0.00.10.20.30.40.5mrem m0\nrem (nA m2)\n(ffc)0.22\n(ffc)0.42\nfc=5×1015 cm2\nfc=2.5×1016 cm2\n200 eV\n1000 eVFIG. 13. Remanent magnetic moment mremat zero field, measured at T= 300 K after setting a magnetic\nfieldB= 5T, as a function of the irradiation fluence f. The background remanence m0\nremof the unirradi-\nated samples was subtracted. The symbols represent experimental data of samples S200 ( \u000f) and S1000 ( \u0007).\nThe dashed lines represent fits to mrem/(f\u0000fc)\f, with the critical fluences ( fc) and critical exponents\n(\f) as indicated.\npredicted the emergence of a magnetic bilayer in sample S200, while in sample S1000 the FM\nphase spans over 8 layers.\nV . DIMENSIONALITY AND SURFACE EFFECTS, EMERGENCE OF A PERPENDICULAR\nMAGNETIC ANISOTROPY\nTwo dimensional long ranged magnetic order has long been thought to be impossible at finite\ntemperatures, as stated by the Mermin-Wagner (MW) theorem [49]: In bulk 3D ferromagnets, the\nexchange interaction asserts a long range magnetic order up to the Curie temperature, TC, where\nthermal fluctuations become strong enough to randomize the spin orientation. In 2D magnetic\nsystems with isotropic exchange interaction, the dimensionality effect leads to an abrupt jump in\nthe magnon dispersion and therefore strong spin excitations at any finite temperature, destroying\nthe magnetic order. The presence of a strong uniaxial local magnetic anisotropy opens a gap\nin the magnon dispersion, counteracting the Mermin-Wagner theorem in 2D and restoring long\n24range order. This has been demonstrated experimentally in ultrathin transition metal films [50,\n51] and 2D magnetic van der Waals materials [10, 52, 53]. 2D magnetic structures are not only\ninteresting from a fundamental physics perspective, but also regarding their possible applications\nin 2D spintronics, magnonics or spin-orbitronics [1–3, 10, 53]. In the following section, we shall\nshow the role of the magnetic anisotropy and of the surface to stabilize the artificial FM phase at\nroom temperature in TiO 2hosts, even in two dimensions.\nA. Measuring the magnetic anisotropy of the emerging FM phase in TiO 2hosts\nFigure 14 shows measurements of the magnetic hysteresis loops obtained by applying an exter-\nnal magnetic field parallel to the film surface (blue) and perpendicular to the surface (orange) of\nthe three TiO 2samples. In panels (a,c), the raw signals are shown and a clear magnetic anisotropy\nis visible. After subtracting the linear diamagnetic contribution (Figure 14(b,d)), the total mag-\nnetic anisotropy energy (MAE) can be calculated as the area difference between the two curves.\nHere, we calculated the area difference by first fitting Equation (1) to the experimental data and\nintegrating the result analytically. To compensate differences in the resulting saturation moments\nmsatfor the two field orientations, e.g. due to fitting error or finite sample size effects (see Sec-\ntion IV .A), we rescaled the values of msatandmrem, such thatmsatcoincides. The sign is defined\nsuch that positive MAE indicates a magnetic easy in-plane direction, parallel to the film surface\nwhile a negative MAE indicates an out-of-plane easy axis. More details are given in the supporting\ninformation. At the selected irradiation fluences, sample S200 (panels (a,b)) shows a perpendicular\nmagnetic anisotropy while sample S500 (panels (c,d)) shows an in-plane anisotropy.\nFigure 15 shows the total MAE as a function of the irradiation fluence, of the samples S200 (a)\nand S500 (b). In sample S200, where the thickness of the FM phase is estimated to only two layers\nof the host lattice, the MAE is negative throughout all irradiation fluences, indicating a magnetic\neasy axis normal to the film surface. In sample S500, the MAE is positive and its magnitude is\nroughly four times larger than that of sample S200. For sample S500, the magnetic phase diagram\n(Figure 9) predicts a FM phase spanning the first four layers of the film surface, i.e. twice as many\nas for sample S200, hinting towards the role of the surface in the emergence of a perpendicular\nmagnetic anisotropy.\n250.2\n0.1\n0.0\n0.1\n0.2\n10\n5\n0\n5\n10Magnetic Moment (nA m2)\n(a)\nB || Surface\nB Surface\n0.0\n0.5\n1.0\n1.5\n0.0\n2.5\n5.0\n7.5\n10.0\n(b)\nB || Surface\nB Surface\n0.2\n0.1\n0.0\n0.1\n0.2\n10\n5\n0\n5\n10Magnetic Moment (nA m2)\n(c)\nB || Surface\nB Surface\n0.0\n0.5\n1.0\n1.5\n0.0\n2.5\n5.0\n7.5\n10.0\n(d)\nB || Surface\nB Surface\nMagnetic Field (T) Magnetic Field (T)FIG. 14. Magnetic hysteresis loops measured with the external magnetic field applied in the plane of the\nTiO2films (blue) and out-of-plane (orange), measured at T= 300 K. The left panels show the total mag-\nnetic moment measured with the SQUID, the right panels show the ferromagnetic signal after subtracting\nthe background diamagnetic and paramagnetic signals. Panels (a),(b) show the results obtained for sample\nS200 after irradiation with a fluence of 2:5\u00021016cm\u00002and panels (c),(d) for sample S500 at a fluence of\n6:0\u00021016cm\u00002.\nB. DFT electronic structure calculations of the defective TiO 2surface\nTo understand the origin of the magnetic anisotropy shown in Figure 15, we performed\nDFT electronic structure calculations using the full potential linearized augmented plane wave\n(FLAPW) method implemented in the FLEUR code, including spin-orbit interaction and a Hub-\nbard termU= 4:0eV , on a 3\u00023\u00021supercell of anatase TiO 2, containing one dFP defect\n260.00.51.01.52.02.53.03.5\nIrradiation Fluence (1016 cm2)\n0.075\n0.050\n0.025\n0.000MAE (mJ cm2)\n(a)\nS200\n0246810\nIrradiation Fluence (1016 cm2)\n0.000.050.100.15\n(b)\nS500FIG. 15. The magnetic anisotropy energy (MAE) obtained from the magnetic hysteresis curves measured\nafter each irradiation step for the TiO 2sample S200 (a) and S500 (b). The MAE is defined such that a\nnegative value indicates an out of plane easy-axis (along the film surface normal), while a positive value\nindicates an easy in-plane magnetization direction (parallel to the film surface). The shaded area indicates\nthe confidence margin within 5% significance level.\n(the defect labeled “di-FP1” in Ref. [45]). We used a planewave cutoff times muffin tin radius\nKmax\u0002aMT= 7:0. The atomic structure was relaxed using a 2\u00022\u00022k-point grid and the\nfinal charge and spin density was calculated using a 6\u00026\u00026k-point grid. Figure 16(a) shows\nthe relaxed bulk atomic structure, with the two Ti interstitials colored in pink. The isosurface at a\nspin density of 0:005\u0016Ba\u00003\n0is shown in yellow and is mainly located around the two interstitials\nand their neighboring Ti lattice atoms in the (010) plane, having dxzcharacter, as confirmed by\nthe density of states (Figure 16(a)). We find a total magnetic moment of 2\u0016B/dFP defect. These\ncalculations match well with those presented by Stiller et al. [45]. We also calculated the MAE\nfrom the total energy difference between the in-plane and out-of-plane magnetization state and\nfindMAE = 11 \u0016eV/atom with an (001) easy-plane in the bulk, as opposed to the out-of-plane\neasy axis found by Stiller et al [45].\nWe then calculated the electronic structure of the dFP defect at the (001) anatase surface, using\na3\u00023supercell containing four layers of anatase, as shown in Figure 16(b). Only the lower\ntwo layers were relaxed, while the upper two were held fixed at the bulk atomic positions. After\nstructural relaxation, the surface layer shows displacements comparable to values found in the\nliterature [54] ( \u000b= 142\u000e, Ti5c–O2c(short) = 1:813 ˚A, Ti 5c–O2c(long) = 2:010 ˚A, Ti 5c–O3c=\n270.004\n0.0Spin Density (µBa0\n-3)z\ny\nx(a)\n(b)Tii2Tii1\nTii1\nTii2c-axisFIG. 16. FLAPW-DFT calculations for (a) a 3x3x1 bulk anatase supercell and (b) a 3x3x1 supercell of\nthe (001) anatase surface, each containing one dFP defect per supercell. The relaxed atom positions are\nrepresented by spheres (blue: Ti, red: O, purple: Ti i). The isosurface at 0.005 \u0016Ba\u00003\n0spin density is shown\nin yellow. The spin density in the (010)-plane through the two interstitials is indicated by shades of gray\naccording to the scale on the right.\n1:941 ˚A). As visible in Figure 16(b), the spin density (shown in yellow) has similar structure\nas in the bulk around the interstitial Ti i1 on the second layer. On the first layer, on the other\nhand, the spin density at interstitial Ti i2 changes strongly owing to the reduced coordination.\nThere, the Ti 3d z2orbital is mainly spin polarized, as reflected by the DOS (Figure 17(b)). The\nmagnetic anisotropy at the (001) surface results in an out-of-plane easy axis with a large MAE =\n\u0000137\u0016eV/atom.\nThese results together with the calculated defect distribution (Figure 9) explain the measured\nMAE (Figure 15): At low ion energy of Eion= 200 eV (sample S200), the FM phase emerges in\nthe first two layers at the film surface, resulting in a large negative MAE. At higher ion energies,\nthe FM phase emerges in a larger volume and also has contributions from bulk states, which favor\nan in-plane easy-axis. Depending on the local distribution of the defects, either the surface or the\n281.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00DOS (arb. units)(a)\nO 2p\nTi 3d\nTii 3d\nTotal\n6\n 4\n 2\n 0 2\nEnergy (eV)1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00DOS (arb. units)(b)\nO 2p\nTii (bulk) 3d\nTii (surface) 3d\nTi 3d\nTotalFIG. 17. Density of states (DOS) (a) of the bulk anatase structure and (b) of the (001)-surface, each con-\ntaining 5:5% dFP. The total DOS is shown in gray; the partial DOS (PDOS) of O 2p states is shown in blue;\nthe PDOS of Ti 3d states is shown in orange. In (a), the PDOS of the two equivalent interstitial 3d states is\nshown in red; in (b), the PDOS of the bulk (Ti i1) and surface (Ti i2) interstitials are shown in green and red,\nrespectively.\nbulk states dominate the total MAE.\n29VI. CONCLUSIONS\nThe computational methods proposed in this work serve as a viable route toward the systematic\ndiscovery of new artificial functional magnetic materials that can be created experimentally by\nion irradiation techniques. With a minimal amount of input parameters, the scheme provides\nexcellent quantitative predictions in a large range of ion energies. The information gained from\nfirst principles helps to understand existing experimental results and notably solve the inherent\nproblem of the experimental uncertainty regarding the magnetic volume, which has been a major\nsource of controversy.\nBy revisiting experimental results from the literature of a FM phase emerging in SiC upon high\nenergy ion irradiation and comparing them to our computational predictions, we found that the\nmain process limiting the evolution of the artificial FM phase at high ion energies is the amor-\nphization of the host lattice.\nIn the case of low ion energies, sputtering of lattice atoms at the surface plays an important role\nand limits the degree of amorphization, as we demonstrated experimentally in anatase TiO 2hosts.\nWhen the defect production and sputtering processes reach an equilibrium, high defect densities\n(up to 17 at. %) can be created, which allows full magnetic percolation. We have shown that at\nlow enough ion energies ( Eion= 200 eV in the TiO 2), ultrathin ferromagnetic films down to a\nmagnetic bilayer can be created.\nIn the ultrathin artificial FM layers created in TiO 2hosts, we have investigated the magnetic\nanisotropy and showed that a perpendicular magnetic anisotropy emerges, depending on the thick-\nness of the magnetic phase. We could identify the origin of this PMA in the contribution of\nmagnetic surface states, as shown by DFT calculations.\nACKNOWLEDGMENTS\nThe authors thank A. Setzer and M. Stiller for fruitful discussions; Hichem Ben Hamed and\nWolfram Hergert for the cooperation and support. Part of this study has been supported by the\nDFG, Project Nr. 31047526, SFB 762 “Functionality of oxide interfaces”, project B1. Compu-\ntations for this work were done (in part) using resources of the Leipzig University Computing\n30Centre.\n[1] A. Hirohata, K. Yamada, Y . Nakatani, I.-L. Prejbeanu, B. Dieny, P. Pirro, and B. 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Hysteresis loops measured in the TiO 2sample S200 at T= 300 K after each Ar+irradiation step.\nThe linear diamagnetic background was subtracted.\n350.2\n0.1\n0.00.10.2\nMagnetic Field (T)6\n3\n036Magnetic Moment (nA m2)S500\nAs Grown\n1.50×1016 cm2\n2.25×1016 cm2\n3.00×1016 cm2\n4.50×1016 cm2\n6.00×1016 cm2\n7.50×1016 cm2\n7.88×1016 cm2\n8.25×1016 cm2\nFIG. 19. Hysteresis loops measured in the TiO 2sample S500 at T= 300 K after each Ar+irradiation step.\nThe linear diamagnetic background was subtracted.\n0.2\n0.1\n0.00.10.2\nMagnetic Field (T)6\n4\n2\n0246Magnetic Moment (nA m2)S1000\nAs Grown\n0.62×1016 cm2\n1.25×1016 cm2\n2.50×1016 cm2\n3.13×1016 cm2\n5.00×1016 cm2\n6.25×1016 cm2\n7.50×1016 cm2\n8.75×1016 cm2\n9.84×1016 cm2\n11.09×1016 cm2\n12.34×1016 cm2\n18.28×1016 cm2\nFIG. 20. Hysteresis loops measured in the TiO 2sample S1000 at T= 300 K after each Ar+irradiation\nstep. The linear diamagnetic background was subtracted.\n36Appendix B: Magnetic Anisotropy Energy\nFigure 21 shows the magnetic anisotropy energy (MAE) of sample S200 and S500 as a function\nof the irradiation fluence. The MAE was obtained from magnetic hysteresis curves such as those\nshown for some example fluences. Hysteresis curves were measured once with the magnetic field\napplied in-plane (blue bullets) and perpendicular to the sample surface (orange bullets). Each\ncurve was then fit using Equation (B1)\nm(B;ms;mr;Bc) =2ms\n\u0019arctan\u0014B\u0006Bc\nBctan\u0012\u0019mr\n2ms\u0013\u0015\n; (B1)\nto recover the saturation moment ms, the remanent moment mrand the coercive field Bc. The\nmagnetic energy was then calculated along the virgin curve as\nE(ms;mr;Bc) =ZBmax\n0m(B;ms;mr;Bc)dB: (B2)\nThe integration cutoff Bmaxwas set to 5 T. The MAE was then calculated as\nMAE =1\nA\u0002\nE(mk\ns;mk\nr;Bk\nc)\u0000E(mk\ns;m?\nrmk\ns=m?\ns;B?\nc)\u0003\n; (B3)\nwithAthe sample surface area. We note that the saturation and remanent moments obtained in\nthe perpendicular configuration were rescaled, so that the two saturation moments match.\n370.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0\n3.5\nIrradiation Fluence (1016cm2)\n0.075\n0.050\n0.025\n0.000MAE (mJ cm2)\nS200\n0\n2\n4\n6\n8\n10\nIrradiation Fluence (1016cm2)\n0.00\n0.05\n0.10\n0.15\nS500\n1.00\n0.75\n0.50\n0.25\n0.00\n0.25\n0.50\n0.75\n1.00\n0H(T)\n6\n4\n2\n0\n2\n4\n6Moment ( emu)\n1.00\n0.75\n0.50\n0.25\n0.00\n0.25\n0.50\n0.75\n1.00\n0H(T)\n6\n4\n2\n0\n2\n4\n6Moment ( emu)\n1.00\n0.75\n0.50\n0.25\n0.00\n0.25\n0.50\n0.75\n1.00\n0H(T)\n4\n2\n0\n2\n4Moment ( emu)\n1.00\n0.75\n0.50\n0.25\n0.00\n0.25\n0.50\n0.75\n1.00\n0H(T)\n6\n4\n2\n0\n2\n4\n6\n8Moment ( emu)\nIn-plane\nPerpendicularMagnetic FieldFIG. 21. Hysteresis loops, measured at T= 300 K with a magnetic field applied in-plane and perpendicular\nto the film surface, corresponding to the magnetic anisotropy energy (MAE) values marked by arrows.\nThe dashed lines show the fits to Equation (1) in the main text. The method to obtain the MAE from the\nhysteresis curves is described in Section V .A of the main text.\n38"    },    {        "title": "2103.09029v1.Complete_mapping_of_magnetic_anisotropy_for_prototype_Ising_van_der_Waals_FePS__3_.pdf",        "content": "Complete mapping of magnetic anisotropy for prototype \nIsing van der Waals FePS 3 \n \nMuhammad Nauman†1, Do Hoon Kiem†2, Sungmin Lee3, Suhan Son3,4, Je-Geun Park3,4, \nWoun Kang5, Myung Joon Han*2, and Younjung Jo*1 \n1Department of Physics, Kyungpook National University, Daegu 41566, Korea.  \n2Department of Physics, Korea Advanced Institute of Science and Technology (KAIST), \nDaejeon 34141, Korea.  \n3Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea.  \n4Center for Quantum Materials, Se oul National University, Seoul 08826, Korea.  \n5Department of Physics, Ewha Womans University, Seoul 03760, Korea.  \n†These two authors contributed equally to this work  \nE-mail: mj.han@kaist.ac.kr, jophy@knu.ac.kr  \n \nABSTRACT  \nSeveral Ising -type magnetic van der Waals (vdW) materials exhibit stable \nmagnetic ground states. Despite these clear experimental demonstrations, a \ncomplete theoretical and microscopic understanding of their magnetic \nanisotropy is still lacking. In particular, the validity limit of identifyi ng their \none-dimensional (1 -D) Ising nature has remained uninvestigated in a \nquantitative way. Here we performed the complete mapping of magnetic \nanisotropy for a prototypical Ising vdW magnet FePS 3 for the first time. \nCombining torque magnetometry measure ments with their magnetostatic \nmodel analysis and the relativistic density functional total energy calculations, \nwe successfully constructed the three -dimensional (3 -D) mappings of the \nmagnetic anisotropy in terms of magnetic torque and energy. The results  not \nonly quantitatively confirm that the easy axis is perpendicular to the ab plane, \nbut also reveal the anisotropies within the ab, ac, and bc planes. Our approach \ncan be applied to the detailed quantitative study of magnetism in vdW \nmaterials.  \n \nKeywords:  FePS 3, Torque measurement, Magnetic anisotropy energy, Ising -type \nmagnetic structure   1. Introduction  \n Two-dimensional (2 -D) van der Waals (vdW) materials, including those with atomic -layer \nthickness, provide an exciting platform for the study of 2-D magnetism [1 –10]. These 2 -D magnets \nare increasingly receiving attention owing to their intriguing magnetic properties and potential for \napplication as intercalation materials [11, 12] and in spintronic devices [13, 14]. Among the focuses \nin this li ne of study is magnetic anisotropy, which offers the possibility of stabilizing spin ordering \nunder circumstances that are close to the ideal 2 - D limit [6, 15 –17]. Therefore, detailed information \non magnetic anisotropy and its energy scale has been crucia l in unveiling the fundamental nature of \nmagnetism and in the vdW material based spintronics.  \nAmong the many magnetic vdW materials, the layered transition metal trichalcogenide family, \nnamely, TMPS3 (TM = Fe, Co, Mn, V, Zn, or Ni), provides a unique plat form for conducting \nfundamental studies. In these antiferromagnetic (AFM) materials, whose magnetic and \ncrystallographic structures are both 2 -D [1, 5, 18 –23], the vdW gap between the 2 -D layer facilitates \nan indirect exchange interaction along  the c-axis,  resulting in a highly anisotropic magnetic behavior. \nGiven that the anisotropy of these materials is closely related to the crystal field effect, orbital \noccupation, and symmetry [24], each of these compounds can serve as a material platform for the \nexplo ration of theoretical spin models in the 2 -D limit [25]. For example, the magnetization axes in \nFePS 3 and MnPS 3 lie perpendicular to the layer planes, while in NiPS 3 they lie within the layer plane \nin the ordered state. Further, it is believed that FePS 3, NiPS 3, and MnPS 3 effectively represent the \nIsing -, XY - (or XXZ -), and Heisenberg -type spin models, respectively [22, 26 –28]. Detailed \ninvestigations of their paramagnetic (PM) regimes have shown that the magnetic susceptibilities of \nMnPS 3, NiPS 3, and FePS 3 are isotropic, weakly anisotropic, and highly anisotropic, respectively \n[24].  \nOur current study is focused on FePS 3, which is unreactive in air; thus, is suitable for studying the \nhitherto unknown phenomena regarding 2 -D magnetism and for realizing antif erromagnet -based \nspintronic functionalities [29 -31]. However, the important details regarding its anisotropy profile \nand the energetic validity limit of its Ising nature are still unclear; i.e., the magnetic anisotropy of \nunstacked monocrystalline FePS 3 has not yet been reported, and its  quantitative Ising nature is still \nlargely unexplored. Particularly, the energy cost of the spin rotation from its easy axis to its hard \naxis alignment is still unknown. Additionally, the in -plane magnetic anisotropy energy , which \nmust be zero in an ideal Ising system, but is likely to be non -zero in practice, has not yet been \naddressed, even though such a quantitative understanding represents a crucial step in fundamental \nresearch and application. Therefore, detailed measur ements and theoretical calculations regarding \nthe trichalcogenide family as well as other 2 -D vdW systems are highly needed.  \nTo investigate the Ising magnetic anisotropy and its validity limit, angle dependent torque \nmagnetometer experiments were carried out using high -quality single crystals. Systematic \ntheoretical analysis and total energy calculation were performed based on the magnetostatic model \nconstruction and the relativistic density functional theory (DFT) calculation. We successfully \nconstructed the complete mapping of magnetic anisotropy for this prototype Ising vdW FePS 3 for \nthe first time. It is expected that the results of this study will pave the way for a better understanding \nof the quantitative aspects of similar systems.  \n  \n2. Results and discussion  \n2.1 Crystal structure, electronic configuration, and anisotropic antiferromagnetism in \nFePS 3 \nFePS 3 is an AFM Mott insulator with an optical gap of ~1.5 eV  [32–34]. Its honeycomb lattice \nstructure consists of Fe2+ [d6] ions, each of which is surr ounded by six sulfur atoms with trigonal \nsymmetry as shown in Fig. 1(a). The sulfur atoms are bonded to two phosphorus atoms, forming a \ndumbbell structure, and [P 2S6]4− provides the local octahedral environment for Fe2+. The elongated \noctahedron induces th e splitting of t2g orbitals into egπ and a1g, and the high spin configuration is \nstabilized. FePS 3 has the ‘zigzag -stripe’ AFM spin ordering; see Fig. 1(b)  [21, 24, 26, 35 –37]. This \nAFM ground state is more stable than ferromagnetic phase by 31.7 meV/Fe2+ (~ 127 T/Fe2+). It has \nattracted significant attention owing to the presence of the same order down to the monolayer limit  \n[26, 37] . Neutron single -crystal diffraction measurements have shown that the moment of Fe atom \nFigure 1 (a) Local atomic structure around Fe atoms and the spin configuration. The yellow, bright violet, and brown \nspheres represent the S, P, and Fe atoms, respectively. The trigonal distortion of the S 6 octahedra around the Fe atoms \nis depicted by black arrows. Honeycomb layers of [P 2S6]4− and Fe2+ alternate along the c-axis. (b) The ground -state \nspin order of FePS 3. (c) The measured magnetic susceptibility as a function of temperature. The black, red, a nd blue \ncurves correspond to the out -of-plane susceptibility, in -plane susceptibility, and their absolute difference, \nrespectively. The inset shows the definitions of axes and field angle rotations. The c-axis is inclined at 108.37°  with \nrespect to the a-axis, and normal to the b-axis. c*- is perpendicular to the ab plane. (d) The experimental torque \nmagnetometer data for various temperatures at H = 5 T. The field rotates out -of-plane. (e) The calculated torque , \n𝝉𝒂𝒄(𝜽), by using Eq. (2).  is ferromagnetically coupled with tho se of the two atoms that are closest to it and \nantiferromagnetically coupled with that of the third. Also, the previous neutron scattering and \nMössbauer spectroscopy showed the Ising nature of the spins in this material  [21, 38, 39]  and the \nmagnon data fit ting on to the spin model found the large magnetic anisotropy  [40].  \n2.2 3-D mapping of anisotropic torques and theoretical interpretation   \nFig. 1(c) shows our magnetic susceptibility data of FePS 3 as a function of temperature. The black \nand red dots represent the measurements along the out -of-plane and the in -plane directions, \nrespectively. The PM -to-AFM phase transition is marked by the supp ression of the susceptibility \nsignals at TN = 118 K. Part icularly, the suppression of the out -of-plane susceptibility ( 𝜒⊥ or 𝜒c) is \nsignificantly more pronounced than that of the in -plane susceptibility ( 𝜒∥ or 𝜒𝑎𝑏). As discussed in \nprevious studies  [24, 26], together with data from Mössbauer, and neutron s cattering  [21, 38, 39], \nthis behavior can be regarded as key evidence for Ising nature of the spins in this material, which \nhas an easy axis along the c*-direction (defined as perpendicular to the ab plane; se e inset of Fig. \n1(c)). Given that the 𝜒⊥ suppression is stronger than the 𝜒∥ suppression, this eventually results in the \nmagnitude  reversal of these susceptibility components, i.e., 𝜒⊥ < 𝜒∥ at T ≤ 108 K. As discussed \nbelow, it generates the intriguing behaviors in our torque measurements.  \nImpo rtant details regarding magnetic anisotropy are revealed via angle -dependent torque \nmeasurements. Fig. 1(d) shows the torque, 𝜏(𝜃), that was measured by changing the azimuthal field \nangle relative to the c*-axis (see the inset of Fig. 1(c)). In this measur ement, the applied magnetic \nfield of H = 5 T was lying within the ac plane. At temperatures below TN, 𝜏(𝜃) exhibits such \nbehavior that its amplitude gradually decreases with increasing temperature. Remarkably, the sign \nof the amplitude  of 𝜏(𝜃)  is suddenly reversed when the temperature changes from  110 K to 120 K, \nand at T = 120 K the amplitude  reach es to the maximum.  Note that this sudden change observed \nacross TN is indicative of the response of the mechanical torque to the magnetic transitio n. \nTo understand this behavior as well as its physical implications, a macroscopic magnetostatic \nmodel was constructed. We note that, in the absence of an external field ( H = 0), the net magnetic \nmoment is zero in both AFM and PM phases. However, there is a key difference between these two \nphases. In the PM phase ( T > TN), spins have the preferred direction along the c*-axis while they are \nnot ordered. Thus, the external field induces a moment along the easy axis direction ( c*-axis; θ = 0°) \nby aligning the originally disordered spins. On the other hand, in the AFM phase ( T < TN), the up \nand down spins are paired with each other, resulting in a zero net moment. The field induces the \nmoment most likely when it is perpendicular to the eas y axis (i.e., θ = 90°) given that the H-field \nstrength is much smaller than the AFM pairing strength. Namely, the net moment in this case is \ninduced mainly via spin tilting. This difference in the direction of the induced moment between the \nPM and AFM phas es can be responsible for the 90° phase shift observed in Fig. 1(d). Taking this \npicture as our working hypothesis, we express the torque induced by the external magnetic field,  𝐻⃗⃗ , \nas follow:  \nτ⃗ =(𝜒⃡⋅𝐻⃗⃗ )×𝐻⃗⃗ ,       (1) \nwhere τ⃗  represents the torque per unit v olume on a system, and 𝜒⃡ represents the magnetic \nsusceptibility tensor. For the moment, we assume that 𝜒⃡ has zero off -diagonal components owing to \nthe crystal symmetry, with only diagonal components contributing to the torque. We will see that \nthis assum ption is useful for understanding most of our experiments. Later, however, this assumption \nwill be refined for understanding the detailed in -plane anisotropy (see below). Now, the directional \ntorques read:  𝜏𝑎𝑐(𝜃)=1\n2(𝜒𝑐𝑐−𝜒𝑎𝑎)𝐻2sin2𝜃     for 𝜙=0°  (2) \n𝜏𝑏𝑐(𝜃)=1\n2(𝜒𝑏𝑏−𝜒𝑐𝑐)𝐻2sin2𝜃     for 𝜙=90°  (3) \n𝜏𝑎𝑏(𝜙)=1\n2(𝜒𝑎𝑎−𝜒𝑏𝑏)𝐻2sin2𝜙   for 𝜃=90°  (4) \nwhere 𝜏(𝜃) and 𝜏(𝜙) represent the torque as measured using the rotation of the magnetic field in \nthe in -plane and out -of-plane field directions, respectively. Note that we distinguish two out -of-\nplane field directions explicitly. 𝜒𝑎𝑎, 𝜒𝑏𝑏, and 𝜒𝑐𝑐 represent each dir ectional component of the \nmagnetic susceptibility tensor, and 𝜒𝑎𝑎,𝑏𝑏 and 𝜒𝑐𝑐 corresponds to 𝜒∥ and 𝜒⊥, respectively. Using the \nmeasured susceptibility as inputs (i.e., 𝜒∥ and 𝜒⊥ as shown in Fig. 1(c)), it is straightforward to \nobtain 𝜏(𝜙) and 𝜏(𝜃) from Eq. (2) –(4). Fig. 1(e) shows the calculation result of the out -of-plane \ntorque, 𝜏𝑎𝑐(𝜃), which is in excellent agreement with Fig. 1(d). This result confirms the relevance of \nour model and simultaneously reveals the detailed magnetic behavior of this material along the out -\nof-plane field direction. In particular, one can conclude that the sudden phase change of 𝜏(𝜃) across \nT = TN is attributed to a 90° change in the induced -moment direction.  \nFig. 2(a) shows the in -plane anisotropy revealed by 𝜏𝑎𝑏(𝜙), i.e., the torque measured via the \nvariation of the in -plane magnetic field at T = 50 K < TN. Emphatically, what we  measured here \nshows the (Ising) spin anisotropy profile in the plane perpendicular to the spin direction. Compared \nwith the out -of-plane torque, 𝜏𝑎𝑐(𝜃), shown in Fig. 1(d), 𝜏𝑎𝑏(𝜙) is order -of-magnitude smaller as \nexpected. Importantly, however, it i s still  clearly non -zero. Once again, the overall behavior of \n𝜏𝑎𝑏(𝜙) can be well understood based on o ur model. The red lines in Fig. 2(a) represent the curves \nobtained from Eq. (4), which are in good agreement with the measured data shown in black. Th e \ndata also show that, for 0°  ≤ 𝜙 ≤ 90°, 𝜏(𝜙) is positive  and 𝜒𝑎𝑎 > 𝜒𝑏𝑏. Given that the torque is \nexerted to minimize the magnetostatic energy, the results show that the induced moment favors the \nalignment along the b-axis rather than the a-axis wit hin the plane. This point will be quantitatively \nconfirmed by our DFT total energy calculations (see below).  \nFigure 2 (a) The measured torque,  𝜏𝑎𝑏(𝜙) (black) along with the fitting lines (red) as functions of the in -plane field \nrotation angle, ϕ, and the field strength. The temperature was set at 50 K. (b) The out -of-plane torque at temperatures \nbelow (solid symbols; left -hand Y -axis; T = 100 K) and ab ove (open symbols; right -hand Y -axis; T = 120 K) TN. The blue \nand red curves correspond to the measured torque in the field H = 5 T on the ac ( ϕ=0°) and bc (ϕ=90°) planes, \nrespectively.  In Fig. 2(b), 𝜏𝑎𝑐(𝜃) (i.e., the field rotation within the ac plane; 𝜙 = 0°) is compared with 𝜏𝑏𝑐(𝜃) \n(i.e., the field rotation within the bc plane; 𝜙 = 90°). Here, the blue (red) color represents the data \ncorresponding to 𝜙 = 0° (90°), and the filled (open) symbols refer to the results of T = 100 K (120 \nK). In 𝜏𝑎𝑐(𝜃) and 𝜏𝑏𝑐(𝜃), the sign changes are clearly observed below and above T = TN (i.e., \ncomparing the filled vs. open symbol data), indicating the tilting of the effective moment by 90° as \ndiscussed above. It is observed that the amplitude of the bc-rotation (in red) is always smaller than \nthat of the ac-rotation (in blue); |𝜏𝑏𝑐(𝜃)| < |𝜏𝑎𝑐(𝜃)|. This finding indicates, as illustrated in Eq. \n(2)–(3), that |𝜒𝑏𝑏−𝜒𝑐𝑐| < |𝜒𝑎𝑎−𝜒𝑐𝑐|. By combining the susceptibility results with this out -of-\nplane torque measurement, it follows that χ𝑎𝑎>𝜒𝑏𝑏>𝜒𝑐𝑐 for T < TN, and χ𝑎𝑎<𝜒𝑏𝑏<𝜒𝑐𝑐 for T > \nTN. It constitutes further evidence that the b-axis is the favored spin alignment direction within the \nplane, while the overall easy axis is aligned along the c*-axis direction.  \n2.3 Further details of magnetic anisotropy – the nonlinear  off-diagonal response and the \ncrystalline symmetry  \nAn important feature revealed in Fig. 2(b) is that, for the ac-rotation ( 𝜙 = 0°; blue circles), the \npositive and negative amplitudes are significantly different, i.e., the crests of the sinusoidal motion  \nof the filled -blue curve reach up to +5.3, while its troughs reach only −4.3. A similar observation is \nmade for the open -blue curve, ranging from −18.9 to +24.7. On the other hand, for the bc-rotation \n(𝜙 = 90°; red circles), the filled -symbol and empty -symbol curves are symmetric in the positive and \nnegative planes. To further elucidate this asymmetry, 𝜏𝑎𝑐(𝜃) was examined by varying the field \nFigure 3 (a) The out -of-plane torque, 𝝉𝒂𝒄(𝜽) (black) measured with the rotating field in the ac plane at various field \nstrengths and at a fixed temperature, T = 50 K. The red lines represent the fitting from the theoretical model of Eq. \n(2). (b) The field -dependent torque measured at various field o rientations. The numbers show the field angles θ in \nthe ac plane as defined by the inset of Fig. 1(c). Temperature was fixed at T = 5 K. (c) The calculated torque, 𝝉𝒂𝒄(𝜽), by \nthe updated formula of Eq. (6). (d) The calculated field -dependent torque bas ed on Eq. (6). The off -diagonal \ncomponents were fitted by 𝝌𝒄𝒂𝟎=𝟎.𝟐×𝝌𝒂𝒂, 𝝌𝒂𝒄𝟎=𝟎.𝟒×𝝌𝒂𝒂. strength at T = 50 K. The results are presented in Fig. 3(a). For a weak field H < 4 T, the measured \n𝜏𝑎𝑐(𝜃) is consistent with the sin2𝜃 curve (red lines). Conversely, for a strong field, H > 4 T, the \npositive values of 𝜏𝑎𝑐(𝜃) are clearly greater than its negative values, indicating that rotating the \nsystem is easier in the range 90° <  𝜃 < 180° (or, e quivalently, 0° < 𝜃 < −90°) than in the range 0° < \n𝜃 < +90°. The same asymmetry is also evident with the field -dependent torque, 𝜏(𝐻), as illustrated \nin Fig. 3(b), which shows that 𝜏(𝐻) follows the parabolic lines in positive and negative  𝜃s at low \nfield. Under a high field, on the other hand, 𝜏(𝐻) significantly d eviates from the parabola and \nappears saturated only at positive  𝜃.  \nWe found that this observed asymmetry carries a significant physical meaning by reflecting the \ncrystalline symmetry of this material. In the model described above, it was assumed that all the off -\ndiagonal components of susceptibility are zero. We first not ed that the asymmetries shown in Fig. \n2(b) and Fig. 3(a) and (b) are indicative of a limit that violates this assumption. Thus, we presume \nthat the asymmetry originates from the nonlinear off -diagonal components of the magnetic \nsusceptibility. Actually, th is non -zero off -diagonal contribution can be expected from such a crystal \nFigure 4  (a)–(c) The crystal structure of FePS 3 as viewed from the direction perpendicular to the (a) ab, (b) bc, and \n(c) ac planes. The solid -black and the magenta -dashed lines indicate the unit cell and the mirror plane, respectively. \nThe gray circular arrows show the spin rotating direction in the D FT calculation for each plane. (d) –(f) The calculated \nmagnetic anisotropy energy profile calculated with the rotating angles of AFM spin alignment for (d) ab-, (e) bc-, and \n(f) ac-rotation. The red arrows in the left panels indicate the angles correspondin g to the specific crystallographic \naxes. The right panels show that calculated total energies in polar coordinates. The scales of radial axes are same with \nthe left panels.  The maximum energy angle in (f) corresponds to 127 °, and it is close to the directi on pointing to the S \nions which are a part of the octahedron cage surrounding Fe ions.   symmetry. Fig. 4(a) –(c) show the  crystal structure viewed from the direction perpendicular to the (a) \nab, (b) bc, and (c) ac planes, respectively. The magenta dashed lines represent the mirror plane. The \nmirror plane is well defined in the ab- and bc-projections, but not in the ac plane which possesses \nonly C 2 rotational symmetry. Owing to the mirror symmetry plane parallel to the ac plane, 𝜒𝑎𝑏=\n𝜒𝑏𝑎=𝜒𝑏𝑐=𝜒𝑐𝑏=0. However, in t he ac plane, the C 2 rotational symmetry without mirror \nsymmetry can lead to non -zero off -diagonal components of susceptibility; namely, 𝜒𝑎𝑐≠0 and \n𝜒𝑐𝑎≠0. Considering this point, Eq. (2) c an be updated as follows:  \n𝜏𝑎𝑐=(𝜒𝑐𝑐−𝜒𝑎𝑎)𝐻𝑎𝐻𝑐+𝜒𝑐𝑎𝐻𝑎2−𝜒𝑎𝑐𝐻𝑐2,   (5) \nwhere 𝐻𝑎 and 𝐻𝑐 indicate the magnetic field components parallel to the a- and c-axes, \nrespectively. This relationship is directly deduced from Eq. (1) with non -zero off -diagonal terms. If \nwe further suppose that these off -diagonal components are nonlinear as 𝜒𝑐𝑎=𝜒𝑐𝑎0𝐻𝑐2 and 𝜒𝑎𝑐=\n𝜒𝑎𝑐0𝐻𝑎2, Eq. (5) then becomes:  \n𝜏𝑎𝑐=1\n2(𝜒𝑐𝑐−𝜒𝑎𝑎)𝐻02sin2𝜃+1\n4(𝜒𝑐𝑎0−𝜒𝑎𝑐0)𝐻04sin22𝜃,    (6) \nwhere 𝜒𝑐𝑎0 and 𝜒𝑎𝑐0 are constant. The calculated torque in Eq. (6) is plotted in Fig. 3(c) and (d) \nwhich are in good agreement with the experimental results shown in Fig. 3(a) and (b), resp ectively. \nOur experimental results together with theoretical interpretations demonstrate the capability of \ntorque magnetometry in revealing the detailed crystalline as well as magnetic anisotropy, which can \nbe generally applicable to other material systems . Hereby we constructed the 3 -D mapping of \nmagnetic anisotropy in terms of torque.  \n \n2.4 Energetics of ma gnetic anisotropy - DFT calculation  \nTheoretically, magnetic anisotropy can also be revealed by performing total energy calculations \nwithin the relativistic non -collinear DFT scheme. Fig. 4 summarizes our results for various spin \ndirections in which we rotate the spin alignment angles while k eeping the AFM order. The \ncomputation details are given in the Methods section. Fig. 4(d) –(f) shows the magnetic anisotropy \nenergy profiles as a function of angle changes within the ab, bc, and ac planes, respectively. From \nFig. 4(d), it is evident that, w ithin the ab plane, the spins align preferably along the b-axis direction \nrather than along the a-axis direction, in agreement with our torque measurements. Additionally, Fig. \n4(e) and (f) show that, in the bc and ac planes, the preferred direction corresp onding  to the minimal  \nrelative  energy  is along the c*-axis. This finding constitutes a direct and independent confirmation \nof the experimentally known magnetic easy axis direction  [21, 24, 26, 39]  in terms of energy. The \nenergetic order of the spin alignme nt direction obtained via the DFT calculations is E //c < E //b < E //a. \nAgain, this finding is consistent with the conclusion stated above regarding the torque measurements \nat T < TN. Therefore, the DFT calculations not only confirm the experimental results,  but also reveal \nthe quantitative energetics of the magnetic anisotropy of this material for the first time. The energy \nprofile is depicted in polar coordinates as shown in the right panel of Fig. 4(d) –(f). A peculiar feature \nof these figures is the non -sinusoidal shape of the anisotropy energy observed only for the ac-rotation. \nThe total energy graph shows a minimum at a rotation angle of approximately 90°. Additionally, the \nincreasing behavior of the relative energy from 𝜃 = 0° to ~60° is not symmetric with the decreasing \nbehavior from 𝜃 = ~120° to 180°. Thus, the magnetic anisotropy energy profile calculated on the ac \nplane exhibits only C 2 rotational symmetry. This mechanism also naturally explains the asymmetric \ntorque observed in the ac-rotation as shown in Fig. 3.  \n3. Conclusions  The detailed magnetic anisotropy in the full 3 -D space and in terms of torque and energy have \nbeen quantitatively revealed for vdW FePS 3. The good agreement between our results of torque \nmeasurement, model an alysis, and DFT total energy calculations is impressive especially \nconsidering the small energy scale of the magnetic anisotropy energy. The observed anisotropy is a \nconcerted result of numerous factors including the local crystal field effect, orbital occ upation and \nshape, crystalline anisotropy, and the long -range pattern of the spin order. Our approach can be \nextended to other vdW magnets for the investigation of their detailed magnetic behaviors. vdW \nmaterials can host various magnetic configurations an d related phenomena such as pressure -driven \nspin-crossover, semiconductor -to-metal transition, and superconductivity  [41]. Given that the \nchange in the magnetic field direction is the key to control magnetism, the intimate coupling of the \nmagnetic anisotro py to the induced phase transitions will enrich the material candidates as well as \ntheir spin functionalities for all vdW material -based spintronics.  \n \n4. Methods  \n4.1 Single -crystal growth and characterization  \nSingle -crystal FePS 3 was synthesized via chemic al vapor transport. The starting materials were \nsealed in a quartz ampule at a pressure below 10−2 Torr. A horizontal two -zone furnace was used, \nwith the temperatures set to 750 and 730 °C for the hot and cold zones, respectively; these \ntemperatures were m aintained for 9 days. The stoichiometry of each single crystal was confirmed \nusing a scanning electron microscope (COXI EM30, COXEM) equipped with an energy -dispersive \nX-ray spectrometer (Quantax 100, Bruker). The quality and orientation of each sample wer e \ncharacterized using two X -ray diffractometers: a Laue diffractometer (TRY -IP-YGR, TRY SE) and \na single -crystal diffractometer (XtaLAB P200, Rigaku). All the samples exhibited C2/m symmetry. \nThe crystallographic directions were distinguished, and the axes  were labeled: a-, b-, and c*-axis. \nSusceptibility was measured in a magnetic field of 1 T applied parallel to the c*- or a-axis using a \ncommercial magnetic property measurement system (MPMS, Quantum Design). The single -crystal \nFePS 3 was mounted on a piezo resistive cantilever such that the torque generated by the \nmagnetization was determined from the change in the resistance of the piezo material. The change \nin resistance was obtained using a Wheatstone bridge circuit.  \n4.2 First -principles DFT calculations  \nDFT calculations were performed within the generalized gradient approximation (GGA)  [42] for \nthe exchange -correlation potential. Projector -augmented wave potentials were used  [43] as \nimplemented in the VASP (Vienna Ab initio simulation package) code  [44]. The wave functions \nwere expanded with plane waves up to an energy cutoff of 450 eV, and gamma -centered 9×9×12 k -\nmeshes were adopted. The known ground state of the zigzag -striped AFM unit cell was obtained \nfrom the experimental crystal structure  [21]. To p roperly describe the on -site electronic correlation \nwithin Fe 3 d orbitals, the DFT+ U method was adopted with a charge -density -based scheme  [45–50]. \nU = 6.80 eV and J = 0.89 eV were used  [37]. To calculate the magnetic anisotropy energy, the spin –\norbit inte raction was taken into account within the relativistic non -collinear scheme as implemented \nin the VASP package. In this scheme, the wave functions and the charge densities of the zigzag -\nstriped AFM FePS 3 were obtained in a collinear DFT scheme and then wer e used to compute the total energy as a function of the spin angle rotation. In this procedure, the size of the moment was \nfixed, and the collinear AFM coupling was retained to align along the given orientation.  \n \nAcknowledgements  \nY.J. was supported by the National Research Foundation of Korea (NRF) grant by the Korea \ngovernment (MSIT) (Nos. NRF -2018K2A9A1A06069211 and NRF -2019R1A2C1089017). M.J.H. \nwas supported by the National Research Foundation of Korea (NRF) grant funded by the Korea \ngovernment (MSIT) (N o. 2018R1A2B2005204 and NRF -2018M3D1A1058754). This research was \nsupported by the KAIST Grand Challenge 30 Project (KC30) in 2020 funded by the Ministry of \nScience and ICT of Korea and KAIST (N11200128). W.K. acknowledges the support by the NRF \ngrant (Nos.  2018R1D1A1B07050087, 2018R1A6A1A03025340).  Work at the Center for Quantum \nMaterials and SNU was supported by the Leading Researcher Program of the National Research \nFoundation of Korea (Grant No. 2020R1A3B2079375). A portion of this work was performed at the \nNational High Magnetic Field Laboratory, which is supported by National Science Foundation \nCooperative Agreement No. DMR -1644779* and the State of Florida.  \n \nAuthor contributions  \nM. N., W. K. and Y. J. performed the torque measurement. D. H. K. and M. J . H. conducted the \ntheoretical analysis and calculation. S. L., S. S. and J. G. P. synthesized the single crystals. M. J. H. \nand Y. J. supervised the project. M. J. H. and Y. J. wrote the manuscript together with D. H. K and \nall the other authors. 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Rodrigues\nIDMEC, Instituto Superior T´ ecnico, University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon ,\nPortugal\n(Dated: 31 March 2021)\nElectrical control of magnetization or magnetic control of polariz ation offers an extra degree of\nfreedom in materials possessing both electric and magnetic dipole mom entsviz.,magnetoelectric\nmultiferroics. Microstructure with polycrystalline configurations t hat enhances the overall polariza-\ntion/magnetization and that outperform single crystalline configur ations are identified. The char-\nacterization of local fields corresponding to the polycrystal confi guration underlines nontrivial role\nplayed by randomness in better cross-coupling mediated by anisotr opic and asymmetric strains.\nThe magnetoelectric (ME) effect manifests in the\nlinear relation between the magnetic and electric\nfields in matter and it causes, for instance, either a\nmagnetizationorelectricalpolarizationproportional\nto applied electric/magnetic fields1. Yet, the ME\neffect is possible only for certain magnetic symme-\ntry classes, akin to piezomagnetism. Set of mate-\nrials classified as magnetoelectric multiferroics pos-\nsess both the magnetic and ferroelectric orders in\nthe same phase2. Apart from exhibiting function-\nalities of both the orders, a coupling between the\nferri-/ferro-magnetic (FM) and ferroelectric (FE)\nstates enable appearance of novel characteristicsnot\npresent in either of the states3. Nonetheless, simul-\ntaneousoccurrenceofmagnetismandferroelectricity\nin materials is constrained by the conflicting classic\nchemical requirements regarding electronic orbital\noccupancy4,5. In such a situation, one should envis-\nage only a moderate ME coupling though6. For in-\nstance, only few examples exist of materials possess-\ning intrinsic ME coupling at room temperature7,8.\nAnunambiguousresolutiontocircumventthischem-\nical ’contraindication’ is by juxtaposing an FE and\nFM material artificially into strain-mediated multi-\nphase materials such as composites combining the\ntwo phases, that thereby yields substantial ME\neffect5,6,9. Moreover, the composites do not need to\nadhere to the symmetry restrictions of single phase\nmaterials and consequently do possess the freedom\nof choice from a wide variety of ferroics existing\nabove room temperature. Crystallites of the two\nphases in an ME composite are assumed to be in\ngood mechanical contact if the two phases are poly-\ncrystalline. The change in shape of the FE grains in\nresponse to an applied electric field causes the ferro-\na)Electronic mail: kpjayachandran@gmail.commagnetic grains to deform, resulting in a change in\nmagnetization10.\nMagneto-elastic effect by means of strain transfer\nacross the interface of an FM/nonmagnetic bilayer\ncould unleash significant changes in the magnetic\nproperties of the ferromagnet. When an FE mate-\nrial is used as the nonmagnetic layer, strain transfer\ncan be reinforced through the induction by an elec-\ntric field thanks to the inverse piezoelectric effect\nprevails in the FE material. This additional strain\ntransfer allows one to manipulate the magnetisation\nby an electric field11. The concept of magnetic in-\nformationrecordingandmemorydevicetechnologies\nas well as electric-field driven magnetoelastic effects\non magnetic anisotropy have attracted much inter-\nest recently from both fundamental and technolog-\nical aspects owing to the said modification accom-\nplished on magnetisation orientation5,11,12. Among\nmany FM/FE hybrid structures, perovskite man-\nganites/BTOarethe well-suitedmagnetoelastichet-\nerostructures since transport, magnetic, and elec-\ntronic properties of perovskite manganites are very\nsensitive to lattice strain and can be easily in-\ntegrated with BaTiO 3(BTO) due to the similar\ncrystal structure13. For instance when an electric\nfield is applied to the nanostructured BTO/CFO\n1–3 composite, the strain created by the piezoelec-\ntric (BTO) matrix couples to the magnetostrictive\nCFO(CoFe 2O4) pillars and hence the strength of the\ncoupling between the ferroelectricity and magnetism\nare critically dependent on the anisotropy and crys-\ntalline structure of the constituents14,15.\nIn this letter, we would be studying the impact\nof an applied electrical/magnetic fields on a mot-\nley ME composite composed of disparate FE and\nFM materials. We underscore the significance of\nheterogeneity of one of the constituent phases, viz.,\nthe FE phase by introducing randomness in the mi-xyz\ny1θ\ny2y3\nψ\nφ\ny2y3Ba\nxyz\nO\nTi\ny1FM Fibre\nFE Matrix\nFIG. 1. Schematic hierarchical diagram of the ME multiferro ic microstructure browsing through the succession of\nmodeling tasks in the study. The microstructure Yis a fibre-reinforced ME composite where an FM fibre made\nof CoFe 2O4is embedded in a perovskite FE matrix of BaTiO 3. BaTiO 3is a polycrystal – an aggregate of grains\nof various orientations (depicted in different colours). Th e orientation of the grains quantified by the Euler angles\n(φ,θ,ψ) is in fact that of the underlying crystallographic unit cel l and is measured against the microscopic ( y1,y2,y3)\ncoordinate system.\ncrostructure. Prior research have shown that intro-\nducing heterogeneity locally can improve the piezo-\nelectric properties of FE materials and the ensuing\ncoupling effect of the ME composite of which the\nformer is a component16,17. Apart from the effect\nof applied fields, the sway of crystal anisotropy on\nthe polarization/magnetization as demonstrated in\nvarious studies18–21on magnetoelectric heterostruc-\ntures is also shown. Another important challenge is\nto identify the optimum texture and stress level and\nefficient controlofthe sameto achievetargetedmag-\nnetoelectric properties22. We demonstrate the effect\nof local texture in a 1–3 composite, tantamount to\nthe architecture used in Zheng et al.,14to map its\nlocal field distributions upon applying an external\nbias field. Moreover, the average polarization and\nmagnetization post application of ample bias fields\nsufficient to saturate the composite would be com-\nputed.\nEstimation of the equilibrium macroscopic\nmagneto-electro-electric properties of the ME\ncomposite,have been done using the mathematical\nhomogenization method23,24. A ME multiferroic\ncomposite occupying a volume Ω of coordinates x\n(orxi) is considered and the material properties are\nassumed to change periodically and that period ǫ\nis characterized by the dimension of an elementary\ncell Y of coordinates y(oryi) of the body (Fig. 1).\nThis lead us to examine fields for unknown physical\nquantities in the form of two-scale asymptotic\nexpansion in powers of ǫas\nζǫ(x) =ζ0(x,y)+ǫζ1(x,y),withy=x/ǫ(1)and the equivalent macroscopic behaviour is esti-\nmated to be first order by the behaviour of ζ0. Here\nζstands in lieu of the fields viz.,u, the displace-\nment,ϕandψ, the scalarelectrical/magneticpoten-\ntials respectively of a multiferroic composite25. Here\nthe functions ζ1(≡u1,ϕ1orψ1) are the local vari-\nations (or field perturbations ) describing the hetero-\ngeneouspartofthe solutionsand areassociatedwith\ny=x/ǫ. Applying calculus of variations, utilising\nthe asymptotic Eq. (1) and its derivatives, the ho-\nmogenization method essentially culminates in the\ncharacterization of effective magneto-electro-elastic\nmoduli whenthe characteristiclength ǫofthe period\ntends to zero. The electrical, magnetic and mechan-\nical fields ( degrees of freedom ) govern the constitu-\ntiveequationsofthelinearMEmultiferroicsolid(see\nSupplementary material). The model quantifies all\nthe local fields and potentials besides stress ( σ) and\nstrain (ε) fields, magnetization (via magnetic flux\ndensity,B), polarization (via electric displacement,\nDas well as von-Mises stress upon the application\nof an external field (either electric/magnetic or me-\nchanical). The macroscopic averages /angbracketleftσij/angbracketright,/angbracketleftDi/angbracketright,\nand/angbracketleftBi/angbracketrightcan be computed once the homogenized\nsolution for u0,ϕ0andψ0and that of the corre-\nsponding fields viz.,∂u0\nj(x)\n∂xk,∂ϕ0(x)\n∂xjand∂ψ0(x)\n∂xjare\nprescribed (see Supplementary material). This pos-\ntulateis equally applicable for the cases when mag-\nnetic field H= 0, the applied macroscopic electric\nfield/angbracketleftEi/angbracketrightgenerates average magnetization1\n/angbracketleftMk/angbracketright=/tildewideαik/angbracketleftEi/angbracketright (2)\nand when electric field E= 0, the applied macro-\n2scopic magnetic field /angbracketleftHi/angbracketrightgenerates average electri-\ncal polarization1\n/angbracketleftPi/angbracketright=/tildewideαik/angbracketleftHk/angbracketright (3)\nThe FE matrix of the ME composite is composed\nof polycrystalline BaTiO 3. The orientation of the\nBaTiO 3crystallographic grains in the composite is\ncharacterized using Euler angles. It is seen from the\nEuler angles ( φ,θ,ψ) (Fig. 1) that it measure the\nrotation of the crystallographic coordinates (x, y,\nz) with reference to the microstructure coordinates\n(y1,y2,y3). The three Euler angles are modelled as\nstatistically distributed in a normal distribution af-\nter poling obeying the probability distribution func-\ntionf(α|µ,σ) = 1/(σ√\n2π)exp−[(α−µ)2/2σ2].\nHereαis the random variable representing the ori-\nentations (i.e., the Euler angles) ( φ,θ,ψ) andµand\nσare the mean and the standard deviation, respec-\ntively. The fibres embedded in the matrix is com-\nposed of the FM material CoFe 2O4which is treated\nas bulk material without assigning any orientation\nwhatsoever. The numerical model developed is im-\nplemented in finite element method. The conver-\ngence of piezoelectric properties with unit cell size\nallowsustodeterminethesimulation-spaceindepen-\ndent, macroscopicmagnetoelectricproperties atvar-\nious distribution of grains (See Supplementary ma-\nterial).\nThe averagemagneticpropertyresponsealongthe\ny3–axis of the local coordinates (which was set to\nalign along the normal to the composite plane) con-\nsequent to an external biasing electrical field is stud-\nied first. Here we apply an electric field ( E3)\nwhich suffices to saturate the polycrystalline BTO–\nCFO composite. The results are summarised in\nFig. 2. In general, the average fields acting through\nout-of-plane (the components along y3–axis) to the\ncomposite plane are greater in magnitude than the\nother components. Unlike the common perception\nof great performance of single crystal piezoelectrics,\nwe see much better magnetization /angbracketleftM3/angbracketrightwhile the\nBTO is still polycrystalline (Fig. 2 and Table I).\nIn Fig. 2 the polycrystalline data of the averages\nare differentiated from single crystal by painting the\nsingle crystal region by a shade. Randomness, intro-\nduced by local microstructural heterogeneity could\npotentially enhance piezoelectricity in relaxor fer-\nroelectric ceramics16. Here, we have incorporated\nrandomness in the ferroelectric BTO phase through\nEuler orientations of the constituent grains kept at\na normal distribution but with a standard devia-\ntion ofµrad. The maximum magnetization /angbracketleftM3/angbracketrightTABLE I. Average values of magnetization /angbracketleftM3/angbracketright(in\nA/m) at constant external electric field, and electric\npolarization /angbracketleftP3/angbracketright(C/m2) at external magnetic field of\n1–3 magnetoelectric composite BaTiO 3(single and poly-\ncrystalline)–ceramic CoFe 2O4.\nBTO phase OrientationaAverages\nor ODPb/angbracketleftM3/angbracketright /angbracketleftP3/angbracketright\nSingle crystalline(0,0,0) 2.63 ×1040.03\n(0,π/4,0) -5.48 ×104-0.07\n(0,π/2,0) -1.39 -2 ×10−6\nPolycrystalline(0,1\n2)/bardbl(0,1\n2)/bardbl(0,1\n2) 1.38×1050.17\n(0,1)/bardbl(0,0)/bardbl(0,0) 2.47 ×1040.03\n(0,0)/bardbl(0,1)/bardbl(0,0) 1.3 ×1050.16\n(0,0)/bardbl(0,0)/bardbl(0,1) 2.47 ×1040.03\n(0,1)/bardbl(0,1)/bardbl(0,1) 1.49 ×1050.19\n(0,π)/bardbl(0,π)/bardbl(0,π) 1.46×1040.02\nExperimentc– ≈3.5×105≈0.23\nExperimentd– ≈3.15×105≈0.1\naEuler angles ( φ,θ,ψ) (rad) of rotation in single crystal\nBTO phase\nbOrientation Distribution Parameters\n(µφ,σφ)/bardbl(µθ,σθ)/bardbl(µψ,σψ) (rad) of polycrystal BTO\nphase\ncCoFe2O4nanopillars embedded BaTiO 3matrix in a 1–3\ncomposite from Ref. [14].\ndCoFe2O4–BaTiO 31–3 nanocomposite from Ref. [26].\nis seen for BTO polycrystal phase with orienta-\ntion distribution parameters ( µφ= 0,σφ= 1,µθ=\n0,σθ= 1,µψ= 0,σψ= 1). Each grain orien-\ntation would be dissected and kept separately and\nthe grains are allowed to chose the combination of\nthree (φ,θ,ψ) but dictated by the corresponding\nmean (µ) and standard deviation ( µ). The value\nof/angbracketleftM3/angbracketright= 1.49×105(A/m) obtained here compares\nwith the order of /angbracketleftM3/angbracketrightmeasured by Zheng et al.,14\non nanostructures with vertically aligned CoFe 2O4\nnanopillars embedded in BaTiO 3matrix. Zheng et\nal. value is the saturation magnetization ( Ms) re-\nsponse against an applied magnetic field (hysteresis)\nofthe compositeratherthanthe crosscouplingmag-\nnetization value as is obtained in the present work\n(Table I) and hence the deviation is obvious. Epi-\ntaxial CFO films on BaTiO 3single crystal substrate\nshows/angbracketleftM3/angbracketright ≈2.5×105(A/m)27under a magnetic\nfield.\nThe property variation at various configurations\n(single crystalline data in shaded area) consequent\nto the application of external magnetic field H3are\ngiven in Fig. 3. Here the electric field is off and\nhence the electrical responses are dictated solely by\nthe piezomagnetic effect of the CFO component of\n3FIG.2. Averagefieldsforthemultiferroic ME 1–3composite B TO–CFO. (a)shows theaveragedielectric displacement\n/angbracketleftDi/angbracketright. (b)istheaveragemagnetic fluxdensity, /angbracketleftBi/angbracketright(c)shows theaveragemagnetization /angbracketleftMi/angbracketrightand(d)showtherelevant\ntensor components of the average stress /angbracketleftσIJ/angbracketrightupon applying an external electric field /angbracketleftE3/angbracketrightalong they3–axis of the\ncomposite (i.e., along the normal to the composite plane). T he data fall in the shaded area corresponds to single\ncrystal BTO matrix–polycrystalline CFO fibre constituting the ME composite. The BTO orientations in Euler angles\n(φ◦,θ◦,ψ◦) are marked along the horizontal axis in the shaded area. Res t of the data correspond to polycrystalline\nME composite with BTO orientation distribution ( µφ,σφ,µθ,σθ,µψ,σψ) (along the horizontal axis). Here the x–\naxes is representative line separating the quantities plot ted on the y–axes. i.e., the data points in the plot have no\nabscissae.\nthe ME composite. More precisely, the emergence\nof electrical displacement /angbracketleftDj/angbracketright(Fig. 3) which bears\nthe signature of polarization and the electrical po-\nlarization /angbracketleftPj/angbracketright(Fig. 3 and Table I) itself. The aver-\nage polarization out-of-plane of the composite layer\n/angbracketleftP3/angbracketrightpeaks at the polycrystal phase of BTO as seen\ninmagnetization /angbracketleftM3/angbracketrightin theprevioussimulationex-\nperiment with applied electric field (Fig. 2). The po-\nlarization /angbracketleftP3/angbracketrightcompares well with the experiments\n(Table I)\nThe causal relationship between local microstruc-\nture and better magnetic response of polycrystalline\nBTO–CFOcompositecanbeexploredfrommapping\nthe local field distribution in response to the exter-\nnal field. The average or macroscopic field appliedto a magnetoelectric composite would permeate the\nmaterial microstructureand would spreadunequally\ninto different points owing to the heterogeneity of\nthe material. The anisotropy due to the underlying\ncrystalline structure of the constituent phases con-\ntribute to this phenomenon. The associated local\nfield distribution exhibit spacial fluctuations critical\ntocouplingphenomena. Thestress/strainmediating\nthe coupling phenomenon is spread non-uniformly\nacross the microstructure as is seen (in Fig. S1) in\nSupplementary material. The equivalent von Mises\nstress indicates a stress concentration across BTO\nmatrix compared to the CFO fibre. This is re-\nflected and underlined in the variation of micro-\nscopic displacement uε(see Fig. S1). The asym-\n4FIG.3. Averagefieldsforthemultiferroic ME 1–3composite B TO–CFO. (a)shows theaveragedielectric displacement\n/angbracketleftDi/angbracketright. (b)istheaveragemagneticfluxdensity, /angbracketleftB3/angbracketright(c)showstheaveragemagnetization /angbracketleftMi/angbracketrightand(d)showtherelevant\ntensor components of the average stress /angbracketleftσIJ/angbracketrightupon applying an external magnetic field /angbracketleftH3/angbracketrightalong they3–axis of\nthe composite (i.e., along the normal to the composite plane ). Here the x–axes is representative line separating the\nquantities plotted on the y–axes. i.e., the data points in th e plot have no abscissae.\nFIG. 4. Pole figures(contourplots) ofthepolycrystalline\nBTO matrix while the ODPs are ( µφ= 0,σφ= 1,µθ=\n0,σθ= 1,µψ= 0,σψ= 1). At this configuration, the\nME composite deliver maximum /angbracketleftM3/angbracketrightand/angbracketleftP3/angbracketright.\nmetry and anisotropy of the local stress/strain dis-\ntributed across the microstructure could be seen in\nhistograms(Fig. S2). The in-planestrain ǫε\n12records\nvalues orders of magnitude greater than other com-\nponents (Fig. S2) conforming the compression of\nthe composite in xandydirections26. The cross-\ncoupling effect of the applied electric field results inappreciable local magnetic potential ψεand associ-\nated magnetism (Fig. S3). The magnetic field in-\nduced local profile of fields are shown in the Supple-\nmentary materials. It reinforces the cross-coupling\nbetween the magnetic and electric degrees of free-\ndom through mechanical stress. The pole figure in\nFig. 4 shows the distribution of grain orientation\nabout they3–axis(orz–axis)of the composite struc-\nture. The <001>axes of the ferroelectric BTO\n(where the spontaneous polarization is oriented )\nis aligned mostly along the y3–axis (here the [001]\npsuedo cubic axes of BTO is directed ou-of-plane).\nIn summary, strong magnetoelectric coupling is\nresulted while a polycrystalline ME composite of\n1–3 BaTiO 3–CoFe 2O4is subjected external elec-\ntric/magnetic fields. In contrast to single crystal\nBaTiO 3–polycrystal CoFe 2O4composite, the aver-\nages of polarization and magnetization of the poly-\ncrystallineBaTiO 3–CoFe 2O4exceedsthatofthesin-\ngle crystal version. The depiction of local fields cor-\nresponding to the polycrystal configuration suggests\nnontrivialrole playedby randomnessin better cross-\n5coupling mediated by anisotropic and asymmetric\nstrains.\nI. ACKNOWLEDGMENTS\nThis work was supported by FCT-Funda¸ c˜ ao para\na Ciˆ encia e a Tecnologia, through IDMEC, under\nLAETA, project UIDB/50022/2020.\n1L. D. Landau and E. M. 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Krumme,\nR. Feyerherm, E. Dudzik, W. Kleemann, and\nH. Wende, “Electric in-plane polarization in multifer-\nroic cofe2o4/batio3 nanocomposite tuned by magnetic\nfields,” Nat Commun 4, 2051 (2013).\n27R. V. Chopdekar and Y. Suzuki, “Magne-\ntoelectric coupling in epitaxial cofe2o4 on\nbatio3,” Appl Phys Lett 89, 182506 (2006),\nhttps://doi.org/10.1063/1.2370881.\n6arXiv:2103.13738v2  [cond-mat.mtrl-sci]  30 Mar 2021Supplementary material to ’Enhancement of polarization an d magnetization in\npolycrystalline magnetoelectric composite’\nK.P. Jayachandran,∗J.M. Guedes, and H.C. Rodrigues\nIDMEC, Instituto Superior T´ ecnico, Universidade de Lisbo a,\nLisboa, Av. Rovisco Pais, 1049-001 Lisbon , Portugal\nThis supporting material for the work on microscopic charac terization of the composite poly-\ncrystalline magnetoelectric multiferroics include Liter ature review, theory of homogenization, its\nnumerical implementation and few key results.\nI. INTRODUCTION\nThe materials classified as magnetoelectric mul-\ntiferroics possess both the magnetic and ferroelec-\ntric orders in one phase[1]. Substantial ME effect\ncan be derived through fabricating composites of a\nferroelectric (FE) and a ferromagnetic (FM) mate-\nrial in the form composites [2]. We have used ho-\nmogenization theory for the analysis of the exter-\nnal field (electrical or magnetic) applied on a mag-\nnetoelectric composite and the resultant polariza-\ntion and magnetization responses which are vital\nin its use as new devices. When the period of the\nstructure is very small, a direct numerical approx-\nimation of the solution to magnetoelectric problem\nmay be prohibitive, or even impossible. Here ho-\nmogenization provides an alternative scheme of ap-\nproximating such solutions by means of a function\nwhich solves the problem corresponding to a “ho-\nmogenized”material. Homogenizationmethoddeals\nwith the asymptotic analysis of partial differential\nequations in heterogeneous materials with a peri-\nodic structure, when the characteristic length ǫof\nthe period tends to zero.\nII. THEORY\nA. Asymptotic homogenization\nA two-scale asymptotic homogenization analysis\ncombined with a variational formulation is devel-\noped for determination of the equivalent material\nproperties of a periodic multiferroic magnetoelec-\ntriccomposite. Localandaverage(global)electrical,\nmagnetic, andmechanicalconstitutivebehaviourare\ncomputed. Foralinearmagneto-electro-elasticsolid,\nconstitutive equationsaregovernedby the electrical,\n∗kpjayachandran@gmail.commechanical and magnetic fields. The model quanti-\nfies the local electrical and magnetic potential, dis-\nplacements, electricalandmagnetic fields, stressand\nstrain fields, magnetization (through magnetic flux\ndensity) and polarization (through electric displace-\nment) and von-Mises stress, besides the effective\nmagneto-electro-mechanical properties. The mathe-\nmaticaltheoryofhomogenizationaccommodatesthe\nphase interaction in characterising both the macro-\nand micro-mechanical behaviours of the composite\nmaterial. i.e., the method permits the introduction\nof different field equations in a microscopic scale to\neach constituent of a composite while following the\nrepresentative volume element (RVE) notion. Let Ω\nbe a fixed domain in x–space. We consider an auxil-\niaryy–space divided into parallelepiped periods Y.\nThe linear constitutive relations for small deforma-\ntions for multiferroics in the absence of heat flux are\ngiven by\nσij=CEH\nijklǫkl−ekijEk−eM\nkijHk(S1)\nDi=eijkǫjk+κǫH\nijEj+αijHj(S2)\nBi=eM\nijkǫjk+αjiEj+µǫE\nijHj(S3)\nHereσ,ǫ,D,Bare stress, strain, displacement,\nbody force, mass density, electric displacement vec-\ntor, and magnetic flux density respectively. CEH,\ne,eM,κǫHandµǫEare stiffness, strain to (electric,\nmagnetic) field coupling constants (or piezo-electric\nand -magnetic coefficients), permittivity (dielectric)\nand (magnetic) permeability respectively. Consid-\nering the standard homogenization procedure, the\nmaterial functions CEH,e,eM,κǫHandµǫE, are\nconsidered to be Y-periodic functions in the unit\ncell domain defined as Y= [0,Y1]×[0,Y2]×[0,Y3]\n[3]. The functions involved in this expansion are\nassumed to be dependent on these two variables,\nwhere one (i.e., x) describing the ”global” or aver-\nage response of the structure and the other (i.e., y)\ndescribing the ”local” or microstructural behaviour\n[4]. Here the two sets of variables xandx/εtake\ninto account the two scales of the homogenization;\nthexvariable is the macroscopic variable, whereasu\u0001\n01.5x10-4\n3x10-4\n4.6x10-4\n6x10-4\n7.6x10-4\n9.1x10-4\n1.1x10-3\n-2.7x109\n-2.1x109\n-1.4x109\n-8.1x108\n-1.9x108\n4.3x108\n1.0x109\n1.7x109\n2.3x109\n2.9x109\n3.5x109\n-2.1x109\n-1.7x109\n-1.2x109\n-8.0x108\n-3.8x108\n4.3x107\n4.7x108\n8.9x108\n1.3x109\n1.7x109\n2.2x109\n-3.9x109\n-3.1x109\n-2.4x109\n-1.6x109\n-8.4x108\n-7.5x107\n6.9x108\n1.5x109\n2.2x109\n3x109\n3.7x109\n-2.9x109\n-2.2x109\n-1.6x109\n-8.9x108\n-2.2x108\n4.6x108\n1.1x109\n1.8x109\n2.5x109\n3.2x109\n3.8x109\n1.2x10-3\n1.4x10-3\n1.5x10-3\n\u0001\u0002\u0003\n11\n \u0001\u0002\u0003\n12\n\u0001\u0002\u0003\n13\n \u0001\u0002\u0003\n22\n\u0001\u0003\n11\n-3.6x109\n-2.9x109\n-2.2x109\n-1.5x109\n-7.8x108\n-6.8x107\n6.4x108\n1.4x109\n2.1x109\n2.8x109\n3.5x109\u0001\u0002\u0003\n23\n\u0001\u0002\u0003\n33\n-4.9x109\n-3.9x109\n-3x109\n-2x109\n-1.1x109\n-9.9x107\n8.6x108\n1.8x109\n2.8x109\n3.7x109\n4.7x109\n-9.6x10-3\n-7.8x10-3\n-6.1x10-3\n-4.3x10-3\n-2.5x10-3\n-7.8x10-4\n9.9x10-4\n2.8x10-3\n4.5x10-3\n6.3x10-3\n8x10-3\n-0.01-8.5x10-3\n-6x10-3\n-3.5x10-3\n-9.2x10-4\n1.6x10-3\n4.2x10-3\n6.7x10-3\n9.3x10-3\n0.0120.014\n\u0000\u0003\n12\n\u0002\u0003\n13\n-7.6x10-3\n-6.1x10-3\n-4.7x10-3\n-3.2x10-3\n-1.8x10-3\n-3.4x10-4\n1.1x10-3\n2.6x10-3\n4x10-3\n5.5x10-3\n6.9x10-3\n\u0003\u0003\n22\n-9.5x10-3\n-7.5x10-3\n-5.6x10-3\n-3.6x10-3\n-1.6x10-3\n3.4x10-4\n2.3x10-3\n4.3x10-3\n6.2x10-3\n8.2x10-3\n0.01\n\u0004\u0003\n23\n-9.9x10-3\n-8x10-3\n-6.1x10-3\n-4.2x10-3\n-2.3x10-3\n-4.4x10-4\n1.5x10-3\n3.4x10-3\n5.2x10-3\n7.1x10-3\n9x10-3\n\u0001\u0002\u0003\nv\n6.1x107\n9.1x108\n1.8x109\n2.6x109\n3.5x109\n4.3x109\n5.2x109\n6x109\n6.9x109\n7.7x109\n8.6x109\n-0.01-9.1x10-3\n-7x10-3\n-4.9x10-3\n-2.8x10-3\n-6.5x10-4\n1.5x10-3\n3.6x10-3\n5.7x10-3\n7.8x10-3\n\u0005\u0003\n33\n9.9x10-3y3y2E\ny1\nFigure S1. Map of local fields (computed at the nodal points of the FEM) of magnetoelectric composite BaTiO 3–\nCoFe2O4,viz., the stressσε\nij(N/m2) , strainǫε\nij, the equivalent von Mises stress σε\nv(N/m2), and displacement uε(m)\nupon applying a global electric field Eof 108V/mon the unit cell. Here the magnetic CoFe 2O4cylindrical pillars are\nsurrounded by ferroelectric BaTiO 3matrix and both are aligned towards the y 3axis of the local coordinate system.2-5 0 5\n10-30200400600\n-0.01 0 0.0105001000\n-5 0 5\n10-30200400600\n-5 0 5\n10-305001000\n-10  -5 0 5\n10-30200400600\n-10  -5 0 5\n10-3050010001500-2 0 2\n109010002000\n-1 0 1\n109050010001500\n-2 0 2\n109010002000\n-2 0 2\n109010002000\n-2 0 2\n109010002000\n-4 -2 0 2 4\n1090100020003000\nFigure S2. Histogram of the local stress ( σε\nij) and strain ( ǫε\nij) values consequent to the application of external electric\nfield (averaged over finite elements).\n3-1.71-1.34-0.96-0.59-0.210.160.540.921.291.662..04\n\u0001D\u0002\n1\n-8.3x107\n-6.6x107\n-4.8x107\n-3x107\n-1.3x107\n4.8x106\n2.2x107\n4x107\n5.8x107\n7.5x107\n9.3x107\n-3.35-2.7-2.05-1.4-0.74-0.090.561.211.862.523.17\n-2.5x103\n-2.2x103\n-1.9x103\n-1.6x103\n-1.3x103\n-783-503-22357.4338\nBTOCFO\n-8.7x106\n-7.3x106\n-6x106\n-4.6x106\n-3.2x106\n-1.9x106\n-5.2x105\n8.4x105\n2.2x106\n3.6x106\n4.9x106\u0003\u0002\n\u0004\u0002\n-1.1x103\nD\u0002\n2\n-1.86-1.47-1.08-0.68-0.290.10.50.891.281.682..07\nD\u0002\n3\n-1.86-1.47-1.08-0.68-0.290.10.50.891.281.682..07-0.310.050.410.781.141.51.862.222.582.943.31\n\u0005\u0002\n1\n\u0005\u0002\n2\n-1.7x108\n-1.4x108\n-1.1x108\n-7.8x107\n-4.8x107\n-1.8x107\n1.2x107\n4.2x107\n7.2x107\n1x108\n1.3x108\n4.6x107\n5.7x107\n6.7x107\n7.8x107\n8.8x107\n9.9x107\n1.1x108\n1.2x108\n1.3x108\n1.4x108\n1.5x108\u0001\u0005\u0002\n3\n \u0001\u0006\u0002\n1\n-3.87-3.16-2.45-1.74-1.03-0.320.41.111.822.533.24\u0001\u0006\u0002\n2\n-2.74-2.04-1.34-0.640.060.761.462.162.853.554.25\u0001\u0006\u0002\n3\n \u0001\u0007\u0002\n1\n-1.9x104\n-1.5x104\n-1.1x104\n-7.6x103\n-3.9x103\n-2703.4x103\n7.1x103\n1.1x104\n1.4x104\n1.8x104\n-1.8x104\n-1.4x104\n-1.1x104\n-6.8x103\n-2.9x103\n9024.7x103\n8.6x103\n1.2x104\n1.6x104\n2x104\u0001\u0007\u0002\n2\n \u0001\u0007\u0002\n3\n-1.3x104\n-1x104\n-7.2x103\n-4.4x103\n-1.6x103\n4x103\n6.8x103\n9.6x103\n1.2x104\n1.5x104\n1.2x103y3y2E\ny1\nFigure S3. Map of local magnetic scalar potential ψε(A), electric potential ϕε(V), electric field Eε\nj(V/m), magnetic\nfield Hε\nj(A/m), electric displacement Dε\n3(C/m2), and magnetic flux Bε\n3(Wb/m2) computed at the nodal points of\nthe FEM, upon applying a global electric field Eof 108V/mon a magnetoelectric composite of BaTiO 3–CoFe 2O4.\n400.0030.0050.0080.010.0130.0150.018\n -8.2x108\n3.1x109\n7x109\n0.020.0230.025\nє\u0001\n11\n-8.1x109\n-6.5x109\n-4.9x109\n-3.2x109\n-1.6x109\n-2.6x107\n1.6x109\n3.2x109\n4.8x109\n6.4x109\n8x109\n-0.18-0.15-0.12-0.1-0.07-0.04-0.010.010.040.070.1\n-0.12-0.09-0.07-0.04-0.020.010.030.060.080.110.13\n-0.02-0.017-0.01-0.008-0.0044.1x10-4\n0.0050.0090.0130.0170.022\n-0.19-0.16-0.13-0.1-0.07-0.04-0.010.010.040.070.1\n-0.03-0.02-0.018-0.013-0.009-0.0046.5x10-4\n0.0050.010.0150.02\n2.8x109\n7.5x109\n1.2x1010\n1.7x1010\n2.1x1010\n2.6x1010\n3.1x1010\n3.5x1010\n4x1010\n4.5x1010\n4.9x1010\n-0.017-0.014-0.01-0.007-0.004-0.0010.0020.0050.0080.0110.014\n\u0002\u0003\u0001\n11\n-4.1x1010\n-3.5x1010\n-2.9x1010\n-2.4x1010\n-1.8x1010\n-1.2x1010\n-6x109\n-1.7x108\n5.7x109\n1.2x1010\n1.7x1010\n\u0002\u0003\u0001\n12\n-2x1010\n-1.7x1010\n-1.3x1010\n-8.7x109\n-4.7x109\n1.1x1010\n1.5x1010\n1.9x1010\n\u0002\u0003\u0001\n13\n-8x109\n-6.3x109\n-4.5x109\n-2.8x109\n-1.1x109\n6.5x108\n2.4x109\n4.1x109\n5.8x109\n7.6x109\n9.3x109\n\u0002\u0003\u0001\n22\n-4.1x1010\n-3.5x1010\n-3x1010\n-2.4x1010\n-1.8x1010\n-1.2x1010\n-6.7x109\n-1x109\n4.7x109\n1x1010\n1.6x1010\n\u0002\u0003\u0001\n23\n\u0002\u0003\u0001\n33\n-5.1x1010\n-4.5x1010\n-4x1010\n-3.4x1010\n-2.9x1010\n-2.3x1010\n-1.8x1010\n-1.2x1010\n-6.7x109\n-1.2x109\n4.3x109\nє\u0001\n12\nє\u0001\n13\n є\u0001\n22\n є\u0001\n23\nє\u0001\n33\n \u0002\u0003\u0001\nvu\u0001\ny3y2H\ny1\nFigure S4. Map of local fields (computed at the nodal points of the FEM) of magnetoelectric composite BaTiO 3–\nCoFe2O4,viz., the stressσε\nij(N/m2) , strainǫε\nij, the equivalent von Mises stress σε\nv(N/m2), and displacement uε(m)\nupon applying a biasing magnetic field Hof 108A/mon the unit cell. Here the magnetic CoFe 2O4cylindrical pillars\nare surrounded by ferroelectric BaTiO 3matrix and both are aligned towards the y 3axis of the local coordinate\nsystem.5-4 -2 0\n1010010002000\n-2 -1 0 1\n101005001000\n-5 0 5\n109010002000\n-4 -2 0\n10100100020003000\n-5 0 5\n109010002000\n-4 -2 0\n1010020004000\n-0.1 0 0.1010002000\n-0.1 0 0.1050010001500\n0 0.020200400600\n-0.1 0 0.1010002000\n-0.02 00200400\n-0.01 0 0.010200400600\nFigure S5. Histogram ofthe local stress ( σε\nij) and strain ( ǫε\nij)values consequenttothe application ofexternal magnetic\nfield (averaged over finite elements).68.53112215318421524\n-1.9x106\n-1.5x106\n-1.1x106\n-6.5x105\n-2.4x105\n\u0001\u0002\n1.8x105\n457\ny3y2H\ny1\nC\n\u0006\n\u0007B\nT\nO\n-1.4x107\n-1.1x107\n-8.3x106\n-5.4x106\n-2.4x106\n5.2x105\n3.5x106\n6.4x106\n9.4x106\n1.2x107\n1.5x107\n\u0003\u0002\n5.9x105\n1x106\n1.4x106\n1.8x106\n2.3x106\n\u0004D\u0002\n1\n-6.48-5.14-3.81-2.47-1.130.211.552.894.235.576.91\n\b\u0002\n2\n-6.77-5.47-4.17-2.87-1.57-0.271.032.333.634.936.23\n\t\u0002\n3\n-3.84-2.97-2.1-1.22-0.350.531.42.273.144.024.89\n\u0005\u0002\n1\n-3.1x108\n-2.5x108\n-1.9x108\n-1.4x108\n-8x107\n-2.3x107\n3.3x107\n9x107\n1.5x108\n2x108\n2.6x108\n\u0005\u0002\n2\n-4.5x108\n-3.7x108\n-2.9x108\n-2.1x108\n-1.3x108\n-5.5x107\n2.3x107\n1x108\n1.8x108\n2.6x108\n3.3x108\n\u0004\u0005\u0002\n3\n-8.8x107\n-6.5x107\n-4.3x107\n-2x107\n2.3x106\n2.5x107\n4.7x107\n7x107\n9.2x107\n1.2x108\n1.4x108\n\u0004\u0006\u0002\n1\n-507-404-301-198-94.5\n\u0004\u0006\u0002\n2\n-507-405-302-200-984.1106208311413515\n\u0004\u0006\u0002\n3\n2x103\n3.6x103\n5.1x103\n6.7x103\n8.2x103\n9.8x103\n1.1x104\n1.3x104\n1.4x104\n1.6x104\n\u0004\u0007\u0002\n1\n-3.6x107\n-2.9x107\n-2.1x107\n-1.4x107\n-6.7x106\n8.1x106\n1.6x107\n2.3x107\n3x107\n3.8x107\n7.2x105\n\u0004\u0007\u0002\n2\n-3.6x107\n-2.9x107\n-2.2x107\n-1.4x107\n-7x106\n7.7x106\n1.5x107\n2.2x107\n3x107\n3.7x107\n3.4x105\n\u0004\u0007\u0002\n3\n7.8x107\n8.3x107\n8.7x107\n9.2x107\n9.6x107\n1.05x108\n1.1x108\n1.15x108\n1.2x108\n1.24x108\n1x108\nFigure S6. Map of local electric potential ϕε(V), magnetic scalar potential ψε(A), electric field Eε\nj(V/m), magnetic\nfield Hε\nj(A/m), electric displacement Dε\n3(C/m2), and magnetic flux Bε\n3(Wb/m2) computed at the nodal points of\nthe FEM, upon applying a biasing magnetic field Hof 108A/mon a magnetoelectric composite of BaTiO 3–CoFe 2O4.7thex/εvariable takes into account the microscopic\ngeometry.\nThe detailed theoretical analysis is given else-\nwhere in a previous publication [3]. The local strain\nǫε\nij(x), electric field Eε\nj(x) and the magnetic field\nHε\nj(x) too can be obtained once the homogenized\nmacroscopic problem is solved.\nǫε\nij(x) =ǫ0\nij(x)+∂ηmn(x,y)\n∂yj\n×ǫmn(u0(x))+∂Rm(x,y)\n∂yj∂ϕ0(x)\n∂xm\n+∂Ψm(x,y)\n∂yj∂ψ0(x)\n∂xm(S4)\nEε\nj(x) =−/bracketleftbig∂ϕ0(x)\n∂xj+∂ηmn(x,y)\n∂yj\n×ǫmn(u0(x))+∂Rm(x,y)\n∂yj∂ϕ0(x)\n∂xm\n+∂Ψm(x,y)\n∂yj∂ψ0(x)\n∂xm/bracketrightbig\n(S5)\nHε\nj(x) =−/bracketleftbig∂ψ0(x)\n∂xj+∂λmn(x,y)\n∂yj\n×ǫmn(u0(x))+∂Θm(x,y)\n∂yj∂ϕ0(x)\n∂xm\n+∂Qm(x,y)\n∂yj∂ψ0(x)\n∂xm/bracketrightbig\n(S6)\nThe microscopic stress σε\nij(x), electrical displace-\nmentDε\ni(x) and magnetic flux densities Bε\ni(x) at\neach point of the domain can be computed using the\nconstitutive equations (S1)–(S3), and the field equa-\ntions as\nσε\nij(x) =Cε\nijkl(x)/parenleftbig∂u0\nk(x)\n∂xl+∂u1\nk(x,y)\n∂yl/parenrightbig\n−eε\nkij(x)/parenleftbig\n−∂ϕ0(x)\n∂xk−∂ϕ1(x,y)\n∂yk/parenrightbig\n−eM ε\nkij(x)/parenleftbig\n−∂ψ0(x)\n∂xk\n−∂ψ1(x,y)\n∂yk/parenrightbig\n(S7)\nDε\ni(x) =eε\nijk(x)/parenleftbig∂u0\nj(x)\n∂xk+∂u1\nj(x,y)\n∂yk/parenrightbig\n+κǫH ε\nij(x)/parenleftbig\n−∂ϕ0(x)\n∂xj−∂ϕ1(x,y)\n∂yj/parenrightbig\n+αε\nij(x)/parenleftbig\n−∂ψ0(x)\n∂xj−∂ψ1(x,y)\n∂yj/parenrightbig\n(S8)Bε\ni(x) =eM ε\nijk(x)/parenleftbig∂u0\nj(x)\n∂xk+∂u1\nj(x,y)\n∂yk/parenrightbig\n+αε\nji(x)/parenleftbig\n−∂ϕ0(x)\n∂xj−∂ϕ1(x,y)\n∂yj/parenrightbig\n+µǫE ε\nij(x)/parenleftbig\n−∂ψ0(x)\n∂xj\n−∂ψ1(x,y)\n∂yj/parenrightbig\n(S9)\nThe average fields are computed to be\n/angbracketleftσij/angbracketright=/tildewideCEH\nijkl(∂u0\nk(x)\n∂xl)+/tildewideekij(∂ϕ0(x)\n∂xk)\n+/tildewideeM\nkij(∂ψ0(x)\n∂xk) (S10)\n/angbracketleftDi/angbracketright=/tildewideeijk(∂u0\nj(x)\n∂xk)−/tildewideκǫH\nij(∂ϕ0(x)\n∂xj)\n−/tildewideαij(∂ψ0(x)\n∂xj) (S11)\n/angbracketleftBi/angbracketright=/tildewideeM\nijk(∂u0\nj(x)\n∂xk)−/tildewideαji(∂ϕ0(x)\n∂xj)\n−/tildewideµǫE\nij(∂ψ0(x)\n∂xj) (S12)\nThe above macroscopic equations (i.e. they do not\ncontain y) can be computed once the homogenized\nsolution for u0,ϕ0andψ0and that of the corre-\nsponding fields viz.,∂u0\nj(x)\n∂xk,∂ϕ0(x)\n∂xjand∂ψ0(x)\n∂xjare\nprescribed.\nIII. NUMERICAL IMPLEMENTATION\nWe have developed a programming platform\ncalled POSTMAT ( material postprocessing ) in this\nwork for the solution of the microscopic system of\nequationsresultingformhomogenizationandthede-\ntails are given elsewhere [3, 5]. A three-dimensional\n(3D)multiferroicfiniteelementisconceivedwithfive\ndegrees of freedom (DOF)-three DOFs for spatial\ndisplacementsandoneeachforelectricandmagnetic\npotentials. Eight-nodedisoparametricelementswith\n2×2×2 Gauss-point integration are used to ob-\ntain solutions. Altogether there were nine micro-\nscopic equations that should be solved for as much\nnumber of unknowns namely the characteristicfunc-\ntionsχmn\ni,ηmn,λmn,Rm,Φm\ni,Θm,Qm,Ψmand Γm\nk,\nwhere the indices m ,n = 1,2,3 [3]. The problem\nis reduced to standard variational FEM, after the\nusual approximations of finite element formulation\n8and can be expressed concisely as Ku=fwhereK\nis the global stiffness matrix, uis the vector of un-\nknown functions and fis the load vector. Magneto-\nelectric multiferroic material, in general, can be con-\nsidered as an aggregate of single crystalline crystal-\nlites/grains and hence the system altogether would\nbe polycrystalline nature. The unit cell or the rep-\nresentative volume element (RVE) conceived in this\nwork is a volume containing a sufficiently largenum-\nber ofcrystallites orgrainsthat its properties can be\nconsidered as equivalent to that of the macroscopic\nsample. The electric polarization Pas well as the\nmagnetization Minterspersed inside the crystallites\ncould be mapped using using some coordinate sys-\ntem. In a sense the underlying crystal orientation\ncan encompass the orientations of PorM. Thus\nwe introduce the Euler angles ( ϕ,θ,ψ) to quantify\nthe crystal orientations of a multiferroic polycrystal\n(Fig. S1), as the crystallites in an as-grown sample\nare randomly oriented in the lattice space and hence\nrequire three angles to describe its orientation with\nreference to a fixed coordinate system. Here we use\nthe so called x-convention , where the first and third\nrotation is through the y-axis (here it is y′\n2–axis)and\nthe second rotation is through the intermediate x-\naxis(hereitis y′\n1–axis). Thusallthephysicalquanti-\ntiesγ′\nijklmn...(y′) expressed in a crystallographic co-\nordinate system y′would be coordinate-transformed\ntothe localcoordinatesystem yaccordingto the fol-\nlowing scheme\nγijkl...(y) =eimejnekpelq.../tildewideγ′mnpq...(y′) (S13)\nbefore it is introduced for homogenization. (i.e., the\nFE and FM materials’ electromechanical property\ndata entered into the homogenization program are\nobtained with respect to the crystallographic coor-\ndinates.) Here eµνare the Euler transformation ma-\ntrices [6].\nFor FEM simulation of the homogenization, a\nmultiferroic crystallite is represented by a finite el-\nement in the microstructure. Thus we have a poly-\ncrystalline unite cell having as much number of crys-tallites as the number of finite elements by which\nit is discretized. As-grown FE (or for that matter\nFM) polycrystal, often ends up in a near complete\ncompensationofpolarization(ormagnetization)and\nthe material consequently exhibit very small, if any,\nelectric (or magnetic) effect until they are poled by\nthe application of an electric (magnetic) field. The\norientation distribution of the crystallites (grains)\nin such a polycrystalline material would be uniform\nwith a standard deviation σ→ ∞before poling (ap-\nplication of electric/magnetic field) and that after\npoling would best be represented by a distribution\nfunction with σ→0. Thus, any pragmatic configu-\nration of orientation distribution of grains in multi-\nferroic material would fit in a Gaussian distribution\ndefined by the probability distribution function The\nconvergence of magnetoelectric properties with unit\ncell size allows us to determine the simulation-space\nindependent, equivalent magnetoelectric properties\nof the composite. Convergence analyses, on magne-\ntoelectric composites reveal that accuracy one de-\nrives from descretizing the unit cell (in other words\nsamplingoftheunitcellswithmorenumberofgrains\nor less number) is minimal above 1000 elements\n(grains) [7]. Consequently, we kept unit cells’ sizes\nabove 1000 finite elements in this study.\nIV. RESULTS\nPiezomagnetism drives materials to acquire mag-\nnetization upon application of mechanical stress.\nSince in BaTiO 3–CoFe 2O4composite system this\nphenomenon plays a crucial role in the overall mag-\nnetization that it acquires consequent to the electri-\ncal loading. The distribution of local ( microscopic )\nfieldsviz., the stress σε, von Mises stress σε\nvand\nstrainǫεconsequenttothe applicationofanexternal\nelectric field along the y3–axis of the composite mi-\ncrostructure is displayed in Fig. S1. The histograms\nin Fig. S2 show the frequency of stress and strains\nthat spread across the microstructure.\n[1] H. Schmid, Ferroelectrics 162, 317 (1994).\n[2] M. Avellaneda and G. Harshe, J. Intell. Mater. Syst.\nStruct.5, 501 (1994).\n[3] K. P. Jayachandran, J. M. Guedes, and H. C. Ro-\ndrigues, J Intel Mat Syst Str 25, 1243 (2014).\n[4] E. Sanchez-Palencia, Non-homogeneous media and\nvibration theory, Lecture notes in physics 127\n(Springer-Verlag, Berlin, 1980).[5] K. P. Jayachandran, J. M. Guedes, and H. C. Ro-\ndrigues, Sci Rep 10, 1276 (2020).\n[6] H. Goldstein, Classical Mechanics (Addison-Wesley,\nReading, MA, 1978).\n[7] K. P. Jayachandran, J. F. A. Madeira, J. M. Guedes,\nand H. C. Rodrigues, Comp Mater Sci 148, 190\n(2018), ISSN 0927–0256.\n9"    },    {        "title": "2104.00943v1.Highly_Complex_Magnetic_Structures_Resulting_From_Hierarchical_Phase_Separation_in_AlCo_Cr_FeNi_High_Entropy_Alloys.pdf",        "content": "1  Highly Complex Magnetic Structures Resulting From Hierarchical Phase Separation in AlCo(Cr)FeNi High Entropy Alloys Qianqian Lan1,2,∗, András Kovács1, Jan Caron1, Hongchu Du1,3, Dongsheng Song1, Sriswaroop Dasari4, Bharat Gwalani4, Varun Chaudhary5, Raju V. Ramanujan5, Rajarshi Banerjee4,5 and Rafal E. Dunin-Borkowski1 1Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons and Peter Grünberg Institute, Forschungszentrum Jülich, 52425 Jülich, Germany 2School of Materials Science and Engineering, Tsinghua University, Beijing, 100086, China 3Central Facility for Electron Microscopy, RWTH Aachen University, 52074 Aachen, Germany 4Department of Materials Science and Engineering, University of North Texas, Denton, TX 76201, USA 5School of Materials Science and Engineering, Nanyang Technological University, Singapore 639798, Singapore  Abstract Magnetic high entropy alloys (HEAs) are a new category of high-performance magnetic materials, with multi-component concentrated compositions and complex multi-phase structures. Although there have been numerous reports of their interesting magnetic properties, there is very limited understanding about the interplay between their hierarchical multi-phase structures and their local magnetic structures. By employing high spatial resolution correlative magnetic, structural and chemical studies, we reveal the influence of a hierarchically decomposed B2 + A2 structure in an AlCo0.5Cr0.5FeNi HEA on the formation of magnetic vortex states within individual A2 (disordered BCC) precipitates, which are distributed in an ordered B2 matrix that is weakly ferromagnetic. Non-magnetic or weakly ferromagnetic B2 precipitates in large magnetic domains of the A2 phase, and strongly magnetic Fe-Co-rich interphase A2 regions, are also observed. These results provide important insight into the origin of coercivity in this HEA, which can be attributed to a complex magnetization process that includes the successive reversal of magnetic vortices.   2  Introduction Magnetic materials are essential in a wide and growing variety of industrial, commercial, and residential applications, including rotating electrical machines, electric vehicles, wind turbines, transformers, power convertors, electronic article surveillance systems, magnetically coupled devices, sensors, and high frequency electromagnetic devices 1, 2. To illustrate the importance of magnetic materials, we note that the energy utilization of electrical machines is a significant fraction of the total energy consumption in the world 3. Even a 1 % increase in their efficiency would have a great impact on energy efficiency and on the reduction of CO2 emissions. Hence, there is considerable interest in the development of improved magnetic materials, such as magnetic high entropy alloys (HEA), for next generation magnets.  HEAs are formed from near-equiatomic proportions of five or more elements 4, 5. They are attracting significant attention because their rich compositional variety and phase space provides opportunities for discovering alloys that have outstanding mechanical and functional properties 6, 7. HEAs can form single phases, comprising random solid solutions of their constituent elements. However, local variations in chemical composition and short-range order have a strong effect on dislocation movement and planar defect energy, leading to increased yield strength without compromising strain hardening and tensile ductility 8, just as for numerous multiphase alloys and composites in which increased heterogeneity results in improvements in properties 9. Considering the huge number of possible combinations of elements and processing conditions that can lead to the formation of secondary phases, the possibilities for developing multi-phase systems with different functionalities is enormous 10.  The formation of compositional heterogeneities and secondary phases can also alter the magnetic properties of HEAa. In general, ferromagnetic HEAs that have face-centered cubic (FCC) and body-centered cubic (BCC) structures are based on Fe, Co and Ni. An exception is the hexagonal-close-packed (HCP) HoDyYGdTb HEA, which exhibits a rich and complex magnetic phase diagram below room temperature 11. Although the mechanical properties of many HEAs are well studied and can be superior to those of conventional alloys, much less work has been performed to obtain an in-depth understanding of their magnetic properties. 3   Ferromagnetic HEAs have good mechanical properties and a wide range of tunable magnetic properties, ranging from soft to semihard to hard 12-14. The magnetic characteristics of HEAs are highly sensitive to the compositions and morphologies of the constituent phases over various length scales 4, 15-17. For example, the addition of paramagnetic or antiferromagnetic elements can induce phase segregation and decrease saturation magnetization MS 15, The addition of 25% Cr to FeCoNi can make the resulting FeCoNiCr alloy paramagnetic 18, while the introduction of Mn to FeCoNiCr can eliminate the energy difference between FCC and HCP structures, resulting in magnetic frustration 19.  An in-depth understanding of structure-magnetism correlations in HEAs has been limited by a lack of appropriate methods to characterize their phases and properties at the nanoscale and beyond. Most magnetic property studies of HEAs are based on conventional magnetization hysteresis loop (M-H) measurement of bulk forms of the HEAs, and are unable to reveal the local magnetic state and interactions between the constituent phases. The magnetic properties of highly heterogeneous alloys, in which each phase displays chemical and topological disorder, cannot be described as a composition-weighted average of the magnetic properties of the constituent phases 16, 17. Specifically, the origin of the dramatic difference between the magnetic properties of AlCoFeNi and AlCo0.5Cr0.5FeNi HEA is unexplored. An experimental understanding of the relationship between the local structure and magnetic texture of AlCoFeNi and AlCo0.5Cr0.5FeNi HEAs is crucial to control their properties.  Here, we use three-dimensional (3D) atom probe tomography (APT) and transmission electron microscopy (TEM) methods, including the Fresnel mode of Lorentz TEM and off-axis electron holography (EH), to investigate the influence of complex hierarchical phase separation of the A2 phase (disordered BCC) and the CsCl-structured B2 phase (ordered BCC) on local variations in the magnetic structure of AlCo(Cr)FeNi HEAs. We study the local magnetic characteristics of the individual phases, including their saturation magnetic induction and coercivity. Our results provide direct experimental measurements of correlations between structure and magnetic properties in HEAs with nm spatial resolution. This information is essential for understanding complex magnetism in multi-phase AlCo(Cr)FeNi alloys, as well as for the design of new HEAs with unique tailored magnetic properties.  4  Results We first report the magnetic properties of AlCoFeNi and AlCo(Cr)FeNi bulk HEAs, followed by TEM studies of the structure and magnetism of AlCoFeNi. The multi-length-scale structure of AlCo(Cr)FeNi is then investigated. Two regions (R1, R2) were identified, within which a heterogeneous phase distribution was observed. The static and dynamic magnetic behavior of R1 and R2, as well as the phases present in these regions, was studied in detail.  \n Figure 1 Magnetization (M) vs applied magnetic field (H) measured at a temperature of 300 K for AlCoFeNi and AlCo(Cr)FeNi HEAs. The inset shows a magnified view of the central part of the hysteresis loop.  1. Magnetic properties of heat-treated AlCoFeNi and AlCo(Cr)FeNi bulk HEAs. Figure 1 shows magnetization vs magnetic field (M-H) hysteresis loops measured at 300 K of heat-treated AlCoFeNi and AlCo(Cr)FeNi alloys. The loops suggest soft ferromagnetic properties. AlCoFeNi HEA has a saturation magnetization (MS) of 99.8 emu/g and a coercivity (Hc) of \n5  1.45 mT. When half of the Co is replaced by Cr in the AlCo(Cr)FeNi HEA, MS decreases to 46.2 emu/g, while Hc increases to 9.56 mT. The HEAs are almost fully magnetically saturated at fields of 500 mT. Cr substitution and annealing induce complex hierarchical phase morphology in heat-treated AlCo(Cr)FeNi 13, 20, defining the resulting magnetic properties. It is therefore important to understand the influence of each phase on the magnetic properties of AlCoFeNi and AlCo(Cr)FeNi HEAs.  \n Figure 2 Structural and magnetic texture of the AlCoFeNi HEA. (a) HAADF STEM image of two grains, with grain 1 aligned to a zone axis. The orientations of the grains are indicated. (b) Combined Al+Fe elemental map obtained from grain 1 using STEM EDXS imaging. (c) Atomic-resolution HAADF STEM image and (d) corresponding EDX map. (e) Overfocus (100 µm) Fresnel defocus image recorded from grain 2 and (f) corresponding magnetic induction map reconstructed using the transport-of-intensity equation. The color wheel indicates the direction and magnitude of the projected in-plane magnetic induction.   \n6  2. Microstructural and magnetic properties of heat-treated AlCoFeNi. The AlCoFeNi HEA has a polycrystalline structure with a grain size of above 100 µm. Figure 2a shows a high-angle annular dark-field (HAADF) scanning TEM (STEM) image of a grain boundary region between two adjacent AlCoFeNi grains. Grain 1 was aligned in the electron microscope to the closest crystallographic zone axis, at which an (001) direction was parallel to the incident electron beam direction. In this projection, chemically sensitive contrast reveals inhomogeneities. 3D APT studies of AlCoFeNi have previously revealed the presence of nm-sized Fe-Co-rich A2 precipitates in an Al-Ni-rich B2 matrix 13.  Figure 2b shows Al and Fe elemental distributions measured using energy-dispersive X-ray spectroscopy (EDXS) in grain 1, confirming the presence of compositional variations. The average sizes of the Al-rich and Fe-rich regions are approximately 10 nm. Figures 2c and 2d show an atomic-resolution image and a corresponding elemental map of the Al-rich B2 phase and the Fe-rich A2 phase. Chemical ordering in the B2 phase is visible in the HAADF STEM image. The structures are mostly coherent, with occasional dislocations at the interface due to the lattice misfit.  The magnetic structure at remanence was initially studied using Fresnel defocus imaging in Lorentz mode in magnetic-field-free conditions. Figure 2e shows a representative overfocus Fresnel image recorded from grain 2. This region contains large magnetic domains, which are separated by magnetic domain walls that appear as black and white lines in the image. The ripple-like contrast variations in the domains originate from small-angle magnetization variations. Figure 2f shows an approximate representation of the projected in-plane magnetic induction obtained from a pair of such defocused images using the transport-of-intensity equation 21, 22. The magnetic domain configuration in the thin specimen was observed to rearrange upon applying a magnetic field as small as 5 mT perpendicular to the specimen using the conventional electron microscope objective lens. This observation suggests isotropic soft ferromagnetic behavior, in which the phase separated structure is not strong enough to pin magnetic domain walls.   7   Figure 3 Phase separation in the AlCo(Cr)FeNi HEA. (a) HAADF STEM image of the multi-scale hierarchical B2 and A2 structure, containing characteristic R1 and R2 regions. EDXS elemental maps of Al, Cr, Fe, Co and Ni recorded from the marked rectangular area are shown around the main frame. An orange arrow marks an A2 shell around an R2 island. (b) Schematic diagram of the R1 and R2 phase arrangements. In the R1 phase, fine A2 gen-3 and medium A2 gen-2 precipitates form in a B2 matrix. The A2 gen-2 precipitates are covered by an A2 shell. In region R2, B2 precipitates form in an A2 gen-1 matrix covered by an A2 shell.  3. Multi-length-scale structure of AlCo(Cr)FeNi. Figure 3a shows that the annealed AlCo(Cr)FeNi HEA specimen has a strikingly different structure from that of the AlCoFeNi alloy, comprising an Al-Ni-Co-rich B2 phase, Fe-Cr-Co-rich A2 regions and their combinations. The different length scales of the A2 phases are referred to here as a) coarse A2 gen-1 (with an average size of 1-5 µm), b) medium-scale A2 gen-2 (50-150 nm) and c) fine-scale A2 gen-3 (<10 nm). Two characteristic regions are labeled R1 (A2 gen-2 + A2 gen-3 + B2 matrix) and R2 (A2 gen-1 + B2 matrix). An additional A2 phase (A2 shell) is observed between the B2 and A2 phases (see below), taking the form of a few-nm-thick shell around the A2 gen-1 and A2 gen-2 phases. Figure 3b shows a schematic diagram of the constituent phases and regions in the AlCo(Cr)FeNi HEA specimen.  \n8   Figure 4 Microstructure and chemical composition of the annealed AlCo(Cr)FeNi HEA. (a) 3D APT reconstruction of A2 gen-2 and gen-3 precipitates in a B2 matrix in region R1. An Fe-Co-rich shell forms around the A2 gen-2 precipitates. The colors represent Al (red), Ni (green) and Fe (magenta). (b-d) HAADF STEM image and corresponding Fe and Co elemental maps measured using EDXS imaging. Fe and Co enrichment is evident in the interface region of the two phases. (e-g) Atomic-resolution HAADF STEM images of A2 gen-3 in the B2 matrix, the A2 shell around the A2 gen-2 precipitates (marked by a white rectangle in Fig. 4b) and the A2 shell around an A2 gen-1 island (marked by an orange arrow in Fig. 3a), respectively. (h, i) Compositional profiles across the A2 shells around the A2 gen-2 phase and the A2 gen-3 phase, respectively. Fe and Co enrichment is observed in the shells.   Figure 4 shows TEM and APT measurements of the shell A2 phase. Figure 4a shows an APT reconstruction of the elemental distribution in a needle-shaped specimen of region R1 in the AlCo(Cr)FeNi HEA, revealing small Fe-Cr-rich A2 gen-3 precipitates and large A2 gen-2 precipitates surrounded by Fe-Co-rich A2 shells in a B2 matrix. A detailed description of the composition of the shell is presented in the Supplementary Information. Figures 4b-4d show an HAADF STEM image and corresponding Co and Fe EDXS elemental maps of the shell region. The enhancement in Co and Fe at the edges of the A2 gen-2 precipitate appears to be discontinuous, \n9  perhaps because of the geometry of the thin TEM specimen, from which part of the precipitate may have been removed by ion milling. Figure 4e shows an atomic-resolution HAADF STEM image of A2 gen-3 precipitates in the B2 matrix. Figures 4f and 4g show atomic-resolution HAADF STEM images of the A2 structure of the shell, whose thickness is 2 nm around the A2 gen-2 precipitates and 7.5 nm around the A2 gen-1 island. The measurements are consistent between the APT result and the EDXS images shown in Figs 4h and 4i. The A2 shell is rich in Fe and Co, whereas little or no Al or Cr is present inside it. Atomic-resolution HAADF STEM images of the interface region are discussed in detail in the Supplementary Information.  4. Magnetic structure of AlCo(Cr)FeNi (I) – A2 gen-2 precipitates in the B2 matrix (region R1).  4.1 Static magnetic structure Highly heterogeneous systems, in which each region contains both chemical and topological disorder, such as heat-treated AlCo(Cr)FeNi HEAs, typically have complex magnetic properties in the constituent phases that cannot be understood from bulk magnetic measurements alone. We studied the local magnetic properties of the A2 and B2 phases in the R1 and R2 regions of the AlCo(Cr)FeNi specimen using Fresnel defocus imaging and off-axis EH. The latter technique provides a direct quantitative measurement of the phase shift of the electron wave that interacted with the specimen, from which the local magnetic state of the region of interest can be determined with nm spatial resolution 23. The total electron optical phase shift contains electrostatic and magnetic contributions that need to be separated to measure the magnetic field distribution in the specimen. In the absence of electron-beam-induced charging, the electrostatic phase shift is proportional to the projected mean inner potential (MIP) of the specimen, which depends on its composition, density and ionicity.   10   Figure 5 Magnetic structure of region R1 containing A2 precipitates in a B2 matrix. (a) Mean inner potential and (b) magnetic contribution to the phase shift (ϕM ) recorded using off-axis EH with the specimen in a magnetic remanent state. The magnetic phase shift (ϕM ) in the A2 gen-2 precipitates is either bright or dark, as a result of the presence of a magnetic vortex in each precipitate. The marked A2 gen-2 precipitates (A and B) are further analysed in (d). (c) Large-field-of-view magnetic induction map derived from the magnetic contribution to the phase shift, showing clockwise and counterclockwise magnetic vortex states in the A2 precipitates. The phase contour spacing is 2π/24 rad. (d) Projected in-plane magnetization (Mxy) in the A2 precipitates marked in (b) determined from the magnetic contribution to the phase shift using model-based iterative reconstruction. An upper limit for the diameter of the magnetic vortex core, which has an out-of-plane magnetic field orientation, is 8 nm. (e) Sections showing embedding of A2 precipitates in the B2 matrix extracted from a tomographic reconstruction obtained from a tilt series of ADF STEM images.   Figure 5a shows the MIP contribution to the electron optical phase shift in region R1 measured using off-axis EH at magnetic remanence. In this image, the A2 gen-2 precipitates, which have close-to-spherical morphologies and diameters of between 50 and 120 nm, appear brighter than the surrounding B2 matrix, as they have a higher mean atomic number per unit volume. Figure 5b shows the corresponding magnetic contribution to the phase shift ϕM, which \n11  provides a measure of the in-plane component of the magnetic induction within and outside the specimen integrated in the electron beam direction 24. Bright or dark contrast is visible within the boundaries of the A2 gen-2 precipitates, which are each surrounded by a thin Fe-Co-rich A2 shell. Figure 5c shows a magnetic induction map obtained by generating contour lines and colors from the magnetic contribution to the phase shift and its gradient, respectively. This image reveals that each A2 precipitate contains a magnetic vortex, with the magnetic field rotating either clockwise or counterclockwise. Similar 3D magnetic vortex states have been observed in sub-100-nm spherical Fe-Ni particles without strong magnetocrystalline anisotropy 25, 26. In region R1, the A2 precipitates are well separated by the B2 matrix, preventing dipolar interactions between individual crystals. The ratio of clockwise to counterclockwise magnetic vortices is approximately 1:1 at remanence after saturating the specimen using the microscope objective lens, suggesting energetically-independent magnetic states that are only weakly coupled to the surrounding phases. The vortices are not associated with significant stray magnetic fields that could be measured at remanence using either bulk magnetometry or surface-sensitive magnetic characterization techniques.  A model-based iterative reconstruction algorithm 27 was used to convert the measured magnetic phase images ϕM into maps of projected in-plane magnetization Mxy, as shown in Fig. 5d for the two A2 gen-2 precipitates marked in Fig. 5b. The magnetization direction of the vortex core is parallel to the incident electron beam direction in the center of each precipitate and does not contribute to the projected in-plane magnetization map in this region. In general, each magnetic vortex core can point either up or down magnetically, irrespective of the vortex rotation direction 28. The magnetization direction of the vortex core cannot be detected from these images, as EH is sensitive to the component of the magnetic field that is perpendicular to the incident electron beam direction. An upper limit for the magnetic vortex core diameter was measured to be ∼8 nm by fitting a Gaussian function to the projected in-plane magnetization distribution. On the assumption that the specimen is magnetically active through its entire thickness, the magnitude of the in-plane magnetic induction (Fig. 5d) peaks at 0.5±0.1 T in the A2 gen-2 precipitates.   12   Figure 6 Magnetic switching of A2 gen-2 precipitates in the R1 phase in the AlCo(Cr)FeNi HEA. (a, b) Magnetic contribution to the phase shift ϕm recorded after returning to remanence from opposite out-of-plane fields of -500 and 500 mT, respectively. The localized regions of dark and bright contrast correspond to clockwise or counterclockwise magnetization rotation directions in individual A2 precipitates. Contrast reversal of the A2 gen-2 precipitates is associated with a change in the magnetic field rotation direction of the vortices. In (b), marked precipitates (white squares) retained their magnetization rotation direction from that observed in (a). (c) Magnitude and sign of the local change in magnetic phase shift (ϕM) of individual A2 gen-2 precipitates of different size plotted as a function of out-of-plane magnetic field. Positive values are associated with counterclockwise magnetization rotation directions. (d) Line profiles of magnetic phase shift extracted from (a, b) between two A2 gen-2 precipitates, as indicated by red and blue arrows. The step in the phase shift ∆ϕ indicates a projected in-plane magnetic field contribution from the B2 matrix.  \n13   The Fe-Cr-Co-rich A2 gen-2 precipitates are covered by Fe-Co-rich A2 shells (Fig. 4). The measured magnetic signal is therefore a superposition of contributions from the two A2 phases. Non-uniform magnetization distributions in some of the A2 precipitates may result from their “incomplete” morphologies in an ion-milled TEM specimen. 3D tomographic reconstruction was used to clarify the shapes and distributions of the A2 gen-2 precipitates in the B2 matrix from a tilt series of ADF STEM images 29. Figure 5e shows sections through a tomographic reconstruction of region R1 that contains approximately spherical A2 gen-2 precipitates (yellow) in a B2 matrix (blue). Some of the precipitates intersect the TEM specimen surface and are incomplete.  4.2 In-situ magnetic switching of BCC gen-2 precipitates in the B2 matrix (region R1).  The magnetic switching properties of A2 gen-2 precipitates in the B2 matrix (region R1) were studied by applying magnetic fields perpendicular to the specimen plane. In situ magnetization reversal was performed by applying magnetic fields of up to 1.5 T using the conventional microscope objective lens. Figures 6a and 6b show magnetic phase images of region R1 recorded after returning to remanence from opposite out-of-plane fields of -500 and 500 mT, respectively. The proportion of clockwise and counterclockwise magnetic vortices in the A2 gen-2 precipitates in region R1 was measured from the magnetic phase images. A comparison of Figs 6a and 6b shows that the majority of the A2 gen-2 precipitates changed their magnetic field rotation direction. Within the field of view of approximately 1.4 µm × 1.4 µm, 46 counterclockwise and 34 clockwise vortices are visible in Fig. 6a, whereas 35 counterclockwise and 45 clockwise vortices are visible in Fig. 6b. White squares in Fig. 6b mark precipitates that retained the same sense of magnetic rotation as in the initial remanent state.  Figure 6c shows the magnitude of the local change in magnetic phase shift ϕm recorded from representative A2 gen-2 precipitates of different size, plotted as a function of applied magnetic field from 0→500→0 mT. The positive (or negative) sign of ϕm is related to a counterclockwise (or clockwise) magnetic field rotation direction in the vortices. Two large A2 precipitates with diameters of 75 and 80 nm show a continuous decrease in magnetic phase shift close to zero as the applied magnetic field is increased to 500 mT, suggesting that the internal field 14  in the precipitates becomes aligned parallel to the electron beam direction. As the applied magnetic field is decreased (Fig. 6c), the magnetic phase shift recovers, but with opposite sign, indicating that the magnetic vortex now has a rotation sense opposite to the original rotation direction. Different switching behavior was observed for a small A2 precipitate with a diameter of 55 nm (Fig. 6c). The initial rotation sense is counterclockwise at 0 mT, changes sign at 200 mT and decreases gradually to zero as the applied magnetic field increases to 500 mT. A possible scenario is that the magnetic field direction of the vortex core was aligned antiparallel to the saturating magnetic field. At 200 mT, the vortex core switches to become aligned with the saturating field, which also changes the vortex rotation direction. As the applied magnetic field is increased further, this state becomes aligned with the applied field direction and the magnetic phase shift approaches zero. On decreasing the applied magnetic field, a magnetic vortex forms again at 400 mT and remains stable as the applied magnetic field is reduced to zero.  The magnetic nature of the B2 matrix in region R1, which contains more than 60% Fe, Co and Ni according to APT and EDXS measurements (Fig. 4), is now discussed. Figure 6d shows a plot of the magnetic phase shift ϕm in the B2 matrix in region R1 before and after magnetization reversal, revealing a region with a gradient in phase and an associated step ∆ϕm ∼1.3 rad. The greatest intensity maxima and minima in the plot correspond to A2 gen-2 magnetic vortices that changed their rotation direction during switching. These A2 phases serve as a reference for studying the magnetic phase contrast change within the matrix region. Changes in the sign of the magnetic phase shift ϕm indicate that part of the Al-Ni-Co-rich B2 matrix is also magnetized in the plane of the specimen, is ferromagnetic and reverses in sign magnetically. Dipolar or magnetostatic interactions 30 between the A2 precipitates are expected to be affected by the nature of the surrounding magnetic phases and interphase boundaries. It is noteworthy that the same magnetic contrast is observed in the B2 matrix, but with a sign change in the magnetically-switched region R1 (Fig. 6).   The in situ magnetic switching experiments reveal details about the magnetic properties of region R1 in annealed AlCo(Cr)FeNi HEAs that contain A2-type precipitates in a B2 matrix. However, the lack of information about the core direction from off-axis EH experiments limits our understanding of the details of the process. The switching characteristics of the magnetic vortices 15  depend on the sizes and shapes of the A2 gen-2 precipitates, the external magnetic field and coupling to the B2 matrix. Further analysis of this complex system requires comparisons of experimental measurements with atomistic spin dynamics or micromagnetic calculations that are beyond the scope of the present paper.  \n Figure 7 Magnetic structure of a B2 + A2 solid solution mixture in regions R1 and R2 and the A2 shell of the AlCo(Cr)FeNi HEA. (a) Fresnel defocus image recorded at remanence. The defocus value used was -200 µm. (b) Mean inner potential contribution to the phase measured using off-axis EH from the R2 (+A2 shell) region between two R1 regions. (c) Corresponding magnetic induction map. The phase contour spacing is 2π/16 rad. Selected B2 inclusions in the A2 gen-1 matrix are marked with white frames in (b) and (c). The colors and arrows mark the projected in-plane magnetic field direction. (d) Line profile of the magnetic phase shift across an R2 region that includes A2 shell regions and a single B2 precipitate. The dip in the middle of the phase shift profile is associated with a weakly magnetic or non-magnetic B2 phase. (e) Line profile of the magnetic phase shift, in which the slope in an A2 shell region (0.075 rad/nm) is 25% higher than that in an R2 region (0.06 rad/nm).   \n16  5. Magnetic properties of AlCo(Cr)FeNi (II) – B2 precipitates in the A2 gen-1 matrix (region R2).  The AlCo(Cr)FeNi alloy contains regions (R2) of micrometer-sized Fe-Cr-Co-rich A2 gen-1 matrix with B2 precipitates. Microstructural and chemical studies show that an A2 shell is present between the B2 and A2 phases (Fig. 4). Figure 7a shows a Fresnel defocus image of an R2 island surrounded by R1 recorded at a defocus value of -200 µm. The magnetic nature of the R2 phase is apparent from the presence of dark and bright bands of contrast. The image also shows the limitation of Fresnel defocus imaging, as the phase boundaries give rise to strong fringing fields at their edges, making it difficult to interpret the magnetic state.   Figures 7b and 7c show the MIP contribution to the phase and a corresponding magnetic induction map recorded using off-axis EH. In Fig. 7b, the A2 gen-1 and gen-2 (bright contrast) and B2 (dark contrast) phases can be distinguished, as they have different mean atomic numbers per unit volume. Selected B2 precipitates are marked by white frames in both images. The color-coded magnetic induction map shows that the R2 region contains large magnetic domains. The magnetic field lines are either disrupted or missing at the B2 precipitates, suggesting that they are weakly magnetic or non-magnetic. The effect of the smaller (<50 nm) B2 precipitates on the magnetic field is less clear, as it can be masked by the signal from the A2 gen-1 matrix in the ∼100-nm-thick TEM specimen.  Figure 7d shows a line profile of the magnetic phase shift across region R2, which contains a single B2 precipitate at its center. The weakly magnetic or non-magnetic Al-Ni-Co-rich B2 precipitate has a lower contribution to the magnetic phase shift and appears as a dip. Close inspection of the phase profile in Fig. 7e reveals a change in slope at the position of the A2 shell, suggesting a difference in its magnetic properties from those of the A2 gen-1 matrix. The slope of the phase in the A2 shell and the A2 gen-1 matrix in region R2 is 0.075 and 0.06 rad/nm, respectively, based on fitted linear functions. As the magnetic phase shift scales with the projected in-plane magnetic induction in the specimen, it can be inferred that the A2 shell has approximately 25% higher magnetization than the A2 gen-1 matrix in region R2. This difference is thought to result from the higher concentration of Cr in the A2 gen-1 matrix, which decreases the magnetization of the Cr-Fe-Co-rich A2 gen-1 matrix in the R2 region.  17  Discussion Saturation magnetization and coercivity in AlCo(Cr)FeNi HEAs are affected by the presence of non-magnetic Al and antiferromagnetic Cr. The AlCoFeNi HEA contains nm-sized Fe-Co-rich A2 precipitates in an Al-Ni-rich B2 matrix. The saturation magnetic induction of AlCoFeNi of 0.987 T (99.8 emu/g) is associated primarily with ferromagnetic ordering of Fe, Co and Ni.  The effect of Cr substitution and heat treatment of AlCo(Cr)FeNi leads to decomposition of the alloy into an interesting duplex distribution of A2 and B2 phases in regions R1 and R2. Bulk measurements of structure and magnetic properties are not able to resolve and provide an understanding of the separate contributions of each phase. On the other hand, the high spatial resolution analyses presented here reveal the intimate relationship between their structure and magnetic properties. Upon annealing the alloy to 600 °C, interdiffusion of Al and Cr defines its structural and magnetic properties. Cr concentrates in the Fe-Cr-Co-rich A2 phase, which dominates the R2 islands and forms spheroidal magnetic precipitates in the R1 phase. A small amount of Cr can also be found in the Fe-Co-rich A2 shell. It is known from previous work that Cr aligns antiferromagnetically with Fe and Co 31 and reduces the magnetic moment. The new information on Cr accumulation in the A2 phase provides possibilities to vary the Cr concentration and annealing conditions to control the phase assemblies and magnetic properties. The application of stress and magnetic fields during annealing can also be used to control magnetic anisotropy.  The constituent phases in AlCo(Cr)FeNi HEAs have different magnetic properties, which form a rich and complex magnetic structure. Based on the measurements in this work, schematic representations of the deduced saturation magnetic induction and magnetic states are presented in Fig. 8. There are three primary contributions to the saturation magnetic induction from:  (i) Fe-Cr-Co-rich A2 gen-2 and A2 gen-3 precipitates in region R1 and the A2 gen-1 matrix in region R2; (ii) Fe- Co-rich A2 shells between the A2 and B2 phases;  (iii) the Al-Ni-rich B2 phase.   18   Figure 8. Schematic diagrams of (a) magnetic induction and (b) magnetic states in the phase-separated AlCo(Cr)FeNi HEA. The grayscale intensity in (a) corresponds to the deduced saturation magnetic induction in the different phases. Region 2 contains magnetic voids. The colors in (b) represent clockwise/ counterclockwise magnetic field rotation in magnetic vortices. a slowly-varying magnetic field in region 1 (R1), and large magnetic domains in region 2 (R2). The scale bar is 200 nm.   The magnetic state of each phase is distinctly different. The first two contributions are strongly linked, as they form core-shell structures with thin A2 shells around A2 spheres or islands. Off-axis EH measurements reveal that the A2 shell in region R2 (Fig. 7) has a higher magnetic induction than the A2 core (Fig.8a). Magnetic interactions, which can affect magnetization reversal, are expected between the two ferromagnetic A2 phases. The A2 spheres in region R1 support 3D magnetic vortex states in the B2 matrix. Based on the magnetic phase shift measurement and on the result of model-based iterative reconstruction of projected in-plane magnetization, the saturation magnetic induction in the A2 spheres, which have a core-shell structure, is estimated to be approximately 0.5 T (Fig. 5). For a 2.6-nm-thick shell around a core with a radius of 40 nm, the shell occupies almost 20% of the total volume. Therefore, the contribution of the thin Fe-Co-rich A2 shell to the total magnetization is significant. A magnetic contribution is also expected from the B2 phase, which contains more than 50% Fe, Ni and Co. In \n19  addition, the magnetic phase images (Fig. 6) provide evidence for a slowly-varying weak magnetic signal in the B2 matrix (Fig. 8b). The A2 spheres in region R1 support 3D magnetic vortex states in the B2 matrix (Fig. 8b), with a random distribution of clockwise and anticlockwise field rotations. Based on these observations, magnetization reversal and domain wall movement across the different magnetic components are expected to be highly complex processes.  In a multicomponent and multiphase alloy, the coercivity Hc is expected to be sensitive to impurities, deformation, grain size and phase decomposition 32. The AlCoFeNi HEA has a smaller value of Hc than the AlCo(Cr)FeNi HEA. The magnetic structure in the AlCoFeNi HEA is characterized by large magnetic domains with small-angle magnetization variations. Magnetic domain walls move easily in the presence of an applied magnetic field. In contrast, in the AlCo(Cr)FeNi HEA, which has a multi-scale hierarchical B2+A2 decomposed structure, phase boundaries between the R1 and R2 regions can act as pinning sites for magnetic domain walls, thereby restricting their movement and resulting in an increase in coercivity Hc compared to that in AlCoFeNi.   Our magnetic switching study shows that, in region R1, magnetic vortices in A2 gen-2 precipitates can change their rotation direction in the presence of external magnetic fields of 100 to 500 mT. The smaller the diameter of the A2 gen-2 precipitate, the lower is the magnetic field that is needed to change the magnetic vortex rotation direction, suggesting that the coercivity of such an AlCo(Cr)FeNi HEA can be tuned by changing the sizes and separations of the constituent A2 precipitates and islands.  In Al-Ni-Co (Alnico), a duplex nanoscale structure of two phases forms during thermal annealing in the presence of an external magnetic field. This leads to anisotropic growth of a periodic Fe-Co hard magnetic phase in an Al-Ni-rich matrix, resulting in shape anisotropy and enhanced coercivity. It is therefore of interest for future studies of AlCo(Cr)FeNi magnetic HEAs to determine the effect of external magnetic field and/or stress annealing, kinetics and other external stimuli to control magnetic anisotropy and magnetic properties, as already successfully demonstrated for Alnico. For instance, soft magnetic HEAs, in which moderate coercivity is tunable by magnetic nanostructures, are yet to be explored for high-frequency device applications.  20  Conclusions The magnetic structure of AlCoxCr1–xFeNi (x = 0.5 and 1) heat-treated HEAs has been investigated with unprecedented spatial resolution in combined sub-Ångstrom structural and 3D chemical compositional studies. In AlCoFeNi, which contains nm-sized A2 Fe-Co-rich precipitates in a B2 matrix, the magnetic structure is characterised by large magnetic domains and small-angle magnetization variations. In contrast, the substitution of Co by Cr in AlCo(Cr)FeNi results in the formation of two regions: (i) a ferromagnetic A2 phase in a weakly-magnetic B2 matrix and (ii) B2 precipitates in a magnetic A2 matrix.  In the first region, the A2 precipitates are approximately spherical and, surprisingly, contain individual magnetic vortices. In the second phase, the B2 precipitates disrupt otherwise-continuous magnetic domains in the A2 matrix. The presence of an Fe-Co-rich A2 shell between the B2 and A2 phases provides an additional contribution to the overall magnetization. The saturation magnetization of the AlCo(Cr)FeNi HEA is dominated by the Fe-Cr-Co-rich A2 phases in both regions, as well as by the Fe-Co-rich A2 shells, whereas the B2 matrix phase provides a minor contribution. Its value is decreased by the substitution of Co by Cr as a result of the antiferromagnetic ordering nature of Cr.  The increased coercivity of the AlCo(Cr)FeNi HEA, which remains in the soft ferromagnetic range, can be attributed to a complicated magnetization reversal process, which involves the reversal of magnetic vortices in a weakly-ferromagnetic matrix. Our results provide direct local information about the intricate complexity of magnetic remanent states and reversal processes in multicomponent HEAs that contain coexisting magnetic phases and hierarchical structures that span multiple length scales.   21  Methods Specimen preparation. AlCoxCr1–xFeNi (x = 0.5 and 1) bulk specimens were prepared by arc melting Al, Co, Cr, Fe and Ni pellets in an Ar atmosphere, followed by annealing at 600 ◦C for 15 h in an Ar atmosphere and quenching in water, as described elsewhere 20. Bulk magnetometry measurements were performed in a vibrating sample magnetometer (VSM-Lakeshore 7404) using a maximum magnetic field of 1 T. Specimens for TEM and APT were prepared using focused Ga ion beam milling in FEI Helios Nanolab 400s and FEI Nova Nanolab 20 dual beam systems following a standard lift-out method. Electron-transparent (∼100 nm) lamellae were attached to Cu Omniprobe support grids for TEM measurements. Atom probe tomography. APT experiments were conducted in a LEAP 300X local electrode atom probe system (Cameca Instruments, Inc.). All atom probe experiments were conducted in electric field evaporation mode at a temperature of 60 K using an evaporation rate of 0.5% and a pulsing voltage of 20% of the steady-state applied voltage. Data analysis was performed using IVAS 3.6.2 software. Transmission electron microscopy. HAADF STEM imaging, EDXS mapping and electron to- mography were performed in an FEI Titan G2 80-200 electron microscope equipped with a high brightness field emission gun, a probe aberration corrector and an in-column Super-X EDXS system. HAADF STEM images were recorded on a Fischione detector using a beam convergence semi-angle of 24.7 mrad and an inner detector semi-angle of 69 mrad. Off-axis electron holography. The same specimens that were used for microstructural characterization were used to study magnetic texture using Lorentz microscopy and off-axis EH. Off-axis electron holograms were recorded in magnetic-field-free conditions (i.e., in Lorentz mode) in an image-aberration-corrected FEI Titan 80-300 electron microscope equipped with a high brightness field emission gun, an electron biprism, and a (Gatan K2 IS) direct electron counting detector 33 camera using a typical exposure time of 6 s. The biprism voltage was typically set to 100 V, resulting in an overlap interference width of 2.1 µm and a holographic interference fringe spacing of 2.76 nm with a contrast of 48% in vacuum. The objective lens of the microscope was used to apply out-of-plane magnetic fields to the specimen of between 0 and 1.5 T. The 22  electrostatic and magnetic contributions to the phase shift were separated by turning the specimen over inside the electron microscope using a modified Fischione 2050 tomography specimen holder. Off-axis electron holograms were reconstructed numerically using a standard Fourier-transform-based method with sideband filtering using HoloWorks software in the Gatan microscopy suite, as well as using home-written scripts in the Semper image processing language 34. Contour lines and colour maps were generated from recorded magnetic phase images to yield magnetic induction maps.   23  Supplementary information Supplementary Information accompanies this paper. Acknowledgments Technical support from W. Pieper is gratefully acknowledged. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreements 856538 (3D MAGiC) and 823717 (ESTEEM3), as well as from the DARPA TEE program through grant MIPR# HR0011831554 and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)-Project-ID 405553726-TRR 270. This work is also sup- ported by the AME Programmatic Fund of the Agency for Science, Technology and Research, Singapore under Grant No. A1898b0043. Author contributions Q.L. and A.K. conceived and conducted the TEM experiments, S.V., R.B. and V.C. processed the alloys and conducted the APT and bulk measurements. Q.L, J.C., H.D. and D.S. analyzed the results. R.B., R.V.R. and R.E.D.B. provided research guidance. All authors contributed to the writing and reviewed the manuscript. Competing interests The authors declare that they have no competing financial interests. 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Lužnik, J.; Koželj, P.; Vrtnik, S.; Jelen, A.; Jagličić, Z.; Meden, A.; Feuerbacher, M.; Dolinšek, J., Complex magnetism of Ho-Dy-Y-Gd-Tb hexagonal high-entropy alloy. Phys. Rev. B 2015, 92 (22), 224201. 12. Dasari, S.; Chaudhary, V.; Gwalani, B.; Jagetia, A.; Soni, V.; Gorsse, S.; Ramanujan, R. V.; Banerjee, R., Highly tunable magnetic and mechanical properties in an Al0. 3CoFeNi complex concentrated alloy. Materialia 2020, 12, 100755. 13. Chaudhary, V., Gwalani, B., Soni, V.; Ramanujan, R.; Banerjee, R., Influence of Cr 25  substitution and temperature on hierarchical phase decomposition in the AlCoFeNi high entropy alloy. Scientific reports 2018, 8 (1), 1-12. 14. Na, S.-M. Lambert, P. K. Jones, N. J., Hard magnetic properties of FeCoNiAlCuXTiX based high entropy alloys. AIP Advances 2021, 11 (1), 015210. 15. Jung, C. Kang, K. Marshal, A. Pradeep, K. G. Seol, J. B. Lee, H. M. Choi, P. P., Effects of phase composition and elemental partitioning on soft magnetic properties of AlFeCoCrMn high entropy alloys. Acta Materialia 2019, 171, 31-39. 16. Vrtnik, S. Guo, S. Sheikh, S. Jelen, A. Koželj, P. Luzar, J. Kocjan, A. Jagličić, Z.; Meden, A. Guim, H. Kim, H. J. Dolinšek, J., Magnetism of CoCrFeNiZrx eutectic high-entropy alloys. Intermetallics 2018, 93, 122-133. 17. Cieslak, J. Tobola, J. Reissner, M., The effect of bcc/fcc phase preference on magnetic properties of AlxCrFeCoNi high entropy alloys. Intermetallics 2020, 118, 106672. 18. Lucas, M., Mauger, L. Munoz, J. Xiao, Y. Sheets, A. Semiatin, S., Horwath, J. Turgut, Z., Magnetic and vibrational properties of high-entropy alloys. J. Appl. Phys. 2011, 109 (7), 07E307. 19. Niu, C.; LaRosa, C. R.; Miao, J.; Mills, M. J.; Ghazisaeidi, M., Magnetically-driven phase transformation strengthening in high entropy alloys. Nat. Commun. 2018, 9 (1), 1-9. 20. Borkar, T. Chaudhary, V. Gwalani, B. Choudhuri, D. Mikler, C. V. Soni, V. Alam, T. V. Ramanujan, R. Banerjee, R., A Combinatorial Approach for Assessing the Magnetic Properties of High Entropy Alloys: Role of Cr in AlCoxCr1-xFeNiAdvanced Engineering Materials 2017, 19 (8). 21. Gureyev, T. Roberts, A. Nugent, K., Partially coherent fields, the transport-of-intensity equation, and phase uniqueness. JOSA A 1995, 12 (9), 1942-1946. 22. Gureyev, T. E. Nugent, K. A., Rapid quantitative phase imaging using the transport of intensity equation. Optics communications 1997, 133 (1-6), 339-346. 23. Dunin-Borkowski, R. E., Kovács, A., Kasama, T., McCartney, M. R., Smith, D. J., Electron holography. Springer Handbook of Microscopy (Eds. Hawkes, P. W., Spence, J. C. H.) 2019, Chapter 16, 767–806. 24. Kovács, A., Dunin-Borkowski, R. E., Magnetic  imaging  of  nanostructures  using  off-axis electron holography. Handbook of magnetic materials (Elsevier, ed. Brück, E.). 2018, 27, 59-154  25. Hÿtch, M. J. Dunin-Borkowski, R. E. Scheinfein, M. R. Moulin, J. Duhamel, C. 26  Mazaleyrat, F. Champion, Y., Vortex flux channeling in magnetic nanoparticle chains. Phys. Rev. Lett. 2003, 91 (25), 257207. 26. Kim, M.-K. Dhak, P. Lee, H.-Y. Lee, J.-H. Yoo, M.-W. Lee, J.; Jin, K. Chu, A. Nam, K. T. Park, H. S., Self-assembled magnetic nanospheres with three-dimensional magnetic vortex. Appl. Phys. Lett. 2014, 105 (23), 232402. 27. Caron, J., Model-based reconstruction of magnetisation distributions in nanostructures from electron optical phase images. . RWTH Aachen University PhD Thesis 2017. 28. Shinjo, T. Okuno, T. Hassdorf, R. Shigeto, K. Ono, T., Magnetic vortex core observation in circular dots of permalloy. Science 2000, 289 (5481), 930-932. 29. Midgley, P. A. Dunin-Borkowski, R. E., Electron tomography and holography in materials science. Nat. Mater. 2009, 8 (4), 271-280. 30. Krishnan, K. M., Fundamentals and applications of magnetic materials. Oxford University Press: 2016. 31. Schneeweiss, O. Friák, M. Dudová, M.; Holec, D. Šob, M. Kriegner, D.; Holý, V.; Beran, P. George, E. P. Neugebauer, J. Dlouhý, A., Magnetic properties of the CrMnFeCoNi high-entropy alloy. Phys. Rev. B 2017, 96 (1). 32. Herzer, G., Grain size dependence of coercivity and permeability in nanocrystalline ferromagnets. IEEE Transactions on magnetics 1990, 26 (5), 1397-1402. 33. Chang, S. L.; Dwyer, C.; Barthel, J.; Boothroyd, C. B.; Dunin-Borkowski, R. E., Performance of a direct detection camera for off-axis electron holography. Ultramicroscopy 2016, 161, 90-97. 34. Saxton, W. Pitt, T. Horner, M., Digital image processing: the Semper system. Ultramicroscopy 1979, 4 (3), 343-353. "    },    {        "title": "2104.11412v1.Interesting_magnetic_response_of_the_nuclear_fuel_material_UO2.pdf",        "content": "arXiv:2104.11412v1  [cond-mat.str-el]  23 Apr 2021Interesting magnetic response of the nuclear fuel\nmaterial UO 2\nSudip Pal1, L. S. Sharath Chandra2, M. K. Chattopadhyay2,3\nand S. B. Roy1\n1UGC DAE Consortium for Scientific Research, University Campus\nKhandwa Road, Indore 452001, India\n2Free Electron Laser Utilization Laboratory, Raja Rammana Centre for Advanced\nTechnology, Indore-452 013, India\n3Homi Bhabha National Institute, Training School Complex, Anusha kti Nagar,\nMumbai 400 094, India\nE-mail:sbroy@csr.res.in\nAbstract. Magnetic response of uranium dioxide (UO 2) has been investigated\nthrough temperature and magnetic field dependent dc magnetizat ion measurements.\nUO2is a paramagnet at room temperature. The magnetic susceptibility, however,\ndeviates from Curie-Weiss (CW) like paramagneticbehaviorbelow T= 280 K. Further\ndown the temperature UO 2undergoes phase transition to an antiferromagnetic state\nbelowTN= 30.6 K. The zero field cooled (ZFC) and field cooled (FC) magnetizatio ns\nexhibit some distinct thermomagnetic irreversibility below TN. The temperature\ndependence of the FC magnetization is more like a ferromagnet, whe reas ZFC\nmagnetizationexhibitsdistinctstructuresnotusuallyobservedint heantiferromagnets.\nIn low applied magnetic field this thermomagnetic irreversibility in magne tization\nexists in a subtle way even in the paramagnetic regime above TNup to a fairly high\ntemperature, but vanishes in high applied magnetic fields. Deviation f rom CW law\nand irreversibility between ZFC and FC magnetization indicate that th e paramagnetic\nstate above TNis not a trivial one. Magnetic response below TNchanges significantly\nwith the increase in the applied magnetic field. Thermomagnetic irreve rsibility in\nmagnetization initially increases with the increase in the strength of a pplied magnetic\nfield, but then gets reduced in the high applied fields. A subtle signatu re of a magnetic\nfield induced phase transition is also observed in the isothermal magn etic field vartaion\nof magnetization. All these experimetal results highlight the non-t rivial nature of the\nantiferromagnetic state in UO 2\n1. Introduction\nUranium di-oxide (UO 2) is a well known nuclear fuel material and used worldwide in\nnuclear reactors for electrical power generation and research. UO2is also recognized as\na Mott-Hubbard insulator [1, 2], and it promises other technological applications apart\nfromanuclear fuel [3, 4]. Thermal conductivity isvery importantfortheremoval ofheat\nproduced ina nuclear reactor by fission inthenuclear fuel materials . As a result thermalUnusual magnetism of UO 2 2\nproperties of UO 2particularly have drawn much attention over the years [1, 5, 6, 7, 8] .\nUO2crystallizes in face centred cubic (fcc) calcium fluorite structure ( Fm3m), in which\nU4+ionsare surrounded by eight O2−ions forming a cube [9]. Therefore, theanisotropic\nthermal conductivity reported in this compound is rather unexpec ted and emphasizes\non the relevance of spin-phonon coupling, which is associated with th e magnetic state of\nthe system [8]. The Mott insulating state in UO 2further highlights the importance of\nstrong electron-electron correlation in the system [2, 4]. Various techniques, including\nneutron scattering and nuclear magnetic resonance (NMR) have r evealed a complex 3k-\nnon-collinear antiferromagnetic (AFM) spin ordering below TN= 30.6 K. The transition\nis first order in nature and is accompanied by a small lattice distortion , predominantly\nin the oxygen cage [10, 11, 12] . In cubic crystal field, the nine fold degenerate (5 f2,\nJ= 4) state splits up with a 3-fold degenerate ground state, resultin g into Jahn-Teller\n(JT) instability [13].\nSpin-orbit coupling, Coulomb interaction, antiferromagnetic excha nge interaction\nand JTdistortion are ofcomparable strength inUO 2. BelowTN, a quadrupoler ordering\nis established together with AFM spin ordering, facilitated by the inte raction between\ncooperative JT distortion, antiferromagnetic exchange interact ion and 5 fquadrupoles\n[14, 15, 16, 17] . While, there is some understanding of the physical properties of U O2\nin the antiferromagnetic state below TN, characteristics of the high temperature state\nare not quite clear. The nature of the high temperature state, alt hough considered\nto be paramagnetic, is not so straightforward. Temperature dep endence of thermal\nconductivity exhibits a minimum at TNand a maximum around T= 220 K, which\nclearly highlight unusual physical stateabove TN. Moreover, apositive magnetostriction\nhas been observed above TN, whereas magnetostriction is negative in the AFM state.\nNeutron scattering has provided evidence for dynamic JT distortio n of the oxygen\nsublattice above TN[18, 19, 20, 21] . Temperature dependence of elastic constant above\nTNalso exhibit unconventional behavior [11]. UO2has been reported to be showing\nsome other interesting physical properties, such as piezomagnet ism and magnetoelastic\nmemory driven by spin-lattice interaction [9].\nHere we present a detailed study of the temperature ( T) and magnetic field\n(H) dependence of magnetization ( M) in UO 2. The results of our study highlight\nvarious hitherto unknown interesting aspects of the magnetic res pose of UO 2. We\nshow that below T= 280 K the low field magnetization or susceptibility deviates\nfrom standard Curie-Weiss paramagnetic behavior. Abrupt chang es in both the zero\nfield cooled (ZFC) and field cooled (FC) magnetization are observed a tTN= 30.6 K\nwhere UO 2undergoes a phase transition to antiferromagnetic state along wit h distinct\nthermomagnetic irreversibility i.e. MZFC(T)/negationslash=MFC(T) inside the antiferromagnetic\nstate. The magnetic response in the antiferromagnetic state cha nges considerably with\nthe increase in the applied magnetic field. The observed behaviour ma y be correlated\nto a magnetic field induced phase transition. The thermomagnetic irr eversibility in the\nantiferromagnetic state continues to exist in the temperature ra nge above TNand below\nT= 150 K, but it gets suppressed completely at higher applied magnetic fields.Unusual magnetism of UO 2 3\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s49/s48/s50/s48/s51/s48\n/s49/s51 /s49/s52 /s49/s53/s48/s50/s48\n/s40/s53/s53/s49/s44/s32/s55/s49/s49/s41\n/s40/s54/s52/s48/s41\n/s40/s54/s52/s50/s41/s40/s52/s52/s52/s41/s40/s54/s50/s50/s41/s40/s53/s51/s51/s41/s40/s54/s50/s48/s41/s40/s54/s48/s48/s44/s32/s52/s52/s50/s41/s40/s53/s51/s49/s41/s40/s52/s52/s48/s41/s40/s53/s49/s49/s44/s51/s51/s51/s41/s40/s52/s50/s50/s41/s40/s52/s50/s48/s41/s40/s51/s51/s49/s41/s40/s52/s48/s48/s41/s40/s50/s50/s50/s41/s40/s51/s49/s49/s41/s40/s50/s50/s48/s41/s40/s50/s48/s48/s41\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s50 /s32/s40/s100/s101/s103/s41/s120/s49/s48/s51\n/s40/s49/s49/s49/s41\n/s40/s50/s48/s48/s41/s40/s49/s49/s49/s41\n/s32/s32\n/s50 /s32/s40/s100/s101/s103/s41/s120/s49/s48/s51\nFigure 1. X-ray diffraction pattern of UO 2powder sample recorded at room\ntemperature using wavelength, λ= 0.7121 ˚A. The data have been fitted using rietveld\nrefinement method. Indexes of the planes corresponding to peak s are also mentioned.\nThe blue line at the bottom shows the difference between experiment al data and fitted\ncurve.\n2. Experimental details\nPowder sample of UO 2has been prepared by reducing UO 3in hydrogen atmosphere\nat 700◦C at Bhabha Atomic Research Centre, Trombay. Room temperatur e X-ray\ndiffraction data have been recorded at the wavelength of λ= 0.7121 ˚A in the beamline-\nBL12 in Indus-2 synchrotron radiation source at Raja Ramanna Ce ntre for Advanced\ntechnology (RRCAT), Indore, India. Magnetic measurements hav e been carried out in\na MPMS-3 SQUID-VSM magnetometer (M/S Quantum Design, USA). T emperature\ndependence measurements of magnetization have been performe d in temperature sweep\nmode of measurement at 0.5 K/min cooling and heating rate.\n3. Results and discussion\nFig.1shows the X-ray diffraction data of UO 2measured at room temperature in θ−2θ\ngeometry. In the inset, we have shown a magnified view of the data a round (111) peak,\nwhich is the most intense peak in this case. Sharp diffraction peaks an d flat background\nreveal the good crystalline nature of the sample. The data have be en analyzed by\nRietveld refinement method using Fullprof software package. All th e peaks can be\nindexed in cubic fcc structure (space group Fm3m), which rules out the presence of any\nsecondary phase in the sample. The optimized lattice parameter obt ained from fitting\nis,a= 5.4594 ˚A. Lattice parameter of UO 2is sensitive to oxygen stoichiometry of the\nsample and follows an empirical law given by, a= 5.4705−0.132x, wherexis defined\nas UO 2+x[22]. In our case, the obtained lattice parameter corresponds to, x= 0.08.Unusual magnetism of UO 2 4\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s49/s48/s46/s48/s48/s50/s48/s46/s48/s48/s51/s48/s46/s48/s48/s52\n/s52/s48 /s54/s48/s48/s46/s48/s48/s49/s52/s52/s48/s46/s48/s48/s49/s53/s50\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s48/s46/s48/s50/s48 /s50/s48 /s52/s48 /s54/s48/s48/s46/s54/s57/s48/s46/s55/s50/s48/s46/s55/s53/s48/s46/s55/s56\n/s48 /s50/s48 /s52/s48 /s54/s48/s49/s46/s48/s48/s49/s46/s48/s52/s49/s46/s48/s56/s32\n/s32\n/s32/s32\n/s32/s90/s70/s67\n/s32/s70/s67/s67\n/s32/s70/s67/s87/s72/s32/s61/s32/s49/s48/s48/s32/s79/s101\n/s32/s32/s77/s32/s40/s101/s109/s117/s47/s103/s41/s84/s32/s40/s75/s41\n/s32/s90/s70/s67\n/s32/s70/s67/s67\n/s32/s70/s67/s87/s84/s32/s40/s75/s41\n/s84/s32/s40/s75/s41\n/s84/s32/s40/s75/s41/s77/s32/s40/s101/s109/s117/s47/s103/s41/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s77/s32/s40/s101/s109/s117/s47/s103/s41/s40/s97/s41\n/s32/s32/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s84/s32/s40/s75/s41/s72/s32/s40/s107/s79/s101/s41/s40/s99/s41\n/s72/s32/s61/s32/s49/s32/s107/s79/s101\n/s40/s101/s109/s117/s47/s103/s41/s32/s90/s70/s67\n/s32/s70/s67/s67\n/s32/s70/s67/s87/s32\n/s32/s84\n/s100\n/s40/s98/s41/s72/s32/s61/s32/s53/s48/s32/s107/s79/s101\n/s40/s100/s41/s84\n/s105/s114/s114\n/s32/s90/s70/s67\n/s32/s70/s67/s67\n/s32/s70/s67/s87/s32\n/s32/s84\n/s100/s84\n/s105/s114/s114\n/s72/s32/s61/s32/s55/s48/s32/s107/s79/s101\nFigure 2. (a) Magnetization ( M) versus temperature ( T) plots measured in applied\nfields of (a) H= 100 Oe, (b) H= 1 kOe, (c) H= 50 kOe and (d) H= 70 kOe in\nthe ZFC, FCC and FCW modes. FCC and FCW curves overlap at all temp eratures.\nTheMZFC(T) andMFC(T) curves show bifurcation below TN. The bifurcation also\npersists above TNat least upto 150 K in the applied fields of 100 Oe. The inset in\nFig. (a) shows the magnified view of ZFC and FCW curve above TNin applied field\nof 100 Oe. The inset in Fig. (b) shows ∆ M=MFC−MZFCin fields of H= 100\nOe and 1 kOe. In applied fields of 50 kOe and 70 kOe MZFC(T) andMFC(T) curves\nshow a sudden and discontinuous fall in magnetization at TN.MZFC(T) andMFC(T)\nbifurcate below temperature Tirr< TN(see Figs. c and d).\nFig.2(a)shows the temperature dependence of magnetization ( M) in UO 2\nmeasured at an applied field of H= 100 Oe in the zero field cooled (ZFC) warming,\nfield cooled cooling (FCC) and field cooled warming (FCW) modes. In ZFC mode the\nmeasuring field H=100Oeisappliedaftercoolingthesampletothelowest temperature\n(here 2 K) of measurement in zero external field. ZFC magnetizatio n (MZFC) is then\nmeasured while warming the sample. After reaching the highest temp erature (room\ntemperature) of measurement the sample is subsequently cooled b ack to 2 K in same\nfield while measuring the FCC magnetization ( MFCC). After FCC measurements the\nFCW magnetization ( MFCW) is measured while by warming the sample again gradually\nto room temperature in the same field. Temperature variation of ma gnetization in the\nhightemperatureregimeinallthemodes(ZFC,FCCandFCW) indicate thepresence ofUnusual magnetism of UO 2 5\nmagnetic moment in UO 2, which is in line with the Mott insulator status of the sample.\nBelowT= 50 K,Mtends to flatten while the temperature is further reduced and the n\nMchanges abruptly around TN= 30.6 K. This temperature matches well with the\nreported temperature where UO 2supposedly undergoes a phase transition to a complex\nantferromagnetic state [12]. Figure 2(a) highlights the presence of a very distinct\nthermomagnetic irreversibility i.e. MZFC(T)/negationslash=MFC(T) in the antiferromagnetic state.\nSuch thermomagnetic irreversibility is not expected in any standard antiferromagnet\n[23], and to the best of our knowledge has not been reported earlier fo r UO2. MFCC(T)\nandMFCW(T) curves overlap at all temperatures of measurement and now onw ords we\nwill designate the field cooled magnetization as MFC. Interestingly in an expanded scale\n(see the inset of Fig. 2(a)) one can see that the MZFC(T) andMFC(T) curves actually\nstart to bifurcate below T= 150 K, which is well over TNand nearly equal to 5 TN; the\nbifurcation gradually increases with decrease in temperature. The bifurcation between\ntheMZFC(T) andMFC(T) curves below the antiferromagnetic transition temperature\nTNismuch largerthanthebifurcationabove TN. Just belowthetransitiontemperature,\nMZFC(T) curve starts to rise sharply with increase in temperature, which is followed\nby a rapid fall resulting into a narrow peak around T= 29 K. Below about T= 10 K,\nMZFC(T) curve increases slowly with decreasing temperature. All these features are not\ntypical of a collinear antiferromagnet. Field cooled magnetization MFC(T) on the other\nhandrises sharply with decrease intemperature below TNandtends tosaturate below T\n= 15 K. Such a sharp rise in magnetization followed by a tendency towa rds saturation is\nnot expected in the case of a collinear AFM state and is usually a chara cteristic feature\nof a ferromagnetic state. This kind of behaviour possibly arises due to non-collinear spin\nordering within the AFM state of UO 2, which results into weak ferromagnetic response\nbelowTN. It may be noted here that, although the antiferromagnetic tran sition in\nUO2is reported to be first order in nature, MFCC(T) andMFCW(T) do not show\nany thermal hysteresis acrosss TN. It means that the supercooling and superheating\nphenomena, which are usually observed across a first order phase transition in many\nsystems[24,25], are absent across the antiferromagnetic transition in UO 2.\nFig.2(b)presents the results of the magnetization measurements as a fun ction\nof temperature carried out in an applied magnetic field of H= 1 kOe. MZFC,MFCC\nandMFCWcurves appear to be qualitatively similar to those obtained at H= 100 Oe.\nHowever, there are important differences, which are noted below:\n(i) The difference between MZFCandMFCcurves i.e. the thermomagnetic\nirreversibility above TNis completely erased.\n(ii) The thermomagnetic irreversibility below TNincreases by a large extent than that\natH= 100 Oe. The value of ∆ M=MFCW−MZFCatH= 100 and 1 kOe is\nshown in the inset of Fig. 2(b).\n(iii) Bifurcation of MZFCandMFCcurves starts just below TNand this temperature\ndoes not seem to change with increase in applied magnetic field from 10 0 Oe to 1\nkOe.Unusual magnetism of UO 2 6\n/s45/s54/s48 /s45/s51/s48 /s48 /s51/s48 /s54/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s54/s48 /s45/s51/s48 /s48 /s51/s48 /s54/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s48 /s53/s48/s46/s48/s48/s46/s49\n/s48 /s49/s48 /s50/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48 /s51/s48 /s54/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s32/s32/s50/s57/s32/s75\n/s72/s32/s40/s107/s79/s101/s41/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s32/s32/s50/s32/s75/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s72/s32/s40/s107/s79/s101/s41/s32/s32/s50/s57/s32/s75\n/s32/s108/s105/s110/s101/s97/s114/s32/s102/s105/s116\n/s40/s98/s41/s40/s97/s41/s40/s99/s41\n/s77/s32/s40/s101/s109/s117/s47/s103/s41/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s72/s32/s40/s107/s79/s101/s41/s72/s32/s40/s107/s79/s101/s41/s32\n/s32/s40/s100/s41\n/s32/s50/s32/s75\n/s32/s76/s105/s110/s101/s97/s114/s32/s102/s105/s116/s51/s50/s32/s75/s77/s32/s40/s101/s109/s117/s47/s103/s41/s32 \n/s72/s32/s40/s107/s79/s101/s41\nFigure 3. Isothermalfield( H)variationofmagnetization( M)intheantiferromagnetic\nregime at T= 29 K and 2 K: (a) and (b) show the virgin curves at T= 29 and 2 K\nnear origin. M(H) in the low field region is fitted and extrapolated to show the change\nin slope in the virgin M−Hcurve. (c) and (d) show isothermal M−Hloops at T=\n29 and 2 K. In the Inset of (c) M−Hcurve at T= 32 K has been shown only in the\nfirst quadrant.\n(iv) The low temperature increase in MZFCis more pronounced.\nThermomagnetic irreversibility is a characteristic feature of spin gla ss or cluster\nglass systems, where it arises due to freezing of spins or magnetic m oments\noriginating from the competing magnetic interactions and associate d frustration\n[26,27]. Thermomagnetic irrersivility is also observed in ferromagnets, w hich is\nassociated with domain wall pinning [28]. Even some antiferromagnet s with ThCr 2Si2\nstructure exhibits thermomagnetic irrersivility, which was attribut ed to the stacking\nfaults in those compunds with layered structures [29]. However, in all these magnetic\nsystems the thermomagnetic irreversibility gets suppressed with in crease in the applied\nmagnetic field. This is in contrast with the observed increase in therm omagnetic\nirreversibility in UO 2with the increase in applied fields from 100 Oe to 1 kOe. Such\nbehaviour in UO 2, however, changes in the region of high applied fields. Fig. 2(c)\nand 2(d) present the temperature dependence of magnetization at H= 50 kOe and\n70 kOe measured in ZFC, FCC and FCW modes. In this case the thermo magneticUnusual magnetism of UO 2 7\nirreversibility as well as Tirrget reduced as the applied magnetic field is increased from\n50 kOe to 70 kOe, whereas the antiferromagnetic transition tempe ratureTNremains\nlargely unaffected. It may also be noted that the thermomagnetic ir reversibility in the\nparamagentic regime is totally absent in this high applied field regime.\nFrom the results discussed above it is clear that magnetic response in low applied\nmagnetic fields in both ZFC and FC states below TNin UO 2is distinctly different from\nthat at highapplied magnetic fields. Moreevidences in this direction em erge if one looks\ncarefullytothetemperature dependence ofhighfieldmagnetizatio n. AstheUO 2sample\nis cooled from high temperature, magnetization increases and tend s to saturate around\n35K.However, justabove TN, magnetizationdecreasesbyasmallamount, resulting into\na small hump prior to TN. The magnetic response at TN, and also at low temperatures,\nchange drastically both in the ZFC and FC states, as compared to ma gnetic behavior at\nlowfields, shown inFig. 2(a)and2(b) . AtH=50kOe, boththe MZFC(T)andMFC(T)\ncurves undergo anabruptanddiscontinuous fallat TN, which is immediately followed by\na small rise inmagnetization with decrease intemperature. Then, th ey show a relatively\nflat region over a temperature region of 28-22 K. In the lower temp erature region, the\nMZFCandMFCCcurves bifurcate: the MZFCcurve starts to decrease, whereas the\nMFCCcurve slowly increases. Note that, the MZFCalso shows an additional shallow\ndip around Td= 15 K. Note that, the bifurcation between MZFC(T) andMFC(T) curves\nappears only below a temperature Tirr(see Fig. 2(c)), which is lower than the transition\ntemperature TN. AtH= 70 kOe, magnetization shows larger drop at TNand all the\nthree curves gradually decrease with further decrease in temper ature (see Fig. 2(d)).\nBelowTirr, theMZFCcurve continues to decrease, whereas the MFCC/MFCWcurve\nincreases. This is in clear contrast to the temperature dependenc e of magnetization\nobtained in applied magnetic fields of 100 Oe and 1 kOe (see Fig.2(a) and 2(b)).\nTo investigate further on the antiferromagnetic state in UO 2, we present in Fig.\n3the isothermal M−Hcurves at T= 29 and 2 K measured starting from the ZFC\nstate. Here, the sample is initially cooled to the temperature of meas urement in the\nabsence of any external field. Then Mis measured while increasing Hisothermally to\n70 kOe to record initial (virgin) curve, which is shown in Figs. 3(a)and3(b)forT=\n29 and 2 K, respectively. After recording the virgin curve, Mis measured while varying\nHbetween ±70 kOe to record the envelope curves, which are shown in Figs. 3(c)and\n3(d). The virgin curves at both temperature lie within the envelope curve s. Note that,\natT= 29 K, the M−Hcurve shows a small change in the slope above Ha= 1.2\nkOe and continues to increase at higher field up to the highest applied field ofH= 70\nkOe. Whereas, the virgin M−Hcurve at T= 2 K deviates from linearity above Ha\n= 15 kOe. The envelope M−Hcurve at T= 29 K shows a small hysteresis with a\ncoercive field of around HC= 2.2 kOe, which increases to 4.5 kOe at 2 K. The M−H\ncurves do not show any tendency of saturation till H= 70 kOe. The nonsaturating\nM−Hcurve highlights the antiferromagnetic state but the presence of hysteresis is\nrather unexpected in AFM state. At further lower temperature, both coercive field and\nremanant magnetization further increase as evident from the M−Hcurve at T= 2 K.Unusual magnetism of UO 2 8\nThis change in the slope of M−Hcurves above H=Ha, may indicate some sort of\nfield induced transition of the zero field cooled AFM state.\nTheincreaseinthedifferencebetween MZFC(T)andMFC(T)withmagneticapplied\nmagnetic field is one of the important signatures of a kinetically arres ted first order\ntransitions [25, 31, 32, 33, 34] . In this case, the dynamics of a first order phase transition\ngets arrested in a H−Twindow. The difference between MZFC(T) andMFC(T)\nincreases because cooling in different magnetic fields produce differe nt volume fractions\nof the high and low temperature magnetic phases. As stated earlier , the transition\nin UO 2is first order in nature, and like a kinetically arrested system thermo magnetic\nirreversibility in the low field regime increases with the applied magnetic fi eld. In this\ncontext, the M−Hcurves shown in Fig. 3, can be rationalized in the following manner:\nthe ZFC state undergoes a field induced transition above a critical fi eldHa, as observed\nfrom the change in slope. However, in the field decreasing cycle, the reverse transition\nis not observed, so that the magnetic state induced by applied magn etic field persists\nduring the entire envelope curve and gives rise to hysteresis loop. S uch behaviour has\nbeen reported inthe literature inthe cases of kinetic arrest offirs t order phase transition\nin various magnetic systems, where the zero field cooled state may b e either equilibrium\nlow-Tphase, or the kinetically arrested high- Tphase, depending on the nature of the\nground state [35, 34]. Now, after zero field cooling, in the first case, when magnetic\nfield is increased, above certain field, the equilibrium phase undergoe s a field induced\ntransition to the high- Tphase as the superheating band is crossed [32, 34, 35] . On\nthe other hand, in the second case, the kinetically arrested state devitrify into the\nequilibrium low- Tphase while increasing the field. In either cases, the envelope curve\ndoes not show the reverse transition.\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50\n/s50/s53/s48 /s50/s55/s53 /s51/s48/s48/s49/s46/s49/s53/s49/s46/s50/s48/s49/s46/s50/s53/s32/s40/s79/s101/s32/s103/s47/s101/s109/s117 /s41\n/s32\n/s84/s32/s40/s75/s41/s32/s76/s105/s110/s101/s97/s114/s32/s102/s105/s116/s32 /s32/s84/s32/s40/s75/s41/s32/s45/s49\nFigure 4. χ−1versusTdata measured at H= 100 Oe in the ZFC mode. The inset\nshows the CW law fitting above 280 K.\nThe unusual dependence of magnetization on temperature and th e finite\ndifference between MZFC(T) andMFC(T) curves much above TNclearly highlight\nan unconventional paramagnetic state at high temperatures. Th erefore, to furtherUnusual magnetism of UO 2 9\nunderstand the magnetic property of UO 2at high temperatures, χ−1versusTatH\n= 100 Oe measured in ZFC mode is plotted in Fig. 4. Theχ−1versus T data appears\nto be grossly linear in the temperature range from T= 300 K to 70 K. We tried to\nfit the results in this temperature regime by using Curie-Weiss (CW) la w, given by\nχ=C\nT+T0, whereC=Nµ2\neff\n3kB=Ng2J(J+1)\n3kB. However, when looked carefully, the CW\nlaw does not fit the experimental data in the entire temperature re gime. In fact, it\ndeviates from linearity below around T= 280 K (see the inset of Fig. 4). The effective\nmagnetic moment µeffobtained to be 2.82 µB/f.u., which is smaller than the expected\nvalue of 3.57 µB/f.u. for J= 4 (L =5 and S= 1). However, it matches extremely well\nwith value of effective moment expected for the threefold degener ate ground state in\nUO2in a cubic crystal field [12].T0is obtained to be around 171 K, which is around\n5.6 times higher than the transition temperature. The deviation of t he susceptibility\nfrom CW law, the presence of irreversibility much above TN, higher value of T0than\nTNare interesting. There exist some earlier reports of the unusual c haracteristics above\nTN[8]. In inelastic neutron scattering experiments, magnetic inelastic re sponse has\nbeen observed above TNup to as high as T= 200 K [20]. It has been suggested that\nthe coherent motion of the neighbouring oxygen cages produces u ncorrelated 1-k type\ndynamical JT distortion in the paramagnetic state of UO 2[20]. With the lowering in\ntemperature correlation builds up and a static 3-k type distortion c ondenses at TN. The\ntendency of Mto saturate as TNis approached during cooling, the deviation from the\nCW law, existence of bifurcation between the ZFC and FC curves high light the unusual\nparamagnetic state at high temperature, which may arise due to sh ort range 1-k type\ndynamic JT distortion.\n4. Conclusion\nSummarizing, we can say from the results of temperature and magn etic field dependent\nstudies of dc magnetization that the low temperature antiferroma gnetic state in UO 2is\nnon-trivial in nature and is accompanied with large thermomagnetic ir reversibility. The\nfield cooled magnetization MFC(T) measured in low fields of 100 Oe and 1 kOe indicates\nthe presence of ferromagnetic character in UO 2. This inference is further corroborated\nby the presence of hysteresis in isothermal field variation of magne tization alongwith\nappreciable coercive field. The nature of the thermomagnetic irrev ersibility changes\nin the presence of high applied magnetic field. In the low magnetic field r egion the\nthermomagnetic irreversibility irreversibility increases with the incre ase in applied field,\nwhich can be rationalized in terms of the kinetic arrest of magnetic fie ld induced first\norder phase transition. The thermomagnetic irreversibility in the hig h magnetic field\nregime deceases with the increase in applied field. Prima facie this beha viour is quite\nsimilar to that observed in some ferro and antiferromagets, where it was attributed to\nthe hindrance in domain wall motion. Furthermore the paramagetic s tate in UO 2is\nalso unusual with a deviation of Curie-Weiss law and the presence of t hermomagnetic\nirreversibility in the temperature region much above TN. These results indicate theUnusual magnetism of UO 2 10\nexistence of some short range magnetic correlations well inside the paramagnetic region\nof UO 2. Overall the results of our present study are expected to stimula te further\nmicroscopic measurements involving neutron scattering and muon s pin rotation ( µSR)\nmeasurents to find out the exact microscopic nature of magnetic s tates in various\nmagnetic field ( H) - temperature ( T) regime of UO 2.\n5. Acknowledgment\nWe acknowledge Dr. Vinay Kumar of Bhabha Atomic Research Centre , Trombay for\nproviding us with powdered UO 2sample and Dr. A. Sagdeo and Dr. A. K. Sinha of\nRaja Ramanna Centre for Advanced Technology, Indore for help in X-ray diffraction\nmeasurement. S B Roy acknowledges financial support from Depar tment of Atomic\nEnergy, India in the form of Raja Ramanna Fellowship. Sudip Pal and S B Roy thank\nDr. A J. Pal, Director, UGC-DAE CSR and Dr. D. M. Phase, Centre Dir ector, UGC-\nDAE CSR, Indore Centre for suuport and encouragement.\n6. References:\n[1] Quan Yin and Sergey Y. Savrasov, Phys. Rev. Lett 100, 225504 (2008).\n[2] S. -W. Yu et al, Phys. Rev. B 83, 165102 (2011).\n[3] J. Schoenes, J. Appl. Phys. 491463 (1978).\n[4] S. B. Roy, Physics of Mott insulator: Physics and applications , IOP Publishing (2019).\n[5] Jack Leland Daniel, J. Matolich, H. W. Deem, Thermal conductivity of UO 2, Hanford\natomic products operation (1962).\n[6] J. H. Harding and D. G. 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Pourovskii and Sergii Khmelevskyi, Phys. Rev. B 99094439 (2019).\n[18] J. Faber, G. Lander and B. Cooper, Phys. Rev. Lett. 35, 1770 (1975).\n[19] J. Faber and G. Lander, Phys. Rev. B 14, 1151 (1976)\n[20] R. Caciuffo, G. Amoretti, P. Santini, G. H. Lander, J. Kulda, and P. de V. Du Plessis,\nPhys. Rev. B 59, 13 892 (1999).\n[21] R. Caciuffo, P. Santini, S. Carretta, G. Amoretti, A. Hiess, N. M agnani, L.-P. Regnault,\nand G. H. Lander, Phys. Rev. B 84, 104409 (2011).\n[22] K. Teske, H. Ullmann, and D. Rettig, J. Nucl. Mater. 116, 260 (1983).\n[23] S. J. Blundell, Magnetism in Condensed Matter (Oxford Universit y Press, 2001).Unusual magnetism of UO 2 11\n[24] S. B. Roy, G. K. Perkins, M. K. Chattopadhyay, A. K. Nigam, K. J. S. Sokhey, P.\nChaddah, A. D. Caplin and L. F. Cohen, Phys. Rev. Lett,. 92147203 (2004).\n[25] S. B, Roy, J. Phys.: Condens. Matter 25183201 (2013).\n[26] J. A. Mydosh, Rep. Prog. Phys. 78, 052501 (2015).\n[27] Sudip Pal, Kranti Kumar, A. Banerjee, S. B. Roy and A. K. Nigam , J. Phys.: Condens.\nMatter33(2021) 025801 (2021) and references therein.\n[28] S.B. Roy, A.K. Pradhan, P. Chaddah and B.R. Coles, Solid St. Comm un.99563(1996).\n[29] S. B. Roy, A.K. Pradhan and P. Chaddah, J. Phys.: Condens. Ma tter65155 (1994).\n[30] S.B. Roy, A.K. Pradhan, P. Chaddah and E. V. Sampathkumaran , J. Phys.: Condens.\nMatter92465 (1997).\n[31] M. K. Chattopadhyay, S. B. Roy, and P. Chaddah, Phys. Rev. B72, 180401(R) (2005).\n[32] A. Banerjee, K. Mukherjee, Kranti Kumar, and P. Chaddah, Phys. Rev. B 74, 224445\n(2006).\n[33] S. B. Roy and M. K. Chattopadhyay, Phys. Rev. B 79, 052407 (2009).\n[34] P. Chaddah, First Order Phase Transitions of Magnetic Materia ls (Taylor and Francis,\n2017).\n[35] Kranti Kumar et al, Phys. Rev. B 73184435 (2006)."    },    {        "title": "2104.12334v1.Highly_sensitive_spin_flop_transition_in_antiferromagnetic_van_der_Waals_material_MPS3__M___Ni_and_Mn_.pdf",        "content": "1   Highly sensitive spin -flop transition  in antiferromagnet ic van-der \nWaals material MPS3 (M = Ni and Mn)  \nRabindra Basnet1, Aaron Wegner1, Krishna Pandey2, Stephen Storment1, Jin Hu1,2* \n \n1Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA  \n2Materials Science and Engineering Program, Institute for Nanoscience and Engineering, \nUniversity of Arkansas, Fayetteville, Arkansas 72701, USA  \n \n \nAbstract  \nRecent  developments  in two-dimensional (2D) magnet ism have  motivated  the search for \nnovel van -der Waals  (vdW s) magnetic materials to explore new magnetic phenomenon in the 2D \nlimit . Metal thiophosphates , MPX3, is a class of magnetic vdW s materials with antiferromagnetic \n(AFM) ordering persisting down to the atomically thin  limit . The magnetism in this material family \nhas been found  to be highly dependent on the choice of transition metal  M. In this work,  we have \nsynthesized the intermediate compounds Ni 1-xMn xPS3 (0 ≤ x ≤ 1) and investigated their magnetic \nproperties . Our study reveals  that the variation of Ni  and Mn content  in Ni1-xMn xPS3 can efficiently  \ntune the spin-flop transition, likely due to the modulat ion of  the magnetic anisotropy.  Such \neffective tunning offers a promising candidate  to engineer 2D magnetism for future device \napplications.  \n \n* jinhu@uark.edu  2  \n \nRecent breakthroughs in two -dimensional (2D) magnetic materials open a new route in \nexploring intrinsic magnetism in the 2D limit. The discoveries of novel 2D magnets  [1–6] provide \nopportunit ies not only to understand the mechanism of low dimensional magnetism but  also to \ndesign next -generation devices. For example, advances  in devices and heterostructures involving  \n2D magnets enable the effective tuning of their electronic and magnetic properties, providing a \nfascinating platform to study fundamental physics  [7–18]. These  breakthrough s have greatly \nenriched our understanding of magnetism in the 2D limit .   \nAmong 2D magnetic materials,  one model system is metal thiophosphates  MPX3 (M = \ntransition metal ions, X = chalcogen ions). MPX3 materials crystallize in a monoclinic  layered  \nstructure with C2/m space group, in which the transition metal ions form a honeycomb layer and \ncarry localized magnetic moments  [19–22]. The van der Waals  (vdW) -type crystal structure allows \nfor the inter -layer intercalation  [23–27] and mechanical exfoliations down to atomically thin \nlayer s [5,28 –36]. The magnetic and electronic properties of the MPX3 family are strongly \ndependent on the choice of the transition metal M. MnPS 3, NiPS 3, and FePS 3 have been found to \nexhibit Heisenberg -type, XY or XXZ-type, and Ising -type antiferromagnetism, respectively  [37–\n42]. The Neel temperature ( TN) also varies with transition metal M, increasing from 78 K for \nMnPS 3, to 118,  122, and 155 K for Fe PS3, CoPS 3, and NiPS 3, respectively  [38,40,42 –45]. Besides \nmagnetism, the electronic properties  of MPX3 are also tunable with M, exhibiting a wide range of \nband gaps from 1.3 eV for FePSe 3 to 3.5 eV for ZnPS 3 [22]. Recent studies on NiPS 3 and MnPS 3 \nhave established  these material s as platform s to study correlated  electron s in 2D magnetic \nmaterials  [32,46,47] . Furthermore,  theoretical and experimental high -pressure studies revealed 3 insulator -to-metal transitions in MnPS 3 [47,48] , MnPSe 3 [48], FePS 3 [49–52], NiPS 3 [47,53]  and \nV0.9PS3 [51,54] , and even the emergence of superconductivity in FePSe 3 [55].  \nThe structur al similarity  of the MPX3 materials allows for the synthesis  of polymetallic \n“mixed” compounds with substitution of transition metal M  [56–66]. Varying lattice constant and \nmagnetic moments through elemental substitution is an effective way to tune magnetism and prob e \nthe underlying physics of magnetic materials. Coupling of magnetism with  lattice constants  and \nsymmetries  in MPX3 has been theoretical ly predicted  [67]. Tuning of magnet ism has also been \nobserved in mixed  compounds such as Mn xFe1-xPS3, Mn 1-xFexPSe3, Fe0.5Ni0.5PS3 and Mn 1-\nxZnxPS3 [56–66]. For example, magnetic order and spin orientation s in MnPS 3 can be \nsystematically tuned by Zn substitution  for Mn  [56–58]. Spin glass is also found to arise from the \ncompeting 3 d magnetism in Mn substituted FePS 3 [59–61].  \nIn this work, we report the systematic study on pr eviously unexplored Ni1-xMn xPS3 (0 ≤ x \n≤ 1). We synthesized single crystals and characterized the evolution of magnetic properties with \nsubstitutions. In addition to the known field-induced spin -flop (SF) transition in Mn PS3, we have \nalso discovered the previously unreported  SF transition s in Ni PS3. Furthermore, we found  these \nSF transitions are extremely sensitive to magnetic  substitutions, likely due  to the re -orientation of \nthe magnetic moments of 3 d elements  controlled by  trigonal distortion of MS6 octahedral . Such \ntunable magnetism offers a promising platform  for studying new phenomenon arising from 2D \nmagnetism and future device applications.  \nSingle crystals  of Ni1-xMn xPS3 (0 ≤ x ≤ 1) used in this work were synthesized by the \nchemical vapor transport method using I 2 as the transport agent. Elemental powders with \nstoichiometric ratio were sealed in a quartz tube and placed in a two -zone furnace with a \ntemperature gradient from 750 to 550 °C for 1 week. Millimeter  to centimeter  size single crystals  4 with various color s have been obtained , as shown in the insets of Fig. 1a . The se crystal s are thin \nplate -like and easily  exfoliable  with hexagonal facets  which is consistent with the vdW structure \nof MPX3 shown in Fig. 1b. The elemental compositions examined by energy -dispersive x -ray \nspectroscopy (ED S) reveal successful Ni -Mn substitution. We have carefully characterized t he \ncompositions of all the Ni1-xMn xPS3 single crystals used in this paper. The Mn contents x \nthroughout this paper are measured values.  In addition to composition analysis, the x-ray \ndiffraction (XRD)  on single crystals also indicates successful substitution . As shown in Fig. 1 (a), \nthe (00L) diffraction peaks exhibit a systematic low -angle shift with increasing x in Ni1-xMn xPS3, \nwhich is consistent with the greater ionic radius of Mn2+ as compared with Ni2+, respectively.   \nIn Fig. 2 we present the magnetic  property  characterizations for pristine  MnPS 3 and NiPS 3. \nFor MnPS 3, below the N éel temperature TN ~ 78 K , a spin -flop (SF) transition can be observed in \nthe isothermal magnetization measured under out-of-plane magnetic field ( H⊥ab) [Fig. 2( a)] but \nis absent under  in-plane field ( H//ab) [Fig. 2( b)], which is consistent with the previous \nreports  [5,34,65,68,69] . Although the SF transition is largely explored in both  bulk and atomically \nthin MnPS 3, it has not been directly  discovered in other members of this family except FePS 3, in \nwhich a metamagnetic transition occurs at very high field ( μ0H > 35 T)  [70]. Here we report  the \nfirst discovery of SF transition in NiPS 3. Unlike  in MnPS 3 in which the SF transition occurs when \nH⊥ab, the metamagnetic transition in NiPS 3 takes place with in -plane field, which is characterized \nby a clear upturn in magnetization above a critical spin-flop field μ0HSF ≈ 6 T below the N éel \ntemperature TN ≈ 155 K, a s shown in Fig s. 2(d) and 2(e). The need for relatively high magnetic \nfield could be the reason that prevented  the discovery of SF transition in earlier studies  [38,71] . \nOn the other hand, the recent magneto -photoluminescence experiment implies a spin re -orientation \nin bulk NiPS 3 under a much higher in -plane magnetic field of 15 T [32]. Such a high critical field , 5 which is 2.5 times higher th an that in our magnetization measurement s, might be attributed to the \nnature of photoluminescence as an indirect probe.  \nIn MnPS3 and Ni PS3, the SF transition occurring under different field directions is in line \nwith their magnetic structures: The Mn moments in MnPS 3 are aligned along out -of-plane \ndirection  [58,69] , whi le the Ni moments in NiPS 3 mostly lie within the plane  [31,38,71] , as shown \nin the insets of Figs. 2b and 2c, respectively. In collinear AFM systems, a magnetic field along the \neasy axis  exceed ing a critical spin-flop field HSF forces the magnetic moments  to rotate  [72,73] . \nIn such a SF state, the moments  re-orient themselves to a canted configuration perpendicular to \nthe field direction, resulting in a net moment along the easy axis  [72,73] . Therefore, the SF \ntransition behaves differently in Mn PS3 and Ni PS3, as illustrated in Figs. 2g and 2h . Furthermore, \nbecause the in -plane projection of Ni moment s in Ni PS3 form s a collinear AFM structure along \nthe a-axis (Fig. 2c, lower inset), the SF transition is expected to show in -plane anisotropy. To \nexamine this, we measured  magnetization with the magnetic field applied along or perpendicular \nto the hexagonal edges  of a NiPS 3 single crystal , as shown in Fig. 2(e). Indeed, magnetization and \nSF transition  are found to be dependent on  in-plane field-orientation s.  \nThe scenario o f SF transition in NiPS 3 is also supported by the temperature dependence of \nmagnetic susceptibility . As shown in Fig. 2( f), susceptibility measured with in-plane field  (χ||) \ndisplays a low temperature upturn , which becomes more significant  at high er fields ( μ0H > 5 T). \nA similar  low temperature upturn has been observed in MnPS 3 [65]. In addition,  the zero -field \ncooling (ZFC , solid lines ) and field cool ing (FC, dashed lines ) data display  weak but clear \nirreversib ility above 5 T, which also become s more visible at higher fields.  The development of \nlow temperature upturn  and irreversib ility can be understood in terms of  the ferromagnetic \ncomponent from  the uncompensated  canted  moments  along the easy axis  in the SF state , as 6 illustrated in Fig. 2h.  This irreversibility disappears at 140 K  when applying 9 T field, coincid ing \nwell with the tempe rature a bove  which the SF transition vanishes as seen in the isothermal  \nmagnetization [Fig. 2(d), inset]. Such  140 K “disappearing temperature” is lower than the magnetic \nordering temperature TN ≈ 155 K , which  can be attribu ted to the fact that a greater field is needed \nfor SF transition at higher temperatures , particularly when approaching TN. HSF enhancement upon \nincreasing temperature is widely  seen in  other SF systems  [5,74 –78], which  can be interpreted \nusing  the molecular field theory  [73,78] : In a weakly anisotropic antiferromagnet , spin-flop field \nHSF can be estimated by ( HSF)2 = 2K/(χꓕ - χ||), where K is the anisotropy constant, χꓕ and χ|| are the \nperpendicular and parallel susceptibilities, respectively  [58,73,78,79] . Generally, in AFM systems  \nthe difference between χꓕ and χ|| reduces  more quickly  than the magnetic anisotropy constant K \nupon increasing temperature, leading to enhanced HSF [78]. \nThe distinct SF transitions  in NiPS3 and MnPS3 due to their different magnetic structures  \n(Figs. 2g and 2h)  motivate us to  further study the “mixed” compounds Ni1-xMn xPS3. As shown i n \nFig. 3, the magnetism is highly tunable with  Ni-Mn substitution. Under  both in -plane ( H||ab, Fig. \n3a) and out -of-plane ( H⊥ab, Fig. 3b) magnetic fields, the magnetization exhibits  a systematic \nenhancement with increasing Mn content x, which is consistent with the much larger  magnetization \nof the pristine MnPS 3 than that of NiPS 3 [Figs. 2(a-d)] and can be ascribed to greater magnetic \nmoment of Mn2+ than Ni2+. Interesting ly, we found that the SF transitions are extremely  sensitive  \nto Ni-Mn substitution.  The SF transition in NiPS 3 under in -plane field  disappears with 5% Mn \nsubstitution ( i.e., x = 0.05  in Ni1-xMn xPS3). Similarly, under out-of-plane  field, HSF in MnPS 3 is \nreduced by half with 5% Ni substitution ( i.e., x = 0.95 ) and  disappears upon 10% substitution  (x = \n0.9).   7 Further insights can be gained from the careful comparison between  the isothermal \nmagnetization s. In Fig. 3d we show isothermal magnetization M(H) of NiPS 3 (i.e., x = 0) and 5% \nMn-substituted  (i.e., x = 0.05)  sample s, reproduced from Figs. 3a and 3b. With Mn substitution, in \naddition to the enhancement of magnetization and the absence of SF transition as mentioned above, \nanother interesting behavior is the sublinear field dependence at low fields.  Such nonlinear M(H) \nis more pronounced under in -plane field  [Inset, Fig. 3(d)] , i.e., the magnetic field direction that SF \ntransition occurs for NiPS 3. Similarly, at the MnPS 3 side, 5% Ni -substitution ( x = 0.95) also \nintroduces remarkable low field nonlinearity in M(H) under out -of-plane field , as shown in the \ninset of Fig. 3e. Such nonlinearity has been verified with multiple samples and careful r emoval of \nbackground signal from the sample holder. The observed low field sublinear M(H) in lightly \nsubstituted compounds is in sharp contrast with the linear M(H) in pristine NiPS 3 and MnPS 3, \nimplying the development of ferromagnetic component with substitution. Such behavior has also \nbeen observed in Zn -substituted MnPS 3. The previous study  [58] on Zn substitution for Mn in \nMnPS 3 suggests the breakdown of long -range magnetic order due to non -magnetic impurities in \nlocal substituted regions, which leads to “weakly bound” Mn moments. The polarization of these \nMn moments causes the low -field nonlinear M(H). Another study  [57] also propose that for MnPS 3 \nin which the dipolar anisotropy dominates, the local dipole field is along the out -of-plane direction. \nAs the consequence, when Mn -magnetism is diluted by Zn substitution, the absence of the \nmagnetic moment of one Mn site would affect the closest Mn in the neighboring layers and cause \ntheir magnetic moments to be canted, leading to an average staggered  magnetic  moment  in a larger \nscale  [57]. Similarly, the nonlinear M(H) at low fields in our Ni1-xMn xPS3 samples can likely be \nattributed to the polarization of canted moment  originat ing from substitution.  This scenario is \nfurther supported by the t emperature dependence of susceptibilit y measurements. As shown in Fig. 8 4a, consistent with the nonlinear M(H) seen under H||ab in the x = 0.05 sample (Fig. 3d, inset), the \nin-plane susceptibilit y (χ||) display s a clear upturn at low temperatures  that suggest s the \ndevelopment of a ferromagnetic component due to moment canting . Likewise , close to the MnPS 3 \nside, for x = 0.95 sample in which the nonlinear M(H) is significant (Fig. 3e, inset), a susceptibilit y \nupturn is also seen, as shown in Fig. 4 c. The upturn become more obvious with increasing the Ni \ncontent to 10% ( x = 0.9), as shown in Fig. 4 b. \nThe tunable SF transition and the presence of a ferromagnetic component by Ni -Mn \nsubstitution is expected  because of  the distinct magnetic structures in Ni PS3 and MnPS3. However, \nthe observed high sensitiv ity to light substitution is surprising. For example, only 5% Ni \nsubstitution can substantially reduce HSF by half  in MnPS 3, which is much less than the amount of \nZn needed  (~20%)  to reduce HSF by the same amount  [58,65] . The HSF suppression by non-\nmagnetic Zn substitution has been ascribed to the reduction of magnetic anisotropy with magnetic \ndilution  [58]. The Ni -Mn substitution in our Ni1-xMn xPS3, however, induces magnetic impurities. \nHence , it is necessary to consider the magnetic interaction to understand the observed sensitive \ndoping dependence  in Ni1-xMn xPS3. In MPX3 compounds, magnetism has been found to be related \nto a structur al distortion [23,80 –83]. As illustrated in Fig. 3 c, every metal atom M in MPX3 is \nlocated at the center of an octahedr on form ed by six X atoms. Such MX6 octahedra, however, \npossess  a trigonal distortion that is characterized by the angle θ between the trigonal axis \n(perpendicular to  the ab-plane) and the M-S bond. Therefore, magnetism in MPX3 can be described \nby an isotropic Heisenberg Hamiltonian  with additional single -ion anisotropy  due to  the combined \neffect of crystal field and spin -orbit splitting: 𝐻=−2∑𝐽𝑆𝑖𝑆𝑗+𝐷𝑆𝑖𝑧2, where J and D are the \nexchange and crystal field parameters, respectively  [37]. The relative strength of J and D leads to \nvarious magnetic structure s in MPX3, so t he trigonal distortion plays critical role in determining 9 the nature of magnetic interactions in  MPX3 [37,80] . In NiPS 3 and MnPS 3, θ has been found to be \n51.05° and 51.67°, respectively  [80]. Compared with θ ≈ 54.75°  for the undistorted octahedra, the \ngreater trigonal distortion in NiPS 3 leads to much stronger single -ion anisotropy (0.3 meV) than \nthat of MnPS 3 (0.0086 meV), as revealed by inelastic scattering measurements  [42,80,81] . This \ncauses the Ni moments to be aligned within the basal p lane perpendicular to the trigonal axis, with \na small out -of-plane component likely due to weak dipolar anisotropy , as shown in the insets of \nFigs. 2c  [37,43,82] . In contrast, the effect of crystal field and spin -orbit splitting is found to be \nnegligible for the high spin ground state of Mn2+ in MnPS3 [37], so the magnetism of the less \ndistorted MnPS 3 is gove rned by the dipolar anisotropy that leads to out -of-plane moment direction \nwith small tilt towards a-axis  [42,69,82] , as shown in the inset of Fig. 2b. Therefore, the tuning \nof trigonal distortion by Ni -Mn substitution would signi ficantly affect the magnetic isotropy and \nfurther efficiently modify the SF transitions.  A similar mechanism has also been proposed  for the \nsuppression of HSF in MnPS 3 under  pressure  [82].  \nIn addition to magnetic anisotropy, magnetic exchange interaction may also play an \nimportant role. The magnetism i n MPX3 is mainly mediated through  superexchange  interaction , \nthereby affected by the M-X-M bonding angle  [37,81] . Additionally, t he d-electron occupancy of \nM is important in determin ing the sign and the nature of the superexchange  [37,80,81] . The 3 d \norbitals of  M2+ ion consist of high energy eg and low energy t2g groups, and their occupanc ies \ndetermine the strength of the exchange interactions  [80,83] . This has recently been demonstrated \nby the inelastic neutron scattering measurement s which reveal s reduced exchange interactions in \nCoPS 3 as compared to NiPS 3 due to distinct occupancy of t2g orbital  [80]. A similar scenario should \nalso occur in Ni1-xMn xPS3, in which the Ni -Mn substitution  modif ies the magnetic exchange.  \nIndeed, t he magnitudes of all the exchanges, except the weakest second nearest -neighbor  10 interaction ( J2), are found to systematic ally incre ase with decreasing spin on M2+ ion and \nincreasing M-S-M bond angles from MnPS 3 to NiPS 3 [80,81,83] . \nAmong the magnetic anisotropy and magnetic exchange interactions, the former may \ngovern the SF transitions in Ni1-xMn xPS3. The magnetic ordering temperature  be related to the \nstrength of  the magnetic interaction . We have extracted TN for Ni1-xMn xPS3 from  magnetic \nsusceptibility  and heat capacity measurements , as summarized in Fig. 4 ( e). For both NiPS 3 and \nMnPS 3, TN decreas es with substitution until reach ing to a minimum value of ~12 K in x = 0.5  \nsample . A similar trend has also been observed in other mixed MPX3 systems such as Mn 1-\nxZnxPS3 [65], Mn 1-xFexPS3 [61], and Mn 1-xFexPSe 3 [66]. Several mechanisms  [65] [61] [66]  have \nbeen proposed for the suppression of TN, such as the attenuati on of the magnetic interaction due to \ndisordered arrangements of the mixed metals ions  with dissimilar ionic radius and outer shell \nelectrons , and magnetic frustration among the 3d metal ions in the magnetic sites  owing to \ncompetition between two different AFM structures . The strong variation of TN in heavily \nsubstituted Ni1-xMn xPS3 samples may also share similar scenarios. However, for lightly substituted \nsamples with x close to 0 or 1, TN only changes slightly as compare d to the  parent compound s \nNiPS 3 and MnPS 3. It is quite interesting to find that the light Ni -Mn substitution only weakly alters \nthe magnetic ordering temperature but drastically suppresses  the SF transitions. This suggest s that \nthe efficient suppression of the SF transition with light magnetic substitution in Ni1-xMn xPS3 can \nbe attributed to the tuning of single ion isotropy rather than exchange interaction . \nIn conclusion, we have demonstrated  very efficient tunning of the  SF transition by light \nNi-Mn substitution , which is likely attribute d to single ion isotropy tuned by trigonal distortion.  \nSuch  strong sensitivity suggest s that magnetic substitutions can be an effective technique to control \nmagnet ism in MPX3 vdWs magnetic materials , leading to a  deep er understanding of low 11 dimensional magnetism and providing insight into strategies for  future magnetic  device \ndevelopment .  \n \nAcknowledgements  \nThis work is primarily suported by US Department of Energy, Office of Science, Basic Energy \nSciences program under Award No. DE -SC0019467 . R. 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(b) Crystal structure of  MPS3 (M = Ni/Mn).   \n24  \n \n \n             \n             \n             \n             \n             \n             \n             \nFIG. 2.  Field induced spin flop transition s in pristine MnPS 3  and NiPS 3 . (a-b) Isothermal  \nmagnetization of MnPS 3 at different temperatures under (a)  H⊥ab and (b) H||ab. Inset  in (b) : \nmagnetic  structure of MnPS 3 . (c -d) Isothermal  magnetization of NiPS 3  at different \ntemperatures under  (a) H⊥ab and (d)  H||ab. Upper inset  in (c) : 3D view of magnetic  structure \nof NiPS 3. Lower inset  in (c) : Top view of magnetic  structure of NiPS 3. Inset  in (d) : Isothermal  \nmagnetization at 50 – 140 K measured with  H||ab. The same color code is used indicate \ntemperatures  for (a -d). (e) Isothermal  magnetization at 2 K under  H||ab with different in -plane \nfield orientation s. Inset: Optical microscope image of NiPS 3 single crystal with arrows pointing \nthe applied field direction. (f) Temperature dependence of susceptibility of NiPS 3 measure with  \n1, 3, 5, 7, 8, and 9 T  fields, measured with  H||ab. Inset: Temperature dependence  of \nsusceptibility at 9 T under  H||ab. The solid and doted lines represent zero -field cooled (ZFC) \nand field -cooled (FC) data, respectively.   25              \n       \n \n \n \n \n \n \n \n \n \n FIG. 3.  (a-b) Isothermal  magnetization of Ni1-xMn xPS3 samples (0 ≤ x ≤ 1) at T = 2 K under \n(a) H||ab and (b) H⊥ab. The same color code is used  to indicate temperatures  for (a) and (b). \n(c) Trigonal distortion in MPS3 compounds. z-axis is the trigonal axis.  (d) Isothermal  \nmagnetization of  Ni1-xMn xPS3(x = 0 and 0.05) at T = 2 K under  H||ab (Left panel) and H⊥ab \n(Right panel). Inset: zoom in of the l ow-field magnetization. (e) Isothermal  magnetization of \nNi1-xMn xPS3  (x = 0.95 and 1) at T = 2 K under  H||ab (Left panel) and H⊥ab (Right panel). \nInset: zoom in of the l ow-field magnetization.  \n26  \n \n \n \n \n \n \n \n \n  \n \n \n \nFIG. 4.  (a-c) Temperature dependence of out -of-plane ( H⊥ab, red, upper panels)  and in-plane \n(H||ab, black, lower panels) susceptibilit ies of (a) Ni0.95Mn 0.05PS3, (b) Ni0.1Mn 0.9PS3, and (c) \nNi0.05Mn 0.95PS3  samples.  The dashed lines denote TN. (d) Temperature dependence of heat \ncapacity of Ni1-xMn xPS3 samples with x = 0, 0.1, 0.2 , 0.8 and 1 . The black  triangles denote TN. \nData  are shifted for clarity . (e) Magnetic phase diagram of Ni1-xMn xPS3  (0 ≤ x ≤ 1). The \ntransition temperatures are determined by susceptibility measurements on single - (χ-single \ncrystal) and poly -crystals ( χ-polycrystal), and heat capacity ( C). \n"    },    {        "title": "2105.00396v1.Magnetic_textures_in_a_hexaferrite_thin_film_and_their_response_to_magnetic_fields_revealed_by_phase_microscopy.pdf",        "content": "1 \n Magnetic textures in a hexaferrite thin film and their response \nto magnetic fields revealed by phase microscop y \nAtsuhiro Kotani1, Ken Harada2, Marek Malac3,4, Hiroshi Nakajima1, Kosuke Kurushima5 \nand Shigeo Mori1 \n \n1Department of Materials Science, Osaka Prefecture University, Sakai, Osaka 599 -8531, Japan  \n2Center for Emergent Matter Science, The Institute of Physical and Chemical Research (RIKEN), \nHatoyama, Saitama 350 -0395, Japan  \n3Nanotechnology Research Centre, National Research Council (NRC), Edmonton, Alberta T6G 2M9, \nCanada  \n4Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada  \n5Toray Research Center, Otsu, Shiga 520 -8567, Japan  \n \nWe investigated magnetic textures in a Sc-doped  hexaferrite film by means of phase \nmicroscopy (PM) with a hole -free phase plate in a transmission electron microscope. In \na zero magneti c field, the stripe -shaped magnetic domains coexist with magnetic bubbles. \nThe magnetization in both magnetic domains was oriented perpendicular to the film and \nthe domain walls have an in -plane magnetization. In the remnant state at 9.2 mT,  several  \nmagnetic bubbles were formed with the formation of stripe -shaped magnetic domains, \nand the out -of-plane component in the stripe -shaped domains gradually appeared as the \nfilm thickness increased. As the film thickness increases further, the magnetic bubbles \nwith clockwise or counter -clockwise spin helicities formed a triangular lattice. These \nresults in the remnant state suggest that the domain wall energy in the magnetic bubble \ndomains is lower in the thicker region.  \n  2 \n 1. Introduction  \nPhase microscopy (PM), especially with a hole -free phase plate (HFPP) in a transmission \nelectron microscope (TEM), has been used to enhanc e phase contrasts of materials \nconsisting of the light elements in biological fields  [1–3]. PM has a potential advantage \nthat the high ly magnified  images can be obtained in-focus , thus not  suffering from Fresnel \nfringes caused by defocusing  [4,5] . Unlike in electron holography, PM observation  does \nnot require a reference wave. Therefore, PM has been utilized for imaging the \nmagnetization distribution of magnetic textures  [6]. For example, skyrmions  [7–10], \nvortex -like magnetic textures were observed using PM with an HFPP, and the semi -\nquantitative magnetic distribution was re produced from the acquired PM image  [11]. \nIn this paper, w e report that PM with an HFPP can be applied to observ e the \nnanoscale magnetic texture s, such as stripe -shaped magnetic domains and magnetic \nbubbles  [12–16], in a uniaxial ferromagnet Sc -substituted M-type hexaferrite, \nBaFe 10.35Sc1.6Mg 0.05O19 (BFSMO). It has been revealed using Lorentz micros copy (LM) \nthat magnetic bubbles, which have the vortex -like spin configuration similar to skyrmions, \nare formed by the application of the external magnetic field along the magnetic easy axis \nparallel to the c axis in BFSMO  [17–19]. It can be, however, difficult to detect and \ninterpret the spatial variations in the magnetization of samples using LM. The detailed \nmagnetization distributions of magnetic domain structures are required to understand the \nformation mechanisms of magnetic bubbles. PM observations revealed semi -quantitative \nmagnetization distributions of stripe -shaped magnetic domains, magnetic bubbles, and \ntheir responses to  applied external magnetic fields. Moreover, the thickness dependence \nof the magnetization distributi on in the magnetic textures in the remnant state was \nrevealed  in the PM observation . \n \n2. Experimental methods   \nA single crystal of BFSMO specimens was synthesized via the floating zone method  [20]. \nSpecimens for TEM observation were thinned using Ar ion milling. The observations \nusing PM with an HFPP were performed in a 300 kV TEM (Hitachi HF -3300). We utilize d \na 13 nm thick amorphous carbon film prepared by electron beam evaporation as an \nHFPP  [1]. After the installation of the HFPP to the microscope column, the HFPP was 3 \n heated to approximately 200 °C and kept at that temperature during the experiment to \nprevent the HFPP film from contamination  [21,22] .  \nWe used the PM optics constructed in previous studies  [11], as illustrated in Fig . \n1. The HFPP was placed at the selected -area ap erture plane, and the condenser lens was \nadjusted to construct the crossover at the HFPP. The application of magnetic fields \nperpendicular to the thin film was achieved by ex citing the objective lens. The deviation \nof the crossover from the selected apertu re position caused by exciting the objective lens \nwas compensated by using the condenser lens. Image focusing was achieved by adjusting \nthe excitation of the intermediate lens.  \nThe phase shift  is expressed as follows  [23]: \nx,y = CEV0(x,y)t(x,y) – 2πe\nh∫B(x,y)dS.    (1) \nHere, CE and V0 and t, are the interaction constant which is 0.00652 rad V-1 nm-1 for 300 \nkeV electrons, the mean inner potential, and the specimen thickness, respectively.  B is the \nmagnetic flux of the specimen magnetization. The phase shift depends on both the mean \ninner potential and the specimen thickness. Assum ing that the spatial change in the \nthickness can be negligible for the change in magnetization, t he magnetization maps can \nbe obtained from gradients of the phase distribution on the PM image acquired with an \nHFPP as follows  [11]: \n   B = h\n2πex, y ∝ h\n2πe(x,y) = h\n2πe(𝜕𝐼𝑥\n𝜕𝑥, 𝜕𝐼𝑦\n𝜕𝑦).\nHere, B,  and I, are the magnetic flux of the specimen magnetization, the phase shift due \nto the magnetic flux , and the intensity of the phase image. The absolute value of the in -\nplane magnetization can be obtained from the following equation,  \n         |B| = h\n2πe√(∂I\n∂x)2\n+(∂I\n∂y)2\n.          (3) \nNote that the above equations are valid when an image recording device has a linear \nrelationship between the number of detected electrons on the detector and the output \nintensity on the display, providing that the phase shift is proportional to the image contrast. \nUsing these methods, we obtained magnetization maps from the PM images under \nmagnetic fields applied externally.  \n 4 \n 3. Results and discussion  \nFirst, magnetic textures of BFSMO in zero magnetic field at room temperature were \nexamined by PM observation s. Figure  2(a) shows  a PM image, showing the stripe -shaped \nmagnetic domains and several magnetic bubbles. To extract the magnetization map of the \nBFSMO specimen in Fig . 2(b), differential images of the inte nsity in Fig . 2(a) were \ncalculated according to Eq. (2). Figures 2(c) and (d) show the gradient images of the x \nand y directions, respectivel y. The white and black indicate positive and negative \ndifferential values, respectively. The magnetization vector map in Fig . 2(b) was obtained \nfrom Fig . 2(c) and (d ). The color map indicates the direction of magnetization, coded \naccording to the color wheel while the color saturation indicates the in -plane component \nof the magnetization intensity calculated using eq. (3).  \nIn Fig . 2(b) the in -plane magnetization is indicated by white arrows . It shows the \nmagnetization in the domain walls between the stripe -shaped magnetic domains is \noriented parallel to the in -plane direction . In-plane component of the magnetization was \nnot detected in the stripe -shaped magnetic domains  and the magnetic bubbles , which \nsuggest s the magnetization in those domains is mainly oriented parallel to the easy axis \n(perpendicular to the thin film). Furthermore, it appears that the magnetization i n the \ncircular domain walls of the magnetic bubbles rotate s clockwise (CW) or \ncounterclockwise (CCW) in the plane  of the thin film . The magnetization distribution of \nthe magnetic bubbles obtained with PM is similar to that obtained by phase \nreconstructions  through an iterative calculation using a series of 32 defocused images  [24]. \nThe magnetic texture varied depending on the strength of the applied external \nmagnetic field. Figures 3(a) and (b) show the PM image and the magnetization map at 80 \nmT of the magnetic field applied, respectively. It can be seen in Fig . 3(a) that the width \nof the stripe -shaped magnetic domains is decreased and simultaneously the diameter of \nthe magnetic bubbles is reduced as the strength of the magnetic field applied is increasing \ndue to the Zeeman effect.  \nA Bloch line is formed, as indicated by the white arrowhead in Fig . 3(b). Note that \na Bloch line is characteristic for the reversal of the domain -wall chirality, in which it has \nbeen accepted that the directions of the  in-plane magnetization are reversed by gradual \nrather than abrupt rotation . The magnetization distribution obtained from Fig . 3(b) is 5 \n schematically shown in Fig . 3(c). It has been recently recognized that the Bloch line plays \nan important role in the form ation of magnetic bubbles  [19]. As understood by comparing \nFigs. 2(b) and 3(b), a Bloch line was formed from the stripe -shaped magnetic domains by \napplying an external magnetic field.  \nAs the strength of the magnetic field increases up to 120 mT, magnetic bubbles \nare formed from the stripe -shaped magnetic domains. Figure  3(d) shows the PM image \nof magnetic bubbles formed at 120 mT. Figure 3(e) is the magnetization map of magnetic \nbubbles indicated by the white dotted lines in Fig . 3(d). The magnetic bubbles have CW \nor CCW  rotation of the magnetization. From the magnetization map s, the diameter of the \nmagnetic bubble inside the circular domain wall can be estimated to be approximately \n230 nm at 0 mT, 150 nm at 80 mT, and 100 nm at 120 mT, respectively. These results \nshow  that the diameters of magnetic bubbles decrease as the strength of the magnetic field \nincreases . \nAfter the external magnetic field up to 2 T  was applied by exciting the objective \nlens, the lens was turned off to reduce the magnetic field quickly down to  9.2 mT . Figure \n4 shows that in the remnant state numerous magnetic bubbles were formed in the region \nfar from the edge of the specimen and the magnetic stripe domain exists in the region near \nthe edge. The specimen in this study was made by ion milling  and has a wedge -like \nthickness profile with progressively  lower thickness closer to the edge. The thickness \nvalues in the region indicated by I, II, and III in Fig . 4 measured using electron energy -\nloss spectroscopy (EELS) are 39 nm, 96 nm, and 163 nm, re spectively. Therefore, it is \nshown that the stripe -shaped magnetic domains are formed in the thinner region and \nmagnetic bubbles are formed in the thicker region.  \nHere, the magnetization distribution of stripe -shaped magnetic domains and \nmagnetic bubbles a re discussed. PM observations revealed that changes in the \nmagnetization distribution of the stripe -shaped magnetic domains and the transformation \nfrom the domains into magnetic bubbles with increasing the film thickness. Figure 5 \nshows the three magnetiza tion maps of the regions I, II, III in Fig . 4 calculated using eq . \n(3). The white arrows indicate the magnetization direction. The magnetization map \nobtained from I shows that the magnetization in the stripe -shaped magnetic domain \nstructure tends to the in -plane direction. Figure 4 shows  that the out -of-plane component 6 \n of magnetization in the magnetic domain increases as the film thickness increases. In the \nmagnetization map II, the dark region where the magnetization is oriented perpendicular \nto the thin f ilm in the stripe -shaped magnetic domain expands. These results show that \nthe thickness dependence of the magnetization distributions of stripe -shaped magnetic \ndomains can be clarified using PM with an HFPP.  \nFor stripe -shaped magnetic domains, it has been reported that the thickness -driven \nreorientation of the magnetization from in -plane to perpendicular is caused  [25,26]  \nbecause the demagnetization field whose contribution favors an in -plane preferential \norientation fo r the magnetization decreases as the thickness increases. The critical \nthickness, tC, of the reorientation is given as follows:  \ntC ~ 27.2 MS2 / KU3/2,     (4) \nhere A is exchange stiffness constant, MS is saturation magnetization, and KU is uniaxial \nanisotropy constant. The parameters of the BFSMO specimen were reported to be  A = \n1.3×10-6 erg/cm, MS = 286 emu/cm3, KU = 5.3 ×105 erg/cm3 [27], resulting in tC ~ 67 nm. \nThe PM observations in Fig . 5(I) and (II) experimentally show the reorientation of the \nmagnetization across tC. \nAs the film thickness increases to more than 100 nm, magnetic bubbles are formed. \nFigure 5(III) shows a magnetization map of the region III in Fig . 4. The magnetization \ndistribution of magnetic bubbles with the CW or CCW spin rotation in the magnetic \ndomain wall can be seen clearly, and those magnetic bubbles are formed locally in a \ntriangular lattice. As shown in the magnetization maps in Figs . 2 and 3, black and white \nballs in Fig . 4 correspond to the magnetic bubbles with CW and CCW rotation of the \nmagnetization, respectively. It can be seen in Fig . 4 that the same quantity of those \nmagnetic bubbles exist s and the magnetization helicities  are oriented randomly . Therefore,  \nit is suggested that the energies in the magnetic bubbles with the CW or CCW helicities \nare equivalent in the remnant state. It appears that the formation of magnetic bubbles in \nthe remnant state depends on the film thickness.  \nIt was reported that the magnetic energ y of the stripe -shaped magnetic domain is \nhigher than that of the magnetic bubble by magnetic domain -wall energy  [28]. The wall \nenergy W of a Bloch domain wall in a uniaxial ferrimagnet is expressed as follows  [29]: \nW = √AK 7 \n The effective magnetic anisotropy  K related to W decrease as the thickness increases  [30]. \nThus, judging fr om eq . (4), the formation of magnetic bubbles in the remnant state is \ninduced by lower W in the thicker region as shown in Fig . 5(III). The mechanism of \nthickness dependence in the formation of magnetic bubbles in the remnant state will be \nrevealed in the  future by the experiments with thickness -controlled films and theoretical \nsimulations.  \n \n4. Conclusions  \nIn conclusion, PM with an HFPP was utilized to reveal the magnetization distributions of \nmagnetic textures, such as stripe -shaped magnetic domains and magnetic bubbles in \nBFSMO and their response to external magnetic fields applied perpendicular to the thi n \nfilm. The stripe -shaped magnetic domains and a few magnetic bubbles coexisted in a zero \nmagnetic field. PM observation revealed that the magnetic domain structures consist of \nthe domain wall with the in -plane magnetization and the domain with the magneti zation \nperpendicular to the film. In the remnant state after the external magnetic field was \napplied up to 2 T and was decreased, many magnetic bubbles with the stripe -shaped \nmagnetic domains were observed. It was revealed the changes in magnetizati on \ndistribution in the stripe -shaped magnetic domains as the film thickness increased. In \naddition, it was revealed that magnetic bubbles with the CW or CCW spin helicities were \nformed in a triangular lattice as the film thickness increased furthermore. Th e PM with \nan HFPP will be one of the powerful tools to analyze the magnetic distribution of complex \nmagnetic textures.  \n \nAcknowledgments  \nThe authors greatly acknowledge Kai Cui and Mark Salomons  for technical support \nduring PM experiments and Jean Nassar for support of programming using Python code s. \nThis work was partially supported by JSPS KAKENHI (No. 18J12180) and Graduate \nProgram for System -inspired Leaders in Material Science (SiMS) of Osaka  Prefecture \nUniversity . This work was also supported by  the National Research Council (NRC) in \nCanada. The experiments were made possible by outstanding support from Hitachi Hig h \nTechnologies for the Hitachi HF -3300 microscope at NRC -Nano.  \n  8 \n References  \n[1] M. Malac, M. Beleggia, M. Kawasaki, P. Li, and R. F. Egerton, Ultramicroscopy 118, 77 \n(2012).  \n[2] R. Danev, B. Buijsse, M. Khoshouei, J. M. Plitzko, and W. Baumeister, Proc. Natl. Acad. \nSci. 111, 15635 (2014).  \n[3] Y. Fukuda, U. Laugks, V. Lučić, W. Baumeister, and R. Danev, J. Struct. Biol. 190, 143 \n(2015).  \n[4] M. Malac, S. Hettler, M. Hayashida, M. Ka wasaki, Y. Konyuba, Y. Okura, H. Iijima, I. \nIshikawa, and M. Beleggia, Micron 100, 10 (2017).  \n[5] M. Hayashida, K. Cui, A. M. Najarian, R. McCreery, N. Jehanathan, C. Pawlowicz, S. \nMotoki, M. Kawasaki, Y. Konyuba, and M. Malac, Micron 116, 54 (2019).  \n[6] S. Pollard, M. Malac, M. Beleggia, M. Kawasaki, and Y. Zhu, Appl. Phys. Lett. 102, \n192101 (2013).  \n[7] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. 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Mori, Microscopy 65, 473 (2016).  \n[17] X. Yu, M. Mostovoy, Y. Tokunaga, W. Zhang, K. Kimoto, Y. Matsui, Y. Kaneko, N. \nNagaosa, and Y. Tokura, Proc. Natl. Acad. Sci. 109, 8856 (2 012).  \n[18] X. Z. Yu, K. Shibata, W. Koshibae, Y. Tokunaga, Y. Kaneko, T. Nagai, K. Kimoto, Y. \nTaguchi, N. Nagaosa, and Y. Tokura, Phys. Rev. B 93, 134417 (2016).  \n[19] H. Nakajima, A. Kotani, K. Harada, Y. Ishii, and S. Mori, Phys. Rev. B 94, 224427 (2016).  \n[20] Y. Tokunaga, Y. Kaneko, D. Okuyama, S. Ishiwata, T. Arima, S. Wakimoto, K. Kakurai, \nY. Taguchi, and Y. Tokura, Phys. Rev. Lett. 105, 257201 (2010).  \n[21] S. Hettler, M. Dries, P. Hermann, M. Obermair, D. Gerthsen, and M. Malac, Micron 96, 9 \n 38 (2017).  \n[22] S. Hettler, E. Kano, M. Dries, D. Gerthsen, L. Pfaffmann, M. Bruns, M. Beleggia, and M. \nMalac, Ultramicroscopy 184, 252 (2018).  \n[23] H. S. Park, X. Yu, S. Aizawa, T. Tanigaki, T. Akashi, Y. Takahashi, T. Matsuda, N. \nKanazawa, Y. Onose, D. Shindo, A. To nomura, and Y. Tokura, Nat. Nanotechnol. 9, 337 \n(2014).  \n[24] T. Tamura, Y. Nakane, H. Nakajima, S. Mori, K. Harada, and Y. Takai, Microscopy 67, \n171 (2018).  \n[25] N. Saito, H. Fujiwara, and Y. Sugita, J. Phys. Soc. Japan 19, 1116 (1964).  \n[26] M. Hehn, S. Pa dovani, K. Ounadjela, and J. P. Bucher, Phys. Rev. B 54, 3428 (1996).  \n[27] K. Kurushima, K. Tanaka, H. Nakajima, M. Mochizuki, and S. Mori, J. Appl. Phys. 125, \n53902 (2019).  \n[28] J. A. Cape and G. W. Lehman, J. Appl. Phys. 42, 5732 (1971).  \n[29] A. Hubert a nd R. Sch¨afer, Magnetic Domains: The Analysis of Magnetic \nMicrostructuresTitle  (Springer Science & Business Media, 1998).  \n[30] K. Chesnel, A. S. Westover, C. Richards, B. Newbold, M. Healey, L. Hindman, B. Dodson, \nK. Cardon, D. Montealegre, J. Metzner, T.  Schneider, B. Böhm, F. Samad, L. Fallarino, \nand O. Hellwig, Phys. Rev. B 98, 224404 (2018).  \n \n  10 \n  \nFig. 1.  A schematic illustration of the optic for PM using an HFPP. The red arrows indicate \na magnetic field applied by a weakly excited objective lens.  \n \n  \n11 \n  \nFig. 2 . (a) A PM image in a zero magnetic field  at room temperature . (b) The \nmagnetization map calculated fr om the differential images of (a) using the areas \nsurrounded by the white dotted lines.  The color and the white arrows indicate \nmagnetization direction as described by the color wheel. (c) and (d) showing the \ndifferential images in (a) in the x and y direc tions, respectively.  \n12 \n  \nFig. 3.  (a) A PM image and (b) a magnetization map calculated by the differentiation of \nthe intensity at 80mT using the region surrounded by the white dotted lines in (a). The \nregion is the same as that in Fig. 2(b). The white arrows indi cate the magnetization and \nthe white arrowhead shows a Bloch line. (c) The schematic image of the magnetization \ndistribution in (b), showing the Bloch line formed in the domain wall. The red arrows \nindicate the magnetic moments. (d) and (e) showing a PM image and a magnetization \nmap of the area surrounded by the white dotted lines in (d) at 120mT.  \n  \n13 \n  \nFig. 4.  The image of the magnetic bubbles formed in the remnant state after a rapid field \nchange from 2T field saturation to 9.2 mT.  \n \n \n  \n14 \n  \nFig. 5.  The magnetization maps of the regions I, II, and III in Fig. 4.  \n \n"    },    {        "title": "2105.04523v1.Model_studies_of_topological_phase_transitions_in_materials_with_two_types_of_magnetic_atoms.pdf",        "content": "Model studies of topological phase transitions in materials\nwith two types of magnetic atoms\nZhuoran He1and Gang Xu1,\u0003\n1Wuhan National High Magnetic Field Center &School of Physics,\nHuazhong University of Science and Technology, Wuhan 430074, China\nWe study the topological phase transitions induced by Coulomb engineering in three triangular-\nlattice Hubbard models AB 2,AC3andB2C3, each of which consists of two types of magnetic\natoms with opposite magnetic moments. The energy bands are calculated using the Schwinger\nboson method. We \fnd that a topological phase transition can be triggered by the second-order\n(three-site) virtual processes between the two types of magnetic atoms, the strengths of which are\ncontrolled by the on-site Coulomb interaction U. This new class of topological phase transitions\nhave been rarely studied and may be realized in a variety of real magnetic materials.\nI. INTRODUCTION\nTopological phase transitions [1{4] play a key role in\ncondensed matter physics. Especially, magnetic topolog-\nical systems [5{7] often exhibit rich topological phases\ndue the complicated interplay between electron-electron\ninteractions, magnetic moments and spin-orbit coupling,\nwhich have been attracting intensive research interests\nfor years [8{10]. A general model for the description of\nmagnetic topological insulators is the spin-orbit coupled\nHubbard model [11, 12] with on-site Coulomb interac-\ntionU. Previous works on Coulomb engineering and\ncorrelation-driven e\u000bects in magnetic topological systems\nhave studied various aspects of this topic including\nthe Hartree-Fock mean-\feld theory [6, 13], dynamical\nscreening e\u000bects [14], and phase transitions due to the\nmagnetic exchange coupling [15{17] using the Schwinger\nboson method. These works mostly focus on systems\nwith one type of magnetic atom, while the topological\nphase transitions in systems with two types magnetic\natoms are comparatively less studied.\nIn this paper, we study systems with two types of mag-\nnetic atoms [18{21] with opposite magnetic moments. In\nsuch systems, the two types of magnetic atoms form two\nsets of Chern bands separately, which then interact via a\ntype of second-order virtual process of order O(t1t2=U).\nThese processes involve the hopping from one type of\nmagnetic atom ito atomjvia the other type of magnetic\natomkas an intermediate site. We call these 1 =U-\ncontrolled virtual processes the three-site terms, which\ncan induce interesting topological phase transitions. We\nstudy their e\u000bect in a 2D hexagonal Hubbard model\nwith 3 types of lattice sites A;B;C forming triangular,\nhoneycomb and Kagome sublattices, respectively. By\nputting spin-up and spin-down electrons on two of the\nthree types of lattice sites, we consider AB2,AC3\nandB2C3models and realize 1 =U-controlled topological\nphase transitions as characterized by changes in the\nChern numbers of the spin-up and spin-down bands. Our\n\u0003Electronic address: gangxu@hust.edu.cnresults demonstrate the interplay between band topology\nand correlation e\u000bects, and present Coulomb engineering\nas a powerful tool to manipulate the topological phases of\nmatter, with potential applications in various solid-state\nphysical systems.\nThe rest of the paper is organized as follows. In\nSec. II, we give the general formalism of our downfolding\ntechnique in the Schwinger boson representation and\nobtain the low-energy e\u000bective Hamiltonian containing\nthe three-site terms. In Sec. III, we apply our formalism\nto theAB2,AC3andB2C3lattice structures to demon-\nstrate the 1 =U-controlled topological phase transitions.\nSection IV is a summary and conclusion, with discussions\nof potential materials to realize the topological phase\ntransitions found in our model studies.\nII. FORMALISM\nSuppose an insulating magnetic material is described\nby the Hubbard model\nH=X\nij\u000b\ft\u000b\f\nijcy\ni\u000bcj\f+UX\nini\"ni#; (1)\nwherei;jare the site indices and \u000b;\flabel the spin. In\nthe largeUlimit, electrons try to avoid double occupancy\nand thus each site becomes spin-polarized so as to\nform di\u000berent long-range orders such as ferromagnetism,\nantiferromagnetism, ferrimagnetism, etc. [22, 23] In the\nSchwinger boson representation [24, 25], the electron\noperator can be represented as\ncy\ni\u001b=by\ni\u001bhi+\u001bdy\nibi\u0016\u001b; (2)\nwhere\u001b=\"(+1);#(\u00001) is the spin index, hianddi\nare the fermionic holon and doublon operators, bi\u001b;bi\u0016\u001b\nare the Schwinger boson operators, and \u0016 \u001b=\u0000\u001bis the\nopposite spin of \u001b. By using the downfolding formula\nHe\u000b=PHP\u00001\nUPH\u0016PHP +O\u00121\nU2\u0013\n; (3)arXiv:2105.04523v1  [cond-mat.str-el]  10 May 20212\nwherePis the projection operator into the Hilbert\nspace of no doublons, we obtain the low-energy e\u000bective\nHamiltonian of the chargeons\nHe\u000b=X\nij~tijhihy\nj=X\nij~tijfy\nifj: (4)\nA particle-hole transformation has been done from the\nholonshi7!fy\nito the chargeons, with the e\u000bective\nhopping amplitudes ~tijgiven by\n~tij=X\n\u000b\fby\ni\u000b0\n@t\u000b\f\nij\u00001\nUX\nk\r\u000e\r\u000et\u000b\u000e\nikt\r\f\nkjby\nk\u0016\u000ebk\u0016\r1\nAbj\f:(5)\nThe derivation of Eqs. (4){(5) will be given in Appendix\nA. For magnetically ordered systems, the bosonic oper-\nators can be viewed as c-numbers in the Bose-Einstein\ncondensation (BEC) approximation [26, 27]. Previous\nworks on topological phase transitions mostly focus on\nthose transitions induced by changes of the electronic\nhopping amplitudes t\u000b\f\nij, which may give rise to gap\nclosing, band inversion [28, 29], etc. Here with Eq. (5), we\ncan study two more types of topological phase transitions\nin terms of ~tij, i.e., a) those induced by changing the\nmagnetic structure and b) those induced by 1 =Uthat\ncontrols the strengths of the three-site virtual processes.\nThis paper focuses on the latter situation. We consider\ngap closing of the chargeon bands induced by the change\nof Hubbard Uwithout changing the magnetic structure.\nFor simplicity, we consider a special case for 2D sys-\ntems that the bare hopping t\u000b\f\nij=t\u000b\nij\u000e\u000b\fconserves spin\nand that the magnetic structure is collinear ferrimagnetic\nin thezdirection. Since the magnetic moments have zero\nx;ycomponents, and no double occupancy is allowed in\nthe largeUlimit, every site can be occupied by either\nthe spin-up electrons or the spin-down electrons only. In\nsuch a situation, Eq. (5) simpli\fes to\n~tij=X\n\u001bz\u0003\ni\u001bzj\u001b \nt\u001b\nij\u00001\nUX\nkt\u001b\nikt\u001b\nkjjzk\u0016\u001bj2!\n;(6)\nwhere the bosonic operators bi\u001b7!zi\u001bhave been mapped\ntoc-numbers. Now we have a Hamiltonian with two sets\nof bands formed by electrons on spin-up sites and spin-\ndown sites, which interact via the second-order virtual\nprocesses described by the three-site O(1=U) terms. In\nthis paper, we use Eq. (6) as our simpli\fed formula.\nOther magnetically ordered systems with more complex\nspin con\fgurations such as noncollinear and spiral spin\nstructures can be studied using Eq. (5).\nIII. RESULTS\nTo study the topological phase transitions within the\nframework of Eq. (6), we construct a 2D lattice structure\nAB2C3with hexagonal symmetry (see Fig. 1). The A\nFIG. 1. The 2D hexagonal lattice structure AB 2C3. The A\nsites form a triangular lattice, the Bsites form a honeycomb\nlattice, and the Csites form a Kagome lattice, all sharing the\nsame lattice vectors ~ a1and~ a2.\nsites form a triangular lattice with one band, which is\ntopologically trivial. The Bsites form a honeycomb\nlattice with two bands, which realize the Haldane model.\nTheCsites form a Kagome lattice with three bands. We\nwill put opposite magnetic moments on two of the three\ntypes of lattice sites and consider electronic phases in the\nAB2,AC3andB2C3structures, respectively.\nA. The AB 2structure\nWe consider an electronic phase with N\"=N#= 1\nper unit cell. In case that the on-site orbital energy of\nan emptyAsite is lower than that of an empty Bsite,\none of the spin species (e.g. the #electrons) would \frst\nsingly occupy the Asites. Then the other spin species\n(the\"electrons) would not occupy the Asites because\nof the Hubbard U, but instead occupy the Bsites at\noccupancy 0 :5. When the SOC is considered, the Bsites\nbecome gapped and the \"electrons realize the Haldane\nmodel with real nearest-neighbor hopping t1and complex\nnext-nearest-neighbor hopping t2. We also consider a real\npara-position hopping t3among the Bsites and denote\nthe real nearest A-Bsite hopping as t. Following Eq. (6),\nthe e\u000bective hoppings ~t1\u00003are given by\n~t1=1\n2\u0012\nt1\u00002t2\nU\u0013\n;~t2;3=1\n2\u0012\nt2;3\u0000t2\nU\u0013\n:(7)\nHere we assume the boson \felds zA#= 1,zA\"= 0 on\ntheAsites andzB\"= 1=p\n2,zB#= 0 on the Bsites.\nIn Eq. (6), when the i;jlabels are on the Bsites, we\nhave\u001b=\"and thus the klabel must be on the Asites,\nwhich are occupied by \u0016 \u001b=#, so as to mediate a three-site\nvirtual process j!k!i. All three hoppings ~t1\u00003are\nrenormalized by such three-site virtual processes. Due to\nthe three-site-enhanced hopping ~t3, theAB2model can3\nFIG. 2. The Chern bands of the Bsites (honeycomb) in the\nAB 2model. Hopping amplitudes t1=\u00000:15 eV, t2= (0:06+\n0:04i) eV, t3=\u00000:01 eV, t= 0:8 eV. Hubbard U= 10 eV in\n(a) and U= 4 eV in (b). The Chern numbers C1;2indicate a\ntopological phase transition (critical U= 5:3 eV).\nnow realize beyond-Haldane phases with occupied-band\nChern numbers\u00062.\nIn terms of the e\u000bective hoppings ~t1\u00003, the spin-up\nHamiltonian (i.e., chargeon Hamiltonian restricted to B\nsites) in atomic gauge takes the form\nHB(~k) =\"\n2Re[ ~t2\u0010\u0003\n2(~k)] ~t1\u0010\u0003\n1(~k) +~t3\u00101(2~k)\n~t1\u00101(~k) +~t3\u0010\u0003\n1(2~k) 2Re[ ~t2\u00102(~k)]#\n;(8)\nwhere the functions \u00101;2(~k) are given by\n\u00101(~k) =ei~k\u0001~ a1\u0000~ a2\n3+ei~k\u0001~ a1+2~ a2\n3+e\u0000i~k\u00012~ a1+~ a2\n3; (9a)\n\u00102(~k) =ei~k\u0001~ a1+ei~k\u0001~ a2+e\u0000i~k\u0001(~ a1+~ a2): (9b)\nA topological phase transition can be realized as shown in\nFig. 2. In Fig. 2a, the Hubbard U= 10 eV is large. The\ntwo spin species are clearly separated by the Hubbard\ninteraction with almost forbidden three-site virtual hop-\npings. The spin-up electrons form a Haldane phase on\ntheBsites with occupied-band Chern number C1= +1\nand unoccupied-band Chern number C2=\u00001. The\nFIG. 3. The Chern bands of the Csites (Kagome) in the\nAC3structure. Hopping amplitudes t1=\u0000(0:6 + 0 :2i) eV,\nt2= (0:1+0:1i) eV, t3=\u00000:25 eV, t= 0:8 eV. Hubbard U=\n10 eV in (a) and U= 4 eV in (b). The Chern numbers C1\u00003\nindicate a topological phase transition (critical U= 4:9 eV).\nspin-down electrons fully occupy the triangular sites ( A\nsites) and form a topologically trivial band (not plotted)\nwith Chern number 0. As Ugets smaller, the three-\nsite virtual processes \u0018O(1=U) become stronger and the\npara-position hopping ~t3is signi\fcantly enhanced. The\nband gap in Fig. 2a then closes at the M point at critical\nU= 5:3 eV and reopens as Uis further reduced to form a\nbeyond-Haldane phase with C1=\u00002 andC2= +2 (see\nFig. 2b for U= 4 eV). Since the contribution t2=Uof\nthe second-order virtual processes is real, the imaginary\npart Im ~t2= Imt2remains una\u000bected by U. Therefore,\nthe system can undergo topological phase transitions\nbetweenC1= +1$\u00002 (if Imt2>0) orC1=\u00001$+2\n(if Imt2<0), but not in between the C1=\u00061 (or\u00062)\nphases by tuning U.\nB. The AC 3structure\nConsider an electronic phase in which the Asites are\nsingly occupied by the #electrons and the Csites are4\noccupied by the \"electrons at occupancy 1 =3. The\nsituation is similar to AB2, except that the Csites form\na Kagome lattice. We consider the nearest-neighbor\nand next-nearest-neighbor hoppings t1,t2, and real para-\nposition hopping t3of theC-site hexagons. Both t1and\nt2can be complex. The real nearest-neighbor A-Csite\nhopping is denoted as t. From Eq. (6), we have\n~t1\u00003=1\n3\u0012\nt1\u00003\u0000t2\nU\u0013\n; (10)\nassumingzA#= 1,zA\"= 0 for the Asites andzC\"=\n1=p\n3,zC#= 0 for the Csites. In terms of ~t1\u00003, the\nC-site Kagome Hamiltonian takes the form\nHC(~k) =3X\n\u0017=1H(\u0017)\nC(~k); (11)\nwhere the nearest-neighbor hopping H(1)\nC(~k), the next-\nnearest-neighbor hopping H(2)\nC(~k) and the para-position\nhoppingH(3)\nC(~k) Hamiltonians are given speci\fcally in\nAppendix B. A topological phase transition analogous to\ntheAB2situation is realized in Fig. 3. In Fig. 3a, the\nHubbardU= 10 eV is large and the three-site virtual\nhoppings are almost forbidden. As Ugets smaller, the\npara-position hopping ~t3is signi\fcantly enhanced. The\noccupied-band Chern number changes from C1= +1 (see\nFig. 3a) to C1=\u00002 (see Fig. 3b) when the gap closes\nat the M point at critical U= 4:9 eV. In the mean time,\nthe Chern number C2of the middle band changes from\n0 to +3 and the Chern number of the \rat band on the\ntopC3=\u00001 remains unchanged.\nIn Secs. III A{III B, we have studied the enhancement\ne\u000bect of para-position hopping ~t3due the three-site\nvirtual processes proportional to 1 =U. We \fnd that in\nboth the honeycomb and the Kagome lattices, the three-\nsite processes can lead to topological phase transitions\nofC1= +1$\u0000 2 (or symmetrically C1=\u00001$+2)\nby closing the band gap at the M point. Because t2=U\nis real, we cannot realize topological phase transitions\nbetween the C1=\u00061 phases. We will demonstrate in\nSec. III C that the +1 $\u0000 1 transitions can be realized\nin the B 2C3model by making the contributions of the\nthree-site processes O(tt0=U) complex.\nC. The B 2C3structure\nIn this section, we consider an electronic phase with\nN\"=N#= 2 per unit cell. Let the two Bsites in a\nunit cell be singly occupied by the #electrons and the\nthreeCsites be occupied by \"electrons at occupancy\n2=3. We consider the hoppings t;t0between the B-C\nsites and hoppings t1;t2among the Csites as shown in\nFig. 4. Since the total Chern number of the two spin-\ndown bands on the Bsites is zero (c.f. Fig. 2a), we focus\non the topological properties of the Kagome bands, which\nFIG. 4. The hoppings considered in B 2C3. Here t; t0are\nbetween the B-Csites and t1; t2are the nearest-neighbor and\nnext-nearest-neighbor hoppings of the C-site Kagome lattice.\nAll hoppings except tcan be complex due to the SOC. The\npara-position hoppings are ignored.\nare controlled by 1 =U. From Eq. (6), we have\n~t1=2\n3\u0012\nt1\u0000t2\nU\u0013\n;~t2=2\n3\u0012\nt2\u00002tt0\nU\u0013\n; (12)\nassumingzB#= 1,zB\"= 0 andzC\"=p\n2=3,zC#= 0.\nThe para-position hopping ~t3= 0 of the Kagome lattice is\nignored. Even though ~t3can be mediated by t02=U, these\ncontributions are small assuming jtj\u001djt0j. Notice that\ntheB-Csite hopping tis real, while t0can be complex\ndue to SOC. We de\fne for \"electrons that the blue-line\nhoppings in Fig. 4 is t0in clockwise directions and ( t0)\u0003in\ncounter-clockwise directions. The Hamiltonian HC(~k) is\nstill given in Appendix B with the e\u000bective hoppings ~t1;2\nnow given by Eq. (12). A topological phase transition is\nrealized as shown in Fig. 5.\nIn Fig. 5a, the Hubbard U= 10 eV and the occupied-\nband Chern number C1+C2= +1, which is determined\nby the imaginary parts Im ~t1;2of the e\u000bective hoppings\nin the Kagome lattice. As Ugets smaller, since t2=Uis\nreal, Im ~t1remains unchanged, so only the tt0=Uterm\nin Eq. (12) can a\u000bect Im ~t2. The band gap closes at\nthe K point at critical U= 5:3 eV and then reopens\nto give rise to a C1+C2=\u00001 phase as Ufurther\ndecreases to 4 eV (see Fig. 5b). The Chern number of\nthe \rat band at the bottom C1= +1 remains unchanged\nthroughout the process. Because the imaginary part of\nthe hopping amplitudes can be tuned by 1 =U, the phase\nseparation between Chern numbers \u00061 is broken. A\ntopological phase transition between the \u00061 phases can\nnow be realized by tuning the Hubbard Udue to the\ncomplex virtual hopping O(tt0=U).5\nFIG. 5. The Chern bands of the Csites in the B2C3model.\nHopping amplitudes t1= (0 :6\u00000:1i) eV, t2=\u0000(0:1 +\n0:02i) eV, t= 0:8 eV, t0= (0:1+0:1i) eV. Hubbard U= 10 eV\nin (a) and U= 4 eV in (b). The Chern numbers C1\u00003indicate\na topological phase transition (critical U= 5:3 eV).\nIV. CONCLUSION\nWe have demonstrated in this paper that the three-\nsite virtual processes in the large Ulimit of the Hubbard\nmodel can exhibit interesting renormalization e\u000bects of\nthe hopping amplitudes and give rise to topological phase\ntransitions in the low-energy e\u000bective theory. We con-\nstructed 2D lattice models to realize the 1 =Ucontrol of\nthe honeycomb and Kagome lattices. In the AB2model,\na topological phase transition between the Haldane phase\nand beyond-Haldane phase is realized by considering the\nenhancement e\u000bect of the para-position hopping ~t3due\nto theA-site mediated virtual hoppings proportional to\n1=U. TheAC3model realizes a similar phase transition\non the Kagome lattice. Both transitions close the band\ngap at the M point. In the B2C3model, we also realize\ntopological phase transitions on the Kagome lattice, but\nthe band gap closes at the K point. The contribution\nO(tt0=U) of the three-site processes can be complex\nand drives the system across the phases boundary of\noccupied-band Chern number = \u00061.\nFIG. 6. Possible realizations of the AB 2C3lattice in a 3D\nhexagonal crystal structure with (a) alternating AB 2andC3\nlayers and (b) alternating AC3andB2layers. Both structures\nhave the P6/mmm space group symmetry.\nThe phase transitions found in our model studies\nare realized using collinear antiferromagnetic (or ferri-\nmagnetic) spin con\fgurations. The spin-up and spin-\ndown electrons occupy inequivalent lattice sites. In the\nexamples shown in this paper, for simplicity, we let one\nspin species fully occupy one type of lattice site so as\nto be topologically trivial, and use them to control the\ntopological phase of the other spin species via the three-\nsite terms. Interesting directions for further studies could\nbe that both spin species exhibit topological properties\nand mutually in\ruence each other via the three-site\nterms, or the realization of similar Coulomb engineering\ne\u000bects in non-collinear spin systems.\nFinally, we would like to discuss the possible real-\nizations of our model in real materials. The 1 =U-\ncontrolled topological phase transitions can be realized\nwithout restricting the atoms to the same 2D plane.\nTwo possible 3D structures are shown in Fig. 6, both\nwith P6/mmm symmetry. Examples of materials with\nthe structure of Fig. 6a are RCo3B2[30] withR= rare-\nearth elements, and also GdNi 3Ga2[31], etc, which are\npotential candidates for the AC3model of two types\nof magnetic atoms. Coplanar AC3candidates include\nthe TiNi 3-type compounds [32{34] with shifted layers of\nclose-packed AC3structures. Candidates for the AB2\nmodel include e.g. UNi 2Al3[35] and EuCo 2Al9[36], etc.\nWe expect our work to be interesting to the \felds of\nmagnetism in alloys, ferrimagnets and other materials\nwith multiple types of magnetic atoms.6\nACKNOWLEDGMENTS\nThis work is supported by the National Key Research\nand Development Program of China (2018YFA0307000),\nand the National Natural Science Foundation of China(11874022). Z. H would thank the support of the 66th\nChinese Postdoc Fellowship. We would also like to thank\nthe helpful discussions with Prof. Biao Lian at Princeton\nCenter for Theoretical Science in Princeton University at\nthe early stage of this work.\nAppendix A: Derivation of the low-energy e\u000bective Hamiltonian\nBy plugging Eq. (1) into Eq. (3), one obtains\nHe\u000b=X\nij\u000b\ft\u000b\f\nijPcy\ni\u000bcj\fP\u00001\nUX\nijklX\n\u000b\f\r\u000et\u000b\f\nijt\r\u000e\nklPcy\ni\u000bcj\f\u0016Pcy\nk\rcl\u000eP: (A1)\nThen plugging in Eq. (2), one \fnds that the projections Ppick out the following terms\nHe\u000b=X\nij\u000b\ft\u000b\f\nijhihy\njby\ni\u000bbj\f\u00001\nUX\nijklX\n\u000b\f\r\u000e\f\rt\u000b\f\nijt\r\u000e\nklby\ni\u000bby\nj\u0016\fbk\u0016\rbl\u000ehi(Pdjdy\nkP)hy\nl: (A2)\nThe bosonic operators are automatically normal ordered. Notice that Pdjdy\nkP=\u000ejk, because the doublon created\nmust also be the doublon destructed so as to go back to the no-doublon subspace. One may then set j=kand\nrename the dummy indices l7!jand\f$\u000eto obtain\nHe\u000b=X\nijhihy\nj2\n4X\n\u000b\fby\ni\u000b0\n@t\u000b\f\nij\u00001\nUX\nkX\n\r\u000e\r\u000et\u000b\u000e\nikt\r\f\nkjby\nk\u0016\u000ebk\u0016\r1\nAbj\f3\n5: (A3)\nThis result agrees with Eqs. (4){(5) in the main text by de\fning the quantity in the square bracket as ~tij. AllO(1=U)\nrenormalizations of ~tijare considered in this formalism. Then we do a particle-hole transformation hi7!fy\nito the\nholon operators and map the bosonic operators bi\u001b7!zi\u001btoc-numbers and obtain\nHe\u000b=X\nijfy\nifj2\n4X\n\u000b\fz\u0003\ni\u000bzj\f0\n@t\u000b\f\nij\u00001\nUX\nk\r\u000e\r\u000et\u000b\u000e\nikt\r\f\nkjz\u0003\nk\u0016\u000ezk\u0016\r1\nA3\n5=X\nij~tijfy\nifj: (A4)\nIn the special case that the bare hopping t\u000b\f\nij=t\u000b\nij\u000e\u000b\fconserves spin, we have\n~tij=X\n\u000b\fz\u0003\ni\u000bzj\f \nt\u000b\nij\u000e\u000b\f\u00001\nUX\nk\u000b\ft\u000b\nikt\f\nkjz\u0003\nk\u0016\u000bzk\u0016\f!\n: (A5)\nThen the collinear ferrimagnetic structure in the zdirection (perpendicular to the 2D lattice plane) with no double\noccupancy eliminates the \u000b6=\fterms because site kcan only be occupied by one type of spin species. Therefore,\none obtains Eq. (6) in the main text.\nAppendix B: Kagome Hamiltonian in terms of ~t1\u00003\nIn terms of the e\u000bective hoppings ~t1\u00003, the full Kagome Hamiltonian HC(~k) contains 3 parts as de\fned by Eq. (11):\nthe nearest-neighbor hopping Hamiltonian as given by\nH(1)\nC(~k) =2\n66640 2 ~t1cos\u0010\n~k\u0001~ a1+~ a2\n2\u0011\n2~t\u0003\n1cos\u0010\n~k\u0001~ a2\n2\u0011\n2~t\u0003\n1cos\u0010\n~k\u0001~ a1+~ a2\n2\u0011\n0 2 ~t1cos\u0010\n~k\u0001~ a1\n2\u0011\n2~t1cos\u0010\n~k\u0001~ a2\n2\u0011\n2~t\u0003\n1cos\u0010\n~k\u0001~ a1\n2\u0011\n03\n7775; (B1)7\nthe next-nearest-neighbor hopping Hamiltonian as given by\nH(2)\nC(~k) =2\n66640 2 ~t2cos\u0010\n~k\u0001~ a1\u0000~ a2\n2\u0011\n2~t\u0003\n2cosh\n~k\u0001\u0000\n~ a1+~ a2\n2\u0001i\n2~t\u0003\n2cos\u0010\n~k\u0001~ a1\u0000~ a2\n2\u0011\n0 2 ~t2cosh\n~k\u0001\u0000~ a1\n2+~ a2\u0001i\n2~t2cosh\n~k\u0001\u0000\n~ a1+~ a2\n2\u0001i\n2~t\u0003\n2cosh\n~k\u0001\u0000~ a1\n2+~ a2\u0001i\n03\n7775; (B2)\nand the para-position hopping Hamiltonian as given by\nH(3)\nC(~k) =2\n66642~t3cos\u0010\n~k\u0001~ a1\u0011\n0 0\n0 2 ~t3cos\u0010\n~k\u0001~ a2\u0011\n0\n0 0 2 ~t3cosh\n~k\u0001(~ a1+~ a2)i3\n7775; (B3)\nall written in the atomic gauge. 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The only input parameters in MagneticTB are the (magnetic) space group\nnumber and the orbital information in each Wyckoff positions. Some useful functions including getting\nthe matrix expression for symmetry operators, manipulating the energy band structure by parameters\nand interfacing with other software are also developed. MagneticTB can help to investigate the physical\nproperties in both magnetic and non-magnetic system, especially for topological properties.\nProgram summary\nProgram title: MagneticTB\nLicensing provisions: GNU General Public Licence 3.0\nProgramming language: Mathematica\nExternal routines/libraries used: ISOTROPY (iso.byu.edu)\nDeveloper’s repository link: https://github.com/zhangzeyingvv/MagneticTB\nNature of problem: Construct the symmetry adopted tight-binding model for the system with arbitrary\nmagnetic space group.\nKeywords: Tight-binding method, Representation theory, Magnetic space group, Mathematica\n1. Introduction\nTight-binding method is a powerful tool to investigate the novel properties in condensed mater physics\n[1–5]. Compared with first-principles method, tight-binding method can greatly simplify calculations. More-\nover, after considering (magnetic) space group symmetry, the tight-binding model can give more reliable\nresults. For example, in topological materials, symmetry plays an important role to protect the topological\nproperties, such as Z2topological insulator protected by time reversal symmetry [6], topological crystalline\ninsulators and topological nodal semimetals protected by space group symmetries [7, 8] and magnetic topo-\nlogical crystalline insulator protected by magnetic space group symmetries, i.e. the combination of space\ngroup operations and time reversal [9, 10].\nAt present, a lot of researchers using symmetry adopted tight-binding model to investigate physical\nproperties of electronic system [11–21]. However, most of the software packages are mainly focused on the\ntight-binding model for non-magnetic materials [22–26] and only a few of them can be used to construct\nthe symmetry adopted tight-binding model automatically. For the first-principles level, Wannier90 can\ngenerate the Wannier-tight-binding model by interfacing with first-principles software, but the symmetry\n∗Corresponding author\nEmail addresses: gbliu@bit.edu.cn (Gui-Bin Liu), ygyao@bit.edu.cn (Yugui Yao)\nPreprint submitted to Elsevier May 21, 2021arXiv:2105.09504v1  [cond-mat.mtrl-sci]  20 May 2021adopted Wannier function can not be applied to other first-principles software (such as VASP, ABINIT)\nexcept Quantum-Espresso [27–30]. FPLO can generate symmetry adopted tight-binding model with proper\nparameters for given structures [31]. Meanwhile, both Wannier90 and FPLO do not support magnetic\nsymmetry. For the model level, GTPack, Qsymm and MathemaTB can generate the tight-binding model\nwith space group symmetry but does not support magnetic symmetry directly [32–34]. WannierTools can\ndo the symmetrization of non-magnetic tight-binding model but cannot generate the tight-binding model\nby itself [35].\nTherefore, itisnecessarytodevelopapackagewhichcanconstructthetight-bindingmodelwithmagnetic\nspace group symmetry automatically. Here we introduce a software package: MagneticTB, a tool for\ngenerating the tight-binding model for the system with arbitrary magnetic space group. The required\ninput information of this package is only the magnetic space group number and the orbital information in\neach Wyckoff positions. It can help to investigate the physical properties of the given symmetry. We also\npresent some useful functions including get the matrix expression for symmetry operators, manipulate the\nband structure by parameters and interface with other software.\nThis paper is organized as follows. In Sec. 2, we give an introduction of symmetry adopted tight-binding\nmethods. In Sec. 3, The usage of MagneticTB are given, including how to install and run MagneticTB. In\nSec. 4, we give three concrete examples, such examples show the specific capabilities of the MagneticTB.\nFinally, conclusions are given.\n2. Symmetry adopted tight-binding method\nIn periodic system, the bases of tight-binding model can be written as Bloch sum [11]\nψn\nlmk(r) =1√\nN/summationdisplay\nRjeik·(Rj+dn\nl)ϕn\nm(r−Rj−dn\nl) (1)\nwhereNis the number of unit cells in the crystal, Rjis the translation vector of the Bravais lattice, dn\nl\nis the position of n-th Wyckoff position’s l-th atom in the unit cell (for each Qanddn\nlthere exists one\nand only one pair of dn\nl/primeandR/primewhich satisfies Qdn\nl=dn\nl/prime+R/primewhereQis arbitrary group element in the\n(magnetic) space group and R/primeis a lattice vector), ϕn\nm(r)is them-th atomic orbital basis for position dn\nl\nbut located at coordinate origin. Then Eq. (1) satisfies the Bloch theorem ψn\nlmk(r+Rj) =eik·Rjψn\nlmk(r).\nThe tight-binding Hamiltonian can be written as:\nHnn/prime\nlml/primem/prime(k) =/summationdisplay\nRjeik·(Rj+dn/prime\nl/prime−dn\nl)Emm/prime(dn\nl,Rj+dn/prime\nl/prime)\nEmm/prime(dn\nj,Rj+dn/prime\nl/prime) =/angbracketleftϕn\nm(r−dn\nl)|ˆH|ϕn/prime\nm/prime(r−dn/prime\nl/prime−Rj)/angbracketright(2)\nfor simplicity, we rewrite the atomic orbitals in vector form: Φn(r−Rj−dn\nl) ={ϕn\nm(r−Rj−dn\nl)},(m=\n1,...,Mn). ThenEmm/prime(dj,Rj+dl/prime) (m= 1,...,Mn;m/prime= 1,...,Mn/prime)form anMn×Mn/primematrix:\nE(dn\nl,Rj+dn/prime\nl/prime) =/angbracketleftΦn(r−dn\nl)|ˆH|Φn/prime(r−Rj−dn/prime\nl/prime)/angbracketright (3)\nThen the Hamiltonian can be rewritten as:\nHnn/prime\nll/prime(k) =/summationdisplay\nRjeik·(Rj+dn/prime\nl/prime−dn\nl)E(dn\nl,Rj+dn/prime\nl/prime) (4)\nE(dn\nl,Rj+dn/prime\nl/prime)is the hopping matrix between n-th Wyckoff position’s l-th atom to n/prime-th Wyckoff position’s\nl/prime-th atom. When the lattice is in invariant under some symmetry E(dn\nl,Rj+dn/prime\nl/prime)may ne not independent\nfor arbitrary dn\nlanddn/prime\nl/prime. Fortunately, for symmetry operation Q, group representation theory gives us\nexplicit expression for the relationship between E(Qdn\nj,Q(Rj+dn/prime\nl/prime))andE(dn\nj,Rj+dn/prime\nl/prime).\n2(a) (b)Structure\nSymmetry  \noperators\nBasis Functio nsBondHamilt onian \nw/o symmetry\nD(R) and D(R T) P(Q)Symm etry adopted \nHami ltonianQdl’\nxy\ndl’C4\n22Figure 1: (a) Sketch of relationship between E(dn\nj,Rj+dn/prime\nl/prime)andE(Qdn\nj,Q(Rj+dn/prime\nl/prime)), in this example we set Q=C4T,\nΦ1(r) ={s}locate atd1\nl= (0,0),Φ2(r) ={px,py}locate atd2\nl/prime= (λ,0)(λ/negationslash= 0). Then we have D1(C4T) = 1,D2(C4T) =\n−iσy, andE(Qd1\nl,Qd2\nl/prime) =E∗(d1\nl,d2\nl/prime)×(−iσy). (b) Workflow of MagneticTB.\nFor the case that symmetry operation does not contain the time reversal T, i.e.Q={R|t}, whereR\nandtare the rotation and translation part of Qrespectively, we have\nE(Qdn\nl,Q(Rj+dn\nl/prime)) =Dn(R)E(dn\nl,Rl+dn/prime\nl/prime)Dn/prime†(R) (5)\nFor the case that operation Qcontains time reversal symmetry T, i.e.Q={R|t}T, we have\nE(Qdn\nl,Q(Rj+dn\nl/prime)) =Dn(RT)E∗(dn\nl,Rj+dn/prime\nl/prime)Dn/prime†(RT) (6)\nWhereDn(R)(Dn(RT))are theMn×Mnrepresentation matrices of R(RT)under atomic orbital bases\nΦn(r)(not necessarily irreducible representations), E∗(dn\nl,Rj+dn/prime\nl/prime)is complex conjugate of E(dn\nl,Rj+dn/prime\nl/prime)\n(see Fig. 1 for example). It is clear that for spinless cases with time reversal symmetry T, when the basis\nfunctions are real, D(T)is equal to identity matrix, indicating that E(dn\nl,Rj+dn/prime\nl/prime)are real matrices.\nThe next step is to get the analytical expressions of Dn(R)(Dn(RA)). For a fixed n, we don’t have to\nworry about mixing superscripts of ninDn(R)because transformations can only occur under the same n.\nSo we temporarily use D(R)rather than Dn(R)in this step. Consider the following four cases:\ni. Spinless system and Qdoes not contain T.\nii. Spinless system and QcontainT.\niii. Spinful system and Qdoes not contain T.\niv. Spinful system and QcontainT.\nIn case. i,D(R)can be obtained simply by solve the linear equation [36]\nˆRΦ(r) =Φ(R−1r) =Φ(r)D(R) (7)\nin which ˆRis the function operator for the rotation R. In case. ii, we define Φ(r) = ˆTΦ(r), for spinless\nsystemT=K, hence,Φ(r) =Φ∗(r)and then solve the linear equation\nˆRˆTΦ(r) =Φ(R−1r) =Φ(r)D(RT) (8)\nIn case. iii, since the spin matrix is orbital independent we define the basis function as\nΦs(r) ={Φ(r)↑,Φ(r)↓} (9)\n3The two spinors↑= (1,0)T, and↓= (0,1)T,and under the rotation ˆRthey are transformed according to\nˆR(↑,↓) = (↑,↓)D1\n2(R) (10)\nFor proper rotation R,D1\n2(R) =exp(−1\n2iαn·ˆσ), whereαis the rotation angle of R,nis the unit vector\nalong rotation axis, for improper rotation S,R=IS,Iis the inversion symmetry, D1\n2(S) =D1\n2(R)[37, 38].\nThenD(R)can be obtained by solving the linear equation\nˆRΦs(r) =Φs(r)D(R) (11)\nCase. (iv) is similar to case. (ii), the only difference is replace the time reversal operator T=KbyT=iˆσyK\nand consider the spin rotation matrices. The above four cases cover all the possibilities of D(R)andD(RT).\nThen the operator (or representation matrix) for Qcan be defined as\nPnn/prime\nll/prime(Q) =\n\nδnn/prime˜δdn\nl,Qdn/prime\nl/primeDn(R)Q does not contain T\nδnn/prime˜δdn\nl,Qdn/prime\nl/primeDn(RT)Q containsT(12)\nwhere ˜δdn\nl,Qdn/prime\nl/primeis equal to 1 only when dn\nlandQdn/prime\nl/primediffer by a lattice vector and to 0 otherwise (it can also\nbe written as ˜δdn\nl,Qdn/prime\nl/prime=δdn\nl,Qdn/prime\nl/prime+Rsif a suitable lattice vector Rsis choosen). The Hamiltonian under\nconstraint of Qcan be written as\nP(Q)−1H(k)P(Q) =H(R−1k) (13)\nforQ={R|t}, and\nP(Q)−1H(k)P(Q) =H∗(−R−1k) (14)\nforQ={R|t}T[See Appendix A for proof of Eqs.(5–6) and Eqs.(13–14)). Eqs.(13–14) are key point to\ngenerate the symmetry adopted tight-binding model. In MagneticTB we first generate the Hamiltonian with\nonly translation symmetry and then use Eqs.(13–14) to the simplify the Hamiltonian (see Fig. 1(b)) for the\nworkflow of MagneticTB]. The database for tight-binding model of 1651 magnetic space group can be found\nin our later work [39].\n3. Capabilities of MagneticTB\n3.1. Installation\nTo install the MagneticTB, unzip the \"MagneticTB.zip\" file and copy the MagneticTB directory to any\nof the following four paths:\n•FileNameJoin[{ $UserBaseDirectory , \"Applications\"}]\n•FileNameJoin[{ $BaseDirectory , \"Applications\"}]\n•FileNameJoin[{ $InstallationDirectory , \"AddOns\", \"Packages\"}]\n•FileNameJoin[{ $InstallationDirectory , \"AddOns\", \"Applications\"}]\nThen one can use the package after running Needs[\"MagneticTB‘\"] . The version of Mathematica should higher\nor equal to 11.0.\n43.2. Running\n3.2.1. Core module\nTo initialize the program one should identify the magnetic space group and the orbital information in\neach Wyckoff positions. Here we provide a function msgopto show the symmetry information of an arbitrary\nmagnetic space group. The only input of the msgopis the magnetic space group number, one can get the\nsymmetry information by the following code:\nmsgop[gray[191]]\nmsgop[bnsdict[{191, 236}]]\nmsgop[ogdict[{191, 8, 1470}]]\nMagnetic space group (BNS): {191.236,P6’/mm’m}\nPrimitive Lattice vactor: {{a,0,0},{ −a/2,( Sqrt[3] a)/2,0},{0,0,c}}\nConventional Lattice vactor: {{a,0,0},{ −a/2,( Sqrt[3] a)/2,0},{0,0,c}}\n{{\"1\" ,{{1,0,0},{0,1,0},{0,0,1}},{0,0,0}, F},\n{\"3z\",{{0, −1,0},{1, −1,0},{0,0,1}},{0,0,0},F},\n{\"3z −1\",{{ −1,1,0},{ −1,0,0},{0,0,1}},{0,0,0},F},\n{\"2x\",{{1, −1,0},{0, −1,0},{0,0, −1}},{0,0,0},F},\n...\nheregray[191] return the magnetic space group code of gray space group 191, bnsdict[{191,236}] return the\nmagnetic space group code of BNS No. 191.236, ogdict[{191,8,1470}] return the magnetic space group code\nof OG No. 191.8.1470. Then the msgopwill print the standard lattice vector and the symmetry operations\n(for primitive cell) of the corresponding magnetic space group, which can be the input of the initfunction.\nNotice the magnetic space group code is build-in constant in MagneticTB, users should use gray, bnsdict,\nogdictfunctions rather than inputting the magnetic space group code directly.\nThen one can feed the above information to initfunction then the basic results of input structure\ncan be generated. The initfunction have five mandatory options neamly, lattice,lattpar,wyckoffposition ,\nsymminformation andbasisFunctions (in ordinary Wolfram language the options for functions are optional, but\nsuch five options must be specified in MagneticTB in order to make the input clear). The latticeis the\nlattice vector of magnetic system which can contain parameters, lattparis the parameters in lattice vector to\ndetermine the bond length of magnetic system,and wyckoffposition is a list to designate atomic position and\nmagnetization-direction for each Wyckoff positions in the magnetic system. The format of wyckoffposition is:\n{{a1,m1},{a2,m2},...}\nwhereaiandmirepresent one of the atomic positions and its magnetization-directions of the i-th Wyckoff\nposition, respectively. The symminformation contain the elements of the coset of magnetic space group with\nrespect to translation group, which can direct use the output of msgop(notice that the output of msgopis\nstandard symmetry operation from ISOTROPY [40, 41]. However, users can also use the non-standard\nstructure as input, not limit to the output of msgop). The format of symminformation is:\n{{n1,R1,t1,A1},{n2,R2,t2,A2},...}\nwhereniis the name of symmetry operation, Riandtiare the rotation and translation part of symmetry\noperation, and Airepresents whether the symmetry operation is combined with time reversal symmetry\n(\"T\"for true and \"F\"for false). Finally, the basisFunctions is the basis function for each Wyckoff position, The\nformat of basisFunctions is:\n{b1,b2,...}\nwherebiis the list of basis functions of the i-th Wyckoff position. The build-in basis functions for spinless\ncase in MagneticTB is shown in Table 1.\nWhen spin is considered, for build-in basis functions, add \"up\"or\"dn\"after basis functions string, e.g.\nfor|px↑/angbracketright, the basis function for spin-up case is \"pxup\". However, users may use other basis functions such\nasf,|3/2,1/2/angbracketrightorbitals. In such cases, users can directly input the analytical expression of basis functions.\nFor example, if only consider fxyzorbital, one should input basisFunctions −> {{x ∗y∗z}}]. The analytical\nexpressions of basis functions can be simply obtained from quantum mechanics or group theory books\n[32, 42, 43].\n5Table 1: String codes representing basis functions and available values for basisFunctions\nBasis function String Basis function String\ns \"s\" px \"px\"\npy \"py\" pz \"pz\"\npx+ipy\"px+ipy\" px−ipy\"px−ipy\"\ndz2 \"dz2\" dxy \"dxy\"\ndyz \"dyz\" dxz \"dxz\"\ndx2−y2 \"dx2 −y2\"\nsgop=msgop[gray[191]];\ninit [\nlattice −> {{a,0,0},{ −(a/2),( Sqrt[3] a)/2,0},{0,0,c}},\nlattpar −> {a −> 1, c −> 3},\nwyckoffposition −> {{{1/3, 2/3, 0}, {0, 0, 0}}},\nsymminformation −> sgop,\nbasisFunctions −> {{\"pz\"}}];\nTable 2: Basic results of init\nproperties illustrate of properties\natompos atomic position and magnetization-direction for each atom\nwcc dlfor each basis function\nreclatt reciprocal lattice vector for given structure\nsymmetryops PQfor each symmetry operation\nunsymham generate the Hamiltonian with only translation symmetry\nsymmcompile summary of P(Q), see main text for detail\nbondclassify summary of bonds information, see main text for detail\nAfter inputting the above five options appropriately, one can run initand obtain the basic results. Here\nwe introduce two important basic results: symmcompile andbondclassify , the other properties are given in\nTable. 2. The format of symmcompile is\n{{N1,{n1,R1,t1,A1},P1,Rk\n1},{N2,{n2,R2,t2,A2},P2,Rk\n2},...}\nwhereNi,Pi,Rk\niare the label, the symmetry operator (Eq.(12)) and the rotation acting on kspace of the\ni-th symmetry operation, respectively. For example\nsymmcompile\n{{1,{\"1\" ,{{1,0,0},{0,1,0},{0,0,1}},{0,0,0}, F},{{1,0},{0,1}},\n{{1,0,0},{0,1,0},{0,0,1}}},\n{2,{\"6z\" ,{{1, −1,0},{1,0,0},{0,0,1}},{0,0,0}, F},{{0,1},{1,0}},\n{{0,−1,0},{1,1,0},{0,0,1}}},\n...}\nThe format of bondclassify is\n{{li,ni,{{pi,j,pi,k},...}},...}\nwhereliis the bond length of the (i−1)-th neighbour hopping ( i= 1for on-site hopping), niis the number\nof the (i−1)-th neighbour’s bonds, {pi,j,pi,k}is the atomic position of nicorresponding bonds.\nTill this moment, symmetry adopted tight-binding model for magnetic system is ready to be generated.\nBy using the symham[n] function, one can obtain the symmetry adopted tight-binding model. When n= 1,\nsymham[1] return the Hamiltonian with only on-site hopping, n= 2return the Hamiltonian with only nearest-\nneighbourhoppingandsoon. Bydefault, MagneticTBwillcheckalltheinputsymmetryoperationstoensure\nthe Hamiltonian is correct. However, it may last long time when the structure is complex. In principle, only\n6the generators of the (magnetic) space group are enough to get the Hamiltonian. Therefore, one can specify\nthe symmetry operations by symmetryset −>listinsymhamwhere listis the list of indexes of the symmetry\noperations. For example, symham[2,symmetryset −>{2}]will generate the Hamiltonian with only C6zsymmetry\nfor nearest-neighbour hopping. symmetryset can not only save computing resource but can also investigate\nthe Hamiltonian for symmetry breaking cases. The parameters for each neighbour in MagneticTB are given\nin Table. 3.\nTable 3: String codes representing n-th neighbour hoppings for symham\nOn-site energy Nearest Second-nearest Third-nearest (k−1)-th nearest\ne1,e2,.. t1,t2 ,... r1,r2 ,... s1,s2 ,... pkn1,pkn2,...\n3.2.2. Plot module\nAfter tight-binding model being generated, there may exist many parameters, one can use bandManipulate\nfunction to manipulate the band structure to investigate the relationship between band structure and\nparameters. The format of bandManipulate is\nbandManipulate[{{{k1,k2},{name of k1, name of k2}},...},np,Hamiltonian]\nwherenpis the number of kpoints per line. Then one can easily check the band structure of different\nparameters. When the proper parameters are obtained, one can use bandplotto plot the band structure\nbandplot[{{{k1,k2},{name of k1, name of k2 }},...}, np,Hamiltonian,parameters]\nsee section. 4 for concrete example.\n3.2.3. IO module\nIn MagnetTB, one can get the tight-binding model for magnetic system. However, MagneticTB do not\ncalculate the other properties (such as surface states, finding the gap-less point and so on) directly, since\nit will generally cost too much computing resources. It is better to do such heavy calculations by Fortran,\nPython or C. Therefore we develop hoppfunction to convert the symmetry adopted tight-binding model to\n\"wannier90_hr.dat\" format, which is convenient to interface with WannierTools [35], Z2Pack [44], PythTB\n[45] and our home-made package Wannflow [46, 47]. The \"wannier90_hr.dat\" in Wannier90 use the following\nconvention [27], namely conventions II:\n˜ψn\nlmk(r) =1√\nN/summationdisplay\nRjeik·Rjϕn\nlm(r−Rj−dn\nl)\n˜Hnn/prime\nlml/primem/prime(k) =/summationdisplay\nRjeik·RjEmm/prime(dn\nl,Rj+dn/prime\nl/prime)(15)\nwhich is different form MagneticTB in Eq. (2), the relationship between two conventions is\n˜H(k) =V(k)H(k)V†(k)\nVnn/prime\nll/prime(k) =eik·dn\nlδll/primeδnn/prime(16)\nIn MagneticTB (convention I) the operation matrix defined in Eq. (12) is kindependent while the Hamil-\ntonian is non-periodic by shifting the reciprocal vector G\nH(k+G) =V†(G)H(k)V(G) (17)\nBy contrast, in conventions II the Hamiltonian is periodic. i.e ˜H(k+G) = ˜H(k). The format of hopp\nfunction is\n7hopp[Hamiltonian,parameters]\nSee section. 4 for concrete example. One can also use symmhamII[ham] to generate the Wolfram expression for\nHamiltonian in convention II. Notice that hoppfunction (but not symmhamII ) is only applied to the output of\nsymhamfunction, and that expressions explicitly including SinorCosmay not work well. Be careful to use it.\n4. Examples\n4.1. Three-band tight-binding model for MoS 2\nMoS 2monolayer has direct bandgap in the visible range, strong spin-orbit coupling, and rich valley\nrelated physics, which make it an candidate for nanoelectronic, optoelectronic, and valleytronic applications\n[48, 49]. The space group of MoS 2isP6m2(space group No. 187). Considering the Mo atom at 1aWyckoff\nposition and using the dz2,dxy, anddx2−y2orbitals, the model can be obtained by\nsgop = msgop[gray[187]];\ntran = {{1, −1, 0}, {0, 1, 0}, {0, 0, 1}};\nsgoptr = MapAt[FullSimplify [tran.#.Inverse@tran] &, sgop, {;; , 2}];\ninit [\nlattice −> {{1, 0, 0}, {1/2, Sqrt[3]/2, 0}, {0, 0, 10}},\nlattpar −> {},\nwyckoffposition −> {{{0, 0, 0}, {0, 0, 0}}},\nsymminformation −> sgoptr,\nbasisFunctions −> {{\"dz2\", \"dxy\", \"dx2 −y2\"}}];\nmos2 = Sum[symham[i, symmetryset −> {9, 11, 13}], {i, {1, 2}}];\nmos2Liu = mos2 /. Thread[{kx, ky, kz} −> ({kx, ky, kz} (2 Pi)). Inverse@reclatt ];\nThis gives exactly the same results as in Ref. [14]. The relationship of parameters between Ref. [14] and\nMagneticTB are\nRef. [14] /epsilon11/epsilon12t0t1t2t11t12t22\nMagneticTB e1 e2 t1 t2 t4 t3 t5 t6\n4.2. Graphene\nGraphene with linear dispersion around Fermi level is one of the most important materials in spintronics\n[50]. The magnetic space group of graphene is P6/mmm 1/prime(BNS No. 191.234). There are two C atoms at\n2cWyckoff position, and the bands near Fermi energy are mainly from pzorbital. The above information is\nenough to establish the tight-binding model near Fermi energy of graphene. The model can be obtained as\nfollow\nNeeds[\"MagneticTB‘\"]\nsgop = msgop[gray[191]];\ninit [\nlattice −> {{a,0,0},{ −(a/2),( Sqrt[3] a)/2,0},{0,0,c}},\nlattpar −> {a −> 1, c −> 3},\nwyckoffposition −> {{{1/3, 2/3, 0}, {0, 0, 0}}},\nsymminformation −> sgop,\nbasisFunctions −> {{\"pz\"}}];\nham = Sum[symham[i], {i, 3}]; MatrixForm [ham]\noutput:\n/bracketleftBigg\ne1+ 2r1(cos(kx+ky) + coskx+ cosky)t1ei/parenleftbig\n−2kx\n3−ky\n3/parenrightbig\n+t1ei/parenleftbigkx\n3−ky\n3/parenrightbig\n+t1ei/parenleftbigkx\n3+2ky\n3/parenrightbig\n† e1+ 2r1(cos(kx+ky) + coskx+ cosky)/bracketrightBigg\nFor spin-orbital coupling case, the only thing which needs to change is the basis functions\nbasisFunctions −> {{\"pzup\", \"pzdn\"}}\n8and the corresponding Hamiltonian reads\n\ne1+h−0h/prime0\n0e1+h+0h/prime\n† 0e1+h+0\n0† 0e1+h−\n\nwhereh±=±2r1(sinkx+ sinky−sinkx+ky) + 2r2(cos(kx+ky) + coskx+ cosky),h/prime=t1ei/parenleftbig\n−2kx\n3−ky\n3/parenrightbig\n+\nt1ei/parenleftbigkx\n3−ky\n3/parenrightbig\n+t1ei/parenleftbigkx\n3+2ky\n3/parenrightbig\n. Such model is corresponding to the first Z2topological insulator [6]. After get\nthe Hamiltonian, we can use bandManipulate andbandplotto plot the band structure, and click the \"ExportData\"\nbutton to print the value of parameters.\npath={\n{{{0,0,0},{0,1/2,0}},{\"\\[CapitalGamma]\",\"M\"}},\n{{{0,1/2,0},{1/3,1/3,0}},{\"M\",\"K\"}},\n{{{1/3,1/3,0},{0,0,0}},{\"K\",\"\\[CapitalGamma]\"}}\n};\nbandManipulate[path, 20, ham]\nbandManipulate[path, 20, hamsoc]\nbandplot[path, 200, ham, {e1 −> 0, t1 −> 0.5, r1 −> 0}]\nbandplot[path, 200, hamsoc, {e1 −> 0, t1 −> 0.5, r1 −> 0.02, r2 −>0}]\nΓ M K Γ-0.80.1.7\n0.8\n(a) (b)\n(d) (c)\nΓ M K Γ-1.5-0.80.1.5\n0.8\nFigure 2: Output of bandManipulate andbandplot for graphene without considering spin (a-b) and with spin (c-d).\nMoreover we can get the wannier90_hr.dat by hopfunction for this model.\nhop[hamsoc, {e1 −> 0, r1 −> 0.02, r2 −> 0, t1 −> 0.5}]\nGenerated by MagneticTB\n4\n7\n91 1 1 1 1 1 1\n−1 −1 0 1 1 0.00000000 0.02000000\n−1 −1 0 2 1 0.00000000 0.00000000\n−1 −1 0 3 1 0.00000000 0.00000000\n−1 −1 0 4 1 0.00000000 0.00000000\n−1 −1 0 1 2 0.00000000 0.00000000\n−1 −1 0 2 2 0.00000000 −0.02000000\n...\n4.3. Magnetic C-3 Weyl point\nThe charge-3 (C-3) Weyl point is a 0D two-fold band degeneracy with Chern number |C|= 3. Encyclo-\npedia of emergent particles tell us that the C-3 Weyl point always appear at least in a pair or coexist with\nnodal surface in nonmagnetic systems [51]. Here we confirm that in magnetic system, due to the breaking of\ntime reversal symmetry T, the C-3 Weyl point can uniquely coexist with conventional Weyl points. Consider\nthe type IV magnetic space group Pc3(BNS No. 143.3). The generator of the group is C3zand{E|001\n2}T,\nPut|px+ipy↑/angbracketright,|px−ipy↓/angbracketrightbasis functions at Wyckoff position 2a, and then the symmetry operator for\nC3zandE{001\n2}Tare\nC3z=−σz\n{E|001\n2}T=iσy\nUnder this bases the effective Hamiltonian at Γpoint can be written as\nHC-3 WP =/epsilon1+αkzσx+ck2\n/bardbl+β(kx+e−iπ\n3ky)3σz+h.c. (18)\nwhere/epsilon1,care real parameters and α,βare complex parameters. Besides, there are another three essential\nWeyl points locate at (π,0,0),(0,π,0),(π,π, 0). Since the C3zsymmetry does not change the Chern number\nof Weyl points, the Chern number at Mhas to be±1. According to no-go theorem, the Chern number of\nΓis∓3. One can easily check that the Chern number of Eq. (18) is ±3. The degeneracies of ΓandMare\nbecause ({E|001\n2}T)2=−1at(0/π,0/π,0). The model can be obtained as follow\nsgop = msgop[bnsdict[{143, 3}]];\ninit [ lattice −> {{ Sqrt[3]/2, −( 1/2), 0}, {0, 1, 0}, {0, 0, 2}},\nlattpar −> {},\nwyckoffposition −> {{{0, 0, 0}, {0, 0, 1}}},\nsymminformation −> sgop,\nbasisFunctions −> {{{x + Iy, 0}, {0, x −Iy}}}];\nc3w = Sum[symham[i], {i, {2, 4}}];\nc3w2band = Table[c3w[[i, j ]], {i, {1, 4}}, {j, {1, 4}}];\nThe band structure of c3w2band is shown in Fig. 3(a).\n4.4. Magnetic cubic nodal-line\nTopological high order nodal line is that the energy difference between the bands are non-linear,and\nthe order of energy dispersion around the degeneracy points plays an important role in different physical\nproperties, such as density of states, Berry phase and Landau-level [52]. Recently, Zhang et. al. proposed\nhigh order nodal-line in magnetic system [53]. In this example, we use MagneticTB to generate magnetic\ncubic nodal-line. Generally, the magnetic cubic nodal-line is protected by C6zandMxsymmetries, there\nare many magnetic space groups which contains the above two symmetries. Consider the type IV magnetic\nspace group Pc6cc(BNS No. 184.196), and put |px+ipy↑/angbracketright,|px−ipy↓/angbracketrightbasis functions at Wyckoff position\n2a. Then the tight-binding model can be generated by\nsgop = msgop[bnsdict[{184, 196}]];\ninit [ lattice −> {{ Sqrt[3]/2, −1/2, 0}, {0, 1, 0}, {0, 0, 2}},\nlattpar −> {},\n10(a) (b)\nΓ M K Γ A L H A|L M|K H-1.-0.50.1.\n0.5Figure 3: (a) Output of bandplot for magnetic C-3 Weyl point, (b) Output of bandManipulate for magnetic cubic nodal-line.\nwyckoffposition −> {{{0, 0, 0}, {0, 0, 1}}},\nsymminformation −> sgop,\nbasisFunctions −> {{{x + Iy, 0}, {0, x −Iy}}}];\ncnl = Sum[symham[i, symmetryset −> {2, 7, 13}], {i, 1, 5}];\ncnl2band = Table[cnl[[ i, j ]], {i, {1, 4}}, {j, {1, 4}}];\nMatrixForm [cnl2band]\npath = {{\n{{0, 1/10, 1/4}, {0, 0, 1/4}}, {\"Q\", \"P\"}},\n{{{0, 0, 1/4}, { −1/10, 1/10, 1/4}}, {\"P\", \"Q\"}}};\nbandManipulate[path, 20, cnl2band]\nOne can check that no mater how the parameters change, the dispersion of arbitrary point along Γ-Aon\nkx-kyplane are non-linear, see Fig. 3(b) which is consistent with Ref. [53].\n5. Conclusion\nIn conclusion, we have developed a software package to generate the symmetry-adopted tight-binding\nmodel for arbitrary magnetic space group. The input parameters for MagneticTB are clear and easy to set\nand both spinless and spinful Hamiltonian can be generated automatically. Besides, some useful functions\nsuch as manipulating the band structure, interfacing with other software are implemented, which can be\nused for further study on the magnetic systems. Moreover, MagneticTB can not only be used to investigate\nphysical properties of electronic systems, but also be used to study photonics, ultracold, acoustic and\nmechanical systems [54–56]. Finally, an exciting direction for future is to apply the magnetic field for the\ntight-binding model in MagneticTB [57].\nAcknowledgments\nZZ acknowledges the support by the NSF of China (Grant No. 12004028), the China Postdoctoral\nScience Foundation (Grant No. 2020M670106), the Fundamental Research Funds for the Central Univer-\nsities (ZY2018). GBL acknowledges the support by the National Key R&D Program of China (Grant No.\n2017YFB0701600). YY acknowledges the support by the National Key R&D Program of China (Grant No.\n2020YFA0308800), the NSF of China (Grants Nos. 11734003, 12061131002), the Strategic Priority Research\nProgram of Chinese Academy of Sciences (Grant No. XDB30000000).\n11Appendix A.\nInthisappendix,weusethetranslationoperation T(d). ForQ={R|v},wehaveQT(d) ={R|v}{E|d}=\n{R|Rd+v}=T(Qd)R. Thus\nE(dn\nj,(Rj+dn/prime\nl/prime)) =/angbracketleftˆT(dn\nl)Φn(r)|ˆQ†ˆHˆQ|ˆT(dn/prime\nl/prime+Rj)Φn/prime(r)/angbracketright\n=/angbracketleftˆQˆT(dn\nl)Φn(r)|ˆH|ˆQˆT(dn/prime\nl/prime+Rj)Φn/prime(r)/angbracketright\n=/angbracketleftˆT(Qdn\nl)RΦn(r)|ˆH|ˆT(Q(dn/prime\nl/prime+Rj))RΦn/prime(r)/angbracketright\n=Dn†(R)/angbracketleftˆT(Qdn\nl)Φn(r)|ˆH|ˆT(Q(dn/prime\nl/prime+Rj))Φn/prime(r)/angbracketrightDn/prime(R)\n=Dn†(R)E(Qdn\nj,Q(Rj+dn/prime\nl/prime))Dn/prime(R)(A.1)\nwhich completes the proof of Eq.(5).\nForQ={R|v}T, notice time reversal symmetry does not change the real space coordinates, i.e. Td=d,\nwe haveQT(d) ={R|v}T{E|d}={R|Rd+v}T=T(Qd)RT. Use the fact that for anti-unitary operator\nˆA,/angbracketleftˆAψ|ˆAφ/angbracketright=/angbracketleftψ|φ/angbracketright∗, then\nE∗(dn\nj,(Rj+dn/prime\nl/prime)) =/angbracketleftˆQˆT(dn\nl)Φn(r)|ˆQˆH|ˆT(dn/prime\nl/prime+Rj)Φn/prime(r)/angbracketright\n=/angbracketleftˆQˆT(dn\nl)Φn(r)|ˆH|ˆQˆT(dn/prime\nl/prime+Rj)Φn/prime(r)/angbracketright\n=/angbracketleftˆT(Qdn\nl)RTΦn(r)|ˆH|ˆT(Q(dn/prime\nl/prime+Rj))RTΦn/prime(r)/angbracketright\n=Dn†(RT)/angbracketleftˆT(Qdn\nl)Φn(r)|ˆH|ˆT(Q(dn/prime\nl/prime+Rj))Φn/prime(r)/angbracketrightDn/prime(RT)\n=Dn†(RT)E(Qdn\nj,Q(Rj+dn/prime\nl/prime))Dn/prime(RT)(A.2)\nwhich completes the proof of Eq.(6).\nProof of Eq.(13),\n[H(k)P(Q)]nn/prime\nll/prime=/summationdisplay\nµνH(k)nν\nlµPνn/prime\nµl/prime(Q)\n=/summationdisplay\nµν/summationdisplay\nRjeik·(Rj+dν\nµ−dn\nl)E(dn\nl,Rj+dν\nµ)δνn/primeδdνµ,Qdn/prime\nl/prime+RsDn/prime(R)\n=/summationdisplay\nRjeik·(Rj+Qdn/prime\nl/prime+Rs−dn\nl)E(dn\nl,Rj+Qdn/prime\nl/prime+Rs)Dn/prime(R)\nRj+Rs→RRj= = = = = = = = = = =/summationdisplay\nRjeik·R(Rj+dn/prime\nl/prime−Q−1dn\nl)E(dn\nl,Q(Rj+dn/prime\nl/prime))Dn/prime(R)\nuse Eq. (5)= = = = = = = = =/summationdisplay\nRjeiR−1k·(Rj+dn/prime\nl/prime−Q−1dn\nl)Dn(R)E(Q−1dn\nl,Rj+dn/prime\nl/prime)(A.3)\n12[P(Q)H(R−1k)]nn/prime\nll/prime=/summationdisplay\nµνPnν\nlµ(Q)H(R−1k)νn/prime\nµl/prime\n=/summationdisplay\nµν/summationdisplay\nRjδnνδdn\nl,Qdνµ+RsDn(R)eiR−1k·(Rj+dn/prime\nl/prime−dν\nµ)E(dν\nµ,Rj+dn/prime\nl/prime)\n=/summationdisplay\nRjDn(R)eiR−1k·(Rj+dn/prime\nl/prime−Q−1(dn\nl−Rs))E(Q−1(dn\nl−Rs),Rj+dn/prime\nl/prime)\n=/summationdisplay\nRjeiR−1k·(Rj+dn/prime\nl/prime−Q−1dn\nl+R−1Rs)Dn(R)E(Q−1dn\nl−R−1Rs,Rj+dn/prime\nl/prime)\n=/summationdisplay\nRjeiR−1k·(Rj+dn/prime\nl/prime−Q−1dn\nl+R−1Rs)Dn(R)E(Q−1dn\nl,Rj+dn/prime\nl/prime+R−1Rs)\nRj+R−1Rs→Rj= = = = = = = = = = = = =/summationdisplay\nRjeiR−1k·(Rj+dn/prime\nl/prime−Q−1dn\nl)Dn(R)E(Q−1dn\nl,Rj+dn/prime\nl/prime)(A.4)\nIn the above derivation we use the relation Q−1(dn\nl−Rs) =Q−1dn\nl−R−1Rs(Q−1is not linear). Compare\nthe last lines of the above two equations and we can find they are equal to each other, i.e.\n[H(k)P(Q)]nn/prime\nll/prime= [P(Q)H(R−1k)]nn/prime\nll/prime⇒H(k)P(Q) =P(Q)H(R−1k) (A.5)\nProof of Eq.(14): similar to Eq.(A.3) and Eq.(A.4) we have\n[H(k)P(Q)]nn/prime\nll/prime=/summationdisplay\nµνH(k)nν\nlµPνn/prime\nµl/prime(Q)\n=/summationdisplay\nµν/summationdisplay\nRjeik·(Rj+dν\nµ−dn\nl)E(dn\nl,Rj+dν\nµ)δνn/primeδdνµ,Qdn/prime\nl/prime+RsDn/prime(RT)\n=/summationdisplay\nRjeik·(Rj+Qdn/prime\nl/prime+Rs−dn\nl)E(dn\nl,Rj+Qdn/prime\nl/prime+Rs)Dn/prime(RT)\nRj+Rs→RRj= = = = = = = = = = =/summationdisplay\nRjeik·R(Rj+dn/prime\nl/prime−Q−1dn\nl)E(dn\nl,Q(Rj+dn/prime\nl/prime))Dn/prime(RT)\nuse Eq. (6)= = = = = = = = =/summationdisplay\nRjeiR−1k·(Rj+dn/prime\nl/prime−Q−1dn\nl)Dn(RT)E∗(Q−1dn\nl,Rj+dn/prime\nl/prime)(A.6)\n[P(Q)H∗(−R−1k)]nn/prime\nll/prime=/summationdisplay\nµνPnν\nlµ(Q)H∗(−R−1k)νn/prime\nµl/prime\n=/summationdisplay\nµν/summationdisplay\nRjδnνδdn\nl,Qdνµ+RsDn(RT)eiR−1k·(Rj+dn/prime\nl/prime−dν\nµ)E∗(dν\nµ,Rj+dn/prime\nl/prime)\n=/summationdisplay\nRjDn(RT)eiR−1k·(Rj+dn/prime\nl/prime−Q−1(dn\nl−Rs))E∗(Q−1(dn\nl−Rs),Rj+dn/prime\nl/prime)\n=/summationdisplay\nRjeiR−1k·(Rj+dn/prime\nl/prime−Q−1dn\nl+R−1Rs)Dn(RT)E∗(Q−1dn\nl−R−1Rs,Rj+dn/prime\nl/prime)\n=/summationdisplay\nRjeiR−1k·(Rj+dn/prime\nl/prime−Q−1dn\nl+R−1Rs)Dn(RT)E∗(Q−1dn\nl,Rj+dn/prime\nl/prime+R−1Rs)\nRj+R−1Rs→Rj= = = = = = = = = = = = =/summationdisplay\nRjeiR−1k·(Rj+dn/prime\nl/prime−Q−1dn\nl)Dn(RT)E∗(Q−1dn\nl,Rj+dn/prime\nl/prime)(A.7)\nCompare the last lines of the above two equations and we can find they are equal to each other, i.e.\n[H(k)P(Q)]nn/prime\nll/prime= [P(Q)H∗(−R−1k)]nn/prime\nll/prime⇒H(k)P(Q) =P(Q)H∗(−R−1k)(A.8)\n13It is easy to verify that for A,Bnot containingTandC,DcontainingT,P(Q)obey the following\ncorepresentation algebra:\nP(A)P(B) =P(AB)\nP(A)P(C) =P(AC)\nP(C)P∗(A) =P(CA)\nP(C)P∗(D) =P(CD)(A.9)\nReferences\n[1] P. 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URL:\nhttp://link.springer.com/10.1007/BF01342591 . doi:10.1007/BF01342591.\n16"    },    {        "title": "2106.00370v2.Impact_of_magnetic_domains_on_magnetic_flux_concentrators.pdf",        "content": "Impact of magnetic domains on magnetic flux concentrators  \n \nFederico  Maspero1, Simone Cuccurullo2, Dhavalkumar Mungpara3, Alexander Schwarz3 and Riccardo \nBertacco1,2 \n1CNR Istituto di Fot onica e Nanotecnologie, Milano, Italy \n2Politecnico di Milano , Milano , Italy and  \n3Institute of Nanostructure and Solid State Physics, University of Hamburg, Germany . \n \nAbstract  \n \nThe impact of magnetic domains in magnetic flux con-\ncentrators is studied using the simulation software \nMuMax3 . First, the simulation parameters are validated us-\ning experimental results from magnetic force microscopy ; \nsecond the simulation output is benchmarked with the one \nobtained using Comsol Multiphysics. Finally, the impact of \nmagnetic domain is assessed, show ing how micromagnetic \neffects can become relevant, if not domin ant, when scaling \nthe gap between the MFC and the sensor.    \n \nKeywords  \n \nMFC, MFM , MuMax3 . \n \n1 Introduction  \n \nIntegrated magnetic flux concentrators (MFCs) are \nmade of thin films of ferromagnetic materials with high \npermeability and low coercive field, e.g. permalloy \n(Ni 80Fe20) or Co based alloy s like CoZrNb/CoZrTa . They \nare used to amplify [1, 2] and/or modulate [3] an external \nmagnetic field nearby a magnetic field sensor, e.g. a mag-\nneto-resistive sensor, which is placed in the gap between \ntwo MFCs.  \nConventionally, the design of MFCs is optimized by \nsolving the magnetostatic Maxwell’s equations through fi-\nnite ele ment methods [4, 5]. This approach allows to simu-\nlate relatively large geometries (MFCs have generally pla-\nnar dimensions of hundreds of µm) but neglects micromag-\nnetic effects, i.e. formation of magnetic domains and mag-\nnetic domain walls, which can be impor tant at the µm scale.  \nThe study of the effect of micromagnetic domains on \nMFC field amplification was partially assessed in the work \nof Trindade et al. [2, 6], yet no modelling or tool to study \nthis effect was given. In this paper, therefore, micromag-\nnetic  simulations  and experimental measurements  are per-\nformed to understand the effect of magnetic domains on the \namplifi cation  behaviour of different MFC shapes.  \nThe paper is divided as follows: first, a description of \nthe employed simulation /experimental  methods , specify-\ning the adopted parameters.  Second, the comparative inves-\ntigation, both by magnetic force microscopy (MFM) and \nby simulations (using MuMax3) of the micromagne tic con-\nfiguration vs. applied field for rectangular MFC, to tune the \nparameters to be used for micromagnetic simulation s [7]. \nFinally, the compar ison between magnetostatic and micro-\nmagnetic simulations for various shapes of MFC, to assess \nthe effect of magnetic domains on the field amplification \ncurve of MFC.   \n2 Simulation Meth ods \n \n2.1 MuMax3  \n \nMicromagnetic simulations of the MFCs were performed \nwith MuMax3 [ 7], which is a GPU -accelerated software \nthat offers a significant speedup compared to other CPU -\nbased computer programs.  \nMuMax3 employs the finite difference method to simul ate \nthe time and space dependent magnetization evolution at \nthe micrometric scale by solving the Landau -Lifshitz -Gil-\nbert equation [ 8, 9]: \n \n𝜕𝒎\n𝜕𝑡=𝛾𝑯𝑒𝑓𝑓×𝒎+𝛼𝒎×𝜕𝒎\n𝜕𝑡 \n \nwhere 𝒎 is the reduced magnetization, 𝛾 is the gyromag-\nnetic ratio, 𝛼 is the damping constant and  𝑯𝑒𝑓𝑓 is the ef-\nfective field comprising the exchange, the Zeeman and the \ndemagnetizing contributions.  \nMagnetic  parameters of permalloy are employed, i.e. satu-\nration magnetization 𝑀𝑠=8×105 A/m, exchange stiff-\nness 𝐴=1.0×10−12 J/m and null anisotropy constant. \nThe typical damping constant for permalloy is 𝛼=0.01. \nHowever, since only stationary conditions are considered, \n𝛼 was set to 0.1 to speed up the simulations.  \nThe size of the discretizati on cell is bounded by the ex-\nchange length 𝜆𝑒𝑥=(2𝐴/𝜇0𝑀𝑠2)1/2≈5 nm. Neverthe-\nless, because of the relatively  large  geometries simulated in \nthis work , cell s sizes ranging from 8 to 25 nm have been \nused. This choice results in a loss of resolution for magn etic \ndomain walls, which is not critical for the scope of the pa-\nper as confirmed by experimental results.  \n \n2.2 COMSOL  Multiphysics  \n \nMagnetostatic simulations of the MFCs were performed \nwith COMSOL Multiphysics by solving the Maxwell’s \nequations in stationary c onditions. The simulations com-\nprised an air volume at whose centre are placed two MFCs \nseparated by a gap , immersed in a uniform magnetic field.  \nBoth air and MFCs were modelled as linear and isotropic \nmagnetic materials with relative permeability equal to 1 \n(air) and 1000  (MFCs). The latter was chosen in accord-\nance with measurements on permalloy thin films used in \nthis work and those found in liter ature  [10]. \n  3 Experimental method  \n \n3.1 Fabrication  \n \nMagnetic field concentrators were  patterned  through opti-\ncal lithography  on a 1 x 1 cm2 silicon substrate . The resist \nadopted was the AZ 5214 E image reversal photoresist (Mi-\ncrochemicals ). \nPermalloy was deposited by electron beam evaporation  \n(Evate c BAK 640). The base pressure was 10-7 mbar and \nthe deposition rate was set to 0.2 nm/s . \n \n3.2 MFM c haracterization  \n \nWe performed m agnetic force microscopy (MFM) meas-\nurements [11] with a commercial instrument [ 12] in ambi-\nent conditions  using amplitude modulation [ 13] and the lift \nmode technique.  In the MFM lift mode , the tip is moved at \na constant height above the surface without ever touching \nthe surface by following the topography path recorded dur-\ning the first scan. The phase (x,y) is recorded, which re-\nflects  the long -range magnetostatic tip-sample interaction \nand is called MFM image.  \nWe used commercially available CoCr coated tips with a \nspring constant cz = 2.8 N/m and a resonance frequency fr \n= 69.3 kHz [14].  Due to the shape anisotropy energy  of the \nthin magnetic film on the tip pyramid, the tip magnetization \nis oriented nearly perpendicular to the sample surface  [15]. \nThis results in a predominantly out -of-plane sensitivity \n[16]. For samples like permalloy thin films with in -plane \nmagnetiz ed domains , out -of-plane components are only \npresent at domain walls. The situation is sketched in Fig. \n1a-b for Bloch and Neel type 180° domain walls, in which \nthe magnetization rotates out -of-plane and in -plane, re-\nspectively.  The characteristic contrast change across a do-\nmain wall allows identifying the type of domain wall pre-\nsent in the sample.   \nTo apply in -plane magnetic fields of up to 20 mT, we used \na calibrated electromagnet (two Cu coils with yokes in a \nHelmholtz configuration).  Even at the maximum field, the \ntip-magnetization remained stable and only the domain \nstructure of the permalloy sample changed.  \n         \n \nFigure 1 – a)-b) MFM contrast across (a) Bloch and (b) Neél -type \ndomain wall. Shape anisotropy energy forces the tip magnetiza-\ntion to be nearly normal to the sample surface. Therefore, in \nMFM images Neél walls can be easily identified by a dark -bright \ncontrast pattern between neighboring magnet ic domains.   \n \n 4 Results  \n \n4.1 Investigat ion of rectangular MFC by MFM and \nMuMAx3 simulations  \n \nFig. 2 and Fig. 3 show the comparison between the experi-\nmental results obtained by M FM and micromagnetic simu-\nlations by MuMax3 for rectangular  MFC , with 20  x 26 mm \narea and 150 nm thickness . It should be pointed out that \ntwo different physical quantit ies are represented in Fig.2 \nand Fig.3. The result of the simulation is shown by plotting \nthe x -component of the computed magnetization. This \ngives an idea of the domain  configuration and highlights \nthe component which is relevant for the calculation of field \namplification produced by the MFC . On the other hand, the \nMFM measurement shows the interaction force between \nthe probe tip and the stray field of the sample in  the out-of-\nplane direction . This gives information on the presence of \ndomain walls and th e type of wall , but it does not directly \nshows the magnetization of the domain. Nevertheless, t he \nextrapolated domain map and its evolution under applied \nfield can be used  for comparison of the two methods.  \nExperimentally we found two characteristic micromagnetic \nflux-closure configurations  at zero applied field : (i) a Lan-\ndau pattern with two vort exes and a 180°  central domain \nwall (Fig. 2d), (ii) a diamond pattern with a central hexag-\nonal domain and four vortices (Fig 3d). The magnetization \ndirection within the domain are indicated by arrows  on the \nzero-field configurations . All 90° and most 180° domain \nwalls are of Nèel -type, clearly iden tifiable by the dark -\nbright contrast between neighbouring domains . However, \nsometimes we also observe more complex 180°  cross -tie \nwalls. MFM results show that the initial domains configu-\nrations can change from one MFC to another likely due to \npresence of d efects in the film.  In any case, applying an \nexternal field always increase s the size of those domains, \nwhich are parallel to the field direction.  In both cases, an \nexternal magnetic flux density of 10 mT does not fully sat-\nurated the elements  and resulte s in an S -state.   \nTo simulate the behaviour of MFCs with a Landau pattern \nthe zero-field magnetization of the MFC was obtained \nupon  relaxation of a single central vortex magnetic config-\nuration , which is typical ly found in  rectangular  dots of  \nPermalloy of this size  [17].  \nAfter re aching the equilibrium state  an external field in the \nx direction ranging from 1 to 10 mT was applied  in simu-\nlations, together with a cross -axis field of 1  % both in y and \nz direction  to avoid instabilities of the solution  and account \nfor misalignment in the experimental setup . The equilib-\nrium magnetization  was recorded at each step and com-\npared with the experimental results , as shown in Fig. 2 and \nFig. 3 . In both experiment  and simula tions,  the same mo-\ntion of domain walls is observed , corresponding to the ex-\npansion of the left domain aligned to the external field.    \n \nFigure 2 – MFM measurement  (bottom images)  and x -axis \ncomponent of the simulated magnetizaton  (top images) . \nMumax simulation were performed with a 1024  x 1024  x \n16 cells grid, the size the rectang ular shap e is 20  x 26 x \n0.15 µm. This lead s to a miximum cell size of 25nm.   The \nmagnetization was acquired in the central plane of the \nsimulated film , however t he same domain s configuration  \nwas observed in all horizontal planes.  \nGood agreement among  the simulation and the experi-\nmental results can be seen also in the case of the diamond  \nconfiguration . This configuration was less common in the \narray of fabricated permalloy MFC,  but still present.  Such \nconfiguration was reproduced in MuMax3 by placing two \nmagnetization vortex es with anticlockwise and clockwise \nrotation in the left and right  region  of the MF C respec-\ntively . This time, domain walls motion takes place together \nwith coherent rotation and domain reconfiguration.  \nOverall, this comparative study allowed to tune the simu-\nlation parameters (see methods) and find a very good \nagreement between simulations and MFM exper iments. \nThe very same parameters have been used also for other \ngeometries, to assess the impact of the MFC micromag-\nnetic structure on their functional properties.  \n  \nFigure 3 – Sequence of MFM  images  and MuMax \nsimulation  results starting with two vortex es configuration. \nWith  no applied field the two vortex es relax into an \nexagonal central domain. The applied field has the same \ndirection of the external domains magnetization. T his \ncauses the centreal exagon  to shrink  until a large central \ndomain is formed  due to the opposite direction of field and \nmagnetization . The two domains colored in b lue which are \nevident in the simulation are not so clearly visible in the \nexperimental measurement  where they progressivly move \nto the boarder of the MFC and disappear.   \n4.2 MuMax3 as a tool to model MFC  \n \nIn the second part of this work, MuMax 3 and COMSOL  \nwere used to model the magnetization of three different ge-\nometries of MFC under varying magnetic field and evalu-\nate the demagnetizing field produced by the three shapes. \nThe three chosen geometries are  typical MFC geometries \n[4]: funnel -like, T -shape and rectangular.  \nThe size of the MFC (5x8 micron) was chosen to be smaller \nthan those used for the previous simulation (Fig.  2 and Fig. \n3) to save computational time and simulate pairs of MFC s \ninstead of a single MFC . \nFig. 4 shows an example of the magnetizat ion map ob-\ntained with the two software s.  \nThe x component of the magnetization obtained with \nCOMSOL  Multiphysics  follows the results observed in lit-\nerature [5]: the magnetization  reaches the maximum value \nin the center of the MFC  and decreases towards the edges.  \nA different result is found in  MuMax3, where the magnet-\nization map shows magnetic domains. Again, a vortex -like \nconfiguration is seen  in both concentrators, with the upper \n \nFigure 4 – Magnetization (M x ) of the funnel -like geometry \ncomputed with COMSOL ( a) and MuMax3 ( b-c) when ap-\nplying 2  mT in the x direction.  This time MuMax simulation \nwas started from a saturated magnetization and the n the  \nmagnetic field was swept from  -80 mT to +80 mT  and  vi-\nceversa . Figure b) and c) show the magnetization map ob-\ntained under 2 mT for different sweep order . The magneti-\nzation asymmetry of the MFC is mirrored  when the applied \nfield sweeping order is reversed . The grid was set to 1024  \nx 1024  x 16. The largest cell size was around 12  nm. Scale \nis reported for size evaluation. The thickness was set to 150  \nnm as in the experimental case. Comsol magnetization is \ngradually increasing towards the center of the geometry \nand it is linearly proportional to the e xternal field.  MuMax \nshows a more realistic domains magnetization.  \nand lower parts showing opposite magnetization. However ,  \nin the left one there is just a  180° DW, while in the right -\none a diamond -like configuration appears . The difference \nbetween the l eft and right concentrators depends on the \nsweep order  of the field . Indeed, the same simulation  with \nopposite sweep order  (+80 mT to -80 mT instead of -80 mT \nto +80 mT) leads to an almost  specular result  (Figure 4 b-\nc). This is due to the asymmetric shape of the MFC along \nx, which together with the applied field  direction , has an \nimpact on the magnetic domain s formation.   \nNote, however, that here what is crucial is the magnetiza-\ntion close to the central poles, in a regi on where left and \nright MFCs have a similar domain.  \nThe second crucial aspect studied in the simulations is the \nfield produced in the region between the two MFC.  Fig. 5 \ncompares the demagnetizing field vs. applied external field \nat the center of the gap. The results obtained with the two simulation tools agree  only in the central region . The ef-\nfects of DW motion under external field and magnetic sat-\nuration appears only in  micromagnetic simulations, clearly \nshowing deviations from the linear behaviour at a few mT.  \n \nFigure 5 – d)   Demagnetizing fie ld versus applied field at \nthe center of the constriction  computed using COMSOL  \nand MuMax3  for funnel -like (a), T -shaped (b) and rectan-\ngular (c) MFCs . The field was averaged over a rectangular  \narea in the center of the constriction. x=100nm and \ny=2um.  Only half hysteresis cycle (sweep -up) is shown to \nbetter compare the slope of the curves.    \n \nFigure 6 – a) Comparison between the hysteresis cycle s of \nthe three MFC shapes. The simulation was computed only \nby sweeping the field from -80 to 80  mT and assuming a \nsymmetric response  of opposite sign  for the down -sweep.  \nb) zoomed image around zero applied field to evaluate the \ncoercivity and the slope of  the three curves.  \n \nFigure 7 – Demagnetizing field produced by the MFC \nalong y -axis cutline taken at  different distances from the \ncenter (represented in figure insert). The graph is obtained \nwith 2  mT applied. It is visible how the demagnetizing field \nis related to the domains configuration. First, the maximum \nis not found at the center of the cutline. S econd, close to the \nMFC the field distribution shows lobes which depend on \nthe magnetic domain distribution.  \nIn Fig. 6 the simulated hysteresis cycles of the three MFC \nshapes are reported.  \nThe slope of the curves for small fields, i.e. the MFC am-\nplificati on factor, increases proportionally  to the ratio \n𝐿2/𝐿1 (see Fig. 5). Therefore, the funnel -like and T -shape \ngeometries have the largest amplification factor.  \nFurthermore, the funnel -like and rectangular geometries \nshow smaller coercivities. This is likel y due to the absence \nof pinning sites for magnetic domains, which are present in \nthe T -shape MFC for geometric reasons.  \nThe funnel -like configuration thus presents the best char-\nacteristics in terms of coercivity, amplification factor and \nlinear range.  \nThe final aspect which was investigated was the effect of \nthe domains on the uniformity of the field produced by the \nmagnetic flux concentrator. In Fig.  7, the value of the de-\nmagnetizing field along the y -axis is plotted at different \ndistances from the center  of the constriction.  \nAs it can be seen, w hen moving close to the MFC edges , it \nis important to consider  that the field produced by the MFC \nis not uniform and related to the domain distribution.  For \ninstance, the sensitivity of a sensor with a square active \narea of 1x1um could be affected by misalignment error in \nthe fabrication process . The same sensor at different y -axis \ncoordinate would respond differently to an external field.  \n \n5 Conclusions  \n \nMagnetic flux concentrators are magnetic objects used to \namplify and shape the magnetic field. They are currently \nused in several magnetic sensors.  \nIn this work, different MFC s geometries were  studied con-\nsidering  micromagnetic effects , in particular the fragmen-\ntation into magnetic domains.   \nFirst, the parameters to be used in the micromagnetic sim-\nulation tool  (MuMax3)  were tuned by comparison with ex-\nperimental  result from MFM measurement of permalloy \nMFC under applied field.  Second, the field amplificati on of three different MFC ge-\nometries  was studied using MuMax 3 and COMSOL  Mul-\ntiphysics . Our results show  good agreement between the \nsimulation platforms only in the linear regime, i.e. within a \nfew mT . With respect to COMSOL , MuMax3 enables the \npossibility of studying micromagnetic effects  giving more \nrealistic modelling of the MFC behavior , including hyste-\nresis, saturation and impact of the micromagnetic configu-\nration on the spatial profile of the field within small gap \nMFCs .  \nOn the other hand,  COMSOL  offers a much faster simula-\ntion tool and is more suited for simulation of large size \nMFC (hundreds of microns).   \n \nAcknowledgmen ts \n \nThis work was carried out in the frame of the European \nFET-Open Project OXiNEMS. This project has received \nfunding from the European Union’s Horizon 2020 research \nand innovation programme under grant agreement No \n828784.  \n \nOpen data  \n \nThe data that support the findings of this study and \nsupplementary material are openly available in a Zenodo \nrepository with DOI: 10.5281/zenodo.4446957 .  \n \nReferences  \n \n[1] Guedes, A., Almeida, J. M., Cardoso, S., Ferreira, R., \n& Freitas, P. P. (2007). Impro ving magnetic field de-\ntection limits of spin valve sensors using magnetic flux \nguide concentrators. IEEE transactions on magnetics, \n43(6), 2376 -2378.  \n \n[2] Trindade, I. G., Fermento, R., Sousa, J. B., Chaves, R. \nC., Cardoso, S., & Freitas, P. P. (2008). Lin ear field \namplification for magnetoresistive sensors. Journal of \nApplied Physics, 103(10), 103914.  \n \n[3] Edelstein, A. S., Fischer, G., Bernard, W., Nowak, E., \n& Cheng, S. F. (2006). The MEMS Flux Concentrator: \nPotential Low -Cost, High -Sensitivity Magnetome ter. \nARMY RESEARCH LAB ADELPHI MD.  \n \n[4] Drljača, P. M., Vincent, F., Besse, P. A., & Popović, R. \nS. (2002). Design of planar magnetic concentrators for \nhigh sensitivity Hall devices. Sensors and Actuators A: \nPhysical, 97, 10 -14. \n \n[5] Sun, X., Jiang, L., & Pong, P. W. (2013). Magnetic flux \nconcentration at micrometer scale. Microelectronic en-\ngineering, 111, 77 -81. \n \n[6]  I. G. Trindade et al. , «Control of hysteretic behavior in \nflux concentrators», Appl. Phys. Lett. , vol. 94, n. 7, \npag. 073501, feb. 2009 . \n \n[7] Vansteenkiste, A., Leliaert, J., Dvornik, M., Helsen, M., \nGarcia -Sanchez, F., & Van Waeyenberge, B. (2014). \nThe design and verification of MuMax3. AIP ad-\nvances, 4(10), 107133.  \n \n[8] Landau, L. A. L. E., & Lifshitz, E. (1992). On the theory \nof the dispers ion of magnetic permeability in ferro-\nmagnetic bodies. In Perspectives in Theoretical Phys-\nics (pp. 51 -65). Pergamon.  \n \n[9] Gilbert, T. L. (1955). A Lagrangian formulation of the \ngyromagnetic equation of the magnetization field. \nPhys. Rev., 100, 1243.  \n \n[10] Gardner, D. S., Schrom, G., Paillet, F., Jamieson, B., \nKarnik, T., & Borkar, S. (2009). Review of on -chip in-\nductor structures with magnetic films. IEEE Transac-\ntions on Magnetics, 45(10), 4760 -4766.  \n \n[11] Y.   Martin   and   H.   K.   Wickramasinghe,   “Mag-\nnetic   imaging   by   \"force   microscopy\" with 1000 \nA resolution ”, Appl. Phys. Lett. 50, 1455 (1987) . \n \n[12] Nanoscope IIIa with extender box, originally from \nDigital Instruments then Veeco now Bruker Co opera-\ntion. \n \n[13] Y. Martin , C. C. Williams , and H. K. Wick-\nramasinghe , “Atomic force microscope –force map-\nping and profiling on a sub 100 -Å scale “ J. Appl. \nPhys. 61, 4723,  (1987) . \n \n[14] PPP -MFMR from Nanosensors from Nanoworld AG, \nSwitzerland.  \n \n[15] U. Ha rtmann  “Magnetic force microscopy”, Annu. \nRev. Mater. Sci. 29, 53 ( 1999 ). \n \n[16] D. Rugar, H. J. Mamin, P. Guethner, S. E. Lambert, J. \nE. Stern, I. McFadyen, and T. Yogi , “Magnetic force \nmicroscopy: General  principles and application to lon-\ngitudinal  recording media “. J. Appl. Phys. 68, 1169 \n(1990) ,  \n \n[17] D. Goll, G. Schütz, and H. Kronmüller, “Critical \nthickness for high -remanent single -domain configura-\ntions in s quare ferromagnetic thin platelets,” Phys. \nRev. B , vol. 67, no. 9, p. 094414, Mar. 2003, doi: \n10.1103/PhysRevB.67.094414.  \n \n \n \n "    },    {        "title": "2106.02906v1.On_the_anomalous_low_resistance_state_and_exceptional_Hall_component_in_hard_magnetic_Weyl_nanoflakes.pdf",        "content": "1 \n On the a nomalous  low-resistance state and exceptional Hall \ncomponent in hard -magnetic Weyl  nanoflakes  \nQingqi Zeng1†, Gangxu Gu1†, Gang Shi1, Jianlei Shen1, Bei Ding1, Shu Zhang2, Xuekui Xi1, \nClaudia Felser3, Yongqing Li1,4, Enke Liu1,4,* \n1. Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Scienc es, \nBeijing 100190, China  \n2. Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA  \n3. Max Planck Institute for Chemical Physics of Solids, Dresden D -01187, Germany  \n4. Songshan Lake Materials Laboratory, Dongguan , Guangdong 523808, China  2 \n Abstract  \nMagnetic topological materials, which combine magnetism and topology, are \nexpected to host emerging topological states and exotic quantum phenomena. In this \nstudy, with the aid of  greatly enhanced coercive fields in high-quality  nanoflakes of \nthe magnetic Weyl semimetal Co 3Sn2S2, we  investigate  anomalous  electronic \ntransport properties that are difficult to reveal in bulk Co 3Sn2S2 or other magnetic \nmaterials. When the magnetization is antiparallel to the applied magnetic  field, the \nlow longitudinal resistance state  occurs,  which is in sharp contrast to the high \nresistance state for the parallel case. Meanwhile, an exceptional Hall component that \ncan be up to three times larger than conventional anomalous Hall resistivity is also \nobserved for transverse transport. These anomalous  transport behaviors can be further  \nunderstood by considering nonlinear magnetic textures and the chiral magnetic field \nassociated with Weyl fermions , extending the longitudinal and transverse trans port \nphysics and providing novel degrees of freedom in the spintronic applications of \nemerging topological magnets.  \n \nKeywords : hard -magnetic material, magnetic Weyl semimetal, magnetoresistance, \nHall effect  3 \n Introduction  \nRecently, the first magnetic Weyl semimetal Co 3Sn2S2 was discovered [1-6], \nwhich has significantly improved the understanding of the interplay of magnetic order \nand topol ogical physics after the experimental realization of the quantum anomalous \nHall effect (QAHE) [7]. Co 3Sn2S2 is a Shandite compound with Weyl nodes at ~60 \nmeV above the Fermi level and shows semi -metallicity with small Fermi surfaces and \nlow carrier concentr ation. Transport properties are dominated by the electrons near \nFermi energy . As the topological features  in this system  are sufficient close to the \nFermi energy and there are few trivial bands,  it is promising to observe the transport \nbehavior dominated b y topological band structures. In Co 3Sn2S2 bulk single crystals, \nlarge intrinsic anomalous Hall conductivity and anomalous Hall angle have been \ndetected [1, 2] . In the two -dimensional (2D) limit, theoretical studies have predicted \nthat the desired high -temp erature QAHE could be realized [8-10], which has been \nindicated experimentally by the observation of chiral edge states in a recent STM \nmeasurement [11]. In addition, regardless of the specific types of magnetic domain \nwalls, there may exist a large magnetor esistance (MR) that can be retained against the \nfinite strength of the disorder in Co 3Sn2S2[10]. \nThe application prospects of Co 3Sn2S2 make it a promising candidate for \npractical applications of topological materials in spintronic or advanced electronic \ndevices. For topology -related spintronic applications and the realization of QAHE,  the \ngrowth and physical study of Co 3Sn2S2 films are highly desired. Epitaxial \npolycrystalline Co 3Sn2S2 films can be grown via molecular beam epitaxy  and \nco-sputtering depositi on methods [12, 13] . Moreover, reports on Co 3Sn2S2 nanoflakes \ngrown via the chemical vapor transport (CVT) method exist [14, 15] . Out-of-plane \nferromagnetic order  and large anomalous Hall effects were observed. These studies \noffer important indications for f urther studies of low -dimensional magnetic Weyl \nsemimetals. In addition to studies on possible QAHE and its practical applications, the \nunexplored properties of this magnetic Weyl low -dimensional system are important . \nAnomalous  transport properties  in Co 3Sn2S2 nanoflakes  have been observed in our \nprevious work [14]. However, these intriguing behaviors still need to be studied and 4 \n analyzed in detail.  \nIn this work, anomalous  transport properties are studied  in high -quality Co 3Sn2S2 \nsingle -crystalline nanoflake s with 30 –100-nm thickness. The system clearly enters a \nlower electrical resistance state once the external magnetic field is antiparallel to the \nmagnetization. Furthermore, an exceptional Hall component besides the normal and \nanomalous Hall effects is obs erved concurrently. Possible magnetic textures induced \nby an antiparallel magnetic field are possibly  responsible for  the anomalous  transport \nbehaviors, as the interaction between Weyl electrons and magnetic textures can result \nin local states and conducti ve modes [16-20]. This study will attract interest in  the \ninterplay of magnetic order and topological transport behavior and offer a platform for \nmagnetic topological material -based applications.  \nMaterials and methods  \nSingle -crystalline Co 3Sn2S2 nanoflakes were grown via CVT using  Co3Sn2S2 \npolycrystalline powder as the precursor. The morphology and thickness were \ninvestigated using atomic force microscopy. Hall bar pattern with a long edge along \nthe a axis of Co 3Sn2S2 for out -of-plane transport measurement w as fabricated using \nmicrofabrication  technology (Supplementary Note 1). Transport data were measured \nusing a Janis system and collected by employing lock -in amplifier technology. \nSymmetric and antisymmetric processes were performed on the longitudinal and \ntransverse resistivity, respectively (details in  Supplementary Note 3). \nResults and discussion  \nThe morphology and thickness of a single -crystalline nanoflake are shown in \nFigure 1(a)  (more details are given in Supplementary Note 1). Samples of thicknesses \n30, 41, 71, and 94 nm were studied in this work. Figure 1(b) shows one of the Hall bar \npatterns for transport measurements. The Curie temperatures of the two selected \nsamples were determined as 181 K through the kink temperature in the \ntemperature -dependen t resistivity curve ( Figure S3 ). The  single -crystalline  Co3Sn2S2 5 \n nanoflakes being studied show consistent basic physical properties with bulk single \ncrystals [1, 21]  (Supplementary Note 4) . Furthermore, the nanoflakes  show high \nquality with a residual resis tivity ratio of ~20, an anomalous Hall conductivity of \n~1000  Ω−1 m−1, and a large coercive field as high as 5.5 T.   \nFigures 1(c –j) show the magnetic field dependence of the longitudinal and \ntransverse resistivity measured at various temperatures for the 30 -nm-thick sample \n(data of other samples are shown in Supplementary Note 4). The longitudinal data \n(Figure s 1(c –f)) show positive parabolic -like MR at all measured temperatures. Apart \nfrom the ordinary MR, hysteresis behavior exists where the data with decr easing and \nincreasing fields are not coincident between zero and coercive fields ( μ0Hc). Thus, the \nvalue of the longitudinal resistivity depends on the magnetization history, being lower \n(50 K) or higher (1.6 K) for the demagnetization compared with the magnetization  \nprocess . \nAs shown in Figure s 1(g–j), the Hall data exhibit hysteresis loop behavior like \nthe shape of the magnetic field dependence of magnetization in a hard -ferromagnetic \nmaterial. The anomalous Hall resistivity, in addition to the normal Hall effect, is \ndetected. The loop behavior and the anomalous Hall effect like that o f the bulk \nsample [1] suggest that the easy axis of Co3Sn2S2 single -crystalline nanoflake is still \nalong the c axis. Clear nonlinear normal Hall curves are observed at low temperatures, \nreflecting that both electron and hole carriers contribute to the trans port properties in \nCo3Sn2S2 single -crystalline nanoflakes. We performed two -band [1, 22, 23]  analyses \non all measured samples. The carrier concentration of holes and electrons are both ~1 \n×  1020 cm−3 in all studied samples (details in Supplementary Note 4), indicating a near \ncompensation of carriers in Co 3Sn2S2 single -crystalline nanoflakes with the \nabovementioned different thicknesses. The abovementioned results reveal the \nconsistent carrier properties of Co 3Sn2S2 single -crystalline nanoflake and bulk \nsampl es[1, 14, 15] . 6 \n \n \nFigure 1 . (a)  Thickness and morphology (inset) of the 94 -nm-thick sample \ndetermined using atomic force microscopy. (b) Optical image of the Hall bar shape \n41-nm-thick sample. A type of electrode connection path is denoted. Current is \nappli ed along the a axis (indicated by a white  arrow) of  Co3Sn2S2. (c–f) Longitudinal \nresistivity measured at 50, 30, 20, and 1.6 K for the 30 -nm-thick sample. The arrows \nindicate the direction of field variation. (g –j) Hall data measured simultaneously with \nthe longitudinal resistivity in (c–j). \nThe additional MR peak or dip emerging in the background of the parabolic -like \nMR (the curves without magnetization reversal, i.e., from ± 9 T to zero fields ) was \nextracted and denoted as \nxx . At 5 0 K, when the field decreased from ± 9 T to zero \nand then increased in the opposite direction, a linear field -dependent positive \nxx  \nappeared immediately and became zero  instantaneously  after the magnetization \nreversal ( Figure 2(a) ). However, a negative linear \nxx  was unexpectedly observed at 7 \n 1.6 K when the applied field approached the coercive field ( Figure 2(b) ). \nWe further analyzed the emerging additional MR when magnetization was \nantiparallel to the applied m agnetic field. In a widely accepted case, a system will be \nmore magnetically ordered when the magnetic field is much higher than  μ0Hc. \nBetween zero and coercive fields, the magnetic field gradually turns the antiparallel \nmoments into parallel moments. Thus , the antiparallel moments become more \ndisordered as the field increases before the magnetization reversal. Hence, the \nspin-dependent scattering [7, 24, 25]  would be strong, increasing the longitudinal \nresistivity. Therefore, positive \nxx , like the behavior in Figure 2(a),  is normally \nobserved in magnetic systems [7, 26] . The difference from the common case is that the \npositive \nxx  in the studied sample appears  immediately  when the antiparallel \nmagnetic field star ts increasing. More widely observed positive MR peaks appear \nonly near the coercive field [27-29]. The linear magnetic field dependence is also a \nunique and unexplained phenomenon.  \nInterestingly, a contrary  situation emerges when the system goes to low \ntemp eratures. The longitudinal resistivity in the case when the applied field is \nantiparallel to the magnetization (antiparallel configuration) is even lower than that of \nthe parallel case ( Figure 2(b) ), i.e.,  \nxx  is negative. This behav ior has been \npreviously observed in our 180 -nm-thick Co 3Sn2S2 single -crystalline nanoflakes [14] \nand can also be recognized in polycrystalline films (although the spin -related positive \nMR is mixed in) [13]. Evidently, the trend is a universal behavior in thi s \nlow-dimensional Weyl system.  Nevertheless, negative \nxx  has scarcely been \nobserved among common magnetic materials or magnetic functional films in the past. \nA distinct mechanism resulting in a low -resistance state against the antip arallel \nexternal field may emerge in this magnetic topological system.  \nIn magnetic information storage technology, the giant MR effect can be produced \nby switching the parallel and antiparallel directions of ferromagnetic and 8 \n antiferromagnetic layers in ma gnetic tunnel junctions, which use different electrical \nresistance states between parallel and antiparallel cases [30-32]. The anomalous  \nnegative MR effect observed in this magnetic nanoflake  may provide a novel degree \nof freedom for spintronic applications . \n \nFigure 2.  (a–b) Parabolic -like MR -subtracted magnetic field dependence of the \nlongitudinal resistivity (\nxx ) at 50 and 1.6 K for the 30 -nm-thick sample. The arrows \nindicate the direction of field variation. (c) \nxx  difference before and after the jump \nat μ0Hc (\n0()xx c H ). (d) Ratio of \nxx  to \nxx  on the parabolic -like MR background \nat μ0Hc (dark blue dots) and separate contributions (green and orange symbols).  9 \n Figures 2(c –d) show \nxx  at μ0Hc (\n0()xx c H ) and the  \n0()xx c H\n-to-\n0()xx c H  ratio on the background of the parabolic -like MR (dark blue \ndots). Above and below 30 K, \n0()xx c H  is positive and negative, respectively. As a \nkink of \n0()xx c H  exists around 100 K, we considered the existence of competing \ncomponents leading to different field dependences of longitudinal resistivity and their \ndominant position var iation  with temperature. For si mplicity, two factors are currently \nassumed in this system. To separate these two parts, we assume that one part, called \nthe low -temperature part (low -T part), occurs only below the kink temperature in \n0()xx c H\n. The other part is the high -temperature part (high -T part), which is fitted \nusing the data just above 100 K. By adopting the form of y = A e−x/B + y0, the \nexperimental data (dark blue dots) can be well fitted (green square symbols) above 80 \nK (see Figure S6  for details). The low -T part is then extracted by subtracting the \nhigh-T part from the experimental results, denoted by orange triangle symbols in \nFigure 2(d) . The current straightforward but effective fitting indicates that the high -T \npart contributes to longitudinal resistivity continuously below the Curie temperature. \nBy contrast, the low -T part appears only below 70 K. Here, we stress that the \nlow-resistance state can be observed directly from the original MR data without \nseparation ( Figures 1(e) and 1(f) ). Thus, the current se paration will not affect the main \nresults.  \nWe next analyze the transverse transport behavior in Co 3Sn2S2 single -crystalline \nnanoflakes. Conventional magnetic materials usually show both normal and \nanomalous Hall effects. For a hard -magnetic material with c oercivity and sharp \nmagnetization reversal, the typical Hall curve is shown in the upper panel of Figure \n3(a). We denote the Hall resistivity curve from +9 T to −9 T as \nyx− . Similarly, the \ncurve from −9 T to +9 T is denoted as \nyx+ . Evidently, \nyx yx−+−  maintains a stable \nplateau (two times as anomalous Hall resistivity (\nAH )) within the coercive field (the 10 \n lower panel of Figure 3(a) ). For Co 3Sn2S2 nanoflakes, the difference between the Hall \ndata with different directions of the magnetic field variation is shown in  Figure 3(b) . \nIn sharp contrast, an increasing \nyx yx−+−  effect exists with increasing applied fields up \nto the coercive field, which is quite different from the conventional case.  \nThe corresponding Hall loop, subtracting the normal Hall component, is depic ted \nin Figure 3(c) . In a conventional hard -magnetic material, the normal Hall -subtracted \nHall curve should be a square loop [7], indicated by the red -dashed lines in Figure 3(c) . \nHowever, unexpected  Hall contribution (highlighted by light -blue-colored filli ng, \ndenoted as exceptional Hall resistivity \nyx  in this text ) exists except normal and \nanomalous Hall components in the current system.  \nFigure 3(d) shows the \nyx  at different temperatures in the negative field region. \nThe \nyx  appears from the zero -field and increases noticeably with increasing \nantiparallel field up to the  coercivity . Once the magnetization and applied field are \nparallel to each other after the magnetization reversal at the coercive field, the \nexceptional Hall resistivity suddenly drops to zero. Compared with the bulk case \nhaving a small coercive field (~0.5 T) [1], the enhanced coerciv ity in Co 3Sn2S2 \nnanoflake makes it possible to clearly  observe \nyx  in a wide antiparallel fie ld range.  \nBy further adopting the function form of \nb\nyxaB =  , \nyx is found to evolve from a \n~B2.5 law at 50 K to a stable ~ B1.8 law below 30 K ( Figure 3(e) ). \nEvidently, \nyx  emerges around 70 K a nd becomes more notable at lower \ntemperatures. A large value of 1.5 µΩ cm is observed at 1.6 K. Because the low -T \nnegative MR ( Figure 2(d) ) also appears around 70 K, we conclude that there should \nbe connections between anomalous  longitudinal and transverse  transport behaviors.  \nFigure 3(f)  shows \nyx  at μ0Hc and its ratio to \nAH . Evidently, the exceptional \nHall component can be more than 2.5 times larger than \nAH , indicating a notable 11 \n effect driven by a distinct mec hanism in Co 3Sn2S2 nanoflakes at low temperatures.  \n \nFigure  3. Transverse resistivity analyses of the 30 -nm-thick sample.  (a) Sketches of \nthe Hall loop in a conventional hard -magnetic material (upper panel). The lower panel \nshows the value of \nyx yx−+− , where the Hall resistivity curve from +9 T  (−9 T) to −9 T  \n(+9 T ) is denoted as \nyx−  (\nyx+). (b) \nyx yx−+−  in the sample being studied. (c) Normal \nHall-subtracted transverse resistivity. (d) Exceptional Hall resistivity, denoted as \nyx . \nThe gray lines are fitting results obtained by adopting \nb\nyxaB =  . (e) Temperature \ndependence of fitting parameter b. (f) \nyx  at the coercive field. The inset shows the \n0()yx c H\n-to-\nAH  ratio. \nFigure s 4(a –b) summarize the anomalous  transport behaviors of four \ndifferent -thick  samples. The Hall conductivity analysis can be found in \nSupplementary Note 6. All samples show similar temperature -dependent transport 12 \n behaviors, exhibiting anom alous  negative MR and exceptional Hall resistivity below \n70 K. According to the data, no obvious thickness dependence exists for these  \nanomalous transport behaviors in the current thickness range.  In addition, the \n41-nm-thick sample shows smaller negative MR and exceptional Hall values \ncompared with other samples. Further, it shows lower mobility ( Figure S5(b) ), \npossibly attributed to the deviation of sample quality.  \nNext, we analyze the underlying origin of the anomalous  transport behaviors. \nThe longitudin al resistance states for a certain magnetic field value are different \nduring field -decreasing and -increasing processes. Thus, the anomalous  transport \nbehaviors could not result from the samples’ basic properties (e.g., carrier mobility), \nwhich may vary wi th the magnetic field. The two -band analysis ( Figure S5(c) ) also \nindicates additional components apart from the normal and anomalous Hall \nconductivities, emerging only under an antiparallel magnetic field.  Considering the \ncoexistence of negative MR and exc eptional Hall effect, these two effects may \nsimultaneously originate from a specific physical state that emerges under an \nantiparallel magnetic field at low temperatures. It has been reported that nontrivial \ntopological magnetic textures may emerge in a ma gnetic Weyl system [19, 33 -35]. \nOnce the magnetic textures (even without topological property) are formed, they will \ninteract with the Weyl conduction electrons and affect the transport behavior in a \nsystem through an emergent chiral magnetic field [16-20]. The fictitious magnetic \nfield may be reminiscent of the real -space Berry curvature  in conventional materials \nwith trivial band structures. However, the chiral magnetic field shows a richer degree \nof freedom as the related chiral gauge potential is closely linked to the Weyl nature \nand depends on the direction of the background magnetization itself [19, 20] . \nTheoretical calculations show that there could be bound states and potential current, \nwhich are highly localized around a magnetic domain wall or texture [16, 17, 20] . In \nexperiments, Co 3Sn2S2 bulk samples indeed showed other possible magnetic \ninteractions besides the out -of-plane ferromagnetic interaction, and thus, nonuniform \nmagnetization behavior below the Curie temperature [1, 21, 36, 37] . A round -shape d 13 \n domain structure has also been observed in the Co 3Sn2S2 system [38]. Based on the \nabove theoretical and experimental reports, a promising possibility responsible for the \nanomalous  negative MR and exceptional Hall effect in Co 3Sn2S2 nanoflakes is the \nnonco llinear magnetic structure. Based on our experiment results, we consider it the \nmagnetic texture induced by the antiparallel magnetic field.  \nFigures 4(c –d) show schematic illustrations of the magnetic structure when an \nexternal magnetic field is antiparall el to the magnetization. As the magnetization \nreversal process is sharp, we consider the antiparallel magnetic domains formed close \nto the coercivity sweeping out the entire sample immediately. Hence, only a slight tilt \nof moments occurs when the applied f ield is less than the coercivity. Moments may \ntilt randomly owing to the antiparallel magnetic field perturbation at temperatures \nhigher than 70 K ( Figure 4(c) ), while the ordered magnetic textures (sketched as red \npatterns in Figure 4(d) ) may emerge at lo w temperatures  due to the competition of \nferromagnetic and other magnetic interactions . \n \nFigure 4. (a) \n0()xx c H -to-\n0()xx c H  ratio of the low -T part. (b) \n0()yx c H -to-\nAH  14 \n ratio. Sketches of magnetic moments (dark blue arrows) and locally ordered magnetic \ntextures (red patterns) at high (c) and low (d) temperatures. The purple arrows \nindicate the scattered electrons. Blue arrows denote possible directions of the \nlocalized chiral magn etic field ( bm) and current ( jm). \nOnce the abovementioned potential magnetic textures are induced by the \nantiparallel magnetic field, a Weyl -fermion -correlated chiral gauge field \n( ) ( )ex\nm\nFJb r M rv=\n[16-20] (where \nexJ is the e xchange interaction strength between \nlocalized magnetic moments and Weyl electrons, and \nFv  is the Fermi velocity) \nemerges around the magnetic textures. Furthermore, this localized chiral gauge field \nmay lead to bound states near the  textures [16, 17, 20] . When an electric field is \napplied to the sample, potential bound states can also offer emergent conductive \nmodes in this system. As anomalous  transport behaviors exist in both longitudinal and \ntransverse channels in Co 3Sn2S2 nanoflak es, a varying \n()Mr\n  with a complicated \norder (not varied within parallel planes, schematic sketched in Figure 4(d) ) may occur, \nleading to jm with both longitudinal (corresponding to low-resistance state ) and \ntransverse (corresponding t o exceptional Hall effect)  components. The specific \nstructure of the possible magnetic textures in Co 3Sn2S2 single -crystalline nanoflakes \nand the confirmation of its influence on the transport behavior s require further study.  \nConclusions  \nHerein, an anomalo us low electrical resistance state and exceptional Hall \ncomponent are reported  in high -quality single -crystalline nanoflakes of magnetic \nWeyl semimetal Co 3Sn2S2 once the magnetization is antiparallel to external magnetic \nfields. Possible antiparallel field -induced magnetic textures and \nWeyl -fermion -associated chiral magnetic fields are expected to account for the \ndecrease and increase in the longitudinal and transverse resistivities, respectively, \nthrough the possible additional conductive modes. Our report  will draw more interest \non the interplay of magnetism and topological transport phenomena, offering new \nelementary states for the applications of magnetic topological materials.  15 \n Supplemental material  \nSupplemental material is available online or from the author.  \n \nAcknowledgments  \nQingqi Zeng and Gangxu Gu contributed equally to this work.  \nThis work was supported by the National Natural Science Foundation of China \n(Nos. 52088101 and 11974394), National Key R&D Program of China (No. \n2019YFA0704900), Beijing N atural Science Foundation (No. Z190009), the Strategic \nPriority Research Program (B) of the Chinese Academy of Sciences (CAS) (No. \nXDB33000000), the Scientific Instrument Developing Project of CAS (No. \nZDKYYQ20210003),  Users with Excellence Program of Hefe i Science Center CAS \n(No. 2019HSC -UE009) , and the Youth Innovation Promotion Association of Chinese \nAcademy of Sciences ( No. 2013002 ). S. 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Nakamura, Phys Rev Mater. 3, \n104421 (2019).  \n \n \n "    },    {        "title": "2106.06941v1.Investigation_of_physical_dose_enhancement_in_core_shell_magnetic_gold_nanoparticles_with_TOPAS_simulation.pdf",        "content": "Investigationofphysicaldoseenhancementin\ncore-shellmagneticgoldnanoparticleswith\nTOPASsimulation\nXiaohanXu1,2,YaoqinXie2,JiananWu1,2,ZhitaoDai1,RuiHu4,LuhuaWang1,3,*\n1DepartmentofRadiationOncology,NationalCancerCenter/NationalClinicalResearchCenterforCancer/CancerHospital&Shenzhen\nHospital,ChineseAcademyofMedicalSciencesandPekingUnionMedicalCollege,Shenzhen,518116,China\n2InstituteofBiomedicalandHealthEngineering,ShenzhenInstitutesofAdvancedTechnology,ChineseAcademyofSciences,Shenzhen,\n518055,China\n3DepartmentofRadiationOncology,NationalCancerCenter/NationalClinicalResearchCenterforCancer/CancerHospital,Chinese\nAcademyofMedicalSciencesandPekingUnionMedicalCollege,Beijing,100021,China\n4DepartmentofRadiationOncology,AffiliatedSuzhouHospitalofNanjingMedicalUniversity,Suzhou215000,China\n*Correspondingauthors.\nE-mail:wlhwq@yahoo.com\nAbstract\nTheapplicationofmetalnanoparticlesassensitizationmaterialsisacommonstrategythatis\nusedtostudydoseenhancementinradiotherapy.Recentinvitrotestshaverevealedthat\nmagneticgoldnanoparticlescanbeusedincancertherapyunderamagneticfieldtoenhance\nthesynergisticefficiencyinradiotherapyandphotothermaltherapy.However,magneticgold\nnanoparticleshaverarelybeenstudiedassensitizationmaterials.Inthisstudy,weobtained\nfurtherresultsofthesensitizationpropertiesofmagneticgoldnanoparticlesusingtheMonte\nCarlomethodTOPASandTOPAS-nBio.Weanalyzedthepropertiesofmagneticgold\nnanoparticlesinmonoenergeticphotonsandbrachytherapy,andweinvestigatedwhetherthe\nmagneticfieldcontributestothesensitizationprocess.Ourresultsdemonstratedthatthedose\nenhancementfactorofthemagneticgoldnanoparticleswas16.7%lowerthanthatofgold\nnanoparticlesinasingleparticleirradiatedbymonoenergeticphotons.Inthecellmodel,the\ndifferencewaslessthan8.1%inthecytoplasm.Werevealedthatthemagneticfieldhasno\ndetrimentaleffectonradiosensitization.Moreover,thesensitizationpropertiesofmagnetic\ngoldnanoparticlesinaclinicalbrachytherapysourcehavebeenrevealedforthefirsttime.\nKeywords:doseenhancementfactor,radiotherapy,MonteCarlo,TOPAS,TOPAS-nBio,magneticgoldnanoparticle\n1.Introduction\nCancerisaseriousdiseasethatcontinuestothreaten\nhumanhealth.Atpresent,morethan50%ofcancerpatients\nhavebeentreatedandcuredbyradiotherapy[1,2].Although\nradiotherapycankilltumorcells,itsimultaneouslythreatens\nhealthytissues.Therefore,simulationstudiesonimproving\nthesensitivityoftumorcellstoradiotherapyandminimizing\nthemortalityofhealthycellstoenhancetheefficiencyof\nradiotherapycanprovideatheoreticalbasisforpromoting\ntheclinicalapplicationofradiotherapy.Withtherapiddevelopmentsinbiotechnologyand\nnanotechnology,theuseofnanomaterialsas\nradiosensitizationmaterialsoffersnewpossibilitiesfor\ncancerradiotherapy[3,4,5].Nanoparticles(NPs)congregate\nintumorsasaresultofenhancedpermeabilityandretention\n(EPR)[6,7].Inradiotherapy,highatomicnumber(Z)\nmaterialscanbeusedtoenhancethedoseintumorsin\ncombinationwiththeEPRproperty.Gold\nNanoparticles(AuNPs)haveexhibitedhighX-raycross\nsection,lowtoxicity,goodbiocompatibility,andeasy\nsynthesis,therebyattractingsignificantattentioninresearch\nontheradiationsensitizationofnanomaterialsinrecentyears.2AuNPshavethepotentialtobeappliedtomedicalimaging,\nmedicaldrugdelivery,photothermaltherapy,andradiation\nsensitizationtherapy.In2004,Hainfeld[8]demonstratedthe\nradiationdoseenhancementeffectofAuNPsthroughanimal\nexperiments,whichlaidthefoundationforresearchon\nAuNPsinradiationsensitization.\nRecentadvanceshaverevealedthehighpotentialof\ntargetedmagneticNPsinradiotherapy[9],wherebya\nmagnetitecorecombinedwithasuitablecoatingcanbe\nbestowedwithbiochemicalanddrug-deliveryproperties[10].\nForthisreason,amagnetitecorecombinedwithagoldshell\nwasproposedtoimprovethestabilization,biocompatibility,\nandsurfacereactivityofsensitizingNPs[11].\nTheMonteCarlo(MC)method[12]isacomputational\napproachthatrepresentsphysicalprocessesbysimulating\nnumerousrandomparticles.CommonlyusedMCcodes\nincludeGeant4,MCNP,andFluka,whichhaveahigh\ncalculationefficiency.Amongthese,theGeant4-DNA\nextensionpackagecanbeusedtosimulatetheinteractionof\neVenergyelectrons.Thispackagehasattractedtheattention\nofmedicalphysicistsowingtoitsuser-friendlyoperation\ninterfaceintheformofTOPAS.TOPAS-nBioisan\nextensionofTOPASthatisbasedonandextendstheGeant4\nSimulationToolkitforradiobiologyapplications[13].Since\n2020,humanliveshavechangeddramaticallyasaresultof\ntheCOVID-19pandemic.In2021,Francis[14]investigated\ntheinfluenceofionizingradiationonSARS-CoV-2using\nGeant4-DNA,therebyprovidinganewconceptforthe\nproductionofaninactivatedvaccine,whichisstillbeing\ndevelopedatpresent,andrevealingtheextensiveapplication\nprospectsandsignificantpotentialoftheMCmethod.\nNevertheless,magneticgoldnanoparticleshaverarely\nbeenstudiedassensitizationmaterials.Toaddressthis\nlimitation,inthisstudy,weusedTOPAS[15,16]and\nTOPAS-nBio[17]tostudytheFe3O4@AuNPpropertiesin\nradiotherapysensitizationcomparedtoanAuNPinasingle\nnanoparticleandacellmodelusingmonoenergeticphotons.\nSubsequently,wecombinedthesimulationwithamagnetic\nfieldtoinvestigatetheinfluenceonthesensitivityprocess.\nFinally,wechangedthephotonbeamswithabrachytherapy\nsourcetoperformthesameprocess.Ourworkcontributesto\ntheresearchonFe3O4@AuNPsinradiotherapyusingtheMC\nmethodandprovidesareferenceforclinicalresearch.2.Materialsandmethods\n2.1CelluptakeofFe3O4@AuNPsbyHeLacellswithor\nwithoutmagneticfield\nTheFe3O4@AuNPthatwasusedinthetestofHu[18]\nconsistedofaFe3O4coreandagoldshell,asshowninFig.\n1(a).ThemeandiameteroftheFe3O4@AuNPswas100nm\naccordingtodynamiclightscatteringanalysis,asillustrated\ninFig.1(b).AccordingtoFigs.(a)and(b),weobtainedthat\ntheFe3O4@AuNPconsistedofa60nmdiameterFe3O4core\nanda20nmthicknessgoldshell.Therefore,weselected100\nnmasthediameteroftheFe3O4@AuNPandusedthesame\nFe3O4andAuratioinoursimulationwork.Huusedconfocal\nlaserscanningmicroscopy(CLSM)toobservethe\ndistributionoftheFe3O4@AuNPsinternalizedbytheHeLa\ncells,asdepictedinFig.1(c).TheFe3O4@AuNPsand\nlysosomeswerelabeledbyfluoresceinisothiocyanateand\nLysoTrackerRed,andexhibitedgreenandredfluorescence,\nrespectively,intheCLSM.Thedistributionsofthe\nFe3O4@AuNPsandlysosomeswereclearlypartially\noverlapped.ItmeanttheFe3O4@AuNPswereinternalizedby\ncellsandcouldbeswallowedbythelysosomesinthe\ncytoplasm.Wealsoobservedthatthegreenfluorescence\nintensitywitha0.2Tmagneticfieldwashigherthanthat\nwithoutamagneticfield.Huusedflowcytometrytoanalyze\nthemeanfluorescenceintensitytocomparethecelluptakeof\nFe3O4@AuNPswithandwithoutamagneticfield\nquantitatively,asillustratedinFig.1(d).Theresults\ndemonstratedthatthefluorescenceintensityofthe\nFe3O4@AuNPsinanexternalmagneticfieldwas1.48times\nhigherthanthatwithoutamagneticfield.Moreover,Hu\ndemonstratedthatFe3O4@AuNPscanbeusedtodecrease\ntheviabilityofHeLacellsinradiotherapywithanexternal\nmagneticfield(0.2T).Theresultsindicatedthatthecell\nviabilitywasaffectedbythemagneticfieldowingtothecell\nuptakepropertiesbeingenhancedunderthemagneticfield.\nInfact,thecellviabilitymaybeaffectedbythecelluptake\nproperties,thephysicaldoseenhancementofFe3O4@AuNPs,\nandotherconditions.Inourresearch,westudiedthedose\nenhancementpropertiesofFe3O4@AuNPswithandwithout\namagneticfieldusingTOPASandTOPAS-nBio.3\nFigure1.(a)TEMimageofFe3O4@AuNPs.(b)DynamiclightscatteringanalysisofFe3O4@AuNPs.(c)DistributionofFe3O4@AuNPsin\nHeLacellswithandwithoutmagneticfieldinCLSMafter3hofincubation.(d)FluorescenceintensityofFe3O4@AuNPswithandwithout\nmagneticfield.\n2.2Calculationmethodsfordoseenhancementfactor:\ntwo-stepandone-stepmethods\nTwomethodsareusedforcalculatingthedose\nenhancementfactor(DEF)withTOPASandTOPAS-nBio.\nLin[19]investigatedthedoseenhancementofprotonand\nphotonirradiationonAuNPsusingTOPAS.Lincalculated\nthedosedistributionaroundasingleAuNPthatchangedwith\nthedistancefromtheparticlesurfaceandobtainedthe\ndistributionoftheDEFatdifferentdistancesfromtheAuNP\nsurface.TheDEFisdefinedastheratioofthedosedeposited\naroundthemetalnanoparticle(MNP)andwaternanoparticle\n(WNP).AstheGeant4-DNAphysicsprocessesareworkable\ninwateronly,thispackagecannotbeusedtocalculatethe\ntracksinAuNPaccurately;thus,Lindividedthedose\ncalculationintotwosteps.First,thephysicsmodule\nPenelopewasactivatedtocalculatetheinteractionbetween\ntheparticlesourceandAuNPinwater,followingwhichthe\ngeneratedsecondaryelectronsthatwereexcitedfromthe\nAuNPsurfacewerestoredinthephasespacefile.Second,\nthephasespacefilewasplacedintoawaterboxtocalculate\nthedosedistributionofthesecondaryelectronsinwaterand\nGeant4-DNAwasactivatedinthewaterboxregion.Sucha\nmethodofcalculatingtheDEFisreferredtoasthe\"two-step\nmethod\"inourresearch.However,thesurfacedose\ndistributionaroundasingleAuNPisnottheexclusivefactor\naffectingthecelllivability,andtheeffectsoftheradiationemergingorscatteringfromanAuNPontheotherAuNPsin\nacellmodelshouldalsobeconsidered.\nScientistsdevelopedTOPAS-nBiotosimulate\nradiobiologicalexperimentsonnanometerscalecells\nconsideringthephysics,chemistry,andbiologyeffects.\nTOPAS-nBiosupportstheassignmentofdifferentphysical\nmodelstodifferentgeometrycomponents.Rudek[20]\nestablishedtheAuNPsthatwereinternalizedinacellmodel\nirradiatedbyphotons,protons,andcarbonionsrespectively\nusingTOPAS-nBio.Todefinesuitablephysicalmodulesin\ndifferentregions,RudeksettheLivermorephysicalprocess\nintheAuNPregionandtheGeant4-DNAprocessoutsidethe\nAuNPregion.Thereafter,theDEFsinthecytoplasmand\nnucleuswerecalculated.ThismethodofcalculatingtheDEF\nisreferredtoasthe\"one-stepmethod\"inthiswork.\nThetwomethodsmentionedaboveareaimedatasingle\nnanoparticleandasinglecell.Asinglecellincludesthe\nphysicalinteractionbetweentheprimarybeamandMNPs.\nTherefore,inthesimulationstudyofFe3O4@AuNPs,we\nconsideredthecalculationresultsofboththetwo-stepand\none-stepmethodtoevaluatethesensitivityenhancement\nperformanceinasinglenanoparticleaswellasinacell.To\ncomparethetwo-stepmethodandone-stepmethod,we\nmodeledthesamegeometrytosimulatetheinteraction\nprocessofthephotonsandAuNPinwater,asillustratedin\nFig.2(a).Figure2.GeometrysketchofMCsimulation(nottheactualscale):(a)A100nmdiameterAuNPwasplacedina20μmdiametersphere\nfilledwithwater.AphotonbeamwiththesamesizeastheAuNPindiameterwasplacedupstreamtotheAuNP.Thedosewasscoredin\nthegridswithavolumesizeof1×1×1nm3for0to150nm,10×10×10nm3for150nmto1.95μm,and100×100×100nm3for1.95\nμmto19.95μmfromtheAuNPsurface.(b)The10μmdiameterwaterspherecontaineda5μmwatersphereinthecentertomodelthecell\ncontaininganucleus.The100nmdiameterNPswererandomlydistributedinthecytoplasm.Thephotonbeamhadthesamediameteras\nthecellandwasplacedupstreamtothecell.\nForthetwo-stepmethod,wedividedthesimulationinto\ntwosteps,asdescribedabove.Inthefirststep,theAuNPwas\nplacedinabox(100×100×100nm3)filledwithvacuum\nandirradiatedbya50keVphotonbeam,followingwhichthe\noutputelectronphasespacefilewasobtainedfromtheAuNP\nsurface.Inthesecondstep,theelectronphasespacefilewas\nusedasaparticlesourceandplacedinthecenterofthewater\nsphere(20μmdiameter).Thedepositeddosewasscoredin\nthegridsfrom0to150nm,150nmto1.95μm,and1.95μm\nto19.95μmfromtheAuNPsurfacewithdifferentprecisions.\nWesettheLivermorephysicsprocessesforthefirststepand\ntheGeant4-DNAphysicsforthesecondstep.Thede-\nexcitationwasactivatedtoincludeAugerproductionand\nparticleinducedX-rayemission.\nIntheone-stepmethod,theAuNPwasplacedinthecenter\nofthewaterbox(20μmdiameter).Thereafter,the50keV\nphotonbeaminteractedwiththeAuNPandthedosewas\nrecordedatdifferentdistancesfromtheAuNPsurface.The\nAuNPregionwasassignedwithLivermorephysicsprocesses,\nwhereasalloftheotherregionsweresetwithGeant4-DNA\nphysicsprocesses.Inbothmethods,werecordedthedose\ndistributionthatwasproducedbytheelectronsandphotonstoinvestigatewhichparticletypesmainlycontributedtothe\ndepositeddose.\n2.3Photonenergydependenceofsingle\nFe3O4@AuNPdoseenhancementusingtwo-step\nmethod\nWeusedanFe3O4@AuNPwiththesamesizeand\ncompositionasinHu’stestinoursimulation.Five\nmonoenergeticphotonbeams(50,100,150,200,and250\nkeV)wereusedasparticlesourcestoirradiatethesingle\nFe3O4@AuNP,AuNP,andWNP.Thephotonsourcewas\nplaneparallelwitha100nmdiameterandstartedatthe\nnanoparticlesurface,asillustratedinFig.2(a).Toevaluate\nthepropertiesoftheFe3O4@AuNPatdifferentphoton\nenergies,wecomparedtheDEFsoftheFe3O4@AuNPand\nAuNPthatwereirradiatedbythesamefivemonoenergetic\nphotonbeamswiththesamesimulationparameters.\n2.4PhotonenergyandNanoparticles\nconcentrationdependenceofcelldose\nenhancementusingone-stepmethod5Theradiationprocesseswereimplementedinasimplified\ncellmodel.The10μmdiametercellcontaineda5μm\ndiameternucleusinthecenterandthecellwasplacedina\nwaterbox.Boththecytoplasmandnucleuswerefilledwith\nwatertomodelthecellularenvironment.Themonoenergetic\nphotonsource(50,100,150,200,and250keV)wasplane\nparallelwitha10μmdiameterandstartedfromthecell\nsurface,asillustratedinFig.2(b).ConsideringthatNPsare\npredominantlydispersedinthecytoplasmwhenNPsenter\nthecell[21],the100nmdiameterFe3O4@AuNPsand100\nnmdiameterAuNPswereplacedinthecytoplasmrandomly\nrespectivelyinthesimulationtodrawacomparison.To\ncoverthedesireddoserangeonthecelllevel,theNPsmass\nconcentrationwasincrementedintherangeof1to50\nmg/mL[22].Subsequently,weselected1,5,10,and50\nmg/mLastheconcentrationweightsoftheFe3O4@AuNPs\nandAuNPsinthecytoplasm.ThecorrespondingNPs\nnumbersarelistedinTable1.\nTable1.Numberof10μmdiameterFe3O4@AuNPsandAuNPsin\ncytoplasmforfiveconcentrationweights(units:mg/mL).\nMass/volume\n(mg/mL)1 5 10 50\nNumberof\nFe3O4@AuNPs\nincytoplasm62 307 615 3074\nNumberof\nAuNPsin\ncytoplasm52 259 518 2588\n2.5Magneticfielddependenceofsingle\nnanoparticleandcelldoseenhancement\nWiththeincreasinguseofMRI-guidedradiotherapy,itis\nnecessarytoinvestigatetheinfluenceofthemagneticfield\nontheradiotherapy.TheinvitrotestsperformedbyHu[18]\ndemonstratedthatcore-shellFe3O4@AuNPscanbeusedto\ndecreasetheviabilityofHeLacellsbyimprovingtheir\ninternalizationbythecellsinanexternalmagneticfield(0.2T).Bug[23]andLazarakis[24]demonstratedthatthe\nmagneticfieldaffectedthechargedparticletrajectoryonly;\nthephysicalcrosssection,DNAstrandbreaks,andcluster\nsizedistributioncouldnotbechangedbythemagneticfield\ninGeant4.\nScientistshaveshownthatmagnetictargetingisa\npromisingtechnologyamongpassivetumoraccumulationin\nradiotherapy.MagneticNPscanbefocusedonthetumors\nunderthemagneticfieldoutsidethebody[25].However,the\nmagnetictargetingpropertyformagneticmaterialina\nmagneticfieldcannotbesimulatedwiththeMCmethod.\nTherefore,weusedfourconcentrationsofFe3O4@AuNPsto\nsimulatethetargetingfocusoftheFe3O4@AuNPsinfour\nmagneticfields,asdiscussedinsection2.4.\nWeinvestigatedtheinfluenceofthechangedparticle\ntrajectoryunderthemagneticfieldonthesensitization\nprocessoftheFe3O4@AuNPandAuNP.Thesimulationwas\nperformedonasinglenanoparticleandacellmodelusingthe\ntwo-stepandone-stepmethods,withirradiationbya50keV\nmonoenergeticphotonbeam.Themodelsusedwerethesame\nasthosedescribedinsections2.3and2.4.Anexternal\nmagneticfieldwithastrengthof0.1to2Twasusedinthe\nsimulations.TheNPsconcentrationwas50mg/mLinthe\ncellmodel.\n2.6DEFsofFe3O4@AuNPandAuNPinteracted\nwithbrachytherapysource\nIntheinvitrotestsofHu[18],theHeLacellswere\nirradiatedbyphotonsfromaVarianlinearaccelerator(True\nBeam).Inthisstudy,wefurtherevaluatedthesensitization\npropertiesoftheFe3O4@AuNPandAuNPunderaclinically\nappliedsource.WeimplementedtheVarianGammaMed\nPlusHDR192Irbrachytherapysourcemodel[26]using\nTOPAStoexploretheDEFoftheFe3O4@AuNPirradiated\nbyabrachytherapysource.Theparticlenumbersthatwere\nemittedfromthebrachytherapysourcemodelperkeVforthe\nperinitialphotons,whichwererecordedonaparallelplane\nata2cmdistancefromthesourcecenter,arepresentedin\nFig.3.Atotalof108initialphotonswereusedasthesource\nbeamandthesourcewasplacedinan80×80×80cm3\nwaterboxtocalculatethedosedistributioninwater.With\ntheexceptionoftheradiationsource,allparameterswere\nconsistentwiththosedescribedinsection2.5.6\nFigure3.Particleenergyspectraemittedfrombrachytherapysourcemodel,recordedonparallelplaneat2cmdistancefromsourcecenter.\n3.Results\n3.1Comparisonoftwo-stepandone-step\nmethods\nFigure4presentstheresultsofthecomparisonbetween\ntheone-stepandtwo-stepmethodsaswellasthedose\ncontributionsfromtheelectronsandphotons.Itisobvious\nthatthefourcurvesinFig.4exhibitsimilartrends.The\nresultsrevealtwosignificantconclusions.First,thedeposited\ndosewasmainlycontributedbythesecondaryelectrons.The\ndoseproducedbythephotonswasveryslightcomparedto\ntheelectronsinboththeone-stepandtwo-stepmethods.\nAccordingtotheresults,onlytheelectronsneedtobe\nconsideredinirradiationsimulationtoimprovethe\ncalculationefficiency,regardlessofwhethertheone-stepor\ntwo-stepmethodisused.Second,thedosedistributionsof\ntheone-stepandtwo-stepmethodsexhibitednosignificant\ndifferences.Thus,wecanusethetwo-stepmethodto\ncalculatetheDEFaroundasinglenanoparticleandtheone-\nstepmethodtocalculatetheDEFinacellmodel.Moreover,\ntheinfluenceofthetwomethodsonthesimulationresults\ndoesnotneedtobeconsidered.\nFigure4.Dosedistributionperincidentphotonvs.distancefrom\nAuNPsurface.\n3.2Photonenergydependenceofsingle\nFe3O4@AuNPdoseenhancement\nTheresultsofthephotonirradiationsaredepictedinFig.5.\nFigures5(a),(b),and(c)presentthedosedistributionsat\ndifferentdistancesfromthesurfaceofthesingle\nFe3O4@AuNP,singleAuNP,andsingleWNP,respectively,\nperincidentphoton.Itisclearthatthefivedosedistribution\ncurvesinbothFigs.5(a)and(b)exhibitedsimilartrends.\nHigherenergyofthephotonledtoahigherdosedistribution\nintheenergyrangefrom150to250keV.However,the\ndepositeddoseofthe100keVphotonwashigherthan50\nkeVwithintherangeof1.3×103to7×103nm.Accordingto\nFig.5(c),the100keVphotondosedistributionwashigher\nthanthatofthe50keVphotonintherangeof1×103to1.47×104nm.TheDEFsoftheFe3O4@AuNPandAuNPwere\ncalculatedbasedonFigs.5(a),(b),and(c)andtheresultsare\nplottedinFigs.5(d)and(e).ThefivecurvesinFigs.5(d)and\n(e)alsoexhibitedsimilartrends.TheDEFsofbothNPs\nincreasedwithanincreaseinthephotonenergyforthe150,\n200,and250keVphotons.Thecurvesofthe50and100keV\nphotonscrossedat103and1.8×104nm.Tocomparethe\ntotaldosedepositionintherangeof1to2×104nmintuitively,thedosesthatweredistributedatdifferent\ndistancesweretotaledforeachphotonenergy,asillustrated\ninFig.5(f).Accordingtothefigure,theDEFofthe\nFe3O4@AuNPwas16.7%lowerthanthatoftheAuNPon\naverage.Moreover,thepeakoftheDEFverusphotonenergy\ncurveappearedat100keV.ForthousandsofkeVenergy\nphotonsirradiatedwithFe3O4@AuNPandAuNP,theDEF\nwasthehighestnear100keV.\nFigure5.Relationshipbetweendosedistributionperincidentphotonanddistancefromsurfaceof(a)Fe3O4@AuNP,(b)AuNP,and(c)\nWNPfor50,100,150,200,and250keVphotons.DEFdistributionsaround(d)Fe3O4@AuNPand(e)AuNPasfunctionofdistancefrom\nthenanoparticlesurface.(f)TotalDEFaroundFe3O4@AuNPandAuNPinrangeof1to2×104nmvs.photonenergy.\n3.3PhotonenergyandNanoparticles\nconcentrationdependenceofcelldose\nenhancement\nTheDEFsoftheFe3O4@AuNPsandAuNPsinthe\ncytoplasmandnucleusarepresentedinFigs.6(a)and(b),\nrespectively.Inthecytoplasm,theDEFsofthe\nFe3O4@AuNPsandAuNPsdecreasedwithanincreaseinthe\nphotonenergy.ThemaximumDEFsoftheFe3O4@AuNPs\nandAuNPsfor50mg/mLwere3.69and3.83.The\nmaximumdifferencewaswithin2.1%,1%,2.2%,and8.1%\nwhencomparingtheDEFsoftheAuNPsandthe\nFe3O4@AuNPsforthe1,5,10,and50mg/mLNPs\nconcentrations,respectively.Inthenucleus,theDEFofthe\nAuNPsreachedthemaximumat100keVwiththe5,10,and50mg/mLweightconcentrationsand50keVwith1mg/mL.\nThemaximumdifferencewaswithin1.9%,5%,13%,and\n3.1%whencomparingtheAuNPsandFe3O4@AuNPsforthe\n1,5,10,and50mg/mLNPconcentrations,respectively.\nTheDEFdifferencesbetweentheFe3O4@AuNPsand\nAuNPsinthecytoplasmandnucleusaresummarizedin\nTable2.Itcanbeobservedthattheenergyhadagreater\ninfluenceontheDEFwhentheparticleconcentrationwas\nhigher.Inthecytoplasm,theDEFoftheAuNPswashigher\nthanthatoftheFe3O4@AuNPsexceptfor1mg/mL.\nHowever,thedifferencewasnotobviousinthenucleus.\nFurthermore,ingeneral,ahigherNPsconcentrationledtoa\nhigherDEFinthecytoplasmandnucleus.Thismeansthat\nthehighmagneticfocuspropertycanachievebetterdose\nenhancementforradiotherapy.8\nFigure6.FunctionofDEFandphotonenergyin(a)cytoplasmand(b)nucleus.\nTable2.DEFdifferencesbetweenFe3O4@AuNPsandAuNPsincytoplasmandnucleuswithdifferentphotonenergiesandNPs\nconcentrations.\nPhotonenergy\n(keV)Concentrationinweight(mg/mL)\n1 5 10 50 1 5 10 50\nDEFdifferencesbetweenAuNPsandFe3O4@AuNPsincytoplasm(%) DEFdifferencesbetweenAuNPsandFe3O4@AuNPsinnucleus(%)\n50 -0.16 0.68 0.90 3.51 0.34 2.67 1.53 0.87\n100 0.19 0.41 2.11 8.09 1.45 4.91 4.56 3.09\n150 -2.05 0.98 1.46 3.74 -0.50 -1.40 7.35 -2.04\n200 -1.09 1.00 1.76 4.19 -1.88 0 12.54 -1.45\n250 -0.65 0.15 -0.03 2.36 0.98 0.67 7.43 -1.71\n3.4Magneticfielddependenceofdose\nenhancement\nTherelationshipbetweenthemagneticfieldandDEFof\ntheNPsispresentedinFigs.7and8.Weusedthetwo-step\nmethodonasinglenanoparticleandfoundthattheDEFof\ntheAuNPwas15%higherthanthatoftheFe3O4@AuNP\nunderthemagneticfield,asillustratedinFig.7.TheDEF\nwasstableabove0.5Tandthevaluewasslightlyhigherthan\ntheDEFwithoutamagneticfield.TheminimumDEFvalue\nappearedat0.2Tandthiswas2.5%lowerthantheconstantvalue.Thissimulationresultindicatesthatthemagneticfield\ndidnotcontributesignificantlytotheDEF.\nTheDEFsoftheAuNPandFe3O4@AuNPinthe\ncytoplasmandnucleusareillustratedinFig.8.Weusedthe\none-stepmethodinacelltosimulatetheinfluenceofthe\nmagneticfieldontheDEF.TheFe3O4@AuNPDEFwas\n3.9%and3.1%lowerthanthatoftheAuNPinthecytoplasm\nandnucleus,respectively.TheDEFinthecytoplasmwas\n11.7%and12.4%higherthanthatinthenucleusforthe\nFe3O4@AuNPandAuNP.Ingeneral,themagneticfielddid\nnotcontributesignificantlytotheDEFinthecellmodel.In\nthisstudy,weconcludedthatamagneticfieldwithastrength\nof0.1Tto2Twouldnothaveanegativeeffectonthe\nsensitizationprocess.9\nFigure7.RelationshipbetweenmagneticfieldandDEFofasingleFe3O4@AuNPandAuNP.\nFigure8.RelationshipbetweenmagneticfieldandDEFofFe3O4@AuNPsandAuNPsincytoplasmandnucleus.\n3.5DEFsofFe3O4@AuNPandAuNPirradiatedby\nbrachytherapysource\nTheresultsofthebrachytherapysourceirradiationsare\ndepictedinFig.9.ForthesinglenanoparticlemodelinFig.\n9(a),theDEFoftheFe3O4@AuNPwas9.26%lowerthan\nthatoftheAuNP.ForthecellmodelinFig.9(b),theDEFsoftheFe3O4@AuNPwere6.3%and2.7%lowerthanthose\noftheAuNPinthecytoplasmandnucleus,respectively,\nwhereastheDEFsoftheFe3O4@AuNPandAuNPinthe\ncytoplasmwere26.75%and31.62%higherthanthoseinthe\nnucleus.10\nFigure9.TotalDEFofFe3O4@AuNPandAuNPin(a)singlenanoparticleinrangeof1to2×104nm,and(b)incytoplasmandnucleus\nwhenirradiatedbybrachytherapysource.\n4.Discussion\nAuNPsarestudiedextensivelyinradiosensitizationowing\ntotheirpropertiesofhighX-rayabsorption,hypotoxicity,\nandeasysynthesis.Magnetitecanbeusedastargeting\nmaterialtoimprovetumordrugdeliverybecauseofthe\nmagnetictargetingpropertyinthemagneticfield[25].Asa\nnovelnanoparticle,theFe3O4@AuNPcombinesthe\npropertiesofgoldandmagnetite,andithasbeenusedinin\nvitroexperimentstodecreasethecellsurvivalrate[18].\nInthisstudy,weexploredtheDEFofanFe3O4@AuNPin\nasinglenanoparticleandinacellmodelcomparedtothe\nAuNP.TheDEFaroundthesingleFe3O4@AuNPwas16.7%\nlowerthanthatoftheAuNP,andthedifferencesbetweenthe\nAuNPandFe3O4@AuNPinthecytoplasmandnucleusare\ndetailedinTable2.WiththeincreaseintheNPs\nconcentration,theDEFresidualsbetweentheFe3O4@AuNPs\nandAuNPsincreasedinthecytoplasm.Itwasexpectedthat\ntheDEFoftheFe3O4@AuNPwouldbelowerthanthatof\ntheAuNPbecausethephotoelectriccrosssectionofironand\noxygenislowerthanthatofgold.Wequantifiedthe\ndiscrepancybetweentheFe3O4@AuNPandAuNPtoprovide\nananalysisofcore-shellmagneticNPsthatareusedas\nsensitivitymaterials.Itiswellknownthattheclustering\npropertyofAuNPswilldecreasetheDEFinacellin\nradiotherapy.However,littleresearchhasbeenconductedon\nhowtoavoidmagneticNPclustering.Significantanalytical\npotentialexistsforimprovingthemagneticNPsstabilityby\nmodifyingtheextramagneticfieldsoastoincreasetheDEF.Weinvestigatedtheinfluenceofthemagneticfieldonthe\nDEFanddemonstratedthatthemagneticfielddidnothavea\nsignificanteffectonthesensitizationprocess.Theresults\nrevealedthatthechangedelectrontrajectorywasinsufficient\ntoinfluencethedoseenhancement,ortheelectrontrajectory\nwasinsufficienttobechangedwithsuchelectronenergyand\nthemagneticfield[23].Therefore,themagneticfieldwould\nnotriskphysicalenhancementbecausetheelectronenergy\nwastoohighaccordingtothemagneticfield.Combinedwith\ntheinvitroexperimentcarriedoutbyHu,weverifiedthatthe\nradiosensitizationmainlybenefitedfromthephysical\nenhancementofFe3O4@AuNPinadditiontothecelluptake\ninthemagneticfield.\nFurthermore,weconstructedabrachytherapysourcefor\nirradiationwithasinglenanoparticleandacellmodel.The\nresultsofthebrachytherapyirradiationshowedtheresiduals\nbetweentheFe3O4@AuNPandAuNPinasingle\nnanoparticleandacellmodel.TheDEFdifferencesbetween\ntheAuNPandFe3O4@AuNPwere9.26%,6.3%,and2.7%\nforthesingleparticle,cytoplasm,andnucleus,respectively.\nTheresultsclarifiedthedoseenhancementofthe\nFe3O4@AuNPsunderthebrachytherapysource.Inthefuture,\nresearchonguidingtheFe3O4@AuNPstofocusontumors\nthroughthemagneticfieldwillbequitebeneficial.For\nexample,thesourceapplicatormaybemagnetizedtoguide\nmagneticNPsorthesensitizationmaybecombinedwith\nMRI-guidedbrachytherapytofocusthemagneticNPs.This\nresearchmayraiseconcernsregardingMRI-guided\nbrachytherapycombinedwithmagneticNPs.115.Conclusions\nInthiswork,wecomparedtheone-stepandtwo-step\nmethodsforcalculatingtheDEFtoverifythattherewasno\nsignificantdifferencebetweenthemethods.Thereafter,we\nappliedthetwomethodstoasingleparticleandacellmodel\ntoinvestigatetheDEFsoftheFe3O4@AuNPandAuNP.The\nDEFoftheFe3O4@AuNPwas16.7%lowerthanthatofthe\nAuNPinasingleparticle.Inthecellmodel,theDEF\ndifferencebetweentheFe3O4@AuNPandAuNPwasbelow\n8.1%inthecytoplasmwithanNPsconcentrationof1to50\nmg/mL.Wealsodemonstratedthatthemagneticfieldhasno\ndetrimentaleffectontheNPsradiosensitization.Furthermore,\nweappliedabrachytherapysourceforinteractionwiththe\nFe3O4@AuNPandAuNPinasinglenanoparticleandacell\nmodeltoobtaintheDEFinbrachytherapysourceirradiation.\nInsummary,thisstudyhasrevealedtheFe3O4@AuNP\npropertiesinradiotherapydoseenhancementusingtheMC\nmethodforthefirsttime.Moreover,wedemonstratedthat\nthephysicaldoseenhancementoftheFe3O4@AuNPis\nindependentofthemagneticfield.Finally,wedeterminedthe\nDEFofFe3O4@AuNPsinabrachytherapysourcetoprovide\nsimulationresultsforclinicalresearch.Infutureresearch,\nFe3O4@AuNPsmaybecombinedwithamagneticfield\n(suchasMRI)toovercomethechallengeofNPsclustering\nandtoimprovetheNPsconcentrationinthecell.Thiswillbe\ndesirableforfutureinvitrotestsonradiosensitizationaswell\nasclinicalresearch.\nAcknowledgements\nThisworkissupportedbySanmingProjectofMedicinein\nShenzhen(No.SZSM201612063).Theauthorswouldliketo\nthankHDVideoR&DPlatformforIntelligentAnalysisand\nProcessinginGuangdongEngineeringTechnologyResearch\nCenterofCollegesandUniversities(GCZX-A1409),\nShenZhenUniversityforprovidingaccesstotheirhigh\nperformanceworkstations.\nReferences\n[1]Joiner,MichaelC.,andAlbertJ.vanderKogel,eds.Basic\nclinicalradiobiology.CRCpress,2018.\n[2]Delaney,Geoff,etal.\"Theroleofradiotherapyincancer\ntreatment:estimatingoptimalutilizationfromareviewof\nevidence‐basedclinicalguidelines.\"Cancer:Interdisciplinary\nInternationalJournaloftheAmericanCancerSociety104.6(2005):\n1129-1137.\n[3]Wang,Zheng,etal.\"Berberine‐loadedJanusnanocarriersfor\nmagneticfield‐enhancedtherapyagainsthepatocellular\ncarcinoma.\"Chemicalbiology&drugdesign89.3(2017):464-469.[4]Shao,Dan,etal.\"Janus“nano-bullets”formagnetictargeting\nlivercancerchemotherapy.\"Biomaterials100(2016):118-133.\n[5]Shao,Dan,etal.\"FacileSynthesisofCore–shellMagnetic\nMesoporousSilicaNanoparticlesforpH‐sensitiveAnticancer\nDrugDelivery.\"Chemicalbiology&drugdesign86.6(2015):\n1548-1553.\n[6]Maeda,Hiroshi,etal.\"TumorvascularpermeabilityandtheEPR\neffectinmacromoleculartherapeutics:areview.\"Journalof\ncontrolledrelease65.1-2(2000):271-284.\n[7]Torchilin,Vladimir.\"Tumordeliveryofmacromoleculardrugs\nbasedontheEPReffect.\"Advanceddrugdeliveryreviews63.3\n(2011):131-135.\n[8]Hainfeld,JamesF.,DanielN.Slatkin,andHenryM.Smilowitz.\n\"Theuseofgoldnanoparticlestoenhanceradiotherapyin\nmice.\"PhysicsinMedicine&Biology49.18(2004):N309.\n[9]Peng,Xiang-Hong,etal.\"Targetedmagneticironoxide\nnanoparticlesfortumorimagingandtherapy.\"Internationaljournal\nofnanomedicine3.3(2008):311.\n[10]Chomoucka,Jana,etal.\"Magneticnanoparticlesandtargeted\ndrugdelivering.\"Pharmacologicalresearch62.2(2010):144-149.\n[11]Wang,Lingyan,etal.\"Core@shellnanomaterials:gold-coated\nmagneticoxidenanoparticles.\"JournalofMaterials\nChemistry18.23(2008):2629-2635.\n[12]Cho,SangHyun.\"Estimationoftumourdoseenhancementdue\ntogoldnanoparticlesduringtypicalradiationtreatments:a\npreliminaryMonteCarlostudy.\"PhysicsinMedicine&\nBiology50.15(2005):N163.\n[13]Wu,Jianan,etal.\"MonteCarlosimulationsofenergy\ndepositionandDNAdamageusingTOPAS-nBio.\"Physicsin\nMedicine&Biology65.22(2020):225007.\n[14]Francis,Ziad,etal.\"MonteCarlosimulationofSARS-CoV-2\nradiation-inducedinactivationforvaccinedevelopment.\"Radiation\nresearch195.3(2021):221-229.\n[15]Perl,Joseph,etal.\"TOPAS:aninnovativeprotonMonteCarlo\nplatformforresearchandclinicalapplications.\"Medical\nphysics39.11(2012):6818-6837.\n[16]Faddegon,Bruce,etal.\"TheTOPAStoolforparticle\nsimulation,aMonteCarlosimulationtoolforphysics,biologyand\nclinicalresearch.\"PhysicaMedica72(2020):114-121.\n[17]Schuemann,J.,etal.\"TOPAS-nBio:anextensiontothe\nTOPASsimulationtoolkitforcellularandsub-cellular\nradiobiology.\"Radiationresearch191.2(2019):125-138.\n[18]Hu,Rui,etal.\"Core-shellmagneticgoldnanoparticlesfor\nmagneticfield-enhancedradio-photothermaltherapyincervical\ncancer.\"Nanomaterials7.5(2017):111.\n[19]Lin,Yuting,etal.\"Comparinggoldnano-particleenhanced\nradiotherapywithprotons,megavoltagephotonsandkilovoltage\nphotons:aMonteCarlosimulation.\"PhysicsinMedicine&\nBiology59.24(2014):7675.12[20]Rudek,Benedikt,etal.\"Radio-enhancementbygold\nnanoparticlesandtheirimpactonwaterradiolysisforx-ray,proton\nandcarbon-ionbeams.\"PhysicsinMedicine&Biology64.17\n(2019):175005.\n[21]Peckys,DianaB.,andNielsdeJonge.\"Visualizinggold\nnanoparticleuptakeinlivecellswithliquidscanningtransmission\nelectronmicroscopy.\"Nanoletters11.4(2011):1733-1738.\n[22]Gadoue,SherifM.,PiotrZygmanski,andErnoSajo.\"The\ndichotomousnatureofdoseenhancementbygoldnanoparticle\naggregatesinradiotherapy.\"Nanomedicine13.8(2018):809-823.\n[23]Bug,M.U.,etal.\"Effectofamagneticfieldonthetrack\nstructureoflow-energyelectrons:aMonteCarlostudy.\"The\nEuropeanPhysicalJournalD60.1(2010):85-92.\n[24]Lazarakis,P.,etal.\"Comparisonofnanodosimetricparameters\noftrackstructurecalculatedbytheMo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   },    {        "title": "2106.13678v1.Long_Time_Magnetic_Relaxation_in_Antiferromagnetic_Topological_Material_EuCd__2_As__2_.pdf",        "content": "Long-Time Magnetic Relaxation in Antiferromagnetic Topological\nMaterial EuCd 2As2\nYang Wang1;2], Cong Li1;2], Yong Li1;2], Xuebo Zhou1;2, Wei Wu1;2, Runze Yu1;2,\nJianfa Zhao1;2, Chaohui Yin1;2, Youguo Shi1;2;3, Changqing Jin1;2;3, Jianlin\nLuo1;2;3, Lin Zhao1;2;3, Tao Xiang1;2;4, Guodong Liu1;2;3\u0003and X. J. Zhou1;2;3;4\u0003\n1Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n2University of Chinese Academy of Sciences, Beijing 100049, China\n3Songshan Lake Materials Laboratory, Dongguan 523808, China\n4Beijing Academy of Quantum Information Sciences, Beijing 100193, China\n]These people contributed equally to the present work.\n\u0003Corresponding authors: gdliu ARPES@iphy.ac.cn and XJZhou@iphy.ac.cn\n(Dated: May 21, 2021)\nAbstract\nMagnetic topological materials have attracted much attention due to the correlation between\ntopology and magnetism. Recent studies suggest that EuCd 2As2is an antiferromagnetic topolog-\nical material. Here by carrying out thorough magnetic, electrical and thermodynamic property\nmeasurements, we discover a long time relaxation of the magnetic susceptibility in EuCd 2As2. The\n(001) in-plane magnetic susceptibility at 5 K is found to continuously increase up to \u001810% over the\ntime of \u001814 hours. The magnetic relaxation is anisotropic and strongly depends on the temperature\nand the applied magnetic \feld. These results will stimulate further theoretical and experimental\nstudies to understand the origin of the relaxation process and its e\u000bect on the electronic structure\nand physical properties of the magnetic topological materials.\n1arXiv:2106.13678v1  [cond-mat.mtrl-sci]  25 Jun 2021Searching for new topological states with novel exotic properties has become an im-\nportant subject in condensed-matter physics[1{11]. Magnetic topological materials, as a\nnew category of topological materials, have attracted much recent attention[10{53]. Com-\npared with non-magnetic topological materials, the introduction of magnetism can break\nthe time-reversal symmetry and produce new quantum topological phases such as the quan-\ntum anomalous Hall state[12, 18], the axion state[40{50] and even high-order topological\nphase[52{55]. These new physical states greatly broaden the \feld of the topology research\nas well as have a great application potential in many \felds such as spintronics and topological\nquantum computing[6, 51]. Much e\u000borts have been devoted to looking for intrinsic magnetic\ntopological materials because they can reduce disorder e\u000bect and achieve quantized states\nat higher temperature[6, 38, 39]. Although some intrinsic magnetic topological materials\nhave been predicted, very few have been experimentally veri\fed[10, 11, 20, 23{37, 56].\nRecent work has shown that EuCd 2As2is an interesting magnetic topological material\nwhere multiple topological states can be realized by considering the magnetic structure\nand crystal symmetries[56{60]. Various magnetic structures in EuCd 2As2are proposed[56,\n61], experimentally detected[59, 60, 62{66] and manipulated[67{70]. When EuCd 2As2is in\nthe inter-layer antiferromagnetic state, it hosts only one pair of Dirac points at the Fermi\nlevel[56]. Inducing ferromagnetism along the c axis can generate a single pair of Weyl points\nthat provides an ideal case to study the unique properties of Weyl semimetals[57, 59, 67].\nDi\u000berent magnetic structures combined with rotational or inversion symmetry breaking in\nEuCd 2As2, can result in a plethora of other topological states including antiferromagnetic\ntopological insulators[60], a triple-point magnetic topological semimetal[56] and a quantum\nanomalous Hall insulator[58]. EuCd 2As2has become a fertile playground for studying the\ninterplay between magnetism and topology and the magnetic structure plays a key role in\ndictating its topological properties.\nIn this paper, we carried out thorough magnetic, electrical and thermodynamic property\nmeasurements on EuCd 2As2. We discover a long time relaxation of the (001) in-plane\nmagnetic susceptibility in EuCd 2As2. The in-plane magnetic susceptibility at 5 K is found\nto continuously increase up to \u001810% over the time of \u001814 hours. We also \fnd that this\nslow magnetic relaxation does not have an obvious e\u000bect on the bulk properties including\nthe crystal structure, the resistivity and the speci\fc heat. Our unexpected \fnding of the\nslow magnetic relaxation process in EuCd 2As2will simulate further studies to understand\n2its microscopic origin and its manifestation in other physical properties.\nFigure 1 shows the crystal structure and the magnetic structure of EuCd 2As2. EuCd 2As2\ncrystallizes in a trigonal structure with a P \u00163m1 space group (no.164)[62]. The double-\ncorrugated Cd-As layers are sandwiched between the hexagonal Eu layers, as shown in Fig.\n1a. It has been shown that EuCd 2As2undergoes a magnetic transition at 9.5 K from a\nparamagnetic state to an antiferromagnetic (AFM) state[62]. The magnetism of EuCd 2As2\noriginates from the localized 4f electrons of the Eu atoms in the Eu layers. According to\nthe theoretical calculations[56, 61], EuCd 2As2may have A-type antiferromagnetic structure\nwith in-plane or out-of-plane magnetic moments at low temperatures, as shown in Fig. 1b\nwith three possible A-type AFM structures on Eu sites with magnetic moments along b,c\nandxdirections. While some measurements suggest that the magnetic moment is along the\nc direction[64], most experiments support that it is located within the a-b plane[59, 60, 65,\n68, 69].\nHigh quality single crystals of EuCd 2As2were grown by Sn \rux method[64]. The single\ncrystals were characterized by Laue di\u000braction and X-ray di\u000braction (XRD). Fig. 1c shows\nthe Laue pattern of EuCd 2As2crystal for the (001) plane. It shows sharp di\u000braction spots\nwith a three-fold symmetry. The X-ray di\u000braction was measured by using a rotating anode\nX-ray di\u000bractometer with Cu K\u000bradiation (\u0015= 1.5418 \u0017A). All the observed peaks can be\nattributed to the Miller indices (00 l) and the extracted lattice constant c=7.322 \u0017A.\nWe further characterized our samples of EuCd 2As2single crystal by powder X-ray di\u000brac-\ntion measured at di\u000berent temperatures. To this end, the powder sample was obtained by\ngrinding the EuCd 2As2single crystals. Fig. 2a shows the XRD patterns of the EuCd 2As2\npowder at di\u000berent temperatures. All the observed peaks can be attributed to EuCd 2As2\nwith a trigonal structure in P \u00163m1 space group, as marked on the 300 K data except for\nthe two peaks (marked by asterisks in Fig. 2a) that are from the copper sample holder.\nOver the temperature range between 12 K and 300 K, the XRD patterns exhibit no obvious\nchange in terms of the number of the observed peaks, consistent with the absence of struc-\nture transition in this temperature range. The peak position shows a clear variation with\ntemperature, as exempli\fed by the (110) peak shown in Fig. 2b, which indicates that the\nlattice constants change with temperature. The structural re\fnement of the observed XRD\npatterns in Fig. 2a gives the lattice constants a(b) and c as di\u000berent temperatures, as shown\nin Fig. 2c and 2d, respectively. From 300 K to 12 K, the lattice constants a and c decrease\n3by 0.22% and 0.34%, respectively.\nFigure 3 shows magnetic susceptibility measurements of EuCd 2As2. The magnetic mea-\nsurement was carried out by using magnetic property measurement system-3 (MPMS-3).\nThe temperature-dependent magnetic susceptibility is measured in two modes (zero-\feld-\ncooled, ZFC, and \feld-cooled, FC) under di\u000berent applied magnetic \felds parallel (Fig. 2a)\nand perpendicular (Fig. 2b) to the (001) plane. The residual magnetic \feld at the sample\nposition is obtained by measuring and renormalizing the magnetic susceptibility at a high\ntemperature in the paramagnetic state under di\u000berent magnetic \felds. The magnetic \feld\nshown in this work is corrected after considering the residual \feld. First, the magnetic\nsusceptibility displays an obvious antiferromagnetic transition at T N\u00189.3 K in all the mea-\nsurements (marked by red arrows in the upper-left insets of Fig. 3a and 3b). Second, the\nmagnetic susceptibility measured under di\u000berent magnetic \felds bifurcates at a temperature\n(Tc) higher than T N; it is\u001816 K for the in-plane measurement (marked by black arrow in\nthe upper-left inset of Fig. 3a) and \u001814 K for the out-of-plane measurement (marked by\nblack arrow in the upper-left inset of Fig. 3b). This is consistent with the previous mea-\nsurements and such behaviors are attributed to a ferromagnetic transition[62]. Third, in the\nparamagnetic state at high temperature, the magnetic susceptibility is the same when mea-\nsured under di\u000berent magnetic \felds. It also satis\fes the Curie-Weiss law \u001f=C=(T\u0000\u0012), as\nshown in the upper-right insets of Fig. 3a and 3b. Fourth, in the ferromagnetic or antifer-\nromagnetic state at low temperature, the magnetic susceptibility di\u000bers when measured in\ndi\u000berent modes between ZFC and FC measurements. The di\u000berence is signi\fcant when the\napplied magnetic \feld is small and gets suppressed when the applied \feld is large. Fifth,\nthe magnetic susceptibility exhibits a strong anisotropy; the (001) in-plane susceptibility\n\u001fabat the N\u0013 eel temperature T Nin Fig. 3a is nearly 4 times the out-of-plane susceptibility\n\u001fcin Fig. 3b. It indicates that the magnetic moments lie mainly in the (001) plane. Our\nmagnetic measurements are consistent with the previous results[62, 63, 65]. The in-plane\nA type AFM structure was directly determined by resonant elastic x-ray scattering in Ref.\n[65]. Therefore, our EuCd 2As2samples should have the magnetic ground state because the\nsamples we used are the same as those used in Ref. [65] and they come from the same\nsample provider. Our measured temperature- and magnetic-\feld- dependent magnetization\ncurves are consistent with those in Ref. [65].\nThe electrical resistivity measurement of EuCd 2As2was carried out by the standard four-\n4probe method (lower-right inset in Fig. 4a) using a physical property measurement system-\n14 (PPMS-14). Fig. 4a shows the temperature dependence of the (001) in-plane resistivity\nmeasured under di\u000berent applied magnetic \felds normal to the (001) plane. Without the ap-\nplied magnetic \feld, the resistivity exhibits a metallic-like behavior between 50 K and 300 K.\nBelow 50 K, the resistivity shoots up with decreasing temperature, reaches a maximum at\n\u00189.3 K and drops again with further decrease of the temperature. The occurrence of the\nresistivity peak is closely related to the antiferromagnetic transition at T N\u00189.3 K. Upon\napplying the magnetic \feld, the resistivity peak drops in its magnitude, broadens in its peak\nwidth, and shifts to higher temperature in its peak position. These results are consistent\nwith the previous measurements[65]. We further \fnd that the resistivity peak temperature\nincreases linearly with the applied magnetic \feld, as shown in the upper inset in Fig. 4a. It\nis clear in Fig. 4a that, at low temperature, the resistivity of EuCd 2As2exhibits a strong\ndependence on the applied magnetic \feld. To investigate the magnetoresistance e\u000bect quan-\ntitatively, we measured the magnetic \feld dependent resistivity at di\u000berent temperatures\nwith the magnetic \feld perpendicular to the (001) plane, as shown in Fig. 4b. At 5 K in\nthe antiferromagnetic state, the resistivity initially rises rapidly with the applied magnetic\n\feld, reaches a maximum at \u00180.1 T, then drops precipitously till about 0.5 T before it levels\no\u000b at higher magnetic \feld. The dominant resistivity change is con\fned within a narrow\nrange of the applied magnetic \feld. Right at the antiferromagnetic transition temperature\nof 9.3 K, the resistivity starts to drop right after the magnetic \feld is applied and then the\nresistivity change spreads to a wider range of the magnetic \feld. At 15 K that is above the\nantiferromagnetic transition temperature, the resistivity increases with the applied magnetic\n\feld and reaches a maximum at \u00180.65 T, then drops gradually with the magnetic \feld. The\nmagnitude of the resistivity change at 15 K is obviously smaller than those at 5 K and 9.3 K,\nand the change spreads over a much wider range of the applied magnetic \feld. The strongest\nnegative magnetoresistance signal appears at 9.3 K which indicates that the magnetoresis-\ntance behaviors of EuCd 2As2are intimately related with its magnetic structure.\nWe also carried out measurements of the speci\fc heat, the Seebeck coe\u000ecient and the\nthermal conductivity on EuCd 2As2. These measurements were performed in the physical\nproperty measurement system-9 (PPMS-9). Fig. 5a shows the measured speci\fc heat as a\nfunction of temperature. The speci\fc heat drops from 300 K to 13 K, reaches a minimum\nat around 13 K and exhibits a peak at 9.3 K that is apparently related with the antifer-\n5romagnetic transition. Fig. 5b shows the Seebeck coe\u000ecient of EuCd 2As2as a function\nof temperature measured under two di\u000berent magnetic \felds. The Seebeck coe\u000ecient is\npositive over the entire temperature range, indicating that the dominant charge carriers of\nEuCd 2As2is hole-like. It decreases with decreasing temperature between 300 K and \u001840 K\nand exhibits a sharp peak at 9.3 K (black curve in Fig. 5b) which is suppressed when a high\nmagnetic \feld is applied (red curve in Fig. 5b). Fig. 5c shows the temperature-dependent\nthermal conductivity of EuCd 2As2measured under two di\u000berent magnetic \felds. It increases\nwith the decrease of the temperature and exhibits a sharp peak at 9.3 K (black curve in Fig.\n5c). This peak is slightly suppressed when a magnetic \feld of 8 T is applied (red curve in Fig.\n5c). Our results indicate that the transport and thermodynamic properties of EuCd 2As2\nare intimately related to its magnetic structure at low temperature.\nUnexpectedly, we \fnd that EuCd 2As2exhibits a long-time relaxation in its magnetic\nsusceptibility. Fig. 6 shows the variation of the magnetic susceptibility with time after\nthe sample is quickly cooled down from the room temperature to 5 K and the magnetic\nsusceptibility is immediately measured under a magnetic \feld of 3.7 Oe. When the magnetic\n\feld is parallel to the (001) plane (Fig. 6a), the in-plane magnetic susceptibility \u001fabshows\na quick increase initially, followed by a gradual increase later on. Over a period of \u001814\nhours, it increases by about \u001810% (black curve in Fig. 6c). The out-of-plane magnetic\nsusceptibility \u001fc(Fig. 6b) also exhibits an obvious increase with the relaxation time. The\nrelative change over the same period time of \u001814 hours is about \u00183% (red curve in Fig.\n6c) which is much smaller than that of the in-plane magnetic susceptibility. We note that,\neven though the magnitude of the relative change is di\u000berent, the curve shape of the in-\nplane and out-of-plane magnetic susceptibilities is identical when they are scaled at the\ndata points of long relaxation time. When trying to \ft the data with an usual formula\ninvolving one relaxation time \u001c:a+b\u0003exp(\u0000t=\u001c) (blue curve in Fig. 6c), we \fnd that it\nis impossible to match the data. When using a formula with two relaxation time \u001c1and\u001c2:\na+b\u0003exp(\u0000t=\u001c 1) +c\u0003exp(\u0000t=\u001c 2) (red and green curves in Fig. 6c), we \fnd that the \ftted\ndata can capture the main characteristic of the measured data although there are still some\ndeviations. These results indicate that the slow magnetic relaxation process in EuCd 2As2is\nunusual and may involve multiple time scales.\nIn order to check how the magnetic relaxation process in EuCd 2As2depends on the tem-\nperature, we carried out the magnetic relaxation experiments at three typical temperatures,\n65 K below the N\u0013 eel temperature, 9.3 K at the N\u0013 eel temperature and 15 K above the N\u0013 eel\ntemperature. Fig. 7(a-c) shows the relaxation results at these three temperatures under\nan applied magnetic \feld of 3.7 Oe. The relative change of the magnetic susceptibility at\nthe three temperatures is summarized in Fig. 7d. It is found that the magnetic relaxation\ndepends strongly on the sample temperature. While the magnetic relaxation is signi\fcant\nat low temperature (5 K), it gets weaker with increasing temperature and becomes nearly\nnegligible at 15 K. Fig. 7e shows the comparison of the magnetic susceptibility before and\nafter the relaxation at di\u000berent temperatures measured under a magnetic \feld of 3.7 Oe. The\nmagnetic property of EuCd 2As2has been changed after the relaxation process and the e\u000bect\nis most prominent at low temperature below the antiferromagnetic transition at 9.3 K (Fig.\n7e). Such an e\u000bect is observable when the magnetic \feld used to determine the magnetic\nsusceptibility is small, such as 3.7 Oe used in Fig. 7e. The e\u000bect becomes invisible when the\nmagnetic \feld is large, as shown in Fig. 7f. When the magnetic \feld is 3.7 Oe (Fig. 7e), the\nmagnetic susceptibility shows a clear dependence on the relaxation temperature in the ZFC\nmeasurement mode but it exhibits little di\u000berence when measured in the FC mode. When\nthe magnetic \feld is 500 Oe (Fig. 7f), the magnetic susceptibility shows little dependence\non the relaxation temperature in both the FC and ZFC modes. These results indicate that\nthe magnetic relaxation occurs in the magnetic state; the lower is the sample temperature,\nthe stronger is the magnetic relaxation.\nWe also checked the e\u000bect of the applied magnetic \feld on the magnetic relaxation process\nin EuCd 2As2. Fig. 8(a-d) shows the relaxation results under di\u000berent magnetic \felds of\n3.7 Oe (a), 20 Oe (b), 100 Oe (c) and 500 Oe (d) measured at 5 K. The relative change of\nthe magnetic susceptibility under these four magnetic \felds is summarized in Fig. 8e. It\nis clear that the magnetic relaxation also depends sensitively on the applied magnetic \feld.\nWhen the magnetic \feld is small like 3.7 Oe, the relaxation process is strong with a relative\nmagnetic susceptibility change up to 10% over a period of time 14 hours (red curve in Fig.\n8e). When the magnetic \feld gets larger, the relaxation process becomes weaker (blue and\ngreen curves in Fig. 8e). The relaxation process is strongly suppressed when the applied\nmagnetic \feld is large like 500 Oe (yellow curve in Fig. 8e). The magnetic property of\nEuCd 2As2relaxed at di\u000berent magnetic \feld is changed and the change is obvious at low\ntemperature in the magnetic state (Fig. 8f). Again, this change is observable when the\nmagnetic \feld used to determine the magnetic susceptibility is small, such as 3.7 Oe in Fig.\n78f, but becomes invisible when the magnetic \feld is large, as shown in Fig. 8g. These results\nindicate that for the magnetic relaxation in EuCd 2As2to occur in the magnetic state, the\nsmaller is the applied magnetic \feld, the stronger is the relaxation e\u000bect.\nThe magnetic relaxation process we have observed indicates that the magnetic structure\nchanges with time in the bulk EuCd 2As2. It is natural to ask whether the bulk crystal struc-\nture and physical properties also vary with time in EuCd 2As2. To check on the resistivity\nchange, we cooled down quickly the sample from the room temperature to 5 K and mea-\nsured the in-plane resistivity at 5 K after di\u000berent time (Fig. 9a). Little change ( <0.5%) is\nobserved in the resistivity over a period of 78 hours (Fig. 9a). In order to check the change\nof the speci\fc heat, we cooled down the sample quickly from the room temperature to 9.3 K\nand measured the speci\fc heat at di\u000berent time. We also did the same measurement by\ncooling down the sample to 5 K. As shown in Fig. 9b, at both temperatures, the speci\fc\nheat keeps nearly constant over a period of \u001820 hours. To check the e\u000bect of the magnetic\nrelaxation on the crystal structure, we carried out powder XRD measurement at di\u000berent\ntime after the sample is quickly cooled down from the room temperature to 12 K, as shown\nin Fig. 9c. Over a period of 18 hours, the measured XRD patterns are identical and no\nobvious change is found with time (Fig. 9c).\nIt is the \frst time to observe the slow magnetic relaxation process in EuCd 2As2. The\nrelaxation is unusual because the process is slow that can last for hours and the e\u000bect is\nsigni\fcant that the change of the magnetic susceptibility with time can reach \u001810%. The\nsluggish magnetic relaxation in EuCd 2As2is rather similar to that recently discovered in\n(MnBi 2Te4)(Bi 2Te3)n[71], in terms of the bifurcation between the ZFC and FC measurements\n(Fig. 3), the relaxation curve shape that can not be described by a simple Arrhenius law\n(Fig. 6), and the dependence of the relaxation on the applied magnetic \feld (Fig. 8).\nThe slow magnetic relaxation observed in (MnBi 2Te4)(Bi 2Te3)nwith n \u00152 is considered as a\nsignature of two-dimensional magnet where the inter-layer magnetic coupling is approaching\nzero[71]. In EuCd 2As2, the inter-layer magnetic coupling is strong that gives rise to the\nantiferromagnetic transition below 9.3 K. Therefore, our \fnding of the similar relaxation\nbehaviors in EuCd 2As2strongly indicates that two dimensionality is not a necessary factor\nto induce the slow magnetic relaxation process. As in (MnBi 2Te4)(Bi 2Te3)n, the microscopic\norigin of the slow magnetic relaxation remains to be investigated as it involves complex\ninteractions of many factors such as the inter-layer coherent spin rotation and intra-layer\n8domain wall movement[71]. Our new \fnding of the magnetic relaxation in EuCd 2As2can\nprovide new insight in uncovering the relaxation mechanism in these magnetic topological\nmaterials.\nIn summary, by carrying out magnetic, electrical transport and thermodynamic mea-\nsurements on EuCd 2As2, a long-time relaxation of the magnetic susceptibility in EuCd 2As2\nis observed for the \frst time. It is anisotropic that the (001) in-plane relaxation is stronger\nthan the out-of-plane case. The e\u000bect gets stronger with decreasing temperature in the\nmagnetic state. It is prominent under low magnetic \feld and gets suppressed when\nthe applied magnetic \feld is large. 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Wang et al., Anisotropic transport and optical spectroscopy study on antiferromag-\nnetic triangular lattice EuCd 2As2: An interplay between magnetism and charge transport\nproperties, Phys. Rev. B 94, 045112 (2016).\n[65] M. C. Rahn et al., Coupling of magnetic order and charge transport in the candidate Dirac\nsemimetal EuCd 2As2, Phys. Rev. B 97, 214422 (2018).\n[66] J. R. Soh et al., Resonant x-ray scattering study of di\u000buse magnetic scattering from the\ntopological semimetals EuCd 2As2and EuCd 2Sb2, Phys. Rev. B 102, 014408 (2020).\n[67] J. R. Soh et al., Ideal Weyl semimetal induced by magnetic exchange, Phys. Rev. B 100,\n201102(R) (2019).\n[68] N. H. Jo et al., Manipulating magnetism in the topological semimetal EuCd 2As2, Phys. Rev.\nB101, 140402(R) (2020).\n[69] L. D. Sanjeewa et al., Evidence of Ba-substitution induced spin-canting in the magnetic Weyl\nsemimetal EuCd 2As2, Phys. Rev. B 102, 104404 (2020).\n[70] Y. Xu et al., Unconventional transverse transport above and below the magnetic transition\ntemperature in Weyl semimetal EuCd 2As2, Phys. Rev. Lett. 126, 076602 (2021).\n[71] J. Z. Wu et al., Toward 2D magnets in the (MnBi 2Te4)(Bi 2Te3)nbulk crystal, Adv. Mater.\n32, 2001815 (2020).\nAcknowledgement\nWe thank Changjiang Yi for the EuCd 2As2crystal growth, single XRD measurement and\nthe discussion in data process. This work is supported by the National Key Research and\nDevelopment Program of China (Nos. 2016YFA0300600 and 2018YFA0305600), the Na-\ntional Natural Science Foundation of China (No. 11974404), the Strategic Priority Research\nProgram (B) of the Chinese Academy of Sciences (No. XDB33000000), and the Youth\n13Innovation Promotion Association of CAS (No. 2017013).\n14FIG. 1: Crystal structure, magnetic structure and crystal characterization of\nEuCd 2As2.(a) Crystal structure of EuCd 2As2. (b) Proposed possible magnetic structures in\nEuCd 2As2. They represent A-type antiferromagnetic structure on Eu sites with the magnetic mo-\nment along b(i),c(ii) or x(iii) directions[61]. (c) Laue di\u000braction pattern of EuCd 2As2single\ncrystal from (001) cleavage plane. (d) Single crystal XRD pattern of EuCd 2As2measured at 300K.\nThe photo of EuCd 2As2crystal with (001) plane is shown in the upper-right inset.\n15FIG. 2: Temperature-dependent powder XRD measurements of EuCd 2As2. (a) Powder\nXRD patterns of EuCd 2As2measured at di\u000berent temperatures. All the observed peaks can be\nindexed as shown on the 300 K data except for the two peaks marked by asterisks which can be\nattributed to copper from the sample holder. (b) Zoom-in (110) peaks at di\u000berent temperatures.\n(c-d) Temperature dependence of the lattice constants a (c) and c (d) extracted from (a).\n16FIG. 3: Magnetic susceptibility measurements of EuCd 2As2.(a-b) Temperature-dependent\nmagnetic susceptibility measured with the applied magnetic \feld parallel (a) or perpendicular (b)\nto the (001) plane. They are measured at di\u000berent external magnetic \felds using \feld-cooled (FC)\nand zero-\feld-cooled (ZFC) modes. The upper-left insets show the zoom-in magnetic susceptibility\nmeasured at low temperature. The upper-right insets show the inverse magnetic susceptibility as a\nfunction of temperature. We note that there is a transition at \u00183.7 K in some measurements which\nmay be caused by the residual tin (superconducting with a T cat 3.7 K) that was used during the\ngrowth of the EuCd 2As2single crystals by \rux method.\n17FIG. 4: Resistivity measurements of EuCd 2As2.(a) Temperature-dependent resistivity mea-\nsured on the (001) plane under di\u000berent magnetic \felds that are applied perpendicular to the\n(001) plane. The lower-right inset shows the measured sample with the four electrodes. The upper\ninset plots the resistivity peak temperature as a function of the applied magnetic \feld. (b) Mag-\nnetic \feld-dependent resistivity measured at, below and above the N\u0013 eel temperature of 9.3 K. The\nmagnetic \feld is applied perpendicular to the (001) plane.\n18FIG. 5: Thermodynamic measurements of EuCd 2As2.(a) Temperature dependence of the\nspeci\fc heat C pmeasured at zero magnetic \feld. (b) Temperature dependence of the Seebeck coef-\n\fcientSmeasured without (black curve) and with (red curve) magnetic \feld applied perpendicular\nto the (001) plane. (c) Temperature dependence of the thermal conductivity \u0014measured without\n(black curve) and with (red curve) magnetic \feld applied perpendicular to the (001) plane.\n19FIG. 6: Slow magnetic relaxation in EuCd 2As2.(a) In-plane magnetic susceptibility as a\nfunction of the relaxation time measured after the sample is quickly cooled down from the room\ntemperature to a low temperature of 5 K. The applied magnetic \feld is 3.7 Oe that is parallel to the\n(001) plane. (b) Out-of-plane magnetic susceptibility as a function of the relaxation time measured\nafter the sample is quickly cooled down from the room temperature to a low temperature of 5 K.\nThe applied magnetic \feld is 3.7 Oe and perpendicular to the (001) plane. (c) Comparison of the\nrelative change between the in-plane and out-of-plane magnetic susceptibility as a function of the\nrelaxation time. The measured data are \ftted by two forms. The \frst one is a+b\u0003exp(\u0000t=\u001c) by\nconsidering only one relaxation time \u001c. The second one is a+b\u0003exp(\u0000t=\u001c1) +c\u0003exp(\u0000t=\u001c2) by\nconsidering two relaxation time \u001c1and\u001c2.\n20FIG. 7: Magnetic relaxation at di\u000berent temperatures in EuCd 2As2.(a-c) In-plane mag-\nnetic susceptibility as a function of the relaxation time measured after the sample is quickly cooled\ndown from the room temperature to temperature of 5 K (a), 9.3 K (b) and 15 K (c). The three\nmeasurements are independent and each measurement starts with the sample at the room temper-\nature. The applied magnetic \feld is 3.7 Oe and parallel to the (001) plane. (d) Comparison of the\nrelative change of the in-plane magnetic susceptibility measured at di\u000berent temperatures. The\ndata are obtained from (a-c). (e-f) Temperature-dependent magnetic susceptibility of EuCd 2As2\nbefore and after the relaxation at di\u000berent temperatures, measured with the applied magnetic\n\feld parallel to the (001) plane. They are measured at di\u000berent external magnetic \felds of 3.7 Oe\n(e) and 500 Oe (f) using both the \feld-cooled (FC, upper curves) and the zero-\feld-cooled (ZFC,\nlower curves) modes. The relative magnetic susceptibility change before and after the relaxation\nat di\u000berent temperatures, \u001fab(After)/\u001fab(Before), is also plotted in (e) for the ZFC measurements.\n21FIG. 8: Magnetic relaxation under di\u000berent magnetic \felds in EuCd 2As2.(a) In-plane\nmagnetic susceptibility as a function of the relaxation time. It is measured after the sample is\nquickly cooled down from the room temperature to a low temperature of 5 K and then the magnetic\n\feld of 3.7 Oe is applied parallel to the (001) plane and the magnetic susceptibility is measured\nat di\u000berent relaxation time under the same magnetic \feld 3.7 Oe. (b-d) Same as (a) but using\ndi\u000berent magnetic \felds of 20 Oe (b), 100 Oe (c) and 500 Oe (d), respectively. These measurements\nare independent and each measurement starts with the sample at the room temperature. (e)\nComparison of the relative change of the in-plane magnetic susceptibility for the samples relaxed\nunder di\u000berent magnetic \felds. The data are obtained from (a-d). (f-g) Temperature-dependent in-\nplane magnetic susceptibility of EuCd 2As2before and after the relaxation under di\u000berent magnetic\n\felds. They are measured at di\u000berent external magnetic \felds of 3.7 Oe (f) and 500 Oe (g) using\nboth the \feld-cooled (FC, upper curves) and the zero-\feld-cooled (ZFC, lower curves) modes.\n22FIG. 9: Time-dependent physical property and crystal structure measurements of\nEuCd 2As2.(a) Time-dependent measurement of the (001) in-plane resistivity measured at 5 K.\nThe sample was quickly cooled down from the room temperature to 5 K before the resistivity was\nmeasured at di\u000berent time. (b) Time-dependent measurement of the speci\fc heat measured at 5 K\n(black circles) and 9.3 K (blue circles). The sample was quickly cooled down from the room tem-\nperature to 5 K and then the speci\fc heat was measured at di\u000berent time. After this, the sample\nwas warmed up to 200 K and then cooled down quickly to 9.3 K. The speci\fc heat was measured at\n9.3 K at di\u000berent time. (c) Time-dependent measurement of the powder XRD measured at 12 K.\nThe sample was quickly cooled down from the room temperature to 12 K and the powder XRD\npattern was measured at di\u000berent time.\n23"    },    {        "title": "2106.13912v1.Bias_tunable_two_dimensional_magnetic_and_topological_materials.pdf",        "content": "1 \n Bias-tunable two-dimensional magnetic and topological materials  \nJie Li and Ruqian Wu * \nDepartment of Physics and Astronomy, University of California, Irvine, California 92697-4575, \nUSA. \n \nSearching for novel two-dimensional (2D) materials is crucial for the development of the next \ngeneration technologies such as electronics, optoelectronics, electrochemistry and biomedicine. In \nthis work, we designed a series of 2D materials based on endohedral fullerenes, and revealed that \nmany of them integrate different functions in a single system, such as ferroelectricity with large \nelectric dipole moments, multiple magnetic phases with both strong magnetic anisotropy and high \nCurie temperature, quantum spin Hall effect or quantum anomalous Hall effect with robust \ntopologically protected edge states. We further proposed a new style topological field-effect \ntransistor. These findings provide a strategy of using fullerenes as building blocks for the \nsynthesis of novel 2D materials which can be easily controlled with a local electric field.  \n \n \n \n \n \n \n \n \n \n* E-mail: wur@uci.edu. \n  2 \n 1. INTRODUCTION \nThe discovery of excellent functional materials is the foundation for major technological \ninnovations, and this has been manifested again in the new wave of research interest in diverse \ntwo-dimensional (2D) materials. Ever since the successful exfoliation and synthesis of graphene,1 \na great deal of efforts have been dedicated to fabricate and manipulate new 2D materials, from \nsemiconducting transition metal dichalcogenides2,3 2D van der Waals (vdW) magnetic \nmonolayers,4,5 to 2D topological insulators6,7. Due to their unique physical properties, many 2D \nmaterials are promising for the design of the next generation electronics, optoelectronics, \nelectrochemistry and biomedicine devices.8-13 During the course of the development of emergent \nfunctional 2D materials, theoretical predictions via density functional theory (DFT) calculations \nhave played a major role by predicting possible material combinations and providing insights for \nexperimental observations. They inspired tremendous experimental interest and a decent number \nof hypothetical materials have been actually synthesized in experiments. Remarkable examples in \nthis category include metal-organic frameworks (MOFs),14-18 covalent-organic frameworks \n(COFs),19-21 graphene-based systems,3, 22-24 thin metal nanostructures,25,26 and Mxenes,27,28 to \nname a few. Obviously, reliable theoretical predictions for new functional material systems are \ncrucial to propel this exciting field forward. \nEndohedral fullerenes with metal atoms, ions, or clusters embedded in carbon cages can give \nrise to exotic properties by adjusting their ingredients, size and symmetry. Superconductivity,29,30 \nferroelectricity,31-33 and single-molecule magnetism34-36 have been reported for these systems. \nBuoyed by the ongoing effort of the synthesis of 2D fullerenes on graphene or transition metal \nsurfaces by self-assembly process,37-39 we proposed that W@C 28 may serve as building blocks for 3 \n the synthesis of multifunctional 2D materials in our previous work.40 W@C 28 has structural \nbistability, ferroelectricity, multiple magnetic phases, and excellent valley characteristics. \nFurthermore, it can be functionalized as a valleytronic material by integrating with magnetic \ninsulators such as the MnTiO 3 substrate or by switching a small amount of them to the magnetic \nmetal stable state. This inspires us to examine C 28 fullerenes with different 3d transition metal \ninsertions and search for their potentially useful properties. \nIn this paper, we construct various 2D materials with M@C 28 endohedral fullerenes (M=3d \ntransition metals) as building blocks. Through systematic ab-initio calculations and model \nsimulations, we find that many of them are ferroelectric, with the core metal atoms taking two \nstable sites. Some of them simultaneously have multiple magnetic phases with large magnetic \nanisotropy energies and high Curie temperatures, ideal for magnetoelectric operations. \nFurthermore, Ti and Cr cores in the honeycomb lattice produce standalone 2D topological \ninsulators, with robust quantum spin Hall effect (QSHE) and quantum anomalous Hall effect \n(QAHE), respectively. These results indicate that 3d M@C 28 endohedral fullerenes may form \nexcellent 2D multifunctional materials for technological innovations. To show the possible \napplication of their unique physical properties, we further propose a design of topological \nfield-effect transistor. \n \n2. METHODOLOGY \nAll density functional theory calculations in this work are carried out with the Vienna ab-initio \nsimulation package (VASP) at the level of the spin-polarized generalized-gradient approximation \n(GGA) with the functional developed by Perdew-Burke-Ernzerhof.41 The interaction between 4 \n valence electrons and ionic cores is considered within the framework of the projector augmented \nwave (PAW) method.42,43 The energy cutoff for the plane wave basis expansion is set to 500 eV. \nAll atoms are fully relaxed using the conjugated gradient method for the energy minimization \nuntil the force on each atom becomes smaller than 0.01 eV/Å, and the criterion for total energy \nconvergence is set at 10-5 eV. The one-dimensional (1D) band of nanoribbon is calculated with a \ntight-binding (TB) model based on the maximally localized Wannier functions (MLWFs), as \nimplemented in the Wannier90 code.44 \n3. RESULTS AND DISCUSSION  \nAs many endohedral fullerenes have been successfully synthesized,29-36,45 it is attractive to use \nthem as building blocks for the construction of 2D materials. In this work, we focus on \nendohedral M@C 28, M=Ti-Zn that are the smallest stable endohedral fullerenes synthesized in \ncarbon vapor.45 Typical single endohedral C 28 molecules have inherent electric dipoles as the core \natoms shift away from the center of the carbon cage, as sketched for the up- and down \npolarizations in Fig. 1(a), denoted as phase I and phase II below. They also have the C 3v symmetry \nabout the vertical axis, and we hence perceive that highly symmetric 2D covalent crystals may \nform in either the close-packed or honeycomb lattice as shown in Fig. 1(b) and 1(c), respectively. \nAs a result, each M@C 28 has six possible lattice-polarization combinations. As shown by the \nrelative energies with respect to the close-packed M@C u case (set as zero) in Fig.  1 (d), ab-initio \ncalculations indicate that Ti (V, Cr, Fe)@C 28 prefer the close-packed structure, while others prefer \nthe honeycomb lattice in their ground state. Nevertheless, energy differences among these three \nlattices are not large for most fullerenes (˂ ~0.5 eV per molecule) except for the Ti cases. The \nenergy differences for several Cr cases (C u, Cd, H1-u and H 2-u) are even smaller than 0.1 eV. We 5 \n may assume that all of them can be synthesized in experiments, depending on the substrate and \ngrowth conditions. Fig. S2 shows that the honeycomb structure overtakes the close-packed one \nfor Cr@C 28 as the density of fullerenes is reduced. To keep the following discussions more \ngeneral, we mostly give properties of these 2D materials in all three lattices below. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1 (a) Schematic structures of M@C 28 in phases I and II, two colored arrows represent the \n6 \n direction of their dipoles. 2D fullerene crystals in (b) close-packed lattice (M@C) and (c) \nhoneycomb lattice with either the C 3v symmetry (M@H 1) or C3 symmetry in which the molecule \nat the A site rotates by ~62 °(M@H 2). (d) The relative total energies of 2D endohedral C 28 with \nrespect to that of M@C u. (e) The difference of electric dipole moment between the up- and down \npolarizations. (f) The magnetic moments per molecule in different lattice-polarization \ncombinations.   \n \nThe structural stability of these lattices is indicated by their binding energies in Table S1, \nwhich is defined as 𝐸=𝐸−𝐸ଶ𝑛⁄. Here, 𝐸 and 𝐸ଶ are the total energies of the \nisolated M@C 28 molecule and the 2D crystal; and n is the number of M@C 28 molecules per unit \ncell. Large positive values of 𝐸suggest strong interactions among M@C 28 molecules, helpful for \nsynthesizing M@C 28 2D covalent crystals in experiments. Furthermore, the phonon calculations \nand ab initio molecular dynamics (AIMD) simulations are used to check their thermal and \ndynamic stability. Here, we take Cr@C u, Cr@H 1-u and Cr@H 2-u as examples. The corresponding \nphonon dispersions in Fig. S1 show the absence of imaginary frequency branch, indicating that \nthese systems are dynamically stable. In the AIMD simulations with a 6×6 (4×4) supercell of \n1044 (928) atoms for Cr@C u (Cr@H 1-u/2-u), the structures are not noticeably deformed after 5000 \nsteps (10ps) at 300K. The total energies fluctuate around their equilibrium values without sudden \njump as shown in Fig.  S1. Therefore, we believe that these 2D covalent crystals are thermally \nstable at least up to room temperature.  \nInterestingly, we find that almost all M@C 28 2D crystals have two stable structural phases as \ndepicted in Fig. 1 (a), except for Mn, Cu, Zn which has no unpaired d electrons in the carbon cage.   \nBy switching between their two structural phases, the metal cores shift along the vertical direction \nand the corresponding atomic displacements are list in Table S2. Nevertheless, the carbon cage 7 \n noticeably deforms while forming the close-packed lattice, as suggested by the decrease of height \n(cf. data in Table S3 for h) and hence the displacements are small for M@C geometries, except \nCo and Ni which have more localized d-orbitals and smaller size. As listed in Table S4, most \nenergy barriers between these two stable structural phases are smaller than 0.5 eV, so the phase \nchange can be driven by bias. As metal cores donate electrons to the carbon cage, there are net \nelectric dipoles in these molecules, and the 2D lattice may manifest strong ferroelectricity, as we \ndiscussed before for W@C 28.37 Quantitatively, the electric dipole moments are calculated. The \nresults are shown in Fig. 1(e). Taking Cr@H 1 as an example, the Bader charge analysis shows that \neach Cr atom donates 1.26 e (1.21e) to the carbon cage in phase I (II) (see charge redistribution in \nFig. S3), and the Cr atom shifts down by 1.22Å as switching from phase I to phase II which have \ndipole moments of -0.268 eÅ and 0.292 eÅ per molecule, respectively.  These large and opposite \nelectric dipole moments indicate that Cr@H 1 has exceedingly strong ferroelectric polarization \n(36.5μC cm-2) compared to other 2D ferroelectric materials (e.g., 1.6μC cm-2 for Sc 2CO2, 18.1μC \ncm-2 for SnSe, and 19.4μC cm-2 for SnTe).46-48  \nMeanwhile, DFT calculations reveal that most of these fullerene lattices have large local \nmagnetic moments in Fig. 1(f) and also ferromagnetic couplings (see in Fig. S4). The \ncorresponding exchange parameters (see in  Table S5) are obtained by mapping the DFT total \nenergies of different magnetic configurations to the classical Heisenberg Hamiltonian: \n𝐻=𝐻−𝐽ଵ∑𝑆ழ,வ∙𝑆                   (2) \nwhere J1 represent the nearest neighbor exchange interactions. We only consider J1 as the \nseparations between fullerene molecules are large. To determine the thermal stability of their \nmagnetic phase, we calculate their magnetic anisotropy energies (MAEs) by using the torque 8 \n method (see in Fig. S4).49,50 One may see that Cr, Mn, Fe, Co and Ni cases have large MAEs, \nespecially for Fe@H 2-d (1.02meV per molecule), Co@C d (-2.47meV per molecule) and Ni@C u \n(-2.36meV per molecule). With the exchange parameters in Table S5 and magnetic anisotropy \nenergies in Fig. S4(a), we further calculate Curie temperatures ( Tc) of these new 2D magnetic \nmaterials by using the renormalized spin-wave theory (RSWT)51,52 and results are shown in Fig. \nS4(b). Assuming the Curie temperature as the renormalized magnetization drops zero, we find \nthat several these systems have comparably higher Curie temperatures than many existing 2D \nmagnetic materials, such as CrI 3 (45K) and Cr 2Ge2Te6 (66K).53,54 Especially, Fe@H 1-d is \nhalf-metallic (cf. the DOS curves Fig. S5) and has a Curie temperature as high as 147K, which \nmakes it very attractive as an efficient spin filter for spintronic applications. \nIt is intriguing to investigate if these systems may have nontrivial topological properties for the \nrealization of quantum spin Hall effect and quantum anomalous Hall effect which may find \napplications in quantum information technologies. Through systematic search from their band \nstructures, we find that the nonmagnetic Ti@H 1-u has a SOC-induced band gap of 10.8 meV right \nat the Fermi level, a desirable feature for topological materials (see in Fig. 2(a)). To verify if this \ngap is topologically nontrivial, the Z 2 number of Ti@H 1-u is calculated. In  Fig. 2b, one may see \nthat Z2=1 for Ti@H 1-u by counting the positive and negative n-field numbers over half of the torus. \nThis undoubtedly indicates that Ti@H 1-u is a new 2D topological insulator that may manifest the \nQSHE.As a further evidence, we construct a zigzag Ti@H 1-u nanoribbon about 30nm in width \n(see in Fig. 2(c)) and check if it has the topologically protected edge states. Due to large number \nof atoms in the nanoribbon, we build a TB model with parameters obtained by fitting the DFT \nband structure of 2D Ti@H 1-u. Using s and p orbitals for C and s and d orbitals for Ti in the bases, 9 \n this TB model gives excellent band structures near the Fermi level comparing with the DFT \nresults (see in Fig. S6). In Fig. 2(d), one may see two edge bands intercepting the Fermi level and \nconnecting the bulk conduction and valence bands in the two sides of the Г point. This further \nsupports that Ti@H 1-u is a new 2D topological insulator and may offer a possible platform for the \nrealization of the quantum spin Hall effect. In contrast, the Ti@H 1-d lattice is a trivial insulator, \nand the topological phase transition therefore can be controlled by the ferroelectric polarization. \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 2 (a) The band structures of Ti@H 1-u without and with SOC. (b)  The n-field configuration \nwith solid, hollow circles and blank boxes denoting n= -1, n= 1 and n= 0. (c) The schematic \nstructure of a zigzag Ti@H 1-u nanoribbon about 30nm in width (along y-axis) and keeps the \nperiodicity along the x-axis. (d) The corresponding 1D band structure and the top edge states. \n10 \n  \nAs many fullerene lattices are magnetic, it is intriguing to examine if they may manifest the \nQAHE. Due to the difficulty of controlling the distribution and magnetic order of dopants or the \nstrong interfacial hybridization, very few successful observations of QAHE have been reported \nusing the conventional approaches that magnetize the topological surface states with doping or \ninterfacial proximity.55-57 Here, our calculations show that Cr@H 1-u has a SOC-induced spin \npolarized band gap of 7.4 meV right at the Fermi level (see in Fig.  3(a)), and its local magnetic \nmoment is 2μ B per molecule. To verify if Cr@H 1-u is a Chern insulator, its Berry curvature Ω(k) is \ncalculated in the Brillouin zone as \n        Ω(k)= 2Im∑ ∑⟨நౡ|୴౮|நౣౡ⟩ൻநౣౡห୴౯หநౡൿ\n(கౣౡିகౡ)మ ୫{୳} ୬{୭}                (4) \nHere, {𝑜} and {𝑢} are the sets of occupied and unoccupied states;  𝜓 and 𝜀 are the Bloch \nwave function and eigenvalue of the nth band at the k point; and  𝑣௫ and 𝑣௬ are the velocity \noperators, respectively. One may see that large positive Berry curvature presents around the Г \npoint, indicating that the gap is truly topologically nontrivial (see in Fig.  3(b)). Meanwhile, the \nChern number (C), which gives the Hall conductance as 𝜎௫௬=𝐶(𝑒ଶ/ℏ), is directly calculated by \nintegrating the Berry curvature in the Brillouin zone as \nC =ଵ\nଶ∫Ω(k)dଶ \nk                               (5) \nIn Fig. 3(c), one may see that a small terrace with C = 1 presents in the SOC induced band gap, \nwhich is the hallmark of the QAHE materials. As further evidence, the top and bottom edge states \nof a zigzag Cr@H 1-u nanoribbon can be seen in the bulk gap according to the TB calculations with \nparameters extracted from the DFT bands. The top edge state moves along the positive x-axis \nwhile the bottom edge state moves along the opposite direction (see in Fig.  3(d)). Obviously, a net  11 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 3 (a) The band structures of Cr@H 1-u without and with SOC (the middle are the \ncorresponding zoom-in band structures near the Fermi level around the Γ point). (b) and (c)  The \ndistribution of Berry curvature in the 2D Brillouin zone and Fermi level-dependent anomalous \nHall conductance (σ xy) for Cr@H 1-u, respectively. (d) The corresponding 1D band structure and \nthe top and bottom edge states. \n \nquantized transverse Hall current can be expected when zigzag Cr@H 1-u nanoribbons are used in \nnanodevice. Since this material has a Tc of 14.3 K, we may expect that the QAHE can be \n12 \n observed with the liquid Helium4 temperature, which is attainable in most labs. The discovery of \nthese standalone 2D topological insulators is exciting as they may avoid technical complexities in \ndealing with doping and unwanted interfacial hybridization with the conventional approaches. We \nbelieve it is particularly rewarding for experimentalists to try synthesizing Ti@C 28 and Cr@C 28 \nhoneycomb lattices and using their topological properties.  \n \n \n \n \n \n \n \n \n \nFig. 4 (a) The relative energy of Cr@H 1-u/d as functions of the external electric field. (b) \nSchematic topological field-effect transistor based on Cr@H 1-u/d, (top and side view). Two Au \nelectrodes are for the detection of Hall voltage. The nano-electrode buried in HfO 2 are for gate \ncontrol. (c) The band structure of Cr@H 1-u with a phase transition concentration of 50%. \n \nFinally, we discuss how to combine the topological and ferroelectric properties of Cr@H 1-u \nfor the design of spintronic devices. By using a gate bias, we may realize a reversible structural \nphase transition as shown in Fig. 4(a), as also drastically change the electronic and magnetic \nproperties of the M@C 28 lattice. This interplay allows us to design a conceptual topological \n13 \n field-effect transistor as shown in Fig. 4(b). We use a doped silicon/SiO 2 film as the substrate and \nfour Au electrodes for the measurement of transvers voltage and longitudinal current. An ultrathin \nHfO2 (or hexagonal BN) overlayer provides a protective environment, with nano-electrodes \nburied underneath as the top gate. As discuss above, the Cr@H 1-u nanoribbon manifests the \nQAHE at a reasonably low temperature, and hence have spin polarized current at the edges. When \nsome Cr@C 28 molecules are converted to the phase II, the ribbon switches to an ordinary \nmagnetic semiconductor, as shown by the bands in Fig. 4(c) and Fig. S7. The transport properties \nare thus highly tunable by the top gate (~100%) is this geometry, which behaves as an ideal \ntopological field-effect transistor. With such an easy controllability and unique topological feature, \nit is perceivable that many different devices can be designed with these new 2D fullerene lattices.  \n \n4. Conclusion  \nIn summary, by using systematic ab-initio calculations and model simulations, we found several \nimportant new 2D functional materials based with 3d M@C 28 molecules in hexagonal or \nhoneycomb lattice. Explicitly, 1) Fe@C 28 honeycomb lattice is a halfmetallic materials with a Tc \nup to 147 k; 2) Ti@C 28 honeycomb lattice is a topological insulator with a gap for 10.8 meV right \nacross the Fermi level; and 3) Cr@C 28 honeycomb lattice is a Chern insulator with a gap of 7.4 \nmeV. The discovery of these standalone functional materials may allow the realization of 100% \nspin filtering, quantum spin Hall effect or quantum anomalous Hall effect, each of them is an \nimportant research subject for the development of spintronics and quantum information \ntechnologies. Another advantage of these 2D materials is the easy controllability via electric field \nas most of them are intrinsically ferroelectric. As an example, we demonstrated a conceptual 14 \n topological field-effect transistor based on the Cr@H 1-u lattice, which can be switched from Chern \ninsulator to ordinary magnetic semiconductor by tuning some Cr@C 28 molecules to metastable \nstructural phase II with a local electric field. This work enriches the family of 2D magnetic and \ntopological materials and points to a way for integrate different functionalities in a single simple \nsystem for diverse technological innovations.   \n \nAuthor contributions   \nJ.L. and R.W. designed the studies. R.W. conceived this project. J.L. performed the calculations \nand analyzed data. 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Lett . 2017, 119, 027201. \n \n \n "    },    {        "title": "2107.04825v1.Multi_Material_Topology_Optimization_with_Continuous_Magnetization_Direction_for_Permanent_Magnet_Synchronous_Reluctance_Motors.pdf",        "content": "Multi-Material Topology Optimization with Continuous\nMagnetization Direction for Permanent Magnet Synchronous\nReluctance Motors\nThomas Gauthey *1,2, Peter Gangl†2, and Maya Hage Hassan‡1\n1Université Paris-Saclay, CentraleSupélec, CNRS, Laboratoire de Génie Electrique et\nElectronique de Paris, 91192, Gif-sur-Yvette, France. ,\nSorbonne Université, CNRS, Laboratoire de Génie Electrique et Electronique de Paris, 75252,\nParis, France.\n2Technische Universität Graz, Institut für Angewandte Mathematik, 8010 Graz, Austria.\nJuly 13, 2021\nAbstract\nPermanent magnet-assisted synchronous reluctance mo-\ntors (PMSynRM) have a significantly higher average\ntorque than synchronous reluctance motors. Thus, deter-\nmining an optimal design results in a multi-material topol-\nogy optimization problem, where one seeks to distribute\nferromagnetic material, air and permanent magnets within\nthe rotor in an optimal manner.This study proposed a novel\ndensity-based distribution scheme, which allows for con-\ntinuous magnetization direction instead of a finite set of\nangles. Thus, an interpolation scheme is established be-\ntween properties pertaining to magnets and non-linear ma-\nterials. This allows for new designs to emerge without in-\ntroducing complex geometric parameterization or relying\non the user’s biases and intuitions. Toward reducing com-\nputation time, Nitsche-type mortaring is applied, allow-\ning for free rotation of the rotor mesh relative to the sta-\ntor mesh. The average torque is approximated using only\nfour-point static positions. This study investigates several\ninterpolation schemes and presents a new one inspired by\nthe topological derivative. We propose to filter the final de-\nsign for the magnetization angle using K-mean clustering\naccounting for technical feasibility constraints of magnets.\nFinally, the design of the electrical motor is proposed to\nmaximize torque value.\nKeywords: Topology optimization, Permanent mag-\nnets machines, Design optimization, Acceleration methods\n*E-Mail:thomas.gauthey(at)geeps.centralesupelec.fr\n†E-Mail: gangl(at)math.tugraz.at\n‡E-Mail: maya.hage-hassan(at)centralesupelec.fr1 Introduction\nSynchronous reluctance machines (SynRM) are standard\nin households and industrial applications, thanks to their\ncheap cost compared to permanent magnet motors and ad-\nvances in manufacturing techniques. Although the deploy-\nment of these machines continues [15], PMSynRM of-\nfers an excellent alternative for both structures, solving for\nSynRM, its poor power factor and, for permanent mag-\nnet machines (PMM), its cost. The design of these ma-\nchines using parametric optimization often necessitates ei-\nther complex analytical models or the use of Finite Ele-\nment Analysis (FEA) relying heavily on experienced engi-\nneers and known good designs [23].\nDensity based optimizations allow for bypassing such\ncumbersome frameworks. Although they were first devel-\noped for two materials application in continuous mechan-\nics [33], a rise in n-materials optimization in the field of\nelectromagnetics has allowed for new PMM and SynRM\nto emerge [14, 18]. In most optimizations where perma-\nnent magnets are involved, magnetization direction are\nfixed [29] or limited to a set of a couple values [6,7,20]. If\ncontinuous directions are considered during the optimiza-\ntion process, the final design is filtered to take into account\nonly a couple of predefined directions to meet manufactur-\ning constraints [17, 36].\nIn this paper we propose a simultaneous density-based op-\ntimization scheme consisting of three-material (air-iron-\nmagnet) with a continuous magnetization direction. The\nproposition is applied to design the rotor of a distributed\nwinding stator as described in [11, 25] to maximize the\nmean torque under constraints. The final designs are fil-\ntered using an unbiased K-means heuristic for accounting\n1arXiv:2107.04825v1  [math.OC]  10 Jul 2021Figure 1: Machine geometry\nTable 1: Geometric parameters\nParameter Value\nSlot number 24\nAxial length 50.0 mm\nOuter rotor radius 18.5 mm\nInner stator radius 26.5 mm\nOuter stator radius 47.5 mm\nAir gap length 8.0 mm\nfor feasibility constraints. Here, we propose also to accel-\nerate the torque calculation through a four-point method.\n2 Problem description\nWe chose to investigate a SynRM described in [11, 25],\nof which the rotor design had proven to be a challenging\nproblem for topology optimization and use it for our PM-\nSynRM optimization problem.\n2.1 Geometry description\nThe electrical machine geometry and current density dis-\ntributions are given respectively in Figure 1 and 2. The di-\nmensions for the considered machine are given in Table\n1.\nThis machine differs from most conventional SynRM\nby its large air gap which constrains the statoric winding\ndistribution to only one pair of poles (cf. Figure 2).\nWe introduce the relationship between the electrical\nangle qelecand the mechanical angle q\nqelec=nppq; (2.1)\nwith nppthe number of pair of poles, here npp=1.\nFigure 2: Statoric winding distribution and current param-\neters\nTable 2: Statoric winding parameters\nParameter Value\nNumber of turn Ns 64\nWinding type Distributed\nConnection type Star\nResistance ( RS;20\u000eC) 7.1 W\nV oltage Ue f f 230 V\nPeak intensity Imax 12 A\nNumber of pole pairs npp 1\nWe define the three phases as follows:\n8\n><\n>:IU(q) =Imaxcos(nppq+j)\nIV(q) =Imaxcos\u0000\nnppq+j\u00002p\n3\u0001\nIW(q) =Imaxcos\u0000\nnppq+j\u00004p\n3\u0001\n:(2.2)\nHere, jis the phase angle. The computational domain W\nconsists of iron, air, permanent magnet and coils,\nW=Wf[Wair[Wmag[Wc (2.3)\nwhere we further subdivide the ferromagnetic and air sub-\ndomains into their rotor and stator parts,\nWf=Wf;stat[Wf;rot; Wair=Wair;stat[Wair;rot:\n(2.4)\nMoreover, we subdivide the coil subdomains according to\nthe distribution shown in Figure 2,\nWc=WU+[WU\u0000[WV+[WV\u0000[WW+[WW\u0000:(2.5)\n2.2 Partial differential equation\nIn the two-dimensional magnetostatic setting, the mag-\nnetic flux density B=curl((0;0;u)>)for rotor position\n2q2[0;2p]can be computed via the solution of the bound-\nary value problem\nFind u2H1\n0(W):Z\nWnq(x;jÑuj)Ñu\u0001Ñvdx=\nZ\nWcj(q)vdx+Z\nWqmagRq\u0014\n\u0000My\nMx\u0015\n\u0001Ñvdx(2.6)\nfor all v2H1\n0(W), see e.g. [26].\nHere, the magnetic reluctivity is a nonlinear function ˆnof\nthe flux densityjBj=jÑujin the ferromagnetic subdomain\nand a constant n0=107=(4p)elsewhere, i.e.,\nnq(x;jÑuj) =(\nˆn(jÑuj)x2Wq\nf\nn0 x2Wq\nair[Wc[Wq\nmag(2.7)\nwith the rotated domains\nWq\nf=Wf;stat[RqWf;rot (2.8)\nWq\nair=Wair;stat[RqWair;rot (2.9)\nWq\nmag=RqWmag (2.10)\nandRqa rotation matrix around angle q,\nRq=\u0014\ncosq\u0000sinq\nsinq cosq\u0015\n: (2.11)\nThe first term on the right hand side of (2.6) represents the\nimpressed current density which is given by\nj(x;q) =cWU+(x)jU(q)+cWV+(x)jV(q)+cWW+(x)jW(q)\n\u0000cWU\u0000(x)jU(q)\u0000cWV\u0000(x)jV(q)\u0000cWW\u0000(x)jW(q);\n(2.12)\nwhere cAdenotes the characteristic function of a set A,\ncA(x) =(\n1x2A;\n0 else :\nHere the current distribution is defined by :\njp(q) =1\nSslotNsIp(q);p2fU;V;Wg (2.13)\nwith Sslotthe cross-sectional area of one coil, Nsthe num-\nber of turns per coil and IU,IV,IWas defined in (2.2).\nThe second term on the right hand side of (2.6) represents\nthe magnetization M= (Mx;My)>coming from permanent\nmagnets which will be added in the course of the multi-\nmaterial optimization procedure.\nIn the following, we will denote by uqthe solution to\nthe state equation (2.6).\nWe present here after the properties of interest of the ma-\nterials (air,ferromagnetic,magnet) used in the machine.Table 3: Material properties\nMaterial Reluctivity [ m:H\u00001] Magnetization [ A:m\u00001]\nAir n0 0\nCopper n0 0\nFerromagnetic ˆn(j~Bj) 0\nMagnet n0 Mmax\nThe maximum norm of the magnetization vector was\nchosen as Mmax=2:33\u0001105A:m\u00002to fit data from [28] on\nferrite magnets. The reluctivity of the magnets and of the\ncopper coils is assimilated to the one of air to simplify fur-\nther material interpolation and avoid complex schemes like\nthe ones found in [37]. The non-linear behaviour of the\nferromagnetic material is modelled with a Marrocco’s BH\ncurve approximation [24].\nˆn(j~Bj) =8\n>>>><\n>>>>:n0(e+(c\u0000e)j~Bj2a\nt+j~Bj2a‘ ifj~Bj\u0014Bmax;\nn0\u0010\n1\u0000Ms\nj~Bj\u0011\nelse ifj~Bj>Bs;\nexp\u0012\ng(j~Bj\u0000b)\nj~Bj\u0013\notherwise ;(2.14)\nwhere Bs=b+log\u0010n0\ng\u0011\ngandMs=Bs+1\ngand the coeffi-\ncient of the Marrocco curve in Figure3 are defined in the\nTable 4.\nTable 4: Marrocco curve coefficient for the ferromagnetic\nmaterial\nParameter Value\na 6.84\nb -1.30\u000110\u00001\ng 4.86\ne 1.57\u000110\u00004\nt 4.14\u0001103\nc 1.90\u000110\u00002\nBmax 1.80 (T)\n2.3 Torque computation method\nFor computing the torque, we chose a method based on\nMaxwell’s stress tensor, Arkkio’s method [30]. While co-\nenergy torque computation methods were proven to be\nmore precise and less prone to error, they are more costly\nin terms of computation time and not fit for optimization.\nUsing Arkkio’s method, the torque can be computed as\nT=Lzn0\nrs\u0000rrZ\nSp\nx2+y2BrBfdS (2.15)\n3(a) Magnetic permeability\n(b) Marrocco BH-curve\nFigure 3: Non-linear characteristic for the ferromagnetic\nmaterial.\nwhere BrandBfdenote the radial and tangential magnetic\ninduction, respectively, Lzdenotes the length of the ma-\nchine in z-direction and Sdenotes the surface between radii\nrsandrrin the air gap (with rs>rr).\nIn the setting of two-dimensional magnetostatics, the\ntorque for the rotor position given by angle qthus amounts\nto\nTq=T(uq) =Lzn0\nrs\u0000rrZ\nSQÑuq\u0001ÑuqdS (2.16)\nwith Q(x;y) =2\n4xyp\nx2+y2y2\u0000x2\n2p\nx2+y2\ny2\u0000x2\n2p\nx2+y2\u0000xyp\nx2+y23\n52R2\u00022:(2.17)\nDetermining the average torque by means of its instan-\ntaneous values can be very expensive. It is shown in [5]\nthat a good approximation to the average torque can be ob-\ntained when evaluating the torque for only suitably chosen\nrotor positions,\n¯T=1\n4\u0010\nT0+Tp\n12+Tp\n6+Tp\n4\u0011\n: (2.18)\nWe compared the average torque obtained by evaluation at\n500 equally distributed rotor positions between 0 and 2 p\nwith the value obtained by the four-point formula (2.18).Table 5: Four static positions method error\nDesign ¯T[N:m] ¯T[N:m] Error\n(500 points) (4points) [%]\nUnbiased starting point* 1.5790 \u000110\u000065.7232\u000110\u00006262.4\nFinal design Table 7 1.1123 1.1129 0.048\nFinal design Table 9 1.4513 1.4516 0.027\n* in this design rn=0:5;rMx=0:5;rMy=0:5\neverywhere in the rotor\nWhen the torque value is not equal to zero, the error\nfound was to be lower than 0.1% as expected and described\nin literature [1]. This is solved beyond the first iteration.\n3 Optimization problem\nIn this section, we define our optimization problem and\nreformulate the forward problem to fit the density-based\ntopology optimization approach. Our goal is to maximize\nthe average torque computed via (2.18),\n(P1):(\nmaximize ¯T=1\n4\u0010\nT(u0)+T(up\n12)+T(up\n6)+T(up\n4)\u0011\ns.t.uqis a solution of (2.6) for q2f0;p\n12;p\n6;p\n4g\n(3.1)\nThis is achieved by finding the optimal material distribu-\ntion consisting of ferromagnetic material, air and perma-\nnent magnets on the one hand, and the optimal magnetiza-\ntion direction of the permanent magnets on the other hand.\nMoreover, we will incorporate a bound on the maximum\nallowed permanent magnet volume.\n3.1 Density based topology optimization\nLet us reformulate the forward problem (2.6), introduc-\ning the three density variables respectively for the ferro-\nmagnetic material and the two components of the per-\nmanent magnets magnetization, rn;rMx;rMydefined in\nWq\nrot=Rq(Wf;rot[Wair;rot[Wmag). Moreover we introduce\nthe rotated design variables\nrq\nn(x;y) =rn(Rq((x;y)>))\nrq\nMx(x;y) =rMx(Rq((x;y)>))\nrq\nMy(x;y) =rMy(Rq((x;y)>))\nwhich represent the design given by rn,rMx,rMy\nafter rotation, and the vector of design variables\nX:=\u0002rn;rMx;rMy\u0003>.\nGiven two interpolation functions\nfn:[0;1]![0;1]; fM:[0;1]![0;1]; (3.2)\n4we define the operator\nKq:(X;u;v)7!Z\nWn(rq\nn;jÑuj)Ñu\u0001Ñv\n\u0000Z\nWrotfn(1\u0000rq\nn)MmaxfM(j~Mqj)\nj~MqjRq\u0014\u0000Mq\ny\nMq\nx\u0015\n\u0001Ñv;\n(3.3)\nwith the reluctivity function\nn(rq\nn;jÑuj) =8\n><\n>:ˆn(jÑuj) inWf;stat\nn0 inWc[Wair;stat\nn0+fn(rq\nn)(ˆn(jÑuj)\u0000n0)inWrot\n(3.4)\nand with the components of the magnetization vector\n~Mq= (Mq\nx;Mq\ny)given in dependence of the two rotated\ndensity variables rq\nMx,rq\nMy,\n(Mq\nx;Mq\ny) = ˜fsd(rq\nMx;rq\nMy) (3.5)\nfor a mapping ˜fsdwhich will be discussed later on. Hence,\nthe state equation (2.6) can be reformulated into\nFind u2H1\n0(W):\nKq(u;v;X) =Z\nWcj(q)vdx;for all v2H1\n0(W):(3.6)\nThe optimization problem ( P1) can then be reformu-\nlated into\n(P2):(\nmaximize\nX¯T=1\n4åq2f0;p\n12;p\n6;p\n4gT(uq)\ns.t.uqis a solution of the (3.6) q2f0;p\n12;p\n6;p\n4g\n(3.7)\n3.1.1 Material interpolation schemes\nIn density based topology optimization, the quality of the\nfinal solution is dependant on the choice of interpolation\nfunctions (equation (3.2)). We present here two existing\nschemes and a novel one based on properties of the topo-\nlogical derivative.\nThe polynomial interpolation scheme\nfn(r) =rnn>0; (3.8)\nalso referred to as SIMP (Solid Isotropic Material with Pe-\nnalization), is the most used material interpolation scheme\nfor topology optimization and allows for easy penalization\nof intermediate materials. However, it presents some sym-\nmetry issues and favors low rassociated material in the\nfinal design. In [31], the authors compared this scheme to\nother schemes and concluded that the final design was not\nas good as many other proposed ones.\nFigure 5: D. Lukàš’s interpolation scheme\nFigure 4: SIMP Polynomial interpolation scheme\nTo solve symmetry issues introduced by the classi-\ncal polynomial interpolation, D. Lukàš introduced a new\nscheme in [21]:\nfl(r) =1\n2\u0012\n1+1\narctan (l)arctan (l(2r\u00001))\u0013\n;l>0:\n(3.9)\nIn this equation the particular invariant point r=0:5 does\nnot promote intermediate materials, grey material depends\nonlvalues (cf. Figure 5).\nHigh lvalues permit to penalize intermediate materials\nbut can lead to a poor convergence of the algorithm. A pa-\nrameter study for lled us to choose l=5. This interpo-\nlation method is chosen for the norm of the magnetization\nvector ( fMin (3.6)). Finally, we propose a new interpo-\nlation scheme as given in Figure 6,which is inspired by\nthe topological derivative as done in [2], see also the the\nSIMP-All method for linear elasticity [9]. Here, we seek to\ndesign a material interpolation function whose derivative\n5Figure 6: Topological derivative inspired interpolation\nscheme\nwith respect to the density variable rcoincides with the\ntopological derivative of the problem at r=0 and r=1.\nWhen interpolating between two linear materials with re-\nluctivity values n0andn1, the conditions for the material\ninterpolation function faccording to [2] would read\n8\n>>><\n>>>:f(0) =0;\nf(1) =1;\nf0(0) =2n0\nn0+n1;\nf0(1) =2n1\nn0+n1:(3.10)\nDue to the involved formula of the topological derivative\nfor nonlinear magnetostatics [3], a mathematically rigor-\nous extension of this method to the nonlinear setting is\nnot straightforward. However, inspired by the particular\nbehaviour of the Marrocco BH-curve where the magnetic\nreluctivity is almost constant for low flux density values,\nsee Figure 3, we simply use this idea for that constant re-\nluctivity value n1:=n0e\u0019124:94. Using cubic Hermite\ninterpolation for the conditions (3.10), we obtain the poly-\nnomial\nf(r) =2n0\nn0+n1r\u0000n0\u0000n1\nn0+n1r2: (3.11)\nNote that the term of order 3 happens to vanish.\n3.1.2 Magnetization vector transform\nWe deal with two magnetization density variables rMx\nandrMyin order to represent the magnetization direc-\ntion(Mx;My). One way of relating these quantities to\neach other would be to have rMxrepresent the first and\nrMythe second coordinate, resulting in a representation\nin Cartesian coordinates, which was also considered in\n[36]. In this case, however, some magnetization directions\nexhibit higher maximum magnetization than others, e.g.\n(a) Interpolated BH curve for r=0:25\n(b) Interpolated BH curve for r=0:75\n(c) Interpolated BH curve for r=0:9\nFigure 7: Interpolated BH curves using material interpola-\ntion scheme (3.11).\n6Figure 8: Square to disk transform for the Magnetization\nvector coordinates\nrMx=rMy=1 would correspond to j(Mx;My)>j=p\n2\nwhereas for the magnetization direction pointing to the\nrightrMx=1,rMy=0:5 would yield a maximum magneti-\nzation ofj(Mx;My)>j=1, thus making the maximum mag-\nnetization angle dependent. As an alternative, one could\nuse polar coordinates and represent the magnetization di-\nrection by just one periodic density function. In this case,\nhowever, the ambiguity of angular values causes problems\nin the gradient computation. We define a means of solv-\ning these issues without resorting to polar coordinates. We\ndecide on two density variables rMx;rMywith values in\n[0;1], but map them onto a disk, thereby avoiding an angle\ndependent maximum magnetization value.\nThere are several mappings that approximately realize\nsuch a square-to-disk transformation. In order to preserve\nthe angle,as much as possible, without being too computa-\ntionally heavy, the elliptic grid mapping.\nfsd(x;y) =8\n<\n:xq\n1\u0000y2\n2\nyq\n1\u0000x2\n2with(x;y)2[\u00001;1]2:(3.12)\nwas chosen as a good compromise out of the methods\ndescribed in [10]. While the associated inverse transforma-\ntion fds=f\u00001\nsdgiven by\nfds(u;v) =8\n<\n:1\n2\u0010p\n2+u2\u0000v2+2p\n2u\u0000p\n2+u2\u0000v2\u00002p\n2u\u0011\n1\n2\u0010p\n2\u0000u2+v2+2p\n2v\u0000p\n2\u0000u2+v2\u00002p\n2v\u0011\n(3.13)\nis computationally more costly, it is only used once per\niteration and in post-processing. The mapping between the\nmagnetization density variables rMx;rMyand the magneti-\nzation vector ~M= (Mx;My)(3.5) is then given by\n˜fsd(rq\nMx;rq\nMy) =fsd(2(rq\nMx\u00000:5);2(rq\nMy\u00000:5))(3.14)\n3.2 Incorporation of volume constraints\nConstraints on iron and magnets volume are added to avoid\nhaving structures with disproportionate volumes of mate-rial. Hence we add new constraints with the operator\nIvol:r7!1\nVWrotZ\nWrotr(x)dx (3.15)\nrepresenting the volume fraction inside the rotor domain\nWrotof a material given by a density function r. Here,\nVrot=R\nWrot1dxdenotes the total area of Wrot. Based on\n(P2) we define a new constrained optimization problem\n(P3):8\n>>><\n>>>:maximize ¯T=1\n4åq2f0;p\n12;p\n6;p\n4gT(uq)\ns.t.uqis a solution of (3.6) ;q2f0;p\n12;p\n6;p\n4g\nIvol(rn)\u0014fv;f\nIvol((1\u0000rn)j~Mj)\u0014fv;mag\n(3.16)\nwith given upper bounds on the allowed ferromagnetic\nand permanent magnet material fv;f,fv;mag2[0;1], respec-\ntively. We reformulate the inequality constraints of ( P3)\nusing the augmented Lagrangian framework as described\nin [27],\n(P4):8\n><\n>:minimize L(X;u) =\u0000¯T(u)+y(hv;f(X);gf;m)\n+y(hv;mag(X);gmag;m)\ns.t.uqis a solution of (3.6) ;q2f0;p\n12;p\n6;p\n4g\n(3.17)\nWith the state vector\nu:= (u0;up\n12;up\n6;up\n4): (3.18)\nHere,\n(\nhv;f(X) =fv;f\u0000Ivol(rn);\nhv;mag(X) =fv;mag\u0000Ivol((1\u0000rn)j~Mj);(3.19)\nwith the scalar function\ny(t;s;m) =(\n\u0000st+1\n2mt2ift\u0000ms\u00140;\n\u0000m\n2s2otherwise :(3.20)\nThe positive scalar multipliers gf;gmag;mare updated ac-\ncording to the LANCELOT-Method of Multipliers pre-\nsented in [27].\n3.3 Adjoint Method\nTo solve the optimization problem (P4)as formulated in\n(3.17) by a gradient descent algorithm, we introduce the\nLagrangian for the PDE-constrained problem (3.17)\nL(X;u;w) =L(X;u)+å\nq2f0;p\n12;p\n6;p\n4gKq(X;uq;wq)\u0000Z\nWstatj(q)wq;\n(3.21)\n7where w= (w0;wp\n12;wp\n6;wp\n4)is a vector of Lagrange\nmultipliers. The adjoint states lqcorresponding to prob-\nlem (3.17) for different rotor positions qare defined by\n¶L\n¶uq(X;uq;lq) =0, i.e. lqis the solution to\nFindlq2H1\n0(W):¶Kq\n¶u(X;uq;lq)(v) =\u00001\n4¶T\n¶u(uq)(v)\n(3.22)\nfor all v2H1\n0(W): (3.23)\nHere, the left and right hand side, respectively, involve\nthe operators\nd\nduKq(X;u;l)(v) =d\ndu\u0012Z\nWn(rq\nn;jÑuj)Ñu\u0001Ñl\u0013\n(v)\n=Z\nWn(rq\nn;jÑuj)Ñv\u0001Ñl+Z\nWd\ndun(rq\nn;jÑuj)(v)Ñu\u0001Ñl\n=Z\nWn(rq\nn;jÑuj)Ñv\u0001Ñl+Z\nWfn(rq\nn)ˆn0(jÑuj)\njÑuj(Ñu\u0001Ñv)(Ñu\u0001Ñl)\nd\nduT(u)(v) =2Lzn0\nrs\u0000rrZ\nSQÑu\u0001Ñv dS:\n(3.24)\nSimilarly to (3.18), we introduce the adjoint vector\nl:= (l0;lp\n12;lp\n6;lp\n4): (3.25)\nHence, the sensitivity associated with (3.21) amounts to\n¶L\n¶X(X;u;l) =å\nq2f0;p\n12;p\n6;p\n4g¶Kq\n¶X(X;uq;lq)\n+¶y(hv;f(X);gf;m)\n¶X+¶y(hv;mag(X);gmag;m)\n¶X\n(3.26)\nwhere we used that¶T\n¶X=0 and¶j\n¶X=0.\n3.4 Update method\nSeveral methods exist to consider the movement in elec-\ntrical motors, such as the Moving Band (MB) technique\ndescribed in [8]. Even when using high order elements in\nthe MB, this method remains less accurate than the mortar\nelement method [4]. A variant of the mortar method, the\nNitsche method, is chosen to take into account the rota-\ntion [16]. The operator Kqwill be replaced by KNM\nqandL\nby ˜Lto fit the new formulation (detailed in Appendix.A).\nWe introduce a triangular mesh with a total of N\nelements, with Nrotmany elements inside the rotor do-\nmainWrot. We use piecewise linear and globally contin-\nuous finite element basis functions to solve the state and\nadjoint equations and we represent the density variables\nrn;rMx;rMyas piecewise constant functions on the mesh\ncorresponding to Wrot. Thus, these density variables can be\nrepresented by a vector of dimension Nrotconsisting of the\nvalues of the discrete functions in each element. We willuse the same notation rn;rMx;rMyfor the vectors repre-\nsenting the discretized density variables.\nTo comply with the bounds of the density variables, we\nintroduce the projected gradient as defined in [27, p. 520]\nG:=h\nPrn;[0;1]\u0010\n¶˜L\n¶rn\u0011\nPrMx;[0;1]\u0010\n¶˜L\n¶rMx\u0011\nPrMy;[0;1]\u0010\n¶˜L\n¶rMy\u0011i>\n;\n(3.27)\nwith the projection operator Pr;[a;b]:RNrot!RNrotdefined\nby\n\u0000\nPr;[a;b](X)\u0001\ni=8\n><\n>:Xi ifri2]a;b[;\nmin(0;Xi)ifri=a;\nmax(0;Xi)ifri=b;(3.28)\nfori=1;:::;Nrot. This in itself allows for finer geometry to\nemerge by amplifying the relative importance of the gradi-\nent where change in the density function can be made.\nFrom the sensitivity we derive the update equation at iter-\nation n\nXn+1\ni=Q[0;1]\u0012\nXn\ni\u0000sGi\njGij\u0013\n;fori=1;:::;Nrot(3.29)\nwith the projection operator Q[a;b]:R3Nrot!R3Nrotdefined\nby\n\u0000\nQ[a;b](v)\u0001\ni=max(a;min(b;vi)) (3.30)\nfor a vector v= (v1;:::; vNrot)>2R3Nrotandi=1;:::; Nrot,\nto enforce the bounds of the density variables. Here, sde-\nnotes the step size which is chosen in such a way that a\ndescent of the augmented Lagrangian is obtained,\nL(Xn+1;un+1)<L(Xn;un) (3.31)\nwith unis the vector of states for the design represented by\nXn.\n3.5 Filtering and projection method\nIn density-based topology optimization, checkerboard pat-\nterns and small isolated elements of one material are\navoided using filtering methods at each step of gradient de-\nscent. While this filtering has a regularizing effect on the\ndensity variables, it may introduce more gray areas. There-\nfore, in the next step, so-called projection methods are ap-\nplied in order to get a more defined shape. This combina-\ntion allows achieving smoother and more defined bound-\naries between the materials in the final design.\nIn our approach, in the first step we perform density\nfiltering by solving the PDE\n\u0000r2Ñ2r+r=rre f r>0: (3.32)\n8Figure 9: Comparison of projection curves for a varying b\nparameter.\nwithrre fthe given density function, which is commonly\nreferred to as Helmholtz filtering [19]. This mesh-\nindependency filter modifies the sensitivity by averaging\non the neighbor cells. The parameter ris the radius of\ninfluence of the filtering and is defined in our case as a\nfactor dof the minimum representative mesh-element\nlength h, i.e. r=dh.\nFor the projection step, we choose the function pro-\nposed in [35],\nfrcut;b(r) =tanh(b(r\u0000rcut))+tanh(brcut)\ntanh(b(1\u0000rcut))+tanh(brcut)\nrcut2[0;1];b>0(3.33)\nwithrcut=0:5 such as not to favor one material. The other\nparameters dandbare chosen to preserve the equilibrium\nbetween the two parts of the filtering step.\nWe applied the filtering technique (3.32) and the pro-\njection technique (3.33) for the material densities. The\nsame PDE-based filter (3.32) is applied for the densities\nrMxandrMy, to favor uniform magnetization direction in a\nmagnet area while the projection (3.33) is only applied on\nj~Mjto avoid scaling issues.\n3.6 Direct penalization of intermediate ma-\nterials\nSome final designs can still present fuzzy boundaries and\nintermediate material, especially if the optimization start-\ning point is near a local minimum. To help overcome this\nissue, we propose to penalize the intermediate materials\ndirectly as done in the phase-field topology optimization\nmethod [13] and add to the cost function the followingterm with a weight g>0:\nIg(r) =4g\nVWrotZ\nWrotr(x)(1\u0000r(x))dx: (3.34)\nThe penalization is only applied on iron density rnand the\nmagnetization norm j~Mj.\n3.7 Post-processing: K-mean heuristic\nIn our optimization problem we look for permanent mag-\nnetization directions which may change continuously in\nspace. In order to obtain designs which comply with feasi-\nbility constraints, we here propose a post-processing step.\nA K-mean heuristic [22] clustering method is applied.\nHere, we suggest adapting it to create clusters of elements\nof similar magnetization direction.\nLet us define a point P= (px;py;pbM)by its coordinates\npx,pyin the 2D plane and its magnetization angle pbM. We\ndefine a set of kpoints C1;:::Ckwhere Cj= (cx;j;cy;j;\u0000),\nwhich we will refer to as centroids, and which are first ran-\ndomly sampled in the 2D plane.\nLetP1;:::; PNrotbe the centroids of the triangles in the ro-\ntor domain Wrot. At each step of the algorithm, we asso-\nciate each point Piwith the closest centroid Cj2Skusing\na modified 3D Euclidean distance da. For two such points\nPi= (px;i;py;i;pbM;i)andCj= (cx;j;cy;j;cbM;j), this modi-\nfied distance function is defined as\nda:R3\u0002R3!R\n(Pi;Cj)7!s\u0012px;i\u0000cx;j\nNx\u00132\n+\u0012py;i\u0000cy;j\nNy\u00132\n+a\u0012pbM;i\u0000cbM;j\n2p\u00132\n(3.35)\nwhere Nx,Ny,aare three weighting constants which can\nbe used to tune the method.\nBy this procedure, we get kpoint clusters S1;:::; Sk\nwith Sj=fPi:da(Pi;Cj)<da(Pi;Cl)for all l6=jgfor\nj=1;:::; k. Thus, all points in cluster Sjare closer to point\nCjin terms of dathan to any other centroid Cl.\nThe positions of the centroids C1;:::; Ckare then up-\ndated according to the mean coordinates of the points as-\nsociated with them,\nCj (mx;j;my;j;mbM;j)where mx;j=1\n#Sjå\nPi2Sjpx;i(3.36)\nandmy;j;mbM;jdefined analogously. Here, # Sjdenotes the\ncardinality of Sj. In the first iteration where cbM;jis not\ndefined, we use the modified 2D Euclidean distance da=0.\n4 Application to the magnetostatics\nproblem\nAll computations were conducted using the NGSolve [12,\n32] framework with its python interface. Three sets of re-\nsults are presented in this section to validate the proposed\n9strategy. For the different optimization sets, the starting\npoints are given in Table 6.\nThe first design in Table 6 is used for the Iron-Air opti-\nmization. The results for the material distribution are given\nin Table 7 for different volume fractions fn;f. These re-\nsults are coherent with the literature of synchro-reluctant\nactuators with distributed winding [34], with a phase an-\nglej=5p\n6and one pair of poles. This phase angle cor-\nresponds to the maximum torque of the machine. Results\ncan also be compared with results found in [11, 25] which\nare validated by means of JMAG .\nFor the multi-material topology optimization including\nmagnets, the final design obtained at 40% of ferromagnetic\nmaterial is used as a starting point. Results for magnets\ndistribution are presented in Table 8. As expected, mag-\nnets are distributed on the air barriers domain. Several\nconstraints on magnets volume were chosen. One should\nnote that the ferromagnetic distribution is modified at the\nouter radius of the rotor. Without magnets, the reluctance\ntorque is equal to 1 :0768 [N.m]. Optimized torque and\npost-processed values after k-mean clustering are given,\nthe number of clusters is fixed to k=5.\nIn order to validate the proposed strategy, a new phase an-\ngle is chosen. It is equal to j=3p\n32. The objective is to find\nthe optimal materials distribution to maximize the torque\nat this phase angle. An unbiased starting point is chosen,\nwhere homogeneous grey materials are set in the rotor\nsuch as rn=0:5;rMx=0:5;rMy=0:5. The optimal rotor\nstructure is given in Table 9. Again, we present optimized\ntorques as well as the torque values after post-processing,\nwhere we used k=5 clusters for the K-mean clustering.\nResults on the torque value are comparable for the previ-\nously obtained but at higher magnets volume, which is also\ncoherent with literature. On the other hand, the first opti-\nmal multi-material model has larger reluctant torque due\nto optimal current supply in the q-axis.5 Conclusion and outlooks\nIn this study we proposed a novel multi-material interpo-\nlation method to determine the optimal distribution of air,\niron, and magnets for PMSynRM. The interpolation takes\ninto account magnetization amplitudes and direction, and\na post-processing clustering method is also suggested to\nhomogenize magnets direction for feasibility constraints.\nThis study also investigated the use of the four statics posi-\ntions method to reduce global computation time. With fur-\nther work, an exhaustive comparison with other interpola-\ntion schemes will be considered, and the extension to mul-\ntiobjective optimization under gradient-based methods.\nA Nitsche-mortaring reformulation\nSeveral methods exist to consider the movement in elec-\ntrical motors, such as the Moving Band (MB) technique\ndescribed in [8], even when using high order elements in\nthe MB, this method remains less accurate than the mortar\nelement method [4]. A variant of the mortar method, the\nNitsche method, is chosen to take into account the rota-\ntion [16]. Let us define the adapted operator KNM\nqto fit the\nnew formulation\nKNM\nq(X;u;h;v;m) =å\ni2frot;statg\u0012Z\nWin(rq\nn;jÑuij)Ñui\u0001Ñvi\u0013\n\u0000Z\nWrotfn(1\u0000rq\nn)MmaxfM(j~Mj)\nj~MjRq\u0014\u0000Mq\ny\nMq\nx\u0015\n\u0001Ñvrot\n+å\ni2frot;statg\u0012\n\u0000Z\n¶Win0¶ui\n¶n(vi\u0000m)\u0000Z\n¶Win0¶vi\n¶n(ui\u0000h))\n+ap2\nhZ\n¶Wi(ui\u0000h)(vi\u0000m)!\n;\n(A.1)\nwhere (u;h;v;m)2V\u0002W\u0002V\u0002WandV;Ware defined\nas\n(\nV=fv= (vrot;vstat)2H1(Wrot)\u0002H1(Wstat);v=0 on¶Wg;\nW=fw2L2(¶Wrot\\¶Wstat)g:\n(A.2)\nHere, a>0 is a stabilization parameter which we chose\nasa=160, p=1 denotes the polynomial degree of the\nfinite element discretization and hthe diameter of the\nlargest element of the mesh. Moreover, recall the implicit\ndependence of ~M= (Mq\nx;Mq\ny)>on the density variables\nrMqx;rMqy(3.5), (3.14).\nHence the state equation (3.6) can be formulated as\nFind(uq;hq)2V\u0002W;KNM\nq(Xr;uq;hq;v;m) =Z\nWstatj(q)vdx for all (v;m)2V\u0002W:\n(A.3)\nIn a similar manner, the adjoint equation (3.22) can be re-\n10Table 6: Starting designs\nStarting Design\nRotor\nTable 7: Designs Iron-Air\nFinal filtered design\nRotor\nfn;f 10 % 20 % 40 %\n¯T[N:m] 0.20897 0.83351 1.1129\n11Table 8: Designs Magnet-Air-Iron\nFinal filtered designs\nRotor\nfn;mag7.5 % 15 % 30 %\n¯T[N:m] 1.2830 1.4729 1.9000\n¯TKmeans [N:m] 1.2646 1.4469 1.8671\nTable 9: Designs Iron-Air-Magnets\nFinal Design\nRotor\nfv;mag 10% 20% Not bounded\n¯T[N:m] 1.2710 1.4513 2.0484\n¯TKmeans [N:m] 1.1097 1.3863 1.6422\n12defined as\nFind(wq;mq)2V\u0002W:¶KNM\nq\n¶(u;h)(X;uq;hq;wq;mq)(ˆu;ˆh) =\u00001\n4¶T\n¶u(uq)(ˆu)\n(A.4)\nfor all (ˆu;ˆh)2V\u0002Wfor each q2f0;p\n12;p\n6;p\n4g.\nThe corresponding Lagrangian reads\n˜L(X;u;h;v;m):=L(X;u)+å\nq2f0;p\n12;p\n6;p\n4gKNM\nq(X;uq;hq;vq;mq)\u0000Z\nWstatj(q)vq;\n(A.5)\nand its gradient is given by\n¶˜L\n¶X(X;u;h;v;m) =0\n@å\nq2f0;p\n12;p\n6;p\n4g¶KNM\nq\n¶X(X;uq;hq;vq;mq)\u00001\n4¶Tq\n¶X1\nA\n+¶y(hv;f(X);gf;m)\n¶X+¶y(hv;mag(X);gmag;m)\n¶X:\n(A.6)\nReferences\n[1] P. 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Struct\nMultidisc Optim , 55(2):477–491, 2 2017.\n14"    },    {        "title": "2107.06883v1.Evolving_Devil_s_staircase_magnetization_from_tunable_charge_density_waves_in_nonsymmorphic_Dirac_semimetals.pdf",        "content": "Evolving Devil's staircase magnetization from tunable charge density waves in\nnonsymmorphic Dirac semimetals\nRatnadwip Singha,1Tyger H. Salters,1Samuel M. L. Teicher,2Shiming\nLei,1Jason F. Khoury,1N. Phuan Ong,3and Leslie M. Schoop1,\u0003\n1Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA\n2Materials Department and Materials Research Laboratory,\nUniversity of California, Santa Barbara, California 93106, USA\n3Department of Physics, Princeton University, Princeton, New Jersey 08544, USA\nWhile several magnetic topological semimetals have been discovered in recent years, their band\nstructures are far from ideal, often obscured by trivial bands at the Fermi energy. Square-net\nmaterials with clean, linearly dispersing bands show potential to circumvent this issue. CeSbTe,\na square-net material, features multiple magnetic \feld-controllable topological phases. Here, it is\nshown that in this material, even higher degrees of tunability can be achieved by changing the\nelectron count at the square-net motif. Increased electron \flling results in structural distortion\nand formation of charge density waves (CDWs). The modulation wave-vector evolves continuously\nleading to a region of multiple discrete CDWs and a corresponding complex \\Devil's staircase\"\nmagnetic ground state. A series of fractionally quantized magnetization plateaus are observed,\nwhich implies direct coupling between CDW and a collective spin-excitation. It is further shown\nthat the CDW creates a robust idealized non-symmorphic Dirac semimetal, thus providing access\nto topological systems with rich magnetism.\nI. INTRODUCTION\nThe quest for \fnding novel quantum phases drives the\ndiscovery of new materials and phenomena in condensed\nmatter physics. The introduction of topological band\nstructure in material classi\fcation, is perhaps the most\nnotable example in the last decade. Starting from\nthe identi\fcation of the topological insulating state\nin a handful of materials [1{4], this \feld has been\nenriched by subsequent realization of a plethora of\nunconventional quantum states such as Dirac [5, 6],\nWeyl [7{9], nodal-line semimetals [10, 11], and topolog-\nical superconductors [12]. While the idea of exploring\nthe dynamics of relativistic particles in low-energy\nelectronic systems is already quite enticing, the prospect\nof probing new quasiparticle excitations beyond the\nrealm of particle physics [13] is even more attractive.\nMotivated by these predictions, researchers have utilized\ndata mining to identify and categorize a huge number\nof compounds hosting di\u000berent types of non-trivial\ntopological band structures [14, 15]. To realize new\ntopological phases, another e\u000bective route is to introduce\nadditional tuning parameters in these materials such\nas strong electron correlations [16, 17] and magnetism.\nIncorporation of magnetism in topological systems is\nparticularly interesting as it can break the inherent time-\nreversal symmetry (TRS) of the crystal structure. TRS\nbroken topological semimetals are extremely rare and\nhave been the major focus of a number of studies [18{21].\nContrary to the approach of scanning through mate-\nrial databases, it has been shown that simple structural\n\u0003lschoop@princeton.edumotifs can provide strong indications of topological\nnon-trivial states in a compound [22]. A great example\nof this are square-net materials. In 2015, Young and\nKane proposed nonsymmorphic symmetry protected\nDirac semimetal states in a two-dimensional (2D)\nsquare-net lattice [23]. In recent times, a vast array of\ntopological semimetals have been discovered, all hosting\nthe 2D square-net motif in their crystal structure [24].\nProbably the most extensively studied member of this\nfamily is ZrSiS, which shows the largest reported energy\nrange (\u00182 eV) of linearly dispersive bands [11]. It\nhosts multiple Dirac nodes near the Fermi energy, which\nform a diamond shaped nodal-line in momentum space\n[11, 25, 26]. In addition, both far below and above\nthe Fermi energy, there are nonsymmorphic symmetry\nprotected Dirac nodes, which, in contrast to other Dirac\ncones, remain gapless even under spin-orbit coupling\n[11]. Thus, ZrSiS presents an unique platform to explore\ndi\u000berent types of topological physics. CeSbTe is an\nisostructural compound, which in addition to having a\nrich band structure similar to ZrSiS, introduces mag-\nnetism as an additional controllable parameter [27]. In\nfact, it has been reported that this material can be driven\nthrough distinct topological phases while changing the\nmagnetic ordering by modulating the strength of an\napplied magnetic \feld, resulting in a tunable topological\nDirac/Weyl semimetal [27]. Therefore, CeSbTe is an\nideal template to investigate the interplay of magnetism\nand di\u000berent topological phases, accessible via Fermi\nlevel engineering by electron \flling [28].\nIt is well established that square-net lattices are\ninherently unstable and only preserved by delocalized\nelectrons as in graphene [29]. Changing the electron\ncount in the square-net, therefore, induces distortion in\nthe crystal structure [29, 30]. Such a distorted latticearXiv:2107.06883v1  [cond-mat.mtrl-sci]  14 Jul 20212\nsupports the formation of charge density waves (CDWs),\nwhich open a band gap at the Fermi level. In a few rare\nearth antimony tellurides ( LnSbTe;Ln=lanthanides),\nthe presence of CDWs has been con\frmed when there is\na deviation from nominal stoichiometry. For example,\nin LaSbxTe2\u0000x, the CDW has been observed to evolve\ncontinuously with antimony substitution [31]. On the\nother hand, in GdSb xTe2\u0000x, the CDW is shown to have\nan important role in designing new Dirac semimetals [1].\nRecently, the signature of a \fve-fold CDW modulation\nhas also been reported in an o\u000b-stoichiometric single\ncrystal of CeSbTe [33]. However, the detailed evolution\nof this phase and its impact on the topological band\nstructure remains unexplored. Such an investigation is\nimportant to reveal the true potential of doped LnSbTe,\nin respect to their magnetic, structural, and topological\nproperties as well as their interplay.\nIn this report, we combine structural and magnetic\nmeasurements with theoretical calculations to probe the\ninterplay of magnetic ordering, CDW, and topological\nband structure in single crystalline CeSb xTe2\u0000x\u0000\u000e(\u000e\nrepresents the vacancy concentration in the crystal).\nWe show that there are two distinct regimes in this\nmaterial with formation of either one or multiple CDW\norderings. Moreover, it is possible to continuously tune\nthe modulation wave-vector by controlling the electron\n\flling at the square-net position. The corresponding\nunique distortions of the square-net motif lead to modi-\n\fcations in the electronic band structure. While CeSbTe\nhas a simple antiferromagnetic type-A ground state, a\ncomplex magnetic structure starts to emerge with chem-\nical substitution. In particular, for a certain electron\n\flling range, fractionally quantized magnetization in the\nform of a \\Devil's staircase\" is observed. We explain\nthe origin of this phenomenon as the consequence of\ndirect coupling between the CDW and a collective\nspin excitation. From \frst-principle calculations, we\ncon\frm that electron \flling and the CDW formation\nresult in an idealized magnetic Dirac semimetal with a\nnonsymmorphic, symmetry-protected Dirac node at the\nFermi energy. This leads to the opportunity to study\nthe interplay of complex magnetism with clean Dirac\nnodes in topological systems.\nII. RESULTS AND DISCUSSION\nA. Evolution of the crystal structure and the\ncharge density wave\nPowder x-ray di\u000braction (XRD) spectra along with\nLeBail \ftting for the crushed single crystals with dif-\nferent compositions are shown in Figure S1 a. Within\nthe experimental resolution, no secondary phases are\ndetected. Figure 1 illustrates the extracted lattice\nparameters as functions of Sb content at the square-net.\nFIG. 1. Powder x-ray di\u000braction measurements on\nCeSbxTe2\u0000x\u0000\u000ecrystals. Lattice constants extracted from\npowder x-ray di\u000braction measured on ground single crystals\nas a function of Sb-content. The dashed black vertical line\ncorresponds to the phase boundary for the transition from an\northorhombic to a tetragonal structure. The color gradient\nrepresents the smooth evolution of the parameters across the\nboundary. The red dashed vertical line represents a transition\nfrom multiple CDW q-vectors to one q-vector region.\nWe could not prepare any CeSb xTe2\u0000x\u0000\u000ecrystal with\nx >0.91. Therefore, the parameters for CeSbTe are\nobtained from an earlier report [27]. Several other\nparameters from Ref. [27] are also used throughout\nthis work in order to put the results in context. CeS-\nbTe crystallizes in the ZrSiS-type tetragonal structure\n(spacegroup P4=nmm ). Our data for CeSb 0:91Te0:91are\nsimilar to those for ideal stoichiometric CeSbTe [27].\nWith decrease in the Sb content, the distorted crystal\nstructure is better described using the orthorhombic\nspacegroup Pmmn . This also explains the orthorhombic\nstructure observed by Wang etal. [34] and Lv etal. [35]\nfor \\CeSbTe\" crystals, which had a signi\fcant vacancy at\nthe Sb-site. From our analysis, we identi\fed a boundary\naroundx=0.70 for the tetragonal to orthorhombic phase\ntransition. We note that the orthorhombic structure has\nalso been reported in few other members of the LnSbTe-\nfamily [31, 36, 37]. The lattice parameters undergo a\nsmooth transition across this phase boundary. Two\nlocal maxima at x=0.91 and 0.34 are observed for both\nlattice constants candb, respectively. While the peak\nincmight indicate a deviation from ideal stoichiometry\nfor CeSbTe crystals in Ref. [27], the maximum in b\nrepresents a phase transition from multiple to one CDW\nmodulation vectors, as we will show later. Signatures of\nsatellite peaks in the XRD spectra, which indicate the\npresence of the CDW, can be seen in all orthorhombic\nsamples.\nThe structural evolution of the CDW in CeSb xTe2\u0000x\u0000\u000e\nis investigated by single crystal XRD. The presence of a\nCDW is con\frmed by the weak intensity satellite peaks3\nFIG. 2. Single crystal x-ray di\u000braction of CeSb xTe2\u0000x\u0000\u000e. a) Precession x-ray di\u000braction image in the h0lplane of a\nCeSb 0:11Te1:90single crystal. Enlarged region shows the superlattice re\rections generated by three charge density wave (CDW)\nmodulation vectors. The modulated crystal structure is also shown. b) Distorted square-net motif for compounds with di\u000berent\nelectron \fllings.\nas observed in precession images ( Figure 2 a and S1b).\nThe solved crystal structures for all the compositions are\nfound to be either commensurately modulated or nearly\ncommensurately modulated (Figure 2a and S1b). In the\nintermediate Sb-composition range, a single modulation\nwavevector qis observed in the b-direction (the longer\naxis of theab-plane) of the parent structure and it corre-\nsponds to a three-fold expansion in CeSb 0:34Te1:67and a\n\fve-fold expansion in the CeSb 0:51Te1:40unit cell. More\ncomplex behavior emerges in the low Sb-composition\nrange. CeSb 0:11Te1:90exhibits three di\u000berent q-vectors\n(one within the ab-plane,q1=1/3 b\u0003; and two having\ncomponent along the c-axis,q2=1/3 a\u0003+1/3 b\u0003+1/2 c\u0003\nandq3=1/3 a\u0003+1/3 b\u0003-1/2c\u0003, where a\u0003,b\u0003, and c\u0003are\nthe reciprocal lattice vectors) corresponding to multiple\ndistinct CDW orderings, which result in an overall\n3\u00023\u00022 expansion of the parent cell. This continuously\nevolving CDW leads to signi\fcant distortions of the\nSb/Te square-net, yielding a diversity of bonding\nmotifs at the hypervalent square-net (Figure 2b) as\nSb-composition corresponds to further a localization\nof bonding in the structure. Considering a maximum\nbond distance of 3.1 \u0017A in the undistorted square-net,\nthe Sb/Te square-net distorts into patterns containing\nzig-zag chains and isolated atoms in CeSb 0:51Te1:40,\nfused 4-member rings in CeSb 0:34Te1:67, and \fsh-like\nfunctionalized 4-member rings with isolated atoms in\nCeSb 0:11Te1:90. As evident from Figure S1b, no CDW\nordering is observed in tetragonal CeSb 0:79Te1:05. The\nresults of the structural solution for all compositions are\nsummarized in Table S1 .\nFIG. 3. Magnetization measurements of CeSb xTe2\u0000x\u0000\u000esin-\ngle crystals. a) Temperature dependent susceptibility ( \u001f)\nfor CeSb 0:91Te0:91at di\u000berent magnetic \felds applied along\nthe crystallographic c-axis. Inset shows the zero-\feld-cooled\n(ZFC) and \feld-cooled (FC) magnetization curves at 100 Oe.\nb) Phase diagram of CeSb 0:91Te0:91, constructed from the\nmagnetization measurements. c) Evolution of the antifer-\nromagnetic N\u0013 eel temperature ( TN) with Sb-content. Inset\nshows a typical plot of the \frst order derivative of \u001f, which is\nused to extract TN. d) Doping dependence of the Curie-Weiss\ntemperature.4\nB. Magnetic phase diagrams\nInFigure 3 a, we have plotted the temperature de-\npendence of the magnetic susceptibility ( \u001f) of tetragonal\n(unmodulated) CeSb 0:91Te0:91for di\u000berent magnetic\n\feld strengths, applied along the crystallographic c-axis.\nIn general the results are very similar to those observed\nin CeSbTe [27], showing that within the tetragonal re-\ngion, the magnetic properties do not change drastically\nwith doping. Just like CeSbTe,\nCeSb 0:91Te0:91is antiferromagnetic at low applied \felds\nwith N\u0013 eel temperature ( TN)\u00182.73 K. From the powder\nneutron di\u000braction measurements, the exact magnetic\nstructure of CeSbTe was found to be type-A AFM,\nwhere moments of Ce3+ions are aligned parallel to\neach other within a layer along the ab-plane and these\nlayers are stacked along the c-axis with AFM coupling\nbetween two consecutive layers [27]. With an applied\nmagnetic \feld along the c-axis, the moments of all the\nCe-layers start to align parallel to the \feld, resulting\nin susceptibility curves similar to a ferromagnetic\n(completely spin-polarized) state. From the transition\ntemperature and \feld, we have constructed a phase\ndiagram for CeSb 0:91Te0:91in Figure 3b, showing three\ndistinct magnetization regions - AFM, completely spin\npolarized [ferromagnetic (FM)], and paramagnetic (PM).\nIn CeSbTe, a group theory analysis revealed that it is\npossible to realize di\u000berent types of topological band\nstructures including Dirac (both type I and II) and Weyl\nnodes as well as higher order band crossings (threefold\nand eightfold degenerate), depending on the nature and\norientation of the magnetic ordering [27]. Therefore,\nby controlling the temperature and magnetic \feld, it\nis possible to tune the electronic band structure of this\nmaterial. Here, we are introducing the electron \flling at\nthe square-net as an additional tuning parameter. Figure\n3c illustrates the evolution of TNwith Sb-content. The\ninset shows the \frst order derivative of the \u001f(T) curve,\nused for extracting TN, for a representative crystal. We\nobserve that TNincreases signi\fcantly with decreasing\nSb-composition, con\frming that the AFM state becomes\nmore stable. For x=0.11,TNis found to be 4.27 K,\nwhich also agrees well with the transition temperature\n(4.4 K) of CeTe 2[38], the other terminal compound.\nTo extract the Curie-Weiss temperature ( \u0012CW), we\n\ft the inverse susceptibility \u001f\u00001(T) curve in the PM\nregion using Curie-Weiss law [ \u001f=NA\u00162\neff\n3kB(T\u0000\u0012CW)] for some\nrepresentative compositions ( Figure S2 ). While\u0012CW\nis negative for higher Sb-compositions as expected for\nAFM ordering, it changes sign below x\u00180.50 indicating\nsome FM component [Figure 3d]. This might not be\nunexpected as we note that the magnetization in CeTe 2\nhas been reported to be quite complex. There are\ncon\ricting reports of a ferrimagnetic ground state with\npositive\u0012CW [39, 40] as well as a long range AFM\nground state coexisting with a short range FM ordering\n[38]. From Curie-Weiss \ftting for all the compositions,the e\u000bective moment ( \u0016eff) is calculated to be close to\nthe moment (2.54 \u0016B) of the free Ce3+ion. Nevertheless,\nsome deviation from the theoretical value is observed,\nwhich is possibly due to the strong crystal electric \feld\ne\u000bect [41].\nTo get the complete picture, we have constructed\nmagnetic phase diagrams from the \frst order derivative\nof\u001f(T) curves with the \feld parallel to the c-axis for\nthe entire electron \flling range ( Figure 4 a-c and S3a).\nFrom the results, it is clear that there are three distinct\nregimes, represented by three compositions in Figure\n4a-c. Forx=0.11 (orthorhombic with three q-vectors),\nin addition to the AFM and FM states that are similar\nto the ones in tetragonal CeSbTe, we observe two new\nphases, namely `Phase I' and `Phase II'. Among these,\nPhase II is a \feld-induced state, whereas the boundary\nfor Phase I extends towards lower \felds and becomes\nindistinguishable from the AFM phase boundary. To\nidentify the true magnetic ground state, we have per-\nformed a zero-\feld heat capacity ( CP) measurement on a\nCeSb 0:11Te1:90crystal ( Figure S4 a). From the enlarged\nview of the low-temperature region in Figure S4b, we\ncan clearly see an additional peak in CPadjacent to the\nAFM transition, con\frming that Phase I coexists even\nat zero-\feld. Upon application of a magnetic \feld, this\nsecond peak becomes more prominent as also evident\nfrom the emergence of an explicit boundary with \feld\nin the magnetic phase diagram. Both of these peaks\ninCPare suppressed completely at 0.5 T, indicating a\nfully spin-polarized state. With increasing Sb-content\natx=0.34 (orthorhombic with three-fold modulation\nalong theb-axis), Phase II disappears, whereas Phase\nI still remains. For tetragonal CeSb 0:74Te1:25, however,\nthe phase diagram becomes much simpler and closely\nrelated to CeSbTe [27]. In Figure 4d, we have plotted\nthe doping dependent phase diagram. From this \fgure,\nwe can correlate the boundaries for both new magnetic\nphases with the structural transitions. Phase II only\nappears in presence of multiple CDW orderings, whereas\nPhase I only exists in orthorhombic structures. The\npreviously published magnetization data for CeTe 2sug-\ngests that Phase I probably corresponds to short range\nFM ordering [38]. On the other hand, Phase II repre-\nsents a more complex magnetic state as we discuss below.\nContrary to the complex magnetic structure for\n\feld along the c-axis, the phase diagrams for all the\ncompositions turn out to be quite simple, when the\nmagnetic \feld is applied along the ab-plane (Figure\nS3b). In this con\fguration, only three states have been\nobserved (AFM, FM, and PM), similar to the c-axis\nphase diagram of CeSbTe [27].\nFigure 5 a shows the magnetization ( M) as a function\nof magnetic \feld ( Hkc-axis) for tetragonal\nCeSb 0:91Te0:91at di\u000berent temperatures both below and\naboveTN. To compare, the M(H) curve at 2 K for H ?5\nFIG. 4. Magnetic phase diagrams of CeSb xTe2\u0000x\u0000\u000e. a)-c) Magnetic phase diagram for three di\u000berent Sb-content samples, show-\ning three distinct magnetization regimes for CeSb xTe2\u0000x\u0000\u000e. The discrete points and red curve, obtained from the magnetization\ndata, represent the boundary between FM and PM states. d) Doping dependent phase diagram.\nFIG. 5. Field dependence of magnetization for\nCeSb 0:91Te0:91. a) Magnetization ( M) at di\u000berent tem-\nperatures as a function of \feld ( H), applied along the c-axis.\nFor comparison, the M(H) curve at 2 K for H ?c-axis is\nalso plotted. b) The low-\feld region of the M(H) curves,\nshowing `spin-\rip' (upper panel) and `spin-\rop' (lower panel)\ntransitions for two di\u000berent applied \feld directions. The\narrows illustrate the spin arrangement for two consecutive\nCe-layers.\nc-axis is also plotted in the same graph. The material\nreaches the FM state at lower \feld, when H kc-axis,\ncon\frming that it is the easy axis of magnetization.\nSimilar to in CeSbTe, we resolve the `spin-\rip' (H k\nc-axis) and `spin-\rop' (H ?c-axis) transitions [Figure\n5b: arrows illustrate the spin arrangements between two\nconsecutive Ce-layers], as well as strong magnetocrys-\ntalline anisotropy.\nInFigure S5 a, we have compared the M(H) curves of\nall Sb-compositions for two measurement con\fgurations.It is evident that the magnetocrystalline anisotropy\nchanges systematically with the electron \flling, whereas\nthe direction of spin-\rip and spin-\rop transitions remain\nunaltered, indicating that the spins are still aligned\nalong thec-axis. Forx=0.11, the nature of anisotropy is\ncompletely opposite to that for x=0.91. This suggests\nthat the electron \flling at the Sb square-net position\ncontinuously tunes the AFM exchange interaction\nbetween two consecutive Ce-layers (nearest neighbors).\nThis is indeed expected, as the square-net motif in\nCeSbxTe2\u0000x\u0000\u000eprovides the conduction electrons, re-\nquired for Ruderman-Kittel-Kasuya-Yosida (RKKY)\ninteraction between localized Ce3+moments at two\ndi\u000berent layers. We \fnd that the anisotropy reverses at\na composition close to x=0.50, where the Curie-Weiss\ntemperature also changes its sign indicating predomi-\nnantly the AFM ground state for x\u00150.50. It is possible\nthat below this critical Sb-content, with higher electron\n\flling, the FM interaction between two next nearest\nneighbor layers starts to dominate and contributes to a\npositive Curie-Weiss temperature.\nC. Fractionally quantized magnetization plateaus\nAn even more surprising magnetic structure appears\nat high electron \flling. Figure S5b shows the low-\feld\nregion (-0.5 T\u0014H\u00140.5 T) of the M(H) curves for\nsamples with x\u00140.51, when Hkc-axis. For x=0.11\nand 0.20 (multiple q-vector region), we observe cascades\nof metamagnetic transitions, represented by a series\nof plateaus and steps in magnetization, leading up to\nthe spin-\rip transition. In addition, there is prominent6\nFIG. 6. Devil's staircase in magnetization of CeSb 0:11Te1:90. a) Normalized magnetization as a function of \feld, revealing\nfractionally quantized plateaus. b) Magnetic phase diagram for plateaus corresponding to di\u000berent quantum numbers. c)\nSchematic showing the magnetic \feld-tunable spin wave, responsible for devil's staircase structure.\nhysteresis between the increasing and decreasing-\feld\nmeasurements, con\frming the presence of a FM com-\nponent. The metamagnetic transitions signi\fcantly\nweaken and then completely disappear at x\u00150.34. In\nFigure 6 a, we have plotted the M(H) curve at 1.8 K\nfor CeSb 0:11Te1:90, normalized by the saturation magne-\ntization (MS). Remarkably, the plateaus are found to\nbe locked to rational fractions identical to the quantized\nHall resistivity in the fractional quantum Hall e\u000bect\n(FQHE). Quantized magnetization plateaus have been\npreviously observed in low-dimensional magnets [42, 43]\nand magnetically frustrated systems [44, 45], forming a\nstructure called \\Devil's staircase\" as each step consists\nof in\fnite number of steps under magni\fcation, and\nso forth [46]. We note that such complex magnetic\nstructure has also been reported in CeSbSe with a\ntetragonal structure [47]. Analogous to the FQHE, a\ndevil's staircase originates from an energy gap in the\nmany body excitation spectra in a compound due to the\ntranslational symmetry breaking [48{51]. The plateau\nappears when a commensurability condition between\nlattice wave vector and localized excitation is satis\fed.\nFor quantum spin systems, this commensurability con-\ndition is found to be n(S-m)=integer, where n,S, and\nmare number of spins in the unit cell, the magnitude\nof spin, and magnetization per site, respectively [48]. In\nprinciple, this staircase structure can be observed fordi\u000berent physical properties in presence of two coupled\nwaves [46]. A continuous variation in frequency (wave-\nlength) of one wave then drives the frequency of other\nwave such that they go through regimes of phase-locked\n(plateau) and non-phase-locked (steps) states.\nBy repeating the magnetization measurements for\nCeSb 0:11Te1:90at several temperatures ( Figure S6 ), we\nhave tracked the evolution of the plateaus corresponding\nto di\u000berent fractions. Using these results, in Figure\n6b, a phase diagram is constructed. We conclude that\nthis staircase structure is a magnetic \feld induced state\nand only appears within the AFM phase. Interestingly,\nit is also sensitive to the crystallographic directions,\nas no metamagnetic transition is observed, when the\n\feld is applied perpendicular to the c-axis ( Figure\nS7). Moreover, the doping range ( x=0.11 and 0.20),\nwhere this complex state is observed also suggests that\nit corresponds to the Phase II in the magnetic phase\ndiagram. We note that only within this region, there\nare multiple CDW modulation wave-vectors, two with\ncomponents along the c-axis and the other one residing\nwithin the ab-plane. Below this electron \flling, the\nCDW wave-vectors along the c-axis disappear. All these\nfeatures help us to construct a microscopic picture of the\norigin of the devil's staircase structure in CeSb xTe2\u0000x\u0000\u000e\nas illustrated in Figure 6c.7\nThe AFM ordered ( \"#\"#) Ce3+-layers in\nCeSbxTe2\u0000x\u0000\u000eform a spin wave along the c-axis.\nWith application of a magnetic \feld perpendicular to\nthe layers, the down spins try to \rip in order to align\nwith the \feld. However, under a low enough \feld\nstrength, this transition does not occur at all down-spin\nlayers simultaneously. Instead, di\u000berent layers undergo\nspin-\rip transitions one after another, thus, e\u000bectively\ncreating a spin-wave with continuously \feld-tunable\nwavelengths. As the magnetic exchange interaction\nbetween these layers is mediated by conduction electrons\nthrough the RKKY interaction, the spin-wave is already\ncoupled to any modulation in the electron density along\nthec-axis. Moreover, the c-axis component of the two\nCDW wave-vectors is commensurate with the lattice,\nhence also with the spin-wave modulation-vector. So, by\ncontrolling the magnetic \feld, these coupled waves can\nbe driven to a series of sequential phase-locked and non-\nphase-locked states. Once all the spins are aligned along\nthe magnetic \feld (fully spin-polarized state), there is\nno longer a spin-wave structure and hence, no more\nsteps are observed. On the other hand, for x >0.2, the\nabsence of a q-vector along the c-axis causes the steps\nto disappear. We note that in cubic CeSb, a continuous\nmodulation of the spin-wave with both temperature and\n\feld, and its commensurability with lattice wave-vector,\nhave been con\frmed to be the origin of the devil's\nstaircase in M(T,H) [52, 53]. In particular, the \\1/3\nplateau\" is observed when the layers are arranged in a\n\"\"#con\fguration. Interestingly, two extremely weakly\ninteracting spin-excitations with one of them having\nanalogous properties to astrophysical \\dark matter\",\nhave recently been observed within the \\1/3 plateau\"\nof CeSb [54], showing the possibility of exploring new\nquantum states in the fractional magnetization plateaus.\nIn the case of CeSb xTe2\u0000x\u0000\u000e, we observe another\nsignature of the spin-wave in the magnetic entropy ( Sm),\ncalculated from the heat capacity data (Figure S4c). For\nx=0.51,Smreaches a saturation value Rln2, expected\nfor localized Ce3+moments, whereas it is smaller for\nx=0.11, indicating a gapped spin excitation spectrum\n[55, 56].\nD. Topological properties of the electronic band\nstructure\nNext, we investigate the topological nature of\nCeSbxTe2\u0000x\u0000\u000e. As mentioned above, CeSbTe has been\nshown to be of interest for several distinct topological\nsemimetal phases, which can be modulated by an applied\nmagnetic \feld. In fact, non-trivial topological states\nare either predicted or con\frmed in di\u000berent members\nof theLnSbTe-family [36, 57{59]. Recently, the in\ru-\nence of electron count on the topological band structure\nhas been studied in GdSb xTe2\u0000x\u0000\u000e, where it was shownthat an idealized nonsymmorphic Dirac semimetal can\nbe achieved in GdSb 0:46Te1:48[1]. So, it is of inter-\nest to also study the band structures of CeSb xTe2\u0000x\u0000\u000e\nwith respect on their topological nature. We have per-\nformed \frst-principle calculations for three di\u000berent com-\npositions, representing di\u000berent regimes of the structural\nphase diagram (orthorhombic structure with multiple\nq-vectors, one q-vector, and tetragonal structure). In\nthe tetragonal state (CeSb 0:91Te0:91), the band structure\n(Figure 7 a) unsurprisingly closely resembles the one of\nCeSbTe [27], with only a slight di\u000berence in electron\n\flling. For the single q-vector range, the band struc-\nture of CeSb 0:51Te1:40is shown in Figure 7b, which fea-\ntures a \fve-fold supercell along the b-axis. Orthorhombic\nCeSb 0:51Te1:40is isostructural to GdSb 0:46Te1:48[1] and\nanalogous to this compound, it also hosts a close to ideal\nnonsymmorphic Dirac semimetal band structure. Non-\nsymmorphic symmetry protected gapless Dirac nodes re-\nside near the Fermi energy ( EF) at high symmetry points\nXandU(yellow arrows), whereas almost all other band\ncrossings gapped out by the CDW and do not contribute\nto the low-energy transport response (Figure 7c). While\nit seems that some bands still cross EF(as CDW band\ngap is smaller than in its Gd-counterpart), there are small\ngaps in their spectra exactly at the Fermi level. This is\ninteresting for two reasons: (i) it suggests that the mech-\nanism reported in Ref. [1] can be universally applied to\nLnSbxTe2\u0000x\u0000\u000ephases that are in the single q-vector re-\ngion with q\u00190.20b\u0003. (ii) it provides a second idealized\nnonsymmorphic Dirac semimetal, which features com-\npletely di\u000berent magnetic properties. Most importantly,\nin CeSbxTe2\u0000x\u0000\u000ethe spins are aligned along the c-axis,\nwhile GdSb xTe2\u0000x\u0000\u000ethey are aligned in plane (perpen-\ndicular to the c-axis) [60]. This provides an additional\ndegree of tunability for magnetically induced new topo-\nlogical phases and opens the door to investigate further\nLnSbxTe2\u0000x\u0000\u000ephases that should also show an idealized\nDirac semimetal band structure.\nFinally, in Figure 7d, we show the band structure\nof CeSb 0:11Te1:90, which has multiple q-vectors. In\nthis compound, due to the in\fnite number of magnetic\nphases that arise from the devil's staircase, the tunability\nof topological phases becomes extremely rich. As can be\nseen in Figure 7d, the band structure of CeSb 0:11Te1:90\nfeatures both nonsymorphically enforced Dirac crossing\n(yellow arrow) as well as nodal-line crossings (green\narrows). Especially, a nodal-line crossing resides exactly\natEFalongS-\u0000. In addition, there are multiple\nnodal-line crossings just above EF, which should be\neasily accessible by \fne tuning of the electron \flling at\nthe Te-site. Future studies could further investigate the\ninterplay of the topological band structure and devil's\nstaircase magnetism in the multiple q-vector region of\nLnSbxTe2\u0000x\u0000\u000e.8\nFIG. 7. Electronic band structure of CeSb xTe2\u0000x\u0000\u000e. Results of the band structure calculations for a) tetragonal CeSb 0:91Te0:91\nand b) orthorhombic CeSb 0:51Te1:40with one q-vector. c) Ideal non-symmorphic Dirac cone in CeSb 0:51Te1:40at the Fermi\nenergy. d) Electronic band structure of orthorhombic CeSb 0:11Te1:90with multiple q-vectors. The yellow and green arrows\nshow the positions of the non-symmorphic Dirac node and nodal-line crossings, respectively.\nIII. CONCLUSION\nTo conclude, we have probed the detailed structural\nand magnetic properties of topological semimetals\nCeSbxTe2\u0000x\u0000\u000eas a function of electron \flling at the\nSb-square-net site. We found that the crystal structure\nincreasingly transforms from a tetragonal structure\nat ideal stoichiometry to an orthorhombic one with\nelectron \flling. This distorted orthorhombic structure\nenables formation of CDWs with continuously tunable\nmodulation wave-vectors. More interestingly, two\ndistinct regimes have been observed, hosting either one\nor multiple CDWs along di\u000berent crystallographic axes.\nComplete solutions of the modulated structures have\nbeen obtained, which also reveal the unique distortions\nof the square-net motif at di\u000berent electron \flling re-\ngions. Contrary to the simple AFM ordering in CeSbTe,\nthe magnetic phase diagram of CeSb xTe2\u0000x\u0000\u000eemerges\nto be quite rich and evolves continuously with chemical\nsubstitution. Speci\fcally, at higher electron \flling, we\nshow that a complex interplay of CDW and a collective\nspin-excitation leads to a series of fractionally quantized\nmagnetization plateaus, which o\u000bers the potential of\nrealizing new quantum phases in these compounds. The\nresults of our electronic band structure calculations\ncon\frm that CeSb 0:51Te1:40is an ideal magnetic Dirac\nsemimetal with a non-symmorphic Dirac node at the\nFermi energy, while almost all other bands are gappedout. Thus we present a unique template material,\nwhere the topological band structure can be controlled\nby electron \flling (tuning chemical potential), CDWs\n(gapping out non-essential band crossings), or magnetic\n\feld (changing the magnetic ordering). This also encour-\nages future studies to design new topological states in\nCeSbxTe2\u0000x\u0000\u000eand other members of the LnSbTe-family.\nIV. EXPERIMENTAL SECTION\nA. Single crystal growth and determination of\nstoichiometry\nSingle crystals of CeSb xTe2\u0000x\u0000\u000ewere grown by\nchemical vapor transport. Stoichiometric amounts of\nhigh purity Ce (Sigma Aldrich 99.9%), Sb (Alfa Aesar\n99.999%), and Te (Alfa Aesar 99.9999%) along with\na few milligrams of iodine (Sigma Aldrich 99.999%)\nwere placed into a quartz tube. The tube was then\nevacuated, sealed, and put into a gradient furnace. The\nfurnace was heated such that the hot end of the quartz\ntube, containing the materials, remained at 950\u000eC and\nthe other end at 850\u000eC for 7 days. After cooling, the\ncrystals were mechanically extracted from the cold end\nof the tube. The elemental composition of the crystals\nwas determined by energy dispersive x-ray spectroscopy\n(EDX) in a Verios 460 scanning electron microscope,9\noperating at 15 keV and equipped with an Oxford EDX\ndetector.\nB. X-ray di\u000braction and magnetic measurements\nThe powder XRD measurements were performed on a\nSTOE STADI P di\u000bractometer operating in transmission\ngeometry using Mo- K\u000b(\u0015=0.71073 \u0017A) source. The sin-\ngle crystals were crushed into powder in an argon-\flled\nglovebox and sealed in glass capillary to use for XRD\nexperiments. The XRD spectra was analyzed by LeBail\n\ftting using FULLPROF software package.\nThe single crystal XRD data were collected at 250(1) K\nwith either a Bruker Kappa Apex2 CCD di\u000bractometer\nor a Bruker D8 VENTURE equipped with a PHOTON\nCMOS detector, using\ngraphite-monochromatized Mo- K\u000bradiation. Raw\ndata were corrected for background, polarization, and\nLorentz factors as well as a multiscan absorption cor-\nrection was applied. Structure solution was carried\nout via either intrinsic phasing as implemented in the\nShelXT program or via charge-\ripping as implemented\nin SUPERFLIP. Commensurately modulated phases for\nCeSb 0:11Te1:90and CeSb 0:34Te1:67; and tetragonal struc-\nture of CeSb 0:79Te1:05were able to be indexed fully and\ncould be re\fned in conventional three-dimensional space\ngroups, using the ShelXL least-squares re\fnement pack-\nage in the Olex2 program. In the modulated phase\nCeSb 0:51Te1:40, the satellite peaks re\fned to a slightly\nincommensurate modulation vector q=0.201 b\u0003. Re\fne-\nment was thus carried out using the superspace approach,\nwhere the displacive distortion of atomic positions is ex-\npressed by a periodic modulation function, yielding a 3+1\ndimensional space group [61]. Re\fnements with the su-\nperspace approach were carried out in JANA2006. For\nthe re\fnement, the modulation vector was rounded to a\ncommensurate q=1/5 b\u0003without any signi\fcant changes\nto the \ftting compared to a fully incommensurate treat-\nment. Determination of Sb/Te occupancy and ordering\nwithin the square-net is limited by near-identical scatter-\ning power due to their closeness in atomic number ( Z=51\nand 52, respectively) [62]. At laboratory-accessible x-\nray wavelengths, the two species are indistinguishable\nif present on the same crystallographic site. Thus, for\nall re\fnements, atomic occupancies within the Sb/Te\nsquare-net were constrained to the stoichiometry derived\nfrom EDX. Assignment of ordering in CeSb 0:11Te1:90was\nbased on the chemical bonding intuitions preferring mul-\ntiple bonds to Sb.\nThe magnetic measurements were performed using\nthe vibrating sample magnetometer (VSM) option of\na physical property measurement system (Quantum\nDesign).C. Band structure calculations\nDensity functional theory (DFT) calculations were\nperformed in VASP v5.4.4 [63{65] using the PBE\nfunctional [66]. Localization of the Ce f-orbitals was\ncorrected by applying a Hubbard potential of U=6eV\n[67], as in previous work on tetragonal CeSbTe [27].\nPAW potentials [68, 69] were chosen based on the\nv5.2 recommendations. Simulations approximating\nCeSb 0:91Te0:91, CeSb 0:51Te1:40, and CeSb 0:11Te1:90were\nperformed on the tetragonal CeSb 0:91Te0:91unit cell with\nfull occupancy and supercells of 1 \u00025\u00021 Ce 10Sb10Te10\nand 3\u00023\u00022 Ce 36Te68Sb4with Fermi levels adjusted\nbased on the electron counts of the true experimental\ncells. Calculations employed a plane wave energy cuto\u000b\nof 400 eV and a k-mesh density, `= 30 (corresponding\nto 7\u00027\u00023 and 7\u00021\u00023 and 2\u00022\u00022 \u0000-centered k-meshes\nfor the tetragonal CeSb 0:91Te0:91subcell and the\nCeSb 0:51Te1:40and CeSb 0:11Te1:90supercells, respec-\ntively). Unfolded spectral functions for the supercells\nin the subcell BZ were calculated using the method of\nPopescu and Zunger [70] in VaspBandUnfolding .\nThe \u0000-Xhigh symmetry line in CeSb 0:51Te1:40(the only\nelongated cell for which this is ambiguous) was chosen\nto lie along the distortion axis. Crystal structures were\nvisualized with VESTA [71]. Spin-orbit coupling e\u000bects\nwere not incorporated. See additional computational\ndetails in the supporting material.\nAcknowledgements\nThis research was supported by the Princeton Cen-\nter for Complex Materials, a National Science Founda-\ntion (NSF)-MRSEC program (DMR-2011750). Addi-\ntional support for property characterization was provided\nby the Air Force O\u000ece of Scienti\fc Research (AFOSR),\ngrant number FA9550-20-1-0282. The authors acknowl-\nedge the use of Princeton's Imaging and Analysis Center,\nwhich is partially supported by the Princeton Center for\nComplex Materials. Work at UC Santa Barbara was sup-\nported by the National Science Foundation though the Q-\nAMASE-i Quantum Foundry, (DMR-1906325). We ac-\nknowledge use of the shared computing facilities of the\nCenter for Scienti\fc Computing at UC Santa Barbara,\nsupported by NSF CNS-1725797, and the NSF MRSEC\nat UC Santa Barbara, NSF DMR-1720256. S.M.L.T.\nhas been supported by the NSF Graduate Research Fel-\nlowship Program under Grant no. DGE-1650114. Any\nopinions, \fndings, and conclusions or recommendations\nexpressed in this material are those of the authors and\ndo not necessarily re\rect the views of the NSF.\nCon\rict of interest\nThe authors declare no con\rict of interest.\nAuthor contributions\nR.S. and L.M.S. initiated the project. R.S. synthesized\nsingle crystals and characterized them with input from\nS.L. R.S. measured and analyzed magnetization data,\nwith input from N.P.O. and L.M.S. T.H.S. performed10\nsingle crystal x-ray di\u000braction experiments and solved\nstructures with input from J.F.K. S.M.L.T. performedthe band structure calculations. L.M.S. supervised the\nproject. All authors discussed the results and contributed\nto writing the manuscript.\n[1] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava,\nM. Z. Hasan, Nature 2008 , 452, 970.\n[2] Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K.\nMo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang,\nS. C. Zhang, I. R. Fisher, Z. Hussain, Z.-X. 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The\nsuperlattice re\rections in precession di\u000braction images\ncon\frm the formation of three-fold and \fve-fold modu-\nlated charge density waves (CDWs) along the ab-plane\nfor orthorhombic CeSb 0:34Te1:67and CeSb 0:51Te1:40,\nrespectively. No signature of a CDW is observed\nfor tetragonal CeSb 0:79Te1:05. The corresponding\nsolved crystal structures are also shown. The results\nof the structural re\fnements are summarized in Table S1.\nFigure S2 illustrates the Curie-Weiss \fts of the\ninverse magnetic susceptibility \u001f\u00001(T) curve in the\nparamagnetic region for di\u000berent compositions. The\nextracted Weiss temperature from the \ftting is shown\nin Figure 3d in main text.\nFigure S3 a shows the magnetic phase diagram of\nthree representative CeSb xTe2\u0000x\u0000\u000ecrystals for magnetic\n\felds applied along the crystallographic c-axis. A\ncomplex magnetic ground state is observed for higher\nelectron \flling. On the other hand, the phase diagram\nfor \felds along the ab-plane, is found to be quite simple\nthroughout the entire doping range (Figure S3b).\nThe zero-\feld speci\fc heat ( CP) of CeSb 0:11Te1:90is\nplotted in Figure S4 a as a function of temperature.\nThe enlarged low-temperature region in Figure S4b\nclearly shows the presence of two magnetic transitions.\nWith an applied \feld of 0.5 T, the system reaches a\nfully spin-polarized state. In order to calculate the\nmagnetic entropy ( Sm), we have \ftted the heat capacity\ndata with Debye model, which gives the lattice contri-\nbution toCP. Subtracting the lattice component from\ntotal speci\fc heat and assuming a negligible electronic\ncontribution, magnetic component Cmwas obtained.\nThe magnetic entropy was calculated using the relation\nSm=RT\n0Cm\nTdT. As the lowest measured temperaturewas 1.8 K, the speci\fc heat data were interpolated using\nCP=0 atT=0 K. This interpolation might introduce\na small uncertainty in the calculated value of Sm.\nThe magnetic entropy for both CeSb 0:11Te1:90and\nCeSb 0:51Te1:40is plotted in Figure S4c.\nFigure S5 a illustrates the evolution of magnetocrys-\ntalline anisotropy with electron \flling in CeSb xTe2\u0000x\u0000\u000e.\nThe low-\feld region of the M(H) curves reveal a series\nof metamagnetic transitions at higher electron \flling\n(Figure S5b).\nThe quantized magnetization plateaus in\nCeSb 0:11Te1:90are shown in Figure S6 at di\u000ber-\nent temperatures.\nInFigure S7 , the low-\feld region of the M(H)\ncurve for CeSb 0:11Te1:90with \felds applied along the\nab-plane, con\frms no metamagnetic transition for this\nmeasurement con\fguration.\nAdditional computational details:\nInFigure S8 a, we present the full spin-polarized band\nstructure of CeSb 0:91Te0:91. While compounds in the\nCeSbxTe2\u0000x\u0000\u000efamily generally have an antiferromag-\nnetic ground state, the spin-splitting near the Fermi level\nis relatively small, with the result that the band struc-\nture is computationally well-approximated by displaying\nonly one spin-channel from the ferromagnetic calculation\nas we have done in the main body of the text. While sim-\nple, this ferromagnetic model has also previously demon-\nstrated good agreement with angle resolved photo emis-\nsion spectroscopy measurements of the electronic struc-\nture of the sister compound GdSb xTe2\u0000x\u0000\u000ewhich also\nundergoes similar Peierls-like distortions [1]. In Figure\nS8b, we present the same band structure with inclusion\nof spin-orbit coupling e\u000bects. Spin-orbit coupling makes\nonly small adjustments to the overall band dispersion.\nWhile spin-orbit coupling has not been included in the\nsupercell calculations due to computational expense, it is\nunlikely to a\u000bect our \fndings about the large Peierls-like\nband-gapping in these distorted structures.\nAs a \fnal comment for computationalists interested in\nthese compounds - instead of the true magnetic ground\nstate, metastable magnetic states (with the lowest DFT\nenergy and 0.9-1 \u0016Bmoments, consistent with experi-\nment) were frequently encountered when studying these\nsystems using DFT. A magnetic energy landscape with\nmany local minima is unsurprising in light of the rich\nmagnetic phase diagram discussed in the main text.\n[1] S. Lei, S. M. L. Teicher, A. Topp, K. Cai, J. Lin,\nF. Rodolakis, J. L. McChesney, M. Krivenkov, D.\nMarchenko, A. Varykhalov, C. R. Ast, R. Car, J. Cano, M.G. Vergniory, N. P. Ong, L. M. Schoop, arXiv:2009.00620\n2020 .13\nFigure S 1. Powder and single crystal x-ray di\u000braction of CeSb xTe2\u0000x\u0000\u000e. a) Powder x-ray di\u000braction spectra for di\u000berent\nSb-compositions. The experimental data are analyzed using LeBail \ftting. b) Precession di\u000braction image in the 0 klplane of\nthe single crystals with di\u000berent electron \fllings at the square-net. Superlattice re\rections can be clearly seen for x=0.34 and\n0.51. The corresponding solved crystal structures are also shown.14\nFigure S 2. Curie-Weiss \ftting. Curie-Weiss \ftting of the inverse magnetic susceptibility \u001f\u00001(T) curve in the paramagnetic\nregion for di\u000berent Sb-compositions.\nFigure S 3. Magnetic phase diagram of CeSb xTe2\u0000x\u0000\u000e. Magnetic phase diagram of di\u000berent crystals with varying Sb-content\nfor \feld applied along the crystallographic a) c-axis and b) ab-plane. The discrete points and red curve, obtained from the\nmagnetization data, represent the boundary between FM and PM states.15\nTable S I. Results of single crystal structural re\fnement.\nCeSb 0:11Te1:90 CeSb 0:34Te1:67 CeSb 0:51Te1:40 CeSb 0:79Te1:05\nFormula weight 395.95 394.61 380.86 370.29\nTemperature 250.15 K 293.0 K 293 K 298.15 K\nWavelength 0.71073 \u0017A 0.71073 \u0017A 0.71073 \u0017A 0.71073 \u0017A\nCrystal system Orthorhombic Orthorhombic Orthorhombic Tetragonal\nSpace group Pnma Pmmn Pmmn (0\f0)00s P 4=nmm\nModulation type Commensurate Commensurate Incommensurate\nModulation vectors q1=1/3 b\u0003q=1/3 b\u0003q=1/5 b\u0003\nq2=1/3 a\u0003+1/3 b\u0003+1/2 c\u0003\nq3=1/3 a\u0003+1/3 b\u0003-1/2c\u0003\nUnit cell dimensions a=13.4641(5) \u0017A a=4.4255(7) \u0017Aa=4.3925(1) \u0017Aa=4.3853(2) \u0017A\nb=13.4214(5) \u0017A b=13.4684(3) \u0017Ab=4.4392(1) \u0017Ab=4.3853(2) \u0017A\nc=18.2375(7) \u0017A c=9.1926(11) \u0017Ac=9.3053(3) \u0017Ac=9.4000(5) \u0017A\n\u000b=\f=\r=90\u000e\u000b=\f=\r=90\u000e\u000b=\f=\r=90\u000e\u000b=\f=\r=90\u000e\nVolume 3295.6(2) \u0017A3547.92(11) \u0017A3181.446(8) \u0017A3180.770(19) \u0017A3\nZ 36 6 2 2\nDensity (calculated) 7.159 g/cm37.152 g/cm37.1822 g/cm36.803 g/cm3\nAbsorption coe\u000ecient 27.702 mm\u0000127.635 mm\u0000127.722 mm\u0000126.385 mm\u00001\nF(000) 5828 970 314 306\n\u0012range for data collection 1.884-29.997\u000e2.215-44.980\u000e2.19-72.86\u000e2.167-36.496\u000e\nIndex ranges 0 \u0014h\u001425 -26 \u0014h\u001426 -11 \u0014h\u001411 -7 \u0014h\u00147\n-18\u0014k\u001418 -8 \u0014k\u00148 -12 \u0014k\u001411 -7 \u0014k\u00147\n-18\u0014l\u001418 -18 \u0014l\u001418 -23 \u0014l\u001424 -15 \u0014l\u001415\n-1\u0014m\u00141\nRe\rections collected 18530 61201 33837 8258\nIndependent re\rections 4988 2524 11239 309\nCompleteness to \u0012=25.242\u000e100% 99.8% 99% ( \u0012=72.86\u000e) 100%\nRint 0.0231 0.1058 0.0534 0.0545\nGoodness-of-\ft 1.076 1.124 2.48 1.179\nFinalRindices [I >2\u001b(I)]Robs=0.0596 Robs=0.0519 Robs=0.0718 Robs=0.0386\nwRobs=0.2715 wRobs=0.1461 wRobs=0.0944 wRobs=0.1038\nRindices [all data] Rall=0.0879 Rall=0.0622 Rall=0.0961 Rall=0.0386\nwRall=0.3366 wRall=0.1680 wRall=0.0979 wRall=0.1038\nExtinction coe\u000ecient 0.000026(10) 0.0012(4) 4130(150) 0.021(4)\nLargest di\u000b. peak and hole 4.793 and -6.749 e\u0017A\u000036.926 and -11.907 e\u0017A\u0000312.64 and -14.55 e\u0017A\u000032.577 and -3.190 e\u0017A\u00003\nRe\fnement method: Full-matrix\nleast-squares on F2.\nR=\u0006jjFoj\u0000jFCjj\n\u0006jFoj\nwR= [\u0006[w(jFoj2\u0000jFcj2)2]\n\u0006[w(jFoj4)]]1=2\nFigure S 4. Heat capacity measurements. a) Temperature dependence of the zero-\feld heat capacity ( CP) for CeSb 0:11Te1:90.\nb) Low-temperature region of the heat capacity curve for di\u000berent magnetic \feld, applied along the c-axis. c) Magnetic entropy\n(Sm) calculated from heat capacity data for x=0.11 and 0.51.16\nFigure S 5. Field dependence of magnetization. a) Magnetization vs. \feld curves for di\u000berent CeSb xTe2\u0000x\u0000\u000ecrystals at\n2 K with \feld applied along two mutually perpendicular crystallographic directions, showing reversal of magnetocrystalline\nanisotropy. b) Low-\feld region of the magnetization curves for di\u000berent crystals with \feld applied along the c-axis, showing a\nseries of metamagnetic transitions for higher electron \flling.\nFigure S 6. Quantized magnetization plateaus in CeSb 0:11Te1:90. Evolution of magnetization plateaus with temperature for\n\feld applied along the c-axis.17\nFigure S 7. Magnetization curve along the ab-plane for CeSb 0:11Te1:90. Low-\feld region of the M(H) curve at 2 K for \feld\napplied along the ab-plane, showing no metamagnetic transition in this measurement con\fguration.\nFigure S 8. Full band structure of FM CeSb 0:91Te0:91. a) Full spin-polarized calculation with spin-up and spin-down populations\nin orange and blue, respectively. b) Full band structure after the inclusion of spin-orbit coupling."    },    {        "title": "2107.13810v1.Magnetic_frustration_in_a_van_der_Waals_metal_CeSiI.pdf",        "content": "Magnetic frustration in a van der Waals metal CeSiI  Ryutaro Okuma,1,4 Clemens Ritter,2 Gøran J. Nilsen,3 Yoshinori Okada1 1Okinawa Institute of Science and Technology Graduate University, Onna-son, Okinawa, 904-0495, Japan 2Institut Laue-Langevin, 71 avenue des Martyrs, 38042 Grenoble, France 3ISIS Neutron and Muon Source, Science and Technology Facilities Council, Didcot OX11 0QX, United Kingdom 4Present address: Clarendon Laboratory, University of Oxford, Oxford, OX1 3PU, UK  The realization of magnetic frustration in a metallic van der Waals (vdW) coupled material has been sought as a promising platform to explore novel phenomena both in bulk matter and in exfoliated devices. However, a suitable material platform has been lacking so far. Here, we demonstrate that CeSiI hosts itinerant electrons coexisting with exotic magnetism. In CeSiI, the magnetic cerium atoms form a triangular bilayer structure sandwiched by van der Waals stacked iodine layers. From resistivity and magnetometry measurements, we confirm the coexistence of itinerant electrons with magnetism with dominant antiferromagnetic exchange between the strongly Ising-like Ce moments below 7 K. Neutron diffraction directly confirms magnetic order with an incommensurate propagation vector k ~ (0.28, 0, 0.19) at 1.6 K, which points to the importance of further neighbor magnetic interactions in this system. The presence of a two-step magnetic-field-induced phase transition along c axis further suggests magnetic frustration in the ground state. Our findings provide a novel material platform hosting a coexistence of itinerant electron and frustrated magnetism in a vdW system, where exotic phenomena arising from rich interplay between spin, charge and lattice in low dimension can be explored. The exploration of novel quantum phenomena in van der Waals (vdW) coupled materials is a promising and rapidly growing field motivated by both fundamental physics interest and, thanks to recent advances in exfoliation technologies, device applications1-4. An essential component to accelerate the development of the field is to increase the number of materials that host states whose properties are described by multiple coupled degrees of freedom, such as charge, spin, orbital, and lattice5. Among these, a particularly interesting playground for discovering such emergent phenomena has been identified in systems which manifest an interplay between itinerant electrons and magnetism, and intensive studies have been pursued and reported6-17. However, a promising area of this playground which simultaneously remains relatively unexplored is that of vdW materials, where itinerant electrons coexist with low-dimensional and potentially frustrated magnetism. Magnetic frustration can result in non-collinear spin textures and field-induced transitions to exotic quantum phases18,19. This includes emergent topologically non-trivial spin textures which lead to exotic couplings between the magnetism and the motion of itinerant electrons20,21. Furthermore, non-collinear spin textures can couple to lattice degrees of freedom to produce phenomena like multiferroicity22.  \n FIG.1. | a, Crystal structure of CeSiI. Red, blue, and purple spheres indicate Ce, silicon, and iodine atoms, respectively. Red, green, and blue arrows indicate the a, b, and c axes, respectively. b, Optical microscope image of a single crystal of CeSiI. The image is taken in a globe box filled with pure Ar gas. c, Temperature dependence of magnetic susceptibilities from a single crystal of CeSiI and zero-field resistivity of a polycrystalline sample of CeSiI. The magnetization measurements were carried out upon cooling in a magnetic field of 1 T applied along the a and c axes.   Part of the reason why magnetic metallic vdW materials are so scarce is that most vdW materials studied so far have been chalcogenides, which tend to be nonmagnetic. Another important class of materials with vdW structures is the halides. The large ionic radius and small number of chemical bonds of halides leads to low dimensional vdW bonded structures23. While typical transition metal halides24-27 are low-dimensional Mott-insulators, this is not the case for reduced rare-earth halides LnX2. In rare-earth elements with a strong tendency to stabilize the trivalent state, an electron produced by reduction occupies the outer 5d orbital of lanthanide rather than inner 4f orbital and metallicity results28-30. Moreover, the finding of giant magnetoresistance and \nCe3+\nSi2-I-\n1cmvdW gapa\nbc0.120.100.080.060.040.020.00M/H (cm3mol-1)\n6050403020100T (K)3.02.52.01.51.00.50.0ρ (10-4Ω•cm) powderB // cB // abTN ~ 7 Kferromagnetic order above room temperature in GdI2 motivates us to expect that 4f magnetism strongly couples to conduction electrons in rare earth halide materials31,32.  The ternary system LnAI (Ln = La, Ce, Pr, Gd. A = Al, Si, Ga, Ge) is a class of materials closely related to the reduced rare earth halides33-35. CeSiI appears as a cleavable material in 2DMatpedia36, and its crystal structure is depicted in Fig.1a. Ce atoms form a triangular bilayer in the ab plane sandwiching a honeycomb net of Si. These blocks are terminated by a vdW-coupled layer of iodine. While its synthesis and crystal structure have been reported previously, its physical properties, especially the effect of geometrical frustration and the influence of the Fermi surface on magnetism, have never been investigated. In order to demonstrate CeSiI as a promising host of itinerant electrons with frustrated magnetism, we have synthesized crystalline CeSiI by a high-temperature-solid-state reaction33 in a bulk form (Fig.1b) and performed resistivity, magnetometry, heat capacity, and neutron diffraction measurements.  The coexistence of metallicity and antiferromagnetic interactions in a CeSiI is confirmed. Figure 1c shows the temperature-dependence of the susceptibility χ(T) along the a and c axes (left axis), together with the temperature dependence of the resistivity (right axis). χ(T) exhibits strongly anisotropic behavior below 100 K due to the trigonal crystalline electric field splitting of the J = 5/2 manifold of the Ce3+ 4f electrons. The lowest Kramers doublet dominates the magnetism below ~10 K because the other J = 5/2 levels are typically located above 102~103 K in the trigonal crystalline electric field37. Below 7 K, the susceptibility component along the c axis drops, while that along a axis slightly increases, which indicates long-range antiferromagnetic ordering at TN ~ 7 K. The presence of a magnetic phase transition is also confirmed by the drop of resistivity below 7 K due to suppression of electron-spin scattering.   \n FIG.2. | Neutron diffraction study of CeSiI. (a) Temperature dependence of magnetic diffraction in the temperature range from 16 to 3.8 K. Nuclear reflections were removed from the data by subtracting data obtained at 17 K. The positions of the six observed magnetic Bragg peaks are labeled θi (i = 1, …, 6). The color scales of the neutron intensity are shown on the right side. (b). The evolution of the integrated intensity around the magnetic Bragg peaks derived from (a). The temperature is scaled by TN = 7 K. The integration intervals for θi (i = 1, …, 6) are (9.96, 11.56), (13, 14.8), (27.31, 28.5), (28.81, 30.00), (34.13, 35.42), and (35.76, 36.7), respectively. A linear background was assumed in integration. In order to elucidate the magnetic structure, powder neutron diffraction was performed below 20 K on the D20 c-20-10010Idiffuse (arb. unit)403020100θ (˚)161412108642T (K)403020100θ (˚)\n8004000-400I (arb.  unit)θ1θ2θ3θ4θ5θ6a\n1.00.50.0-0.5IT – I17K (arb.  unit)2.01.51.00.5T/TNθ1θ2θ3θ4θ5θ6binstrument at ILL. Whereas single crystal neutron diffraction would be ideal for pinning down the magnetic structure, the required sample volume is difficult to obtain. Nevertheless, as we describe hereafter, powder diffraction still provides essential information to understand this relatively new compound. In Figure 2a, we plot the temperature dependence of the low-angle magnetic scattering of CeSiI obtained by subtracting a high-temperature intensity from each 2q. Six magnetic peaks, which are labelled by q1 ~ q6 in Fig. 2a, are visible below 40˚ (see Supplementary Fig. 1 for the nuclear refinement). As shown in Figure 2b, the positions of these peaks are temperature-independent, while their intensity gradually increases below TN, which is consistent with the presence of a second order phase transition. The quality of the data did not allow for a determination of the critical exponent, however. The increase in the intensity of q5 above TN is affected by a ferromagnetic impurity CeSi1.7 with Tc ~ 11K38. We note that the change of background below TN is due to suppression of diffuse scattering by paramagnetic Ce3+.  We employed a long scan of the 1.6K data subtracted by that of the 7.7 K data for the magnetic structure analysis in order to disentangle the effect of diffuse scattering and magnetic Bragg peak.  IR BV mx my mz Γ1 ψ1 0 1 0 Γ2 ψ2 1 0 0 ψ3 0 0 1 Table 1 | Irreducible representations (IRs) and basis vectors (BVs) for the space group P–3m1 with k = (0.28, 0, 0.19). The decomposition of the magnetic structure representation for the 2c site (0, 0, z) is Γmag = Γ1 +2Γ2. The BV components of one orbit along a*, b, and c are shown by mx, my, and mz, respectively. The other orbit (0, 0, -z) has same BVs.  Model IR mx,1 my,1 mz,1 mx,2 my,2 mz,2 Rp (%) Rwp (%) χ2 SDWb  Γ1 0 1.00(1) 0 0 1.00(1) 0 39.2 28.8 6.15 SDWa*c  Γ2 0.14(1) 0 1.08(1) 0.14(1) 0 1.08(1) 26.9 18.2 2.59 Cycloid1  Γ2 0.07(2)i 0 1.11(1) 0.07(2)i 0 –1.11(1) 26.7 18.6 2.76 Cycloid2  Γ2 0.71(2)i 0 1.024(9) 0.71(2)i 0 1.024(9) 21.8 15.9 2.26 Table 2 | Results of Rietveld refinement of neutron diffraction pattern. The magnetic structure models are described in the main text. The mi,j represents the component of j site along  direction (i = x, y, z and j = 1, 2). Rp, Rwp, and χ2 represent profile factor, weighted profile factor, and reduced chi square, respectively.  Indexing the magnetic peaks produced a unique solution with an incommensurate wavevector of k ~ (0.28, 0, 0.19). In the space group P–3m1, the identity and mirror perpendicular to b axis render the propagation vector invariant, and these make up the little group Gk. The magnetic representation of a crystallographic site at 2c site (0, 0, z) can be decomposed into the irreducible representation Γ1 + 2Γ2, for which the projected basis vectors are listed in Table 1. While the Ce site is separated into two orbits (0, 0, z) and (0, 0, –z) in Gk, the symmetry operation that maps k to -k in the full group relates these two; the real and imaginary coefficients of the basis vectors have opposite and same values between the two orbits, respectively. Inversion and two-fold rotation about the b axis are added to Gk in the full group that also includes time reversal (Supplementary Note 2).  To determine the magnetic structure below TN, we refined the neutron diffraction pattern using all possible magnetic structures described by a single irreducible representation: SDW along the b axis (SDWb), which belongs to Γ1 (in BasIreps notation), and SDW (SDWa*c) and counter-rotating cycloid (Cycloid1) in the a*c plane, which belong to Γ2. Cycloid1 has opposite chiralities between the two orbits to maintain the inversion symmetry. We also consider a structure that breaks sublattice symmetry but is an element of the little group and is more common than the counter-rotating cycloid; a co-rotating cycloid (Cycloid2) in the a*c plane. In all cases, impurity phase CeSi1.7 is included in all the refinement and we confirmed that the existence of the secondary phase does not have any impact on the main conclusions of this study. Further details of the magnetic structures and fits are described in Supplementary Note 3. We plot the results of the Rietveld refinement in Fig.3a-d and Table 2. As shown in Fig. 3a, SDWb is clearly excluded from the candidate magnetic structure because of the poor agreement around the (010) –, (011)–, (011)+, and (01-1)+ reflections. By contrast, SDWa*c, Cycloid1, and Cycloid2 reproduce the overall behavior of the observed diffraction pattern as indicated by Figs. 3b-d. Thus, based on diffraction experiments, we confirm that a common feature of these three magnetic structures, all of which belong to Γ2, are a dominant out-of-plane component of the order of 1.0-1.1μB. Two of the magnetic structures that yielded the smallest R factors are schematically shown in Fig. 4a and b as representatives of possible magnetic structures with and without relatively large in-plane magnetic moment.   \n Figure 3 | Powder neutron diffraction pattern of CeSiI and a Rietveld fitting to the magnetic spin structures Γ1 SDWb (a), Γ2 SDWa*c (b), Γ2 Cycloid1 (c), and Γ2 Cycloid1 (d). The red circles represent data taken at 1.6 K after the subtraction of the 7.7 K data as a reference of nuclear contributions. The black, blue, and green line represent a Rietveld fit, a residual of fitting, and background, respectively. The red thin and thick green bar represents position of magnetic Bragg peaks of CeSiI and CeSi1.7, respectively. The arrows indicate the indices of magnetic Bragg peaks, where (hkl)± represents a Miller index of (hkl)±(0.28, 0, 0.19) reflection.    Finally, we point out that magnetization process is consistent with the presence of magnetic frustration. Figure 4c shows magnetization process along out-of-plane (//c) and in-plane (//ab) direction. Above 4 T, the magnetization along the c axis reaches ~1.1µB, which is close to the expected saturated magnetization of SDWa*c and Cycloid2 model and suggests localized nature of the Ce 4f moment. The magnetization along the a axis does not saturate below 7 T because this is the hard axis. The key feature is seen at 2 K for out-of-plane 1.00.50.0Imag (arb. units)605040302010θ (˚)Γ2 Cycloid11.00.50.0Imag (arb. units)605040302010θ (˚)Γ2 SDWa*c1.00.50.0Imag (arb. units)605040302010θ (˚)(000)\n±\n(010)–(011)–(010)+(01-1)+Γ1 SDWb\n1.00.50.0Imag (arb. units)605040302010θ (˚)Γ2 Cycloid2a\nbcd(//c) case. Between magnetic fields of 2 T and 4 T, two metamagnetic transitions are clearly observed. These anomalies are visible only below TN and are therefore thought to be associated with the magnetic order. It is important to note that the anomalous behavior should be distinct from a simple spin flop transition in a collinear antiferromagnet, which is usually a single-step process. A similar magnetization process with multiple steps is seen in layered triangular lattice antiferromagnets in insulating systems39-43, which include MnI2, NiI2, and CoI2. In these compounds, magnetic frustration is supposed to occur due to competing nearest neighbor and further neighbor magnetic coupling, leading to a helical order with in-plane periodicity of (a, 0). The magnetic structure in these insulating iodides is analogous to Cycloid2 case (see table 2) and competing further neighbor couplings between localized spins are naturally present in metallic systems. As far as we recognize, the observation of multiple magnetization process in a vdW metal is the first report in CeSiI.  \n Figure 4 | Magnetic frustration in CeSiI  (a,b) Magnetic structures are schematically drawn for the case of  Cycloid2 and SDWa*c. Red spheres and magenta arrows indicate the magnetic Ce ions and their magnetic moments, respectively. Red, green, and blue arrows indicate the a, b, and c axes, respectively. (c) Magnetization process of CeSiI. The red, green, and blue circles and red and blue triangles represent magnetization along c axis at 2, 4, and 7 K and along ab plane at 2 and 7 K. respectively. The red dotted lines are derivative of magnetization at 2K, where the peaks indicate the two-step metamagnetic transition.  Our experimental findings collectively point towards CeSiI being a novel material platform hosting magnetic frustration coexisting with itinerant electrons within a vdW material family. In the case of the out-of-plane SDW (Fig. 4b), the realization of a quantum disordered state by frustration and reduced dimensionality will be an interesting question to be explored with the help of the exfoliation technique. Further exotic phases can be expected in the case of the co-rotating cycloid case (Fig. 4a). For example, the point group of this structure is m1’, which allows a finite electric polarization perpendicular to b axis. Even if polarization is allowed by symmetry, the conduction electrons are expected to screen polarization in metallic compounds. However, in the van der Waals metal WTe2, which features a polar-nonpolar structural transition, an out-of-plane electric field can switch the displacement of W when atomically-thin flakes are used44.  In summary, we investigated CeSiI using resistivity, magnetometry measurements and neutron diffraction experiments on both powder and single crystal samples. Through these, we demonstrated that CeSiI is a novel material platform to search exotic phenomena arisen from rich interplay between spin, charge and lattice degree of freedom, for example, by using exfoliation-based device fabrication techniques.   Acknowledgement We thank Takeshi Yajima, and Hajime Ishikawa for advice on the synthesis of CeSiI, Dmitry Khalyavin for help with the determination of the magnetic structure, and Rieko Ishii, Daichi Ueta, and Wonjong Lee for helping the experiments. This work was carried out by the joint research in the Institute for Solid State Physics, the University of Tokyo.  Method Single crystal growth  Single crystals of CeSiI were synthesized by a high-temperature-solid-state reaction following ref. 33. The handling of the elements and compounds was performed in an argon-filled glovebox. CeI3 was made by reaction of stoichiometric amounts of each element: Ce and I2 were sealed in an evacuated quartz tube in vacuo and reacted at 300˚C for 24 hours followed by subsequent heating at 900˚C for 24 hours to complete the cab\n1.61.20.80.40.0dM/dB (μB/Ce-T)76543210B (T)1.21.00.80.60.40.20.0M (μB/Ce)B // c2 K4 K7 KB // ab2K7KAntiferromagneticFerro-magneticreaction. The quartz tube was then placed vertically with one end outside the furnace and heated again at 900˚C in order to facilitate the sublimation of polycrystalline CeI3. Yellow needle-like crystals of CeI3 were formed at the colder end of the quartz tube. Stoichiometric amounts of CeI3, Ce, and Si in a total weight of 1g were sealed in a Nb tube with a diameter of 1 cm and length of 15 cm by arc welding. The sealed Nb tube was sealed in a quartz tube to prevent oxidation of Nb at elevated temperatures. The sealed quartz tube was then placed horizontally in a two-zone furnace. The cold and hot ends were kept 800˚C and 1000˚C, respectively. After a week, shiny bronze plate-like crystals with a maximum dimension of 2 x 2 x 0.05 mm3 formed at the hot end. The synthesis of powder CeSiI was performed at 1000˚C for four days.   X-ray diffraction, magnetometry, heat capacity, and resistivity measurement Single crystals of CeSiI could be indexed by a trigonal space group P-3m1 with lattice constants of a = 4.1872(1) Å and c = 11.7036(4) Å at 300 K using a single crystal X-ray diffractometer Bruker Venture D8 (Mo Kα, λ = 0.71069 Å). DC SQUID Magnetometry measurements were performed using MPMS-3 (Quantum Design). A single crystal was placed on a diamagnetic sapphire substrate and coated with paraffin. Resistivity was measured by conventional four-probe method using the resistivity option of PPMS (Quantum Design). An as-grown polycrystalline chunk of CeSiI was attached to indium electrodes and gold wires.  Powder Neutron Diffraction experiment and data analysis. Powder neutron diffraction of was performed on the high-flux D20 thermal diffractometer at Institut Laue-Langevin at temperatures from 20 K to 1.6 K using a constant wavelength λ1 ~ 2.4 Å45. The powder sample was sealed in a vanadium can with an indium seal to prevent oxidation. The magnetic peaks were indexed by the k-search software in the FULLPROF suite46. The analysis of the irreducible representations is performed using SARAh suite software (little group)47 and ISODISTORT software (full group)48,49. Structural and magnetic refinement of powder diffraction data was performed by Fullprof46.  Data availability The data files are also available from the corresponding author upon request.  References 1. Geim, A. K., & Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 419 (2013). 2. Burch, K. S., Mandrus, D., & Park, J.-G.  Magnetism in two-dimensional van der Waals materials. Nature 563, 47–52 (2018). 3. Gong, C. & Zhang, X. 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The D20 instrument at the ILL: a versatile high-intensity two-axis neutron diffractometer. Meas. Sci. Technol. 19, 034001 (2008). 46. Rodríguez-Carvajal, J. Recent advances in magnetic structure determination by neutron powder diffraction. Physica B 192, 55-69 (1993). 47. Wills, A. S. A new protocol for the determination of magnetic structures using simulated annealing and representational analysis (SARAh). Physica B: Condens. Matter 276, 680-681 (2000). 48. Stokes, H. T., Hatch, D. M., & Campbell, B. J. ISODISTORT, ISOTROPY Software Suite, iso.byu.edu. 49. Campbell, B. J. et al. ISODISPLACE: An Internet Tool for Exploring Structural Distortions. J. Appl. Cryst. 39, 607-614 (2006). Supporting information for “Magnetic frustration in a van der Waals metal CeSiI”  Ryutaro Okuma,1,4 Clemens Ritter,2 Gøran J. Nilsen,3 Yoshinori Okada1 1Okinawa Institute of Science and Technology Graduate University, Onna-son, Okinawa, 904-0495, Japan 2Institut Laue-Langevin, 71 avenue des Martyrs, 38042 Grenoble, France 3ISIS Neutron and Muon Source, Science and Technology Facilities Council, Didcot OX11 0QX, United Kingdom 4Present address: Clarendon Laboratory, University of Oxford, Oxford, OX1 3PU, UK  Supplementary Note 1 Nuclear structure refinement of powder neutron diffraction data We show here that neutron analysis for drawing main conclusion is not altered due to impurity phases in our samples. Figure S1 shows the result of nuclear Rietveld refinement in Fullprof1 using data at 19.4 K considering four phases according to reported structural data. It successfully converged to Rwp = 8.76% and Rp = 7.45%. The refined value of molar ratio of the sample was CeSiI : CeI3 : CeSi1.7 : CeOI = 56.4(6) : 25.3(4) : 14.4(3) : 3.9(1). As CeI3 and CeOI are paramagnetic above 2K and CeSi1.7 is ferromagnetic below ~11 K2, magnetic scattering away from nuclear scattering should originate purely from magnetic moments on CeSiI.  \n  Supplementary Fig. 1 | Powder neutron profile of CeSiI at 19.4 K fitted by the Rietveld method, showing observed (black circle), calculated (red line), and difference (blue line). The green ticks represent the positions of the Bragg reflections of CeSiI and impurity phases: CeI3, CeOI, and CeSi1.7.            2.01.51.00.50.0I (105cts)150100500θ (˚)CeSiICeI3CeOICeSi1.7\nFomula CeSiI Space group P-3m1 a / Å 4.1697(1) c / Å 11.6581(4) V / Å3 175.537(9) Z 2 Atom Wyck. x y z Occ. Ce1 2c 0 0 0.1751(6) 1 I1 2d 1/3 2/3 0.3548(4) 1 Si1 2d 1/3 2/3 0.9924(5) 1 Fomula CeI3 Space group Cmcm a / Å 4.3636(3) b / Å 13.917(1) c / Å 9.9333(7) V / Å3 603.23(8) Z 4 Atom Wyck. x y z Occ. Ce1 4c 0 3/4 1/4 1 I1 4c 0 0.0803(7) 1/4 1 I2 8f 0 0.3541(5) 0.0700(7) 1              Supplementary Table 1 | Structural parameters of CeSiI, CeSi1.7, CeI3, CeOI, and obtained from the Rietveld refinements. Isotropic displacement parameter Biso is fixed to 0.5/Å2 for all atoms.   Supplementary Note 2 Symmetry analysis of the magnetic structure  Details of the symmetry analysis of the magnetic structure is given in this section. CeSiI crystallizes in a trigonal space group P-3m1 with lattice constants a = 4.1697(1) Å and c = 11.6581(4) Å at 20 K and Ce ions occupy single Wyckoff position 2c as described in the Supplementary Note 1. To comply with the symmetry of the single-k magnetic structure described by k = 0.28a* + 0.2c*, we use (a* b c) as principal axes in the following discussion. The magnetic moment at the position r = (x, y, z) can be expressed as S(r) = eik•r(mx, my, mz)+ e–ik•r(mx*, my*, mz*). Here mx, my, mz, and * correspond to the component along a*, b, and c and a complex conjugate operator, respectively. Symmetry of the magnetic structure can be characterized by the symmetry operations in P-3m11’ that transform k•r to ±k•r, namely, 1’: (x, y, z, mx, my, mz) → (x, y, z, –mx, –my, –mz), mb: (x, y, z, mx, my, mz) → (x, –y, z, –mx, my, –mz), 2b: (x, y, z, mx, my, mz) → (–x, y, –z,  –mx*, my*, –mz*), where 1’, mb, and 2b describe time-reversal operation, mirror operation perpendicular to the b axis, and two-fold rotation about the b axis, respectively. Superspace groups of the candidate magnetic structures are obtained by ISODISTORT3,4 as shown in Supplementary Table 2 and Fig. 2.  IR Superspace group Magnetic point group S+ S– Types of magnetic structures Γ1 C2/m1’(α, 0, β)00s 2/m1’ (0, m, 0) (0, m*, 0) SDW // b axis Cm1’(α, 0, β)0s m1’ (0, m, 0) (0, u, 0) SDW // b axis Γ2 C2/m1’(α, 0, β)0ss 2/m1’ (m, 0, u) ‒(m*, 0, u*) SDW and counter-rotating cycloid // a*c Cm1’(α, 0, β)ss m1’ (m, 0, u) (v, 0, w) SDW and co-rotating cycloid // a*c Supplementary Table 2 | Superspace groups of single-k magnetic structures with the incommensurate propagation vector k = 0.28a* + 0.2c*. S± denotes the allowed (mx, my, mz) at (0, 0, ±z) in the given superspace group and m, u, v, and w are complex numbers. The s in superspace groups denotes the shift of the internal coordinate by a factor of π.   In C2/m1’(α, 0, β)00s and C2/m1’(α, 0, β)0ss, two-fold rotation related the two orbits inside a unit cell and inversion symmetry is retained whereas in Cm1’(α, 0, β)0s and Cm1’(α, 0, β)ss, two orbits are not related by symmetry and only mirror symmetry is present as shown in Supplementary Fig. 2. In terms of types of magnetic structure, C2/m1’(α, 0, β)00s allows only a collinear spin density wave order parallel to b axis and both orbits have same amplitude. Cm1’(α, 0, β)0s also allows only a collinear  SDW but the amplitude between the sites can be different. C2/m1’(α, 0, β)0ss in general describes elliptic counter rotating cycloids in the sense that the chirality defined as b•Sr×Sr+a takes the opposite sign and same amplitude between the two orbits, which includes SDW parallel to the b axis. Cm1’(α, 0, β)ss can describe more general elliptic cycloids including co-rotating cycloids.   Fomula CeSi1.7 Space group Imma a / Å   4.1195(4) b / Å 4.1835(4) c / Å 13.883(1) V / Å3 239.26(4) Z 4 Atom Wyck. x y Z Occ. Ce1 4e 0 1/4 0.124(1) 1 Si1 4e 0 1/4 0.533(1) 0.85 Si2 4e 0 1/4 0.702(1) 0.85 Fomula CeOI Space group P4/nmm a / Å 4.1051(5) c / Å 9.100(2) V / Å3 153.35(5) Z 2 Atom Wyck. x y z Occ. Ce1 2c 1/4 1/4 0.131 1 O1 2a 3/4 1/4 0 1 I1 2c 1/4 1/4 0.672 1  Supplementary Fig. 3 | Typical magnetic structure allowed in the superspace group given in Supplementary Table 2. The red sphere and purple and green arrows represent Ce ions and magnetic moments with different sites, respectively. Sky-blue bond and shaded plane indicate two-fold rotation axis and mirror plane, respectively. The sign indicates phase shift of the internal coordinate; it is multiplied to the magnetic moment after of the symmetry operation is performed to keep the magnetic structure unchanged.  SDW, counter-rotating cycloid, and co-rotating cycloids are sketched as examples for C2/m1’(α, 0, β)00s and Cm1’(α, 0, β)0s, C2/m1’(α, 0, β)00s, and Cm1’(α, 0, β)ss, respectively.  Supplementary Note 3  Magnetic structure refinement  To construct physical models of the magnetic structure in CeSiI, we considered typical magnetic structures allowed in the superspace groups of Supplementary Table 2, which is discussed in the main text. Each parameter used to describe the magnetic structures is shown in Supplementary Table3.  Model Superspace group S+ S– SDWb C2/m1’(α, 0, β)00s e–iφ/2 (0, My, 0) eiφ/2 (0, My, 0) SDWa*c C2/m1’(α, 0, β)0ss e–iφ/2 (Mx, 0, Mz) eiφ/2 (Mx, 0, Mz) Cycloid1 C2/m1’(α, 0, β)0ss e–iφ/2 (iMx, 0, Mz) eiφ/2 (iMx, 0, –Mz) Cycloid2 Cm1’(α, 0, β)ss e–iφ/2 (iMx, 0, Mz) eiφ/2 (iMx, 0, Mz) Supplementary Table 3 | Magnetic structure models used in the Rietveld refinement. S± denotes the allowed (mx, my, mz) at (0, 0, ±z) in the given superspace group and Mx, My, Mz, φ, and θ are real numbers.   We note that there are three features that require careful treatment in the magnetic Rietveld refinement of the powder diffraction data while they do not affect the main conclusion that the magnetic structure has an incommensurate dominant easy-axis component: ferromagnetic impurity phase CeSi1.7, asymmetric peak broadening and diffuse scattering at low angles. First, the ferromagnetic reflections of CeSi1.7 were included in all the analysis assuming that the Ce moment points in the <100> direction1. Second, asymmetric peak broadening, which manifests itself at (000)± and (001)– peaks, indicates finite correlation length along the out-of-plane direction due to two-dimensional character of the magnetism. This effect is simulated by plate-like magnetic domains via Size-model function in Fullprof1. Finally, presence of diffuse scattering is inferred from the residual intensity around 15˚, between (000)± and (001)– reflections, which cannot totally be reproduced by plate-like domains. For the background, we initially used a linear interpolation of fixed points in the regions without magnetic Bragg peaks. Then to reproduce the actual background containing broad peaks of diffuse scattering, we also refined the background intensity of intermediate positions, which include magnetic Bragg peaks. The results of the refinement with the fixed background points and the refined background points are presented in Supplementary Tables 4 and Supplementary Fig. 4 and Supplementary Table 5 and Supplementary Fig.5, respectively.  In the fitting without refinable background, the obtained Mz exceeds the saturation field of 1.1µB and χ2 is much larger than 1 due to poor agreement near (000)± and (001)– peak. By contrast, in the fitting with refinable background, Mz is closed to the observed value in Γ2 models and χ2 is greatly improved. The refined CeSi1.7 moment also agrees well with the reported value of 0.5µB2 regardless of the background refinement. Therefore, fittings with the refined background are more likely to capture the feature of the magnetic structure and Cycloid2 is the primary candidate of the magnetic structure. However, small difference in the agreement factor between the collinear and noncollinear orders certainly calls for polarized neutron diffraction experiments using single crystals in order to measure the chiral term iMqxMq*, which is proportional to spin chirality. Although the limitation of the crystal size C2/m1’(α, 0, β)00s\n+mb+2b\n+2b–mbC2/m1’(α, 0, β)0ss\n–mbCm1’(α, 0, β)ssbca\nP-3m11’\n+mbCm1’(α, 0, β)0sΓ1\nΓ2hinders such a measurement at the current stage, a large single crystal will uncover the magnetic structures even under magnetic fields and thus shed light on the origin of the frustration in CeSiI.  Model Mx or My Mz φ ξ ka* kc* MCeSi1.7 Rp Rwp Rexp χ2 SDWb 1.157(7) ‒ 0.351(3) 15.7(5) 0.2784(1) 0.2029(8) 0.65(2) 36.5 31.7 9.18 11.9 SDWa*c 0.25(1) 1.262(6) 0.363(2) 14.0(3) 0.27807(5) 0.2032(4) 0.43(1) 24.7 18.4 9.17 4.0 Cycloid1 0.22(3) 1.288(7) 0.187(1) 13.6(3) 0.27824(6) 0.2059(5) 0.46(1) 26.6 20.8 9.17 5.1 Cycloid2 0.40(4) 1.27(1) 0.179(2) 12.5(3) 0.27800(6) 0.2047(5) 0.42(2) 27.9 21.6 9.17 5.5 Supplementary Table 4 | Magnetic structure refinement with fixed background. M, θ, and φ are parameters that define the magnetic structure in Supplementary Table3. ξ is anisotropic Lorentzian contribution of particle size for a plate-like coherent domain3. ka*, kc* are a* and c* component of the propagation vector. MCeSi1.7 is length of ferromagnetic moment of CeSi1.7. Rp, Rwp, and Rexp, χ2 represent profile factor, weighted profile factor, expected weighted profile factor, and reduced chi square, respectively.  Model Mx or My Mz φ ξ ka* kc* MCeSi1.7 Rp Rwp Rexp χ2 SDWb 1.00(1) ‒ 0.344(5) 10.4(6) 0.27833(8) 0.1936(6) 0.37(3) 39.2 28.8 11.6 6.2 SDWa*c 0.14(1) 1.08(1) 0.355(3) 7.5(3) 0.27809(5) 0.1935(4) 0.52(1) 26.9 18.2 11.3 2.6 Cycloid1 0.07(1) 1.11(1) 0.175(2) 6.5(3) 0.27808(5) 0.1938(4) 0.53(1) 26.7 18.6 11.2 2.8 Cycloid2 0.71(2) 1.024(9) 0.161(2) 8.1(3) 0.27806(4) 0.1936(3) 0.55(1) 21.8 15.9 10.6 2.3 Supplementary Table 5 | Magnetic structure refinement with refined background. The refinable background consists of 36 points between 9 and 99˚ in equal spacing of 2.5˚.   Supplementary Figure 4 | Powder neutron diffraction pattern of CeSiI and a Rietveld fitting to the magnetic spin structures Γ1 SDWb (a), Γ2 SDWa*c (b), Γ2 Cycloid1 (c), and Γ2 Cycloid1 (d) with fixed background parameters. Fitting regions are between 8˚ and 100˚. The red circle represents data taken at 1.6 K after the subtraction of the 7.7 K data as a reference of nuclear contributions. The black, blue, and green line represent a Rietveld fit, a residual of fitting, and background, respectively. The red thin and thick green bar represents position of magnetic Bragg peaks of CeSiI and CeSi1.7, respectively. The arrows indicate the indices of magnetic Bragg peaks, where (hkl)± represents a Miller index of (hkl)±(0.28, 0, 0.20) reflection.      1.00.50.0Imag (arb. units)100755025θ (˚)Γ2 Cycloid10.50.0Imag (arb. units)100755025θ (˚)Γ2 SDWa*c1.00.50.0Imag (arb. units)100755025θ (˚)(000)±\n(000)±(001)–Γ1 SDWb\n1.00.50.0Imag (arb. units)100755025θ (˚)Γ2 Cycloid2a\nbcd Supplementary Figure 5 | Powder neutron diffraction pattern of CeSiI and a Rietveld fitting to the magnetic spin structures with refined background parameters.    Supplementary References 1. Rodríguez-Carvajal, J. Recent advances in magnetic structure determination by neutron powder diffraction. Physica B 192, 55 –69 (1993). 2. Yashima, H. & Satoh, T. Nonmagnetic-magnetic transition in Ce-Si system. Solid State Commun. 41, 723-727 (1982). 3. Stokes, H. T., Hatch, D. M., & Campbell, B. J. ISODISTORT, ISOTROPY Software Suite, iso.byu.edu. 4. Campbell, B. J. et al. ISODISPLACE: An Internet Tool for Exploring Structural Distortions. J. Appl. Cryst. 39, 607-614 (2006). 1.00.50.0Imag (arb. units)100755025θ (˚)Γ2 Cycloid10.50.0Imag (arb. units)100755025θ (˚)Γ2 SDWa*c1.00.50.0Imag (arb. units)100755025θ (˚)(000)±\n(000)±(001)–Γ1 SDWb\n1.00.50.0Imag (arb. units)100755025θ (˚)Γ2 Cycloid2a\nbcd"    },    {        "title": "2108.07305v3.Flavors_of_Magnetic_Noise_in_Quantum_Materials.pdf",        "content": "Flavors of Magnetic Noise in Quantum Materials\nShu Zhang\u0003and Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n(Dated: September 7, 2022)\nThe complexity of electronic band structures in quantum materials o\u000bers new charge-neutral de-\ngrees of freedom stable for transport, a promising example being the valley (axial) degree of freedom\nin Weyl semimetals (WSMs). A noninvasive probe of their transport properties is possible by ex-\nploiting the frequency dependence of the magnetic noise generated in the vicinity of the material.\nIn this work, we investigate the magnetic noise generically associated with di\u000busive transport using\na systematic Langevin approach. Taking a minimal model of magnetic WSMs for demonstration,\nwe show that thermal \ructuations of the charge current, the valley current, and the magnetic or-\nder can give rise to magnetic noise with distinctively di\u000berent spectral characters, which provide a\ntheoretical guidance to separate their contributions. Our approach is extendable to the study of\nmagnetic noise and its spectral features arising from other transport degrees of freedom in quantum\nmaterials.\nIntroduction. |Many recently discovered novel quan-\ntum materials are featured by the complexity of their\nelectronic band structures, as a result of spin-orbit cou-\npling [1], magnetic order [2], twist engineering [3], etc.\nThese structures may exhibit new degrees of freedom sta-\nble for transport, in addition to charge and spin, due to\nthe protection either by symmetry or topology, which can\nbe explored for the next-generation information devices.\nOne prominent example is the valley degree of freedom\npresent in some hexagonal two-dimensional semiconduc-\ntors or Weyl semimetals (WSMs), the possibility of ma-\nnipulating which gives rise to the \feld of valleytronics [4].\nTransport is generally noisy. As the Johnson-Nyquist\nnoise (thermally excited electric currents) is important in\nelectronic devices, understanding the generation of noise\nfrom these new degrees of freedom is of practical rele-\nvance in spintronic or valleytronic devices. At the same\ntime, the electromagnetic noise emitted by a material\ninto its environment encodes rich information about its\nintrinsic excitation dynamics and transport properties.\nFor example, the magnetic noise in the vicinity of a con-\nductor is directly related to its impedance [5, 6]. The\nrecent development of magnetic noise spectroscopy us-\ning single qubits, especially the nitrogen-vacancy (NV)\ncenters in diamond [7], has provided a nanoscale probe\nto access such information noninvasively, and with high\nfrequency resolution [8{10]. NVs have also turned out to\nbe useful in the study of magnetic insulators [11{14] by\nprobing the magnetic noise generated by spin excitations.\nWhile thermodynamic valley \ructuations have been ac-\ncessed by optical methods [15], the noise associated with\nvalley transport in low-frequency regimes is so far rarely\nexplored.\nIn this work, we o\u000ber a qualitative perspective to the\nstudy of the magnetic noise in quantum materials, which\ncan be \ravorful due to the presence of various transport\ndegrees of freedom, focusing on their generic di\u000busive\naspect. To this end, we take an example of a magnetic\nWSM, which naturally involves three sources of noise,\nfrequencycharge−espinvalleyAmpère’s Lawcharge currentvalley currentspin density}“magnetic charge”magnetic  noiseFermi  surfacemagnetic  ordermagnetic field  in the environment FIG. 1. An illustration of the three \ravors of magnetic noise\nin a magnetic WSM and their di\u000berent spectral characters.\nnamely, charge, spin, and valley. Intriguingly, each \ravor\ncan contribute a distinct spectral character, as shown in\nFig. 1. Our perspective can be extended to consider other\npseudospin degrees of freedom in general and may inspire\nfuture experimental work in the NV probe of quantum\nmaterials in light of its advantage in frequency resolution\nin the GHz regime.\nWSMs are a family of topological quantum materials\npromising for valleytronic applications, because the band\ncrossing at Weyl points is topologically protected, and\nthe valley relaxation time can be very long in a clean\nsystem [16]. Magnetic WSMs, such as those in magnetic\nHeusler compounds [17{19], allow the existence of a sin-\ngle pair of energy-degenerate Weyl valleys [20], and are\nthus ideal for the study of valley transport. The interplay\nbetween charge, valley, and spin [21, 22] also makes them\nattractive for spintronics. The detection of valley trans-\nport often relies on the conversion from valley excitations\nto optic or electric signals, for instance, with the help of\nthe chiral anomaly e\u000bect in a nonlocal geometry [23]. In\nmagnetic WSMs, however, this is not totally unambigu-\nous due to the presence of spin excitations, which usually\nhave a long di\u000busion length as well. The magnetic noise\nspectroscopy can serve as a direct probe of the intrinsic\ntransport properties in the absence of external perturba-arXiv:2108.07305v3  [cond-mat.mes-hall]  3 Sep 20222\n23\nNV\nMagnetic WSMVacuum\nFIG. 2. Quantum-impurity relaxometry of a magnetic Weyl\nsemimetal. A three-dimensional sample of a magnetic WSM\nwith a band structure given by Eq. (1) has a magnetic order\nMin the zdirection and a surface lying in the xyplane. An\nNV center is placed at distance dfrom the surface, with a\ntunable resonance frequency !.\ntions and electric contacts. As we will show, it is possible\nto distinguish the valley and spin contributions to the\nmagnetic noise based on their spectral characteristics.\nMain concepts. |We \frst brie\ry summarize our con-\nceptual understanding of why the three \ravors contribute\ndi\u000berently, even all under a di\u000busive treatment. Fo-\ncusing on the scenario with a nonvanishing carrier den-\nsity, where the electric charge density \ructuations are\nscreened by Coulomb interactions, transverse \ructua-\ntions of the charge current dominate the charge channel.\nIn contrast, longitudinal \ructuations of the valley current\nare important due to its charge neutrality and hence the\nabsence of screening of the axial charge density. In our\nmodel (see below), the Weyl nodes are induced by the\nbroken time-reversal symmetry associated with the mag-\nnetic order, which dictates the response of valley currents\nto magnetic \felds and thus the generation of magnetic\nnoise by valley \ructuations. Consequently, only the lon-\ngitudinal component of the valley current generates mag-\nnetic \felds in the environment: The valley current jvbe-\nhaves as a magnetization, and determines a \\magnetic\ncharge\" distribution in the bulk \u001aM/ \u0000r\u0001jv, while\nthat on the surface vanishes due to the boundary condi-\ntion\u001bM/n\u0001jv= 0. In the spin channel, however, the\nspin density (rather than the spin current) plays such a\nrole. See Fig. 1 for comparison.\nModel. |The following minimal model is considered for\na magnetic WSM with four bands [24]:\nH=v\u001cz\n(\u001b\u0001~k) + \u0001\u001cx+J\u001b\u0001M; (1)\nwhere \u001band\u001care vectors of the Pauli matrices in the\nspin and valley spaces, respectively, vis the Fermi veloc-\nity, \u0001 is a Dirac mass, and Jis the magnetic exchange\nbetween the itinerant electrons and the magnetic order\nM. WhenjJjM >j\u0001j, this model realizes a single pairof Weyl nodes with opposite chiralities separated in the\nmomentum space. Here after, jJjM\u001dj\u0001jis assumed\nto approximately conserve the valley index [22]. We also\nassume a small but \fnite Fermi surface, and thus a \fnite\ncarrier density at the valleys, i.e., a Weyl metal.\nDe\fning j+(\u0000)as the number \rux density operator of\nelectrons belonging to the valley with positive (negative)\nchirality, the valley current becomes\njv=j+\u0000j\u0000=\u001cz@kH=~=v\u001b; (2)\nat a single-particle level. The assumptions about the spin\nand parity symmetries [22] in our model (1), therefore, es-\ntablish a proportionality between the valley current and\nthe itinerant spin density at the Fermi surface. Since\na Zeeman term \u001b\u0001Bis allowed, the valley current can\ndirectly couple to magnetic \felds, with its \ructuations\ngenerating magnetic noise.\nWe employ a simple setup of magnetic noise measure-\nment using an NV center placed at a nanoscale distance\ndfrom a \rat surface of the three-dimensional material,\nas shown in Fig. 2. We choose the magnetic order Mk^z\nand the surface plane to be the xyplane. With this\ngeometry, we focus on the contributions from bulk trans-\nport, separated from the Fermi arcs. The central object\nof our study is the magnetic noise tensor in the frequency\ndomain at the position of the NV center rNV, expressed\nin the symmetrized correlation functions of the magnetic\n\feld operators,\nBii0(!) =1\n2Z\ndtei!thfBi(rNV;t);Bi0(rNV;0)gi:(3)\nIts components determine the NV relaxation rate de-\npending on the NV orientation [11, 25]. Keeping the\ntypical GHz frequency of NV centers in mind, we focus\non the magnetostatic limit, where the wavelength \u0015of the\n\ructuating electromagnetic \feld and the skin depth \u0015sof\nthe material are both much larger than d. This frequency\nalso puts us in the classical limit ( ~!\u001ckBT), with the\nexception only for very low temperatures. Accordingly,\nwe develop a simpli\fed but systematic treatment of the\nmagnetic-noise generation from charge, valley, and spin\ndegrees of freedom, which we now turn to.\nCharge. |To introduce our approach, we \frst repro-\nduce the established results for the Johnson-Nyquist\nnoise. The Langevin dynamics [26, 27] of the electron\ncurrent (the number \rux density) jc=\u0000\u001br\u0016c+\u000fcis\nconsidered, where \u001bis the conductivity (neglecting the\nHall e\u000bect [28]), \u0016cis the electrochemical potential, and\n\u000fcis a Gaussian white noise with h\u000fc\ni(r;t)i= 0 and\nh\u000fc\ni(r;t)\u000fc\ni0(r0;t0)i= 2\u001bkBT\u000eii0\u000e(r\u0000r0)\u000e(t\u0000t0). The co-\ne\u000ecient in the correlation follows from the equipartition\ntheorem [29], and is consistent with the \ructuation dis-\nsipation theorem [30]. This treatment assumes the dif-\nfusive regime, focusing on dynamics on the length scale\nmuch larger than the mean free path of the electrons,3\nand at frequencies much lower than the momentum re-\nlaxation rate. Focusing on the transverse components\nof the charge current, r\u0001jc= 0, we have the following\ndi\u000berential equation for the electrochemical potential \u0016c:\n\u001br2\u0016c=r\u0001\u000fc: (4)\nThe charge current does not \row across the surface\njc\nz(r;t)\f\f\f\nz=0= 0, giving the Neumann boundary condi-\ntion\u001b@z\u0016c(r;t)\f\f\f\nz=0=\u000fc\nz(r;t)\f\f\f\nz=0. Solving the di\u000beren-\ntial equation yields\njc(r;t) =\u000fc(r;t) +Z\n\nd3r0\u000fc(r0;t)\u0001rr0rrGL(r;r0);\n(5)\nwhereGL(r;r0) is the Green function for the Lapla-\ncian compatible with the Neumann boundary condi-\ntion [31].\nThe magnetic \feld generated by the charge current is\ngiven by the Biot-Savart law,\nBc(r;t) =\u0000e\ncZ\n\nd3r0jc(r0;t)\u0002(r\u0000r0)\njr\u0000r0j3; (6)\nwherecis the speed of light, and we have taken the\ncharge carriers to have charge \u0000e. Inserting the magnetic\n\feld (6) into the de\fnition (3), we obtain the magnetic\nnoise due to charge \ructuations,\nBc\nii0(!) =\u0019e2kBT\u001b\nc2d\u0003ii0; (7)\nwhere \u0003 = diag(1 =2;1=2;1). This reproduces exactly the\nresult from the formulation in terms of transmission and\nre\rection of electromagnetic \felds at a metal surface [33{\n35], in the magnetostatic limit.\nValley. |We next apply this approach to the valley\ndegree of freedom. Di\u000berent from the charge current,\nthe valley \ructuations are neutral and thus compressible,\nwhich, as we have pointed out, turns out to be crucial for\ngenerating magnetic noise. The weak intervalley scatter-\ning allows us to consider electron conservation in each\nvalley separately: For the valley p(p=\u0006),\n@t\u001ap+r\u0001jp= 0; (8)\nwhere\u001ap= (\u0017=2)\u0016pandjp=\u0000(\u001b=2)r\u0016p+\u000fp. The to-\ntal density of states \u0017at the Fermi surface includes both\nvalleys, consistent with the convention of the total charge\ndensity\u001ac=\u001a++\u001a\u0000and the average electrochemical po-\ntential\u0016c= (\u0016++\u0016\u0000)=2. The Langevin noise here obeys\nh\u000fp\ni(r;t)\u000fp0\ni0(r0;t0)i=\u001bkBT\u000epp0\u000eii0\u000e(r\u0000r0)\u000e(t\u0000t0), suppos-\ning \ructuations in the two valleys are uncorrelated. We\ntherefore arrive at the stochastic di\u000busion equation\n@t\u0016p\u0000Dr2\u0016p=\u00002\n\u0017r\u0001\u000fp; (9)whereD=\u001b=\u0017is the di\u000busion coe\u000ecient, as given by\nthe Einstein relation. Under the boundary condition\njp\nz(r;t)\f\f\f\nz=0= 0, we solve for \u0016pto obtain the currents\njp(r;t) =\u000fp(r;t)\u0000DZ\n\nd3r0Zt\n\u00001dt0\n\u000fp(r0;t0)\u0001rr0rrGD(r;r0;t;t0);(10)\nwhereGD(r;r0;t;t0) is the Green function for the di\u000bu-\nsion equation satisfying the homogeneous boundary con-\ndition [31].\nOur model dictates that the valley current (2) gener-\nates magnetic \felds equivalently to a local magnetization,\nvia the demagnetization kernel,\nBv(r;t) =gv\u0016B\nvZ\n\nd3r0\u0012\n\u0000rrrr01\njr\u0000r0j\u0013\n\u0001jv;(11)\nwheregvis the e\u000bective g factor characterizing the cou-\npling of the valley degree of freedom to an external mag-\nnetic \feld, and \u0016Bis the Bohr magneton. The valley\ncontribution to the magnetic noise can then be calcu-\nlated, yielding\nBv\nii0(!) =\u0010gv\u0016B\nv\u001124\u0019kBT\u001b\nd3\u0003ii0Z\nd\u0018\u00182e\u00002\u0018Iv(\u0018;\u0010);\n(12)\nwith dimensionless quantities \u0018=Kd,\u0010=!d2=D, and\na=p\n\u0000i!=DK2+ 1 =p\n\u0000i\u0010=\u00182+ 1,\nIv(\u0018;\u0010) = 1 +1 +aa\u0003\n(a+a\u0003)aa\u0003\u00001\na\u0003\u00001\na: (13)\nThe integration is essentially taken over the magnitude\nof the wavevector K=jKjin thexyplane, with \u00182e\u00002\u0018\nas a form factor. The anisotropy tensor \u0003 is the same\nas inBc(7). The frequency dependence of the magnetic\nnoise is contained in Iv, which scales as !2for!!0\nand approaches 1 for !!1 , 1\u0000Iv\u0018!\u00001=2. The mag-\nnetic noise vanishes at zero frequency, consistent with our\nunderstanding that transverse components of the valley\ncurrent do not contribute, in contrast to the charge cur-\nrent.\nWe estimate the magnitude of the valley contribution\nrelative to the charge contribution,\nBv\nzz\nBczz\u0018g2\nv\u0010c\u000b\nv\u00112\u0010a0\nd\u00112Z\nd\u0018\u00182e\u00002\u0018Iv(\u0018;\u0010); (14)\nwhere we have reduced the result to physical constants,\nnamely the \fne-structure constant \u000band the Bohr radius\na0. The e\u000bective g factor gvcan be greatly enhanced by\nthe strong spin-orbit coupling, especially in topological\nsemimetals [36{38]. Taking gv\u0018100, the Fermi velocity\nv\u0018105m/s [39], the NV distance d\u0018100 nm, and\nevaluating the integral numerically at \u0010= 5, we obtain\nBv\nzz=Bc\nzz\u00180:2. Recalling that Bc(7) is spectrally \rat, in4\nour treatment,Bvcan be easily recognized, for example,\nby taking a frequency derivative of the total measured\nnoise@!B, exploiting the high frequency resolution [13]\nof NV probes. Moreover, this will not be spoiled by the\npresence of spin contribution, which will be seen to have\na distinct frequency behavior.\nSpin. |Fluctuations of the magnetic order, in the form\nof thermal magnon excitations, constitute a third ingre-\ndient of the magnetic noise in magnetic WSMs. Consid-\nering the magnon gap in magnetic WSMs can be several\nmeV [40, 41], we focus on the subgap magnetic noise gen-\nerated by the longitudinal (relative to the order parame-\nterM) spin \ructuations [25, 42]. The Langevin dynamics\nof the longitudinal spin density szcan be described by\na stochastic di\u000busion equation, sometimes referred to as\nthe Cahn-Hilliard model [26, 43]:\n@tsz\u0000Dsr2sz=\u000fs; (15)\nwhereDsis the spin di\u000busion constant. Here, the\nGaussian white noise \u000fshas the correlation function\nh\u000fs(r;t)\u000fs(r0;t0)i=\u00002\u001bskBTr2\u000e(r\u0000r0)\u000e(t\u0000t0), where\nthe spin conductivity \u001bsis related to the spin di\u000busion\nconstantDs=\u001bs=\u001f0by the static longitudinal spin sus-\nceptibility\u001f0. The corresponding boundary condition\njs\nz(r;t)\f\f\f\nz=0= 0 is speci\fed in terms of the spin current\njs=\u0000Dsrsz. The solution is\nsz(r;t) =Z\n\nd3r0Zt\n\u00001dt0Gs\nD(r;r0;t;t0)\u000fs(r0;t0);(16)\n1050100500\n2510501005000.51.05.010.01.05.010.050.0\n50.0\nspinvalleyvalley+spindip\n1002000.40.81.24810020012\nFIG. 3. The frequency-dependent magnetic noise generated\nby valley (green) and spin (red) \ructuations in a linear (top)\nand a log-log (bottom) plot. The total magnetic noise (black)\nexhibits a \\dip\" feature at frequency !dip. Two NV distances\nd= 100 nm (solid lines) and d= 150 nm (dashed lines) are\nplotted. Inset: !dipas a function of d. The vertical axis of\nthe bottom plot spans three orders of magnitude (in arbitrary\nunits).whereGs\nD(r;r0;t;t0) is simply the Green function of\nEq. (10) with Dreplaced by Ds.\nThe spin density generates demagnetization \felds, and\ntheir the correlation function gives the magnetic noise\nBs\nii0(!) = (gs\u0016B)22\u0019kBT\u001f0\n!d3\u0003ii0Z\nd\u0018\u00182e\u00002\u0018Is(\u0018;\u0011);\n(17)\nwheregsis the g factor of localized spins, and\nIs(\u0018;\u0011) =\u0000i\u00141\nb(b+ 1)\u00001\nb\u0003(b\u0003+ 1)\u0015\n(18)\nwhereb=p\n\u0000i\u0011=\u00182+ 1,\u0018=Kdand\u0011=!d2=Ds. As\n!!0,Isscales as!andBsconverges to a \fnite value.\nFor!!1 ,Is\u00181=!andBsvanishes asymptotically as\n1=!2.\nAccording to recent experiments around room temper-\nature, the resultant NV transition rate from charge \ruc-\ntuations in a good conductor [8] and that from longitu-\ndinal spin \ructuations in a magnetic insulator [44] are\nboth of ms\u00001order. In magnetic WSMs with relatively\nlow electrical conductivity, spin \ructuations may domi-\nnate the magnetic noise at low frequencies. However, the\nspin contribution quickly drops a few orders of magnitude\nwith increasing frequency. Consequently, the total mag-\nnetic noise coming from valley and spin exhibits a \\dip\"\nfeature in its frequency dependence, as shown in Fig. 3,\non top of the \rat charge contribution. The frequency\n!dipat the bottom of the dip shifts lower at a larger dis-\ntanced. For the plot, we have assumed similar charge\nand spin conductivities \u001b\u0018\u001bs[45], and taken both elec-\ntron and magnon mean free paths \u001830 nm, Fermi veloc-\nity\u0018105m/s and magnon velocity \u0018103m/s, yielding\nD\u001810\u00003m2/s andDs\u001810\u00005m2/s.\nDiscussion. |It should be remarked that the valley\nnoise derived here is based on a speci\fc model (1), as\na demonstration of our formalism. The resulted predic-\ntion is not universal for all WSMs. Di\u000berent symmetry\nassumptions about spin and parity in the band model\ncan lead to a di\u000berent coupling scheme of the valley de-\ngree of freedom to magnetic \felds. The frequency and\ndistance dependence of the magnetic noise can therefore\nalso re\rect the applicability of the e\u000bective theories. For\nspeci\fc materials, one may need to construct the cou-\npling scheme from \frst-principles calculations.\nWe have considered the magnetic noise primarily\ndriven by the di\u000busive transport, neglecting the valley\nand spin relaxation e\u000bects. This applies to NV distances\nmuch smaller than the corresponding di\u000busion lengths\n(while larger than the mean free paths), which is of-\nten the case in a clean material. A \fnite valley relax-\nation time modi\fes the valley di\u000busion equation, and in\nour Langevin approach, is accompanied by an interval-\nley \ructuation of the axial charge density. Likewise, spin\ndensity relaxation and \ructuation can be added to the\nCahn-Hilliard model. We refer to the discussion in the5\nsupplemental material [31]. Furthermore, our treatment\nfocuses on the low-frequency regime, where dynamic ef-\nfects of the demagnetization \felds, such as the Faraday\ninduction, are negligible.\nThe Fermi arc states are universal in WSMs and are\ngenerally dissipative [46, 47]. The magnetic noise they\ngenerate, which is not detected in our choice of geome-\ntry, may also be interesting to look into, by placing the\nNV near material surfaces parallel to the magnetic order\nM. The Fermi-arc contribution may be recognized by\nits distance dependence [48], which is di\u000berent from the\nbulk one. For the spin channel, the change in the surface\norientation relative to the order parameter would acquire\na geometric factor [49]. 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Phys.\n14, 1125 (2018).7\nSupplemental Material for\n\\Flavors of Magnetic Noise in Quantum Materials\"\nShu Zhang and Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nIn this document, we present the details on the generation of the magnetic noise by the \ructuations of the charge,\nvalley, and spin degrees of freedom in a magnetic Weyl semimetal, via di\u000berent generating kernels. Speci\fcally, we\nlook into the magnetic \feld B(rNV;t) at the position of the NV center rNV= (0;0;d) generated by the material\noccupying the half space \n with z\u00140, assuming the material dimensions are much larger than d.\nThe following identity and its variants will be frequently used, which is convenient for the broken translational\ninvariance in the zdirection,\n1\njr\u0000r0j=Zd2K\n(2\u0019)22\u0019\nKeiK\u0001(R\u0000R0)\u0000Kjz\u0000z0j; (S.1)\nwhere R= (x;y;0), and K= (kx;ky;0) denotes the wavevector in the xandydirections,K=jKj.\nI. Transport\nA. Charge\nWe take a stochastic approach and study the response of the charge current to a Langevin noise, and verify that our\nmethod reproduces the known result for the Johnson-Nyquist noise from the thermal \ructuations of electric charge\ncurrent in a metal. The charge transport obeys the continuity equation\n@t\u001ac+r\u0001jc= 0: (S.2)\nIn the low frequency limit, the charge density \ructuation is e\u000eciently screened by Coulomb interactions. We thus\nconsider the transverse current \ructuations only, r\u0001jc= 0, where jc=\u0000\u001br\u0016c+\u000fc. Here\u001bis the conductivity,\n\u0016cis the chemical potential, and \u000fcis a Gaussian white noise satisfying h\u000fc\ni(r;t)i= 0 andh\u000fc\ni(r;t)\u000fc\ni0(r0;t0)i=\n2\u001bkBT\u000eii0\u000e(r\u0000r0)\u000e(t\u0000t0), as a result of the \ructuation dissipation theorem at high temperatures. We \fnd the\nresponse of the charge current jcto the thermal noise \u000fcby solving the following di\u000berential equation\n\u001br2\u0016c=r\u0001\u000fc: (S.3)\nIn a free space, it is easy to show by the Fourier transform that \u001bk2\u0016c=k\u0001\u000fcandjc=\u000fc\u0000kk\u0001\u000fc=k2, which gives\nthe familiar transverse current correlation functions hjc\ni(r;t)jc\ni0(r0;t0)i=\u0000\n\u000eii0\u0000kiki0=k2\u0001\n2\u001bkBT\u000e(r\u0000r0)\u000e(t\u0000t0).\nHere, since the current does not \row out of the material, we solve Eq. (S.3) in the half space \n with the Neumann\nboundary condition \u001b@z\u0016c(r;t)\f\f\f\nz=0=\u000fc\nz(r;t)\f\f\f\nz=0. The Green function for the Laplacian with the homogeneous\nNeumann boundary condition is [the time dependent factor is simply \u000e(t\u0000t0)]\nGL(r;r0) =\u00001\n4\u0019\"\n1p\n(x\u0000x0)2+ (y\u0000y0)2+ (z\u0000z0)2+1p\n(x\u0000x0)2+ (y\u0000y0)2+ (z+z0)2#\n: (S.4)\nTherefore,\n\u0016c(r;t) =Z\n\nd3r0GL(r;r0)rr0\u0001\u000fc(r0;t)\n\u001b\u0000Z\n@\nd2r0GL(r;r0)\u000fc\nz(r0;t)\n\u001b; (S.5)\nand integrating by parts\njc(r;t) =\u000fc(r;t) +Z\n\nd3r0\u000fc(r0;t)\u0001rr0rrGL(r;r0): (S.6)\nUsing the Fourier transform of the Green function (S.4) in the xandydirections with R= (x;y;0),\nGL(K;z;z0) =\u00001\n2K\u0010\ne\u0000Kjz\u0000z0j+e\u0000Kjz+z0j\u0011\n; (S.7)8\nwe take the Fourier transfrom of jc(r;t) in the following form,\njc(K;t) =Z0\n\u00001dzeKzZ\nd2Re\u0000iK\u0001Rjc(r;t)\n=Z0\n\u00001dzeKzZ\nd2Re\u0000iK\u0001R[\u000fc(r;t)\u0000\u000fc\nz(r;t)^z]\n\u00001\n2KZ0\n\u00001dz0eKz0Z\nd2R0e\u0000iK\u0001R0\u000fc(r0;t)\u0001\u0012\nK\u0014\u00031\nK\u0000\u0014\u0014z0\u0013\n;(S.8)\nwhere \u0014=K+iK^z. The second term in the square brackets comes from the Dirac \u000e-function in rr0rrGc(r;r0).\nIn the magnetostatic limit, where the wavelength of the electromagnetic \feld and the skin depth are both much\nlarger than the NV distance d, the Biot-Savart law gives the magnetic \feld generated by the electric charge current\ndistribution,\nBc(rNV;t) =\u0000e\ncZ\n\nd3r jc(r;t)\u0002rr1\njrNV\u0000rj\n=\u0000e\ncZd2K\n(2\u0019)22\u0019\nKe\u0000Kd(i\u0014)\u0002jc(K;t)\n=\u0000e\ncZd2K\n(2\u0019)22\u0019\nKe\u0000Kd\u0014\ni\u0014\u0002\u000fc(K;t) +1\nK(^z\u0002K)\u000fc(K;t)\u0001\u0014\u0015\n;(S.9)\nwhere \u000fc(K;t) =R0\n\u00001dzeKzR\nd2Re\u0000iK\u0001R\u000fc(r;t). We have used \u0014\u0002\u0014\u0003=i2Kz\u0002Kand\u0014\u0002\u0014= 0. This result\nreproduces that in Ref. [35] derived from Maxwell equations with proper boundary conditions, and is consistent with\nthe approach of \fnding the Fresnel coe\u000ecients [33, 34]. Finally, we obtain the magnetic noise tensor contributed from\ncharge \ructuations, according to Eq. (3),\nBc\nii0(!) =\u0010e\nc\u00112Z\nd2K2\u001bkBT\n2K3e\u00002Kd\u0012\n\u000fijk\u000fi0j0k\u0014j\u0014\u0003\nj0+i\nK\u000fijk\u000fi0zk0\u0014jKk0\u0014\u0003\nk\u0000i\nK\u000fi0j0k0\u000fizk\u0014\u0003\nj0Kk\u0014k0+ 2\u000fizk\u000fi0zk0KkKk0\u0013\n=\u0010e\nc\u00112Z\ndKe\u00002Kd2\u0019\u001bkBT\u0001ii0\n=\u0019e2kBT\u001b\nc2d\u0001ii0;\n(S.10)\nwhere the tensor \u0001 = diag(1 =2;1=2;1) and Einstein summation is taken over repeated indices.\nB. Valley\nFollowing the same approach, we next look into the contribution from the valley current. We \frst focus on the\nparticle conservation in a single valley p, assuming the interaction and scattering between the two Weyl nodes are\nweak. We do not consider the chiral-anomaly related perturbations here.\n@t\u001ap+r\u0001jp= 0; (S.11)\nwhere\u001ap= (\u0017=2)\u0016pandjp=\u0000(\u001b=2)r\u0016p+\u000fp. Here\u0017refers to the total density of states at the Fermi surface,\nincluding both valleys. Assuming no correlation between \u000f+and\u000f\u0000,h\u000fp\ni(r;t)\u000fp0\ni0(r0;t0)i=\u001bkBT\u000epp0\u000eii0\u000e(r\u0000r0)\u000e(t\u0000t0).\nThe di\u000berential equation to be solved is then\n@t\u0016p\u0000Dr2\u0016p=\u00002\n\u0017r\u0001\u000fp(S.12)\nwhere the di\u000busion coe\u000ecient D=\u001b=\u0017, as is consistent with the Einstein relation and the Drude law. The boundary\nconditionjp\nz(r;t)\f\f\f\nz=0= 0 is similar to the case with charge. We use the Green function for the di\u000busion equation9\nsatisfying the homogeneous Neumann boundary condition, and its Fourier transform\nGD(r;r0;t;t0) =\u0002(t\u0000t0)\n[4\u0019D(t\u0000t0)]3=2e\u0000[(x\u0000x0)2+(y\u0000y0)2]=4D(t\u0000t0)\u0010\ne\u0000(z\u0000z0)2=4D(t\u0000t0)+e\u0000(z+z0)2=4D(t\u0000t0)\u0011\n;\nGD(K;z;z0;!) =1\n2DaK\u0010\ne\u0000aKjz\u0000z0j+e\u0000aKjz+z0j\u0011\n;(S.13)\nwhere \u0002(t\u0000t0) is the Heaviside step function. Here, the dimensionless parameter a=p\n\u0000i!=DK2+ 1 carries the\nfrequency dependence in the following context, and is evaluated in the branch with Re a>0. The current distribution\nis therefore\njp(r;t) =\u000fp(r;t)\u0000DZ\n\nd3r0Zt\n\u00001dt0\u000fp(r0;t0)\u0001rr0rrGD(r;r0;t;t0); (S.14)\nand\njp(K;!) =Z0\n\u00001dzeKzZ\nd2RZ\ndtei!t\u0000iK\u0001Rjp(r;t)\n=Z0\n\u00001dzeKzZ\nd2RZ\ndtei!t\u0000iK\u0001R[\u000fp(r;t)\u0000\u000fp\nz(r;t)^z]\n\u00001\n2aK2Z0\n\u00001dz0Z\nd2R0Z\ndtei!t0\u0000iK\u0001R0\u000fp(r0;t0)\u0001\"\nk?k?eKz0\na+ 1+kkeKz0\u0000eaKz0\na\u00001+kk?eaKz0\na+ 1#\n;(S.15)\nwhere k=K+iaK^z, andk?=K\u0000iaK^z.\nThe valley current density behaves like a local magnetization in generating the magnetic \feld:\nBv(rNV;!) =gv\u0016B\nvZ\n\nd3r\u0012\n\u0000rrNVrr1\njrNV\u0000rj\u0013\n\u0001jv(r;!)\n=\u0000gv\u0016B\nvZd2K\n(2\u0019)22\u0019\nKe\u0000Kd\u0014\u0014\u0001\u0002\nj+(K;!)\u0000j\u0000(K;!)\u0003\n=\u0000gv\u0016B\nvZd2K\n(2\u0019)22\u0019\nKe\u0000KdZ0\n\u00001dzZ\nd2RZ\ndtei!t\u0000iK\u0001R\u0014\u0012\neKz\u0014\u0000eaKz\nak\u0013\n\u0001\u0002\n\u000f+(r;t)\u0000\u000f\u0000(r;t)\u0003\n;\n(S.16)\nwheregvis the e\u000bective g factor describing the coupling of the valley degrees of freedom to an external magnetic \feld\nandvis the Fermi velocity. We have used \u0014\u0001k=\u0000K2(a\u00001) and \u0014\u0001k?=K2(a+ 1). With the help of\nDZ0\n\u00001dzec1zZ\nd2RZ\ndte\u0000i!t+iK\u0001R\u000fp\ni(r;t)Z0\n\u00001dz0ec2\u0003z0Z\nd2R0Z\ndt0ei!0t0\u0000iK0\u0001R0\u000fp0\ni0(r0;t0)E\n=\u001bkBT\nc1+c2\u0003(2\u0019)3\u000epp0\u000eii0\u000e(K\u0000K0)\u000e(!\u0000!0);(S.17)\nwe obtain the magnetic noise tensor generated by valley \ructuations\nBv\nii0(!) =1\n2\u0019hBi(rNV;!)B\u0003\ni0(rNV;!)i\n=\u0010gv\u0016B\nv\u00112Zd2K\n(2\u0019)2(2\u0019)2\nKe\u00002Kd2\u001bkBT\u0014i\u0014\u0003\ni0\u0014\n1 +1 +aa\u0003\n(a+a\u0003)aa\u0003\u00001\na\u0003\u00001\na\u0015\n=\u0010gv\u0016B\nv\u001124\u0019kBT\u001b\nd3\u0003ii0Z\nd\u0018\u00182e\u00002\u0018Iv(\u0018;\u0010);(S.18)\nwhere\nIv(\u0018;\u0010) = 1 +1 +aa\u0003\n(a+a\u0003)aa\u0003\u00001\na\u0003\u00001\na; (S.19)\nwith dimensionless quantities \u0018=Kd,\u0010=!d2=D, anda=p\n\u0000i\u0010=\u00182+ 1. The anisotropy tensor \u0001 is the same as in\nBc.10\nC. Spin\nWe invoke the Cahn-Hilliard model [43] to describe the Langevin dynamics of the longitudinal spin density sz\n@tsz\u0000Dsr2sz=\u000fs; (S.20)\nwhereDsis the spin di\u000busion constant and \u000fsis a scalar Gaussian white noise with the correlation function\nh\u000fs(r;t)\u000fs(r0;t0)i=\u00002\u001bskBTr2\u000e(r\u0000r0)\u000e(t\u0000t0), containing the Laplacian operator. The spin conductivity \u001bsis\nrelated to the spin di\u000busion constant Ds=\u001bs=\u001f0by the static longitudinal spin susceptibility \u001f0. The boundary\nconditionjs\nz(r;t)\f\f\f\nz=0= 0 is speci\fed in terms of the spin current js=\u0000Dsrsz. The solution is\nsz(r;t) =Z\n\nd3r0Zt\n\u00001Gs\nD(r;r0;t;t0)\u000fs(r0;t0): (S.21)\nThe Green function Gs\nD(r;r0;t;t0) is simply Eq. (S.13) with D!Ds,\nGs\nD(r;r0;t;t0) =\u0002(t\u0000t0)\n[4\u0019Ds(t\u0000t0)]3=2e\u0000[(x\u0000x0)2+(y\u0000y0)2]=4Ds(t\u0000t0)\u0010\ne\u0000(z\u0000z0)2=4Ds(t\u0000t0)+e\u0000(z+z0)2=4Ds(t\u0000t0)\u0011\nGs\nD(K;z;z0;!) =1\n2DbK\u0010\ne\u0000bKjz\u0000z0j+e\u0000bKjz+z0j\u0011\n;(S.22)\nwhereb=p\n\u0000i!=DsK2+ 1. Taking the Fourier transform,\nsz(K;!) =Z0\n\u00001dzeKzZ\nd2RZ\ndtei!t\u0000iK\u0001Rsz(r;t)\n=Z\n\nd3r0Zt\n\u00001dt0ei!t0\u0000iK\u0001R01\nDsbK2beKz0\u0000ebKz0\n(b+ 1)(b\u00001)\u000fs(r0;t0)(S.23)\nThe spin density generates stray \felds via the demagnetization kernel,\nBs(rNV;!) =gs\u0016BZ\n\nd3r\u0012\n\u0000rrNVrr1\njrNV\u0000rj\u0013\n\u0001sz(r;!)^z\n=gs\u0016BZd2K\n(2\u0019)22\u0019\nKe\u0000Kd\u0014Z0\n\u00001dzZ\nd2RZt\n\u00001dtei!t\u0000iK\u0001R 1\niDsbKbeKz\u0000ebKz\n(b+ 1)(b\u00001)\u000fs(r;t):(S.24)\nUsing the correlation relation of the Gaussian noise, we obtain the magnetic noise tensor\nBs\nii0(!) = (gs\u0016B)2Zd2K\n(2\u0019)2(2\u0019)2\nKe\u00002Kd\u0014i\u0014\u0003\ni0\u001bskBT\niDs!\u00141\nb(b+ 1)\u00001\nb\u0003(b\u0003+ 1)\u0015\n= (gs\u0016B)22\u0019kBT\u001f0\n!d3\u0003ii0Z\nd\u0018\u00182e\u00002\u0018Is(\u0018;\u0011);(S.25)\nwheregsis the g factor of localized spins, and\nIs(\u0018;\u0011) =\u0000i\u00141\nb(b+ 1)\u00001\nb\u0003(b\u0003+ 1)\u0015\n(S.26)\nwith\u0018=Kd,\u0011=!d2=Dsandb=p\n\u0000i\u0011=\u00182+ 1.\nII. Relaxation\nA. Spin\nIn the Langevin approach, the spin relaxation can be handled by replacing the correlation functions of the Gaussian\nnoise byhe\u000fs(r;t)e\u000fs(r0;t0)i= 2kBT\u001f0(1=\u001cs\u0000Dsr2)\u000e(r\u0000r0)\u000e(t\u0000t0) in\n\u0012\n@t+1\n\u001cs\u0013\nsz\u0000Dsr2sz=e\u000fs; (S.27)11\nwhere\u001csis the spin relaxation time. This is because the consideration of the spin relaxation has to be accompanied\nby a spin density \ructuation, as dictated by the \ructuation-dissipation theorem. Here, the Green function and its\nFourier transform are, respectively,\neGs\nD(r;r0;t;t0) =\u0002(t\u0000t0)\n[4\u0019Ds(t\u0000t0)]3=2e\u0000[(x\u0000x0)2+(y\u0000y0)2]=4Ds(t\u0000t0)\u0010\ne\u0000(z\u0000z0)2=4Ds(t\u0000t0)+e\u0000(z+z0)2=4Ds(t\u0000t0)\u0011\ne\u0000(t\u0000t0)=\u001cs;\neGs\nD(K;z;z0;!) =1\n2DsebK\u0010\ne\u0000ebKjz\u0000z0j+e\u0000ebKjz+z0j\u0011\n;\n(S.28)\nwhereeb=p\n(\u0000i!+ 1=\u001cs)=DsK2+ 1. The magnetic noise thus becomes\neBs\nii0(!) = (gs\u0016B)2Zd2K\n(2\u0019)2(2\u0019)2\nK3e\u00002Kd\u0014i\u0014\u0003\ni02kBT\u001f0\nDs(1 +eb+eb\u0003)(eb+ 1)(eb\u0003+ 1) + [2(eb+eb\u0003+ 1)2+ebeb\u0003(eb+eb\u0003)]=2K2Ds\u001cs\nebeb\u0003(eb+ 1)2(eb\u0003+ 1)2(eb+eb\u0003)\n= (gs\u0016B)2\u0019kBT\u001f0\nDs\u0001ii0Z\ndKe\u00002Kd\"\n2 +eb\neb(eb+ 1)2+2 +eb\u0003\neb\u0003(eb\u0003+ 1)2#\n:\n(S.29)\nIn the limit 1 =\u001cs!0, Reh\n(2 +eb)=eb(eb+ 1)2i\n!Im\u0002\n(2DsK2=!)=b(b+ 1)\u0003\n, we arrive at the same result as in Eq. (S.25).\nIn the opposite limit \u001cs!0, (2 +eb)=eb(eb+ 1)2\u0019DsK2\u001cs,Bs\nii0(!)!(gs\u0016B)2\u0019kBT\u001f0\u001cs\u0001ii0=2d3is independent of\nDs, as expected from the relaxation dominated equation of motion.\nAlternatively, the noise can be computed from the dynamic spin susceptibility, with the help of the \ructuation\ndissipation theorem. We derive dynamic spin susceptibility from the di\u000busion equation for the longitudinal spin\ndynamics\n@tsz+r\u0001js=\u0000\u001f0\n\u001cs\u0016s; (S.30)\nwhere js=\u0000\u001bsr\u0016sis the spin current, \u0016s=sz=\u001f0\u0000his the spin chemical potential, and his a force thermody-\nnamically conjugate to sz, given by the external magnetic \feld. The spin distributionr relaxes towards \u001f0h(\u0016s= 0\nin equilibrium), characterized by the spin relaxation time \u001cs. The di\u000berential equation\n\u0012\n@t+1\n\u001cs\u0013\n\u0016s\u0000Dsr2\u0016s=\u0000@th: (S.31)\nagain has the boundary condition @z\u0016s(r;t)\f\f\f\nz=0= 0. From the solution of \u0016s(r;t), we have\nsz(r;t) =\u001f0Z\n\nd3r0Zt\n\u00001dt0h\n\u000e(r\u0000r0)\u000e(t\u0000t0)\u0000eGs\nD(r;r0;t;t0)@t0i\nh(r0;t0); (S.32)\nwhich de\fnes the longitudinal susceptibility \u001f(r;r0;t;t0). The Fourier transform of the dynamic spin susceptibility is\ntherefore\n\u001f(z;z0;K;!) =\u001f0\u0014\n\u000e(z\u0000z0) +i!\n2DsebK\u0010\ne\u0000ebKjz\u0000z0j+e\u0000ebKjz+z0j\u0011\u0015\n: (S.33)\nThe spin induced magnetic noise is then\neBs\nii0(!) = (g\u0016B)2Zd2K\n(2\u0019)2(2\u0019)2\nK2e\u00002Kd\u0014i\u0014\u0003\ni0\u0014j\u0014\u0003\nj0Z0\n\u00001dzZ0\n\u00001dz0eK(z+z0)2kBT\n!Im\u001f(z;z0;K;!)\n= (g\u0016B)22\u0019kBT\u001f0\nDs\u0001ii0Z\ndKe\u00002KdRe\"\n2 +eb\neb(eb+ 1)2#\n;(S.34)\nwhich reproduces the result (S.25) from the Langevin approach.12\nB. Valley\nIn the same spirit, we consider the following di\u000busion and relaxation of the valley degree of freedom\n\u0012\n@t+1\n\u001cv\u0013\n\u001av+r\u0001jv=e\u000fv; (S.35)\nwhere\u001av= (\u0017=2)\u0016v,\u0017is the total density of states at the Fermi surface, \u0016v= (\u0016+\u0000\u0016\u0000)/2 is the valley chemical\npotential, jv=\u0000(\u001b=2)r\u0016v+\u000fv, and\u001cvis the valley relaxation time. The intravalley noise has the same correlation\nas beforeh\u000fv\ni(r;t)\u000fv\ni0(r0;t0)i= 2\u001bkBT\u000eii0\u000e(r\u0000r0)\u000e(t\u0000t0), and we also account for an intervalley \ructuation of the\naxial charge density he\u000fv(r;t)i= 0 andhe\u000fv(r;t)e\u000fv(r0;t0)i= (\u0017=\u001cv)2kBT\u000e(r\u0000r0)\u000e(t\u0000t0). The di\u000berential equation to\nbe solved is then\n\u0012\n@t+1\n\u001cv\u0013\n\u0016v\u0000Dr2\u0016v=2\n\u0017(\u0000r\u0001\u000fv+e\u000fv); (S.36)\nwith the Neumann boundary condition. The Green function in this case is\neGD(r;r0;t;t0) =\u0002(t\u0000t0)\n[4\u0019D(t\u0000t0)]3=2e\u0000[(x\u0000x0)2+(y\u0000y0)2]=4D(t\u0000t0)\u0010\ne\u0000(z\u0000z0)2=4D(t\u0000t0)+e\u0000(z+z0)2=4D(t\u0000t0)\u0011\ne\u0000(t\u0000t0)=\u001cs;\neGD(K;z;z0;!) =1\n2DeaK\u0010\ne\u0000eaKjz\u0000z0j+e\u0000eaKjz+z0j\u0011\n;\n(S.37)\nwhereea=p\n(\u0000i!+ 1=\u001cv)=DK2+ 1. Following Eqs. (S.14-S.18),\nejv(r;t) =\u000fv(r;t)\u0000DZ\n\nd3r0Zt\n\u00001dt0[e\u000fv(r0;t0) +\u000fv(r0;t0)\u0001rr0]rreGD(r;r0;t;t0); (S.38)\nejv(K;!) =Z0\n\u00001dzeKzZ\nd2RZ\ndtei!t\u0000iK\u0001R[\u000fv(r;t)\u0000\u000fv\nz(r;t)^z]\n\u00001\n2eaK2Z0\n\u00001dz0Z\nd2R0Z\ndtei!t0\u0000iK\u0001R0\u000fv(r0;t0)\u0001 \nek?ek?eKz0\nea+ 1+ekekeKz0\u0000eeaKz0\nea\u00001+ekek?eeaKz0\nea+ 1!\n\u0000i\n2eaK2Z0\n\u00001dz0Z\nd2R0Z\ndtei!t0\u0000iK\u0001R0e\u000fv(r0;t0) \nekeKz0\u0000eeaKz0\nea\u00001+ek?eKz0+eeaKz0\nea+ 1!\n;(S.39)\nwhereek=K+ieaK^zandek?=K\u0000ieaK^z.\neBv(rNV;!) =\u0000gv\u0016B\nvZd2K\n(2\u0019)22\u0019\nKe\u0000Kd\u0014\u0014\u0001ejv(K;!)\n=gv\u0016B\nvZd2K\n(2\u0019)22\u0019\nKe\u0000KdZ0\n\u00001dzZ\nd2RZ\ndtei!t\u0000iK\u0001R\u0014\u0014\u0012\n\u0000eKz\u0014+eeaKz\neaek\u0013\n\u0001\u000fv(r;t) +ieeaKz\neae\u000fv(r0;t0)\u0015\n:\n(S.40)\nThe magnetic noise turns out to be\neBv\nii0(!) =\u0010gv\u0016B\nv\u00112Zd2K\n(2\u0019)2(2\u0019)2\nKe\u00002Kd(2\u001bkBT)\u0014i\u0014\u0003\ni0\u0014(ea\u00001)(ea\u0003\u00001)(1 +ea+ea\u0003) + 1=D\u001cvK2\n(ea+ea\u0003)eaea\u0003\u0015\n=\u0010gv\u0016B\nv\u00112\n(4\u0019\u001bkBT)\u0001ii0Z\nK2dKe\u00002Kd\u0014\n1\u0000ea+ea\u0003\n2eaea\u0003\u0015\n;(S.41)\nwhich goes back to Eq. (S.18) in the limit 1 =\u001cv!0. In the opposite limit \u001cs!0, the equation for the chemical\npotential is dominated by the intervalley \ructuation and relaxation, and essentially decouples from the intravalley\n\ructuations and the valley current transport. The latter continues to contribute to the magnetic noise eBv\nii0(!)!\n(gv\u0016B=v)(\u0019\u001bkBT=d3)\u0001ii0:13\nIII. Simple metal\nFor a paramagnetic metal, the spin \ructuations are associated with electrons on the Fermi surface and can be\ncompared with the charge \ructuations on an equal footing. The spin di\u000busion equation remains a valid description\nin this scenario. We follow the discussion below Eq. (S.29), replacing Dsand by the charge di\u000busion constant D. For\na system with a long spin relaxation time, the magnetic noise induced by spin transport yields\nBs\nzz\nBczz\u0018(gs\u0016B)22c2\u001f0\ne2\u001b!d2Z\nd\u0018\u00182e\u00002\u0018Is(\u0018;\u0010): (S.42)\nFor an estimation, we take \u001f0to be the Pauli paramagnetic susceptibility \u001fp\u0018\u0017the density of states at the Fermi\nsurface,\nBs\nzz\nBczz\u0018g2\ns\n2\u0010c\u000b\nv\u00112\u00123a0\n`mfp\u00132Z\nd\u0018\u00182e\u00002\u0018Is(\u0018;\u0010)=\u0010; (S.43)\nresulting in a ratio of 0 :001 withgs\u00182,v\u0018105m/s,`mfp\u001830 nm,D\u0018v`mfp=3 and\u0010\u00180:5. The presence of\nstrong spin-orbit interactions may lead to an extremely short spin relaxation time, such as in Pt, and thus the spin\ndecay length can be even shorter than the electron mean free path [32]. For v\u001cs\u0018`mfp, an estimation of\nBs\nzz\nBczz\u0018(gs\u0016B)2c2\u001f0\u001cs\n2e2\u001bd2\u001f0\u0018\u0017\u00183g2\ns\n8\u0010c\u000b\nv\u00112\u0012v\u001cs\n`mfp\u0013\u0010a0\nd\u00112\n(S.44)\nis also typically much smaller than 1. Therefore, the charge \ructuations dominate the magnetic noise in most circum-\nstances for simple metals. However, the spin susceptibility \u001f0can be greatly enhanced near the Stoner instability,\nleading to a larger contribution of the spin \ructuations to the magnetic noise."    },    {        "title": "2108.10622v1.Spin_orbit_torque__Moving_towards_two_dimensional_van_der_Waals_heterostructures.pdf",        "content": "1 \n Spin-orbit torque: Moving towards two-dimensional  van \nder Waals heterostructures  \nR. C. Sahoo1,2, Dinh Loc Duong1,3, Jungbum Yoon1, Pham Nam Hai4,5,6,* and Young Hee \nLee1,3,7,*  \n1Center for Integrated Nanostructure Physics (CINAP), Institute for Basic Scien ce (IBS), \nSuwon 16419, South Korea  \n2Department of Chemical Science and Engineering, Tokyo Institute of Technology, 2 -12-1 \nOokayama, Meguro, Tokyo 152- 8552, Japan  \n3Department of Energy Science, Sungkyunkwan University, Suwon 16419, South Korea  \n4Department of Elect rical and Electronic Engineering, Tokyo Institute of Technology , 2-12-\n1 Ookayama, Meguro, Tokyo 152- 0033, Japan  \n5Center for Spintronics Research Network (CSRN), The University of Tokyo , 7-3-1 Hongo, \nBunkyo, Tokyo 113- 8656, Japan  \n6CREST, Japan Science and Technology Agency , 4-1-8 Honcho, Kawaguchi, Saitama 332-\n0012, Japan  \n7Department of Physics, Sungkyunkwan University, Suwon 16419, South Korea  \n \n*) Electronic addresses: pham.n.ab@m.titech.ac.jp and leeyoung@skku.edu  \n \n \n \n  \n \n \n \n \n \n \n \n \n 2 \n Abstract:   \nThe manipulation of magnetic properties using either electrical currents or gate bias is \nthe key of future high -impact nanospintronics applications such as spin-valve read \nheads, non -volatile logic, and random -access memories. The current technology for \nmagnetic switching with spin -transfer torque requires high current densities, whereas \ngate-tunable magnetic materials such as ferromagnetic semiconductors and \nmultiferroic materials are still far from practical applications. Recently, magnetic \nswitching in duced by pure spin currents using the spin Hall and Rashba effects in \nheavy metals, called spin -orbit torque (SOT), has emerged as a candidate for \ndesigning next -generation magnetic memory with low current densities. The recent \ndiscovery of topological mat erials and two -dimensional (2D) van der Waals (vdW) \nmaterials provides opportunities to explore versatile 3D -2D and 2D -2D \nheterostructures with interesting characteristics. In this review, we introduce the emerging approaches to realizing SOT nanodevices i ncluding techniques to evaluate \nthe SOT efficiency as well as the opportunities and challenges of using 2D topological materials and vdW materials in such applications.  \n \n \n   \nKeywords: Two -dimensional  materials,  Topological Insulators, Spin orbit torque, La yered \nheterostructures, Future nanodevices.  3 \n  \nContents  \nIntroduction \nSpin- torque mechanism  \nSpin- transfer torque  \nSpin- orbit torque   \nProbing the SOT  \nHall measurements  \nSpin- torque ferromagnetic resonance  \nMagneto -optical Kerr effect  \nTraditional materials for S OT \nNonmagnetic metals for SOT  \nAntiferromagnets for SOT  \nTopological mate rials for SOT  \n2D materials for SOT  \n2D materials as torque layers  \n2D materials as FM layers  \n2D materials for both torque and FM layers  \nChallenges and opportunities for 2D SOTs  \n2D topolog ical insulators  \nNew 2D magnetic materials  \n2D magnetic semiconductors  \nHeterostructure growth of 2D materials  \nEnhancement of conductivity of 2D TMDs  \nAdditional out -of-plane SOT  \nMechanism of SOT  \n \n \n 4 \n 1. Introduction \nThe discovery of spin ordering modulation by me ans of carrier injection has paved the \nway for spin -based electronics since 1857 1. Spin- based electronics received a second -boost \nwith the revolutionary findings of giant magnetoresistance (GMR) and tunneling \nmagnetoresistance (TMR), where the magnetic st ates of nanoscale elements can be \naccurately determined from their electrical resistance in the presence of an external \nmagnetic field 2–5. The replacement of existing inductive read heads by GMR and TMR \nhard-drive read heads in data storage has resulted i n breakthroughs in storage capacity and \ndesign of random -access memory 2,5–8.  \nThe electrical resistance in GMR and TMR -based devices can be efficiently tuned by \nmodulating the magnetization directions of the materials. However, an external magnetic field (H\next), created by currents flowing through a nearby conductor, is required to control \nthe material magnetization in standard GMR and TMR memories. This limits the ability to scale down to nanosized devices \n5,9,10. In this scenario, one of the easiest and most effective \nways to actively manipulate magnetization of magnetic layers is to use a spin- polarized \ncurrent through  spin angular momentum transfer from the charge current, called spin- torque. \nThe spin angular momentum of the electron current exerts an e ffective field and torque on \nthe local  magnetization of nanoscale -thick magnetic layers. T he performance of GMR and \nTMR -based devices has recently been improved by applying the spin -torque technology. \nThe spin- torque -induced magnetization dynamics are inve stigated primarily using \nmicrolithography in various multilayers, especially nanopillar structures such as Co/Cu 11, \nCo/Cu/NiFeCo/Cu 12, NiFeCo/Cu/CoFe/Cu 13, and Co/Cu/Co 14. In these spin- transfer \ntorque (STT) devices, which consist of two ferromagnetic (FM) layers sandwiched by a non-magnetic (NM) thin layer, the spins of one FM layer experience a spin -torque due to \nangular momentum transfer from the other FM layer when spin- polarized electrons scatter \nat the interface \n15,16. However, this configuration requires a high current density because of \nthe low spin- transfer efficiency, which induces significant Joule -heating energy loss 17–19. \nAn emerging approach that uses the lateral current in FM/NM bilayer structures without an \nadditional magnetic layer to c reate spin -polarized currents called spin -orbit torque (SOT), 5 \n can be a promising solution for STT devices. It provides further opportunities for future \nnanospintronics through the enhanced efficiency of torque transfer between the FM and \nNM or torque layer s 20.  \nMeanwhile, the efficiency  of SOT devices can be further improved by choosing \nappropriate materials of ferromagnetic or non -magnetic layers. Such devices using three-\ndime nsional  (3D) materials have a long history but have a few drawbacks such as low \nefficiency, single dominating component of SOT on magnetization switching , and Joule \nheating or thermal fluctuation 21–24. These limitations should be overcome for a desired low \npower - and energy -efficient SOT devices for realistic applications. An alternative is to use \ntopological insulators (TIs) and van der Waals (vdWs) layered materials. Recently, TIs have \nbeen proposed as a promising candidate for efficient torque layer in SOT devices due to the \nunique behavior of charge carriers 25,26. They have spin- momentum locking feature in \nwhich the spin of the protected topological surface states lies in in -plane and is locked at \nright angles to the momentum of carriers (Fig. 1a). This feature results in the preferential \ngeneration of pure spin current in the dir ection perpendicular to the film plane 27–29. \nConsequently, an in- plane charge current flowing on the surface of TI can effectively \ngenerate a spin current perpendicular to the topological surface (Fig. 1b) 30. This spin \ncurrent can exert a spin -torque on the adjacent FM layer for magnetization switching. The \nimprovement of SOT efficiency of TI -based devices strongly depends on the carrier density \nof topological surface states as well as the position of Fermi level, which can be tuned by the impurity doping  in Bi\n2Te3 family (Fig. 1c) 31,32. Furthermore, SOT efficiency can be \nenhanced by modulating the layer thickness as well as applying external gate bias in TIs \n(Fig. 1d) 33,34.  \nAnother promising classes of SOT materials are two -dimensional  (2D) vdWs materi als, \nwhich have been demonstrated recently. The 2D materials with sufficiently low crystal \nsymmetry 35–37, crystal strain and strong Rashba or spin Hall -type SOC 38,39 are efficient to \ngenerate strong SOT, which is much energy resourceful compared to 3D ma terials. One of \nthe limits  of SOT applications (i.e., Joule heating or thermal fluctuation) can also be \nreduced by using 2D vdWs in monolayer limit. Additional advantages of 2D vdWs 6 \n materials are their strong SOC in atomically -thin layer 40, perpendicular magnetic \nanisotropy 41,42, tunable conductivity of topologically protected surface states and easy \ndevice fabrication. The intrinsic SOT in 2D vdWs (Fig. 1e), which strongly depend on the \nSOC -controlled electron spin, can be tuned by modulating crystal sym metry  38, external \nelectric field 43 and proximity effect with transition metal dichalcogenides (TMDs) 44. The \nSOC in graphene, for example, can be enhanced by the proximity effect of WS 2 44. \nFurthermore, the magnetism of the 2D FM layer can be modulated b y different approaches \nsuch as impurity doping (Fig. 1f), electrostatic gating (Fig. 1g), and layer thickness 45–47, \nrather than the traditional magnetic field. For example, magnetism can be introduced or \ntuned deliberately by magnetic dopants in monolayer  semiconductors (e.g. V -doped WSe 2, \nFe-doped SnS 2) 45,48, or by electro -gating using a liquid electrolyte in monolayer Fe 3GeTe 2 \n47. Layer -dependent magnetic properties are also observed in CrI 3 49,50, further implying the \ndimensionality effect in the 2D vdW materials. In the 3D, the distance between dopants is proportional to 1/n\n1/3, whereas it is proportional to 1/n1/2 in the 2D (n stands for percent \nconcentration of dopants) 51. The difference in the scaling rule between 2D and 3D gives \nrise to  different  interaction strength between dopant s in 2D and 3D forms  at a given doping \nconcentration. Furthermore, the pd- d hybridization in 2D TMD DMSs may be more \nadvantageous in applications than the weak sp -d hybridization in III -V semiconductors (e.g. \nMn-doped GaA s) 52. The stronger pd- d hybridization presents opportunities to overcome \nthe Tc limit in 3D semiconductors.  Meanwhile, proximity  exchange interactions between \n2D-FMs and 2D -SOC play a key role to improve SOT strength in 2D -based SOT devices \n(Fig. 1h) 53.  \nAppearing as an emerging field, SOT phenomenon and devices were reviewed \nextensively from basic principles, experimental methods, and materials 37,54 –57.  However, \nthe quick explorations and developments in this research field, especially in topological \ninsulators and 2D materials 31–34,37,57, require a comprehensive updated review. Here, our \npurpose of this review article is to bring the basic advantages of 2D materials such as \ntopological insulators and van der Waals materials, operation principle of spi n-orbit torque \nin general, the physical mechanism behind it. The three  well-known methods to probe SOT 7 \n efficiency are introduced. We summarize the state- of-art SOT efficiency of different \nmaterials for spin -orbit torque devices and explain the reasons why topological insulator \nand 2D van der Waals materials are promising candidates for improving the spin- orbit \ntorque devices. In the end, we list some challenges and perspectives of using topological \ninsulators and 2D materials for this research field.  \n \n2. Spin- torque mechanism  \nTwo important phenomena in generating the spin- torque at the interface of the materials \nare STTs and SOTs. Both phenomena are explained by the similar mechanism, although the \nconfigurations of the device structures are different. The key  mechanism is the conservation \nof angular momentum during the scattering process of spin -polarized electrons in magnetic \nmaterials. Consequently, the spin- torque is generated and transferred to the adjacent \nmaterials, which allows for efficient switching  of the magnetic moment. The efficient spin -\ntorque mechanism using the least power consumption is the key to modern \nnanosprintronics. We will discuss each of these spin- torque  mechanisms in  detail  below .  \n2.1 Spin -transfer torque  \nA typical structure of the STT device consists FM/NM/FM trilayer heterostructure (Fig. \n2a) 19,58,59. Here, the  two FM layers have non- collinear magnetizations. The magnetization \nof the first FM layer is fixed ( Mfixed), whereas the second FM layer has a free magnetization \n(Mfree), which can be easily switched. A spin -unpolarized charge current, j c (red arrow), \nflows from the free to the fixed FM layer  (right to left) in the FM/NM/FM STT structure, \nwhich is equivalent to an electron flow from the fixed to the free FM layer (left to right).  \nAfter passing through the first ( Mfixed) FM layer (FM fixed), the unpolarized electrons become \npolarized along the magnetization direction of FM fixed layer. The spin- polarized electrons \nare injected into the second (switchable) FM layer (FM free) through the NM layer (tunnel \nbarrier). This spin- polarized current contains an angular momentum j s = (ℏ/2e) pcjc (where \npc=(n up-ndown)/(n up+ndown) is the net polarization of conduction electrons) and undergoes \nspin- dependent scattering processes 60. Because the total angular momentum \n(electron+lattice) is conserved in these scattering processes, a trans verse component of the 8 \n spin angular momentum is ejected into the free FM layer (FM free), exerting a torque on \nMfree to conserve angular momentum 15,60. This spin- torque  forcefully changes the \norientation of M free when the charge current reaches a critical limit. Therefore, a stable \nparallel spin configuration between the two FM layers is achieved. The switching efficiency \ndepends on the magnitude ratio of j s to Mfree as well as the thickness ( tfree) of the free layer, \ni.e. js/(Mfree×tfree) or (j c×pc)/(M free×tfree) 60. Obviously, a high p c and a thin tfree will give rise \nto a high efficiency of angular momentum transfer.    \nWhen the charge current flows in the reverse direction (FM fixed → FM free) (Fig. 2b), the \nunpolarized electrons become polarized by the FM free layer and are then incident on the \nFM fixed layer. Similarly, the transverse spin component of this incident angular momentum \ntries to orient the magnetization of the FM fixed layer towards its direction by the spin torque. \nHowever, the exerted torque on the FM fixed is insufficient to switch the magnetization \nbecause of the high anisotropy of M fixed. Instead, the scattering process at the FM fixed/NM  \ninterface causes a reflected electron flow, which is spin -polarized in the direction opposite \nto that of t he FM fixed owing to the conservation of total angular momentum. This reflected \nelectron flow with reverse spin angular momentum travels back to the FM free layer. The \ntransverse spin component of the incident angular momentum due to the reflected electron \nis now absorbed by the FM free layer and exerts spin- torque  on the layer to rotate M free \ntowards the reflected spins. Hence, an antiparallel configuration between the magnetization \nof FM free and FM fixed is now stabilized for the reverse current flow. Note that similar \nangular momentum reflection also happens in Fig. 2a, but the spin- torque  is insufficiently \nexerted on the FM fixed. Therefore, the STT can generate parallel or antiparallel spin \nconfigurations between the two FM layers. The I -dV/dI characteristic of a typical STT \nnanopillar structure (e.g. Co/Cu/Co) is shown in Fig. 2c , where an asymmetric response can \nbe seen 8. Typically, the critical current to induce magnetic switching is approximately 1010 \nA m-2. Both parallel and antiparallel spin switching with the current flow direction can be \nuseful for read -write applications 8,58. \n2.2 Spin -orbit torque    \nThe SOT is an alternative approach with a higher spin- charge conversion efficiency 9 \n than STT, reducing critical current densities for switching devices. Unli ke STT using a \nvertical current, an in -plane charge current is utilized in SOT without the additional FM \npolarizer layer (Fig. 3a) . A traditional SOT device consists of an FM/NM bilayer \nheterostructure. When an in- plane charge current passes through the NM  layer or the \nFM/NM interface, SOT is developed at their interface because of the coupling between the \nelectron spin and its orbital motion, i.e . spin- orbit coupling (SOC) 1. As a consequence, \nnon-equilibrium transverse spin currents are generated along the out -of-plane direction and \naccumulated at the interface, which is responsible for the spin -torque acting on the FM \nmagnetization. Simultaneously, a spin current polarization, p s, is generated perpendicular to \nthe plane defined by both j c and js. Compared with the STT devices, the design of SOT \ndevices is easier as the magnetization (in -plane and out -of-plane) of the FM layer can be \ncontrolled by a smaller in -plane write current. Another advantage of the SOT devices is that \nthe orientation of the magnetic state can be identified by passing a small out- of-plane read \ncurrent 19. Therefore, SOT -based devices such as SOT -Magnetoresistive random -access \nmemory (MRAM) and SOT - Magnetic tunnel junction (MTJ) are more effective for \nspintronics owing to their robust design, low power consumption, and faster speed than \nSTT-based MRAM and MTJ devices 61.  \nFigure 3b shows a simple model of the generated spin- torque , wherein H eff is the \neffective magnetic field including of applied external, dipolar and anisotropy fields;  M is \nthe local magnetization. Assuming that there is no anisotropy, H eff contains the external \nmagnetic field and magnetic dipolar field induced by the magnetic layers. Without \nelectrical  current, if local  magnetization ( M) is tilted away from an effectiv e magnetic field \n(Heff), two types of torques appear, namely an effective field torque ( Tfield = –M×Heff) and a \ndamping torque ( Tdamping ∝M×dM/dt) 62. The former induces a precessional movement of M, \nwhereas the latter rotates M  to align  it along the Heff. In the presence of an applied current, \na new torque can be generated, which can be decomposed into two components, namely \ndamping -like torque ( TDL) and field -like torque ( TFL) 58–60,62 –65. TDL acts either along or \nopposite to the direction of T damping , depending on the direction of the current. As a \nconsequence, it behaves as an additional damping or antidamping source. Notably, T DL is 10 \n an important factor for the reversal of M  towards its final equilibrium position. When T DL is \nin the same direction as T damping, the spin- polarized current enhances the magnitude of the \ntotal effective damping, leading to the rapid dissipation of the M  oscillation energy. \nSimilarly, the direction of T FL also depends on the direction of the current. Note that the \nspin- torque m echanisms (both damping -like and field -like torques) shares similar basic \nconcepts for STT and SOT. The total magnetization in the final equilibrium state due to \nspin- torque (e.g. STT and SOT) is thus due to the combined effects of both T DL and TFL 65,66. \nThere are two key mechanisms for inducing SOT: spin Hall effect 66 and Rashba -\nEdelstein effect 67. Both phenomena generate non- equilibrium spin accumulation at the \nFM/NM heterostructure interface and modify the magnetization dynamics of FM. The spin \nHall e ffect mechanism represents a collection of SOC phenomena at the NM layer, in \nwhich unpolarized charge currents can produce transverse spin currents and vice versa (Fig. 3c). This effect was first proposed by Dyakonov and Perel in 1971 and more recently by \nHirsch in 1999 \n42,68. The basic mechanism of spin Hall effect in FM is closely interlinked \nwith the anomalous Hall effect (AHE). However, unlike FM materials, NM materials have equal number of spin- up and spin- down electrons in equilibrium without any char ge \nimbalance. When an in -plane charge current is applied in NM materials, asymmetric spin -\ndependent scattering occurs because of the SOC. Consequently, spin- up and spin- down \nelectrons are deflected in opposite directions, inducing a transverse spin current , j\ns ∝ θSH(jc \n× ps); where θ SH is known as spin hall ang le or SOT efficiency 69,70. Such a deflection gives \nrise to a spin Hall voltage at the FM/NM interface.  Moreover, both the sign and magnitude \nof θSH and the Hall voltage are indicative of the intrinsic characteristics of the NM such as \nthe spin- current generation efficiency, SOC, and asymmetric carrier density.   \nIn some FM/NM heterostructure systems, the interfacial SOC generated from both the \ncrystal structure of the NM and broken crystal symmetry gi ves rise to the SOT mechanism, \nnamely the Rashba- Edelstein effect (Fig. 3d) 19,67,71 –73. This symmetry breaking lifts the \nspin degeneracy in k -space and shifts the valence band maxima and/or conduction band \nminima from their symmetry points. The modification of the band structure by the Rashba \nsplitting energy in k -space can have a strong impact on the NM charge carriers 61,73. At the 11 \n same time, an electric field also generates symmetry breaking in the NM. Consequently, the \nmoving conduction electron experi ences an effective force perpendicular to the direction of \nboth the electric field and spin angular momentum. This force now couples with the spin angular momentum of the conduction electrons and acts on up-  and down- spins along \nopposite directions. The re sultant polarized spin accumulation at the FM/NM interface \ngives rise to SOT and magnetization switching. Detailed description on the microscopic \norigin of the spin Hall effect and Rashba -Edelstein effect can be found elsewhere \n74–80. It is \nworth noting that the functionalities of SOT devices can be tuned to meet the requirements \nof spin- based electronics to a large extent by adjusting the material properties with the help \nof impurity doping and structural modulation of the NMs, conventional device geometry  \ndesign, or by modifying the injected current and output voltage response . \n3. Probing the SOT  \nThe SOT effect can be detected via magnetization orientation of the switchable FM \nlayer under different excitation conditions (e.g. amplitude of the applied current or Hext) 81. \nTo evaluate the efficiency of the SOT, the spin Hall angle ( θSH = js/jc) should be extracted. \nNote that the current injection may increase the magnetothermal effect caused by the Joule \nheating, which can also affect the magnetization dynamics. Therefore, special care should \nbe taken during the experiments. Keeping the SOT device inside a liquid nitrogen cryostat \nduring the measurements is one possible solution to reduce the Joule heating 82. Here, we  \nwill explain in detail three conventional techniques to investigate the SOT phenomena and efficiency: i) Hall measurements (e.g. anomalous Hall effect ( R\nxy)) 83, ii) spin -torque \nferromagnetic resonance (ST -FMR) measurements using the anisotropic magnetoresistance \n(RAMR) 10,84, and iii) optical measurements using the magneto optical Kerr effect (MOKE) \n85,86. \n3.1 Hall measurem ents \nA conventional Hall bar geometry can be used to characterize the SOT (Fig. 4a) 24,87. \nThe key concept is to detect the anomalous Hall effect  signal (strongly dependent on \nmagnetization of the FM layer) of the structure which can be modulated by applyi ng \ndifferent currents. To evaluate the SOT efficiency, the Hall resistance ( RH) on the xy -plane 12 \n is investigated as functions of out -of-plane Hext at different amplitudes of DC currents (in -\nplane) (Fig. 4b). Here, the saturated hysteresis loops originate fr om the rapid reorientation \nof the magnetic domains along H ext, which results in the effective field and RH reaches their \nmaximum values at the saturation field. The measured field -dependent resistivity in this \nSOT heterostructure shows a hysteresis loop shift, which exhibits coercive field changes \n(∆Hc) at different values of  currents. In this situation, θ SH can be easily calculated by using \n, where H so is the out -of-plane spin- orbit field, tFM is the thickness of \nthe FM layer, and h is the Planck constant. One can consider a macrospin switching model \nto determine H so at different currents 82,88. When the FM magnetization has only an in- plane \ncomponent and no out -of-plane component at H c, the induced H so(ẑ), which counters H ext(-\nẑ) (inset of Fig. 4b), significantly affects ∆ Hc. Therefore, the measured ∆Hc is almost equal \nto Hso according to mic romagnetic simulations, which can be used to estimate the θ SH in the \nHall bar structure 82,88. It is noted that the in -plane magnetic anisotropy contributes to the \nhysteresis loop if in -plane magnetic anisotropy or remanent magnetization along in -plane \ndirection is not completely zero. Nevertheless, the in -plane magnetic anisotropy is normally \nquite small in many magnetic materials and can easily reorient magnetization with a small \nout-of-plane external magnetic field during SOT measurements. For example, t he MnGa \nlayer in BiSb/MnGa bilayer has a uniaxial easy axis along the z direction, and biaxial easy axes at φ = ±45° from the z direction with no in- plane magnetic anisotropy \n82. \nTo determine direction and amplitude of the torque, which cannot be evaluated by a \nsimple Hall measurement with DC current, the harmonic Hall measurement technique is used by applying AC current under different directions of in- plane H\next. The total Hall \nvoltage VH with AC current consists of the anomalous Hall effect  and planar Ha ll effect \n(PHE) (Hall signal generated by an applied in -plane magnetic field) signals across the xy -\nplane in the Hall bar structure. It can be expressed as V H(Iac)=RAHE×Iac×cosθ + \nRPHE×Iac×sin2θ×sin2 ϕ, where R PHE is the Hall resistance induced by the planar Hall effect  \nsignal 21,87. By measuring V H at ϕ = 0° and 45°, one can easily determine both the RAHE and \nRPHE at a fixed current. Because both θ  and ϕ are functions of I ac(t), the Hall resistance can 13 \n be written as \n , where \n  and \n are the first - \nand second- harmonic Hall resistances, respectively. In general, the observed \n is almost the same as the DC -RH in a similar \nHall-bar structure. However, \n  is preferably used to analyze the SOT by adopting the \nconvenient approximation so that the rotating  Hext is applied in -plane and M  rotates with \nthe constant in- plane Hext (> perpendicular anisotropy field, HA) 21,87. In this scenario, H ext \nand local magnetization  point towards random directions with ϕ . The change in \n  as a \nfunction of ϕ  is measured at different constant Hext (Fig. 4c). These curves individually \nreflect the sum of R DL (damping -like SOT component), RFL (field -like SOT component), \nand ROe (Oersted field SOT component). It has been established that \n contains both cos ϕ \nand cos(3ϕ ) terms according to the relation \n  \n39,89. A theoretical simulation of the experimental data provides the strengths of all the \nactive SOT components. Additionally, analysis of the dependence of \n on Hext needs to be \nperformed to separate all the SOT components to calculate the acting SOT fields (e.g. \ndamping -like field ( HDL), field -like field ( HFL), and Oersted field ( HOe)). In this case, \ncan be expressed as \n . \nBy comparing the coefficients of cos ϕ and cos(3 ϕ) from the two expressions of \n , the \neffective SOT fields can be easily estimated. It should be noted that the sets of \n  vs. ϕ or \nHext measurements required depend on the number of unknown components present. \nAccordingly , the SOT efficiency can be estimated by using \n  21,87.  \n3.2 Spin -torque ferromagnetic resonance  \nThe ST -FMR relying on the ferromagnetic resonance (FMR) technique is another \napproach for investigating the magnetization dynamics of FM materials as well as SOT \nefficiency with the help of microwave- frequency charge current ( jrf) 10,84.The sensitivity of \nthe ST -FMR method is high enough to detect the output SOT signals from microsized or \neven nanosized spintronic devices. The method was first utilized to demonstrate the SOT 14 \n efficiency in a Pt/NiFe heterostructure planar spin Hall bar device in 2011 84. \nFigure 4d shows a schematic of the ST -FMR measurement setup and its results. An \noscillatory jrf injected into the SOT heterostructure can generate an oscillatory current that \ninduces non- equilibrium spin accumulation at the interface due to the SOC. The \naccumulated spins diffuse into the adjacent FM layer and exert oscillatory SOTs ( TDL \nand/or TFL) on the local magnetization. As a result, the local magneti zation precesses \naround the H eff, leading to an R AMR in the SOT device. The bias -tee is used in this setup to \napply a jrf and simultaneously measure the output voltage. The voltage drop in the \nheterostructure device is detected by either a DC voltmeter or a lock -in amplifier. The DC \nand AC techniques can detect a signal resolution on the order of microvolts and 10 \nnanovolts, respectively 10,90,91. The detected output voltage depends on the amplitude of j rf, \nthe RAMR of the device, ϕ  (azimuthal- angle of M), θc (cone angle of the M ), and φ \n(resonance phase between T DL and/or TFL and M) 10. The output voltage is a combination of \na symmetric function and an asymmetric Lorentzian function, i.e.  Vmix=VsFs+VaFa, where \n‘s’ an d ‘a’ stand for the symmetric and asymmetric parts of the signal, respectively (Fig. 4e) \n84. The symmetric amplitude ( Vs) of the Lorentzian function is related to the T DL by the \nrelation Vs ∝ jsh/(4π eµ0M×t FM), while the asymmetric amplitude ( Va) is directl y connected \nto the oscillatory torques on M  (i.e. TFL+ rf-driven Oersted field torque) by the formula V a \n∝ HOe×[1+(4π Meff/Hext)]1/2, where Meff is the effective magnetization of the FM layer 10,84. \nFurthermore, the SOT efficiency can be easily estimated fr om the ratio of the Lorentzian \ncomponents (dotted curve), i.e . θSH=(Vs/Va)(2πeµ0MstFMtNM/h)[1+(4π Meff/Hext)]1/2 10,33,92. \nCurrently, this technique is extensively used to explore SOT in various magnetic heterostructures such as MTJs \n93, spin valves 91, and magnetic semiconductors 94.  \n3.3 Magneto- optical Kerr effect  \nPolarized light interacts with the magnetic order of materials. Magneto -optic techniques \nhelps to determine the magnetic structure or the optically active magnetization 85,86, and \ndomain structures or spin density of states 95,96. However, both current -induced in- plane and \nout-of-plane magnetization dynamics and SOT efficiency of magnetic heterostructures can \nbe quantified by analysing the light polarization rotation (either reflected or transmitted 15 \n light) due to the MOKE in either harmonic 96–98 or pump- probe techniques 99. When \nlinearly -polarized laser light is incident on the surface of a magnet, the polarizations of the \nreflected and transmitted light change according to the magnetization orientation of the \nmagnet 97. This polarization change in SOT devices is directly affected by the \nmagnetization dynamics of the FM layer due to the spin- torque . Here, we will discuss polar \nMOKE measurements where the laser light is normally incident on the sample surface and \nthe polarization of the reflected laser is sensitive only to the out -of-plane magnetization of \nthe FM.   \nFigure 5a shows the schematic of a typical polar MOKE setup. Orientation of the \nmagnetization at the laser spot is modulated by the SOT when  an in -plane jac is injected into \nthe NM layer. Each component of the switched magnetization can be detected through the \nKerr rotation angle ( θK), which can be recorded by a lock -in amplifier coherent with the \nfrequency of j ac 96,97. Here, the balanced detector is used to analyse the voltage difference \n(∆V) between two separate linearly polarized beams with orthogonal polarization of the \nreflect ed laser signal after the laser passes through a Wollaston prism (beam splitter). The \nobserved Kerr signal exhibits a sharp change near H ext → Hc due to magnetization reversal \nwhile the other portion of the signal depends weakly on H ext 97,100. Changing the current \nmodulates the magnetization of the FM layer, which can be detected from the MOKE signal. The dynamics of the SOT can be detected by adding another lock -in amplifier (figure not \nshown) 97. \nIn addition, time -resolved (TR) MOKE can probe the SOT -induced magnetization \nswitching and domain wall motion, as shown in the schematic of a TR MOKE setup in Fig. \n5b 101. The pump- probe stroboscopic technique is useful for investigating regularly \nrepeating motions in the magnetization dynamics due to SOT. The electrical pulse (pump) plays a role in generating the current flow for SOT. The MOKE signals in the reflected \npicosecond laser beam (probe) from the FM layer are used for detecting the corresponding \nmagnetization dynamics. The electrical pulse is synchroniz ed to the picosecond laser beam \nwith a well- defined delay time, which determines the temporal resolution. Furthermore, it is \nexpected that the switching mechanisms of the SOT, including T\nFL and TDL, would be more 16 \n obvious from the temporal analysis of the S OT-driven magnetization dynamics. It is \nimportant to note that the magnetization of FM layer returns to its initial equilibrium state \nbetween the pump pulses of current. To date, there are only a few reports on probing SOT \nby the optical technique for vari ous systems like metallic ultra -thin Ta/CoFeB/MgO 97,102, \nY3Fe5O12/Pt 97 and Ni 80Fe20/Pt 103. \n4. Traditional materials for SOT  \nThe SOT device can be improved by choosing efficient and appropriate materials for \nboth the torque and FM layers 81,104. The torque  layers are NM with high SOC and/or \ninduced structural inversion asymmetry at the interface of the FM layer that can generate \nstrong current -driven torque via the Spin Hall or the Rashba -Edelstein effects. A few \nantiferromagnetic (AFM) materials have also been utilized instead of NMs owing to their own anomalous Hall effect \n105,106. The key characteristic for the SOT efficiency of the \ntorque layer is θ SH, which  represents the spin -charge conversion efficiency. A high θ SH is \nrequired  for an energy efficient SOT device. Furthermore, a high conductivity ( σ) in the \ntorque layer is as important as θ SH. It ensures that the current flows through the torque layer \ninstead of the FM layer at FM/NM heterojunction. Otherwise, the applied j c cannot generate \njs in the FM/ NM (when σ FM > σNM) heterostructure system because the j c is shunted through \nthe FM layer from the NM layer and then passes through the FM layer without producing \nany js in the NM layer 107. The power consumption ( P ∝ 1/(σ×θSH2)) for switching a unit \nmagne tic volume in the FM, is also an essential parameter in choosing the best torque \nmaterials for SOT devices with low power consumption 108. Besides these key parameters, \nthe spin Hall conductivity ( σSH), which reflects the magnitude of the spin accumulation at \nthe interface, is also an important figure of merit in realistic SOT applications. Obviously, \nthe operating temperature of the device should be near room temperature (RT) for practical \napplications. Table 1 summarizes θ SH, σ, σSH, P, and working temper ature of various torque \nmaterials. FMs for SOT must have low coercive fields (e.g. a few tens of Oersted) for easy \nmagnetization switching and lower conductivities than the torque layers for the generation \nof j s 107.  \n4.1 Nonmagnetic materials for SOT  17 \n Among the various NMs, heavy element -based NM metals (5 d-transition metals such as \nTa, W, and Pt) are more attractive because of their stronger SOC compared with light \nelement -based NM metals (Ti and Cu) 22–24. Heavy metals such as Pt 23,84,109, β-W 110, β-Ta \n107,111, Hf 112, and W(O) 110 are widely investigated for SOT applications. Among these NM \nmetals, only W in both β -phase and oxidized state shows large SOT efficiencies. Other NM \nmetals exhibit low SOT efficiency because of the opposite si gns of their bulk and \ninterfacial SOC effects 22,111,112. Besides these single NM metals, bilayer NM systems such \nas W/Hf 113, Pt/Hf 114, Pt/Ta 115, Pt/W 116, and Pt/Ti 117, as well as Rashba interface materials \nsuch as LaALO 3/SrTiO 3 11,118, and Bi/Ag 106, are also widely investigated. The \ncorresponding studies confirm that the SOT efficiency can be amplified by combining two-\nmaterial structures 119,120. A significantly higher SOT efficiency in a 2D electron gas formed \nat the LaAlO 3/SrTiO 3 interface because of the direct Rashba -SOC that has been observed at \nRT 11,36,118. Interestingly, the SOC of these NM metals can be significantly enhanced \nthrough natural oxidation, such as in Cu(O) 10,121 and/or impurity doping, such as in Cu 1-\nxPtx 122, Cu 99.5Bi0.5 123 and Cu 97Ir3 124. For example, the alloy Cu 99.5Bi0.5 shows a larger θ SH \nthan the heavy metals Pt and Ta 84,107,123. Recently, a new type of NM/FM/NM multilayer \n(e.g. Ta/Au/FM/Au/AlO x) has been found to exhibit very strong T DL owing to the \ngeneration of the pl anar Hall current in the FM layer when one of the NM/FM interfaces in \nthe NM/FM/NM multilayer behaves like a good spin -transfer source and the other interface \nlike a spin sink 125.  \n4.2 Antiferromagnets for SOT  \nPromising SOT results (Table 1) have recently be en observed when non- collinear AFMs \n(such as PtMn and IrMn) are used instead of NMs because of the self spin Hall or \nanomalous Hall effects in the AFMs 105,106. This approach uses an antiferromagnet as the \ntorque layer because the AFM can exert an internal  exchange bias field on the adjacent FM \nlayer without the assistance of any external field 126,127. As a result, it can easily switch the \nmagnetization of FM layer along the out- of-plane direction via the SOT that emerges from \nthe direct spin Hall effect in these torque l ayers. A collinear IrMn AFM structure has \nexhibited a comparatively large SOT efficiency of approximately 0.6 compared to the 18 \n heavy metal β -W (θSH ~ 0.33) 102 and polycrystalline IrMn ( θSH ~ 0.08) 106. Magnetization \nreversal in AFM/FM heterostructure systems through SOT without H ext can be a future core \ntechnology for read- write and memristor applications 53,59,94,105.    \n5. Topological mate rials for SOT  \nAs a part of the exploration for new materials that exhibit significantly large current-\ndriven toque, topological insulators (TI) are found to be potential candidates for future \nSOT-MRAM 30,128. TIs are quantum materials that have insulating bulk states and metallic \nsurface states 1. TIs were extensively studied in the early 1980s in research on the quantum \nHall effect, which originates from the non- trivial topology of the two -dimensional electron \nwavefunctions under a strong magnetic field 25. It was later predicted that the quantum Hall \neffect can be realized without external magnetic fields (quantum - anomalo us Hall effect), \nbased purely on topology arguments 26. It is worth noting that the quantum spin Hall effect \nis another important aspect of topological insulators for the low -dissipation transport 129–132.  \nThe quantum spin Hall effect is quantization of t he spin Hall effect, which is analogue to \nthe quantum Hall effect (quantization of Hall effect). Quantum spin Hall effect can be considered as the spin version of quantum Hall effect, wherein the carrier transport at the \nedge is spin- polarized \n131,132.  \nThe first 2D TI with spin -polarized edge states was predicted in a heterostructure \nconsisting of a HgTe quantum well sandwiched by two (HgCd)Te barriers 133 and was \nimmediately confirmed experimentally 134. Three -dimensional (3D) TIs were proposed \n135,136 and first confirmed in BiSb 137, followed by several Bi -based chalcogenides, such as \nBi2Se3, Bi 2Te3, and (BiSb) 2Te3 138–141. TIs have many different characteristics including \nquantum Hall states that make them attractive for RT spintronic applications: i) Ex istence \nof the one (two) -dimensional edge (surface) states of TIs is ensured by the non -trivial \ntopologies of their band structures and can emerge without the application of large external \nmagnetic fields, ii) the surface states have Dirac -like band disper sions, which promise a \nlarge intrinsic spin Hall effect due to Berry phase mechanism, and iii) a unique spin -\nmomentum locking feature of the topological surface states prioritizes pure j s generation in \nthe direction perpendicular to the film plane 27–29.  19 \n Recent experiments have demonstrated large θ SH at RT in Bi 0.9Sb0.1/Mn 0.5Ga0.55 \nheterostructure because of both surface and bulk spin Hall effect 82. The large θSH was \nestimated from Hall bar measurements of a Bi 0.9Sb0.1 (10 nm)/Mn 0.5Ga0.55 (3 nm) \nheterostructure with a small tilted magnetization an d the change in Hall resistance at \ndifferent jc and θ = 2° (angle between H ext and z-axis) (Fig. 6a). The changes in ∆ Hc at \ndifferent given values of j c in RH vs. Hext plot are exactly equal to H so, which counters Hext \nand is aligned opposite to H ext (as m entioned in the inset of the R H vs. Hext plot). The ratio \nof Hso/jc (~∆Hc/jc) is used to calculate θ SH as discussed in probing section. The estimated \nθSH at RT is 52 82. In addition to Bi 0.9Sb0.1, strong spin Hall effect has been observed in \nvarious TIs (T able 1) such as Bi 2Se3 (θSH = 2-3.5 at RT) 30, Bi xSe1-x (θSH = 18.8 at RT) 142, \n(Bi 0.5Sb0.5)2Te3 (θSH = 25 at 200 K) 31, and (Bi 0.5Sb0.5)2Te3 (θSH = 140- 410 at 1.9 K) 128. SOT \nmagnetization switching with ultralow current densities has been demonstrated at RT in \nBi2Se3/CoTb, (BiSb) 2Te3/CoTb 108, Bi 2Se3/NiFe 33, Bi xSe1-x/Ta/CoFeB/Gd/CoFeB 142, and \nBiSb/MnGa 82. \nAlthough giant spin Hall angles and low SOT switching current densities have been \nconfirmed in various TI/FM bilayers, the origin of the giant spin H all effect in TIs is still \nunclear because the fact that the current may flow in both the surface and bulk of TIs. To \ndefinitively determine the origin of the giant spin Hall effect in TIs, the spin Hall effect in \n(Bi 1-xSbx)2Te3/Ti/CoFeB was investigated a t various Sb compositions with different Fermi \nlevels (Fig. 6b) 32. When the Sb compositions are about 85% and 93%, the Fermi levels are \nin the band gap of (Bi 1-xSbx)2Te3 and approach the Dirac point 32,141. Near the Dirac point, \nthe switching current dens ity is minimized and the SOT -induced effective field maximized. \nThis unambiguously demonstrates the surface state origin of the giant spin Hall effect. \nFurthermore, SOT efficiency depends on the layer thickness of TIs. The TI -based SOT \ndevice shows large θ SH when the thickness of TI layer goes to a monolayer limit, resulting \nfrom the formation of topological surface states and reduction of 2D electron gas states as \nwell as bulk states (e.g. Bi 2Se3/Py heterostructure) 33. Recently, gate- tunable θ SH is obser ved \nin Cr impurity - doped TI -based SOT device 34. \nFor realistic applications in SOT -MRAM, both θ SH and σ are equally important for 20 \n reducing the writing power consumption. With respect to this, it was observed that the \nnarrow bandgap TI BiSb is the most promising material because it satisfies both the above \ncriteria (high σ  ~ 2.5×103 Ω-1cm-1 and large θ SH ~ 52) 82,143. In addition, a strong interfacial \nDzyaloshinskii -Moriya interaction in BiSb/MnGa 144 and a giant unidirectional spin Hall \nmagnetoresistance o f 1.1%, which is three orders of magnitude larger than those in metallic \nbilayers, have been observed in a BiSb/GaMnAs bilayer 145. Thus, BiSb has been \nconsidered as one of the best SOT materials for realistic SOT devices.  \n6. 2D materials for SOT  \nThe first ke y expectation from 2D magnetic materials is the low density of their spin -\npolarized carriers, which can be tuned by different approaches (e.g. electrostatic gating 47, \nand external electric field 47,49,50) rather than the traditional magnetic field. Intere stingly, the \nweak interaction between two adjacent layers introduces antiferro -ferro switching by an \nexternal electric or magnetic field 50.  Another importance of 2D magnetic materials is their \npossibilities  to shed light on the proximity effect between t wo different quantum materials \nthat occur at the interfaces of two bulk or thin films, especially in magnetic and superconducting materials. Many exotic phenomena appear across a wide range of \ncondensed matter physics such as magnetic exchange bias, enhanc ement of \nsuperconducting transition temperature, and superspin- current generation \n73,146. However, \nsuch physics at the interfaces are hidden from surface analysis tools. This problem can be \nresolved if 2D magnetic materials are utilized. Furthermore, the p roximity effects, which \nnormally appear at a very thin layer at heterostructure interfaces, occur throughout the entire 2D thin layers, strengthening the effect at the interface. The SOT can be a \nrepresentative example of this advantage of 2D magnetic materials. Here, we will briefly \ndiscuss about the 2D layered materials for SOT devices as a torque, FM and both layers.     \n6.1 2D materials as torque layers  \nTransition metal dichalcogenides (TMDs) consisting of one heavy transition  metal and \ntwo chalcogen atoms exhibit strong SOC band structures, which can generate current -\ndriven out -of-plane spin polarization \n147–149. Such a strong SOC can induce a large spin Hall \neffect as well as Rashba -Edelstein effect at the interface with an adjacent material .  SOT 21 \n has been  demonstrated  in recent experiments in which 2D TMDs were used as the spin \nsource layers owing to their tunable structural symmetry 38,39,102, atomically flat surfaces, \nbroken inversion symmetry even in monolayer range 150,151, gate -modulated SOC 152,153, \nadjustable out -of-plane magnetic anisotropy 82, and tunable self - anomalous Hall and spin \nHall effects 153,154. The 2D Weyl semimetal WTe 2 is extensively used in SOT devices as a \ntorque layer owing to its topological nature governed by its reduced crystal symmetry \n38,102,155102. Interestingly, while almost- thick SOT devices reveal in -plane TDL, out-of-plane \nTDL and magnetization switching have been reported in a WTe 2 (1.8- 15 nm)/Py (6 nm) \nheterostructure at RT 38. Figure 7a shows a ST -FMR measurement setup and the results in a \nWTe 2/Py heterostructure. The result represents the ST -FMR signals at RT and a frequency \nof 9 GHz with two different directions of magnetization rotated 180°  with respect to each \nother (e.g. ϕ 1 = 40° and ϕ2 = 220°). The signals at these two 180° rotated directions will be \nsymmetrical if the system has two -fold rotational symmetry. However, the voltage signal in \nthe WTe 2/Py bilayer is completely asymmetric (e.g . Vmix(ϕ = 40 °) ≠ -Vmix(ϕ = 220°)), \nindicating a low -symmetry SOT current induced by the low -symmetry crystal WTe 2. \nFurthermore, it has been confirmed that the SOT efficiency  in the WTe 2/FM heterostructure \ncan be tuned with crystal symmetry 38 as well as s mall critical current 48. However, a \ncomparable θSH of approximately 0.013 with Pt/Py heterostructure 23 was estimated in \nWTe 2/Py bilayer from the variation of V mix with Hext using the V s/Va ratio method (details in \nprobing section) 38. Also, the SOT effic iency of this device structure weakly depends on the \nthickness of WTe 2 layer but strongly depends on the different FM layer 155.  Owing to the \nout-of-plane TDL and strong SOC, this 2D material has the potential to replace existing 3D \ntorque layers for SOT devices.  \nA high SOT efficiency of approximately 0.14, which can exert both T DL and TFL on the \nFM layers (e.g. CoFeB, Py at RT 39,156), has also been observed in the layered 2H -MoS 2. \nThese two torques are strong enough to excite FMR in the FM layer 156. Ho wever, the \nobserved TDL is much larger than the T FL from  ST-FMR measurements in huge difference \nbetween symmetric and asymmetric ST -FMR peaks. Recently, the layered TMD WSe 2 was \nused as a torque layer because of its strong Rashba -Edelstein -type SOC owing t o low 22 \n crystal symmetry 39,157. This material can generate out -of-plane TDL with Py at RT. It was \nsuggested that monolayer WSe 2 in a Ta/WSe 2 heterostructure has the ability to enhance the \nSOT efficiency of Ta 157. The low crystal symmetry β -MoTe 2 shows out -of-plane TDL and \nlayer dependent SOT efficiency with a low conductivity as that of WTe 2 158. Experimentally, \nA fully metallic and highly symmetric TMD NbSe 2 exhibits out -of-plane Oersted torque  as \nwell as in -plane TDL with comp eratively high conductivity ~103 Ω-1cm-1 (comparable with \nWTe 2 38) 159, which can be slightly tuned by the thickness modulation and crystal strain. \nTherefore, both crystal  symmetry breaking and crystal strain are the possible origin of SOT  \nin 2D TMDs. Recently, a type- II Dirac semimetal PtTe 2 shows high SOT strength ~ 0.15 \nwith a high conductivity ~104 Ω-1cm-1 with a giant spin Hall effect, as the largest value \ncompared to existing TMDs 39. This TMD has also thickness -dependent SOT strength  due \nto topological surface state like spin texture, suggesting future energy efficient and low -\npower SOT material.   \n6.2 2D materials as FM layers  \nAs mentioned above, the switching efficiency depends on the magnitude ratio of the \nspin current to the total mag netization of the free layer, i.e. j s/(Mfree×tfree) or \n(jc×pc)/(M free×tfree) 60. Furthermore, the torque occurs at the interface only within the spin \ndiffusion length, which is usually quite short 160,161. Therefore, an FM layer consisting of \none or up to a few monolayers can improve the efficiency of SOT devices. Obviously, the \nintroduction of 2D magnetic materials is the ultimate choice.  \n2D vdW FMs are emerging as an interesting research field owing to their layer -\ndependent properties and gate tunable Curie temperature ( Tc) 162–164. 2D vdW FMs also \nexhibit clean interfaces and strong out -of-plane magnetic anisotropy even at monolayer \nscales without any dead layers when such 2D FMs are stacked with different torque layers \n11,38,165. Among the various 2D FM s (e.g. VSe 2 166, Cr 2Ge2Te6 167,168, and CrI 3 136), Fe 3GeTe 2 \nhas attracted considerable attention owing to its high T c (approximately 230 K for bulk \ncrystals and 130 K for single atomic layers), out -of-plane magnetic anisotropy (a significant \nadvantage for  generating SOT), low coercive field ( Hc ~ 0.65 kOe), and a gate tunable T c of \nup to RT 162–164. Figure 7b illustrates the harmonic Hall measurements and results in a 23 \n Fe3GeTe 2 (15–23 nm)/Pt (5 nm) heterostructure 11. The second -harmonic Hall signals (\n ) \nwith rotating and constant magnetic fields ( ≥ anisotropy field of Fe 3GeTe 2) at the AC \ncurrent amplitude of 2.4 mA are shown here for the Fe 3GeTe 2 (23 nm)/Pt (5 nm) \nheterostructure. Traditionally, \n  as a function of the AC current is a  mixture of both cos ϕ  \nand cos 3ϕ  terms 11,89,165. Interestingly, the observed \n  is well- fitted with only cos ϕ terms, \nindicating a small contribution by the field -like torque and the effect of the Oersted field 11. \nThis confirms that only T DL dominates the torque for switching the magnetization of \nFe3GeTe 2 because the cos(3 ϕ) term is zero. A very large T DL (approximately 0.14 at j c ~ \n2.5×107 A/cm2 and 180 K) without any assistive of H ext compared with those in \nTm 3Fe5O12/W (TDL ~ 0.02 at j c ~ 6 × 106 A/cm2 and RT) 169, Tm 3Fe5O12/Pt (TDL ~ 0.014 at j c \n~ 1.8 × 107 A/cm2 and RT) 148,169, and WTe 2/Py ( TDL ~ 0.013 at RT) 38 has been observed \nfrom \n  analysis 11. The large TDL is attributed to the local domain wall motion in the \nFe3GeTe 2/Pt heterostructure.  \nAnother significant result was observed in layered Cr2Ge2Te6 2D FM. The material has \nreceived immense scientific interest because of its semiconducting nature, gate- tunable \nFermi level, and magnetic properties induced by the anomalous Hall  effect, which is the \nsource of spin Hall effect with different torque layers (e.g. Pt, Ta) 11,170. It was also \nobserved that the required j c of approximately 5×105 A/cm2 for switching out -of-plane \nmagnetization in a Cr 2Ge2Te6/Ta heterostructure is approxi mately two orders of magnitude \nsmaller than that in a Fe3GeTe 2/Pt heterostructure 11,170,171. Recent observations of gate -\ntunable magnetic domains at RT 45 suggest that V -doped WSe 2 can be a future FM layer \nmaterial for SOT devices owing to its long -range FM ordering along with semiconducting \nnature near RT.   \n6.3 2D materials for both torque and FM layers  \nTo date, all SOT experiments have been performed using bulk- NM/bulk- FM and/or \nbulk- FM/2D -NM, and/or 2D -FM/bulk- NM heterostructures. A combination of 2D mater ials \nmay bring benefits to both the torque and magnetic layers. These heterostructures may also \nprovide smooth and transparent interfaces so that comparatively small switching currents 24 \n are required for magnetization reversal.  \n  With respect to the use of 2D TMDs for both the torque and FM layers, the \nmagnetization switching of CrI 3 (FM insulator) driven by SOT from TaSe 2 (NM metal) has \nbeen theoretically  predicted  in a bilayer -CrI 3/monolayer -TaSe 2 heterostructure (Fig. 7c) 118. \nCrI 3 is one of the best choi ces for the 2D FM, as it exhibits layer -dependent magnetism 172. \nFor example, bilayer CrI 3 shows AFM ordering, while its monolayer behaves as a FM 172. \nThe weak exchange interaction between the layers of CrI 3 can be tuned from the AFM \nground state to an FM  state by using external magnetic or electric fields.  It is therefore \npossible to use the spin- torque to manipulate this antiferro- ferro coupling switching. \nMeanwhile, TaSe 2 shows a huge broken crystal symmetry -driven SOC 148. In this \nCrI 3/TaSe 2 heterostr ucture, passing an unpolarized charge current through the monolayer \nTaSe 2 generates an interfacial SOT that is strong enough to switch the magnetization of the \nfirst CrI 3 monolayer next to the TaSe 2 (the dynamics of magnetization switching are shown \nby the  sphere for both the layers in the upper panel of Fig. 7c ). As a result, the \nantiferromagnetically -coupled state of the bilayer CrI 3 is changed to the ferromagnetically -\ncoupled state. The microscopic physical mechanism of this conventional SOT is described  \nat the lower panel of Fig. 7c using the results of a first -principles quantum transport \ncalculation. The torque acting on CrI 3 can only manipulate the magnetization of the bottom \nmonolayer, which is the nearest neighbor  to TaSe 2. The magnetization of the top CrI 3 \nmonolayer away from the TaSe 2 remains unchanged because the spin density is zero in this \nlayer (lower panel of Fig. 7c).  The switching dynamics of the first CrI 3 monolayer can be \nverified by measuring the out -of-plane tunneling magnetoresistance and second -harmonic \nHall responses experimentally, as suggested in Ref. 11.  \n7. Challenges and opportunities for 2D SOTs  \n      For a decade from the discovery of SOT phenomenon, a lot of achievements in \ndevelopment of the SOT device have been made . The most i mportant success of SOT \ndevice is to reduce the critical switching current down to ~ 105 Acm-2 by using topological \nmaterials and 2D materials 82,102102, which is totally adaptable for industrial requirements  \nfor memory applications . Nevertheless, an appropriate process to fabricat e devices 25 \n involving topological and 2D materials is still under develop ment . Furthermore, the target \nof reduction of critical currents moves so fast that the investigation of underline \nmechanisms is overlooked. Therefore, the next  step of the SOT research will focus more in \ngrowth techniques for materials and exploration of physical mechanisms . 2D vdW \nmaterials will be good candidate s for such purposes. A search for materials with better \nperformance will also be a mainstream in nex t decades. Together with memory applications, \nSOT devices showed possibilities for other important novel devices such as nano -\noscillators  173,174, p-bit for probabilistic computing 175 for future neuron network hardware. \nFor the last but not least, it is w orth noting that the SOT -based devices use electrical current \nas the main driving force, which generates unavoidable energy los s due to joule heating. \nNew multifunctional spintronic devices using gate bias or combining electrical current with \ngate bias ed should be focused in the near future 176.  \n7.1 2D topological insulators  \nMost of the TI thin films studied so far were epitaxially grown on dedicated III -V \nsemiconductor substrates by molecular beam epitaxy (MBE), which is not feasible for \nmass -production. The refore, it is essential to investigate the performance of non -epitaxial \nTI thin films deposited on silicon substrates by an industry -friendly technique, such as \nsputtering and/or chemical vapor deposition and/or pulse laser deposition technique. A \nrecent attempt to investigate the performance of sputtered Bi xSe1-x TI thin films shows a \npromisingly large θ SH ~ 18.6 but a low σ ~ 7.8×101 Ω-1cm-1 due to the poor crystal quality \n142. On the other hand, it has been demonstrated that high- quality BiSb thin films similar to \nthose grown by MBE can be fabricated by sputtering deposition on sapphire substrates 177. \nFurther research will be needed to produce BiSb thin films with high crystal quality and \nspin Hall performance on Si substrates by sputtering deposition for ultralow -power SOT -\nMRAM.  \n7.2 New 2D magnetic materials  \nAlthough a library of 2D magnetic materials ha s been established, all of these materials \nhave quite low T c and high H c. The requirements of stable magnetization with \nperpendicular magnetic anisotropy and suitable Hc at RT are not met yet. Exploration of 26 \n new 2D magnetic materials is still a challenging  task. Beyond the conventional working \nprinciple, the weak interlayer magnetic interaction in 2D vdW materials provides \nopportunities for SOT devices with new designs, such as SOT -induced switching between \ninterlayer magnetic states 118,172. Recently notic ed RT magnetism in monolayer WSe 2 \nsemiconductor via vanadium dopant with reasonable H c can help to design new SOT \ndevices for realistic and RT applications 45.   \n7.3 2D magnetic semiconductors  \nTo prevent the applied current from flowing through the FM layers, one option is to use \n2D FM semiconductors, which have high resistivities as the conventional 3D FM layers. The challenge is the limited availability of RT ferromagnetic semiconductors. The answer \ncould be found in 2D diluted magnetic semiconducting TMDs \n45,46. This research field is \nunder investigation.  It is also important to note that the magnetic ordering and thermal fluctuation in SOT devices using 2D magnetic materials can be manipulated artificially by \nreducing layer thickness toward the monolayer li mit \n149.    \n7.4 Heterostructure growth of 2D materials  \nSimilar to other spin devices, SOT devices require clean interfaces between the layers. \nNevertheless, atomic- thick 2D materials are quite sensitive to the ambient environment. \nThe incorporation of 2D mater ials by existing 3D fabrication techniques (e.g. sputtering, \nthermal/e -beam evaporation) without exposing the 2D materials to the ambient \nenvironment is still challenging. The in -situ growth of heterostructures could be a solution \nto minimize residues and environmental effects. SOT devices based on only 2D materials \nrequire vdW layer -by-layer growth, which is still premature at the current stage 35,178. \n7.5 Enhancement of conductivity of 2D TMDs  \n2D semiconducting TMDs that exhibit high SOC at their valence bands can be used for \ntorque layers. Their resistivities are often too high for incorporation with FM metals in SOT devices because all the current passes through the metallic layers instead of the TMD layers, \nwhich reduces the SOT efficiency \n107. Heavily dope d TMDs may be a solution to this issue. \nSeveral candidates for p -doped TMDs such as V , Nb, and Ta have been demonstrated 45,179. \nShifting the Fermi level down to the valence band may enhance the spin Hall effect of the 27 \n TMDs.  \n7.6 Additional out -of-plane SOT  \nThe modern society magnetic memory technology needs out -of-plane magnetic \nanisotropy or perpendicular magnetization controlled by current or electric field instead of \nan external magnetic field. Magnetic switching devices with a perpendicular magnetization \n(e.g. perpendicular magnetic anisotropy) is strongly required to achieve large densities and \nthermal stability. However, almost- 3D SOT devices exhibit high efficiencies only for in-\nplane torque on magnetic layers with in -plane magnetic anisotropy. Several approaches \nwere demonstrated to couple the in- plane spin- torque with perpendicular magnetization by \nbreaking the symmetry of the device structure or torque symmetry such as tilted magnetic \nanisotropy 180,181, interlayer exchange bias (e.g. using antiferroma gnet/ferromagnet/oxides \nheterostructure) 105,106,182 –184. Incorporation of an additional in- plane ferromagnetic layer \ninto the NM/FM (out -of-plane) junction can induce the vertical torque by changing the \nmagnetization of the bottom layer 56.  \nNevertheless,  an out -of-plane spin- torque is required to high- efficiency perpendicular \nmagnetization switching. The main approach in this research stream is to search materials with low symmetry point groups for the bilayer interface \n185–187. Recently, an L 11-ordered \n(fcc staking order along [111] direction) CuPt (NM metals)/CoPt (FM) bilayer exhibits \nfield-free out -of-plane magnetization switching of CoPt layer. The actual reason for such \nfield-free effect is the low -symmetry point group ( 3m1) at the NM -metal/FM interf ace 185. \nFurthermore, the incorporation of 2D materials into the SOT devices (e.g. WTe 2 for the \ntorque layer 38,102,155102, MoS 2 as a substrate for Pt  188) can also generate the additional out -\nof-plane torque exerted perpendicularly to the magnetic layers along with in -plane torques, \ngiving rise to more efficient magnetization manipulation in SOT devices with perpendicular magnetic anisotropy. Understanding the generation mechanism of the out -of-plane spin \ntorque is also an interesting subject.  \n7.7 Mechanism of  SOT  \nThe spin Hall and Rashba -Edelstein band -type structures are the two accepted \nmechanisms for SOT. However, experimental evidence indicates unknown origins of SOT 28 \n using harmonic study of the anomalous Hall effect and planar Hall effect in AlO x/Co/Pt and  \nMgO/CoFeB/Ta devices 87. It was confirmed by space and time inversion symmetry \narguments that SOT devices consisting of  NM/FM heterostructure  exhibit even and odd \ntwo different SOTs. The mechanism of this SOT has another origin rather than spin Hall \nand R ashba -Edelstein effects. Recently, several approaches to induce SOT have been \nproposed including orbital Hall effect 189–191, thermal gradients 191 and magnons 191,192. The \ndetail of discussion for these effects can be found in recent road map 191. These effects are \nstill under investigated. 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Sahoo and Dinh Loc Duong\n \ncontributed equally in this work.  \nCorrespondence should be addressed to PNH (Email: pham.n.ab@m.titech.ac.jp) and \nYHL (Email: leeyoung@skku.edu).  \nCompeting interests statement  \nThe authors declare no competing  interests.  \n  \n 40 \n   \n \n \n \nFig. 1 Opportunities in topological insulators and two- dimensional layered materials.  a \nDirac cone of the TI surface state with helical spin texture. The spin and momentum are \nlocked at right angles to each other in the topologicall y protected surface states.  b Bottom \nview of the Fermi surfaces on the k x – ky plane. An applied charge current along kx \ngenerates an effective net momentum of the topological surface states (TSSs) in the same \ndirection and an effective surface spin accu mulation along the ky direction due to the spin -\nmomentum locking of the charge carriers. The arrows indicate the spin magnetic moment \ndirections of the carriers.  c Schematic of the Fermi levels ( EF) modulation with doping \neffect in TIs. At the critical d oping compositions, the E F are in the bulk bandgap of TIs and \napproach toward the Dirac point. d  Schematic of layer dependent band structure in TIs. e  \nSpin- current generated by a charge- current in  the proximity  of strong  spin -orbit  coupling \n2D vdWs mat erials.  f-h Owing to the nature of the atomically thick layer, the properties of \n2D layered vdWs materials can be easily tuned by various means such as external electric \nfields, strain, different magnetic dopant, and proximity with other materials. 2D ma gnetic \nmaterials also reveal the layer -dependent magnetic exchange coupling between the layers. 41 \n \nThe magnetic interaction between dopants in 2D materials may also differs from their 3D \nconfigurations.  \n \n \n \n \n \n \n \n \n \nFig. 2 Concept of spin- transfer torque mechan ism. a Schematic of a conventional spin-\ntransfer torque  structure consisting of fixed and free ferromagnetic layers (FM fixed and \nFM free) sandwiching a nonmagnetic (NM) tunnel layer. M fixed and Mfree are the \nmagnetizations of FM fixed and FM free, respectiv ely. The unpolarized conduction electrons \nare polarized in the direction of M fixed, and their transverse polarization components are \nabsorbed by M free when charge current ( jc) passes through the FM free/NM/FM fixed \nheterostructure. As a result, M free becomes  parallel to the M fixed owing to the spin- torque. b \nA reverse current flow ( FM fixed → FM free) switches M free to become antiparallel to M fixed. c \nOutput I-V characteristics of a typical nanopillar STT device. Electron flow (positive bias) \nabove the thre shold current from the bottom FM fixed to the top FM free layer or vice versa can \nswitch the magnetization direction and show different resistance states and hysteresis \nbehaviours.  \n \n 42 \n  \n \n \n \nFig. 3 Concept and mechanism of spin- orbit torque . a Spin- orbit tor que (SOT) device \ncomposed of an FM layer and an NM layer. A  longitudinal jc passing through the NM layer \ninduces a transverse spin current ( js) due to the strong spin- orbit coupling of the NM layer. \njs flows into the FM layer and exerts a torque on the loc al magnetization of the FM layer \nthat then precesses about the effective field ( Heff). b The enlarged view of the total spin-\ntorque acting on the local magnetization of FM. The total torque is the sum of the damping -\nlike ( TDL, parallel or antiparallel to  the damping) and field- like ( TFL, orthogonal to the plane \nshared by both FM magnetization and H eff) torques. Two key mechanisms ( c spin Hall \neffect and d  Rashba -Edelstein effect) that induce SOT at the NM/FM interface.  \n  \n 43 \n  \n \n \n \n \nFig. 4 Electrical meas urement method for probing SOT . a Hall-bar measurement geometry \nfor SOT. Schematics of the experimental  nanodevice structure with conventional contact \npads. b External magnetic field ( Hext) dependence of DC Hall resistance ( RH). c Second -\nharmonic Hal l resistance ( RH2f) as a function of applied field angle ( ϕ°) with different Hext. \nd Schematics of the experimental geometries for spin -torque ferromagnetic resonance (ST -\nFMR). The bilayer sample is integrated into a terminated coplanar waveguide along w ith \nBias-Tee circuit. e  Output ( Vmix) of the ST-FMR  measurement  as a function of H ext. The \ndotted lines are fitted to Lorentzian function,  showing symmetric and antisymmetric \nresonance components.  \n \n \n \n \n \n 44 \n  \n \n \n \nFig. 5 Optical measurement method for probing SOT . a Schematic illustration of the \nexperimental setup of MOKE -based magnetometer with in -plane AC current j ac \n(=jc0×sin( ft)) at frequency f and scattering geometry of the probe laser beam. The change in \nthe polarization of the reflected laser beam is  analyzed using a λ/2-wave plate at 45°, \nWollaston prism (beam splitter) and balanced detector. Lock- in amplifier is used to measure \nthe Kerr angle θ K or voltage change ∆V by jac with the presence of magnetic field. Output of \nthe lock -in amplifier from MOK E measurement as a function of H ext is depicted in the right \npanel. b The time-resolved MOKE setup for SOT -driven magnetization switching and \ndomain- wall-motion measurement. Corresponding output is shown in the right panel.  \n \n \n \n \n \n 45 \n  \n \n \n \nFig. 6  Topologica l materials for SOT . a Left panel: a 3D schematic of a SOT device \ncomprising Bi 0.9Sb0.1 (10nm) and Mn 0.6Ga0.4 (3nm) layers. Hext is the externally applied \nmagnetic field on the zx -plane at an angle θ  to the z -axis. The yellow arrow along the x -axis \nrepresents the applied electric current direction. Right panel: a top view of the Hall bar \ndevice structure of the same heterostructure used for the corresponding SOT measurements. \nLower panel: room temperature DC Hall resistance ( RH) of the Bi 0.9Sb0.1 \n(10nm)/Mn 0.45Ga0.55 (3nm) Hall bar device with a small tilted magnetization ( M) as a \nfunction of Hext, measured at θ  = 2° and different jc (13.8×105 to -7.7×105 A/cm2). (Inset) \na macrospin model where the spin -orbit field ( Hso) is perpendicular to M  at Hext=-Hc \n(coercive magnetic field at j c). Reprinted figures are taken from Ref.82. b Upper panel: \nSchematic of the Fermi levels ( EF) at different Sb compositions in (Bi 1−xSbx)2Te3 (x = 0, 0.7, \n0.78, 0.85, 0.93, 1.0) TIs. When the Sb compositions are about 85% and 93%, the E F is in \nthe bulk bandgap of (Bi 1-xSbx)2Te3 and approach the Dirac point. Lower panel: | j c | \n(switching current) and | χSOT | (SOT -driven effective field) as a function of Sb ratio. This 46 \n implies that both insulating bulk and conducting surface states are responsible for large \nχSOT and small j c in (Bi 1-xSbx)2Te3. Reprinted figur es are taken from Ref.32 \n \n \n \nFig. 7 Two-dimensional materials for SOT. a Upper panel : Micrograph of the WTe 2 (5.5 \nnm)/Py (6 nm) heterostructure device with contact pads and ST -FMR measurement setup. \nLower panel: Output resonances of ST -FMR measurements for the SOT heterostructure at \nRT and a frequency of 9 GHz with two different magnetization directions rotated by 180° \n(e.g. ϕ = 40° and 220°) with respect to the applied current direction. Reprinted figures are \ntaken from Ref.38. b Upper panel: Schemati c device structure of a Fe 3GeTe 2 (15–23 nm)/Pt \n(5 nm) bilayer and the corresponding coordinate system used for second- harmonic Hall \nsignal measurements, where M  is the magnetization of Fe 3GeTe 2, Iac is the in -plane injected \nAC current, and H 0 is the  in-plane external magnetic field or Hext. Lower panel: the second -\nharmonic Hall resistance \n of a Fe 3GeTe 2 (23 nm)/Pt (5 nm) bilayer as a function of ϕ  \n(azimuthal angle between the applied AC current ( Iac=2.4 mA) and H ext direction) recorded \nat different fixed H ext. Reprinted figures are taken from Ref.  11. c Upper panel: Bilayer -\nCrI 3/monolayer -TaSe 2 hybrid device for SOT switching. In the presence of a small bias \nvoltage ( Vb), an unpolarized current through the monolayer -TaSe 2 generates a SOT strong \nenough to switch the magnetization ( m1) of the first CrI 3 layer, whereas the magnetization \n(m2) of the second CrI 3 layer remains unchanged because of the zero -spin density in this \nlayer, which is clearly observed in the current -induced non- equilibrium spin density \n(SCD=SCDx, y, z) in the linear response region (lower panel). Reprinted figures are taken from 47 \n Ref. 118.  \n \n \n \nTable 1  \nComparison of spin Hall angle, θ SH; conductivity, σ; spin Hall conductivity, σ SH; power \nconsumption, P ; and operating temperature of various torque layers.  \n \nCritical \nCurrent or \njc (A cm-2)  SOT  \nefficiency  \nor |θSH| σ \n(Ω-1cm-1) σSH = \n(ℏ/2e)( θSH×σ) \n(ℏ/2e Ω-1cm-1) P ∝ 1/(σ×θ SH2) \n(a.u.)  Working  \ntemperature  \nLight metal  Cu(O) 121 - 0.08 1.2×104 0.96×103 0.13×10-1 RT \nHeavy metal  β-Ta 107 5.5×106 0.15 5.3×103 0.79×103 0.83×10-2 RT \nβ-W 110 3.2× 109 0.33 3.8×103 1.25×103 0.24×10-2 RT \nW(O) 110 4.57×106 0.49 6.0×103 2.94×103 0.69×10-3 RT \nPt 23,84 3.35×107 0.013 - 0.16 5.0×104 2.50×103 0.80×10-2 RT \nAlloy  Cu 72Pt28 122 - 0.054  1.6×104 0.86×103 0.21×10-1 RT \nCu 99.5Bi0.5 123 - 0.24 1.0×105 0.24×104 0.17×10-3 10 K  \nCu 97Ir3 124 - 0.027  7.1×104 1.92×103 0.19×10-1 10 K  \nRashba  \ninterface  STO/LAO 36 105 6.3 1.1×102 0.68×103 0.23×10-3 RT \nBi/Ag 106 - 0.18 1.7×105 3.06×104 0.18×10-3 RT \nAntiferromagnet  PtMn 105 1.0 × 109 0.10 4.4×103 0.4×103 0.23×10-1 RT \nIrMn 106 - 0.60 1.2×104 7.2×103 0.23×10-3 RT \nTopological  \ninsulator  Bi2Se3 30 2.8×106 2-3.5 (5.5-5.7)×102 (1.1-2)×103 (0.45 -\n0.14)×10-3 RT \n(Bi, Sb) 2Te3 108 2.5×106 0.40 2.5×102 0.1×103 0.25×10-1 2 K \nBixSe1-x 142 4.3×105 18.8 7.8×101 1.47×103 0.36×10-4 RT \n(Bi 0.5Sb0.5)2Te3 31 - 25 1.7×103 4.25×104 0.94×10-6 <200 K  \n(Bi 0.5Sb0.5)2Te3128 8.9×104 140-410 2.2×103 (3.08 -\n9.02)×105 (2.3-0.27)×10-8 1.9 K  \nBi0.9Sb0.182 1.5×106 52 2.5×103 1.30×105 0.48×10-6 RT \nBi2Te3 32,48 2.43×106 1.76 8.35×102 1.47×103 3.86×10-4 RT \nBixSe1−x/Ta 142 2.0 × 107 6 - - - RT \nCr-BixSb2-xTe3 15 2.57 × 105 0.3-90 - - - RT-2.5 K  \n2D material  MoS 2 39 - 0.14 2.1×102 2.88×101 0.25×10-2 RT \nWTe 2 38,102  2.96×105 0.013  6.1×103 8.0×101 0.97 RT \nWSe 2 39 - – 6.3×103 5.52×101 – RT \nβ-MoTe 2 158 - 0.032  1.8×103 5.8×101 0.05 RT \nNbSe 2 159 - 0.005 -0.013 (6-6.15) ×103 (3-8)×101 6.6-0.96 RT \nPtTe 2 39 3.1×105 0.05-0.15 (0.3-3)×104 (0.2-1.6) ×103 0.13-0.14×10-2 RT \nTaTe 2193 - - 1.4×104 (10-20)×101 - RT \nTaS 2194 5.1×105 0.25 5.96×104 14.9×103 2.68×10-4 RT \n "    },    {        "title": "2109.04107v2.Unraveling_the_magnetic_softness_in_Fe_Ni_B_based_nanocrystalline_material_by_magnetic_small_angle_neutron_scattering.pdf",        "content": " Unraveling the magnetic softness in Fe -Ni-B based nanocrystalline \nmaterial by magnetic small -angle neutron scattering  \n \nMathias Bersweilera*, Michael P. Adamsa, Inma Perala, Joachim Kohlbrecherb, Kiyonori Suzukic, and Andreas \nMichelsa*  \n \naDepartment of Physics and Materials Science, University of Luxembourg, 162A Avenue de la Faïencerie,  L-1511 \nLuxembourg, Grand Duchy of Luxembourg  \nbLaboratory  for Neutron Scattering, ETH Zurich & Paul Scherrer Institut, 5232 Villigen PSI, Switzerland  \ncDepartment of Materials Science and Engineering, Monash University, Clayton, VIC 3800, Australia  \n \nCorrespond ence  emails : mathias.bersweiler@uni.lu and andreas.mi chels@uni.lu  \n \nSynopsis    We employ m agnetic -field-dependent small -angle neutron scattering  to analyze the mesoscale magnetic \ninteractions in a soft magnetic HiB-NANOPERM -type alloy and relate the parameters to the experimental coercivity . \n \nAbstract      We employ magnetic small -angle neutron scattering to investigate the magnetic interactions in (Fe 0.7Ni0.3)86B14 \nalloy, a HiB -NANOPERM -type soft magnetic nanocrystalline material, which exhibits an ultrafine microstructure with an \naverage grain size below 10 nm. The neutron data reveal a significant spin -misalignment scattering, which is mainly related to \nthe jump o f the longitudinal magnetization at internal particle -matrix interfaces. The field dependence of the neutron data can \nbe well described by the micromagnetic small -angle neutron scattering theory. In particular, the theory explains the “clover -\nleaf-type” an gular anisotropy observed in the purely magnetic neutron scattering cross section. The presented neutron -data \nanalysis also provides access to the magnetic interaction parameters, such as the exchange -stiffness constant, which plays a \ncrucial role towards the optimization of the magnetic softness of Fe -based nanocrystalline materials.  \n \nKeywords:  small -angle neutron scattering,  micromagnetic theory,  soft magnetic  material s, nanocrystalline alloys   \n \n  \n1. Introduction  \n \nSince the pioneering work of Yoshizawa et al.  (Yoshizawa et \nal., 1988) , the development of novel Fe -based nanocrystalline \nsoft magnetic materials raised considerable interest owing to \ntheir great potential for technological applications  (Petzold, \n2002; Makino et al. , 1997) . The most well -known examples \nare FINEMET - (Yoshizawa et al. , 1988) , VITROPERM -\n(Vacuumschmelze GmbH, 1 993), and NANOPERM -type \n(Suzuki et al. , 1991)  soft magnetic alloys, which find \nwidespread application as magnetic cores in high -frequency \npower transformers or in interface transformers in the ISDN -\ntelecommunication network. For a brief review of the \nadvances in Fe -based nanoc rystalline soft magnetic alloys, we \nrefer the reader to the article by Suzuki  et al.  (Suzuki et al. , \n2019) . \n \nMore recently, an ultra -fine-grained microstructure combined \nwith excellent soft magnetic properties were obtained in HiB -\nNANOPERM -type alloys  (Li et al. , 2020) . The magnetic \nsoftness in such materials is due to the exchange -averaging \neffect of the local magnetocrystalline anisotropy K1. This \nphenomenon has been successfully modeled within the \nframework o f the random anisotropy model (RAM)  (Herzer, \n1989, 1990; Suzuki et al. , 1998; Herzer, 2007) , and becomes \neffective w hen the average grain size D is smaller than the \nferromagnetic exchange length 𝐿0=𝜑0√𝐴ex𝐾1⁄, where   𝐴ex  \nis the exchange -stiffness constant and 𝜑0 is a proportionality \nfactor  of the order of unity  which reflects the symmetry of K1. \nIn this regime, the RAM predicts that the coercivity 𝐻𝐶  scales \nas  𝐻𝐶  ∝(𝐷𝐿0⁄)𝑛, where n = 3 or n = 6 depending on the nature of the magnetic anisotropy (see , e.g., Ref s. (Suzuki et \nal., 1998, 2019)  for details). Therefore, an improvement of the \nmagnetic softness comes about by either reducing D and/or \nincreasing  𝐿0.  \n \nIn the context of increasing  𝐿0, the quantitative knowledge of  \n 𝐴ex  could help to further develop novel Fe -based soft \nmagnetic nanocrystalline materials. However, up to now, most \nof the research activities  in this field  are focused on the overall \ncharacterization, e.g., via hysteresis -loop measurements \n(coercivity, saturation magnetization, and permeability) and \nmagnetic -anisotropy determination (crystalline, shape, or \nstress related)  (McHenry et al. , 199 9; Herzer, 2013; Suzuki et \nal., 2019) . One reason for this might be related to the fact that \nmany of the conventional methods for measuring   𝐴ex  (e.g., \nmagneto -optical, Brillouin light scattering, spin -wave \nresonance , or inelastic neutron scattering ) require thin -film or \nsingle -crystal samples.  \n \nIn the present work, we employ magnetic field -dependent \nsmall -angle neutron scattering (SANS) to determine the \nmagnetic interaction parameters in (Fe 0.7Ni0.3)86B14 alloy, \nspecifically, the exchange -stiffness constant and the strength \nand spatial structure of the magnetic anisotropy and \nmagnetostatic fields. The particular alloy under study is  a \npromising HiB -NANOPERM -type soft magnetic material, \nwhich exhibits an ul tra-fine microstructure with an average \ngrain size below 10 nm  (Li et al. , 2020) . Magnetic SANS is a \nunique and powerful technique to investigate the magnetism \nof materials  on the mesoscopic length scale of ∼1–300 nm  \n(e.g., nanorod arrays  (Grigoryeva et al. , 2007; Günther et al. ,  2014; Maurer et al. , 2014) , nanoparticles  (Bender et al. , 2019; \nBersweiler et al. , 2019; Zákutná et al. , 2020; Kons et al. , 2020; \nBender et al. , 2020; Köhler et al. , 2021) , INVAR  alloy \n(Stewart et al. , 2019)  or nanocrystalline materials  (Ito et al. , \n2007; Mettus & Michels, 2015; Titov et al. , 2019; Oba et al. , \n2020; Bersweiler et al. , 2021) ). For a summary of the \nfundamentals and the most recent applications of the magnetic \nSANS technique, we refer the reader to Refs.  (Mühlbauer et \nal., 2019; Michels, 2021) . \n \nThis paper is organized as follows: Section 2 provides some \ndetails of the sample characterization and the neutron \nexperiment. Section 3 summarizes the main expressions for \nthe magnetic SANS cross section and describes the data -\nanalysis procedure to obtain the exchange constant and the \naverage magnetic anisotropy field and magnetostatic field. \nSection 4 presents and discusses the experimen tal results, \nwhile Section  5 summarizes the main findings of this study.  \n \n \n2. Experimental details  \n \nUltra -rapidly annealed (Fe 0.7Ni0.3)86B14 alloy (HiB -\nNANOPERM -type) was prepared according to the synthesis \nprocess detailed in Ref.  (Li et al. , 2020) . The sample for the \nneutron experiment was prepared by employing the low -\ncapturing isotope 11B as starting material. The average \ncrystallite size was estimated by wide -angle x -ray diffraction \nusing a Bruker D8 diffractometer in Bragg -Brentano geometry \n(Cu Kα radiation  source). The magnetic measurements were \nperformed at room temperature using a Cryogenic Ltd. \nvibrating sample magnetometer equipped with a 14 T \nsuperconducting magnet and a Riken Denshi BHS -40 DC \nhysteresis loop tracer. The crystallization and Curie \ntempe ratures were determined by means of differential \nthermal analysis (DTA) and thermo -magneto -gravimetric \nanalysis (TMGA) on Perkin Elmer DTA/TGA 7 analyzers \nunder a constant heating rate of 0.67 K/s.  For the neutron \nexperiments, six (Fe 0.7Ni0.3)86B14 ribbons with a surface area of \n12 x 20 mm and a thickness of ≈ 15 μm were stacked together, \nresulting in a total sample thickness of ≈ 90 μm. The neutron \nmeasurements were conducted at the instrument SANS -1 at \nthe Swiss Spallation Neutron Source at the Pa ul Scherrer \nInstitute, Switzerland. We used an unpolarized incident \nneutron beam with a mean wavelength of λ = 6.0 Å and a \nwavelength broadening of Δλ/λ = 10% (full width at half \nmaximum). All neutron measurements were conducted at \nroom temperature and wit hin a q-range of about 0.036 nm-1 ≤ \nq ≤ 1.16 nm-1. A magnetic field H0 was applied perpendicular \nto the incident neutron beam ( H0⊥k0). Neutron data were \nrecorded by decreasing the field from the maximum field \navailable of 8.0 T to 0.02 T following the magnetization curve \n(see Fig. 2 ). The internal magnetic field 𝐻𝑖 was estimated as \n𝐻𝑖=𝐻0−𝑁𝑑𝑀𝑆, where 𝑀𝑆 is the saturation magn etization \nand 𝑁𝑑 is the demagnetizing factor, which was determined \nbased on the analytical expression given for a rectangular \nprism  (Aharoni, 1998) . The neutron -data reduction \n(corrections for background scattering and sample \ntransmission) was conducted using the GRASP software \npackage  (Dewhurst, 2018) .  \n  \n3. Micromagnetic SANS theory  \n \n3.1. Unpolarized SANS  \n \nBased on the micromagnetic SANS theory  for two-phase \nparticle -matrix -type ferromagnets developed by Honecker et \nal. (Honecker & Michels, 2013) , the elastic total (nuclear + \nmagnetic) unpolarized SANS cross section dΣ/dΩ at \nmomentum -transfer vector q can be formally written as \n(H0⊥k0): \n \n𝑑Σ\n𝑑Ω(𝒒,𝐻𝑖)=𝑑Σres\n𝑑Ω(𝒒)+𝑑Σmag\n𝑑Ω(𝒒,𝐻𝑖)        (1) \n \nwhere  \n \n𝑑Σres\n𝑑Ω(𝒒)=8𝜋3\n𝑉𝑏𝐻2(𝑏𝐻−2|𝑁̃|2+|𝑀̃𝑆|2sin2(𝜃))                 (2) \n \ncorresponds to the (nuclear and magnetic) residual SANS \ncross section, which is measured at complete magnetic \nsaturation, and  \n \n𝑑Σmag\n𝑑Ω(𝒒,𝐻𝑖)=8𝜋3\n𝑉𝑏𝐻2(|𝑀̃𝑥|2+|𝑀̃𝑦|2cos2(𝜃)\n+(|𝑀̃𝑧|2−|𝑀̃𝑆|2)sin2(𝜃)\n−(𝑀̃𝑦𝑀̃𝑧∗+𝑀̃𝑦∗𝑀̃𝑧)sin(𝜃)cos(𝜃))     (3) \n \ndenotes the purely magnetic SANS cross -section.  In Eqs. \n(1)(3), V is the s cattering volume, bH = 2.91  108 A-1m-1 \nrelates the atomic magnetic moment to the atomic magnetic \nscattering length,  𝑁̃(𝒒) and 𝑴̃(𝒒)=[𝑀̃𝒙(𝒒),𝑀̃𝒚(𝒒),𝑀̃𝒛(𝒒)] \nrepresent the Fourier transforms of the nuclear scattering \nlength density N(r) and of the magnetization vector field M(r), \nrespectively,  θ specifies the angle between H0 and q  q{0, \nsin(𝜃), cos(𝜃)} in the small -angle approximation, and the \nasterisks “*” denote the complex conjugated quantities. 𝑀̃𝑆(𝒒) \nis the Fourier transform of the saturation magnetization profile \nMS(r), i.e., 𝑀̃𝑆(𝒒)=𝑀̃𝑧(𝒒) at complete magnetic saturation  \n[compare Eq. (2) ]. For small -angle scattering, the component \nof the scattering vector along the incident neutron beam, here \nqx, is smaller than the other two components qy and qz, so that \nonly correlations in the plane perpendicular to the incoming \nneutron beam are probed.  \n \nIn our neutron -data analysis, to experimentally access the \ndΣmag/dΩ, we have subtracted the SANS cross section dΣ/dΩ \nmeasured at the largest available field (approach -to-saturation \nregime; compare Fig. 2 ) from the dΣdΩ⁄ measured at lower \nfields . This specific subtraction procedure eliminates the \nnuclear SANS contribution  ∝|𝑁̃|2, which is field \nindependent, and therefore  \n  𝑑Σmag\n𝑑𝛺(𝒒,𝐻𝑖)=8𝜋3\n𝑉𝑏𝐻2(∆|𝑀̃𝑥|2+∆|𝑀̃𝑦|2cos2(𝜃)\n+∆|𝑀̃𝑧|2sin2(𝜃)\n−∆(𝑀̃𝑦𝑀̃𝑧∗+𝑀̃𝑦∗𝑀̃𝑧)sin(𝜃)cos(𝜃)) ,(4) \n \nwhere the “ ” represent the differences of the Fourier \ncomponents at the two selected fields (low field minus highest \nfield).  \n \n \n3.2. Approach -to-saturation regime  \n \nIn the particular case of the approach -to-saturation regime,  \nwhere  𝑀̃𝑧≅𝑀̃𝑆, and which implies therefore ∆|𝑀̃𝑧|2→0 in \nEq. (4) , dΣ/dΩ can be re -written as:  \n \n𝑑Σ\n𝑑Ω(𝒒,𝐻𝑖)=𝑑Σres\n𝑑Ω(𝒒)+𝑆𝐻(𝒒)×𝑅𝐻(𝒒,𝐻𝑖)     \n+𝑆𝑀(𝒒)×𝑅𝑀(𝒒,𝐻𝑖)                               (5) \n \nwhere 𝑆𝐻(𝒒)×𝑅𝐻(𝒒,𝐻𝑖) and 𝑆𝑀(𝒒)×𝑅𝑀(𝒒,𝐻𝑖) correspond \nto the magnetic scattering contributions due to perturbing \nmagnetic anisotropy fields and magnetostatic fields, \nrespectively. More specifically, the anisotropy -field scattering \nfunction  \n \n𝑆𝐻(𝒒)=8𝜋3\n𝑉𝑏𝐻2|𝐻̃𝑝(𝒒)|2                           (6) \n \ndepends of the Fourier coefficient 𝑯̃𝑝(𝒒) of the magnetic \nanisotropy field, whereas the scattering function of the \nlongitudinal magnetization  \n \n𝑆𝑀(𝒒)=8𝜋3\n𝑉𝑏𝐻2|𝑀̃𝑧(𝒒)|2                          (7) \n \nis related to the Fourier coefficient   𝑀̃𝑧∝Δ𝑀. For an \ninhomogeneous material of the NANOPERM -type, the latter \nquantity is related to the magnetization jump M at internal \n(e.g., particle -matrix) interfaces. We would like to emphasize \nthat the q dependence of 𝑆𝐻  and  𝑆𝑀  can often be described \nby a particle form factor ( e.g., sphere) or a Lorentzian -squared \nfunction. The corresponding (dimensionless) micromagnetic \nresponse functions 𝑅H  and 𝑅M  are given by:  \n \n𝑅H(𝒒,𝐻𝑖)=𝑝2\n2[1+cos2𝜃\n(1+𝑝sin2𝜃)2]          (8) \n \nand \n \n𝑅M(𝒒,𝐻𝑖)=𝑝2sin2𝜃cos4𝜃\n(1+𝑝sin2𝜃)2+2𝑝sin2𝜃cos2𝜃\n1+𝑝sin2𝜃.     (9) \n \nThe dimensionless function   𝑝(𝑞,𝐻𝑖)=𝑀𝑆[𝐻𝑖(1+𝑙𝐻2𝑞2)] ⁄  \ndepends on the internal magnetic field 𝐻𝑖 and on the exchange \nlength   𝑙𝐻(𝐻𝑖)=√2𝐴ex(𝜇0𝑀𝑆𝐻𝑖) ⁄ . \n \n 3.3. Estimation of the magnetic interaction \nparameters  \n \nMost of the time it is more convenient to analyze  the (over 2π) \nazimuthally -averaged SANS cross sections instead of the two -\ndimensional ones. By performing an azimuthal average of the \nresponse functions [ Eqs. (8)  and (9)] with respect to the angle \nθ, i.e.,  1(2𝜋)∫(…)𝑑𝜃2𝜋\n0⁄  and by assuming 𝑆𝐻 and 𝑆𝑀  to be \nisotropic ( -independent), the SANS cross section dΣ/dΩ can \nbe written as:  \n \n       𝑑Σ\n𝑑Ω(𝑞,𝐻𝑖)=𝑑Σ𝑟𝑒𝑠\n𝑑Ω(𝑞)+𝑆𝐻(𝑞)×𝑅𝐻(𝑞,𝐻𝑖)   \n+𝑆𝑀(𝑞)×𝑅𝑀(𝑞,𝐻𝑖)                             (10) \n \nwhere  \n𝑅H(𝑞,𝐻𝑖)=𝑝2\n4[2+1\n√1+𝑝]                  (11) \n \nand \n \n𝑅M(𝑞,𝐻𝑖)=√1+𝑝−1\n2 .                        (12) \n \nFor a given set of parameters 𝐴ex  and  𝑀𝑆, the numerical \nvalues of 𝑅H and  𝑅M are known at each value of q and  𝐻𝑖. \nBecause of the linearity of Eq. (10) in  𝑅H and  𝑅M, one can \nobtain the values of 𝑑Σres𝑑Ω⁄ (as the intercept) and 𝑆𝐻 and 𝑆𝑀 \n(as the slopes) at each q-value by performing a (weighted) non -\nnegative least -squares global fit of the azimuthally -averaged \nSANS cross sections  dΣ/dΩ measured at several   𝐻𝑖. By \ntreating 𝐴ex in the expression for 𝑝(𝑞,𝐻𝑖) as an adjustable \nparameter during the fitting procedure allows us to estimate \nthis quantity. The best -fit value for 𝐴ex  is obtained from the \nminimization of the (weighted) mean -squared devia tion \nbetween experiment and fit:  \n \n𝜒2(𝐴ex)=1\n𝑁∑∑1\n𝜎𝜇,𝜈2𝑁𝜈\n𝜈=1𝑁𝜇\n𝜇=1[𝑑Σexp\n𝑑Ω(𝑞𝜇,𝐻𝑖,𝜈)\n−𝑑Σsim\n𝑑Ω(𝑞𝜇,𝐻𝑖,𝜈)]2\n                               (13) \n \nwhere the indices 𝜇 and 𝜈 refer to the particular q and 𝐻𝑖-\nvalues, the  𝜎𝜇,𝜈2 denote the uncertainties in the experimental \ndata, N = NN corresponds to the number of data points , and \ndΣexp/dΩ and dΣsim/dΩ are the azimuthally -averaged SANS \ncross section determined from the neutron experiments and \nnumerically computed using Eq. (10) , respectively. We would \nlike to point out that the best -fit value for   𝐴ex represents an \naverage over the sample volume.  \n \nFinally, the numerical integration of the determined 𝑆𝐻(𝑞) and \n𝑆𝑀(𝑞) over the whole -q space according to (Honecker & \nMichels, 2013)  \n \n1\n2𝜋2𝑏𝐻2∫𝑆𝐻,𝑀(𝑞)𝑞2𝑑𝑞                              (14)∞\n0   \nyields the mean -square anisotropy field 〈|𝐻𝑝|2〉 and the mean -\nsquare longitudinal magnetization fluctuation   〈|𝑀𝑧|2〉, \nrespectively. Since the neutron experiments are performed \nwithin a finite q-range and since both integrands  𝑆𝐻,𝑀𝑞2  do \nnot exhibit any sign of convergence, one can only obtain a \nlower bound for both quantities by numerical integration. \nMoreover, it is important to realize that the specific neutron -\ndata analysis described above does not represent a \n“continuous” fit of dΣ/dΩ in the conventional sense, but rather \nthe point by point reconstruction of the theoretical cross \nsections based on the experiment al data.  \n \n \n4. Results and discussion  \n \nFigure 1  displays the wide -angle x -ray diffraction (XRD) \nresults of the (Fe 0.7Ni0.3)86B14 ribbons. The XRD pattern \nexhibits only the reflections from the fcc -Fe(Ni) phase, as \nexpected for this particular composition  (Li et al. , 2020), and \ntherefore confirms the high -quality synthesis of the sample. \nThe values of the lattice parameter a and the  average crystallite \nsize D were estimated from the XRD data refinement using the \nLeBail  fit method (LBF) implemented in the Fullprof Suite  \n(Rodríguez -Carvajal, 1993) . The best -fit values are \nsummarized in Table 1. Both values are consistent with data \nin the literature (compare Refs.  (Anand et al. , 2019)  and Ref.  \n(Li et al. , 2020)  for a and D, respectively). As previously \ndiscussed, the origin of the exceptionally fine microstructure \nobserved in (Fe 0.7Ni0.3)86B14 alloys may be qualitatively \nattributed to the ultrafast nucleation kinetics of the fcc -Fe(Ni) \nphase  (Li et al. , 2020) . \nFigure 2(a)  presents the positive magnetization branch on a \nsemi -logarithmic scale (measured at room temperature), while \nthe hysteresis loop on a linear -linear scale, and  between  ± 0.03 \nmT, is displayed in  Fig. 2(b) . The data have been normalized \nby the saturation magnetization   𝑀𝑆, which was estimated \nfrom the linear regression 𝑀(1𝐻𝑖)⁄  for 𝜇0𝐻𝑖∈[10 T−14 T]  \n[see inset in Fig. 2(a )]. The value s of  𝑀𝑆 and 𝐻𝐶 (see Table 1) \nare in agreement with the ones reported in the literature  (Li et \nal., 2020) . Defining th e approach -to-saturation regime by \n𝑀𝑀𝑆≥90 ⁄  %, we can see that this regime is reached for \n𝜇0𝐻𝑖≥65 mT. Moreover, the extremely small value for 𝐻𝐶  \ncombined with the high 𝑀𝑆  confirm s the huge potential of \n(Fe 0.7Ni0.3)86B14 alloy as a soft magnetic material, and suggests \nthat in the framework of the RAM  (Herzer, 2007) ,  𝐻𝐶   should \nfall into the regime where 𝐻𝐶  ∝(𝐷𝐿0⁄)3 (Suzuki et al. , \n2019) . \n \nFigure 3  shows the DTA and TMGA curves for amorphous \n(Fe 0.7Ni0.3)86B14 alloy . Two exothermic peaks are evident on \nthe DTA curve reflecting the well -known two -stage reactions , \nwhere fcc -Fe(Ni) forms at the 1st peak followed by \ndecomposition of the residual amorphous phase at the 2nd peak. \nThe sharp drop of the TMGA signal just before the 2nd stage \ncrystallization corresponds to the Curie temperature of the \nresidual amorphous phase ( 𝑇𝐶am ≈ 720 K).  This value , which \nreflects the exchange integral in our sample  (see below) , is consistent with the one s determined for amorphous Fe 86B14 \nsample s prepared under similar conditions  (Zang et al. , 2020) . \n \nFigure  4 (upper row) shows the experimental two -dimensional \ntotal (nuclear + magnetic) SANS cross sections dΣ/dΩ of the \n(Fe 0.7Ni0.3)86B14 ribbons at different selected fields. As can be \nseen, at 𝜇0𝐻𝑖=7.99 T (near saturation), the pattern  is \npredominantly elongated perpendicular to the magnetic field \ndirection. This particular feature in dΣ/dΩ is the signature of \nthe so -called “ sin2(𝜃)-type” angular anisotropy  [compare Eq. \n(2)]. Near saturation, th e magnetic scattering resulting from \nthe spin misalignment is small compared to the one resulting \nfrom the longitudinal magnetization jump at the internal ( e.g., \nparticle -matrix) inter faces. By reducing the field , the patterns \nremain predominantly elongated perpendicular to the \nmagnetic field, but at the smaller momentum transfers q an \nadditional field -dependent signal is observed “roughly” along \nthe diagonals of the detector, suggesting a more complex \nmagnetizatio n structure. Figure 4 (middle row) presents the \ncorresponding two-dimensional  purely magnetic SANS cross \nsections dΣmag/dΩ determined by subtracting dΣ/dΩ at 𝜇0𝐻𝑖=\n7.99 T from the data at lower fields.  In this way, the maxima \nalong the diagonals of the detector become more clearly \nvisible, thereby revealing the so -called “clover -leaf-type” \nangular anisotropy pattern.  This particular feature was also \npreviously observed in NANOPERM -type soft magnetic \nmaterials  (Honecker et al. , 2013) , and is related to the \ndominant magnetostatic term  𝑆𝑀×𝑅𝑀 in the expression for \ndΣmag/dΩ [compare Eqs. (8) and (9) ]. More specifically, the \njump in the magnitude of the saturation magnetization at the \nparticle -matrix inter faces, which can be of the order of 1  T in \nthese type of alloys  (Honecker et al. , 2013) , results in dipolar \nstray fields which produce spin disorder in the surrounding s. \nFigure 4 (lower row) displays dΣmag/dΩ computed using the \nmicromagnetic SANS theory [ Eqs. (5) –(9)] and the \nexperimental parameters summarized in Table 1. As is seen, \nthe clover -leaf-type angular anisotropy experimentally \nobserved in Fig. 4 (middle row)  can be well reproduced using \nthe micromagnetic theory.  \n \nFigure 5(a) displays the (over 2π) azimuthally -averaged \ndΣ/dΩ, while the corresponding dΣmag/dΩ are shown in Fig. \n5(b). By decreasing   𝜇0𝐻𝑖 from 7.99 T to 10 mT, the intensity \nof dΣ/dΩ increases by almost  two orders of magnitude  at the \nsmallest momentum transfers q. By compari son to  Eqs. (1) –\n(4), it appears obvious that the magnetic -field dependence of \ndΣ/dΩ can only result from the mesoscale spin disorder ( i.e., \nfrom the failure of th e spins to be fully aligned along H0). As \nis seen in Fig. 5(b), the magnitude of dΣmag/dΩ is of the same \norder as dΣ/dΩ, supporting therefore the notion of dominant \nspin-misalignment scattering in (Fe 0.7Ni0.3)86B14 alloy . \n \nFigure 6 shows the magnetic SANS results determined from \nthe field -dependent approach described in Sec. 3.3. In the \npresent case, to warrant the validity of the micromagnetic \nSANS theory, only dΣ/dΩ measured for 𝜇0𝐻𝑖≥65 mT were \nconsidered (i.e., within the approach -to-saturation regime, \ncompare Fig. 2 ). We have also restricted our neutron -data \nanalysis to 𝑞≤𝑞max =√𝜇0𝑀𝑆𝐻0max/(2𝐴ex)=0.65 nm-1, \nsince  the magnetic SANS cross section  is expected to be field -\nindependent for   𝑞≥𝑞max (Michels, 2021) . In Fig. 6(a), we  plot the (over 2π) azimuthally -averaged dΣ/dΩ along with the \ncorresponding fits based on the micromagnetic SANS theory \n[Eq. (10), black solid lines ]. It is seen that the field  dependence \nof dΣ/dΩ over the restricted q-range can be well reproduced \nby the theory. Figure 6(b) displays the (weighted) mean -\nsquared deviation between experiment and fit,   𝜒2, determined \naccording to Eq. (13) , as a function of the exchange -stiffness \nconstant  𝐴𝑒𝑥. In this way, we find 𝐴ex=(10 ± 1) pJ/m (see \nTable 1). The comparison with previous studies is discussed in \nthe next paragraph for more clarity. Figure 6(c) displays the \nbest-fit results for dΣres/dΩ, 𝑆𝐻, and  𝑆𝑀. Not surprisingly, the \nmagnitude of dΣres/dΩ (limit of dΣ/dΩ at infinite field)  is \nsmaller than the dΣ/dΩ at the largest fields [compare Fig. 6(a)], \nsupporting the validity of the micromagnetic SANS theory . \nFurthermore, the magnitude of 𝑆𝐻 is about two orders of \nmagnitude smaller than 𝑆𝑀, suggesting that the magnetization \njump   Δ𝑀  at internal particle -matrix interfaces represents the \nmain source of spin disorder in this material. The estimated \nvalues for the mean -square anisotropy field and the mean -\nsquare magnetostatic field in terms of Eq. (14)  are, \nrespec tively, 0.3 mT and 24 mT. These values qualitatively \nsupport the notion of dominant spin -misalignment scattering \ndue to magnetostatic fluctuations.  The q-dependence of 𝑆𝑀 can \nbe described using a Lorentzian -squared function [blue solid \nline in Fig. 6(c)] from which an estimate for the magnetostatic \ncorrelation length 𝜉𝑀=  2.4 ± 0.2  nm is obtained. This value \ncompares favorably  with the value of 𝑙𝑀=√2𝐴ex/𝜇0𝑀𝑠2=\n 3.7 nm  (using 𝐴ex= 10 pJ/m  and 𝜇0𝑀𝑆=1.34 T [taken from \nTable 1]), which reflects the competition between the \nexchange and magnetostatic energies.  \n \nWe would like to emphasize that our experimental value for \n𝐴ex=10 pJ/m is about 2 –3 times larger than the ones \nreported in NANOPERM -type soft magnetic materials  \n(Honecker et al. , 2013) . Since the Curie temperature of the \nresidual amorphous phase in  our nanocrystalline \n(Fe 0.7Ni0.3)86B14 sample is well above 700 K  (see Fig. 3 and \nTable 1), while the one in the Fe89Zr7B3Cu1 sample used in the \nprevious study  (Honecker et al. , 2013)  was as low as  350 K,  \nthe local  exchange stiffness in the grain boundary amorphous \nphase in HiB -NANOPERM -type alloys is expected to be \nhigher than the one in NANOPERM -type alloys . This finding \ncould explain the origin of the larger 𝐴ex  value  reported in the \npresent study . Therefore, one can expect an improvement of \nthe magnetic softness in HiB -NANOPERM thanks to the \nensuing increase of the ferromagnetic exchange length  𝐿0. It is \nwell established that nonmagnetic and/or ferromagnetic \nadditives and the annealing conditions  strongly affect the \nmicrostructural and magnetic properties of Fe -based \nnanocrystalline materials  (McHenry  et al. , 1999; Herzer, 2007, \n2013; Suzuki et al. , 2019)  and therefore have a strong impact \non their magnetic softness. Using 𝐴ex=10 pJ/m (this study) , \n𝐾1≈9.0 kJ/m3 footnote  1, and 𝜑0 ≈ 1.5  (Herzer, 2007) , we obtain \n𝐿0≈50 nm. This value for L0 is in very good agreement with \nthe typical length scale of ∼30−50 nm previously \nreported  in soft magnetic Fe -based alloys. Moreover, the \ncomparison of the average grain size D = 7 nm with the L0 \nvalue, here  𝐷≪𝐿0, also confirms that in the framework of the \nrandom anisotropy model  (Herzer, 1989, 1990; Suzuki et al. , \n1998; Herzer, 2007) , the exchange -averaged magnetic anisotropy   〈𝐾〉  falls into the regime where   〈𝐾〉∝𝐷3. This \nfinding is also consistent with the  (experimental ) 𝐷3-\ndependence of 𝐻𝐶 reported in Fe-B-based HiB -NANOPERM \nalloys  (Suzuki et al. , 2019; Li et al. , 2020) . \n \n \n5. Conclusions  \n \nWe employed magnetic SANS to determine the magnetic \ninteraction parameters in (Fe 0.7Ni0.3)86B14 alloy, which is a \nHiB-NANOPERM -type soft magnetic material. The analysis \nof the magnetic SANS data suggests the presence of strong \nspin misalignment on a mesoscopic length scale. In fact, the \nmicromagnetic SANS theory provides an excellen t description \nof the field dependence of the total (nuclear + magnetic) and \npurely magnetic SANS cross sections. The clover -leaf-type \nangular anisotropy patterns observed in the magnetic SANS \nsignal can be well reproduced by the theory. The magnitudes \nof the scattering functions SH and SM allow one to conclude \nthat the magnetization jump s at internal particle -matrix \ninterfaces  and the ensuing dipolar stray fields are  the main \nsource of the spin -disorder in this material. Our study \nhighlights the strength of  the magnetic SANS technique to \ncharacterize magnetic materials on the mesoscopic length \nscale. The structural and magnetic results (summarized in \nTable 1) provide  valuable  information on the (Fe 0.7Ni0.3)86B14 \nribbons , and further confirm the strong potential of Fe -Ni-B-\nbased HiB-NANOPERM -type alloys as soft magnetic \nnanocrystalline materials.  In the context of the random \nanisotropy model , we demonstrated  that the  magnetic  softness \nin this system is due  to the combined action of the small \nparticle size (D = 7 nm ) and an increased exchange constant \n(𝐴ex= 10 pJ/m) resulting in an enhanced exchange correlation \nlength  𝐿0. \n \nThe data that support the findings of this study are available \nfrom the corresponding author upon reasonable reque st. \n \n \nAcknowledgements  \n \nThe authors acknowledge the  Swiss spallation neutron source  \nat the Paul Scherrer Institute, Switzerland , for the provision of \nneutron beamtime. A.M. and M.B. acknowledge financial \nsupport from the National Research Fund of Luxembourg.  \n \n \nReferences  \n \nAharoni, A. (1998). J. Appl. Phys.  83, 3432 –3434.  \nAnand, K. S., Goswami, D., Jana, P. P. & Das, J. (2019). AIP Adv.  9, \n055126.  \nBender, P., Honecker, D. & Fernández Barquín, L. (2019). Appl. 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P. (1939). Phys. Rev.  56, 1245.  \nTitov, I., Barbieri, M., Bender, P., Peral, I., Kohlbrecher, J., Saito, K., \nPipich, V., Yano, M. & Michels, A. (2019). Phys. Rev. Mater.  3, \n84410.  \nVacuumschmelze GmbH (1993). Toroidal Cores of VITROPERM,  data \nsheet PW -014. \nYoshizawa, Y., Oguma, S. &  Yamauchi, K. (1988). J. Appl. Phys.  64, \n6044 –6046.  \nZákutná, D., Nižňanský, D., Barnsley, L. C., Babcock, E., Salhi, Z., \nFeoktystov, A., Honecker, D. & Disch, S. (2020). Phys. Rev. X . 10, \n031019.  \nZang, B., Parsons, R., Onodera, K., Kishimoto, H., Shoji, T. , Kato, A., \nGaritaonandia, J. S., Liu, A. C. Y. & Suzuki, K. (2020). Phys. Rev. \nMater.  4, 33404.   \n \n FIGURE 1  \n \nFigure 1 : X-ray diffraction (XRD) pattern for (Fe 0.7Ni0.3)86B14 ribbons, a HiB -NANOPERM -type soft magnetic nanocrystalline \nmaterial ( black c rosses ; Cu K  radiation). Red solid line: XRD data refinement using the Le Bail fit method (LBF) implemented \nin the Fullprof software. The bottom orange solid line represents the difference between the calculated and experimental \nintensities.  \n   \n \n FIGURE 2  \n \nFigure 2 : (a) Normalized positive magnetization branch measured at room temperature (semi -logarithmic scale). Color -filled \ncircles: 𝑀𝑀𝑠⁄ values for which the small -angle neutron scattering measurements have been performed. The approach -to \nsaturation regime, defi ned as 𝑀𝑀𝑠≥ ⁄ 90% , is indicated by the red -shaded area. Inset: Plot of the magnetization as a function \nof 1/𝐻𝑖 (black circles). Red dashed line: Linear regression for 𝜇0𝐻𝑖∈[10 T−14 T] (linear – linear scale). (b) Normalized \nmagnetization curve measured using a  Riken Denshi BHS -40 DC hysteresis loop tracer, revealing a coercivity of  \n𝜇0𝐻𝐶≈ 0.0049 mT (linear -linear scale).  \n  \n \n \n FIGURE 3  \n \nFigure 3 : Results of differential thermal analysis (DTA)  (red solid line)  and thermo -magneto -gravimetric analysis (TMGA) \n(blue solid line) for amorphous (Fe 0.7Ni0.3)86B14 alloy.  The arrows mark the crystallization and Curie temperatures.   \n \n \n FIGURE 4 \n \nFigure 4: Experimental  two-dimensional total (nuclear + magnetic) small -angle neutron scattering (SANS) cross section dΣ/dΩ \nof (Fe 0.7Ni0.3)86B14 alloy  at the selected fields of 7.99, 2.99, 0.59 and 0.29 T (upper row) , and the corresponding purely magnetic \nSANS cross section dΣmag/dΩ (middle row) . The experimental dΣmag/dΩ were obtained by subtracting the dΣ/dΩ at the (near -) \nsaturation field of 7.99 T from the data at the lower fields. The internal magnetic field 𝑯𝑖 is horizontal in the plane of the detector  \n (𝑯𝑖⊥𝒌0). Lower row: Computed dΣmag/dΩ based on the micromagnetic SANS theory [ Eqs. (5 )–(9)] at the same selected field \nvalues as above, and using the experimental parameters summarized in Table 1. Note that dΣ/dΩ and dΣmag/dΩ are plotted in \npolar coordinates with q in nm-1, θ in degree, and the intensity in cm-1. \n   \n \n FIGURE 5 \n \nFigure 5: (a) Magnetic -field dependence of the (over 2π) azimuthally -averaged total (nuclear + magnetic) SANS cross section  \ndΣ/dΩ of (Fe 0.7Ni0.3)86B14 alloy . (b) The corresponding purely magnetic SANS cross section dΣmag/dΩ (log -log scale).  \n \n  \n \n \n FIGURE 6 \n \nFigure 6: Results of the SANS data analysis  of (Fe 0.7Ni0.3)86B14 alloy . (a) Magnetic -field dependence of the (over 2π) \nazimuthally -averaged total (nuclear + magnetic) SANS cross section  dΣ/dΩ plotted in Fig. 5(a) along with the corresponding \nfits (black solid lines) based on the micromagnetic SANS theory [ Eq. (10) ]. (b) Weighted mean -squared deviation between \nexperiment and fit,   𝜒2, determined using Eq. (13)  as a function of the exchange -stiffness constant   𝐴ex. Inset: Fe composition \ndependence of the magnetocrystalline anisotropy K1 in Fe 1-xNix alloys (data taken from Refs.  (Tarasov, 1939; Hall, 1960) ). Black \ndashed line: Linear regression of K1(x). (c) Best-fit results for the residual scattering cross section dΣres/dΩ (red diamonds), the \nscattering function SH (orange open circles), and SM (blue open circles). Blue solid line: Fit of SM assuming a Lorentzian -squared \nfunction for the q-dependence.  \n \n \n  \n \n \n TABLE 1 \nParameter  (Fe 0.7Ni0.3)86B14 alloy  unit \na ≈ 0.359  nm \nD 7 ± 1 nm \n𝜇0𝑀𝑆 1.34 ± 0.20  T \n𝜇0𝐻𝐶 ≈ 0.0049  mT \n𝑇𝑐am 720 K \n𝐴ex 10 ± 1  pJ/m  \n𝜉𝑀 2.4 ± 0.2  nm \n𝐿0 ≈ 50 nm \n𝜇0〈|𝐻𝑝|2〉 ≈0.3  mT \n𝜇0〈|𝑀𝑧|2〉 ≈24 mT \n \nTable 1: Summary of the structural and magnetic parameters for (Fe 0.7Ni0.3)86B14 alloy (HiB -NANOPERM -type soft magnetic \nnanocrystalline material) determined by wide -angle x -ray diffraction, magnetometry,  differential thermal analysis, thermo -\nmagneto -gravimetric analysis , and small -angle neutron scattering.  \n \nFootnote 1  \nEstimated by assuming a linear regression of K1 in Fe 1-xNix alloys for a Fe composition x between 0 and 0.4 at. % (see inset in \nFig. 6(b) , data taken from Refs. (Tarasov, 1939; Hall, 1960) . \n \n \n "    },    {        "title": "2109.05947v1.Gradient_soft_magnetic_materials_produced_by_additive_manufacturing_from_non_magnetic_powders.pdf",        "content": " 1 Gradient soft magnetic materials produced by additive manufacturing from non-magnetic powders   O.N. Dubinin1,2, D. A. Chernodubov3, Y.O. Kuzminova1, D.G. Shaysultanov4, I.S. Akhatov1, N.D. Stepanov4 and S.A. Evlashin1*  1 Center for Design, Manufacturing & Materials, Skolkovo Institute of Science and Technology, 30, bld. 1 Bolshoy Boulevard, Moscow 121205, Russia 2 Saint Petersburg State Marine Technical University, Lotsmanskaya street, 3 Saint-Peterburg 190121, Russia 3 National Research Center \"Kurchatov Institute,\" PI. Kurchatova, 1, Moscow 123182, Russia 4 Laboratory of Bulk Nanostructured Materials, Belgorod State University, Belgorod, 308015, Russia * s.evlashin@skoltech.ru   Abstract  Additive manufacturing (AM) allows printing parts of complex geometries that cannot be produced by standard technologies. Besides, AM provides the possibility to create gradient materials with different structural and physical properties. We, for the first time, printed gradient soft magnetic materials from paramagnetic powders (316L steel and Cu-12Al-2Fe (in wt.%) aluminium bronze)). The magnetic properties can be adjusted during the in-situ printing process. The saturated magnetization value of alloys reaches 49 emu g-1. The changes in the magnetic properties have been attributed to the formation of the BCC phase after mixing two FCC-dominated powders. Moreover, the phase composition of the obtained gradient materials can be predicted with reasonable accuracy by the CALPHAD approach, thus providing efficient optimization of the performance. The obtained results provide new prospects for printing gradient magnetic alloys.   Keywords Direct Energy Deposition (DED), in-situ alloying, soft magnets, gradient structure, magnetic properties  2  1. Introduction Additive manufacturing (AM) comes into play when conventional methods such as casting, rolling, and stamping fail to create parts of sophisticated geometry. Diegel et al. (2019) summarized a wide range of AM applications from producing a trivial guitar stand to a complex metal hydraulic manifold. However, despite the great potential of the AM current state, the printing of multicomponent alloys with gradient properties is not well studied and requires additional research and development. One of the approaches to perform gradient materials printing is the premixing blend of the original materials. Shen et al. (2021) demonstrated the possibility of using innovative combined cable wire arc 3D printing technology to produce parts from multiple filaments composed of 5 elements high entropy alloy (Al-Co-Cr-Fe-Ni) and the possibility of changing mechanical properties of a printed model by varying printing speed. Dobbelstein et al. (2019) used multiple compositions of preblended TiZrNbTa powders to produce graded high entropy samples and to find a composition with better printability for laser material deposition process and better mechanical properties. Chen et al. (2020) used a mixture of high entropy CoCrFeNi pre-alloyed powder with Mn powder for selective laser melting process with varying build parameters and achieved good printability and successful in-situ alloying.  However, this approach does not allow controlling chemical composition during the printing process, while in-situ mixing of the materials during printing allows obtaining gradient materials and creating new alloys. Recently, Melia et al. (2020) presented the homogeneous MoNbTaW alloy produced by AM technique using four powders of pure metals as a feedstock. Moorehead et al., 2020 also demonstrated the possibility to obtain the single solid solution for the MoNbTaW system using Direct Energy Deposition (DED) technique. Their results have a good agreement with a CALPHAD calculation. DED is one of the AM techniques that can produce such graded materials, which allows to evade the use of complex casting technologies or spark plasma sintering. Mixing materials during the printing process is very promising for the manufacturing of magnetic alloys.  Another perspective AM technology for the production of magnetic materials is Laser-Powder Bed Fusion (LPBF). Volegov et al., (2020) demonstrated the possibility to use LPBF to print hard magnets with high coercivity from NdFeB-based magnetic powder. Schönrath et al., (2019) studied the effects of selective laser melting parameters on magnetic properties of premixed Fe and Ni powders. Quite high saturated magnetization values were achieved due to the fact that he used two ferromagnetic powders. Garibaldi et al., (2018) demonstrated the ability to change magnetic properties of 3D printed FeSi samples by heat treatment. Kang et al., (2018)  3 demonstrated the difference in magnetic properties of Fe-Ni-Si samples produced by selective laser melting with different build parameters. The variation of printing conditions in LPBF changes the nitrogen content in high-nitrogen steel and, as a result, leads to the formation of para- or ferromagnetic properties. Arabi-Hashemi et al., (2020) introduced in-situ alloying by precisely controlled selective laser melting parameters to modify magnetic properties inside a single 3D printed part. DED technology is able to print gradient materials AlxCuCrFeNi2, FexCo100-x, FexNi100-x, Fe–Si–B–Nb–Cu. Borkar et al., (2017, 2016) successfully built functionally graded soft-magnets with Fe–Si–B–Cu–Nb and Al-Cr-Cu-Fe-Ni alloys by varying supply rate of elemental powders during the direct energy deposition process. Toman et al., (2018) performed DED printing of magnetic shape memory Ni-Mn-Ga alloy. Magnetic analysis of this alloy showed that heat treatment increases saturated magnetization. Kustas et al., (2019) showed effects of 3D printing parameters on microstructure of soft ferromagnetic FeCo alloy. Mikler et al., (2017) demonstrated effects of laser speed and laser power on magnetic properties of DED ferromagnetic alloy Fe-30at%Ni.  All discussed ways to produce magnetic samples are using ferromagnetic consumable materials. However, utilization of ferromagnetic metal powders in DED technology may lead to clogging of the powder feeding system on some machines due to magnetization of the feeding system and/or metal parts magnetization by the powder. LPBF technology requires demagnetization of build substrate and metal parts inside a build chamber to avoid uneven ferromagnetic powder layering. In-situ production of ferromagnetic parts from paramagnetic powders is the potential way to address these issues. In this paper, for the first time, we demonstrate that the use of in-situ melting in the printing process allows the printing of soft magnets from non-magnetic powders. Two paramagnetic metal powders Aluminum-Bronze and SS 316L were used during this process with variable feed rates to achieve different printing material compositions and different magnetic properties. The results of this work refer to gradient materials as a potential input for the fabrication of parts with magnetization varying from 0 to 49 emu g-1 and quite low coercivity varying from 43 to 81 G.  2. Experimental section The samples were created on the Insstek MX-1000 printer based on direct energy deposition technology. Three feeders with different powder compositions allow printing gradient structures with various concentrations of elements. Aluminum bronze (Cu-12Al-2Fe (in wt.%), denoted as Al-Bronze hereinafter) and 316L stainless steel (SS) powders produced by Praxair  4 and Höganäs were used. The powder size distribution varies in the range of 45-145 µm. The printing regime is as follows: laser power of 420 W, laser speed of 850 mm/min, and hatch spacing of 500 µm. Argon shield gas was used as a protective atmosphere. The structure of the obtained samples was characterized by the Scanning Electron Microscopy (SEM, FEI Quanta 600 FEG), electron-backscattered diffraction (EBSD, FEI Nova NanoSEM with EDAX Hikari detector) analysis, and X-ray diffraction (XRD, Rigaku Ultima IV) analysis. For the characterization of crystalline structures of the samples the using a Bruker D8 ADVANCE was carried out.  The magnetization measurements have been performed with the LakeShore 7410 vibrating sample magnetometer (VSM) at room temperature in the range of fields from -1 T to 1 T. The equilibrium phase diagrams were constructed using CALPHAD approach (ThermoCalc 2020a software, TCFe7.0 and TCHEA3 databases). 3. Results and discussion Fig. 1 demonstrates the SEM images of the powder and particle size distribution (PSD). The average size of 316L is ~ 83 μm, while the average size of the Al-Bronze powder is 95 μm. PSD has a different distribution that is demonstrated in Fig. 1 b), d). The scheme of the experiment is presented in Fig. 2a. The Insstek MX 1000 has 3 feeders for producing the trinary alloys from the initial powders. In our experiments, we used two feeders filled with 316L and aluminium bronze powders without initial pre-mixing. The composition of final alloys changed by varying the powder feed. The ratio between the different powders gradually changed but the final flow was constant 3.5 g min-1. The photo of the sample clearly shows the difference in colors. The yellow color is aluminum bronze, while the \"metal\" color corresponds to the 316L SS. The color gradient along the printing direction can be observed.   5  Fig. 1. a) SEM image of 316L, b) particle size distribution of 316L, c) SEM image of Al-Bronze powder, d) particle size distribution of the Al-Bronze powder. \n 6   Fig. 2. a) Scheme of feeders for printing gradient materials and picture of the gradient alloys after the mechanical treatments, b) EDS mapping of the main element in a different part of the sample.  The EDS mapping demonstrates the evolution of the Fe, Cu, Al (the main elements of the alloy). It is clearly seen that the concentration of Fe has decreased from Zone 1 to Zone 7, while the concentration of Cu and Al is growing.  \n 7  Fig. 3. SEM images of the structures from the bottom, middle, and top of the sample. The pictures were made from both directions: along and perpendicular to the printing.   SEM images of the grain from the different parts of the sample are presented in Fig. 3. The left and right rows of the figure demonstrate structures perpendicular and along the build direction, correspondingly. The structures of the 316L SS are characterized by the grain size with a length of a few hundred microns. In contrast, the grain size of the (Al-Bronze)50(316L)50 is characterized by the length of ten microns. Two phases with distinctive dark and light contrast are found in the bronze steel mixture. The light phase is enriched with Cu and the dark phase - with \n 8 Fe and Cr. Al and Ni are almost evenly distributed between the two phases (Fig. S1 and Table S1). The structure of Al-Bronze has a length of tens microns and reveals the two phases' presence. The Energy Dispersive X-Ray Spectroscopy (EDS) chemical analysis has shown that lighter “grains” have close to nominal chemical composition, whereas the darker phase in-between the “grains” have Cu to Al ratio close to 3:1 (in at.%) (Fig. S1 and Table S1. S2).  \n Fig. 4 (a) XRD spectra of alloys. The black, blue, and red colors of the curves represent the spectra of 316L SS, Al-bronze, and (Al-bronze)50(316L)50, correspondingly. (b)-(c) phase diagrams of alloys. Red and violet colors identify the liquid phase(s) fractions. The olive and black curves are related to FCC and BCC fraction, respectively.   The XRD analysis was performed (Fig. 4a and Fig. S2) to identify the constitutive phases. The XRD pattern of 316L shows only the peaks from the FCC lattice with the parameter of 3.601 A. The single-phase structure is consistent with the SEM observations. The XRD spectra of Cu-12Al-2Fe consist of two different lattices, one of which is the FCC lattice with the parameter of 3.661 Å. Another lattice is orthorhombic and belongs to the space group Pmmn. It has parameters of a=4.424 Å, b=4.514 Å, c=5.181 Å and can be identified as Cu3Al compound Kurdjumov et al., (1938). The intensity of the FCC peaks is higher than that of Cu3Al. Mixing the 316L to Al-bronze \n 9 produces a dual-phase structure with FCC and BCC lattices. The lattice parameters are 3.637 Å and 2.878 Å for FCC and BCC, correspondingly. The intensities of the BCC and FCC peaks are close.  To gain additional insight into the phase formation, the equilibrium phase diagrams were constructed using CALPHAD approach. The phase diagram of the Al-bronze predicts a stable single FCC phase structure after crystallization (Fig. 4b). Note that the formation of Cu3Al compound was not predicted, probably due to limitations of the used database Module et al., (2015). The 316L steel is also expected to have a stable FCC phase structure, except a small amount of the BCC phase that forms during solidification and disappears shortly after (Fig. 4c). Al-Bronze phase diagrams presented in Fig. 4d. In turn, the steel-bronze mixture exhibits a more complex phase transformation scenario. Two liquid phases, one - Cu-rich, and the other one - Fe-rich, exist at the temperature range from 1300 ℃ to 1675 ℃. The Cu-Fe binary is known for limited mutual solubility between the components even in the liquid state Predel, et. al, (1994) . Solidification starts from forming the Fe-rich BCC phase at ~1300 ℃ and ends with the formation of the Cu-rich FCC phase at ~1050 ℃. At cooling, the alloys fall into a metastable condition where Fe-rich and Cu-rich liquids form. Fe-rich phase forms first at the cooling due to higher crystallization temperature Fig. S1 and Table S1. The remaining liquid becomes Cu rich that leads to crystallization and coalescence of the Fe phase at an elevated rate. A similar situation was shown for FeCu powder produced by gas atomization where the cooling rate is comparable with the additive manufacturing process (Abbas and Kim, (2018)). Such phase transformation sequences agree reasonably well with the experimentally observed structure (Fig. 3). The stabilization of the Fe-rich BCC phase can be attributed to the partitioning of some of the FCC forming elements (Ni) from 316L powder to Cu-rich liquid/solid, and, in turn, partitioning of BCC-stabilizing Al from Al-bronze powder to Fe-rich liquid/solid phase. For a more detailed characterization of the structures, Electron-Backscattered Diffraction (EBSD) analysis was used. Fig. 5 demonstrates the EBSD inverse pole figure (IPF) maps, alongside misorientation angle distribution. The phase composition of the alloys and pole figures are shown in Fig. 6 and Fig. S3. The 316L sample has a single austenitic FCC phase structure. The grain size perpendicular to and along the build direction is 130 μm and 250 µm, respectively Fig. S4.  10  Fig. 5. EBSD inverse pole figure (IPF) maps for Al-bronze, 316L SS, (Al-bronze)50(316L)50. The IPF maps are presented for two different orientations of the samples: perpendicular and along the build direction. The misorientation angle distribution is also provided.   \n 11  Fig. 6. EBSD phase fraction map of a selected area of Al-bronze and (Al-bronze)50(316L)50 alloys (red – FCC, green – BCC, blue – Cu3Al).   The Al-bronze has a dual-phase microstructure, composed of the FCC and Cu3Al phases. Note that the EBSD analysis did not adequately recognize the Cu3Al phase due to the absence of corresponding information in the software (EDAX APEX EBSD) used. The volumetric fractions of FCC and Cu3Al are 73 and 27%, correspondingly. According to EBSD analysis, typical grain sizes are 37 and 26 µm for perpendicular and along with printing directions. The size of the grains is ten times smaller than the grain sizes for pure 316L SS.  The alloy of (Al-bronze)50(316L)50 consists of two lattice types, such as FCC and BCC. FCC/BCC ratios for perpendicular and along the build directions are 41/59 and 66/34, respectively. Note that close to the 50:50 phase ratio is predicted by the CALPHAD approach (Fig. 4d). The grain size of the alloy is 8 and 10 µm in different directions. The size of the grains correlates with the results of SEM images (Fig. 3). A strong peak on misorientation angle distribution at ~40-50⁰ can be associated with twinning in the BCC phase (Fig. 5).  \n 12 The perpendicular direction of the 316L characterizes the presence of twins, proved by the peak on misorientation angle distribution around 60⁰. Yadollahi et al., (2015) demonstrated the similar results for the samples produced by the DED. At the same time, the pure Al-bronze did not demonstrate any dominant orientation. For the alloy (Al-bronze)50(316L)50, the two different directions demonstrate the formation of twins according to the misorientation angles ~40-50⁰ (Rusakov et al., (2014), Bertrand et al., (2011)). The detailed parameters of the lattice and their fraction are summarized in Table S4.  Seven (Al-Bronze)x(316L)1-x samples of cylindrical shape with different x values, each with a mass of about 0.6 g, have been measured. The procedure of samples preparation is presented in Fig. S5. The magnetic properties of the 316L SS and Al-bronze mixture showed strong dependence on the composition of the mixture (Fig. 7a). The highest measured value corresponds to the sample with an equal proportion of Al-bronze and 316L. These results correlate with the XRD data Fig. S2 which demonstrates the highest value of BCC phase. Values of saturated magnetizations (Ms) for the samples with x = 0.75, 0.6, 0.5, 0.4, 0.25 were 24.8 emu g-1, 47.9 emu g-1, 49 emu g-1, 35.8 emu g-1, 6.6 emu g-1, respectively. The dependence of MS value on x is presented in Fig. 7(b). While all samples show ferromagnetic behavior, pure 316L and Al-bronze demonstrate the paramagnetic properties. It can be highlighted that this dependence is non-symmetrical - MS of the sample with x = 0.75 is nearly 4 times higher than of x = 0.25. This fact is in accordance with the data obtained on alloy crystalline structure, shown in Table S2 and S4. The main ferromagnetic contribution in our samples comes from the BCC phase, which contains a higher (Al-bronze)50(316L)50 volumetric fraction in alloys. The values of coercivity (Hc) are also different for the measured samples. In the case of x = 0.75, 0.5, and 0.25 they are 43 G, 50 G, and 81 G, respectively (insert in Fig. 7 a).  \n \n 13 Fig. 7. Magnetic characteristics of (Al-Bronze)x(316L)1-x alloy at different values of x. (a) magnetization curves, (b) saturated magnetizations.   The austenite (FCC) phase in the steels is known for its paramagnetic properties, as well as FCC Cu and its alloys. However, the ferrite (BCC) phase is ferromagnetic below the Curie temperature. In the case of the examined (Al-Bronze)x(316L)1-x alloys, the formation of the BCC phase is responsible for variations in magnetic properties. Earlier Arabi-Hashemi et al., (2020) demonstrated that BCC phase formation in the austenitic steel leads to the values of saturated magnetizations of 40 emu/g. A similar observation was made for the AlxCuCrFeNi2 high entropy alloys. In these alloys, the increase in the saturation of magnetization has been associated with the transformation of the FCC-dominated structure to the BCC-based one with the increase of the Al content. Borkar et al., (2016) show a similar transformation in AlxCrCuFeNi2 alloy depending on the aluminum content.  4. Conclusions To sum up, for the first time, we have demonstrated the possibility of printing gradient soft magnetic materials using in-situ melting during direct energy deposition. The paramagnetic powders have been used for the production of soft magnetic materials. The addition of Al-bronze to 316L transforms the single FCC phase structure to the mixture of the FCC and BCC phases. The achieved maximum fraction of the BCC phase was 59% in the perpendicular build direction of the (Al-bronze)50(316L)50 sample. The observed phase transformations are in reasonable agreement with the CALPHAD predictions. Mixing paramagnetic powders during the printing process allows controlling the saturated magnetization. The maximum saturated magnetization reached 49 emu g-1 in the (Al-bronze)50(316L)50 sample. The changes in the magnetic behavior are attributed to the formation of the BCC phase. The obtained data provides new possibilities for the development of materials with gradient magnetic properties by additive manufacturing.  Data availability The data that support the findings of this study are available from the corresponding author upon reasonable request.   14 Declaration of Interest The authors declare no known financial interests or personal relationships that could have appeared to influence the work reported in this paper.  Acknowledgment The research is partially funded by the Ministry of Science and Higher Education of the Russian Federation as part of World-class Research Center program: Advanced Digital Technologies (contract No. 075-15-2020-903 dated 16.11.2020)  References Abbas, S.F., Kim, T.-S., 2018. Microstructural Characterization of Gas Atomized Copper-Iron Alloys with Composition and Powder Size. J. Korean Powder Metall. Inst. 25, 19–24. https://doi.org/10.4150/kpmi.2018.25.1.19 Arabi-Hashemi, A., Maeder, X., Figi, R., Schreiner, C., Griffiths, S., Leinenbach, C., 2020. 3D magnetic patterning in additive manufacturing via site-specific in-situ alloy modification. Appl. Mater. 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A 644, 171–183. https://doi.org/10.1016/j.msea.2015.07.056     17 Supplementary materials for Gradient soft magnetic materials produced by additive manufacturing from non-magnetic powders   O.N. Dubinin1,2, D. A. Chernodubov3, Yu.O. Kuzminova1, D.G. Shaysultanov4, I.S. Akhatov1, N.D. Stepanov4 and S.A.Evlashin1  1 Center for Design, Manufacturing & Materials, Skolkovo Institute of Science and Technology, 30, bld. 1 Bolshoy Boulevard, Moscow 121205, Russia 2 Saint Petersburg State Marine Technical University, Lotsmanskaya street, 3 Saint-Peterburg 190121, Russia 3 National Research Center \"Kurchatov Institute,\" PI. Kurchatova, 1, Moscow 123182, Russia 4 Laboratory of Bulk Nanostructured Materials, Belgorod State University, Belgorod, 308015, Russia  \n Fig. S1. EDS analysis of (Al-bronze)50(316L)50 alloy at different areas.  Table S1. The chemical composition of (Al-bronze)50(316L)50 alloy at different scanning areas, wt. %.   Al K Cr K Fe K Ni K Cu K Area 1 5.4 1.4 6.7 5.2 81.4 Area 2 5.6 3.5 14.5 6.3 70.1 \n 18 Area 3 3.9 16.0 63.2 8.2 8.8 Area 4 4.1 15.8 61.9 8.3 9.9 Area 5 4.4 5.7 26.0 8.8 55.0 Area 6 4.4 10.0 48.9 11.1 25.6   Table S2. EDS analysis of different areas in Fig 2 in wt. %.  Sample\t(316L/Al-Bronze)\tFe\tCu\tAl\tCr\tNi\tMo\tSi\tMn\tZone\t1\t(100/0)\t67.14\t\t\t17.30\t11.81\t1.64\t0.54\t1.57\tZone\t2\t(75/25)\t53.29\t17.75\t1.79\t14.17\t9.72\t1.48\t0.50\t1.30\tZone\t3\t(60/40)\t46.80\t27.08\t2.74\t12.47\t8.07\t1.29\t0.45\t1.10\tZone\t4\t(50/50)\t39.52\t37.24\t3.97\t10.42\t6.31\t1.11\t0.45\t0.98\tZone\t5\t(40/60)\t33.19\t46.13\t4.79\t8.62\t5.18\t0.88\t0.39\t0.82\tZone\t6\t(25/75)\t25.12\t57.12\t5.79\t6.57\t3.58\t0.74\t0.34\t0.74\tZone\t7\t(0/100)\t1.21\t90.17\t8.62\t\t\t\t\t\t \n  Fig. S2. XRD spectra of samples with different concentrations of Al-Bronze. \n 19  Table S3. The phase composition calculated using the Rietveld method. Zone (316L/Al-Bronze) FCC BCC Zone 1 (100/0) 100.00  Zone 2 (75/25) 72.12 27.88 Zone 3 (60/40) 58.80 41.20 Zone 4 (50/50) 43.27 56.73 Zone 5 (40/60) 57.53 42.47 Zone 6 (25/75) 71.33 28.67  Table S4. Lattice parameter for different alloy composition  FCC / Fraction BCC / Fraction Orthorhombic / Fraction 316L 3.6007 / 100 %   Al-Bronze 3.6606 / 73 %  4.4940  / 27 % 5.1890 46.610 (Al-Bronze)50(316L)50 3.6366 / 41% 2.8777 / 59 %     20   Fig. S3. Pole figures of different alloys.  \n 21  Fig. S4. Grain size at different concentration and orientation.   \n Fig. S5. The procedure of sample preparation for the measurement of magnetic properties.  \n"    },    {        "title": "2109.11423v1.Tuning_magnetic_antiskyrmion_stability_in_tetragonal_inverse_Heusler_alloys.pdf",        "content": "Tuning magnetic antiskyrmion stability in tetragonal inverse Heusler alloys\nDaniil A. Kitchaev1,\u0003and Anton Van der Ven1\n1Materials Department, University of California, Santa Barbara, California 93106, USA\nThe identi\fcation of materials supporting complex, tunable magnetic order at ambient tempera-\ntures is foundational to the development of new magnetic device architectures. We report the design\nof Mn 2XY tetragonal inverse Heusler alloys that are capable of hosting magnetic antiskyrmions\nwhose stability is sensitive to elastic strain. We \frst construct a universal magnetic Hamiltonian\ncapturing the short- and long- range magnetic order which can be expected in these materials.\nThis model reveals critical combinations of magnetic interactions that are necessary to approach a\nmagnetic phase boundary, where the magnetic structure is highly susceptible to small perturbations\nsuch as elastic strain. We then computationally search for quaternary Mn 2(X1;X2)Yalloys where\nthese critical interactions may be realized and which are likely to be synthesizable in the inverse\nHeusler structure. We identify the Mn 2Pt1\u0000zXzGa family of materials with X= Au, Ir, Ni as\nan ideal system for accessing all possible magnetic phases, with several critical compositions where\nmagnetic phase transitions may be actuated mechanically.\nA substantial component of spintronic device develop-\nment is the discovery of materials that are capable of\nhosting exotic spin textures over precisely tuned \feld\nand temperature ranges[1]. While spin textures can\nbe controlled by magnetic \felds, dynamically coupling\nthese magnetic phases to other variables such as elec-\ntric \felds or mechanical perturbations allows for new\ncontrol paradigms and device architectures[2]. Magnetic\nskyrmion and antiskyrmion textures have in particular\nattracted attention due to their combination of thermo-\ndynamic stability, unique topological properties, and e\u000e-\ncient transport behavior[3{8]. A number of bulk material\nsystems capable of hosting equilibrium skyrmions[5, 9{\n14] or antiskyrmions[15, 16] have been discovered, and\nrecent reports have indicated that certain materials may\nsupport both topologies as metastable states[17, 18].\nHowever, tuning the geometry and stability windows of\nthese topological phases, at the synthesis stage or in situ ,\nremains a challenge. This is due to the lack of a theoret-\nical understanding of \rexible material systems that are\ncapable of hosting (anti)skyrmion phases at room tem-\nperature, as well as the irreversibility of the structural\ndeformations typically necessary to elicit a substantial\nmagnetic response[14, 19].\nAn attractive model system for realizing chemically-\nand mechanically- tunable (anti)skyrmion states are the\ntetragonal inverse Heusler alloys[15, 16, 20]. These ma-\nterials have the Mn 2XY chemical formula where the X\nsublattice generally consists of late transition metal ele-\nments and Y= Ga, Sn, In[20, 21]. Below their marten-\nsitic transformation temperature, they possess D2dsym-\nmetry. This symmetry is compatible with the forma-\ntion of thermodynamically-stable antiskyrmions[22, 23],\nwhile metastable skyrmions can be nucleated with an\nappropriate history of applied magnetic \felds[17, 18].\nCritically, this symmetry also ensures that the topolog-\nical phases may remain stable from 0 K to Tc(Curie\n\u0003dkitch@alum.mit.edutemperature)[3, 23, 24], which in these materials is of-\nten well above room temperature[21]. Furthermore,\nthe Heusler alloys allow for immense chemical \rexi-\nbility, which has been previously used to tune their\nstructural[25], electronic[26, 27] and magnetic[28, 29]\nproperties. The combination of chemical \rexibility in\ntheXandYsublattices and thermal stability of the\ntopological magnetic states means that the topological\nmagnetism seen in Mn 2XY inverse Heusler alloys may\nin principle be tuned and observed at room temperature,\nas is necessary for device applications.\nHere, we implement a general design strategy for real-\nizing chemically and mechanically tunable antiskyrmions\nusing the Mn 2XYtetragonal inverse Heuslers as a model\nsystem. We \frst derive a universal model for the short-\nand long- range magnetic order of materials with the in-\nverse Heusler structure in terms of computable magnetic\ninteractions. We then enumerate known and hypothet-\nical Mn 2XY inverse Heusler materials and characterize\nthe impact of varying chemistry on the magnetic inter-\nactions and chemical stability of the alloys. We show\nthat with an appropriate choice of composition on the X\nandYsublattices, one can realize all possible magnetic\nphases and create materials where magnetic phase transi-\ntions may be actuated with small, purely elastic mechan-\nical perturbations. Finally, we identify Mn 2Pt1\u0000zXzGa\nwithX=Au, Ir, Ni and z\u00190:1\u00000:2 as an ideal sys-\ntem for realizing this behavior, combining \rexible room-\ntemperature magnetism with chemical and structural\nstability.\nRESULTS\nOur approach to realizing chemically- and\nmechanically- tunable antiskyrmion states is to relate\nthe form of the magnetic phase diagram to variations in\natomistic magnetic interactions, and then characterize\nhow these interactions may be tuned by chemical\nchanges and elastic perturbations. This approach is\nshown schematically in Figure 1. We \frst construct anarXiv:2109.11423v1  [cond-mat.mtrl-sci]  23 Sep 20212\nFIG. 1. Schematic summary of our materials design strategy\nfor obtaining tunable antiskyrmion behavior. We \frst map all\nhypothetical compounds A,B,Cto interaction parameters\nJ1;J2;:::of a quasi-classical atomistic Hamiltonian to deter-\nmine their short-range spin order. We repeat this analysis\nat longer length scales by coarse-graining the magnetocrys-\ntalline anisotropy Kand Dzyaloshinskii-Moriya interaction D\n(DMI). We identify the parameter space where antiskyrmions\n(ASk) may be expected and construct alloys A1\u0000xBxwhich\nfall in the region of ASk stability. Finally, we identify critical\ncompositions xcfalling on magnetic phase boundaries as com-\npounds where the magnetic phase transition may be actuated\nby small perturbations, e.g. reversible elastic strain.\natomistic quasi-classical spin Hamiltonian applicable\nto all tetragonal inverse Heuslers, accounting for local\nexchange, Dzyaloshinskii-Moriya (DMI) and anisotropy\ninteractions. Using this model, we parametrically enu-\nmerate the local spin structures that can be stabilized\nby various combinations of exchange strengths. Next,\nwe coarse-grain these atomistic interactions to produce\na continuum free energy functional that enables a\nparametric exploration of long-range magnetic struc-\ntures such as antiskyrmions. We deduce the magnetic\nbehavior of candidate materials A,B,Cby \ftting\ntheir interaction parameters to density functional theory\n(DFT) data. We then construct alloys A1\u0000xBxbetween\ncompatible materials AandBthat share the same local\nspin order. In the A1\u0000xBxalloy, coarse-grained magnetic\ninteractions vary continuously with composition, making\nit possible to identify critical compositions xcthat reside\nFIG. 2. Idealized structure of a Mn 2XYtetragonal inverse\nHeusler alloy. a.Distinct crystallographic sites which de\fne\nfour interpenetrating face-centered cubic lattices. b.Minimal\nmagnetic interaction model for a prototypical Mn 2XYtetrag-\nonal inverse Heusler. J1represents the coupling of the Mn(1)\nand Mn(2) sublattices. J(1)\n2andJ(1)\n3represent the in-plane\nand out-of-plane interactions respectively within the Mn(1)\nsublattice, and J(2)\n2andJ(2)\n3for the Mn(2) sublattice.\non magnetic phase boundaries where magnetic phase\ntransitions may be actuated by small perturbations such\nas reversible elastic strain.\nShort- and long-range magnetic order in tetragonal\ninverse Heuslers\nThe Mn 2XYtetragonal inverse Heusler alloys are de-\n\fned by the idealized crystal structure shown in Figure\n2a. This structure consists of four tetragonally distorted\ninterpenetrating face-centered-cubic sublattices. Two of\nthese sublattices, Mn(1) and Mn(2), have localized mag-\nnetic moments in the range of 2-3 \u0016Bper atom. The Y\nsublattice generally contains one of Ga, Sn, or In and is\nnon-magnetic. The Xsublattice can be occupied by a\nrange of late transition-metal elements, with previously\nreported compounds having X= Fe, Co, Ni, Rh, Pd, or\nPt[21, 28, 30, 31]. In this study, we supplement these\nelements with other transition metals which could po-\ntentially be doped onto the Xsublattice: Ru, W, Os, Ir,\nAu[21] However, we exclude Fe as it introduces a large\nmagnetic moment on the Xsublattice and cannot be\ntreated with the same magnetic model as systems with\nnon-magnetic Xelements. As both the chemical stabil-\nity and the degree of chemical order vary substantially\nbetween these chemistries, we will discuss which compo-\nsitions are most likely to be synthetically accessible in a\nlater section.\nWe represent the magnetic behavior of Mn 2XY in-\nverse Heuslers with a combination of exchange inter-\nactions, which are the dominant energy scale and con-\ntrol the local spin structure, and coarse-grained spin-\norbit e\u000bects that control the long-range modulation of3\nFIG. 3. Equilibrium phases given by the minimal magnetic Hamiltonian for the Mn 2XYtetragonal inverse Heusler structure.\na.Local spin con\fgurations governed by the relative strength of J1-J2-J3exchange interactions, including collinear ferrimagnetic\n(FiM) and antiferromagnetic (AFM xy, AFM z) phases, as well as a region of frustrated non-collinear order (NCL). b.Long-range\nphases generated as modulations of the FiM order at low- T, which include spin helices (Hx), antiskyrmion lattices (ASk) and\nconical helices (Cx). Long-range structure is governed by the relative strength of uniaxial anisotropy K, Dzyaloshinskii-Moriya\ncouplingD, spin-sti\u000bness Aand applied \feld Halong thec-axis. The phase diagram is evaluated for J2=J3= 0 (red circle in\na.) and an equilibrium helical wavelength of 24 unit cells. c.Extension of the K= 0 region of the long-range phase diagram\nto \fnite temperature. Color denotes the expected number of antiskyrmions per 24x24 unit cell as measured by the topological\nindex density t. Solid lines denote \frst-order phase transitions while dotted lines denote second-order or continuous phase\nboundaries. Tcdenotes the Curie temperature and fd refers to the \ructuation-disordered Brazovskii region[24, 32].\nthe local spin structure. We consider the atomic ex-\nchange interactions in conventional Heisenberg model\nform,Hexchange =P\nij2\u000bJ\u000b(\u0000~Si\u0001~Sj) where the summa-\ntion includes couplings up to the 3rd nearest neighbor as\nshown in Figure 2b. J1represents the strongly antiferro-\nmagnetic direct exchange between the Mn(1) and Mn(2)\nsublattices, while J2andJ3capture the weaker interac-\ntions within the two sublattices. To further simplify the\nmodel, we set J2=J(1)\n2=J(2)\n2andJ3=J(1)\n3=J(2)\n3so\nthat the geometrically-identical interactions within the\nMn(1) and Mn(2) sublattices are assumed to have the\nsame interaction strength. The complete form of this spin\nHamiltonian is given in Supplementary Data 1. Despite\nthe simplicity of this model, we \fnd that it is su\u000ecient to\ncapture the energetics of collinear spin con\fgurations in\nthe Mn 2XY compounds considered in this work, repro-\nducing both the ground state and excited state spectrum\nas computed with density functional theory (DFT). The\nresults of this \ftting procedure and the correspondence\nbetween the model and the electronic structure data is\nquanti\fed in Supplementary Data 2 and 3.\nThe competition between the exchange interactions J1,\nJ2andJ3gives rise to several local spin orderings, as\nshown in Figure 3a. When J2andJ3are ferromagnetic,\nor negligible compared to J1, the spins adopt a ferrimag-\nnetic structure (FiM) with the Mn(1) and Mn(2) sub-\nlattices antialigned with each other. This structure has\na net moment as the local moment on Mn(1) is typi-\ncally larger than that on Mn(2). Antiferromagnetic J2\nandJ3interactions frustrate this order and can lead to\na region of non-collinear order (NCL), or collinear an-\ntiferromagnetic structures with spins either alternatingin thexyplane or along the zaxis (AFM xyand AFM z\nrespectively). Of these structures, we focus on the ferri-\nmagnetic phase as it is the only spin structure with a net\nmagnetic moment at low temperature and \feld.\nThe long-range magnetic texture is de\fned by a grad-\nual rotation of the local spin structure driven by the\nDzyaloshinskii-Moriya component of spin-orbit coupling\nand suppressed by the magnetocrystalline anisotropy and\nspin-sti\u000bness. These phenomena are conventionally de-\nscribed by a coarse-grained magnetic Hamiltonian for the\nD2dpoint group[33, 34]:\nH=Z\ndr3\u0002\nA=2(rm)2+D(w22\u0000w11) +Km2\nz\u0003\nwheremis the unit vector direction of the local mag-\nnetization. Ais the spin-sti\u000bness parameter and rep-\nresents the coarse-grained exchange strength. The re-\nlationship between Aand the atomistic J1,J2,J3pa-\nrameters is given in Supplementary Data 4. Dand\nwkn=\u000fijkmi@mj=@rnrepresent the strength and form\nof the coarse-grained Dzyaloshinskii-Moriya interaction,\nwhere\u000fijkis the Levi-Civita tensor and repeated in-\ndices imply summation[33]. Kparametrizes the uniaxial\nanisotropy with respect to the crystal axes given in Fig-\nure 2a. While higher-order anisotropies are necessary to\naccurately capture the DFT energetics of some Heusler\ncompounds including the Pt and Ir-based systems con-\nsidered here, we have found that these terms are never\nlarge enough to alter the \fnal magnetic phase diagrams\nin our analysis. Here, all spatial dimensions are taken to\nbe in units of the lattice parameter of the conventional\nstructure shown in Figure 2a ( afor thexydirections, c\nfor thezdirection).4\nFIG. 4. Magnetic interactions in ternary Mn 2XY inverse Heusler compounds. a.Local spin order based on competition\nbetween exchange interactions, including only those chemistries which favor the tetragonal inverse Heusler structure at the\nMn2XY stoichiometry. b.Long-range spin textures in chemistries favoring local FiM order, based on the relative strengths\nof the Dzyaloshinskii-Moriya ( D) and mangetocrystalline anisotropy ( K) components of spin-orbit coupling. See caption to\nFigure 3 for phase de\fnitions. Note that D=A = 2\u0019=\u0015where\u0015is the equilibrium wavelength of the helical and antiskyrmion\nphases in units of the basal ( a) lattice parameter.\nWhether or not the local spin structure develops\na long-range texture at equilibrium is determined by\nthe competition between the Dzyaloshinskii-Moriya and\nmagnetocrystalline anisotropy components of spin-orbit\ncoupling (D=A andK=A respectively)[3, 24]. These spin\ntextures include spin helices (Hx), spin cones (Cx) and\nantiskyrmions (ASk) which all have a characteristic en-\nergy scale of D2=2Aand form the phase diagram shown\nin Figure 3b in the low temperature limit. This phase\ndiagram shows that as a function of the normalized\nanisotropy (2 KA=D2) and magnetic \feld along the c-axis\n(2HA=D2), spin helices and conical structures are stabi-\nlized for\u00002\u00142KA=D2\u00143. Antiskyrmions are favored\nunder a small applied \feld for \u00001\u00142KA=D2\u00141:7. The\nchange in this phase diagram at elevated temperature is\nshown in Figure 3c for the case of vanishing anisotropy\nKandJ2=J3= 0. The helical and antiskyrmion phases\npersist at all temperatures up to Tcwith minimal change\nin the phase boundary between them, although the max-\nimum \feld at which antiskyrmions are stable decreases.\nVariation of J2andJ3within the FiM region do not alter\nthe overall shape of this phase diagram, but do signi\f-\ncantly rescale Tcas shown in Supplementary Data 4.\nMagnetic structure and chemical stability of Mn 2XY\ntetragonal inverse Heuslers\nWe now examine where known and hypothetical\nternary Mn 2XY inverse Heuslers fall on the magnetic\nphase diagrams shown in Figure 3. In Figure 4a we plot\nthe exchange interactions in a range of compounds and\ndeduce their local spin structure, focusing only on those\ncompositions that thermodynamically favor the tetrag-\nonal inverse Heusler structure at the Mn 2XY composi-\ntion. The majority of these compounds fall in the FiMregion, with frustrated non-collinear order expected in\nMn2PtSn, Mn 2PtIn and Mn 2RhSn consistent with ex-\nperimental reports[20, 35{38]. For the remaining ma-\nterials that favor FiM order, we compare the coarse-\ngrained spin-orbit coupling to the regions where heli-\ncal, conical, or antiskyrmion long-range phases can be\nexpected. Here we also include the hypothetical com-\npounds Mn 2AuGa and Mn 2WSn for which the inverse\nHeusler chemical order is metastable. As can be seen in\nFigure 4b, most compositions are easy-axis ferrimagnets\n(FiM withK < 0), with only the hypothetical Mn 2AuGa\nand Mn 2WSn materials falling in the easy-plane ferri-\nmagnet region (FiM, K > 0). Non-collinear spin textures\ncan be expected in Mn 2PtGa, Mn 2IrSn, Mn 2PdSn and\nMn2NiSn, where the Dzyaloshinskii-Moriya interaction\nis su\u000eciently large to fall in the antiskyrmion stability\nregion (\u00001\u00142KA=D2\u00141:7).\nOf the various compounds mapped out in Figure 4,\nwe focus on Mn 2XGa forX= Pt, Ni, Ir, as they are\nthe most likely to be synthesizable at equilibrium as\nstoichiometric, well-ordered inverse-Heusler compounds.\nThe synthesis of any inverse Heusler compound can be\nchallenging, as the \fnite-temperature phase diagrams of\nthe binary endpoints are often very complicated and the\nphase diagrams of the full ternary systems are not known.\nFor example, the prototypical compound for the tetrag-\nonal inverse Heusler structure, Mn 3Ga, forms by a low-\ntemperature peritectic reaction with numerous compet-\ning phases that need to be avoided to produce a high-\nquality material[39, 40]. Furthermore, the formation of\nchemical order is complicated by the coupling between\nchemical order and the structural transformation be-\ntween the high-temperature austenite and low- Tmarten-\nsite phase[21, 41, 42]. While assessing the full \fnite-\ntemperature phase diagrams and ordering kinetics of the\nchemistries described here is prohibitive, we can evaluate5\nthe likelihood that any given Mn 2XY may be formed\nby the conventional process of high-temperature mixing\nfollowed by a long low-temperature anneal. We assume\nthat the high-temperature precursor is a disordered state\nprepared at the correct stoichiometry[20]. As this pre-\ncursor is cooled, the formation of an ordered product is\ncharacterized by a driving force \u0001 Eorder-disorder , which\nwe approximate using the di\u000berence in energy between\nthe ordered inverse Heusler product and most favor-\nable disordered state among the common disorder mod-\nels proposed for these systems (L2 1b(Mn(1)/X), BiF 3\n(Mn(1)/Mn(2)/ X))[30]. This ordering reaction competes\nwith phase separation, whose likelihood is correlated with\nthe energy of formation \u0001 Eformation of the target com-\npound from competing phases in each Mn- X-Yternary\nspace[43]. The equilibrium phases used to determine\nchemical stability are given in Supplementary Data 5.\nFigure 5 charts the driving forces for the compet-\ning order-disorder and decomposition reactions for the\nMn2XY chemistries discussed here, excluding the W-\nbased compounds and Mn 2AuSn as they are exception-\nally unstable. All In-based and most Sn-based com-\npounds have a strong driving force for phase separation\nand thus are not likely to retain the desired stoichiome-\ntry after a long anneal. Furthermore, a number of com-\npounds have a minimal driving force to order, or in the\ncase of Pd-based systems and Mn 2AuGa do not favor\nthe ordered states we have considered at all. The sys-\ntems which favor the ordered inverse Heusler structure\nat low-Tequilibrium are Mn 2XGa forX=Ir, Pt, Rh,\nRu, Ni, Co and Mn 2XSn forX=Ru, Rh. From these,\nwe exclude the Ru-based systems and Mn 2RhGa as ex-\nperimental reports of these compounds indicate that the\nordered con\fguration is di\u000ecult to obtain in practice[30],\nMn2RhSn as it does not favor the locally-collinear FiM\nphase, and Mn 2CoGa as it does not produce a tetragonal\ndistortion. We now focus on the remaining synthesizable\nchemistries to identify combinations which, when alloyed,\nmay generate magnetic phase transitions.\nDesigning tunable magnetism in Mn 2Pt1\u0000zXzGa\nalloys\nHaving enumerated the magnetic behavior and chemi-\ncal stability of the ternary Mn 2XYinverse Heuslers, we\nturn to quaternary alloys in this space to tune magnetic\nproperties and fully explore the magnetic phase diagram\nshown in Figure 3b. In a solid solution between two com-\npounds Mn 2X(1)Y(1)and Mn 2X(2)Y(2)with the same lo-\ncal chemical and spin structure, the coarse-grained mag-\nnetic parameters D,KandAmust vary continuously\nwith composition. Graphically, this continuous variation\nmeans that the magnetic parameters of the alloy will fall\non a smooth curve connecting the endpoint compounds\nin Figure 4b. By alloying two materials which are sep-\narated by a magnetic phase boundary in Figure 4b, we\ncan switch the magnetic behavior of the alloy between the\nFIG. 5. Likelihood that an ordered inverse Heusler com-\npound may be obtained by an equilibrium synthesis method\nat the Mn 2XY stoichiometry. \u0001 Eorder-disorder measures the\ndriving force for ordering and is de\fned as the minimum\ndi\u000berence in energy between the ordered structure and the\ncommon types of disorder observed in inverse Heuslers (L2 1b\n(Mn(1)/X), BiF 3(Mn(1)/Mn(2)/ X))[30]. \u0001Eformation mea-\nsures the likelihood of phase separation into other phases in\nthe Mn-X-Ychemical space and is de\fned as the di\u000berence\nin zero-Tenergy between the ordered Mn 2XYphase and an\nequilibrium combination of competing phases given in Sup-\nplementary Data 5.\ntwo phases by varying composition. Furthermore, we can\nidentify the alloy composition that lies on the magnetic\nphase boundary to create a material where the magnetic\nphase transition can be actuated by a small elastic strain.\nOn the basis of the magnetic interaction parameters\nshown in Figure 4 and synthesizability metrics discussed\nin Figure 5, we identify the Mn 2Pt1\u0000zXzGa system for\nX= Au, Ir, Ni as a model system for tunably accessing\nall regions of the magnetic phase diagram. The Mn 2PtGa\nendpoint lies in the ASk region of the long-range mag-\nnetic phase diagram in Figure 4b. The X=Au, Ir, Ni\nendpoints of the alloy are separated from this region\nby the ASk/Hx, ASk/Cx, Hx/FiM and Cx/FiM phase\nboundaries. Thus by varying the dopant element and\ncomposition we expect the alloy to move across the ASk,\nHx, Cx, and FiM regions of the magnetic phase diagram,\nwith several critical compositions corresponding to mag-\nnetic phase boundaries. Furthermore, the majority phase\nMn2PtGa is one of the few compositions that thermo-\ndynamically favors the tetragonal inverse Heusler struc-\nture, which imparts chemical stability to this alloy. Thus,\nwhile Mn 2AuGa for example is metastable in the inverse\nHeusler structure, a modest amount of Au doping into\nMn2PtGa retains the desired structure.\nFigure 6 shows a quantitative evaluation of the mag-\nnetic and chemical behavior of the Mn 2Pt1\u0000zXzGa fam-\nily of alloys for X= Au, Ir, Ni. Assuming that a solid-\nsolution forms across these compositions, the micromag-\nnetic parameters K,DandAmust vary smoothly be-6\nFIG. 6. Evolution of magnetic interactions and phases in Mn 2Pt1\u0000zXzGa alloys for X= Au, Ir, Ni. a.Evolution of mesoscopic\nuniaxial anisotropy Kwith composition, evaluated by linear interpolation between the alloy endpoints, a parametrized cluster\nexpansion (DFT-CE), or explicit calculations of a SQS structure (DFT-SQS). b.Location of various alloy compositions in\nmagnetic phase space, based on the cluster expansion model of anisotropy K, and linear interpolation of spin sti\u000bness Aand\nthe Dzyaloshinskii-Moriya interaction D. Note that D=A = 2\u0019=\u0015 where\u0015is the equilibrium wavelength of the helical and\nantiskyrmion phases in units of the basal ( a) lattice parameter. c.Mixing energy of Mn 2PtGa-Mn 2XGa evaluated using a\nparametrized cluster expansion and the SQS methods. d.Binodal and spinodal regions of the miscibility gap in the X= Au,\nNi alloys based on the DFT-CE mixing enthalpy given in c., and an ideal entropy model for the Xsublattice.\ntween the alloy endpoints. For the magnetocrystalline\nanisotropy K, we compare three models for how this\nquantity varies with alloy composition z, as shown in\nFigure 6a. The simplest model is a linear interpola-\ntionK(z) =K(0) +z(K(1)\u0000K(0)) which neglects\nany new magnetochemical interactions that may appear\nat intermediate compositions of the alloy. A more re-\n\fned model is a cluster expansion (DFT-CE) model,\nK=P\n\u000bJ(K)\n\u000bQ\u001b\u000b, where\u000brepresent two-, three-, and\nfour-body clusters of sites with chemical occupancy de-\nnoted by\u001b, andJ(K)\n\u000bare interaction coe\u000ecients which\ncapture the contribution of each cluster to the total mag-\nnetocrystalline anisotropy. This model is analogous to a\nconventional cluster expansion of the total energy[44, 45]\nand, parametrized using DFT data, captures the impact\nof distinct chemical environments on the magnetocrys-\ntalline anisotropy. The \fnal model is an explicit DFT\ncalculation of the magnetic anisotropy at select composi-\ntions of the alloy using a special quasi-random structure\n(SQS)[46]. As can be seen in Figure 6a, while the DFT-\nCE and DFT-SQS models consistently indicate a degree\nof non-linearity in K(z), the deviation from the simple\nlinear interpolation is small. Thus, in the case of Dand\nA, we assume a simple linear interpolation with compo-\nsitionz,D(z) =D(0)+z(D(1)\u0000D(0)) as accounting for\nany non-linear contribution to these terms is very compu-\ntationally expensive and unlikely to substantially a\u000bect\nour conclusions.\nIn Figure 6b, we combine the DFT-CE model for K(z)\nand linear interpolation models of A(z) andD(z) to eval-\nuate the magnetic phase diagram of Mn 2Pt1\u0000zXzGa al-\nloys. Starting from the ASk region for z= 0, the wave-\nlength\u0015= 2\u0019A=D of the helimagnetic phases increases\nuntil the alloy crosses into new regions of magnetic phasespace atz\u00190:1\u00000:2, depending on the choice of Xel-\nement. In the case of X= Ir, we expect a transition\nto Hx-type behavior at z= 0:22 and easy-axis FiM at\nz= 0:28. ForX= Au, the ASk region instead tran-\nsitions to Cx-type behavior at z= 0:09 and easy-plane\nFiM atz= 0:18. TheX= Ni space contains 3 transi-\ntions, from ASk to Cx at z= 0:14, Cx to easy-plane FiM\natz= 0:22 and easy-plane to easy-axis FiM at z= 0:85.\nClose to the critical z-values for these phase transitions,\nthe magnetic behavior is likely to be highly susceptible\nto mechanical perturbations that would alter KandD,\nsuch as uniaxial strain along the crystallographic c-axis.\nSuch a strain could move the material to either side of the\nmagnetic phase boundary and thus actuate the magnetic\nphase transition.\nFinally, in Figure 6cd we estimate the synthetic acces-\nsibility of the Mn 2Pt1\u0000zXzGa alloys at the compositions\nof interest. We compute the pseudo-binary binodal and\nspinodal curves of these alloys to determine the regions\nof thermodynamic stability and metastability of the solid\nsolution. We obtain the mixing enthalpy of the X= Au,\nIr, Ni alloys from the DFT-CE and DFT-SQS models\nanalogously to the evaluation of the magnetocrystalline\nanisotropy K. As shown in Figure 6c, the X= Ir case\nshows a small negative mixing enthalpy, indicating that\nMn2PtGa and Mn 2IrGa are likely to be miscible in the\ntetragonal inverse Heusler structure at all temperatures\nand compositions. The X= Au, Ni alloys have a posi-\ntive mixing enthalpy indicating that these compositions\nform miscibility gaps. Figure 6d shows the binodal and\nspinodal regions of these miscibility gaps, assuming ideal\nsolution entropy for the Xsublattice. The compositions\nof interestz\u00190:1\u00000:2 are accessible in both X=Au,\nNi spaces but require a relatively high processing tem-\nperature of 800-900oC for initial mixing. These alloys7\ncan then be annealed to induce ordering at lower tem-\nperatures as they resist spinodal decomposition above \u0019\n600oC. Thus, the Mn 2(Pt,Ir)Ga system is readily misci-\nble and synthetically limited primarily by the large mis-\nmatch in the melting temperatures of Ga-rich and Ir-rich\nprecursors. In contrast, synthesizing the Mn 2(Pt,Au)Ga\nand Mn 2(Pt,Ni)Ga alloys is likely to require a careful\noptimization of the processing temperature to form the\nordered inverse Heusler structure while suppressing phase\nseparation into the ternary endpoints.\nDISCUSSION\nWe have surveyed the magnetic phase space of tetrago-\nnal inverse Heusler alloys, focusing on controlling the sta-\nbility of long-range spin textures such as antiskyrmions.\nBy constructing solid-solutions between endpoints with\nthe same short-range spin structure, we are able to tune\nthe e\u000bective Dzyaloshinskii-Moriya interaction and mag-\nnetocrystalline anisotropy in the alloy to vary the long-\nrange magnetic structure. Speci\fc compositions of this\nsolid solution which place the magnetic interactions near\na magnetic phase boundary maximize the magnetoelas-\ntic coupling of the material, as here mechanically-induced\nperturbations can actuate a magnetic phase transition.\nWe demonstrate the power of this design principle by\nidentifying the Mn 2Pt1\u0000zXzGa alloy with X= Au, Ir, Ni\nas a candidate for realizing chemically- and mechanically-\ntunable antiskyrmions. In this material, we predict that\nmoderate levels of doping ( z\u00190:1\u00000:2) can induce\nnumerous magnetic phase transitions, and couple an-\ntiskyrmion stability to small elastic strains at several\ncritical values of z. The speci\fc doping levels where\nthese phase transitions occur are sensitive to the pre-\ncise evolution of the coarse-grained interaction parame-\nters with composition and short-range order, which we es-\ntimate with several state-of-the-art computational meth-\nods. However, independent of these parametrizations, as\nlong as the alloy forms a true solid solution and connects\ncompounds lying on opposite sides of a magnetic phase\nboundary, a critical value of zis guaranteed to exist.\nThis fact suggests that tunable magnetic alloys can be\ndesigned even without detailed knowledge of their mag-\nnetic interaction parameters. As long as candidate alloy\nendpoints can be assigned to distinct regions of the mag-\nnetic phase diagram shown in Figure 4, a critical com-\nposition for realizing the magnetic phase transition and\nlarge magnetoelastic coupling is guaranteed to exist.\nThe primary di\u000eculty with implementing this design\nprinciple for tunable magnetism is ensuring that the al-\nloy maintains the desired crystal structure and chemical\norder at intermediate compositions. We have assumed\nthat the Mn 2XY tetragonal inverse Heuslers maintain\nthe structure shown in Figure 2a, with negligible mix-\ning between the four sublattices. While small amounts\nof intermixing between the sublattices will slightly al-\nter the e\u000bective magnetic interactions and would not af-fect our broad conclusions[47], many Mn 2XY compo-\nsitions are susceptible to substantial disorder and re-\nquire optimized processing to induce ordering. For ex-\nample, mixing between the Mn and Xsublattices cre-\nates an inversion center in the material and eliminates\nthe Dzyaloshinskii-Moriya interaction that drives the for-\nmation of antiskyrmions in this system. Chemical disor-\nder may also suppress the martensitic transition into the\ntetragonal phase that is necessary for DandKto be\nnon-zero. We have identi\fed Mn 2XGa forX=Ir, Pt,\nRh, Ru, Ni and Mn 2XSn forX=Ru, Rh as composi-\ntions that are most likely to form the correct structure\nand chemical order after annealing at moderate temper-\nature as they have a large driving force to order and\nminimal driving force to decompose. Conversely, we\nhave found that most Mn 2XSn and Mn 2XIn compounds\nhave a large driving force for decomposition and thus\nare more likely to phase separate if annealed. While\nthe experimental literature supports our analysis in the\ncase of Mn 2NiGa[48], Mn 2PtGa[49, 50], Mn 2RhSn[37],\nMn2PdSn[51] and Mn 2PtSn[27, 35], the apparent order\nobserved in Mn 2PtIn and Mn 2IrSn[37], and disorder re-\nported in Mn 2RhGa, Mn 2RuGa and Mn 2RuSn[30] sug-\ngest that other processes may need to be considered. Ul-\ntimately, a substantially more detailed understanding of\nthe synthesis process is necessary to quantitatively eval-\nuate the synthesizability of these structures and the fea-\nsibility of controlling their chemical order[52, 53].\nCONCLUSION\nWe have reported a systematic \frst-principles deriva-\ntion of tunable magnetic order in the family of Mn 2XY\ntetragonal inverse Heusler alloys, focusing on design-\ning a robust coupling between room-temperature anti-\nskyrmion stability and elastic deformation. To do so, we\n\frst constructed a universal phase diagram for the lat-\ntice shared by all tetragonal inverse Heuslers, focusing\non the long-range modulation of the common ferrimag-\nnetic spin structure. We characterized the magnetic be-\nhavior of all known stable compounds in this space and\nidenti\fed combinations which, when alloyed, may pro-\nduce magnetic phase transitions as a function of chemi-\ncal composition and mechanical deformation. Finally, we\nperformed an in-depth characterization of the magnetic\nand chemical behavior of Mn 2Pt1\u0000zXzGa withX= Au,\nIr, Ni to demonstrate that for z\u00190:1\u00000:2, this family\nof alloys can transition between all possible long-range\nequilibrium spin textures, including antiskyrmions, he-\nlices and conical phases. At several critical compositions,\nthese magnetic phase transitions may be driven by elastic\nstrain, suggesting that this alloy may exhibit giant mag-\nnetoelastic coupling and serve as a mechanical actuator\nfor the formation of complex magnetic order.8\nMETHODS\nElectronic structure calculations were performed with\nthe Vienna Ab-Initio Simulation Package (VASP) [54]\nusing the Projector-Augmented Wave method[55]. All\nmagnetic interactions (Figure 4, Figure 6) were de-\ntermined using the Perdew-Burke-Ernzerhof (PBE)\nexchange{correlation functional[56], accounting for spin-\norbit coupling. Following previously reported bench-\nmarks, a dense reciprocal-space mesh of 400 k-points per\n\u0017A\u00003was used[21, 31, 43], making sure that all magnetic\ncalculations of the same chemistry and supercell used ex-\nactly the same k-point mesh[14, 24] and converging the\ntotal energy to 10\u00007eV.\nThe relative stabilities of the ordered and disordered\nphases (Figure 5) were determined using the same com-\nputational parameters, but neglecting spin-orbit cou-\npling. Disordered phases were modeled using special\nquasi-random structure (SQS)[46] representations of the\ncommon L2 1b(Mn(1)/X) and BiF 3(Mn(1)/Mn(2)/ X)\ndisorder types in these systems[30], where each SQS rep-\nresentation is relaxed assuming a ferrimagnetic spin con-\n\fguration. To compute global chemical stability within\nthe Mn-X-Ychemical spaces (Figure 5), we rely on struc-\ntures reported in the ICSD[57], Materials Project[58],\nand OQMD[59] databases, with energies computed us-\ning the SCAN exchange-correlation functional[60] as we\nfound that this functional uniquely reproduces the low- T\nbehavior of the known binary phase diagrams and avoids\npreviously reported pathological behavior in e.g. the\nPt-based binaries[61]. These chemical stability calcula-\ntions are converged to 10\u00005eV in total energy and 0.02\neV/\u0017A in forces, and are optimized over likely collinear\nferromagnetic and antiferromagnetic con\fgurations for\nall phases. The equilibrium phases used to determine\nformation energies are given Supplementery Data 5.\nMagnetic Hamiltonians were obtained following\npreviously described methods for generating a com-\nplete basis for quasi-classical spin interactions within\na cluster expansion formalism[24, 33, 45, 62]. Brie\ry,\nfor each symmetrically-distinct group of magnetic\nsites, we construct interaction basis functions con-\nsisting of symmetrized products of spherical Har-\nmonics, e.g. for a pair of spins jl1;l2;L;Mi=\n4\u0019P\nm1;m2cl1;l2;L\nm1;m2;MYl1m1(\u001e1;\u00121)Yl2m2(\u001e2;\u00122) where\ncl1;l2;L\nm1;m2;M are Clebsch-Gordan coe\u000ecients. The\nL= 0 terms correspond to exchange-interactions,\nL= 1 correspond to Dzyaloshinskii-Moriya couplings,\nandL= 2;4;:::correspond to magnetocrystalline\nanisotropies[24]. Here, we consider L= 0 (exchange)\ntwo-spin interactions for \frst, second and third nearest-\nneighbor interactions shown in Figure 2b. L= 1\n(Dzyaloshinskii-Moriya) terms are included for the\nnearest-neighbor interaction ( J1pair in Figure 2a).\nL= 2;4;:::terms are included as an average single-site\nanisotropy summed over the Mn(1) and Mn(2) sublat-\ntices. A full listing of these basis functions is given in\nSupplementary Data 1.We parametrize this Hamiltonian to reproduce the en-\nergies obtained from DFT. We group the basis functions\nby theirL-value and \ft these groups independently to\nmaximally cancel out numerical noise in the DFT calcu-\nlations: (1) we \ft the L= 0 interactions to symmetrically\ndistinct collinear spin con\fgurations, (2) the L= 1 inter-\nactions to di\u000berences in energy between right- and left-\nhanded helical superstructures of the local ferrimagnetic\nspin structure, and (3) the L= 2;4;:::interactions to the\nenergy associated with rotating the ground-state ferri-\nmagnetic spin structure with respect to the crystal axes.\nFinally, we \ft the coarse-grained magnetic parameters\nAandDin the low-Tlimit to the energy of spin helix\ncon\fgurations near the equilibrium wavelength implied\nby the balance of Dzyaloshinskii-Moriya and exchange\nforces, where the spin helix energies are evaluated using\nthe parametrized atomistic cluster expansion.\nCon\fgurational cluster expansions for the total en-\nergy and magnetocrystalline anisotropy (Figure 6ac)\nwere constructed and parametrized following standard\ntechniques[45], including 2-, 3-, and 4- body interac-\ntions. Special quasi-random structures (SQS)[46] based\non these cluster expansions were obtained by Monte\nCarlo optimization targeting the correlations observed in\na random alloy at the desired composition within a 3x3x2\nsupercell of the conventional cell shown in Figure 2a.\nTo determine the \fnite- Tphase diagram (Figure 3c) as\nwell as identify the ground states of the magnetic Hamil-\ntonian (Figure 3ab), we rely on auxiliary-spin dynamics\nHamiltonian Monte Carlo[24, 63] with the No U-Turn\nSampling technique[64], as well as simulated annealing\nand conjugate-gradient optimization. The Monte Carlo\nruns sample 1,000 and 10,000 independent con\fgurations\nfor equilibration and production respectively, where the\ntime between independent samples is estimated from the\ndecay rate of the energy autocorrelation function. Finite-\nTruns are performed for an equilibrium helical wave-\nlength equal to 24 unit cells, and using a 24x42x3 super-\ncell of the conventional structure, approximately com-\nmensurate with a hexagonal antiskyrmion lattice.\nACKNOWLEDGMENTS\nWe are grateful to Justin Mayer and Eve Mozur for\nfruitful discussions. This research was supported by the\nMaterials Research Science and Engineering Center at\nUCSB (MRSEC NSF DMR 1720256) through IRG-1.\nComputational resources were provided by the National\nEnergy Research Scienti\fc Computing Center, a DOE\nO\u000ece of Science User Facility supported by the O\u000ece\nof Science of the U.S. Department of Energy under Con-\ntract No. DE-AC02-05CH11231, as well as the Center for\nScienti\fc Computing at UC Santa Barbara, which is sup-\nported by the National Science Foundation (NSF) Mate-\nrials Research Science and Engineering Centers program\nthrough NSF DMR 1720256 and NSF CNS 1725797.9\n[1] S. Bader and S. 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Here a m odel based on a stretched exponential\ndecay of its magnetization is proposed, which can describe t he main magnetic features of spin glasses\nobserved in experiments as the time-decay of thermoremamen t magnetization, the relaxation of zero\nfield cooled magnetization, the ac and dc magnetization as a f unction of temperature and others.\nIn principle, the here proposed model could be adapted to des cribe other glassy systems.\nINTRODUCTION\nThespin glass(SG) isanothercaseinphysicsforwhich\nthe effect of time ( t) may bring puzzling consequences.\nWhat in the early 1960 decade seemed to be just a dif-\nferent class of dilute magnetic alloys exhibiting unusual\nmagnetic susceptibility and specific heat curves, was a\nfew latter recognized as a complex system, with some of\nits intriguingbehaviorbeing analogousto the mechanical\nproperties of real glasses, showing for instance aging, re-\njuvenationandmemoryeffects[1,2]. Thisdisorderedand\nfrustratedsystemwassoonstablishedasaplaygroundfor\nbothexperimentalistsandtheorists,andthedevelopment\nof models and mathematical tools attempting to explain\nit has found application not only for SG but also in other\ncomplex systems as neural networks, protein folding and\ncomputer science [3].\nThe two mainstream theoretical pictures used to ex-\nplain the SG are the droplet-scaling model [4, 5] and\nthe extensively investigated mean field Sherrington-\nKirkpatrickmodel [1, 6] with its replica symmetry break-\ning derived from the Parisi’s solution [7, 8]. While an-\nalytical investigations suggest a single pair of spin-flip\nrelated states at low temperatures ( T) as described by\nthe first model [9], many computational simulations give\nevidence in favor of the latter with its multitude of pure\nstates [10]. Regarding these and the several other pro-\nposed models, and in spite of the great progress observed\nalong these nearly five decades of investigation, as one\ngoes deeper in these theories it feels that many of the\nresults are poorly (if at all) connected to those obtained\nin laboratory. More importantly, each theory is better\nsuited to describe a sort of SG properties as it contra-\ndicts other features. Consequently, some of the intrigu-\ning properties of SG materials are not well understood,\nin special those related to its dynamics.\nHere an alternative approach is used to describe\nthe magnetic properties of SG. Motivated by experi-\nmental results, a function is proposed to directly de-\nscribe the systems’ magnetization ( M) after the appli-\ncation/removal of an external field ( H). It is the first\nmodel that can, alone, fairly reproduce the main strik-\ning magnetic features of SG, i.e.the thermoremanent M(TRM), the zero field cooled (ZFC) M(MZFC) and the\nac and dc Mas a function of Tcurves [M(T)][2], as well\nas other important experiments.\nRESULTS AND DISCUSSION\nThe model considers that if a SG system was subject\ntoHduring a finite time interval δt=t2−t1, itsMat\na posterior instant twill be given by\nM(t) =/integraldisplayt2\nt1M0e−b(t−t′)ndt′, (1)\nwhere 0 ≤n≤1 and both M0andbdepend on Tand\nHatt′:\nM0=/bracketleftbiggT(t′)\nTg/bracketrightbiggnAH(t′)\nt′−tg;\nb=c·T(t′)\nTg(t′−tg)n,(2)\nwhereAis a constant dependent of the material’s prop-\nerties, as the constituent elements, the density of un-\npaired moments etc. Although a more profound under-\nstanding of the implications of this proposed model is\ndesiredbeforeanyassumptionconcerningitsphysicalori-\ngin, one may speculate, roughly speaking, that the decay\nexpressed in Eq. 1 could be related to the search for\nlower-energy states through the systems’ rugged energy\nlandscape, where the t′−tgterm plays the role of aging,\ni.e.the system is continuously evolving after the tran-\nsition temperature Tgwas achieved at instant tg. Thec\nparameter is expected to depend on H, since changing\nit leads the system to a different position in the energy\nlandscape, thus affecting its relaxation. But as the main\npart of this study is dedicated to situations in which H\nis constant, the discussion of such variable will be post-\nponed to section 2.3. The nparameter, together with\ncandT/Tg, determine the systems’ glassiness, i.e.how\nslowMwill decay.\nAt a first glance, it may look that this model keeps\nclose resemblance with the stretched exponential decay2\nmultiplied by a power law of t[11]\nM(t) =C/parenleftbiggt\ntp/parenrightbigg−α\n·e−(t/tp)n′\n, (3)\nand to its variants that are usually adopted to fit TRM\nand M ZFCcurves [12, 13]. However, there are some re-\nmarkable differences between the here proposed model\nand previous ones, the most significant one being the\nfact that here the magnetization is the outcome of an in-\ntegration along the interval during which Hwas applied.\nMoreover, those previous models are only suitable to fit\nthe TRM and M ZFCcurves whereas the here described\none is proposed to be more general, enabling the descrip-\ntion of other experimental results, as will be discussed.\nThermoremanent Magnetization\nBeginning with the TRM experiment, a typical TRM\ncurveis carriedaftercool the system from above Tgdown\nto a measuring T(Tm) in the presence of H. After keep-\ning the system at this condition for a waiting time tw,H\nis removed (at t= 0) and the remanent Mis recorded as\nafunctionof t(foravisualdescriptionofthisprotocolsee\nthe Supplementary Material - SM [14]). Fig. 1(a) shows\nthe curve calculated at Tm= 0.8Tgwithn= 0.5 (a value\nwithin the range typically found in the fittings of TRM\nwith the stretched exponential Eq. 3), H=A=c= 1\n(arb. units) and tw= 100 s, obtained after cool the sys-\ntem in a constant Tsweep rate |dT/dt|= 0.002 Tg/s.\nIt may be noticed that all parameters are given in arbi-\ntrary units with the exception of t, expressed in seconds\n(s). This is because tis particularly important here in\nthe study of the dynamics of SG, and its description in\ns unit facilitates the comparison of the results obtained\nfrom the model with those referenced from the experi-\nments. The resulting curve shown in Fig. 1(a) is very\nsimilar to those observed experimentally [12, 13].\nFor a quantitative comparison between the here pro-\nposed model and the one largely used to fit experimental\nTRM curves, the solid line in Fig. 1(a) shows a reason-\nably fit of Eq. 3 with the theoretical curve obtained from\nEqs. 1 and 2, yielding tp≃260 s,n′≃0.6, these val-\nues being within the range usually found for canonical\nSG [11]. This clearly demonstrates that the proposed\nmodel is suitable to describe typical experimental TRM\ncurves of SG materials. The fitting is not so good for\nsmallt, as was already observed experimentally at the\nearly stages of investigation of SG systems, which moti-\nvated the search for alternative equations [11–13]. It is\nimportant to note the tendency toward zero in M, con-\ntrasting to the experimental results showing that usually\nthe system reach a finite magnetization value at large t\n[12, 13]. It is thus possible that, in practice, for real SG\nmaterials a fraction of the spins gets pinned toward theHdirection after its removal, while the other part relax.\nThis could be easily adjusted here with the addition of a\nconstant term.\nFig. 1(b) compares TRM curves calculated for differ-\nenttw, where a clear tw-dependence is observed. This\nis better visualized in Fig.1(c) where the modulus of the\nrelaxation rate, S= (1/H)(dM/dlnt ), is computed. As\ncan be seen, a maxima in |S|occurs at tclose to tw,\nagain reproducing the experiments [13]. Such maxima\nis present even for tw= 0, which is due to the finite t\ninterval taken to cool the system from TgtoTm(tcool)\n[15–17]. As twincreases, the relative influence of tcool\ndiminishes and the maxima in Sgets closer to t=tw.\nIf one considers the situation in which the system is im-\nmediately cooled from above TgtoTm(i.e.assuming\nan unrealistic |dT/dt|=∞) thentcool= 0 and the peak\ninSwill shift to the left as shown in the inset of Fig.\n1(d). Interestingly, all TRM curves calculated for tcool=\n0 with different tw, plotted as a function of t/tw, coincide\n[Fig. 1(d)], in agreement with the tendency toward full\naging experimentally found [15].\nThe model can faithfully predict the effect of thermal\nenergy on the TRM curves. Fig. 2(a) compares the tw=\n103s TRM curves obtained with T= 0.8Tgand 0.6Tg,\nwhere it is observed the increase in Mfor the later, while\nFig. 2(b) shows its expected |S|shift to larger tresult-\ning from the fact that the spins get more freezed with\ndecreasing T, turning the decayslower. In spite ofthe re-\nsemblance of Fig. 2(a) with that of the great majority of\nSG materials [12, 13], the T-dependence of M0expressed\nin Eq. 2 is not expected to be universal, in the sense that\nthere were also found materials for which the magnitude\nofMdecreases with T[11]. One can choose other M0(T)\nfunctions leading to different trends for the magnitude of\nMasTchanges without greatly affecting the main SG\nfeatures (see SM [14]).\nZero Field Cooled Magnetization\nBesides the TRM experiments, the here proposed\nmodel can also reproduce the M ZFCcurves, which are\nobtained after ZFC the system down to Tm< Tg, keep it\non this condition for tw, then apply a small H(att= 0)\nand start to capture Mas a function of t(see SM [14]).\nFig. 3(a) shows the curve calculated for tw= 103s atTm\n= 0.8Tgand using the same parameters chosed to pro-\nduce the TRM curves, i.e.n= 0.5,H=A=c= 1 (arb.\nunits), resulting in a fair agreement with the typical ex-\nperimental curves reported for SG system [18]. From a\nlog-linear plot of the curves obtained with different tw,\nFig. 3(b), one can see the expected tw-dependency ob-\nserved experimentally [19]. Fig. 3(c) displays the Sre-\nsulting from these M ZFCcurves. As for TRM, the max-\nima inSfor MZFCoccurs at tlarger than (but close\nto)tw, precisely the same behavior as that of experimen-3\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48\n/s49/s48/s45/s52\n/s49/s48/s45/s51\n/s49/s48/s45/s50\n/s49/s48/s45/s49\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51/s32/s32/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s116/s32/s40/s115/s41/s32/s83/s105/s109/s117/s108/s97/s116/s105/s111/s110/s32/s102/s114/s111/s109/s32/s69/s113/s115/s32/s49/s32/s97/s110/s100/s32/s50\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s70/s105/s116/s32/s119/s105/s116/s104/s32/s69/s113/s46/s32/s51\n/s40/s97/s41/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103\n/s116\n/s119/s32/s61/s32/s49/s48/s50\n/s32/s115/s40/s98/s41\n/s116/s32/s40/s115/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s32\n/s32/s32/s116\n/s119/s32/s61/s32/s48/s32\n/s32/s32/s116\n/s119/s32/s61/s32/s49/s48/s50\n/s32/s115\n/s32/s116\n/s119/s32/s61/s32/s49/s48/s51\n/s32/s115\n/s32/s116\n/s119/s32/s61/s32/s49/s48/s52\n/s32/s115/s40/s99/s41\n/s116/s32/s40/s115/s41\n/s32/s32/s124/s83/s124/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s32/s116\n/s119/s61/s48\n/s32/s116\n/s119/s61/s49/s48/s50\n/s115\n/s32/s116\n/s119/s61/s49/s48/s51\n/s115\n/s32/s116\n/s119/s61/s49/s48/s52\n/s115\n/s40/s100/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s32\n/s116\n/s99/s111/s111/s108/s32/s61/s32/s48/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103\n/s116\n/s119/s61/s49/s48/s50\n/s115\n/s116\n/s119/s61/s49/s48/s51\n/s115\n/s116\n/s119/s61/s49/s48/s52\n/s115\n/s116/s47/s116\n/s119\n/s124/s83/s124/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s116/s32/s40/s115/s41\n/s116\n/s119/s61/s49/s48/s50\n/s115\n/s32/s32\n/s116\n/s99/s111/s111/s108/s61/s32/s49/s48/s48/s32/s115\n/s116\n/s99/s111/s111/s108/s61/s32/s48\nFIG. 1. (a) TRM curve calculated at Tm= 0.8TgwithH=A=c= 1 (arb. units) and tw= 100 s. The red solid line represents\nthe best fit with Eq. 3. (b) Comparison of TRM curves calculate d for different tw. (c) The modulus of the relaxation rate |S|\nfor the TRM curves with different tw. (d)tcool= 0TRMcurves calculated with different tw, plotted as a function of t/tw.\nThe inset compares the |S|fortw= 100 s TRM curves calculated with tcool= 100 s and with tcool= 0.\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s49 /s49/s48 /s49/s48/s48 /s49/s48/s48/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s32/s32/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s116/s32/s40/s115/s41/s32/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103\n/s32/s84\n/s109/s32/s61/s32/s48/s46/s54/s84\n/s103/s32/s32/s40/s97/s41\n/s116\n/s119/s32/s61/s32/s49/s48/s51\n/s32/s115/s40/s98/s41\n/s116/s32/s40/s115/s41/s124/s83/s124/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s116\n/s119/s32/s61/s32/s49/s48/s51\n/s32/s115\n/s32/s32\n/s32/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103\n/s32/s84\n/s109/s32/s61/s32/s48/s46/s54/s84\n/s103\nFIG. 2. (a) Comparison between TRM curves obtained with\ntw= 103s and the same H,a,c, but distinct Tm. (b) The |S|\nfor these curves.\ntal curves [19]. Here, although there is no magnetization\nduring cooling since it occurs at zero H,tcoolstill plays\nits part because according to Eq. 2 the system starts to\nage already after the system passes through Tg(attg).\nAstwincreases, the relative effect of tcooldecreases in\ncomparison to tw, and the maximum in Sgets closer to\nt=tw. As in the case of TRM curves, if we assume tcool\n= 0 in the M ZFCprotocol the plot of Mas a function of\nt/twwill indicate atendency towardfull aging, Fig. 3(d).\nBycomparingFigs. 1(c) and3(c) quantitativelyitcanbe\nnoticed that, as observed experimentally, the relaxation\nrates of TRM and M ZFChave nearly the same absolute\nvalues, indicating a similar aging process for both [20].\nAnother strategy developed to investigate the low T\ndynamics of SG systems is the Tcycling below Tg. Fig.\n4(a) shows the curve resulting from a protocolfirstly pro-\nposed to investigate memory effects in assembly of mag-\nnetic nanoparticles [21, 22], in which the system is ZFC\ndown to Tm< Tg, then a small His applied (at t= 0)\nand the magnetic relaxation starts to be captured. After\nthe lapse of a period t1, however, the system is further\ncooled to a lower T=Tm- ∆T, and kept at this condi-\ntion for a period t2. After the lapse of t2the system is\nheated back to Tmand the magnetization is recorded for\na period t3[14]./s48/s46/s48\n/s52/s46/s48/s120/s49/s48/s51\n/s56/s46/s48/s120/s49/s48/s51\n/s49/s46/s50/s120/s49/s48/s52/s48/s46/s49/s48/s46/s51/s48/s46/s53/s48/s46/s55\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s49/s48/s45/s52\n/s49/s48/s45/s51\n/s49/s48/s45/s50\n/s49/s48/s45/s49\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51/s116\n/s119/s32/s61/s32/s49/s48/s51\n/s32/s115/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103/s40/s97/s41\n/s116/s32/s40/s115/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32\n/s40/s98/s41\n/s116/s32/s40/s115/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s32/s116\n/s119/s32/s61/s32/s48\n/s32/s116\n/s119/s32/s61/s32/s49/s48/s50\n/s32/s115\n/s32/s116\n/s119/s32/s61/s32/s49/s48/s51\n/s32/s115\n/s32/s116\n/s119/s32/s61/s32/s49/s48/s52\n/s32/s115\n/s40/s99/s41\n/s116/s32/s40/s115/s41/s83/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s32\n/s116\n/s119/s32/s61/s32/s48\n/s116\n/s119/s32/s61/s32/s49/s48/s50\n/s115\n/s116\n/s119/s32/s61/s32/s49/s48/s51\n/s115\n/s116\n/s119/s32/s61/s32/s49/s48/s52\n/s115\n/s116\n/s99/s111/s111/s108/s32/s61/s32/s48/s40/s100/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s116/s47/s116\n/s119\n/s32/s32\n/s116\n/s119/s61/s49/s48/s50\n/s32/s115\n/s116\n/s119/s61/s49/s48/s51\n/s32/s115\n/s116\n/s119/s61/s49/s48/s52\n/s32/s115/s83/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s116/s32/s40/s115/s41/s116\n/s119/s61/s49/s48/s50\n/s115\n/s32/s116\n/s99/s111/s111/s108/s61/s32/s49/s48/s48/s32/s115\n/s116\n/s99/s111/s111/s108/s61/s32/s48\nFIG. 3. (a)M ZFCcurve calculated for Tm= 0.8Tg,n= 0.5,\nH=A=c= 1 (arb.units) and tw= 103s. (b) Log-linear\nplots of the curves obtained with different tw. (c) The re-\nlaxation rates, S, of the curves with different tw. (d) M ZFC\ncurves calculated for tcool= 0 and different tw, plotted as a\nfunction of t/tw. The inset compares the Softw= 100 s\nMZFCcurves calculated with tcool= 100 s and tcool= 0.\nThe curve in Fig. 4(a) was produced with Tm= 0.5Tg,\n∆T= 0.2Tg,t1=t2=t3= 4000 s and the same param-\neter values as those used to calculate the conventional\nTRM and M ZFCcurves described above. At t1the curve\nis similar to those of Fig. 3, with an initial jump in the\nmagnetization when His turned on, followed by a slow\nrelaxation. During the temporary cooling at t2, the re-\nlaxation becomes very weak, which can be inferred from\ntheT-dependencies of Eqs. 1 and 2. When the system\nreturns to Tmint3the magnetization comes back to the\nlevel it reached before the Tcycling. The inset shows\nthe curve resulting when the t2interval is removed. It\nmakes clear the fact that during the temporary cooling\nthe relaxation is almost halted, and the memory effect is\nmanifested in t3when the system returns to Tmand the4\n/s48 /s52/s48/s48/s48 /s56/s48/s48/s48 /s49/s50/s48/s48/s48/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57\n/s48 /s52/s48/s48/s48 /s56/s48/s48/s48 /s49/s50/s48/s48/s48/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s48 /s52/s48/s48/s48 /s56/s48/s48/s48/s84/s32/s61/s32/s48/s46/s53/s84\n/s103/s84/s32/s61/s32/s48/s46/s51/s84\n/s103/s116\n/s51/s116\n/s50/s40/s97/s41\n/s116/s32/s40/s115/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s116\n/s49\n/s84/s32/s61/s32/s48/s46/s53/s84\n/s103\n/s84/s32/s61/s32/s48/s46/s53/s84\n/s103/s84/s32/s61/s32/s48/s46/s55/s84\n/s103/s116\n/s51/s116\n/s50/s40/s98/s41\n/s116/s32/s40/s115/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32\n/s116\n/s49\n/s84/s32/s61/s32/s48/s46/s53/s84\n/s103\n/s32/s67/s111/s110/s118/s101/s110/s116/s105/s111/s110/s97/s108/s32/s77\n/s90/s70/s67/s32/s99/s117/s114/s118/s101/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s109/s101/s97/s115/s117/s114/s101/s100/s32/s97/s116/s32/s84\n/s109/s32/s61/s32/s48/s46/s53/s84\n/s103/s84/s32/s61/s32/s48/s46/s53/s84\n/s103/s116\n/s51/s116\n/s49/s32\nFIG. 4. (a) M ZFCcurve calculated at Tm= 0.5Tgwith a\ntemporary cooling of ∆ T= 0.2Tg. The inset shows the curve\nresulting when the data at Tm-∆Tis removed, evidencing the\nmemory effect. (b) M ZFCcurve calculated at Tm= 0.5Tg\nwith a temporary heating of ∆ T= 0.2Tg, where no memory\neffects appear.\nrelaxation is resumed, thus mimicking the experimental\ncurves with precision [21, 22]. Conversely, for a posi-\ntiveTcycling [Fig. 4(b)] the relaxation is hasted in t2,\nand when the system is cooled back to Tmthe magne-\ntization does not restore to the level reached before the\ntemporary heating, also in agreement with experimen-\ntal observations [21, 22]. These results indicate that the\nhere proposed model may be also suitable for magnetic\nnanoparticles.\nThe model has failed, however, to reproduce the mem-\nory and rejuvenation effects for the case of M ZFCexper-\niments in which Tis cycled before the application of H\n[23], as well as the chaotic effect observed in the memory\ndip experiments where the ZFC process is halted prior to\nthe measurement of M(T) [24, 25]. It could not predict\neither the memory and rejuvenation effects in TRM ex-\nperiments where Tis cycled duringthe measurement, be-\ncause in this case Tis changed after the Hcutoff [21, 22].\nFor this last case, such contrast to the experiments sug-\ngests that the internal field may play an important role\non the relaxation, and the here proposed model should\nbe adjusted in order to take this into account. For in-\nstance, a natural attempt could be the replacement of\nT(t′) byT(t) in Eqs. 1 and 2 since one may expect that,\neven in the absence of H, whenTis changed the energy\nlandscape is altered and the decay will be affected (see\nSM [14]). This would lead to Tcycled TRM curvescloser\nto the experimental ones, but would not reproduce the\nmemory dip experiments.\nMagnetization as a function of temperature\nFinally, Eqs. 1 and 2 can also predict the behavior\nof SG systems in ac and dc M(T) experiments. Fig.\n5(a) shows the dc ZFC and FC curves calculated for n\n= 0.5,H=A=c= 1 (arb. units) and |dT/dt|= 0.001\nTg/s. Despite the well known deviation from the Curie-\nWeiss (CW) behavior for the paramagnetic (PM) region\nof SG systems [26], for simplicity it was chosen here a/s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s48/s46/s57/s48 /s48/s46/s57/s53 /s49/s46/s48/s48 /s49/s46/s48/s53/s51/s46/s48/s51/s46/s51/s51/s46/s54/s51/s46/s57\n/s70/s67\n/s72\n/s100/s99/s32/s61/s32/s49\n/s90/s70/s67\n/s84/s47/s84\n/s103/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s40/s97/s41\n/s80/s77\n/s84\n/s103\n/s84/s47/s84\n/s103/s32/s120/s32/s49/s48/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32\n/s40/s98/s41/s80/s77/s72/s40/s116/s41/s32/s61/s49/s43/s99/s111/s115/s40/s50 /s102/s116/s41\n/s32/s48/s46/s49/s32/s72/s122\n/s32/s48/s46/s50/s32/s72/s122\n/s32/s32/s49/s32/s72/s122\n/s32/s80/s77/s32/s40/s67/s87/s41\nFIG. 5. (a) ZFC and FC dcM(T) curves calculated with H\n= 1. (b) ac χ(T) curves calculated for f= 0.1, 0.2 and 1 Hz.\nCW curve for the T > T gregion, which was adjusted to\ncoincide with the T < T gZFC and FC curves at Tg. The\nZFC curve shows a sharp cusp while the FC one shows\na plateau-like behavior, being these striking features of\nSG systems [27]. It is important to notice that the here\nproposed model does not predict the PM-SG transition,\nsince it is only concerned with the SG state, T < T g. The\ncusp-likebehaviorobservedin Fig. 5resultsfromthefact\nthe SG curves were calculated up to Tgand joined to the\nPM ones that were calculated only down to this criti-\ncalT. Concerning the fact that the experimental ZFC\npeaks are usually sharper than that of Fig. 5(a) while\nthe FC ones usually show a small bump close to Tg, it\nmust be stressed that the physics for Tvery close to Tg,\nwhere a divergent behavior is expected, is neither under\nconsideration here.\nAccording to the here proposed model, the ZFC curve\ndepends on the cooling/heating Trates (see SM [14]), as\nexpected for an off-equilibrium condition [2]. Contrast-\ningly, the FC is nearly invariant under changes in |dT/dt|\nand this may be the reason why it is widely believed that\nthe FC is roughly an equilibrium situation [28–30]. How-\never, it is in fact a metastable configuration [31], which\ncan be fairly captured by the here proposed model. Ac-\ncording to the model, if the cooling is halted for a finite t\ninterval below Tgfor instance, the FC magnetization will\nchange [14], as already observed experimentally [32].\nFig. 5(b) shows ac susceptibility curves for some se-\nlectedfrequencies( f), obtainedconsideringanoscillating\nfield of the form H(t) =Hdc+hcos(2πft), where his the\nacfieldamplitude. All curveswerecalculatedintheheat-\ning mode with n= 0.5,h=Hdc=A= 1, and each point\nwasrecordedafteronefieldcycle. The fwerechosenslow\nenoughsothatonecan assumeanearlylinearresponseof\nMinrelationto Handusetheapproximation χ=M/H.\nThe stretched exponential term in Eq. 1 is expected to\ndepend on H, thus for Fig. 5(b) it was used c=|H(t)|,\nbut very similar curves are observed for a constant c(see\nSM [14]). The PM curve was calculated with the same\nslope of that used for the dc field shown in Fig. 5(a),\nand adjusted to coincide with the f= 0.01 Hz curve at\nTg, assumed here as a nearly static situation. The result-5\ning curves are clearly f-dependent, showing a tendency\nof decrease in magnitude with increasing f. Defining the\nfreezingT(Tf) as the point where each curve intercepts\nthe PM curve, one can observe the expected shift of Tf\ntoward higher Tasfincreases. The relative shift δTf\n= ∆Tf/Tf(∆logf) [33] can be computed, yielding in this\ncaseaδTf≃0.003withintherangeexperimentallyfound\nfor canonical SG [2]. Though, care must be taken with\nthis resultsinceit depends onthe choiceofthe PMcurve,\nwhich is known to deviate from CW behavior for SG sys-\ntems [26]. Moreover, it may be also related to the un-\nderline physics around Tg(not considered here), so that\ntheTfvalues may be related to the systems’ behavior at\nboth above and below Tg.\nCONCLUSIONS\nIn summary, the model here proposed, based on a\nstretched exponential decay of the magnetization after\nthe application of Hfor an infinitesimal t, can describe\nthe striking features of TRM, M ZFC, ac and dc ZFC-FC\nM(T) curves and some of the memory experiments. It\ndoes not answer all the questions, thus it must be re-\ngarded as an approximate model. Nevertheless, the fact\nthat it can reproduce several of the main SG features\nis remarkable, and its thorough investigation may give\nimportant insights into its physical origin, resulting in a\nbetter understanding of the microscopic mechanism be-\nhind the glassy behavior. In principle it could be also\napplied to other complex systems after a suitable adjust-\nment of the parameters.\nACKNOWLEDGMENTS\nThis work was supported by Conselho Nacional\nde Desenvolvimento Cient´ ıfico e Tecnol´ ogico (CNPq),\nFunda¸ c˜ ao de Amparo ` a Pesquisa do Estado de Goi´ as\n(FAPEG) and Coordena¸ c˜ ao de Aperfei¸ coamento de Pes-\nsoaldeN´ ıvelSuperior(CAPES).TheauthorthanksWes-\nley B. Cardoso for the computational help.\n[1] M. M´ ezard, G. Parisi, and M. A. Virasoro, Spin Glass\nTheory and Beyond (World Scientific, Singapore, 1987).\n[2] J. A. Mydosh, Rep. Prog. Phys. 78 (2015) 052501.\n[3] D. L. Stein and C. M. Newman, Spin Glasses and Com-\nplexity(Princeton University Press, Princeton, 2013).\n[4] W. L. McMillan, J. Phys. C 17 (1984) 3179.\n[5] D. S. Fisher and D. A. Huse, Phys. Rev. B 38 (1988) 386.[6] D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35\n(1975) 1792.\n[7] G. Parisi, Phys. Rev. Lett. 43 (1979) 1754.\n[8] G. Parisi, J. Phys. A 13 (2008) 324002.\n[9] C. M. Newman and D. L. Stein, Phys. Rev. E 57 (1998)\n1356.\n[10] B. A. Berg and W. Janke, Phys. Rev. Lett. 80 (1998)\n4771.\n[11] M. Ocio, M. Alba, and J. Hammann, J. Physique Lett.\n46 (1985) 23.\n[12] R. V. Chamberlin, G. Mozurkewich, and R. Orbach,\nPhys. Rev. Lett. 52 (1984) 867.\n[13] P. Nordblad, P. Svedlindh, L. Lundgren, and L. Sand-\nlund, Phys. Rev. B 33 (1986) 645.\n[14] See Supplementary Material at ??? for details of the\ncomputational calculation, the protocol and the equa-\ntions used to produce each curve, as well as some results\nnot displayed in the main text.\n[15] G. F. Rodriguez, G.G. Kenning, and R. Orbach, Phys.\nRev. Lett. 91 (2003) 037203.\n[16] V. S. Zotev, G. F. Rodriguez, G. G. Kenning, R. Orbach,\nE. Vincent, and J. Hammann, Phys. Rev. B 67 (2003)\n184422.\n[17] G. F. Rodriguez, G. G. Kenning, and R. Orbach, Phys.\nRev. B 88 (2013) 054302.\n[18] L. Lundgren, P. Svedlindh, P. Nordblad, and O. Beck-\nman, Phys. Rev. Lett. 51 (1983) 911.\n[19] P. Granberg, L. Sandlund, P. Nordblad, P. Svedlindh,\nand L. Lundgren, Phys. Rev. B 38 (1988) 7097.\n[20] P. Nordblad, L. Lundgren and L. Sandlund, J. Magn.\nMagn. Mater. 54-57 (1) (1986) 185-186.\n[21] Y. Sun, M. B. Salamon, K. Garnier, and R. S. Averback,\nPhys. Rev. Lett. 91 (2003) 167206.\n[22] N. Khan, P. Mandal, and D. Prabhakaran, Phys. Rev. B\n90 (2014) 024421.\n[23] P. Granberg, L. Lundgren, and P. Nordblad, J. Magn.\nMagn. Mater. 92 (1990) 228-232.\n[24] R. Mathieu, P. J¨ onsson, D. N. H. Nam, and P. Nordblad,\nPhys. Rev. B 63 (2001) 092401.\n[25] K. Jonason, E. Vincent, J. Hammann, J. P. Bouchaud,\nand P. Nordblad, Phys. Rev. Lett. 81 (1998) 3243.\n[26] A. F. J. Morgownik and J. A. Mydosh, Phys. Rev. B 24\n(1981) 5277.\n[27] S. Nagata, P. H. Keesom, and H. R. Harrison, Phys. Rev.\nB 19 (1979) 1633.\n[28] A. P. Malozemoff and Y. Imry, Phys. Rev. B 24 (1981)\n489.\n[29] R. V. Chamberlin, M. Hardiman, L. A. Turkevich, and\nR. Orbach, Phys. Rev. B 25 (1982) 6720.\n[30] M. Matsui, A. P. Malozemoff, R. J. Gambino, and L.\nKrusin-Elbaum, J. Appl. Phys. 57 (1985) 3389.\n[31] L. Lundgren, P. Svedlindh, and O. Beckman, Phys. Rev.\nB 26 (1982) 3990.\n[32] S. Pal, K. Kumar, A. Banerjee, S. B. Roy, and A. K.\nNigam, Phys. Rev. B 101 (2020) 180402(R).\n[33] C. A. M. Mulder, A. J. van Duyneveldt, and J. A. My-\ndosh, Phys. Rev. B 23 (1981) 1384.arXiv:2110.08625v2  [cond-mat.mtrl-sci]  24 Aug 2022Supplementary Material: “A stretched exponential-based a pproach for the magnetic\nproperties of spin glasses”\nL. Bufai¸ cal\nInstituto de F´ ısica, Universidade Federal de Goi´ as, 7400 1-970, Goiˆ ania, GO, Brazil\nMETHODS\nThe curves displayed in this article were calculated using Maple 17 soft ware (MaplesoftTM, Japan), with the\nexception of the fitting with Eq. 3 shown in Fig. 1(a) of main text and F igs. S1, S3, S5(a), S6(a) and S7(a) showing\nthe protocols adopted to mimic each experiment described in text, w hich were performed on Origin 8.5 software\n(OriginLab Corporation, USA).\nTHERMOREMANENT MAGNETIZATION\nFig. S1(a) shows the protocol used to simulate the thermoremane nt magnetization (TRM) curves. The system is\ncooled in a constant temperature ( T) sweep rate ( |dT/dt|) from above the spin glass (SG) T(Tg) down to a measure\nT(Tm) in the presence of an external magnetic field ( H). It is kept in this condition for a waiting time tw, thenH\nis removed (at tH= 0) and the remanent magnetization ( M) is recorded as a function of time ( t). As can be noticed\nfrom Fig. S1(a), the experimentally usual situation in which |dT/dt|changes in the vicinity to achieve Tmwas not\nconsidered here, as well as the interval taken for the system to r eachH= 0 since this interval is usually very small\nin comparison to the measurement time, and its influence on the resu ltingMis thus negligible.\n/s116\n/s99/s111/s111/s108\n/s116/s116\n/s72/s61/s32/s48/s116\n/s109/s116\n/s103/s116\n/s119\n/s84\n/s109/s72\n/s32/s84\n/s32/s84\n/s32/s72\n/s84\n/s103/s40/s97/s41/s40/s98/s41\n/s116\n/s99/s111/s111/s108/s32/s61/s32/s48\n/s116/s116\n/s72/s61/s32/s48/s116\n/s103/s61 /s32/s116\n/s109/s116\n/s119\n/s84\n/s109/s72\n/s32/s84\n/s32/s84\n/s32/s72\n/s84\n/s103\nFIG. S1. The protocols adopted to produce TRM curves for the c ases (a) in which Tdecreases at a constant finite sweep rate\nand (b) in which Tis immediately quenched to Tm(tcool= 0).\nEach TRM curve shown here results from the integration of Eq. 1 of main text along the whole tinterval at which\nHwas applied. Thus, using Eq. 2 on Eq. 1 one have:\nMTRM(t) =/integraldisplaytH=0\ntg/bracketleftbiggT(t′)\nTg/bracketrightbiggnAH\nt′−tge−cT(t′)\nTg/parenleftBig\nt−t′\nt′−tg/parenrightBign\ndt′, (S1)\nwhich in this case can be divided in two integrals, one for the cooling pro cess and other for the T=Tminterval:\nMTRM(t) =/integraldisplaytm\ntgAH[Tg−|dT/dt|(t′−tg)]n\nTng(t′−tg)e−c[Tg−|dT/dt |(t′−tg)]\nTg/parenleftBig\nt−t′\nt′−tg/parenrightBign\ndt′+/integraldisplaytH=0\ntm/parenleftbiggTm\nTg/parenrightbiggnAH\nt′−tge−cTm\nTg/parenleftBig\nt−t′\nt′−tg/parenrightBign\ndt′.\n(S2)\nIt is clear from Eq. S2 that, although the weight of the first integra l is usually smaller than the second one, the t\ninterval taken to cool the system from TgtoTm(tcool) plays its part in the resulting TRM curve. Fig. S1(b) shows\nthe protocol for the situation in which Tis immediately quenched to Tg(tcool= 0), resulting in the TRM curves\ndisplayed in Fig. 1(d) of main text. In this case, tm=tgand the first integral of Eq. S2 vanishes.\nAccording to Eqs. 1 and 2 of main text, the M-decay will depend on T,Handn. A fundamental difference from\nthis model to other ones is that now nis separated from TandHin the stretched exponential term. In this sense,2\nncan be understood as a characteristic of the material under stud y, giving a measure of its glassiness. Fig. S2(a)\nshows TRM curves calculated for A=H=c= 1,Tm= 0.8Tg,|dT/dt|= 0.002Tg,tw= 103s and different nvalues.\nAs can be noticed from Fig. S2(b) the relaxation becomes slower as ndecreases.\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48\n/s116\n/s119/s32/s61/s32/s49/s48/s51\n/s32/s115/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103/s40/s97/s41\n/s116/s32/s40/s115/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32\n/s32/s32/s110/s32/s61/s32/s48/s46/s51\n/s32/s32/s110/s32/s61/s32/s48/s46/s53\n/s32/s32/s110/s32/s61/s32/s48/s46/s55/s116\n/s119/s32/s61/s32/s49/s48/s51\n/s32/s115/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103/s40/s98/s41\n/s116/s32/s40/s115/s41/s124/s83/s124/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s32/s110/s32/s61/s32/s48/s46/s51\n/s32/s110/s32/s61/s32/s48/s46/s53\n/s32/s110/s32/s61/s32/s48/s46/s55\nFIG. S2. (a) TRM curves calculated with A=H=c= 1,Tm= 0.8Tg,|dT/dt|= 0.002Tg,tw= 103s and different nvalues.\n(b)|S|for these TRM curves.\nZERO FIELD COOLED MAGNETIZATION\nTo produce the zero field cooled (ZFC) Mcurves (M ZFC), the system is cooled in zero Hdown to Tmin constant\n|dT/dt|, it is kept in this condition for a twinterval, then His applied (at tH= 0) and Mstarts to be captured as a\nfunction of t, as shown in Fig. S3(a). For this case, Mwill be given by\nMZFC(t) =/integraldisplayt\ntH=0/parenleftbiggTm\nTg/parenrightbiggnAH\nt′−tge−cTm\nTg/parenleftBig\nt−t′\nt′−tg/parenrightBign\ndt′. (S3)\nDespite the fact the system is cooled in zero H,tcoolstill affects the relaxation due to the ( t′-tg) term. As faster is\n|dT/dt|during cooling, smaller will be the influence of tcoolon the relaxation. Fig. S3(b) shows the protocol used to\nproduce the idealized tcool= 0 M ZFCcurves displayed in Fig. 2(d) of main text.\n/s116\n/s99/s111/s111/s108\n/s116/s116\n/s72/s61/s32/s48/s116\n/s109/s116\n/s103/s116\n/s119\n/s84\n/s109/s72\n/s32/s84\n/s32/s84\n/s32/s72/s84\n/s103/s40/s97/s41/s40/s98/s41\n/s116\n/s99/s111/s111/s108/s32/s61/s32/s48\n/s116/s116\n/s72/s61/s32/s48/s116\n/s103/s61 /s32/s116\n/s109/s116\n/s119\n/s84\n/s109/s72\n/s32/s84\n/s32/s84\n/s32/s72/s84\n/s103\nFIG. S3. The protocols adopted to produce M ZFCcurves for the cases (a) in which Tdecreases at a constant finite |dT/dt|\nand (b) in which tcool= 0.\nAs for the TRM case, M ZFCrelaxation also depends on T, as can be seen in Eq. S3. Fig. S4(a) compares two\ncurves calculated with distinct Tm, 0.8Tgand 0.6Tg, but the same A=H=c= 1,n= 0.5,|dT/dt|= 0.002Tg,tw=\n103s. Fig. S4(b) makes clear that the relaxation becomes slower as Tdecreases, as expected.\nFor the case of the T-cycled M ZFCcurves displayed in Fig. 3 of main text, the initial protocol is similar to t hat\ndescribed above in Fig. S3 for the conventional M ZFC(in this case, with tw= 0). However, after a t1interval of\nrelaxation at Tm,Tis changed to Tm+∆Tfor at2interval, then it returns to Tmfor a period t3, as shown in Fig.3\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s32/s32/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103\n/s32/s32/s84\n/s109/s32/s61/s32/s48/s46/s54/s84\n/s103\n/s116\n/s119/s32/s61/s32/s49/s48/s51\n/s32/s115/s40/s97/s41\n/s116/s32/s40/s115/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32\n/s116\n/s119/s32/s61/s32/s49/s48/s51\n/s32/s115/s40/s98/s41\n/s116/s32/s40/s115/s41/s83/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s32/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103\n/s32/s84\n/s109/s32/s61/s32/s48/s46/s54/s84\n/s103\nFIG. S4. (a) Comparison between two M ZFCcurves calculated with Tm= 0.8Tgand 0.6Tg. The other parameters were kept\nthe same at the values A=H=c= 1,n= 0.5,|dT/dt|= 0.002Tg,tw= 103s. (b)Sfor these M ZFCcurves.\n/s48 /s50/s48/s48/s48 /s52/s48/s48/s48 /s54/s48/s48/s48 /s56/s48/s48/s48/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57\n/s52/s48/s48/s48/s40/s97/s41\n/s84/s116\n/s51/s116\n/s50/s116\n/s49/s116\n/s99/s111/s111/s108\n/s116/s116\n/s109/s32/s61/s32 /s116\n/s72/s61/s32/s48/s116\n/s103/s84\n/s109/s72\n/s32/s84\n/s32/s84\n/s32/s72/s84\n/s103/s40/s98/s41\n/s116\n/s51/s116\n/s49\n/s116/s32/s40/s115/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32\n/s116\n/s51/s116\n/s49/s32\nFIG. S5. (a) Protocol used to produce the T-cycled M ZFCcurves where Tis changed during the relaxation. This figure\nexemplifies the T-cooled experiment (∆ T <0). The T-heated experiment is similar but with ∆ T >0. (b) Resulting M ZFC\ncurve when the t2stretch is removed, calculated for A=H=c= 1,n= 0.5,|dT/dt|= 0.002Tg,tm= 0.5Tg, ∆T= -0.2Tg,\nt1=t2=t3= 4000 s. The inset shows a magnified view of the t1/t3junction.\nS5(a). The equation describing M(t) under this protocol will be\nM(t) =/integraldisplayt1\ntH=0/parenleftbiggTm\nTg/parenrightbiggnAH\nt′−tge−cTm\nTg/parenleftBig\nt−t′\nt′−tg/parenrightBign\n+/integraldisplayt1+t2\nt1/parenleftbiggTm+∆T\nTg/parenrightbiggnAH\nt′−tge−c(Tm+∆T)\nTg/parenleftBig\nt−t′\nt′−tg/parenrightBign\n+/integraldisplayt\nt1+t2/parenleftbiggTm\nTg/parenrightbiggnAH\nt′−tge−cTm\nTg/parenleftBig\nt−t′\nt′−tg/parenrightBign\ndt′.(S4)\nFig. S5(b) shows a magnified view of the inset of Fig. 3(a). This is the M ZFCcurve resulting when the t2stretch is\nremoved, evidencing that the t3stretch seems to be a continuation of t1.\nMAGNETIZATION AS A FUNCTION OF TEMPERATURE\nThe protocol to produce the ZFC M(T) curve is the conventional o ne for which the system is ZFC in a constant\nT-sweep rate (for the main text it was adopted the same |dT/dt|= 0.002Tg/s used for the TRM and M ZFCcurves\ndiscussed above), then a dc His applied and Tis increased, also in a constant |dT/dt|, whileMis recorded. Fig.\n4(a) of main text was calculated with Tincreasing in the continuous mode, as shown in Fig. S6(a). In this cas e, the\nsystems’ Mis\nM(t) =/integraldisplayt\nti=0AH[Ti+|dT/dt|(t′−ti)]n\nTng(t′−tg)e−c[Ti+|dT/dt |(t′−ti)]\nTg/parenleftBig\nt−t′\nt′−tg/parenrightBign\ndt′. (S5)\nSince this is an off-equilibrium situation, the ZFC curve is very sensitive to changes in the T-sweep rate. Fig. S6(b)\nshows remarkable differences between two curves calculated with s lightly different heating T-rates, with all other\nparameters kept the same.4\n/s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s116/s40/s97/s41\n/s84\n/s105/s84\n/s103\n/s116\n/s103/s116\n/s105/s32/s61 /s32/s116\n/s72/s61/s32 /s48/s32\n/s32/s72 /s84\n/s32\n/s32/s32 /s84\n/s32/s32 /s72/s90/s70/s67\n/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s84/s47/s84\n/s103/s90/s70/s67/s40/s98/s41\n/s32\n/s104/s101/s97/s116/s101/s100/s32/s97/s116/s32/s48/s46/s48/s48/s48/s53/s32/s84\n/s103/s47/s115\n/s104/s101/s97/s116/s101/s100/s32/s97/s116/s32/s48/s46/s48/s48/s48/s54/s32/s84\n/s103/s47/s115\n/s116/s40/s99/s41\n/s84\n/s105/s84\n/s103\n/s116\n/s102/s116\n/s103/s32/s61/s32 /s48/s32\n/s32/s72 /s84\n/s32/s32/s32/s84\n/s32/s32 /s72/s70/s67\n/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s84/s47/s84\n/s103/s70/s67\n/s40/s100/s41\n/s32/s32/s99/s111/s111/s108/s101/s100/s32/s97/s116/s32/s48/s46/s48/s48/s48/s53/s84\n/s103/s47/s115\n/s32/s99/s111/s111/s108/s101/s100/s32/s97/s116/s32/s48/s46/s48/s48/s49/s84\n/s103/s47/s115\n/s32/s104/s97/s108/s116/s32/s111/s102/s32/s49/s48/s51\n/s32/s115/s32/s97/s116/s32/s48/s46/s54/s84\n/s103/s32\nFIG. S6. (a) Protocol used to produce ZFC dcM(T) curves, with Tdecreasing and increasing in continuous mode. (b)\nComparison between two ZFC M(T) curves calculated with heat ing|dT/dt|= 0.0005 Tg/s and 0.0006 Tg/s, with A=H=c= 1,\nn=0.5 for both. (c)Protocol used tocalculate theFCM(T) curv es, withTalso decreasingin continuousmode. (d)Comparison\nbetween FC curves calculated with cooling |dT/dt|= 0.0005 Tg/s and 0.001 Tg/s,A=H=c= 1,n= 0.5. It also shows a jump\ninMfor the case in which the cooling is halted for 103s at 0.6Tg.\nConversely, the field cooled (FC) M(T) curves are almost invariant u nder changes in |dT/dt|. Fig. S6(c) shows the\nprotocol used to produce such curves, also calculated with Tvarying in continuous mode, yielding\nM(t) =/integraldisplayt\ntg=0AH[Tg−|dT/dt|(t′−tg)]n\nTng(t′−tg)e−c[Tg−|dT/dt |(t′−tg)]\nTg/parenleftBig\nt−t′\nt′−tg/parenrightBign\ndt′. (S6)\nFig. S6(d) shows that the FC M(T) curves coincide even when |dT/dt|is remarkably changed. However, as discussed\nin the main text this is not an equilibrium configuration. The figure also s hows that, if the cooling is halted for a\nfinitetinterval at T < T g,Mwill increase. When the cooling is resumed Mwill maintain this increased value.\nRegarding the ZFC-FC M(T) curves, there are still some import poin ts to be addressed. The first one is that, as can\nbe noticed from Fig. S6, the initial Mvalue is zero for both ZFC and FC curves. This is obvious because, ac cording\nto Eqs. S5 and S6, at t= 0 there was no time enough for Mto evolve. In practice, however, the situation is a bit\ndifferent. For the ZFC case, experimentally there is an instrumenta ltinterval between the Happlication and the\ninitial increase of T. This certainly affects the systems’ magnetization, leading to a non -zeroMvalue at ti. For the\nFC curves, it must be regarded that the system is coming from a par amagnetic (PM) configuration with non-zero M\ndue to the applied H, which will naturally have its weight in the initial Mvalue of the SG state. Moreover, as stated\nin the main text, the critical behavior in the vicinity of Tgwill also plays its part in this region. The here proposed\nmodel is concerned with the T < T gregion where the critical behavior can be neglected, so it is in fact no t suitable\nto describe the Tclose toTgsituation. These aforementioned details, if considered here, would certainly change the\nslope of the M(T) curves.\nItmustalsobenoticedthatEqs. S5andS6give Masafunctionof t. TocomputetheM(T) curvesonemustperform\na change of variables, which can be easily done since T(t) =Ti+|dT/dt|for the ZFC curve and T(t) =Ti−|dT/dt|\nfor the FC one.\nThe ac susceptibility ( χ=M/H) curves were produced point by point, i.e.it was assumed that the system was\nthermalized during the measuring. The curves were calculated in the heating mode, and from one point to another T\nwas increased in a constant dT/dtwith the system only under the influence of a static dc field. Fig. S7(a ) shows the\nprotocol to obtain each point, leading to the following equation\nM(t) =/integraldisplaytm\nti=0A[Ti+|dT/dt|(t′−ti)]n\nTng(t′−tg)e−[Ti+|dT/dt |(t′−ti)]\nTg/parenleftBig\nt−t′\nt′−tg/parenrightBign\ndt′\n+/integraldisplaytm+1/f\ntmA\nt′−tg/parenleftbiggTm\nTg/parenrightbiggn\n{1+cos[2πf(t′−tm)]}e−/parenleftBig\nTm\nTg/parenrightBig\n|1+cos[2πf(t′−tm)]|/parenleftBig\nt−t′\nt′−tg/parenrightBign\ndt′.(S7)\nIt can be noticed that it was assumed in Eq. S7 that c=|H|, since it was commented in main text that the stretched\nexponential term is expected to depend on H. However, very similar results are observed for the ac curves in th e case\nthat a constant cvalue is adopted. Fig. S7(b) shows the curves obtained for c= 1, for which Eq. S7 can be adjusted5\n/s48/s46/s57/s50 /s48/s46/s57/s54 /s49/s46/s48/s48 /s49/s46/s48/s52/s51/s46/s52/s51/s46/s54/s51/s46/s56\n/s72\n/s100/s99/s43/s32/s72\n/s97/s99\n/s84\n/s109\n/s116/s40/s97/s41\n/s84\n/s105/s84\n/s103\n/s116\n/s103/s116\n/s105/s32/s61 /s32/s116\n/s72/s61/s32 /s48/s32\n/s32/s72 /s84\n/s32\n/s32/s32 /s84\n/s32/s32 /s72/s72\n/s100/s99/s40/s98/s41\n/s32\n/s32/s32\n/s84/s47/s84\n/s103/s32/s120/s32/s49/s48/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s72/s40/s116/s41/s32/s61/s49/s43/s99/s111/s115/s40/s50 /s102/s116/s41/s32/s48/s46/s49/s32/s72/s122\n/s32/s48/s46/s50/s32/s72/s122\n/s32/s49/s46/s48/s32/s72/s122\n/s32/s80/s77/s32/s40/s67/s87/s41\nFIG. S7. (a) Protocol used to produce each point of the acM(T) curves, calculated in the heating mode. (b) χaccurves\ncalculated with A=H=c= 1,n= 0.5,|dT/dt|= 0.002Tg/s andf= 0.1, 0.2 and 1 Hz.\nto give\nM(t) =/integraldisplaytm\nti=0A[Ti+|dT/dt|(t′−ti)]n\nTng(t′−tg)e−[Ti+|dT/dt |(t′−ti)]\nTg/parenleftBig\nt−t′\nt′−tg/parenrightBign\ndt′\n+/integraldisplaytm+1/f\ntmA\nt′−tg/parenleftbiggTm\nTg/parenrightbiggn\n{1+cos[2πf(t′−tm)]}e−/parenleftBig\nTm\nTg/parenrightBig/parenleftBig\nt−t′\nt′−tg/parenrightBign\ndt′.(S8)\nAgain, the frequencies ( f) were chosen slow enough so that one can assume χ=M/Has a good approximation.\nThe relative shift δTf= ∆Tf/Tf(∆logf) obtained for the curves of Fig. S7(b) is ∼0.002, similar to the value found\nwithc=|H|. I must recall that the here proposed model does not predict the PM-SG transition, the Tfwas here\nassumed as the point in which the SG and PM curves intercept, these values being thus closely related to the choice\nof the PM curve. Usually the T > T gcurve is flattened in relation to the CW law in the vicinity of Tg, which would\nlead to a larger δTf.\nDIFFERENT FUNCTIONAL FORMS FOR bANDM0\nEqs. 1 and 2 of main text gives the systems’ Mat an instant tdue to a Hthat was applied at a previous instant\nt’. It may occur the situation in which Tat the instant that Hwas applied is different from that at instant t,\ni.e.T(t′)/ne}ationslash=T(t). This will surely affect the magnetization since changes in Talter the free energy landscape, thus\naffecting the systems’ position in this landscape and the M-decay. A natural step here is to consider the possibility\nofTbeing a function of tinstead of t′in the equations for M0andb. Lets consider first the case in which T(t′) is\nreplaced by T(t) in Eq. 2 for M0, yielding\nM0=/bracketleftbiggT(t)\nTg/bracketrightbiggnAH\nt′−tg. (S9)\nEqs. S5 and S6 can be adapted to give the ZFC-FC M(T) curves for t his case, displayed in Fig. S8(a). Interestingly,\nthe ZFC is similar to that found using T(t′) inM0whereas the FC one decreases with T. The fundamental difference\nbetween these two approaches is that with Eq. S9 we are considerin g that, although Hwas applied at t′, the systems’\nMis immediately related to the M0value at t[and consequently to the T(t) value]. Conversely, with Eq. 2 of main\ntext we compute M(t) as a consequence of the M0(t′) value. In practice, in a dynamic situation like that of M(T)\nmeasurements the systems’ decay may occur continuously betwe ent′andt[i.e.between T(t′) andT(t)], and the\nresulting curve may lies between those computed with T(t′) andT(t).\nFor the ac curves, in this work each point was computed after the s ystem being thermally stabilized. Thus one can\nexpect the same overall behavior independently of using Eq. 2 of ma in text or Eq. S9. For the calculation of TRM,\nit is interesting to note that, in spite of the non-negligible changes in t he equation for the tcoolinterval when T(t′) is\nreplaced by T(t) inM0, the resulting curves displayed in Fig. S8(b)-(d) present the same features of those observed in\nthe main text. Finally, for the M ZFCexperiment, since in this case His applied when the system is already stabilized\natTm, the resulting curves will be precisely the same as those displayed in F ig. 2 of main text.\nThese results indicate that the most important finding of this paper is the stretched exponential Eq. 1 of main text,\nwhich must be integrated along the interval at which the system is un der the influence of H, whereas the equations6\n/s48/s46/s51 /s48/s46/s54 /s48/s46/s57 /s49/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s40/s97/s41\n/s70/s67\n/s90/s70/s67/s72\n/s100/s99/s32/s61/s32/s49\n/s84/s47/s84\n/s103/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s40/s98/s41\n/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32\n/s116/s32/s40/s115/s41/s32/s116\n/s119/s61/s32/s48\n/s32/s116\n/s119/s61/s49/s48/s50\n/s32/s115\n/s32/s116\n/s119/s61/s49/s48/s51\n/s32/s115\n/s32/s116\n/s119/s61/s49/s48/s52\n/s32/s115\n/s124/s83/s124/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s40/s99/s41\n/s32\n/s116/s32/s40/s115/s41/s116\n/s119/s61/s49/s48/s51\n/s115/s40/s100/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s116/s32/s40/s115/s41/s32/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103\n/s32/s84\n/s109/s32/s61/s32/s48/s46/s54/s84\n/s103\n/s116/s32/s40/s115/s41\n/s124/s83/s124/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s32/s32\nFIG. S8. Some curves calculated with Eq. S9, using A=H=c= 1,n= 0.5: (a) ZFC-FC M(T) computed with cooling |dT/dt|\n= 0.001Tg/s. For the ZFC curve it was also used a heating |dT/dt|= 0.0006 Tg/s. (b) TRM curves calculated for different twat\nTm= 0.8Tg. (c) The |S|of the TRM curves for different tw. (d) Comparison between TRM curves at Tm= 0.8Tgand 0.6Tg,\ncalculated with tw= 103s. The inset shows |S|for these curves.\nforM0andbcan in principle be adapted to better describe each material. Anothe r possible functional form for M0,\nfor instance, is that one for which one removes the nexponent in the T(t′)/Tgterm of Eq. 2, yielding\nM0=/bracketleftbiggT(t)\nTg/bracketrightbiggAH\nt′−tg. (S10)\nFig. S9 displays the main results obtained using Eq. S10. The FC M(T) c urve is similar to that obtained with Eq.\nS9,and the evolution of the TRM curves with twis also very similar to those observed when Eqs. 2 and S9 are used,\nalthough the decrease in Mis less pronounced here. But an interesting difference between the results of Eq. S10 and\nthose obtained with the previous functions, shown in Fig. S9(d), is t hat the TRM curve calculated for Tm= 0.6Tg\nstarts below the Tm= 0.8Tgone, as experimentally observedfor some materials. Since the deca y is faster for the 0.8 Tg\ncurve (as expected), the curves will intercept at some point. The acχcurves present the expected f-dependence, and\nthe results for MZFCalso show the expected overall behavior.\n/s48/s46/s51 /s48/s46/s54 /s48/s46/s57 /s49/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s48/s46/s56/s53 /s48/s46/s57/s48 /s48/s46/s57/s53 /s49/s46/s48/s48 /s49/s46/s48/s53/s50/s46/s53/s51/s46/s48\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s84/s47/s84\n/s103/s40/s97/s41\n/s70/s67\n/s90/s70/s67/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s32\n/s116/s32/s40/s115/s41/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103/s40/s98/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s32\n/s32/s116\n/s119/s61/s32/s48\n/s32/s116\n/s119/s61/s49/s48/s50\n/s32/s115\n/s32/s116\n/s119/s61/s49/s48/s51\n/s32/s115\n/s32/s116\n/s119/s61/s49/s48/s52\n/s32/s115\n/s124/s83/s124/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s116/s32/s40/s115/s41/s40/s99/s41\n/s32/s32\n/s84/s82/s77/s116\n/s119/s61/s49/s48/s51\n/s115\n/s116/s32/s40/s115/s41/s40/s100/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s32\n/s32/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103\n/s32/s84\n/s109/s32/s61/s32/s48/s46/s54/s84\n/s103\n/s84/s47/s84\n/s103/s32/s120/s32/s49/s48/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s72/s40/s116/s41/s32/s61/s49/s43/s99/s111/s115/s40/s50 /s102/s116/s41/s40/s101/s41\n/s32/s32\n/s32/s48/s46/s49/s32/s72/s122\n/s32/s48/s46/s50/s32/s72/s122\n/s32/s49/s46/s48/s32/s72/s122\n/s32/s80/s77/s32/s40/s67/s87/s41\n/s116/s32/s40/s115/s41/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103/s40/s102/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s32\n/s32/s116\n/s119/s61/s32/s48\n/s32/s116\n/s119/s61/s49/s48/s50\n/s32/s115\n/s32/s116\n/s119/s61/s49/s48/s51\n/s32/s115\n/s32/s116\n/s119/s61/s49/s48/s52\n/s32/s115/s77\n/s90/s70/s67/s83/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s116/s32/s40/s115/s41/s40/s103/s41\n/s32/s32\n/s116\n/s119/s61/s49/s48/s51\n/s115\n/s116/s32/s40/s115/s41/s40/s104/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s32\n/s32/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103\n/s32/s84\n/s109/s32/s61/s32/s48/s46/s54/s84\n/s103/s124/s83/s124/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s116/s32/s40/s115/s41\n/s32/s32\n/s116/s32/s40/s115/s41\n/s83/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s32\nFIG. S9. Main results obtained with Eq. S10, using A=H=c= 1,n= 0.5: (a) ZFC-FC M(T) computed with cooling |dT/dt|\n= 0.002Tg/s. For the ZFC curve it was also used a heating |dT/dt|= 0.0002 Tg/s. (b) TRM curves calculated for different tw\natTm= 0.8Tg. (c) The |S|for these TRM curves with different tw. (d) Comparison between TRM curves at Tm= 0.8Tgand\n0.6Tg, calculated with tw= 103s. The inset shows |S|for these curves. (e) χ accurves calculated for f= 0.1, 0.2 and 1 Hz. (f)\nMZFCcurves calculated for different twatTm= 0.8Tg, and (g) the Sof these curves. (h) Comparison between M ZFCcurves\natTm= 0.8Tgand 0.6Tg, calculated with tw= 103s. The inset shows Sfor each Tm.7\nOne can also replace T(t′) byT(t) in both M0andb, leading for instance to\nM0=/bracketleftbiggT(t)\nTg/bracketrightbiggAH\nt′−tg;b=c·T(t)\nTg(t′−tg)n. (S11)\nFigs. S10(a) and (b) displays respectively the dc and ac M(T) curve s and the main TRM results obtained when Eqs.\nS11 are used in Eq. 1. The dc FC M(T) curve shows a plateau-like beha vior followed by a small decrease in Mwith\ndecreasing T, while the ac χcurves show the expected f-dependency. The TRM curves are also very similar to those\nobserved for the other functions here discussed, with a maxima in |S|aroundtw, and since for the M ZFCcurvesHis\napplied when the system is already stabilized at Tmthe resulting curves will be precisely the same as those obtained\nwith Eq. S10, Fig. S9.\n/s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s48/s46/s57/s48 /s48/s46/s57/s53 /s49/s46/s48/s48 /s49/s46/s48/s53/s51/s46/s52/s51/s46/s54/s51/s46/s56\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s49/s48/s49\n/s49/s48/s50\n/s49/s48/s51\n/s49/s48/s52/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s90/s70/s67/s70/s67\n/s72\n/s100/s99/s32/s61/s32/s49/s40/s97/s41\n/s84/s47/s84\n/s103/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32\n/s84/s47/s84\n/s103/s72/s40/s116/s41/s32/s61/s49/s43/s99/s111/s115/s40/s50 /s102/s116/s41/s40/s98/s41\n/s32/s32\n/s32/s120/s32/s49/s48/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s32/s48/s46/s49/s32/s72/s122\n/s32/s48/s46/s50/s32/s72/s122\n/s32/s49/s46/s48/s32/s72/s122\n/s32/s80/s77/s32/s40/s67/s87/s41\n/s116/s32/s40/s115/s41/s77/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s40/s99/s41\n/s84\n/s109/s32/s61/s32/s48/s46/s56/s84\n/s103\n/s32/s32/s116\n/s119/s61/s32/s48\n/s32/s116\n/s119/s61/s49/s48/s50\n/s32/s115\n/s32/s116\n/s119/s61/s49/s48/s51\n/s32/s115\n/s32/s116\n/s119/s61/s49/s48/s52\n/s32/s115\n/s124/s83/s124/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s116/s32/s40/s115/s41/s40/s100/s41\n/s32\nFIG. S10. Main results obtained with Eq. S11, using A=H=c= 1,n= 0.5: (a) ZFC-FC M(T) curves computed with\ncooling|dT/dt|= 0.002Tg/s. For the ZFC curve it was also used a heating |dT/dt|= 0.0002 Tg/s. (b)χ accurves calculated\nforf= 0.1, 0.2 and 1 Hz. (c) TRM curves calculated for different twatTm= 0.8Tg. (c)|S|of the TRM curves calculated\nwith different tw.\nSurely, each function here discussed must be carefully investigate d in order to check if it is in fact suitable to\ndescribe SG-like systems and to ensure that it is scientifically sound. Nevertheless, the agreement between the results\nhere obtained and the main experiments reported along these almos t five decades of investigation is remarkable."    },    {        "title": "2110.11158v1.Magnetic_hysteresis_of_individual_Janus_particles_with_hemispherical_exchange_biased_caps.pdf",        "content": "Magnetic hysteresis of individual Janus particles with hemispherical exchange\nbiased caps\nS. Philipp,1B. Gross,1M. Reginka,2M. Merkel,2M. Claus,1M. Sulliger,1A. Ehresmann,2and M. Poggio1\n1)Department of Physics, University of Basel, 4056 Basel, Switzerland\n2)Institute of Physics, University of Kassel, 34132 Kassel, Germany\nWe use sensitive dynamic cantilever magnetometry to measure the magnetic hysteresis of individual magnetic\nJanus particles. These particles consist of hemispherical caps of magnetic material deposited on micrometer-\nscale silica spheres. The measurements, combined with corresponding micromagnetic simulations, reveal\nthe magnetic configurations present in these individual curved magnets. In remanence, ferromagnetic Janus\nparticles are found to host a global vortex state with vanishing magnetic moment. In contrast, a remanent\nonion state with significant moment is recovered by imposing an exchange bias to the system via an additional\nantiferromagnetic layer in the cap. A robust remanent magnetic moment is crucial for most applications of\nmagnetic Janus particles, in which an external magnetic field actuates their motion.\nJanus particles (JPs) are nano- or micronsized parti-\ncles that possess two sides, each having different phys-\nical or chemical properties. There is a multitude of\ntypes of JPs1, differing in shape, material, and func-\ntionalization. As a subgroup of micron and sub-micron\nsized magnetic particles, which are discussed as a multi-\nfunctional component in lab-on-chip or micro-total anal-\nysis systems2,3, magnetic JPs, consisting of a hemispher-\nical cap of magnetic material on a non-magnetic spheri-\ncal template, allow not only a controlled transversal mo-\ntion, but also a controlled rotation by rotating external\nmagnetic fields4,5. Such JPs can be mass-produced via\nthe deposition of magnetic layers on an ensemble of sil-\nica spheres. The transversal and rotary motion of these\nparticles can be controlled via external magnetic fields,\nwhich exert magnetic forces and torques6. This ability to\nexternally actuate magnetic JPs has led to applications\nin microfluidics, e.g. as stirring devices7, as microprobes\nfor viscosity changes8, or as cargo transporters in lab-on-\nchip devices9–11. Magnetic JPs have also been proposed\nas an in vivo drug delivery system12.\nAlthough, in general, a transversal controlled motion\ncan be achieved by both superparamagnetic particles or\nparticles with a permanent magnetic moment, a con-\ntrol over the rotational degrees of freedom can only be\nachieved, if the particles possess a sufficiently large per-\nmanent magnetic moment. Streubel et al.13analyzed\nthe remanent magnetic state of magnetic JPs with fer-\nromagnetic (fm) magnetic caps. Their simulations show\nthat permalloy JPs with diameters larger than 140 nm\nhost a global vortex state at remanence. Because this\nflux-closed state has a vanishing net magnetic moment,\nmagnetic JPs hosting such a remanent configuration are\nunsuited for applications involving magnetic actuation.\nThus, for JPs larger than this critical diameter, strate-\ngies to overcome this limitation need to be developed.\nHere, we make use of exchange bias14–16, which, in\na simplified picture imposes a preferred direction on\nthe magnetic moments of the fm layer and is thereby\nable to prevent the formation of a global vortex at\nremanence. We apply an exchange bias to the fm layer\nby adding an antiferromagnetic (afm) layer beneath thefm layer. In order to verify that this addition leads to a\nremanent configuration with large magnetic moment, we\nmeasure the magnetic hysteresis of individual JPs with\nand without this layer. The measurement of individual\nJPs is necessary in order to eliminate the effects of\ninteractions between neighboring JPs. For this task, we\nemploy dynamic cantilever magnetometry (DCM) and\nanalyze the results by comparison to corresponding mi-\ncromagnetic simulations. This technique overcomes the\nlimitation of earlier measurements, that were restricted\nto ensembles of interacting JPs on a substrate17. These\nmeasurements relied on the longitudinal magneto-optical\nKerr effect and magnetic force microscopy. They found\nan onion state with a large remanent magnetization in\nJPs with an afm layer. Nevertheless, given that the\nmeasurements were done on close-packed ensembles of\nJPs, they do not exclude effects due to the interaction\nbetween the particles and, therefore, cannot be used to\ninfer the behavior of isolated JPs.\nWe fabricate the magnetic JPs by coating a self-\nassembled template of 1 .5µm-sized silica spheres with\nthin layers of different materials via sputter-deposition.\nThe non-magnetic silica spheres are arranged on a silica\nsubstrate using entropy minimization18, which allows\nthe formation of hexagonal close-packed monolayers.\nJPs with two different layer stacks, shown in Fig. 1\n(b), are produced. Ferromagnetic JPs (fmJPs) are\nfabricated by depositing a 10 nm-thick Cu buffer layer\ndirectly on the silica spheres, followed by a 10 nm-thick\nlayer of ferromagnetic CoFe. The film is sealed by\na final 10 nm-thick layer of Si. A second type of JP,\nwhich we denote exchange-bias JPs (ebJPs), includes an\nadditional 30 nm-thick afm layer of Ir 17Mn83between\nthe Cu buffer and the fm layer. Layer deposition is\nperformed by sputtering in an external magnetic field\nof 28 kA/m applied in the substrate plane, i.e. in the\nequatorial plane of the JPs, in order to initialize the\nexchange bias by field growth. This fabrication process\nis described in detail in Tomita et al.17. Individual\nJPs are then attached to the apex of a cantilever for\nmagnetic characterization in a last fabrication step, asarXiv:2110.11158v1  [cond-mat.mes-hall]  21 Oct 20212\n-x\nyz\nx\nyz(a)\n(b)(c)\n(d)\nBuffer CuFM CoFeCapping Si\nAFM IrMnθ\nϕ\nϕθeb\neb\nJP\nJP\n500 nm‘‘\n‘\nBuffer CuFM CoFeCapping SifmJPebJPTruncation\nFigure 1. (a) Sketch of a cantilever with a JP attached to\nits tip and definition of the coordinate system. (b) Cross-\nsectional SEM of a JP showing the gradient of the layer thick-\nness. The two investigated layer stacks of the hemispherical\ncap are shown in the insets. (c), (d) Definition of the angles\nsetting the orientation of the unidirectional anisotropy vector\nused to mimic exchange bias effects ( θeb,ϕeb), and the angles\ndefining the orientation of a JP on the cantilever ( θJPand\nϕJP).\nshown in the scanning electron micrographs (SEMs) of\nFig. 2 (a) and (b).\nNote that the values given for thicknesses are nominal\nand that the film thickness gradually reduces towards\nthe equator of the sphere with respect to the top,\nas shown in Fig. 1 (b), because of the deposition\nprocess17. Furthermore, the touching points of the\nnext neighbors in the hexagonal closed packed arrange-\nment of the silica spheres on the substrate template\nimpose a lateral irregularity on the equatorial line of\nthe capping layers. This is best seen in Fig. 2 (a) and (b).\nWe measure the magnetic hysteresis of each an\nindividual fmJP and an individual ebJP via DCM.\nDCM is a technique to investigate individual, nano- to\nmicrometer-sized magnetic specimens, similar to a stan-\ndard vibrating superconducting quantum interference\ndevice (SQUID) magnetometer. The key differences\nare that DCM is sensitive enough to measure much\nsmaller magnetic volumes than a vibrating SQUID\nmagnetometer and that it measures magnetic properties\nwith respect to rotations of the external magnetic\nfield, rather than modulations of its amplitude as in\nmeasurements of magnetic susceptibility. Details on\nthe technique and measurement setup can be found in\nRefs. 19 and 20.A magnetic specimen is attached to the tip of a can-\ntilever, which is driven in a feedback loop at its resonance\nfrequencyfwith a fixed amplitude, actuated by a piezo-\nelectric transducer. A uniform external magnetic field\nHis applied to set the magnetic state of the specimen\nunder investigation. Hcan be rotated within the plane\nperpendicular to the cantilever’s rotation axis ( xz-plane)\nwith a span of 117 .5°and a maximum field amplitude of\nH=±3.5 T. Its orientation is set by the angle θhas\ndefined and indicated in Fig. 1 (a). The magnetic torque\nacting on the sample results in a deflection of the can-\ntilever as well as a shift in its resonance frequency, which\nis given by\n∆f=f−f0=f0\n2k0l2e/parenleftBigg\n∂2Em\n∂θ2c/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nθc=0/parenrightBigg\n. (1)\nf0andk0are the resonance frequency and spring\nconstant of the cantilever at zero applied field, respec-\ntively,leis the effective length of the cantilever, θc\nthe oscillation angle, and Emthe magnetic energy of\nthe specimen. Properties of the cantilevers used in the\nexperiments can be found in section I of the appendix.\nIn the limit of large applied magnetic field, such that the\nZeeman energy dominates over the anisotropy energy,\nall magnetic moments align along H. In this limit ∆ f\nasymptotically approaches a value determined by the\ninvolved anisotropies, their respective directions, the to-\ntal magnetic moment, θh, and the mechanical properties\nof the cantilever19,20. In particular, the determination\nof these asymptotes allows us to extract the direction of\nmagnetic easy and hard axes by measuring the angular\ndependence of the frequency shift at high field ∆ fhf.\nThe maximum positive (minimum negative) ∆ fhf(θh)\nindicates the easy (hard) axis. For these measurements,\nshown in Fig. 2 (c), we apply µ0H= 3.5 T, where we\nexpect to be in the high field limit. See section VI\nand VII of the appendix for more details. We also\nmeasure the magnetic hysteresis of ∆ f(H) by sweeping\nthe applied field Hup and down for θhfixed to the\nvalues determined for the magnetic easy and hard axes,\nrespectively. This procedure reveals signatures of the\nJPs’ magnetic reversal, as shown in Figs. 2 (d), (e) and 3.\nIn order to analyze both these types of measure-\nments, we perform micromagnetic simulations using the\nfinite-element software Nmag21. We calculate ∆ f(H)\nfrom the micromagnetic state for parameters set by\nthe experiment19,22. Matching these simulations to the\nmeasurements gives us a detailed understanding of the\nprogression of the magnetic configurations present in the\nJPs throughout the reversal process. Events, such as the\nnucleation of a magnetic vortex, can be identified and\nassociated with features in ∆ f(H) measured via DCM.\nFig. 2 (a) and (b) shows false color SEMs of the\nmeasured fmJP and ebJP, respectively, each attached to\nthe tip of a cantilever. The orientation of the particles3\n/gid00132/gid00001/gid00048/gid00040/gid00004/gid00028/gid00041/gid00047/gid00036/gid00039/gid00032/gid00049/gid00032/gid00045ebJP fmJP H\n(a) (b)\n(c)\n(d)\n(e)/gid00132/gid00001/gid00048/gid00040\nH || easy \naxisH || hard\naxis\nFigure 2. False color SEMs of the (a) fmJP and (b) ebJP\nattached to the tip of a cantilever, respectively. The coordi-\nnate system is shown on the right. (c) ∆ fhf(θh) measured at\nµ0H= 3.5 T for the fmJP (blue) and ebJP (brown). Black\ncircles indicate θhof the hysteresis measurements, which are\nshown in (d), (e), and Fig. 3.\nin the images can be correlated with the angle θhof the\nmaxima and minima found in the high-field frequency\nshift ∆fhf(θh) in dependence of the magnetic field angle\nθh, shown in Fig. 2 (c). Doing so, we find a magnetic\neasy direction in the equatorial plane of the particles and\na hard direction along the axis of the pole. The 90 °angle\nbetween easy and hard direction is a clear indication\nthat uniaxial anisotropy is the dominant anisotropy in\nthe system. We ascribe the latter to the shape of the\nJPs, because no other strong anisotropies are expected.\nThe field-dependent frequency shift ∆ f(H) for H\naligned along the easy axis, see Fig. 2 (d), shows a\ntypical hysteretic, V-shaped curve, that approaches\na horizontal asymptote for high field magnitudes19.\nThe fmJP shows a symmetric asymptotic behavior\nforµ0H= 3.5 T and −3.5 T (blue curve). Magnetic\nreversal at low fields, µ0Haround ±20 mT, is symmetric\nupon reversal of the field sweep direction, as shown in\nFig. 2 (e). This behavior is expected for a ferromagnetic\nparticle with a magnetic field applied along its easy axis.In contrast, measurements of the ebJP reveal asymmet-\nric asymptotic behavior with ∆ fhfvalues differing by\nabout 0.9 Hz forµ0H=±3.5 T, as seen in the brown\ncurve of Fig. 2 (d). Furthermore, after a full hysteresis\ncycle, we observe a reduction in the difference of ∆ fhfat\n±3.5 T by about 0 .4 Hz, which is evidence for magnetic\ntraining23,24. Measurements of the ebJP also show a\nhighly asymmetric magnetic reversal, which occurs at\nµ0H=−44 mT when sweeping the field down and at\nµ0H= 12 mT when sweeping the field up. All of these\nfindings are characteristic of an exchange bias imposed\non the fm layer by the afm layer.\nIn order to draw conclusions about the magnetic state\nof the JPs, we establish a micromagnetic model for\neach of the two types of JPs over many iterations of\ncomparison to measured ∆ f(H) and variations of the\nparameters for Happlied along the magnetic easy and\nhard axis, respectively. This iterative process, informa-\ntion from literature, and observations from SEMs lead\nto a final set of parameters (see appendix, section I)\nand a model that reproduces the measured ∆ f(H). The\nmodel assumes that the magnetic JPs are made from a\nhemispherical shell with a thickness gradient from the\npole towards the equator, which accounts for the gradual\nreduction of the shell thickness away from the pole, as\nshown in Fig. 1 (b). The hemisphere is also truncated25\nby a latitudinal belt around the equator, reflecting\nobservations from the SEM images in Figs. 2 (a) and\n(b). For simplicity, in the simulations, we do not account\nfor the magnetic film’s irregular edge at the equator and\na possible change in the crystallographic texturing with\nrespect to the particle surface as a function of position\nwithin the cap. The orientation of a JP with respect to\nthe cantilever rotation axis and His set by infering the\norientation from the SEMs and followed by an iterative\ntuning of the angles ( θJP,ϕJP), as defined in Fig. 1 (d),\nto match the measured ∆ f(H).\nFigs. 3 (a) and (b) shows agreement between the\nmeasured and simulated ∆ f(H) curves for the fmJP\nforHapplied along the magnetic easy and hard axis,\nrepectively. Similar curves are shown for the ebJP\nin Figs. 3 (c) and (d). Details on the progression of\nthe magnetization configurations as a function of H,\nas indicated by the simulations, are found in section\nII of the appendix. Here, we focus on the remanent\nmagnetization configurations in both types of particles,\nas shown on the right of Fig. 3.\nFor the fmJP, an onion state is realized after sweeping\nHalong the easy axis while a global vortex state is found\nafter sweeping along the hard axis. In applications, mag-\nnetic JPs are subject to considerable disturbances from\nthe outside, including thermal activation, interactions\nwith other nearby magnetic particles, and alternating\nexternal magnetic fields for actuation. Hence, we can\nexpect a magnetic JP to relax to its ground state\nconfiguration over time. The simulation of the fmJP4\nfmJP\nebJP\nx\ny\nzhard\neasy\n-1\n mz1-1\n1 mzH || easy \naxis\nH || easy \naxisH || hard\naxis\nH || hard\naxis(a)\n(b)\n(c)\n(d)\nFigure 3. Measured and simulated ∆ f(H) of the fmJP for H\napplied along the (a) easy and (b) hard axis, respectively. A\nvisualization of each corresponding simulated remanent mag-\nnetic state is shown on the right. The same set of data for\nthe ebJP is shown in (c) and (d).\nshows that when the magnetization is in the global\nvortex state, its magnetic energy is 101 aJ lower than\nwhen it is in the onion state. The global vortex state\nis therefore energetically more favorable than the onion\nstate, which is also true if compared to any other\nremanent state that we have found in simulations for\nfmJPs, as discussed in appendix, section IV. This\nanalysis suggests that a remanent global vortex state,\nwhich has a vanishing total magnetic moment, is realized\nin fmJPs over time, independent of magnetic history. If\nwe normalize the magnetic moment of this state by the\nsaturation moment, MsV, we find that the global vortex\nstate hosted by the fmJP has a moment value of 0.03,\nprecluding the use of such particles in applications.\nWe establish a similar micromagnetic model for theebJP. To model the effect of an exchange bias imposing\na preferred direction on the magnetic moments in the\nferromagnet, a unidirectional anisotropy is added to\nthe simulation. Note that other influences of the afm\nlayer are not accounted for, especially contributions\nto the coercive field26, rotational anisotropies27, or\ncontributions arising due to its granular structure28.\nFor this reason, neither the asymmetric values of the\nmagnetic reversal fields nor the observed training\neffects are correctly reproduced by the simulation. The\nunidirectional anisotropy, described by a unit vector ˆ ueb\nwith orientation angles ( θeb,ϕeb), as shown in Fig. 1 (c),\nand an anisotropy constant Keb, is expected to lie\nsomewhere in the equatorial plane of the ebJP. The\nspecific orientation of ˆ uebwithin this plane is induced in\nthe afm by the magnetic field applied during deposition.\nNote that for simplicity, Kebis kept constant within\nthe whole volume of the magnetic cap, despite thickness\nvariations of the afm layer with θeb.\nThe knowledge of where ˆ uebpoints within the equa-\ntorial plane is lost after attaching the JP to the can-\ntilever. In the simulations, we choose to align ˆ uebalong\nthe direction in the equatorial plane that coincides with\nthe applied external field in the easy-axis configuration,\neven though it could point along any direction in this\nplane. This assumption that Handˆ uebare collinear in\nthe easy-axis measurements means that our simulations\npredict the maximum possible ∆ ffor any given choice\nof the unidirectional anisotropy constant Keband hence\ngive a lower bound for Keb.\nWe adjust this anisotropy to match the measured\n∆f(H) along both the easy and hard axes and find\ngood agreement for Keb= 22.5 kJ/m3. In particular, the\nmodel reproduces the asymmetry in ∆ fhffor both the\neasy and hard-axis alignments, as shown in Fig. 3 (c)\nand (d). This value of Kebis also consistent with results\nfrom Ref. 28, once the influence of a reduced sample size\nis considered29.\nAt remanence, the simulations show that the ebJP\nhosts an onion state irrespective of its magnetic history,\nas shown in Figs. 3 (c) and (d). To exclude the presence\nof an equilibrium global vortex state, we test this state’s\nstability by initializing the ebJP in a global vortex\nstate at remanence and then relaxing the system to a\nlocal energetic minimum. Following this procedure, the\nsystem relaxes to the onion state. As a result, we can\nexclude the global vortex state as a possible equilibrium\nremanent state in this ebJP.\nFor the simulations of the ebJP we find a total\nmagnetic moment at remanence, normalized by its\nmaximum value of MsV, of 0.89 and 0.71 depending on\nwhether His applied along the hard or easy direction of\nthe external field, respectively. This remanent moment\nrepresents an increase of more than one order of mag-\nnitude compared to the remanent moment of the fmJP.5\nHence, introducing exchange bias to magnetic JPs, if\nstrong enough, succeeds in stabilizing a high-moment\nonion state in remanence.\nTo conclude, micrometer-sized JPs capped with an\nantiferromagnetic/ferromagnetic or purely ferromagnetic\nthin film system have been mass produced through a\nsputter-deposition process. We have investigated the\nmagnetic reversal and remanent magnetic configurations\nof individual specimens of these JPs using DCM and\ncorresponding micromagnetic simulations. Although\nthe fmJPs host a global vortex state in remanence\nwith a vanishing magnetic moment, the addition of an\nantiferromagnetic layer in ebJPs successfully changes the\nremanent configuration to a stable high-moment onion\nstate. Unlike previous measurements on close packed\nparticle arrays, our measurements on individual JPs\nshow that the stability of this high magnetic moment\ntexture in remanence is a property of the individual\nparticles and present in absence of interparticle interac-\ntions.\nWe thank Sascha Martin and his team in the machine\nshop of the Physics Department at the University of\nBasel for help building the measurement system. We\nacknowledge the support of the Canton Aargau and the\nSwiss National Science Foundation under Grant No.\n200020-159893, via the Sinergia Grant Nanoskyrmionics\n(Grant No. CRSII5-171003), and via the National Cen-\ntre for Competence in Research Quantum Science and\nTechnology. Calculations were performed at sciCORE\n(http://scicore.unibas.ch/) scientific computing core\nfacility at University of Basel.\nThe data that support the findings of this study are\navailable from the corresponding author upon reasonable\nrequest.\nREFERENCES\n1Andreas Walther and Axel H. E. M¨ uller. Janus Particles: Synthe-\nsis, Self-Assembly, Physical Properties, and Applications. Chem.\nRev., 113(7):5194–5261, July 2013.\n2D. Issadore, Y. I. Park, H. Shao, C. Min, K. Lee, M. Liong,\nR. Weissleder, and H. Lee. Magnetic sensing technology for\nmolecular analyses. Lab Chip , 14:2385–2397, 2014.\n3Arno Ehresmann, Iris Koch, and Dennis Holzinger. Manipu-\nlation of superparamagnetic beads on patterned exchange-bias\nlayer systems for biosensing applications. Sensors , 15(11):28854–\n28888, 2015.\n4Brandon H. McNaughton, Karen A. Kehbein, Jeffrey N. Anker,\nand Raoul Kopelman. 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Interrelation between polycrystalline structure\nand time-dependent magnetic anisotropies in exchange-biased bi-\nlayers. Physical Review B , 102(14):144421, October 2020. Pub-\nlisher: American Physical Society.\n29J. Nogu´ es, J. Sort, V. Langlais, V. Skumryev, S. Suri˜ nach, J. S.\nMu˜ noz, and M. D. Bar´ o. Exchange bias in nanostructures.\nPhysics Reports , 422(3):65–117, December 2005.Supplementary Information: Magnetic hysteresis of individual Janus particles with hemispherical exchange\nbiased caps\nS. Philipp,1B. Gross,1M. Reginka,2M. Merkel,2M. Claus,1M. Sulliger,1A. Ehresmann,2and M. Poggio1\n1)Department of Physics, University of Basel, 4056 Basel, Switzerland\n2)Institute of Physics, University of Kassel, 34132 Kassel, Germany\nI. CANTILEVER PROPERTIES AND\nSIMULATION DETAILS\nCantilevers are fabricated from undoped Si. They\nare 75 µm-long, 3.5µm-wide, 0.1µm-thick with a mass-\nloaded end and have a 11 µm-wide paddle for optical\nposition detection. The resonance frequency f0of the\nfundamental mechanical mode used for magnetometry\nis on the scale of a few kilohertz. Spring constant k0\nand effective length leare determined using a finite\nelement approximation1. For the cantilever used for\nthe fmJP we find f0= 5285.8 Hz,k0= 249 µN/m\nandle= 75.9µm. For the cantilever of the ebJP\nf0= 5739.3 Hz,k0= 240 µN/m andle= 75.9µm.\nMicromagnetic simulations are performed with the\nfinite-element software package Nmag2. As an approx-\nimated geometry for the JPs a semi-sphere shell with\na thickness gradient from the pole towards the equa-\ntor is used, that is truncated at the equator, as dis-\ncussed in the main text. The exchange constant is set\ntoAex= 30 pJ/m3.\nIn case of the fmJP, opposing to the SEM image in\nFig. 2 (a) in the main text, which suggests a truncation\nof the fm layer by about 250 nm, it needs to be set to\n350 nm or even more to match the high field progression\nof ∆f(H). For the same reason the nominal thickness of\n10 nm of the fm layer needs to be increased to at least\n12 nm at the pole, which is then gradually reduced to\n0 at the equator. These two geometric constraints are\nnecessary to keep Msat a reasonable value below the bulk\nvalue of 1.95 MA/m4. This suggests, that significantly\nmore fm material than anticipated is deposited on the\nregion around the pole of the JPs, which is the most\ndirectly exposed area of the sphere during deposition.\nFor the simulation of the fmJP we set the following\nparameters: saturation magnetization Ms= 1.8 MA/m,\nsilica sphere diameter of 1 .5µm, truncation d= 350 nm,\nparticle orientation ( θJP,ϕJP) = (91 °,2°) and maximum\nallowed mesh cell size 7 .5 nm.\nFor the ebJP, the geometric parameters have to be ad-\njusted less from their nominal values than for the fmJP,\nin order to match between the micromagnetic model to\nthe experiment. This result suggests that the afm layer,\nwhich is deposited before the fm, acts as an adhesive for\nthe fm, and the ebJP is coated more homogeneously than\nthe fmJP.\nFor the simulation of the ebJP we set the follow-\ning parameters: Ms= 1.44 MA/m, fm layer thickness\nof 10 nm at the pole, gradually reduced to 0 at theequator, silica sphere diameter of 1 .5µm,d= 350 nm,\n(θJP,ϕJP) = (85 °,10°), maximum allowed mesh cell size\n7.0 nm, (θeb,ϕeb) = (−90°,0°), unidirectional anisotropy\nconstantKeb= 22.5 kJ/m3.\nFor the generic simulations in sections V and IV we\nhave usedMs= 1.8 MA/m, , fm layer thickness of 10 nm\nat the pole, gradually reduced to 1 nm at the equator,a\nsilica sphere diameter of 500 nm, ( θJP,ϕJP) = (0 °,0°),\nand (θeb,ϕeb) = (−90°,0°). Parameters that are not\nmentioned here are given in the main text.\nII. PROGRESSION OF THE MAGNETIC\nSTATE WITH EXTERNAL FIELD\nA. Ferromagnetic Janus particles\n∆f(H), measured for Hparallel to the magnetic easy\n(blue data) and hard axis (orange data), respectively,\nas shown in Fig. 1 (a), gives direct information on\nhigh field behavior and magnetic reversal of the JPs.\nForHparallel to the magnetic easy axis, an overall\nV-shape suggest Stoner-Wohlfarth like behavior for\nmost of the field range in Fig. 1 (a). As seen in the\nclose-up in Fig. 1 (b), magnetic reversal appears to take\nplace through a few sequential switching events at small\nnegative reverse fields.\nThe simulated ∆ f(H), also shown in Fig. 1 (green\npoints) together with a few exemplary configurations\nof the simulated magnetic state of the JP, can give\nmore insight into what happens during the field sweep.\nStarting from full saturation, most magnetic moments\nstay aligned with the easy direction down to very low\nreverse fields, nicely seen in Fig. 1, configuration 1 at\n3.5 T and configuration 2 at remanence. The latter is\nan onion state. This progression of configurations is\nconsistent with the Stoner-Wohlfarth-like behavior of\nthe experimental ∆ f(H). Magnetic reversal takes place\nthrough the occurrence of a so-called S-state, for which\nthe magnetization follows the curvature of an S. The\nreversal is shown in configurations 3 and 4 in Fig. 1.\nThen, until full saturation is reached in reverse field,\nonly magnetic moments in proximity to the equator of\nthe JP are slightly canted away from the direction of the\nexternal field (and the easy plane). This progression is\nrobust in simulation, even though sometimes, depending\non slight variations of simulation parameters, a vortex\nappears in reverse field instead of the S-state. The\nobservation of several, individual switching events\nduring magnetic reversal in experiment may originate\n1arXiv:2110.11158v1  [cond-mat.mes-hall]  21 Oct 20211\n3(b)\n5\n 6\n 8 7mz1\n0\n-1\nmz1\n0\n-1mx1\n0\n-1\nmz\n1\n 3 2 41\n0\n-1mz1\n0\n-1mx1\n0\n-1mx1\n0\n-1\nmzEasy direction \nzx\ny\nH\nzx\nyHHard direction \n1\n0\n-1\n627(a)\n4\n58Figure 1. Data for the ferromagnetic Janus particle. (a) Measured ∆ f(H) for easy (blue) and hard (orange) alignment of the\nexternal field, as well as the simulated magnetic configurations in green (easy) and red (hard). (b) Close-up of (a) for low fields.\nData with easy orientation is offset by 2 Hz for better visibility. Colored arrows indicate field sweep directions. Numbers in (a)\nand (b) denote the field values for the configurations of the magnetic state.\nin vortex hopping, or switching of different regions in\nthe JP due to variations in material and geometric\nparameters. Magnetic reversal through a vortex rather\nthan an S-state may also explain the big difference of\nthe coercive fields between experiment ( Hc≈32 mT)\nand simulation ( Hc= 6 mT).\nForHparallel to the magnetic hard axis, the exper-\nimental data has an inverted V-shape, and there is no\neasily identifiable sign of magnetic reversal. Yet, for rel-\natively large fields, around 1 .5 T, switching events are\nobserved, and exist up to negative fields of similar mag-\nnitude. Typically, W-shaped curves are observed for\nmeasurements with the field aligned with the hard direc-\ntion, and can be understood in a simple Stoner-Wohlfarth\nmodel, which is discussed in section III. The inverted V-\nshape instead of the W-shape is a peculiarity of the fact\nthat the fm layer of the JP is curved everywhere. The\nangle between the local surface normal and the direction\nof the external magnetic field is different for every polar\ncoordinate of the JP, which leads to a dependence of the\nlocal demagnetizing field on the polar coordinate. In con-\nsequence, the magnitude of the external magnetic field,\nfor which the local magnetic moments start to rotate to-\nwards their local easy direction depends strongly on the\nposition in the magnetic cap. This leads to the observed\ncurve shape of ∆ f(H), a more detailed discussion can befound in section IV. The magnetic progression in simula-\ntion for external field alignment with the hard direction\ncan be summarized as follows: Starting from full sat-\nuration, the magnetic moments start rotating towards\nthe easy plane with decreasing field magnitude due to\nthe competition between shape anisotropy and Zeeman\nenergy. This takes place for different magnitudes of H\ndepending on where a magnetic moment is located in\nthe JP, as discussed earlier. Configuration 5 in Fig. 1\nshows a state for which magnetic moments at the pole\nhave already started to rotate, while magnetic moments\nin proximity to the equator remain aligned with the ex-\nternal field. Superimposed to this rotation, a minimiza-\ntion of the system’s energy by formation of a magnetic\nvortex localized at the pole for around 1 .8 T takes place,\nwhich grows in size with decreasing field, see configura-\ntion 6. In simulation, this is a gradual evolution, and only\nfor fields below about 300 mT jumps in ∆ fdue to vortex\nmovement are observed. This process is in contrast to\nthe discontinuities that occur at around 1 .5 T in the ex-\nperiment, but can be explained by vortex hopping from\npinning site to pinning site. The latter may be present\ndue to fabrication inhomogeneities in the JPs5. For zero\nfield the vortex dominates the magnetic configuration of\nthe JP and has evolved into a global vortex state, as\nshown in configuration 7. Configuration 8 shows the vor-\ntex in reverse field, which has changed polarity, and has\n2jumped to a slightly off-centered position. The latter is\ntoo small to be visible in the figure. Further decreasing\nthe field, the vortex sits centrally in the JP and shrinks in\nsize, and vanishes around −1.82 T. At the same time, the\nmagnetic moments rotate towards the field direction de-\npending on their position in the JP, as described earlier.\nNote, that by slightly changing simulation parameters,\nwe find that features due to vortex entrance and hopping\nmay manifest themselves in ∆ f(H) with strongly differ-\ning magnitude and for different field values. Introducing\nartificial pinning cites in simulation can be used to adjust\nthe vortex hopping to match the observed signals more\nprecisely5, but consumes vast amounts of computational\ntime and should still be understood only as an exemplary\nprogression of the magnetic state.\nB. Exchange-biased Janus particles\nThe progression of the magnetic state for the ebJP\nis very similar to the fmJP, yet, there are crucial\ndifferences. See Fig. 2 for the DCM data, simulation\nresults, and configurations of some magnetic states. For\nthe field oriented in the magnetic easy direction the\nnearly polarized state, shown in Fig. 2 configuration\n1, is similar to that shown in Fig. 1, configuration 1.\nReducing the field down to remanence, as shown in\nFig. 2 configuration 2, we find an onion state just as\nfor the fmJP. Magnetic reversal occurs again through\nan S-state, rather than via vortex formation, as shown\nin configuration 3. However, the reversal is shifted\ntowards negative fields, and occurs for −15 mT for the\ndown sweep, and for −17.5 mT for the up sweep of the\nmagnetic field. This does not match the experimentally\nobserved values, especially for the latter case, for which\nthe switching occurs for positive field. This is no\nsurprise, since the employed model does not account for\nthe contribution of the exchange bias to the coercivity.\nYet, both simulation and experiment show a shift of the\nhysteresis loop towards negative fields as compared to\nthe fmJP.\nWe only observe a single switching event in experiment\nfor the magnetic reversal, which is consistent with the\nbehavior of the S-state in simulation. For the alternative\nmagnetic reversal process through vortex formation, we\nwould expect several switching events due to vortex\nhopping. We find such a situation e.g. for a few reversal\nprocesses without unidirectional anisotropy, where\ngeometrical parameters of the JPs have been varied, see\nsection V. Yet, it is also possible, that a strong pinning\nsite favors the formation of a vortex, and keeps it in\nplace for all field magnitudes up to the reversal point.\nIf the magnetic field is swept from negative saturation up\nto remanence (not shown here), an onion state is present,\nthat has its total magnetic moment pointing opposite to\nthe exchange bias direction. For applications, this is an\nundesirable state. It is energetically less favorable than\nthe state of parallel alignment, and if the energy barrierbetween the two states is overcome by an external\ninfluence, the JP will switch.\nIf the external field is aligned with the hard direc-\ntion, and starts at full saturation, the magnetic moments\nrotate towards the easy plane depending on their posi-\ntion in the JP for decreasing field just as for the fmJP.\nFurther, the same, superimposed vortex formation takes\nplace, starting for 1 .34 T. Again the vortex occupies more\nand more volume of the JP with further decreasing field.\nSuperimposed, a local vortex forms at the pole of the\nebJP for an applied field of 1 .34 T. As for the fmJP, the\nvortex occupies more and more volume of the JP with\nfurther decreasing field. However, upon further reducing\nthe field, the vortex, rather than inhabiting the whole\nJP as a global vortex centered at the pole of the fmJP,\nit prefers to move to the side of the ebJP, as shown in\nconfiguration 4 of Fig. 2. Moving down from the pole to-\nwards the equator, the vortex exits from the JP through\nthe equator for 5 mT, and an onion state is formed at\nremanence, as shown in configuration 5. The orientation\nof the onion state is governed by ˆ ueb. For a small reverse\nfield a domain wall state forms, as shown in configura-\ntion 6. With further decreasing field, the domain wall\nis rotated with respect to the polar axis of the JP. This\nstate seems to be a precursor of the vortex state, and the\nwall is subsequently replaced by the vortex, sitting again\nin the center of the JP, as shown in configuration 7. The\nvortex vanishes for -1 .36 T. Whether such a domain wall\nstate is indeed realized in the ebJPs for reverse fields, or\nif a vortex enters from the equator and moves back to the\ncenter of the JP, as seen for simulations of smaller JPs\n(see section V), remains an open question. The DCM\nsignal shows in both experiment and simulation many ir-\nregularities for the lower field range, which does not allow\nus to draw clear conclusions on the magnetic state present\nin the JPs. Nevertheless, the simulations clearly suggest\nthat an onion state should be realized at remanence, irre-\nspective of the states present during the hysteresis. This\nsituation is markedly different than that of the fmJP and\nis a direct consequence of the presence of exchange bias.\nIII. STONER-WOHLFARTH MODEL FOR\nEASY PLANE TYPE ANISOTROPY\nThe shape of the magnetic material of the fmJPs is,\nat least in a first approximation, rotationally symmetric\naround the pole axis. Further, the thickness gradient of\nthe CoFe layer, as shown in the cross-sectional SEM in\nFig. 1 (b) in the main manuscript, suggest that the mag-\nnetic material is concentrated in proximity to the pole,\nand that there is less material towards the equator. This\nmaterial distribution suggests that a uniaxial anisotropy\nof easy plane type is imposed on the sample by its shape.\nThe most basic approach to describe such a system is\na Stoner-Wohlfarth model, in which a single macro mag-\nnetic moment replaces the ensemble of distributed mag-\n361\n23Hard direction1\n0\n-1\nEasy direction mz1\n0\n-1\nmz1\n0\n-1mx1\n0\n-1\nmz1\n0\n-1\nmz\n(a) (b)1 2 3\n4 5\n1\n0\n-1\nmz1\n0\n-1\nmz7\nzxy\nHzx\ny\nH\n4\n567\nFigure 2. Data for the exchange biased Janus particle. (a) Measured ∆ f(H) for easy (blue) and hard (orange) alignment of\nthe external field, as well as the simulated pendants in green (easy) and red (hard). (b Close-up of (a) for low fields. Data with\neasy orientation is offset by 2 Hz for better visibility. Numbers in (a) and (b) indicate the field values for the configurations of\nthe magnetic state shown in (c) for hard and in (d) for easy alignment.\nnetic moments6. Following Ref. 7, it is straight forward\nto calculate the magnetic hysteresis and connected DCM\nresponse for easy plane anisotropy, where the latter is\nmanifested in a positive, effective demagnetizing factor\nDu, opposing to a negative Dufor easy axis anisotropy.\nThe model is a good approximation for high fields, where\nall magnetic moments are aligned with the external field,\nand essentially behave like a single macro spin. At lower\nfields deviations from the curve of the SW-model indicate\ninhomogeneous spin orientation. We show the magnetic\nhysteresis of the components mx,myandmzof the nor-\nmalized magnetization mand the DCM response ∆ fin\nFig. 3 for several different orientations ( θu,φu) of the\nuniaxial anisotropy axis ˆ u, which is perpendicular to the\neasy plane. The external field Hhas to be fixed in the\nz-direction in the model for technical reasons, which is\nwhyˆ uis varied rather than H, contrary to the situation\nin experiment. Hence, the angles θJP−θhandφJP, de-\nscribing the equivalent situation in experiment, have to\nbe compared to θuandφu, respectively. However, this\ndoes not limit the validity of the model.\nForθu= 0°, for which His perpendicular to the easy\nplane, we find the typical W-shape for ∆ ffor a magnetic\nhard orientation of H7. The columnar arrangement of\nthe components of the magnetization mx,myandmz\ntogether with ∆ fallow to correlate changes in magneticbehavior with features in ∆ f, such as e.g. the transition\nfrom field alignment to the beginning of a rotation of m\ntowards the easy axis.\nForθu= 90°, for which Hlies in the easy plane, the typi-\ncal V-shaped curve for an easy orientation of His found,\njust as expected7. This V-shape is connected to a perfect\nsquare hysteresis of m. Intermediate values of θulead to\nintermediate curve progression, which are a combination\nof the two extrema described above.\nCompared to experiment (see Figs. 2 and 3 in the main\nmanuscript) we find, that while in the case of the exter-\nnal field Hin the easy plane (last panel in the last row\nin Fig. 3) the model generates relatively similar curve\nshapes for ∆ f, this is not true for Hperpendicular to\nthe easy plane (first panel in the last row). The vertical\nasymptotes for h≈±0.5 are missing in experiment. Fur-\nther, the horizontal asymptote at high field is approached\nfrom negative rather than positive values, as it is seen in\nexperiment. A more detailed analysis in the following\nsection will show, that the latter originates in the cur-\nvature of the magnetic shell, which cannot be captured\nby a single spin model. Even though typically a good\nstarting point, the SW model seems to be of rather lim-\nited validity for the description of the relatively specific\ngeometry of a spherical cap.\n4Figure 3. Components of Mnormalized by Ms(first 3 rows) and ∆ fnormalized byf0µ0VM2\ns\n2k0l2e(last row) vs. normalized magnetic\nfieldh=H\nMsDufor different orientations of the anisotropy axis. Valid for uniaxial anisotropy with Du>0 (hereDu= 0.5).θu\nis increased from 0 °in the first column by 30 °per column up to 90 °.φuis changed in the same steps, given by the different,\ncolor-coded graphs within each column (blue for φu= 0 °, orange for 30 °, green for 60 °and red for 90 °). Arrows indicate\nswitching of the magnetization. ∆ fhas been scaled by a factor 0.5 and offset by 0.5 for θu= 0 °, and scaled by 2 for the other\nvalues ofθu.\nIV. SHAPE ANISOTROPY OF A\nTRUNCATED SPHERICAL HALFSHELL\nAs a measure for the strength of the shape anisotropy\npresent in the JPs, and as a parameter used for SW-\nmodeling, the knowledge of the effective demagnetization\nfactorDuof the geometry is of value. If possible, Duis\ndetermined using an analytical expression8, which, how-\never, is not known for the given geometry. Using micro-\nmagnetic simulations as discussed in the main text, we\ncan extract a good approximation to the demagnetiza-\ntion factor of a given geometry, without necessity for an\nanalytical formula. Here, we analyze a generic, truncated\nspherical halfshell as defined in section I for different de-\ngrees of truncation d.\nWe quickly recall the definition of the effective demagne-\ntization factor Du=Dz−Dx, whereDzandDy=Dx\nare the demagnetization factors of a magnetic object thatis rotationally symmetric in the xyplane.−0.5<D u<1\nis true, and Du<0 is valid for prolate and Du>0 for\noblate bodies. Du= 0 is the case for a perfectly spheri-\ncal body. We find a minimum Duof approximately 0.25\nfor the smallest truncation, and Duincreases with trun-\ncation as shown in Fig. 4 (a). Hence, shape anisotropy\ngets stronger as dis increased, which can be roughly un-\nderstood as the transformation from a spherical halfshell\nto a disc. Note, that a spherical halfshell without thick-\nness gradient and without truncation leads to a Dujust\nslightly large than zero. This implies, that pointing in\nthe easy plane is not very much more favorable that any\nother direction for the magnetization averaged over the\nwhole sample, see the orange dots in Fig. 4.\nThe high field frequency shift, ∆ fhf, which is\nof relevance for extracting anisotropy constants from\nexperiment7,9, first increases with truncation, but later\ndecreases again, see Fig. 4 (c). This is owed to the loss\n50.00.51.0Du\n(a)gradient shell\nhalfshell, no gradient\n012V (nm3)1e6\n(b)\ngradient shell\nhalfshell, no gradient x0.5\n0 50 100 150 200 250\nd (nm)024fhf (Hz)\n(c)\ngradient shell\nhalfshell, no gradientFigure 4. (a) Effective demagnetization factor Duof a trun-\ncated spherical halfshell with a gradient shell thickness in de-\npendence of the truncation d(blue) and of a full halfshell\n(orange). (b) Volume and (c) high field frequency shift ∆ fhf\nof the same geometries as in (a).\n0 -1.4e6\nA/m\nFigure 5. Cut through the geometry of the JP at the position\nof thexzplane showing the xcomponent of the demagnetizing\nfield within the magnetic layer for H/bardblˆ xandµ0H= 20 T.\nof magnetic material for increasing truncation as seen in\nFig. 4 (b).\nIn contrast to the SW model the micromagnetic simu-\nlations are able to correctly reproduce the experimental\ncurve shape of ∆ ffor the hard axis orientation, see Fig. 6\n(d). To understand the reason for this, the xcomponent\nof the demagnetizing field Hdemag,x within the magnetic\nlayer is visualized for a cut through the geometry in Fig. 5\nat high applied field in xdirection. It shows a gradual\nchange of the demagnetizing field magnitude with zpo-\nsition, which is a good measure of the preferred orienta-\ntion of a magnetic moment (top part with Hdemag,x≈0\nprefersxorientation, opposing to bottom part with max-\nimumHdemag,x , which needs maximum external field to\nbe aligned in xdirection). This shows, that magnetic\nmoments in proximity to the pole need the smallest field\nmagnitude to be aligned with a field in xdirection. The\nrequired field magnitude gradually increases the closer a\nmagnetic moment is situated to the equator. This ex-\nplains the gradual change of ∆ fwith increasing field inthis orientation, opposing to what is evident in the SW\nmodel, where all magnetic moments rotate in unison.\nV. GENERIC SIMULATIONS OF\nTRUNCATED SPHERICAL HALFSHELLS\nWITH AND WITHOUT EXCHANGE BIAS\nThe speedup of simulations due to a reduced size of\nthe JPs, as defined in section I, allows us to analyze the\ninfluence of simulation parameters such as the trunca-\ntion on the magnetic hysteresis. In Fig. 6 a set of data\nfor JPs with different truncations is shown. Besides the\ndifferences in high field asymptotes as already discussed\nin section IV, the truncation also significantly changes\nthe curvature of ∆ f(H), see Figs. 6 (a) and (d). By ad-\njusting the truncation, this allows to fit the curvature of\na given experimental ∆ f(H) in the simulation.\nThe magnetic state evolves very much as discussed in\nsection II for field alignment in both easy ( H/bardblˆ x) and\nhard ( H/bardblˆ z) orientation. Yet, especially for the for-\nmer we observe some distinct differences for a truncation\nofd > 100 nm: Rather than through the formation of\nan S-state close to remanence, magnetic reversal occurs\nthrough a vortex state, that only appears for small re-\nverse fields. This manifests itself e.g. in the magnitude\nof the total magnetic moment |µ|, which is significantly\nless for the vortex state as compared to the S-state, as\nshown in Fig. 6 (b).\nFor the external magnetic field perpendicular to the\neasy plane, we find that magnetic reversal takes place via\nvortex formation for all values of the truncation. Con-\nsequently, the total magnetic moment at remanence is\nalways small, as seen in Figs. 6 (b) and (e).\nThere is a big difference between the magnetic\nmoments at remanence depending on through which\nprogression remanence has been reached. Hence, in\norder to find the most stable state, it is instructive to\ncompare the energies of the different progressions as\nshown in Figs. 6 (c) and (f). A state with significant\n|µ|at remanence always seems to possess higher energy\nthan the vortex state. Compare, for example, the energy\nat remanence for the JP with 200 nm truncation for\neasy (no vortex) and hard (vortex state) field alignment.\nThe former has about 40 aJ, while the latter is more\nfavorable with only 7 aJ.\nAs discussed in the main text, exchange bias, which\nforces magnetic moments into a certain direction, can\navoid a global vortex state at remanence. We simulate\nthe magnetic hysteresis of reduced-size JPs with global\nunidirectional anisotropy and vary its strength Keb. As\nan example, we pick a JP with a cut of 100 nm, and match\nthe orientation of the unidirectional anisotropy to the\nparallel field direction, which gives the maximum effect\nin the DCM signal.\nIn Fig. 7, we plot the same set of data for the JP\nwith exchange bias as for the purely ferromagnetic JPs\n60123f (Hz)\n(a)\n0.00.51.01.52.02.5| | (fAm3)\n(b) 40\n20\n02040Etotal (aJ)\n(c)\n0 10\n5\n 510\n0H (T)\n2\n1\n0f (Hz)\n(d)\n100\n 50\n 0 50 100\n0H (mT)\n0.000.250.500.751.001.25| | (fAm3)\n(e)\n50\n 25\n 0 25 50\n0H (mT)\n10\n5\n0510Etotal (aJ)\n(f)\n5\n50\n100\n150\n200Figure 6. Frequency shift ∆ f, total magnetic moment µand total energy Etotal for JPs with different truncation as indicated\nin the legend in nm. Top row: External field applied in the easy plane ( H/bardblˆ x). Bottom row: External field applied in the hard\ndirection ( H/bardblˆ z).\nin Fig. 6. The frequency shift for high fields for H/bardblˆ x\ndevelops an increasing asymmetry for increasing strength\nKuof the unidirectional anisotropy, as shown in Fig. 7\n(a). In turn, for perpendicular alignment this asymme-\ntry is not evident, as shown in Fig. 7 (d). Hence, the\nexperimentally observed asymmetry in the asymptotes\ncan serve as good indicator for the strength of the undi-\nrectional anisotropy, given that its orientation is known.\nThe main influence of the unidirectional anisotropy on\nthe progression of the magnetic state with external field\nfor easy field alignment is to shift magnetic reversal in\nmagnetic field magnitude. This can e.g. be seen in |µ|,\nsee Fig. 7 (b). Magnetic reversal takes place around zero\nfield forKeb= 1 kJ/m3, and is shifted to happen around\n50 mT forKeb= 100 kJ/m3. The dominant state dur-\ning reversal remains an onion state for all values of Keb.\nFor hard alignment, a vortex appears in the JP for all\nvalues ofKeb, just as described in section II B. The vor-\ntex is centered in the JP for larger field magnitudes, but\nmoves to the side of the JP, if the field is reduced. De-\npending on Keb, the vortex moves only very little (small\nvalues forKeb), moves significantly to the side of the JP\n(intermediate Keb), or even escapes the JP through the\nequatorial line (large values of Keb). If the latter hap-\npens, the vortex reenters the JP in reverse field from the\nother side of the JP and then moves back to the cen-\nter for increasing reverse field. The larger the Keb, the\nlarger is the total magnetic moment µat remanence, for\nKeb= 100 kJ/m3we find almost full saturation in direc-\ntion of the anisotropy vector, see Fig. 7 (e).A clear trend is observable, if we consider the energy\nthat sets which remanent state is more likely over long\ntime scales, as shown in Figs. 7 (c) and (f): The difference\nin energy between the remanent states gets smaller for\nincreasingKeb, and hence diminishes the relevance of the\nchosen orientation of an applied field to magnetize the JP.\nThe trend for the total magnetic moment is, irrespective\nof the which orientation for the external field is chosen,\nto be larger in magnitude if Kebis increased.\nVI. DCM IN THE HIGH-FIELD LIMIT\nWITH UNIDIRECTIONAL ANISOTROPY\nThe high-field limit in DCM is reached for |H| /greatermuch\n|/summationtext\niKi/(µ0Ms)|, whereKiare the different anisotropy\ncontributions for a given direction. The frequency shift\nof the cantilever resonance is determined in this limit by\nthe competition of the different anisotropy contributions,\nand can be calculated analytically7,9. For the presence\nof unidirectional anisotropy of strength Kebthis is given\nby:\n∆funidir =−f0VKeb\n2k0l2e·\n/parenleftbigg\ncosθu(sinθhsinθebcosφucosφeb+ cosθhcosθeb)\n−sinθeb(sinθucosθhcosφeb+ sinθhsinφusinφeb)\n+ sinθusinθhcosθebcosφu/parenrightbigg(1)\n70123f (Hz)\n(a)\n0.00.51.01.52.02.5| | (fAm3)\n(b)\n10000\n25000\n50000\n100000\n100\n50\n050Etotal (aJ)\n(c)\n10\n 0 10\n0H (T)\n3\n2\n1\n0f (Hz)\n(d)\n100\n 50\n 0 50 100\n0H (mT)\n0.00.51.01.52.02.5| | (fAm3)\n(e)\n100\n 50\n 0 50 100\n0H (mT)\n100\n50\n0Etotal (aJ)\n(f)Figure 7. Frequency shift ∆ f, total magnetic moment µand total energy Etotal for JPs with 100 nm truncation and different\nstrengths of the unidirectional anisotropy as indicated in the legend in J /m3. Top row: External field applied in the easy plane\n(H/bardblˆ x). Bottom row: External field applied in the hard direction ( H/bardblˆ z).\nHere, (θh,φh= 0) define the orientation of the external\nfield, (θu,φu) of the axis of the uniaxial shape anisotropy\nas defined in Ref. 9, and ( θeb,φeb) of the unidirectional\nanisotropy vector. The latter is oriented first, and ro-\ntated by (θu,φu) in a second step to be consistent with\nthe situation in experiment. Cantilever and magnetic\nparameters are as defined before.\nWe evaluate the high-field limit for the sum of shape\nand unidirectional anisotropy for an exemplary situation\nas it may be present for the exchange biased JPs, using\nsimilar parameters as in section IV. However, we increase\nKebsignificantly to magnify its effects. The angles are\nset to be (θu,φu) = (−3°,0°) and (θeb,φeb) = (−90°,0°),\nwhileθhis varied as in experiment. The result is shown\nin Fig. 8 (a), together with the individual contributions\nfrom shape and unidirectional anisotropy. This shows,\nthat the sum of the two contributions may lead to a pe-\nriodicity that deviates slightly from 180 °, which wold be\ngiven for pure uniaxial shape anisotropy. Furthermore,\nthe magnitude of maxima and minima may differ signif-\nicantly. Here we find 1 .9 Hz for the maxima and 2 .4 Hz\nfor the minima, respectively. We have indeed observed\nsuch large asymmetries in experiment for low tempera-\ntures (not shown here), however, the origin is different,\nas will be discussed in the following. Nevertheless, for\nstrong unidirectional anisotropies these findings should\nbe observable in experiment.\nThe SW model, as described in section III, can be used\nto calculate ∆ f(θh) for a fixed field magnitude, as done\nin experiment for 3 .5 T. This allows to compare the re-\nsult of the SW model with the high field limit as dis-\n2\n02f (Hz)\n(a) sum\nshape\nunidir\n0 50 100 150\nh (°)\n2\n02f (Hz)\n(b)3.5 T\nlimitFigure 8. (a) ∆ f(θh) in the high field limit with shape and\nunidirectional anisotropy. ∆ f(θh) in the SW model for 3 .5 T\napplied field magnitude, and in the high field limit from (a).\ncussed above, see Fig. 8 (b). The curve of the high field\nlimit follows a (negative) cosine with 2 θhin the argument.\nIn turn, for the SW model at 3 .5 T, minima are deeper\nand maxima are shallower in ∆ f, respectively. However,\nthere is no deviation from the 180 °periodicity. Further,\nmaxima are wider than minima, which is a consequence\nof the fact that positive and negative asymptotes are ap-\nproached with a different curvature when ramping up the\nexternal field in the SW model, compare curves in the last\n8panel in the first column with those in the last panel in\nthe last column in Fig. 3. In experiment this behavior\nof the maxima and minima is inverted, which is caused\nby the extreme curvature of the magnetic layer JPs, as\ndiscussed in section IV.\nREFERENCES\n1COMSOL AB. Comsol multiphysics ®, 2021.\n2T. Fischbacher, M. Franchin, G. Bordignon, and H. Fan-\ngohr. A systematic approach to multiphysics exten-\nsions of finite-element-based micromagnetic simulations:\nNmag. IEEE Transactions on Magnetics , 43(6):2896–\n2898, 2007.\n3D. V. Berkov, C. T. Boone, and I. N. Krivorotov. Mi-\ncromagnetic simulations of magnetization dynamics in\na nanowire induced by a spin-polarized current injectedvia a point contact. Physical Review B , 83(5):054420,\nFebruary 2011. Publisher: American Physical Society.\n4J. M. D. Coey. Magnetism and Magnetic Materials .\nCambridge University Press, 2010.\n5N. Rossi, B. Gross, F. Dirnberger, D. Bougeard, and\nM. Poggio. Magnetic force sensing using a self-assembled\nnanowire. Nano Letters , 19(2):930–936, 2019.\n6Allan H. Morrish. The Physical Principles of Mag-\nnetism . Wiley, January 2001.\n7B. Gross, D. P. Weber, D. R¨ uffer, A. Buchter, F. He-\nimbach, A. Fontcuberta i Morral, D. Grundler, and\nM. Poggio. Dynamic cantilever magnetometry of in-\ndividual CoFeB nanotubes. Phys. Rev. B , 93(6):064409,\nFebruary 2016.\n8A. Hubert and R. Sch¨ afer. Magnetic Domains: The\nAnalysis of Magnetic Microstructures . 1998.\n9B. Gross, S. Philipp, E. Josten, J. Leliaert, E. Wetter-\nskog, L. Bergstr¨ om, and M. Poggio. Magnetic anisotropy\nof individual maghemite mesocrystals. Phys. Rev. B ,\n103:014402, Jan 2021.\n9"    },    {        "title": "2111.00422v2.Shape_Programmable_Magnetic_Pixel_Soft_Robot.pdf",        "content": "ShapeProgrammableMagneticPixelSoftRobot∗\nRanZhao1,2,*,HanchenYao1,HoudeDai1,*\n1QuanzhouInstituteofEquipementManufacturingofHaixiInstitutes,\nChineseAcademyofSciences,Quanzhou262000,China\n2Zhongyuan-PetersburgAviationCollege,ZhongyuanUniversityof\nTechnology,Zhengzhou450000,China\n*E-mail:dhd@fjirsm.ac.cn;zhaoran@zut.edu.cn\nAbstract:Magneticresponsesoftrobotrealizesprogrammableshaperegulationwith\nthehelpofmagneticfieldandproducesvariousactions.Theshapecontrolof\nmagneticsoftrobotisbasedonthemagneticanisotropycausedbytheorderly\ndistributionofmagneticparticlesintheelasticmatrix.Intheprevioustechnologies,\nmagneticprogrammingiscoupledwiththemanufacturingprocess,andtheorientation\nofmagneticparticlescannotbemodified,whichbringsrestrictionstothedesignand\nuseofmagneticsoftrobot.Thispaperpresentsamagneticpixelrobotwithshape\nprogrammablefunction.ByencapsulatingNdFeB/galliumcompositesintosilicone\nshell,athermo-magneticresponsefunctionalfilmwithlatticestructurearefabricated.\nBasingonthermal-assistedmagnetizationtechnique,werealizedthediscrete\nmagnetizationregiondistributiononthefilm.Therefore,weproposedamagnetic\ncodingtechniquetorealizethemathematicalresponseactiondesignofsoftwarerobot.\nUsingthesemethods,wepreparedseveralmagneticsoftrobotsbasedonorigami\nstructure.Theexperimentsshowthatthebehaviormodeofrobotcanbeflexiblyand\nrepeatedlyregulatedbymagneticencodingtechnique.Thisworkprovidesabasisfor\ntheprogrammedshaperegulationandmotiondesignofsoftrobot.\nKeywords:Softrobot;Magneticpixel;Magneticencoding;Shapeprogrammable;\nLiquid-metal.1Introduction\nMagneticsoftrobot,asakindofuntetheredrobot,canimplementtasksascell\nmanipulation,medicalimageacquisition,drugdeliveryandnon-invasiveintervention.\nComparingtolight/thermal,chemicalorelectricalactuatedsoftrobots[1-4],magnetic\nsoftrobotshavetheadvantagesoffastresponse,unlimitedenduranceandno\nobstructionrestrictions[5].Therefore,ithasmadegreatprogressinre-centyears.\nThemotionofmagneticsoftrobotcomesfromtheresponseofmagneticparticles\nwrappedinflex-iblematrixwhenapplyingamagneticfield.Theseparticlescanbe\nsoftmagneticmaterials(Fe,NiandFe3O4)[6,7]ormagnetic hard magnetic\nmaterials(CrO2andNdFeB)[8-10].Thesoftrobotbasedonhardmagnetic\nmaterialshashighresidualmagnetization.Itsprogrammableshapecontrolcanbe\nrealizedbycon-figuringthemagneticanisotropy,whichcalledpro-grammable\nmagnetizationtechnique.However,themagnetizingprocessisalwayscoupledwith\ntherobotmanufacturingprocess,andthemagneticanisotropydistributionis\nunchangable.Theintroductionofphase-transitionpolymerhaschangedthisdefect\n(11,12).Byheating,magneticparticlescanbecomefreedomains,bere-oriented,and\nbelockedinthecoolingstate.Buttheheatingtemperaturerequiredistoohighfor\nbiomedicalapplications.\nSofar,magneticprogrammingisusuallyimplementedincontinuousmedia\n[13,14],andthereisnoobviousboundarybetweendifferentmagnetizedre-filmcan\nbeobservedbytheCMOScamera.Theproposedthemo-magneticresponsefunctional\nfilmismadeofNdFeB/galiumcomposites,coatingwithSilicone.AsgiveninFig.1b,\nthefilmhasadiscretelatticestructure,eachbasiccelliscalledmagneticpixel.In\neachmagneticpixel,theNdFeBmicroparticlesarewrappedanduniformlydistributed\nintheliquid-metalmatrixregions.Somecompositeswithlatticestructureare\nproposed,realizingtheindependentprogrammingofdiscretemagnetizationregion.\nThismakesitpossibleformorerefinedmorphologicalcontrolforsoftrobot.Cui\nproposedtheconceptofmagneticencod-ing—atechniquetogeneratemagnetic\nresponsebycodingmagneticvectorsondiscretemagneticunits[15].However,this\ntechniquecanonlydealwithverticalvectorcoding.Insummery,thereisstillalackofsufficienttech-nologytorealizerepeatable\nmagnetizationatlowheatingtemperatureandarbitrarymagneticvectorsencoding.\nThispaperpresentsamagneticpixelrobotbyencapsulatingNdFeB/gallium\ncompositesintosiliconeshell.Wedevelopedathermal-assisted3Dmagnetization\ntechnique,torealizethediscretemagnetizationregiondistributiononthefilm.We\nproposedamagneticencodingtechniquetorealizethemathematicalresponseaction\ndesignforsoftrobot.Thedetailsaregiveninfollowsections.\n2.Principle\n2.1Reprogrammable3Dmagnetization\nFig.1Systemdesignandmechanismofreprogramable3DmagnetizationinMFs-LMfilms:(a)3D\nmagneticvectorprogrammingsystem,(b)StructuresofMFs-LMfilmanditsmagnetic\npixel,(c)WorkingmechanismofMFs-LMcompositesand(d)Programmingamagneticanisotropy\nintheMFs-LMfilmandgeneratingamagneticresponseaction.\nFig.1ashowsthephysicalapparatusforpatterning3Ddiscretemagnetizationin\nNdFeB/Ga/Siliconecompositefilms.Thesystemconsistsofa3-axispositioning\nplatform,a2-DOFrotatingplatform,a3-axisHallsensor,asquaremagnet(N52,24×24×24mm3,surfacemagneticintensityof735mT),asemiconductorcooler,\nandacoaxialopticalsystem(includinga405nmultravioletlaserandaCMOS\ncamera).Basedonthe5-axismotionplatform,a3Dvectormagneticfieldofany\ndirectionandstrengthcanbegenerated.a3Dvectormagneticfieldofanydirection\nandstrengthcanbegenerated.A10W,800µmresolutionUVlaserisusedtoheat\nMFs-LMcomposites..Thesemiconductorcoolerisusedtoreducethetemperatureof\nthenonheatedregionofthefilm.Atlast,thechangeofthefunctionalfilmcanbe\nobservedbytheCMOScamera.Theproposedthermal-magneticresponsefunctional\nfilmismadeofNdFeB/galiumcomposites,coatingwithSilicone.\nAsgiveninFig.1b,thefilmhasadiscretelatticestructure,eachbasiccellis\ncalledmagneticpixel.Ineachmagneticpixel,theNdFeBmicroparticlesarewrapped\nanduniformlydistributedintheliquidmetalmatrix.Fig.1cexhibitsthemechanismof\nreprogramablemagnetizationbasedontheMFs-LMcomposites.Whenheatinga\ncerternregionofthematerialsbythelaser,galliumcanbetransformedfromsolidto\nliquidphase.ThenNdFeBparticleswillbereorientedundertheprogramming\nmagneticfield,andformmacromagneticanisotropyintheheatedregion.Here,the\nrequiredheatingtemperatureisonly40◦C.Whenthemagnetizationprocessis\ncompleted,thelaserstopsworking,andtheliquid-metalisconvertedintosolidphase\nagainundertheactionofsemiconductorcooler.Byrepeatingthisprocess,wecan\nprogramdifferentmagneticanisotropyinthefilm.InFig.1d,weshowshowto\nprogrammagneticanisotropyindifferentregionsofthefilmandproducemagnetic\nresponseactions.ByprogrammingonregionIandregionIIofthefilmrespectively,\ntheresidualmagnetizationwithverticalandhorizontaldirectionsisobtained.Finally,\nundertheexitingofverticalmagneticfield,asimplebendingactionisgenerated.\n2.2Magneticencoding\nHere,weproposedamathematicaldescriptionof3Dmagneticvectorencoding,a\nvectormatrixAisusedtodescribethemagnetizationvectordirectionforeach\nmagneticpixel,whichisgivenas,whereaij=(αij,βij,γij)T,representsthe3Dmagnetizationvectoroftheij-thmagnetic\npixel.AsshowninFig2.a,αij,βijandγijaretheanglesbetweenmagnetizationvector\nandx,yandzaxis,respectively.\nFig.2bshowshowtodesigntheprofileofthesoftrobotonthemagneticpixel\narray,thepartofthematrixwithavalueof0representstheareatoberemoved.Then,\ntherequiredresponseactioncanbeobtainedbymagneticcodingforeachpixelinthe\nreservedarea.Finally,accordingtoequation(1),thefollowingmagnetizationvevtor\nmatrixMcanbeobtained,\n AmM•= (2)\nwheremistheremanentmagnetization.\nFig.2Illustrationofmagneticencoding:(a)Magneticvectoroftheij-thmagneticpixel,(b)\nContouringandencodingonthemagneticpixelarray.\n3Material\n3.1PreparationofMFs-LMcomposites\nHere,theferromagneticliquid-metalfunctionalcompositesispreparedby\nhomogenouslymixingLMofgallium(themeltingpointof29.6°C,purityof99.9%)\nandferromagneticneodymium–iron–boron(NdFeB)microparticles.Theaverage\nparticlesizeofNdFeBparticle(XinnuodeTransmitionDevices,Guangzhou)was6.0\nµm3.AsshowninFig.2a,theunmagnetizedNdFeBpowderandliquidgallium,witha\nvolumeratioof4:6,weremixedbyamechanicalagitator.Themixturewasheatedto50°Candstirredinairenviromentfor5min.TheNdFeB/Gacompositeswas\nmagnetizedinapulsedmagneticfieldwithmagneticstrengthof2T.Themagnetized\nMFs-LMshowshighrheologicalandshapereconfigurableproperties,whichis\nsimilarwiththepreviouslyreportedotherLMcomposites[11-13].\nFig.3Fabricationprocess:(a)Mixingand(b)MagnetizingofMFs-LMcomposites,(c)preparing\nand(d)encapsulatingtheMFs-LMcompositesintothesiliconeshell.\n3.2Fabricationofmagneticpixelfilm\nInFig.2c,abiocompatiblematerial—Silicone(Ecoflex00-50,Smooth-On,\nAmerica)wasusedforcoatingtheFMs-LMcomposites.Anda3D-printedmoldwas\nusedtofabricatetheshellofmagneticpixelfilm.Thedesignedsizeofeachmagnetic\npixelis2×2mm2.TheSiliconesolventwaspouredintothemold,andcuredinvacuum\nfor1h,obtainingashellwiththicknessof100µm.AsshowninFig.2d,theMFs-LM\nplasticinewiththicknessof600µmwasmanuallyencapsulatedintothesiliconeshell.\nThetotalthicknessofthefunctionalfilmis800µm.\n4ExperimentsandDiscussion\n4.1Experimentalresults\nInexperiments,wetestedtheperformanceofthemagneticpixelfilm,and\nmanufacturedseveralmagneticsoftrobotswithdifferentstructures.Byconfiguring\ndifferentmagnetizationregionsonthesoftrobot,weshowedthatthesoftrobotcanperformdifferentmotions,theresultsaregiveninFig.4.Figs.4aand4bexhibitthe\ngraphsoftheproposedmagneticpixelfilm,andthemagneticimagingofencoding\nletters\"CAS\"inthefilm.InFigs.4c-4h,wefabricatedthererobotswithI,Yandring\nshapes,respectively.Whenimplementingdifferentmagneticencodingschemes,they\ncanperformdifferentresponseactions.\nFig.4Resultsofmagneticencodinginthefilm:(a)Photographofmagnticpixelfilm,(b)Magnetic\nimagingofcodingletters,(c)Bendingand(d)TwistingofIshaperobot,(e)Standingand(f)\nTorsionofYshaperobot,(g)Foldingand(h)Higher-orderbendingofringshaperobot.\nAnothercaseisshowninFig.5,aflexiblezaxispositioningplateformwasbuilt\nandtested.Theplateformhasasquareflatandfoursupportingarms,whichcanmove\nalongz-axisdirectionwhenapplyingaverticalmagneticfield(showninFigs.5aand\n5b).TheencodingschemeisgiveninFig.5c.ThepositioningcurveisshowninFig.5d,\nthemaximumdisplacementis3.1mmunderthemagneticfieldof54.5mT.Fig.5Flexiblez-axisplatform:(a)Photographofz-axisplatform,(b)Movingalongz-axisdirection\nwhenapplyingamagneticfield,(c)Magneticencodingand(d)Displacementofz-axis.\nFig.6showsthemagneticresponesofthecrossshapesoftrobot,when\nimplementedthreedifferentmagneticencodes.AsshowninFigs.6aand6b,therobot\nperformsasafanwithfourblades.InFig.6c,whenusinganothermagneticencodes,\ntherobotcanstandup(Fig.6d).Thisrobotalsocanworkasamagneticgraspperor\ncapsule,capturingortransportatarget(giveninFigs.6eand6f).\nFig.6Cross-shaperobot:(a)Encode1,(b)Fan,(c)Encode2,(d)Standing,(e)Encode3,and\n(f)grasping.\nFig.7exhibitsamagneticorigamirobot,consistingofeighttriangularplates.\nFourkindsofencodes(asshowninFigs.7a,7c,7e,and7g)weredesignedto\ngenerateddifferentshapesofparallelfolding,diagonalfolding,pyramidanddiagonal\ndoublefolding(giveninFigs.7b,7d,7f,and7h).ThecasesgiveninFigs.4,5,6and7\nindicatethatthroughtherepeatablemagnetizationtechniques,arobotcanbere-encodedtogeneratedifferentresponseactions.Andthisabilityofreprogrammable\nshapehelpstherobottoperformdifferenttasks.Furthermore,withthereconfigurable\nfunction,thereusabilityofsoftrobotisgreatlyimproved.\nFig.7Magneticorigamirobot:(a)Encode1,(b)Parallelfolding,(c)Encode2,(d)Diagonal\nfolding,(e)Encode3,(f)Pyramid,(g)Encode4,and(h)Diagonaldoublefolding.\n4.2Comparisonofexitingtechniques\nInthissection,weanalyzedandcomparedthereportedmagneticprogramming\ntechniques.ThedetailsareshowninTable1.Ourtechniqueenablesrepeatable3D\nmagnetization,requiringlowerheatingtemperature.Andweprovidesamathematical\nmodelofmagneticvectorencoding.\nTable1Comparisonofexistingmagneticprogrammingtechniques\nRef Materials FabricationmethodsMedium Magneticprogrammingtechnique\n16NdFeB/UVresin UVlithography ContinueUnrepeatable3Dmagnetization\nheating-free\n17 CrO2/PDMS Template-aided ContinueRepeatable3Dmagnetization,\n18\nNdFeB/PCL/Silicone Template-aided Continue110◦Cheatingtemperature,\nrepeatable2Dmagnetization,\n15 NanoFe/PMMA UVlithography Discreet80◦Cheatingtemperature\nUnrepeatable3Dmagnetization\nmagneticencoding\nThisworkNdFeB/Ga/Silicone Template-aided DiscreetRepeatable3Dmagnetization,\nmagneticvectorencoding,\n40◦Cheatingtemperature\n5Conclusion\nInthiswork,wepresentedashapeprogrammablemagneticpixelsoftrobot\nrealizes.TheproposedrobotismadeNdFeB/Ga/siliconecomposites,andhasalatticestructure.Theprogrammablemagnetizationandtheequipmentaredevelopedto\nimplementRepeatablemagnetizationinthemagneticpixelfilm.Amathematical\nmodelofmagneticencodingwasalsoproposed.Theexperimentalresultshave\ntestifiedourconcept.Thecontributionsofthispaperareasfollows:\ni.Wedesignedamagneticpixelfilm,whichhasthermal/magneticresponse\nfunctionandcanberepeatedlymagnetized.\nii.Wedevelopeda3Dmagneticvectorprogrammingequipment,whichcan\nconfigurethemagneticanisotropyinanydirectiontoindependentmagneticpixel.\niii.Wepresentedatechniqueofmagneticencoding,andpresenteditsmathematical\nmodel,whichprovidesabasisforautomaticrobotbehaviordesign.\nThenextstageofourworkistoreducethesizeofmagneticpixels,andto\nmanufacturemilli-/microscalemagneticsoftrobot.\nCompliancewithethicsguidelines\nRanZHAOandHou-deDAIdeclarethattheyhavenoconflictofinterest.\nReferences\n1.LyuLX,LiF,WuK,DengP,SHJeong,WuZ,DingH,2019.Bio-inspired\nuntetheredfullysoftrobotsinliquidactuatedbyinducedenergygradients.\nNationalScienceReview,6(5):970-981.\n2.WieJJ,ShankarMR,WhiteTJ,2016.Photomotilityofpolymers.Nature\ncommunications,7:13260.\n3.WehnerM,TrubyRL,FitzgeraldDJ,MosadeghB,WhitesidesGM,LewisJA,\nWoodRJ,2016.Anintegrateddesignandfabricationstrategyforentirelysoft,\nautonomousrobots.Nature,536:451-455.\n4.ShintakeJ,RossetS,SchubertB,FloreanoD,SheaH,2015.VersatileSoft\nGripperswithIntrinsicElectroadhesionBasedonMultifunctionalPolymer\nActuators.Adv.Mater.,28.\nhttps://doi.org/10.1002/adma.201504264\n5.EbrahimiN,BiC,CappelleriDJ,etal,2020.MagneticActuationMethodsin\nBio/SoftRobotics.Adv.Funct.Mater.,2005137.https://doi.org/10.1002/adfm.202005137\n6.ZhangJ,DillerE,2017.UntetheredMiniatureSoftRobots:ModelingandDesign\nofaMillimeter-ScaleSwimmingMagneticSheet.SOFTROBOTICS,0(0):0126.\nhttps://doi.org/10.1089/soro.2017.0126\n7.GuH,BoehlerQ,CuiH,etal,2020.Magneticciliacarpetswithprogrammable\nmetachronalwaves.NATURECOMMUNICATIONS,11:2637.\nhttps://doi.org/10.1038/s41467-020-16458-4\n8.HuangH,SakarMS,Petruska1AJ,PaneS,NelsonBJ,2016.Soft\nmicromachineswithprogrammablemotilityandmorphology.NATURE\nCOMMUNICATIONS,7:12263.\n9.AlapanY,KaracakolAC,GuzelhanSN,IsikI,SittiM,2020.Reprogrammable\nshapemorphingofmagneticsoftmachines.Sci.Adv.,6:eabc6414.\n10.KimY,ParadaGA,LiuS,ZhaoX,2019.Ferromagneticsoftcontinuumrobots.\nSci.Robot.,4:eaax7329.\n11.ZeZhangF,WangL,ZhengZ,etal,2019.Magneticprogrammingof4Dprinted\nshapememorycompositestructures.CompositesPartA,125:10557.\n12.ZeQ,KuangX,WuS.,2019.MagneticShapeMemoryPolymerswithIntegrated\nMultifunctionalShapeManipulationAdv.Mater.,1906657.\n13.HuW,LumGZ,MastrangeliM,SittiM,2018.Smallscalesoft-bodiedrobotwith\nmultimodallocomotion.NATURE,554:81-85.\n14.ManamanchaiyapornL,XuT,WuX,2020.Magneticsoftrobotwiththe\ntriangularhead-tailmorphologyinspiredbylateralundulation.IEEE/ASME\nTransactionsonMechatronics,25(6):2688-2699.\n15.CuiJ,HuangT,Luoh,2019.Nanomagneticencodingofshape-morphing\nmicromachines.Nature,575:7.\nhttps://doi.org/10.1002/adma.201802595\n16.XuT,ZhangJ,SalehizadehM,OnaizahO,DillerE,2019.Millimeter-scale\nflexiblerobotswithprogrammablethree-dimensionalmagnetizationandmotions.\nSci.Robot.4,eaav4494.\n17.LumGZ,YeZ,DongX,MarviH,ErinO,HuW,SittiM,2016.Shape-programmablemagneticsoftmatter.PNAS,E6007–E6015.\n18.DengH,SattariK,XieY,2020.Laserreprogrammingmagneticanisotropyinsoft\ncompositesforreconfigurable3Dshaping.NATURECOMMUNICATIONS,\n11:6325." 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